CINXE.COM

<!DOCTYPE html> <html lang="en"> <head> <meta content="text/html; charset=utf-8" http-equiv="content-type"/> <title>The Legacy of Henrietta Leavitt: A Re-analysis of the First Cepheid Period-Luminosity Relation</title>  <meta content="width=device-width, initial-scale=1, shrink-to-fit=no" name="viewport"/> <link href="https://cdn.jsdelivr.net/npm/bootstrap@5.3.0/dist/css/bootstrap.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/ar5iv.0.7.9.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/ar5iv-fonts.0.7.9.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/latexml_styles.css" rel="stylesheet" type="text/css"/> <script src="https://cdn.jsdelivr.net/npm/bootstrap@5.3.0/dist/js/bootstrap.bundle.min.js"></script> <script src="https://cdnjs.cloudflare.com/ajax/libs/html2canvas/1.3.3/html2canvas.min.js"></script> <script src="/static/browse/0.3.4/js/addons_new.js"></script> <script src="/static/browse/0.3.4/js/feedbackOverlay.js"></script> <base href="/html/2502.17438v1/"/></head> <body> <nav class="ltx_page_navbar"> <nav class="ltx_TOC"> <ol class="ltx_toclist"> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#S1" title="In The Legacy of Henrietta Leavitt: A Re-analysis of the First Cepheid Period-Luminosity Relation"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">1 </span>Introduction</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#S2" title="In The Legacy of Henrietta Leavitt: A Re-analysis of the First Cepheid Period-Luminosity Relation"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2 </span>Cepheid sample</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#S3" title="In The Legacy of Henrietta Leavitt: A Re-analysis of the First Cepheid Period-Luminosity Relation"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3 </span>Cepheid light curves</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#S4" title="In The Legacy of Henrietta Leavitt: A Re-analysis of the First Cepheid Period-Luminosity Relation"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4 </span>Period-Luminosity relation</span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#S4.SS1" title="In 4 Period-Luminosity relation ‣ The Legacy of Henrietta Leavitt: A Re-analysis of the First Cepheid Period-Luminosity Relation"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4.1 </span>P-L relation with magnitudes from Leavitt & Pickering (1912)</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#S4.SS2" title="In 4 Period-Luminosity relation ‣ The Legacy of Henrietta Leavitt: A Re-analysis of the First Cepheid Period-Luminosity Relation"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4.2 </span>P-L relation with magnitudes from OGLE</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#S5" title="In The Legacy of Henrietta Leavitt: A Re-analysis of the First Cepheid Period-Luminosity Relation"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">5 </span>Discussion</span></a></li> <li class="ltx_tocentry ltx_tocentry_appendix"><a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#A1" title="In The Legacy of Henrietta Leavitt: A Re-analysis of the First Cepheid Period-Luminosity Relation"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">A </span>Note On the Periodicity of BZ Tuc</span></a></li> </ol></nav> </nav> <div class="ltx_page_main"> <div class="ltx_page_content"> <article class="ltx_document ltx_authors_multiline"> <h1 class="ltx_title ltx_title_document">The Legacy of Henrietta Leavitt: <br class="ltx_break"/>A Re-analysis of the First Cepheid Period-Luminosity Relation</h1> <div class="ltx_authors"> <span class="ltx_creator ltx_role_author"> <span class="ltx_personname"><a class="ltx_ref orcid" href="https://orcid.org/0000-0003-3889-7709" title="">Louise Breuval</a> </span><span class="ltx_author_notes"> <span class="ltx_contact ltx_role_affiliation">ESA Research Fellow </span> <span class="ltx_contact ltx_role_affiliation">European Space Agency (ESA), ESA Office, Space Telescope Science Institute, <br class="ltx_break"/>3700 San Martin Drive, Baltimore, MD 21218, USA </span> <span class="ltx_contact ltx_role_affiliation">Department of Physics and Astronomy, Johns Hopkins University, Baltimore, MD 21218, USA </span> <span class="ltx_contact ltx_role_email"><a href="mailto:lbreuval@stsci.edu">lbreuval@stsci.edu</a> </span></span></span> <span class="ltx_creator ltx_role_author"> <span class="ltx_personname"><a class="ltx_ref orcid" href="https://orcid.org/0000-0001-6169-8586" title="">Caroline D. Huang</a> </span><span class="ltx_author_notes"> <span class="ltx_contact ltx_role_affiliation">NSF Astronomy and Astrophysics Postdoctoral Fellow </span> <span class="ltx_contact ltx_role_affiliation">Center for Astrophysics <math alttext="|" class="ltx_Math" display="inline" id="id3.1.m1.1"><semantics id="id3.1.m1.1a"><mo fence="false" id="id3.1.m1.1.1" stretchy="false" xref="id3.1.m1.1.1.cmml">|</mo><annotation-xml encoding="MathML-Content" id="id3.1.m1.1b"><ci id="id3.1.m1.1.1.cmml" xref="id3.1.m1.1.1">|</ci></annotation-xml><annotation encoding="application/x-tex" id="id3.1.m1.1c">|</annotation><annotation encoding="application/x-llamapun" id="id3.1.m1.1d">|</annotation></semantics></math> Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA </span></span></span> <span class="ltx_creator ltx_role_author"> <span class="ltx_personname"><a class="ltx_ref orcid" href="https://orcid.org/0000-0002-6124-1196" title="">Adam G. Riess</a> </span><span class="ltx_author_notes"> <span class="ltx_contact ltx_role_affiliation">Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA </span> <span class="ltx_contact ltx_role_affiliation">Department of Physics and Astronomy, Johns Hopkins University, Baltimore, MD 21218, USA </span></span></span> </div> <div class="ltx_abstract"> <h6 class="ltx_title ltx_title_abstract">Abstract</h6> <p class="ltx_p" id="id5.id1">Henrietta Swan Leavitt’s discovery of the relationship between the period and luminosity (hereafter the Leavitt Law) of 25 variable stars in the Small Magellanic Cloud, published in 1912, revolutionized cosmology. These variables, eventually identified as Cepheids, became the first known ‘standard candles’ for measuring extragalactic distances and remain the gold standard for this task today. Leavitt measured light curves, periods, and minimum and maximum magnitudes from painstaking visual inspection of photographic plates. Her work paved the way for the first precise series of distance measurements that helped set the scale of the Universe, and later the discovery of its expansion by Edwin Hubble in 1929. Here, we re-analyze Leavitt’s first Period-Luminosity relation using observations of the same set of stars but with modern data and methods of Cepheid analysis. Using only data from Leavitt’s notebooks, we assess the quality of her light curves, measured periods, and the slope and scatter of her Period-Luminosity relations. We show that modern data and methods, for the same objects, reduce the scatter of the Period-Luminosity relation by a factor of two. We also find a bias brightward at the short period end, due to the non-linearity of the plates and environmental crowding. Overall, Leavitt’s results are in excellent agreement with contemporary measurements, reinforcing the value of Cepheids in cosmology today, a testament to the enduring quality of her work. <br class="ltx_break"/></p> </div> <section class="ltx_section" id="S1"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">1 </span>Introduction</h2> <figure class="ltx_figure" id="S1.F1"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="468" id="S1.F1.g1" src="x1.png" width="937"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 1: </span><span class="ltx_text ltx_font_bold" id="S1.F1.5.1">Left</span>: Original Period-Luminosity relation (minimum and maximum light) published by <cite class="ltx_cite ltx_citemacro_citet">Leavitt & Pickering (<a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#bib.bib21" title="">1912</a>)</cite> in a ”provisional” scale of magnitudes. Data are taken from their Table 1. <span class="ltx_text ltx_font_bold" id="S1.F1.6.2">Right</span>: Same as left with pulsation periods and apparent magnitudes in the <math alttext="V" class="ltx_Math" display="inline" id="S1.F1.2.m1.1"><semantics id="S1.F1.2.m1.1b"><mi id="S1.F1.2.m1.1.1" xref="S1.F1.2.m1.1.1.cmml">V</mi><annotation-xml encoding="MathML-Content" id="S1.F1.2.m1.1c"><ci id="S1.F1.2.m1.1.1.cmml" xref="S1.F1.2.m1.1.1">𝑉</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.F1.2.m1.1d">V</annotation><annotation encoding="application/x-llamapun" id="S1.F1.2.m1.1e">italic_V</annotation></semantics></math>-band from the OGLE survey <cite class="ltx_cite ltx_citemacro_citep">(Soszyński et al., <a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#bib.bib28" title="">2015</a>)</cite>, uncorrected for reddening. <br class="ltx_break"/></figcaption> </figure> <div class="ltx_para" id="S1.p1"> <p class="ltx_p" id="S1.p1.1">Distances play a crucial role in astronomy, as they establish the absolute scales for a wide range of astronomical objects. While direct geometric distances can be obtained to the nearest objects (<math alttext="\sim" class="ltx_Math" display="inline" id="S1.p1.1.m1.1"><semantics id="S1.p1.1.m1.1a"><mo id="S1.p1.1.m1.1.1" xref="S1.p1.1.m1.1.1.cmml">∼</mo><annotation-xml encoding="MathML-Content" id="S1.p1.1.m1.1b"><csymbol cd="latexml" id="S1.p1.1.m1.1.1.cmml" xref="S1.p1.1.m1.1.1">similar-to</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.1.m1.1c">\sim</annotation><annotation encoding="application/x-llamapun" id="S1.p1.1.m1.1d">∼</annotation></semantics></math>5 kpc with <em class="ltx_emph ltx_font_italic" id="S1.p1.1.1">Gaia</em>), determining distances to extragalactic systems typically necessitates the use of a cosmic distance ladder, where multiple distance measurement techniques are cross-calibrated over the range of distances for which they overlap. Most distance ladders use geometric distances as the first “rung” to calibrate the absolute luminosity of a standard candle – a bright astrophysical source with a known intrinsic luminosity, which can then measure systems out of the range of geometric methods <cite class="ltx_cite ltx_citemacro_citep">(e.g. Riess et al., <a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#bib.bib25" title="">2022</a>)</cite>.</p> </div> <div class="ltx_para" id="S1.p2"> <p class="ltx_p" id="S1.p2.1">Cepheid variables, the first standard candles to be developed, were noticed by Henrietta Swan Leavitt in her seminal paper cataloguing 1777 variables in the Magellanic Clouds <cite class="ltx_cite ltx_citemacro_citep">(Leavitt, <a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#bib.bib20" title="">1907</a>)</cite>. She remarked that for a subset of Small Magellanic Cloud (SMC) variables, <em class="ltx_emph ltx_font_italic" id="S1.p2.1.1">“It is worthy of notice that … the brighter variables have the longer periods. It is also noticeable that those having the longest periods appear to be as regular in their variations as those which pass their changes in a day or two.”</em> The relation between Cepheid luminosities and periods (Period-Luminosity Relation, hereafter P-L relation or Leavitt Law) was further established in <cite class="ltx_cite ltx_citemacro_cite">Leavitt & Pickering (<a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#bib.bib21" title="">1912</a>)</cite> where the periods and magnitudes of 25 SMC Cepheids were reported. Only a few years later, in 1917, Sir Arthur Eddington proposed that this relationship could be driven by fundamental-mode pulsation resulting from the kappa-opacity mechanism, giving this empirical relationship a firm physical explanation <cite class="ltx_cite ltx_citemacro_citep">(Eddington, <a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#bib.bib5" title="">1917</a>)</cite>. The physical size and temperature changes experienced by radially pulsating Cepheids result in periodic variations in their luminosities. By observing the stellar light curves, we can determine the periods of these variations, independent of their distance. Once the absolute scale of the Leavitt Law is calibrated using the geometric distance to the nearest Cepheids, the apparent brightnesses of more distant Cepheids can then be used to infer their true distances.</p> </div> <div class="ltx_para" id="S1.p3"> <p class="ltx_p" id="S1.p3.1">Unfortunately Leavitt, who died in 1921, did not live to see the tremendous impact of her work. The realization that Cepheids could be used to measure distances to a variety of astrophysical systems would change our cosmological paradigm. In 1924, this led directly to Hubble’s transformative discovery of the extragalactic universe. By observing Cepheids in the Andromeda Galaxy (M31) and measuring their distances, he provided definitive evidence that the extragalactic ‘nebulae’ were not part of the Milky Way, but instead were ‘island Universes’ comparable to our own Galaxy. Only five years later, in 1929, he used Cepheids as the foundation of his distance ladder to show that the more distant galaxies were moving away more quickly – evidence that convinced the scientific community that the universe was expanding <cite class="ltx_cite ltx_citemacro_citep">(Hubble, <a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#bib.bib15" title="">1929</a>)</cite>.</p> </div> <div class="ltx_para" id="S1.p4"> <p class="ltx_p" id="S1.p4.1">Hubble’s contributions to cosmology have been revolutionary, but arguably no less important is the fundamental discovery made by Leavitt that enabled his work. However, unlike Hubble’s work, which has previously been re-examined and reconstructed <cite class="ltx_cite ltx_citemacro_citep">(Kirshner, <a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#bib.bib16" title="">2003</a>)</cite>, comparatively little effort has been placed in re-examining Leavitt’s Cepheids and comparing her work with modern methods. Here, we re-examine the Small Magellanic Cloud Cepheids in <cite class="ltx_cite ltx_citemacro_cite">Leavitt & Pickering (<a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#bib.bib21" title="">1912</a>)</cite> using modern datasets and methods and compare with Leavitt’s results. Furthermore, we show that Leavitt’s work is remarkably consistent with contemporary observations for the same sample.</p> </div> <div class="ltx_para" id="S1.p5"> <p class="ltx_p" id="S1.p5.1">First, in Sect. <a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#S2" title="2 Cepheid sample ‣ The Legacy of Henrietta Leavitt: A Re-analysis of the First Cepheid Period-Luminosity Relation"><span class="ltx_text ltx_ref_tag">2</span></a>, we identify the Cepheids adopted in Leavitt’s study, compare their pulsation periods with recent catalogs, and verify their classification and pulsation modes. In Sect. <a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#S3" title="3 Cepheid light curves ‣ The Legacy of Henrietta Leavitt: A Re-analysis of the First Cepheid Period-Luminosity Relation"><span class="ltx_text ltx_ref_tag">3</span></a> we construct Cepheid light curves from the data collected in Leavitt’s notebook <cite class="ltx_cite ltx_citemacro_citep">(Leavitt, <a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#bib.bib19" title="">1905</a>)</cite> and compare them with modern light curves obtained at optical wavelengths. Then, in Sect. <a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#S4" title="4 Period-Luminosity relation ‣ The Legacy of Henrietta Leavitt: A Re-analysis of the First Cepheid Period-Luminosity Relation"><span class="ltx_text ltx_ref_tag">4</span></a> we use recent and well-covered Cepheid light curves to reproduce an updated version of Leavitt’s law in the reddening-free <math alttext="m_{VI}^{W}" class="ltx_Math" display="inline" id="S1.p5.1.m1.1"><semantics id="S1.p5.1.m1.1a"><msubsup id="S1.p5.1.m1.1.1" xref="S1.p5.1.m1.1.1.cmml"><mi id="S1.p5.1.m1.1.1.2.2" xref="S1.p5.1.m1.1.1.2.2.cmml">m</mi><mrow id="S1.p5.1.m1.1.1.2.3" xref="S1.p5.1.m1.1.1.2.3.cmml"><mi id="S1.p5.1.m1.1.1.2.3.2" xref="S1.p5.1.m1.1.1.2.3.2.cmml">V</mi><mo id="S1.p5.1.m1.1.1.2.3.1" xref="S1.p5.1.m1.1.1.2.3.1.cmml">⁢</mo><mi id="S1.p5.1.m1.1.1.2.3.3" xref="S1.p5.1.m1.1.1.2.3.3.cmml">I</mi></mrow><mi id="S1.p5.1.m1.1.1.3" xref="S1.p5.1.m1.1.1.3.cmml">W</mi></msubsup><annotation-xml encoding="MathML-Content" id="S1.p5.1.m1.1b"><apply id="S1.p5.1.m1.1.1.cmml" xref="S1.p5.1.m1.1.1"><csymbol cd="ambiguous" id="S1.p5.1.m1.1.1.1.cmml" xref="S1.p5.1.m1.1.1">superscript</csymbol><apply id="S1.p5.1.m1.1.1.2.cmml" xref="S1.p5.1.m1.1.1"><csymbol cd="ambiguous" id="S1.p5.1.m1.1.1.2.1.cmml" xref="S1.p5.1.m1.1.1">subscript</csymbol><ci id="S1.p5.1.m1.1.1.2.2.cmml" xref="S1.p5.1.m1.1.1.2.2">𝑚</ci><apply id="S1.p5.1.m1.1.1.2.3.cmml" xref="S1.p5.1.m1.1.1.2.3"><times id="S1.p5.1.m1.1.1.2.3.1.cmml" xref="S1.p5.1.m1.1.1.2.3.1"></times><ci id="S1.p5.1.m1.1.1.2.3.2.cmml" xref="S1.p5.1.m1.1.1.2.3.2">𝑉</ci><ci id="S1.p5.1.m1.1.1.2.3.3.cmml" xref="S1.p5.1.m1.1.1.2.3.3">𝐼</ci></apply></apply><ci id="S1.p5.1.m1.1.1.3.cmml" xref="S1.p5.1.m1.1.1.3">𝑊</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.1.m1.1c">m_{VI}^{W}</annotation><annotation encoding="application/x-llamapun" id="S1.p5.1.m1.1d">italic_m start_POSTSUBSCRIPT italic_V italic_I end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_W end_POSTSUPERSCRIPT</annotation></semantics></math> index. In particular, we describe the effect of the Cepheid sampling in the SMC and the choice of reddening law. Additionally, we discuss the P-L slope obtained by Leavitt and find a bias brightward at the short period end, due to the non-linearity of the plates and environmental crowding. Finally, we discuss the remarkable accuracy and precision of Leavitt’s work in Sect. <a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#S5" title="5 Discussion ‣ The Legacy of Henrietta Leavitt: A Re-analysis of the First Cepheid Period-Luminosity Relation"><span class="ltx_text ltx_ref_tag">5</span></a>. Throughout this paper, we use both Cepheid Period-Luminosity Relation (P-L Relation) and Leavitt Law in order to avoid confusion. <br class="ltx_break"/></p> </div> <figure class="ltx_table" id="S1.T1"> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_table">Table 1: </span>Sample of 25 SMC Cepheids adopted by <cite class="ltx_cite ltx_citemacro_citet">Leavitt & Pickering (<a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#bib.bib21" title="">1912</a>)</cite>. <br class="ltx_break"/></figcaption><div class="ltx_flex_figure"> <div class="ltx_flex_cell ltx_flex_size_1"> <table class="ltx_tabular ltx_centering ltx_figure_panel ltx_guessed_headers ltx_align_middle" id="S1.T1.28.28"> <thead class="ltx_thead"> <tr class="ltx_tr" id="S1.T1.3.3.3"> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" id="S1.T1.3.3.3.4">HV ID</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" id="S1.T1.3.3.3.5">OGLE ID</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" id="S1.T1.3.3.3.6">Gaia DR3 ID</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" id="S1.T1.3.3.3.7">RA</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" id="S1.T1.3.3.3.8">DEC</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" id="S1.T1.1.1.1.1"><math alttext="P^{\rm\,(a)}" class="ltx_Math" display="inline" id="S1.T1.1.1.1.1.m1.1"><semantics id="S1.T1.1.1.1.1.m1.1a"><msup id="S1.T1.1.1.1.1.m1.1.2" xref="S1.T1.1.1.1.1.m1.1.2.cmml"><mi id="S1.T1.1.1.1.1.m1.1.2.2" xref="S1.T1.1.1.1.1.m1.1.2.2.cmml">P</mi><mrow id="S1.T1.1.1.1.1.m1.1.1.1.3" xref="S1.T1.1.1.1.1.m1.1.2.cmml"><mo id="S1.T1.1.1.1.1.m1.1.1.1.3.1" stretchy="false" xref="S1.T1.1.1.1.1.m1.1.2.cmml">(</mo><mi id="S1.T1.1.1.1.1.m1.1.1.1.1" mathvariant="normal" xref="S1.T1.1.1.1.1.m1.1.1.1.1.cmml">a</mi><mo id="S1.T1.1.1.1.1.m1.1.1.1.3.2" stretchy="false" xref="S1.T1.1.1.1.1.m1.1.2.cmml">)</mo></mrow></msup><annotation-xml encoding="MathML-Content" id="S1.T1.1.1.1.1.m1.1b"><apply id="S1.T1.1.1.1.1.m1.1.2.cmml" xref="S1.T1.1.1.1.1.m1.1.2"><csymbol cd="ambiguous" id="S1.T1.1.1.1.1.m1.1.2.1.cmml" xref="S1.T1.1.1.1.1.m1.1.2">superscript</csymbol><ci id="S1.T1.1.1.1.1.m1.1.2.2.cmml" xref="S1.T1.1.1.1.1.m1.1.2.2">𝑃</ci><ci id="S1.T1.1.1.1.1.m1.1.1.1.1.cmml" xref="S1.T1.1.1.1.1.m1.1.1.1.1">a</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.T1.1.1.1.1.m1.1c">P^{\rm\,(a)}</annotation><annotation encoding="application/x-llamapun" id="S1.T1.1.1.1.1.m1.1d">italic_P start_POSTSUPERSCRIPT ( roman_a ) end_POSTSUPERSCRIPT</annotation></semantics></math></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" id="S1.T1.2.2.2.2"><math alttext="P^{\rm\,(b)}" class="ltx_Math" display="inline" id="S1.T1.2.2.2.2.m1.1"><semantics id="S1.T1.2.2.2.2.m1.1a"><msup id="S1.T1.2.2.2.2.m1.1.2" xref="S1.T1.2.2.2.2.m1.1.2.cmml"><mi id="S1.T1.2.2.2.2.m1.1.2.2" xref="S1.T1.2.2.2.2.m1.1.2.2.cmml">P</mi><mrow id="S1.T1.2.2.2.2.m1.1.1.1.3" xref="S1.T1.2.2.2.2.m1.1.2.cmml"><mo id="S1.T1.2.2.2.2.m1.1.1.1.3.1" stretchy="false" xref="S1.T1.2.2.2.2.m1.1.2.cmml">(</mo><mi id="S1.T1.2.2.2.2.m1.1.1.1.1" mathvariant="normal" xref="S1.T1.2.2.2.2.m1.1.1.1.1.cmml">b</mi><mo id="S1.T1.2.2.2.2.m1.1.1.1.3.2" stretchy="false" xref="S1.T1.2.2.2.2.m1.1.2.cmml">)</mo></mrow></msup><annotation-xml encoding="MathML-Content" id="S1.T1.2.2.2.2.m1.1b"><apply id="S1.T1.2.2.2.2.m1.1.2.cmml" xref="S1.T1.2.2.2.2.m1.1.2"><csymbol cd="ambiguous" id="S1.T1.2.2.2.2.m1.1.2.1.cmml" xref="S1.T1.2.2.2.2.m1.1.2">superscript</csymbol><ci id="S1.T1.2.2.2.2.m1.1.2.2.cmml" xref="S1.T1.2.2.2.2.m1.1.2.2">𝑃</ci><ci id="S1.T1.2.2.2.2.m1.1.1.1.1.cmml" xref="S1.T1.2.2.2.2.m1.1.1.1.1">b</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.T1.2.2.2.2.m1.1c">P^{\rm\,(b)}</annotation><annotation encoding="application/x-llamapun" id="S1.T1.2.2.2.2.m1.1d">italic_P start_POSTSUPERSCRIPT ( roman_b ) end_POSTSUPERSCRIPT</annotation></semantics></math></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" id="S1.T1.3.3.3.3"><math alttext="m_{VI}^{W}\,{}^{\rm\,(c)}" class="ltx_math_unparsed" display="inline" id="S1.T1.3.3.3.3.m1.1"><semantics id="S1.T1.3.3.3.3.m1.1a"><mrow id="S1.T1.3.3.3.3.m1.1b"><msubsup id="S1.T1.3.3.3.3.m1.1.2"><mi id="S1.T1.3.3.3.3.m1.1.2.2.2">m</mi><mrow id="S1.T1.3.3.3.3.m1.1.2.2.3"><mi id="S1.T1.3.3.3.3.m1.1.2.2.3.2">V</mi><mo id="S1.T1.3.3.3.3.m1.1.2.2.3.1">⁢</mo><mi id="S1.T1.3.3.3.3.m1.1.2.2.3.3">I</mi></mrow><mi id="S1.T1.3.3.3.3.m1.1.2.3">W</mi></msubsup><msup id="S1.T1.3.3.3.3.m1.1.1"><mi id="S1.T1.3.3.3.3.m1.1.1a"></mi><mrow id="S1.T1.3.3.3.3.m1.1.1.1.3"><mo id="S1.T1.3.3.3.3.m1.1.1.1.3.1" stretchy="false">(</mo><mi id="S1.T1.3.3.3.3.m1.1.1.1.1" mathvariant="normal">c</mi><mo id="S1.T1.3.3.3.3.m1.1.1.1.3.2" stretchy="false">)</mo></mrow></msup></mrow><annotation encoding="application/x-tex" id="S1.T1.3.3.3.3.m1.1c">m_{VI}^{W}\,{}^{\rm\,(c)}</annotation><annotation encoding="application/x-llamapun" id="S1.T1.3.3.3.3.m1.1d">italic_m start_POSTSUBSCRIPT italic_V italic_I end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_W end_POSTSUPERSCRIPT start_FLOATSUPERSCRIPT ( roman_c ) end_FLOATSUPERSCRIPT</annotation></semantics></math></th> </tr> </thead> <tbody class="ltx_tbody"> <tr class="ltx_tr" id="S1.T1.28.28.29.1"> <td class="ltx_td" id="S1.T1.28.28.29.1.1"></td> <th class="ltx_td ltx_th ltx_th_column" id="S1.T1.28.28.29.1.2"></th> <th class="ltx_td ltx_th ltx_th_column" id="S1.T1.28.28.29.1.3"></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column" id="S1.T1.28.28.29.1.4">(deg)</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column" id="S1.T1.28.28.29.1.5">(deg)</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column" id="S1.T1.28.28.29.1.6">(days)</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column" id="S1.T1.28.28.29.1.7">(days)</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column" id="S1.T1.28.28.29.1.8">(mag)</th> </tr> <tr class="ltx_tr" id="S1.T1.4.4.4"> <td class="ltx_td ltx_align_center ltx_border_t" id="S1.T1.4.4.4.2">HV 1505</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S1.T1.4.4.4.3">OGLE SMC-CEP-1581</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S1.T1.4.4.4.4">4689056005984060032</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S1.T1.4.4.4.5">12.7421</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S1.T1.4.4.4.6">-72.0740</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S1.T1.4.4.4.7">1.253</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S1.T1.4.4.4.8">1.251</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S1.T1.4.4.4.1"><math alttext="15.960" class="ltx_Math" display="inline" id="S1.T1.4.4.4.1.m1.1"><semantics id="S1.T1.4.4.4.1.m1.1a"><mn id="S1.T1.4.4.4.1.m1.1.1" xref="S1.T1.4.4.4.1.m1.1.1.cmml">15.960</mn><annotation-xml encoding="MathML-Content" id="S1.T1.4.4.4.1.m1.1b"><cn id="S1.T1.4.4.4.1.m1.1.1.cmml" type="float" xref="S1.T1.4.4.4.1.m1.1.1">15.960</cn></annotation-xml><annotation encoding="application/x-tex" id="S1.T1.4.4.4.1.m1.1c">15.960</annotation><annotation encoding="application/x-llamapun" id="S1.T1.4.4.4.1.m1.1d">15.960</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S1.T1.5.5.5"> <td class="ltx_td ltx_align_center" id="S1.T1.5.5.5.2">HV 1436</td> <td class="ltx_td ltx_align_center" id="S1.T1.5.5.5.3">OGLE SMC-CEP-1113</td> <td class="ltx_td ltx_align_center" id="S1.T1.5.5.5.4">4689029862552916736</td> <td class="ltx_td ltx_align_center" id="S1.T1.5.5.5.5">12.0321</td> <td class="ltx_td ltx_align_center" id="S1.T1.5.5.5.6">-72.4331</td> <td class="ltx_td ltx_align_center" id="S1.T1.5.5.5.7">1.664</td> <td class="ltx_td ltx_align_center" id="S1.T1.5.5.5.8">1.656</td> <td class="ltx_td ltx_align_center" id="S1.T1.5.5.5.1"><math alttext="15.784" class="ltx_Math" display="inline" id="S1.T1.5.5.5.1.m1.1"><semantics id="S1.T1.5.5.5.1.m1.1a"><mn id="S1.T1.5.5.5.1.m1.1.1" xref="S1.T1.5.5.5.1.m1.1.1.cmml">15.784</mn><annotation-xml encoding="MathML-Content" id="S1.T1.5.5.5.1.m1.1b"><cn id="S1.T1.5.5.5.1.m1.1.1.cmml" type="float" xref="S1.T1.5.5.5.1.m1.1.1">15.784</cn></annotation-xml><annotation encoding="application/x-tex" id="S1.T1.5.5.5.1.m1.1c">15.784</annotation><annotation encoding="application/x-llamapun" id="S1.T1.5.5.5.1.m1.1d">15.784</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S1.T1.6.6.6"> <td class="ltx_td ltx_align_center" id="S1.T1.6.6.6.2">HV 1446</td> <td class="ltx_td ltx_align_center" id="S1.T1.6.6.6.3">OGLE SMC-CEP-1180</td> <td class="ltx_td ltx_align_center" id="S1.T1.6.6.6.4">4689029931272382848</td> <td class="ltx_td ltx_align_center" id="S1.T1.6.6.6.5">12.1376</td> <td class="ltx_td ltx_align_center" id="S1.T1.6.6.6.6">-72.4194</td> <td class="ltx_td ltx_align_center" id="S1.T1.6.6.6.7">1.762</td> <td class="ltx_td ltx_align_center" id="S1.T1.6.6.6.8">1.872</td> <td class="ltx_td ltx_align_center" id="S1.T1.6.6.6.1"><math alttext="15.679" class="ltx_Math" display="inline" id="S1.T1.6.6.6.1.m1.1"><semantics id="S1.T1.6.6.6.1.m1.1a"><mn id="S1.T1.6.6.6.1.m1.1.1" xref="S1.T1.6.6.6.1.m1.1.1.cmml">15.679</mn><annotation-xml encoding="MathML-Content" id="S1.T1.6.6.6.1.m1.1b"><cn id="S1.T1.6.6.6.1.m1.1.1.cmml" type="float" xref="S1.T1.6.6.6.1.m1.1.1">15.679</cn></annotation-xml><annotation encoding="application/x-tex" id="S1.T1.6.6.6.1.m1.1c">15.679</annotation><annotation encoding="application/x-llamapun" id="S1.T1.6.6.6.1.m1.1d">15.679</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S1.T1.7.7.7"> <td class="ltx_td ltx_align_center" id="S1.T1.7.7.7.2">HV 1506</td> <td class="ltx_td ltx_align_center" id="S1.T1.7.7.7.3">OGLE SMC-CEP-1562</td> <td class="ltx_td ltx_align_center" id="S1.T1.7.7.7.4">4689037898398780544</td> <td class="ltx_td ltx_align_center" id="S1.T1.7.7.7.5">12.7270</td> <td class="ltx_td ltx_align_center" id="S1.T1.7.7.7.6">-72.3104</td> <td class="ltx_td ltx_align_center" id="S1.T1.7.7.7.7">1.875</td> <td class="ltx_td ltx_align_center" id="S1.T1.7.7.7.8">1.886</td> <td class="ltx_td ltx_align_center" id="S1.T1.7.7.7.1"><math alttext="15.634" class="ltx_Math" display="inline" id="S1.T1.7.7.7.1.m1.1"><semantics id="S1.T1.7.7.7.1.m1.1a"><mn id="S1.T1.7.7.7.1.m1.1.1" xref="S1.T1.7.7.7.1.m1.1.1.cmml">15.634</mn><annotation-xml encoding="MathML-Content" id="S1.T1.7.7.7.1.m1.1b"><cn id="S1.T1.7.7.7.1.m1.1.1.cmml" type="float" xref="S1.T1.7.7.7.1.m1.1.1">15.634</cn></annotation-xml><annotation encoding="application/x-tex" id="S1.T1.7.7.7.1.m1.1c">15.634</annotation><annotation encoding="application/x-llamapun" id="S1.T1.7.7.7.1.m1.1d">15.634</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S1.T1.8.8.8"> <td class="ltx_td ltx_align_center" id="S1.T1.8.8.8.2">HV 1413</td> <td class="ltx_td ltx_align_center" id="S1.T1.8.8.8.3">OGLE SMC-CEP-0770</td> <td class="ltx_td ltx_align_center" id="S1.T1.8.8.8.4">4685829218556335616</td> <td class="ltx_td ltx_align_center" id="S1.T1.8.8.8.5">11.4177</td> <td class="ltx_td ltx_align_center" id="S1.T1.8.8.8.6">-73.5713</td> <td class="ltx_td ltx_align_center" id="S1.T1.8.8.8.7">2.174</td> <td class="ltx_td ltx_align_center" id="S1.T1.8.8.8.8">2.154</td> <td class="ltx_td ltx_align_center" id="S1.T1.8.8.8.1"><math alttext="15.473" class="ltx_Math" display="inline" id="S1.T1.8.8.8.1.m1.1"><semantics id="S1.T1.8.8.8.1.m1.1a"><mn id="S1.T1.8.8.8.1.m1.1.1" xref="S1.T1.8.8.8.1.m1.1.1.cmml">15.473</mn><annotation-xml encoding="MathML-Content" id="S1.T1.8.8.8.1.m1.1b"><cn id="S1.T1.8.8.8.1.m1.1.1.cmml" type="float" xref="S1.T1.8.8.8.1.m1.1.1">15.473</cn></annotation-xml><annotation encoding="application/x-tex" id="S1.T1.8.8.8.1.m1.1c">15.473</annotation><annotation encoding="application/x-llamapun" id="S1.T1.8.8.8.1.m1.1d">15.473</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S1.T1.9.9.9"> <td class="ltx_td ltx_align_center" id="S1.T1.9.9.9.2">HV 1460</td> <td class="ltx_td ltx_align_center" id="S1.T1.9.9.9.3">OGLE SMC-CEP-1298</td> <td class="ltx_td ltx_align_center" id="S1.T1.9.9.9.4">4689246260148791680</td> <td class="ltx_td ltx_align_center" id="S1.T1.9.9.9.5">12.3241</td> <td class="ltx_td ltx_align_center" id="S1.T1.9.9.9.6">-72.0299</td> <td class="ltx_td ltx_align_center" id="S1.T1.9.9.9.7">2.913</td> <td class="ltx_td ltx_align_center" id="S1.T1.9.9.9.8">2.913</td> <td class="ltx_td ltx_align_center" id="S1.T1.9.9.9.1"><math alttext="14.689" class="ltx_Math" display="inline" id="S1.T1.9.9.9.1.m1.1"><semantics id="S1.T1.9.9.9.1.m1.1a"><mn id="S1.T1.9.9.9.1.m1.1.1" xref="S1.T1.9.9.9.1.m1.1.1.cmml">14.689</mn><annotation-xml encoding="MathML-Content" id="S1.T1.9.9.9.1.m1.1b"><cn id="S1.T1.9.9.9.1.m1.1.1.cmml" type="float" xref="S1.T1.9.9.9.1.m1.1.1">14.689</cn></annotation-xml><annotation encoding="application/x-tex" id="S1.T1.9.9.9.1.m1.1c">14.689</annotation><annotation encoding="application/x-llamapun" id="S1.T1.9.9.9.1.m1.1d">14.689</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S1.T1.10.10.10"> <td class="ltx_td ltx_align_center" id="S1.T1.10.10.10.2">HV 1422</td> <td class="ltx_td ltx_align_center" id="S1.T1.10.10.10.3">OGLE SMC-CEP-0811</td> <td class="ltx_td ltx_align_center" id="S1.T1.10.10.10.4">4685780668235969152</td> <td class="ltx_td ltx_align_center" id="S1.T1.10.10.10.5">11.5128</td> <td class="ltx_td ltx_align_center" id="S1.T1.10.10.10.6">-73.6807</td> <td class="ltx_td ltx_align_center" id="S1.T1.10.10.10.7">3.501</td> <td class="ltx_td ltx_align_center" id="S1.T1.10.10.10.8">3.501</td> <td class="ltx_td ltx_align_center" id="S1.T1.10.10.10.1"><math alttext="14.930" class="ltx_Math" display="inline" id="S1.T1.10.10.10.1.m1.1"><semantics id="S1.T1.10.10.10.1.m1.1a"><mn id="S1.T1.10.10.10.1.m1.1.1" xref="S1.T1.10.10.10.1.m1.1.1.cmml">14.930</mn><annotation-xml encoding="MathML-Content" id="S1.T1.10.10.10.1.m1.1b"><cn id="S1.T1.10.10.10.1.m1.1.1.cmml" type="float" xref="S1.T1.10.10.10.1.m1.1.1">14.930</cn></annotation-xml><annotation encoding="application/x-tex" id="S1.T1.10.10.10.1.m1.1c">14.930</annotation><annotation encoding="application/x-llamapun" id="S1.T1.10.10.10.1.m1.1d">14.930</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S1.T1.11.11.11"> <td class="ltx_td ltx_align_center" id="S1.T1.11.11.11.2">HV 842</td> <td class="ltx_td ltx_align_center" id="S1.T1.11.11.11.3">OGLE SMC-CEP-2837</td> <td class="ltx_td ltx_align_center" id="S1.T1.11.11.11.4">4688997358257544192</td> <td class="ltx_td ltx_align_center" id="S1.T1.11.11.11.5">14.6204</td> <td class="ltx_td ltx_align_center" id="S1.T1.11.11.11.6">-72.4444</td> <td class="ltx_td ltx_align_center" id="S1.T1.11.11.11.7">4.289</td> <td class="ltx_td ltx_align_center" id="S1.T1.11.11.11.8">4.289</td> <td class="ltx_td ltx_align_center" id="S1.T1.11.11.11.1"><math alttext="14.506" class="ltx_Math" display="inline" id="S1.T1.11.11.11.1.m1.1"><semantics id="S1.T1.11.11.11.1.m1.1a"><mn id="S1.T1.11.11.11.1.m1.1.1" xref="S1.T1.11.11.11.1.m1.1.1.cmml">14.506</mn><annotation-xml encoding="MathML-Content" id="S1.T1.11.11.11.1.m1.1b"><cn id="S1.T1.11.11.11.1.m1.1.1.cmml" type="float" xref="S1.T1.11.11.11.1.m1.1.1">14.506</cn></annotation-xml><annotation encoding="application/x-tex" id="S1.T1.11.11.11.1.m1.1c">14.506</annotation><annotation encoding="application/x-llamapun" id="S1.T1.11.11.11.1.m1.1d">14.506</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S1.T1.12.12.12"> <td class="ltx_td ltx_align_center" id="S1.T1.12.12.12.2">HV 1425</td> <td class="ltx_td ltx_align_center" id="S1.T1.12.12.12.3">OGLE SMC-CEP-0915</td> <td class="ltx_td ltx_align_center" id="S1.T1.12.12.12.4">4689027457371425664</td> <td class="ltx_td ltx_align_center" id="S1.T1.12.12.12.5">11.7102</td> <td class="ltx_td ltx_align_center" id="S1.T1.12.12.12.6">-72.5433</td> <td class="ltx_td ltx_align_center" id="S1.T1.12.12.12.7">4.547</td> <td class="ltx_td ltx_align_center" id="S1.T1.12.12.12.8">4.547</td> <td class="ltx_td ltx_align_center" id="S1.T1.12.12.12.1"><math alttext="14.377" class="ltx_Math" display="inline" id="S1.T1.12.12.12.1.m1.1"><semantics id="S1.T1.12.12.12.1.m1.1a"><mn id="S1.T1.12.12.12.1.m1.1.1" xref="S1.T1.12.12.12.1.m1.1.1.cmml">14.377</mn><annotation-xml encoding="MathML-Content" id="S1.T1.12.12.12.1.m1.1b"><cn id="S1.T1.12.12.12.1.m1.1.1.cmml" type="float" xref="S1.T1.12.12.12.1.m1.1.1">14.377</cn></annotation-xml><annotation encoding="application/x-tex" id="S1.T1.12.12.12.1.m1.1c">14.377</annotation><annotation encoding="application/x-llamapun" id="S1.T1.12.12.12.1.m1.1d">14.377</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S1.T1.13.13.13"> <td class="ltx_td ltx_align_center" id="S1.T1.13.13.13.2">HV 1742</td> <td class="ltx_td ltx_align_center" id="S1.T1.13.13.13.3">OGLE SMC-CEP-2722</td> <td class="ltx_td ltx_align_center" id="S1.T1.13.13.13.4">4685992564797958912</td> <td class="ltx_td ltx_align_center" id="S1.T1.13.13.13.5">14.4179</td> <td class="ltx_td ltx_align_center" id="S1.T1.13.13.13.6">-72.5456</td> <td class="ltx_td ltx_align_center" id="S1.T1.13.13.13.7">4.987</td> <td class="ltx_td ltx_align_center" id="S1.T1.13.13.13.8">4.987</td> <td class="ltx_td ltx_align_center" id="S1.T1.13.13.13.1"><math alttext="14.081" class="ltx_Math" display="inline" id="S1.T1.13.13.13.1.m1.1"><semantics id="S1.T1.13.13.13.1.m1.1a"><mn id="S1.T1.13.13.13.1.m1.1.1" xref="S1.T1.13.13.13.1.m1.1.1.cmml">14.081</mn><annotation-xml encoding="MathML-Content" id="S1.T1.13.13.13.1.m1.1b"><cn id="S1.T1.13.13.13.1.m1.1.1.cmml" type="float" xref="S1.T1.13.13.13.1.m1.1.1">14.081</cn></annotation-xml><annotation encoding="application/x-tex" id="S1.T1.13.13.13.1.m1.1c">14.081</annotation><annotation encoding="application/x-llamapun" id="S1.T1.13.13.13.1.m1.1d">14.081</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S1.T1.14.14.14"> <td class="ltx_td ltx_align_center" id="S1.T1.14.14.14.2">HV 1646</td> <td class="ltx_td ltx_align_center" id="S1.T1.14.14.14.3">OGLE SMC-CEP-2375</td> <td class="ltx_td ltx_align_center" id="S1.T1.14.14.14.4">4685985795927276928</td> <td class="ltx_td ltx_align_center" id="S1.T1.14.14.14.5">13.8376</td> <td class="ltx_td ltx_align_center" id="S1.T1.14.14.14.6">-72.6751</td> <td class="ltx_td ltx_align_center" id="S1.T1.14.14.14.7">5.311</td> <td class="ltx_td ltx_align_center" id="S1.T1.14.14.14.8">5.234</td> <td class="ltx_td ltx_align_center" id="S1.T1.14.14.14.1"><math alttext="14.038" class="ltx_Math" display="inline" id="S1.T1.14.14.14.1.m1.1"><semantics id="S1.T1.14.14.14.1.m1.1a"><mn id="S1.T1.14.14.14.1.m1.1.1" xref="S1.T1.14.14.14.1.m1.1.1.cmml">14.038</mn><annotation-xml encoding="MathML-Content" id="S1.T1.14.14.14.1.m1.1b"><cn id="S1.T1.14.14.14.1.m1.1.1.cmml" type="float" xref="S1.T1.14.14.14.1.m1.1.1">14.038</cn></annotation-xml><annotation encoding="application/x-tex" id="S1.T1.14.14.14.1.m1.1c">14.038</annotation><annotation encoding="application/x-llamapun" id="S1.T1.14.14.14.1.m1.1d">14.038</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S1.T1.15.15.15"> <td class="ltx_td ltx_align_center" id="S1.T1.15.15.15.2">HV 1649</td> <td class="ltx_td ltx_align_center" id="S1.T1.15.15.15.3">OGLE SMC-CEP-2384</td> <td class="ltx_td ltx_align_center" id="S1.T1.15.15.15.4">4685985521049473408</td> <td class="ltx_td ltx_align_center" id="S1.T1.15.15.15.5">13.8447</td> <td class="ltx_td ltx_align_center" id="S1.T1.15.15.15.6">-72.7043</td> <td class="ltx_td ltx_align_center" id="S1.T1.15.15.15.7">5.323</td> <td class="ltx_td ltx_align_center" id="S1.T1.15.15.15.8">5.324</td> <td class="ltx_td ltx_align_center" id="S1.T1.15.15.15.1"><math alttext="14.124" class="ltx_Math" display="inline" id="S1.T1.15.15.15.1.m1.1"><semantics id="S1.T1.15.15.15.1.m1.1a"><mn id="S1.T1.15.15.15.1.m1.1.1" xref="S1.T1.15.15.15.1.m1.1.1.cmml">14.124</mn><annotation-xml encoding="MathML-Content" id="S1.T1.15.15.15.1.m1.1b"><cn id="S1.T1.15.15.15.1.m1.1.1.cmml" type="float" xref="S1.T1.15.15.15.1.m1.1.1">14.124</cn></annotation-xml><annotation encoding="application/x-tex" id="S1.T1.15.15.15.1.m1.1c">14.124</annotation><annotation encoding="application/x-llamapun" id="S1.T1.15.15.15.1.m1.1d">14.124</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S1.T1.16.16.16"> <td class="ltx_td ltx_align_center" id="S1.T1.16.16.16.2">HV 1492</td> <td class="ltx_td ltx_align_center" id="S1.T1.16.16.16.3">OGLE SMC-CEP-1492</td> <td class="ltx_td ltx_align_center" id="S1.T1.16.16.16.4">4689031095213191808</td> <td class="ltx_td ltx_align_center" id="S1.T1.16.16.16.5">12.6072</td> <td class="ltx_td ltx_align_center" id="S1.T1.16.16.16.6">-72.4619</td> <td class="ltx_td ltx_align_center" id="S1.T1.16.16.16.7">6.293</td> <td class="ltx_td ltx_align_center" id="S1.T1.16.16.16.8">6.292</td> <td class="ltx_td ltx_align_center" id="S1.T1.16.16.16.1"><math alttext="13.798" class="ltx_Math" display="inline" id="S1.T1.16.16.16.1.m1.1"><semantics id="S1.T1.16.16.16.1.m1.1a"><mn id="S1.T1.16.16.16.1.m1.1.1" xref="S1.T1.16.16.16.1.m1.1.1.cmml">13.798</mn><annotation-xml encoding="MathML-Content" id="S1.T1.16.16.16.1.m1.1b"><cn id="S1.T1.16.16.16.1.m1.1.1.cmml" type="float" xref="S1.T1.16.16.16.1.m1.1.1">13.798</cn></annotation-xml><annotation encoding="application/x-tex" id="S1.T1.16.16.16.1.m1.1c">13.798</annotation><annotation encoding="application/x-llamapun" id="S1.T1.16.16.16.1.m1.1d">13.798</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S1.T1.17.17.17"> <td class="ltx_td ltx_align_center" id="S1.T1.17.17.17.2">HV 1400</td> <td class="ltx_td ltx_align_center" id="S1.T1.17.17.17.3">OGLE SMC-CEP-0755</td> <td class="ltx_td ltx_align_center" id="S1.T1.17.17.17.4">4689043361630672768</td> <td class="ltx_td ltx_align_center" id="S1.T1.17.17.17.5">11.3614</td> <td class="ltx_td ltx_align_center" id="S1.T1.17.17.17.6">-72.4531</td> <td class="ltx_td ltx_align_center" id="S1.T1.17.17.17.7">6.650</td> <td class="ltx_td ltx_align_center" id="S1.T1.17.17.17.8">6.648</td> <td class="ltx_td ltx_align_center" id="S1.T1.17.17.17.1"><math alttext="13.760" class="ltx_Math" display="inline" id="S1.T1.17.17.17.1.m1.1"><semantics id="S1.T1.17.17.17.1.m1.1a"><mn id="S1.T1.17.17.17.1.m1.1.1" xref="S1.T1.17.17.17.1.m1.1.1.cmml">13.760</mn><annotation-xml encoding="MathML-Content" id="S1.T1.17.17.17.1.m1.1b"><cn id="S1.T1.17.17.17.1.m1.1.1.cmml" type="float" xref="S1.T1.17.17.17.1.m1.1.1">13.760</cn></annotation-xml><annotation encoding="application/x-tex" id="S1.T1.17.17.17.1.m1.1c">13.760</annotation><annotation encoding="application/x-llamapun" id="S1.T1.17.17.17.1.m1.1d">13.760</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S1.T1.18.18.18"> <td class="ltx_td ltx_align_center" id="S1.T1.18.18.18.2">HV 1355</td> <td class="ltx_td ltx_align_center" id="S1.T1.18.18.18.3">OGLE SMC-CEP-0404</td> <td class="ltx_td ltx_align_center" id="S1.T1.18.18.18.4">4688847133137909120</td> <td class="ltx_td ltx_align_center" id="S1.T1.18.18.18.5">10.3501</td> <td class="ltx_td ltx_align_center" id="S1.T1.18.18.18.6">-73.3613</td> <td class="ltx_td ltx_align_center" id="S1.T1.18.18.18.7">7.483</td> <td class="ltx_td ltx_align_center" id="S1.T1.18.18.18.8">7.482</td> <td class="ltx_td ltx_align_center" id="S1.T1.18.18.18.1"><math alttext="13.613" class="ltx_Math" display="inline" id="S1.T1.18.18.18.1.m1.1"><semantics id="S1.T1.18.18.18.1.m1.1a"><mn id="S1.T1.18.18.18.1.m1.1.1" xref="S1.T1.18.18.18.1.m1.1.1.cmml">13.613</mn><annotation-xml encoding="MathML-Content" id="S1.T1.18.18.18.1.m1.1b"><cn id="S1.T1.18.18.18.1.m1.1.1.cmml" type="float" xref="S1.T1.18.18.18.1.m1.1.1">13.613</cn></annotation-xml><annotation encoding="application/x-tex" id="S1.T1.18.18.18.1.m1.1c">13.613</annotation><annotation encoding="application/x-llamapun" id="S1.T1.18.18.18.1.m1.1d">13.613</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S1.T1.19.19.19"> <td class="ltx_td ltx_align_center" id="S1.T1.19.19.19.2">HV 1374</td> <td class="ltx_td ltx_align_center" id="S1.T1.19.19.19.3">OGLE SMC-CEP-0512</td> <td class="ltx_td ltx_align_center" id="S1.T1.19.19.19.4">4685827638001043712</td> <td class="ltx_td ltx_align_center" id="S1.T1.19.19.19.5">10.7716</td> <td class="ltx_td ltx_align_center" id="S1.T1.19.19.19.6">-73.5666</td> <td class="ltx_td ltx_align_center" id="S1.T1.19.19.19.7">8.397</td> <td class="ltx_td ltx_align_center" id="S1.T1.19.19.19.8">8.396</td> <td class="ltx_td ltx_align_center" id="S1.T1.19.19.19.1"><math alttext="13.502" class="ltx_Math" display="inline" id="S1.T1.19.19.19.1.m1.1"><semantics id="S1.T1.19.19.19.1.m1.1a"><mn id="S1.T1.19.19.19.1.m1.1.1" xref="S1.T1.19.19.19.1.m1.1.1.cmml">13.502</mn><annotation-xml encoding="MathML-Content" id="S1.T1.19.19.19.1.m1.1b"><cn id="S1.T1.19.19.19.1.m1.1.1.cmml" type="float" xref="S1.T1.19.19.19.1.m1.1.1">13.502</cn></annotation-xml><annotation encoding="application/x-tex" id="S1.T1.19.19.19.1.m1.1c">13.502</annotation><annotation encoding="application/x-llamapun" id="S1.T1.19.19.19.1.m1.1d">13.502</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S1.T1.20.20.20"> <td class="ltx_td ltx_align_center" id="S1.T1.20.20.20.2">HV 818</td> <td class="ltx_td ltx_align_center" id="S1.T1.20.20.20.3">OGLE SMC-CEP-0351</td> <td class="ltx_td ltx_align_center" id="S1.T1.20.20.20.4">4685813619225612288</td> <td class="ltx_td ltx_align_center" id="S1.T1.20.20.20.5">10.0985</td> <td class="ltx_td ltx_align_center" id="S1.T1.20.20.20.6">-73.6736</td> <td class="ltx_td ltx_align_center" id="S1.T1.20.20.20.7">10.336</td> <td class="ltx_td ltx_align_center" id="S1.T1.20.20.20.8">10.333</td> <td class="ltx_td ltx_align_center" id="S1.T1.20.20.20.1"><math alttext="13.291" class="ltx_Math" display="inline" id="S1.T1.20.20.20.1.m1.1"><semantics id="S1.T1.20.20.20.1.m1.1a"><mn id="S1.T1.20.20.20.1.m1.1.1" xref="S1.T1.20.20.20.1.m1.1.1.cmml">13.291</mn><annotation-xml encoding="MathML-Content" id="S1.T1.20.20.20.1.m1.1b"><cn id="S1.T1.20.20.20.1.m1.1.1.cmml" type="float" xref="S1.T1.20.20.20.1.m1.1.1">13.291</cn></annotation-xml><annotation encoding="application/x-tex" id="S1.T1.20.20.20.1.m1.1c">13.291</annotation><annotation encoding="application/x-llamapun" id="S1.T1.20.20.20.1.m1.1d">13.291</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S1.T1.21.21.21"> <td class="ltx_td ltx_align_center" id="S1.T1.21.21.21.2">HV 1610</td> <td class="ltx_td ltx_align_center" id="S1.T1.21.21.21.3">OGLE SMC-CEP-2230</td> <td class="ltx_td ltx_align_center" id="S1.T1.21.21.21.4">4689007047705199872</td> <td class="ltx_td ltx_align_center" id="S1.T1.21.21.21.5">13.6200</td> <td class="ltx_td ltx_align_center" id="S1.T1.21.21.21.6">-72.4109</td> <td class="ltx_td ltx_align_center" id="S1.T1.21.21.21.7">11.645</td> <td class="ltx_td ltx_align_center" id="S1.T1.21.21.21.8">11.643</td> <td class="ltx_td ltx_align_center" id="S1.T1.21.21.21.1"><math alttext="13.052" class="ltx_Math" display="inline" id="S1.T1.21.21.21.1.m1.1"><semantics id="S1.T1.21.21.21.1.m1.1a"><mn id="S1.T1.21.21.21.1.m1.1.1" xref="S1.T1.21.21.21.1.m1.1.1.cmml">13.052</mn><annotation-xml encoding="MathML-Content" id="S1.T1.21.21.21.1.m1.1b"><cn id="S1.T1.21.21.21.1.m1.1.1.cmml" type="float" xref="S1.T1.21.21.21.1.m1.1.1">13.052</cn></annotation-xml><annotation encoding="application/x-tex" id="S1.T1.21.21.21.1.m1.1c">13.052</annotation><annotation encoding="application/x-llamapun" id="S1.T1.21.21.21.1.m1.1d">13.052</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S1.T1.22.22.22"> <td class="ltx_td ltx_align_center" id="S1.T1.22.22.22.2">HV 1365</td> <td class="ltx_td ltx_align_center" id="S1.T1.22.22.22.3">OGLE SMC-CEP-0423</td> <td class="ltx_td ltx_align_center" id="S1.T1.22.22.22.4">4685821418888717696</td> <td class="ltx_td ltx_align_center" id="S1.T1.22.22.22.5">10.4545</td> <td class="ltx_td ltx_align_center" id="S1.T1.22.22.22.6">-73.7284</td> <td class="ltx_td ltx_align_center" id="S1.T1.22.22.22.7">12.417</td> <td class="ltx_td ltx_align_center" id="S1.T1.22.22.22.8">12.409</td> <td class="ltx_td ltx_align_center" id="S1.T1.22.22.22.1"><math alttext="13.246" class="ltx_Math" display="inline" id="S1.T1.22.22.22.1.m1.1"><semantics id="S1.T1.22.22.22.1.m1.1a"><mn id="S1.T1.22.22.22.1.m1.1.1" xref="S1.T1.22.22.22.1.m1.1.1.cmml">13.246</mn><annotation-xml encoding="MathML-Content" id="S1.T1.22.22.22.1.m1.1b"><cn id="S1.T1.22.22.22.1.m1.1.1.cmml" type="float" xref="S1.T1.22.22.22.1.m1.1.1">13.246</cn></annotation-xml><annotation encoding="application/x-tex" id="S1.T1.22.22.22.1.m1.1c">13.246</annotation><annotation encoding="application/x-llamapun" id="S1.T1.22.22.22.1.m1.1d">13.246</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S1.T1.23.23.23"> <td class="ltx_td ltx_align_center" id="S1.T1.23.23.23.2">HV 1351</td> <td class="ltx_td ltx_align_center" id="S1.T1.23.23.23.3">OGLE SMC-CEP-0387</td> <td class="ltx_td ltx_align_center" id="S1.T1.23.23.23.4">4685841725497006976</td> <td class="ltx_td ltx_align_center" id="S1.T1.23.23.23.5">10.2661</td> <td class="ltx_td ltx_align_center" id="S1.T1.23.23.23.6">-73.5277</td> <td class="ltx_td ltx_align_center" id="S1.T1.23.23.23.7">13.080</td> <td class="ltx_td ltx_align_center" id="S1.T1.23.23.23.8">13.091</td> <td class="ltx_td ltx_align_center" id="S1.T1.23.23.23.1"><math alttext="12.995" class="ltx_Math" display="inline" id="S1.T1.23.23.23.1.m1.1"><semantics id="S1.T1.23.23.23.1.m1.1a"><mn id="S1.T1.23.23.23.1.m1.1.1" xref="S1.T1.23.23.23.1.m1.1.1.cmml">12.995</mn><annotation-xml encoding="MathML-Content" id="S1.T1.23.23.23.1.m1.1b"><cn id="S1.T1.23.23.23.1.m1.1.1.cmml" type="float" xref="S1.T1.23.23.23.1.m1.1.1">12.995</cn></annotation-xml><annotation encoding="application/x-tex" id="S1.T1.23.23.23.1.m1.1c">12.995</annotation><annotation encoding="application/x-llamapun" id="S1.T1.23.23.23.1.m1.1d">12.995</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S1.T1.24.24.24"> <td class="ltx_td ltx_align_center" id="S1.T1.24.24.24.2">HV 827</td> <td class="ltx_td ltx_align_center" id="S1.T1.24.24.24.3">OGLE SMC-CEP-1377</td> <td class="ltx_td ltx_align_center" id="S1.T1.24.24.24.4">4689030549749347456</td> <td class="ltx_td ltx_align_center" id="S1.T1.24.24.24.5">12.4283</td> <td class="ltx_td ltx_align_center" id="S1.T1.24.24.24.6">-72.4959</td> <td class="ltx_td ltx_align_center" id="S1.T1.24.24.24.7">13.470</td> <td class="ltx_td ltx_align_center" id="S1.T1.24.24.24.8">13.464</td> <td class="ltx_td ltx_align_center" id="S1.T1.24.24.24.1"><math alttext="12.788" class="ltx_Math" display="inline" id="S1.T1.24.24.24.1.m1.1"><semantics id="S1.T1.24.24.24.1.m1.1a"><mn id="S1.T1.24.24.24.1.m1.1.1" xref="S1.T1.24.24.24.1.m1.1.1.cmml">12.788</mn><annotation-xml encoding="MathML-Content" id="S1.T1.24.24.24.1.m1.1b"><cn id="S1.T1.24.24.24.1.m1.1.1.cmml" type="float" xref="S1.T1.24.24.24.1.m1.1.1">12.788</cn></annotation-xml><annotation encoding="application/x-tex" id="S1.T1.24.24.24.1.m1.1c">12.788</annotation><annotation encoding="application/x-llamapun" id="S1.T1.24.24.24.1.m1.1d">12.788</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S1.T1.25.25.25"> <td class="ltx_td ltx_align_center" id="S1.T1.25.25.25.2">HV 822</td> <td class="ltx_td ltx_align_center" id="S1.T1.25.25.25.3">OGLE SMC-CEP-0431</td> <td class="ltx_td ltx_align_center" id="S1.T1.25.25.25.4">4685838907998450944</td> <td class="ltx_td ltx_align_center" id="S1.T1.25.25.25.5">10.4812</td> <td class="ltx_td ltx_align_center" id="S1.T1.25.25.25.6">-73.5399</td> <td class="ltx_td ltx_align_center" id="S1.T1.25.25.25.7">16.750</td> <td class="ltx_td ltx_align_center" id="S1.T1.25.25.25.8">16.742</td> <td class="ltx_td ltx_align_center" id="S1.T1.25.25.25.1"><math alttext="12.544" class="ltx_Math" display="inline" id="S1.T1.25.25.25.1.m1.1"><semantics id="S1.T1.25.25.25.1.m1.1a"><mn id="S1.T1.25.25.25.1.m1.1.1" xref="S1.T1.25.25.25.1.m1.1.1.cmml">12.544</mn><annotation-xml encoding="MathML-Content" id="S1.T1.25.25.25.1.m1.1b"><cn id="S1.T1.25.25.25.1.m1.1.1.cmml" type="float" xref="S1.T1.25.25.25.1.m1.1.1">12.544</cn></annotation-xml><annotation encoding="application/x-tex" id="S1.T1.25.25.25.1.m1.1c">12.544</annotation><annotation encoding="application/x-llamapun" id="S1.T1.25.25.25.1.m1.1d">12.544</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S1.T1.26.26.26"> <td class="ltx_td ltx_align_center" id="S1.T1.26.26.26.2">HV 823</td> <td class="ltx_td ltx_align_center" id="S1.T1.26.26.26.3">OGLE SMC-CEP-0574</td> <td class="ltx_td ltx_align_center" id="S1.T1.26.26.26.4">4685823721000358144</td> <td class="ltx_td ltx_align_center" id="S1.T1.26.26.26.5">10.9519</td> <td class="ltx_td ltx_align_center" id="S1.T1.26.26.26.6">-73.6135</td> <td class="ltx_td ltx_align_center" id="S1.T1.26.26.26.7">31.940</td> <td class="ltx_td ltx_align_center" id="S1.T1.26.26.26.8">31.928</td> <td class="ltx_td ltx_align_center" id="S1.T1.26.26.26.1"><math alttext="11.365" class="ltx_Math" display="inline" id="S1.T1.26.26.26.1.m1.1"><semantics id="S1.T1.26.26.26.1.m1.1a"><mn id="S1.T1.26.26.26.1.m1.1.1" xref="S1.T1.26.26.26.1.m1.1.1.cmml">11.365</mn><annotation-xml encoding="MathML-Content" id="S1.T1.26.26.26.1.m1.1b"><cn id="S1.T1.26.26.26.1.m1.1.1.cmml" type="float" xref="S1.T1.26.26.26.1.m1.1.1">11.365</cn></annotation-xml><annotation encoding="application/x-tex" id="S1.T1.26.26.26.1.m1.1c">11.365</annotation><annotation encoding="application/x-llamapun" id="S1.T1.26.26.26.1.m1.1d">11.365</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S1.T1.27.27.27"> <td class="ltx_td ltx_align_center" id="S1.T1.27.27.27.2">HV 824</td> <td class="ltx_td ltx_align_center" id="S1.T1.27.27.27.3">OGLE SMC-CEP-0921</td> <td class="ltx_td ltx_align_center" id="S1.T1.27.27.27.4">4688974886982934016</td> <td class="ltx_td ltx_align_center" id="S1.T1.27.27.27.5">11.7213</td> <td class="ltx_td ltx_align_center" id="S1.T1.27.27.27.6">-72.7144</td> <td class="ltx_td ltx_align_center" id="S1.T1.27.27.27.7">65.800</td> <td class="ltx_td ltx_align_center" id="S1.T1.27.27.27.8">65.937</td> <td class="ltx_td ltx_align_center" id="S1.T1.27.27.27.1"><math alttext="10.111^{\rm\,(d)}" class="ltx_Math" display="inline" id="S1.T1.27.27.27.1.m1.1"><semantics id="S1.T1.27.27.27.1.m1.1a"><msup id="S1.T1.27.27.27.1.m1.1.2" xref="S1.T1.27.27.27.1.m1.1.2.cmml"><mn id="S1.T1.27.27.27.1.m1.1.2.2" xref="S1.T1.27.27.27.1.m1.1.2.2.cmml">10.111</mn><mrow id="S1.T1.27.27.27.1.m1.1.1.1.3" xref="S1.T1.27.27.27.1.m1.1.2.cmml"><mo id="S1.T1.27.27.27.1.m1.1.1.1.3.1" stretchy="false" xref="S1.T1.27.27.27.1.m1.1.2.cmml">(</mo><mi id="S1.T1.27.27.27.1.m1.1.1.1.1" mathvariant="normal" xref="S1.T1.27.27.27.1.m1.1.1.1.1.cmml">d</mi><mo id="S1.T1.27.27.27.1.m1.1.1.1.3.2" stretchy="false" xref="S1.T1.27.27.27.1.m1.1.2.cmml">)</mo></mrow></msup><annotation-xml encoding="MathML-Content" id="S1.T1.27.27.27.1.m1.1b"><apply id="S1.T1.27.27.27.1.m1.1.2.cmml" xref="S1.T1.27.27.27.1.m1.1.2"><csymbol cd="ambiguous" id="S1.T1.27.27.27.1.m1.1.2.1.cmml" xref="S1.T1.27.27.27.1.m1.1.2">superscript</csymbol><cn id="S1.T1.27.27.27.1.m1.1.2.2.cmml" type="float" xref="S1.T1.27.27.27.1.m1.1.2.2">10.111</cn><ci id="S1.T1.27.27.27.1.m1.1.1.1.1.cmml" xref="S1.T1.27.27.27.1.m1.1.1.1.1">d</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.T1.27.27.27.1.m1.1c">10.111^{\rm\,(d)}</annotation><annotation encoding="application/x-llamapun" id="S1.T1.27.27.27.1.m1.1d">10.111 start_POSTSUPERSCRIPT ( roman_d ) end_POSTSUPERSCRIPT</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S1.T1.28.28.28"> <td class="ltx_td ltx_align_center ltx_border_b" id="S1.T1.28.28.28.2">HV 821</td> <td class="ltx_td ltx_align_center ltx_border_b" id="S1.T1.28.28.28.3">OGLE SMC-CEP-0417</td> <td class="ltx_td ltx_align_center ltx_border_b" id="S1.T1.28.28.28.4">4685821487608191104</td> <td class="ltx_td ltx_align_center ltx_border_b" id="S1.T1.28.28.28.5">10.4310</td> <td class="ltx_td ltx_align_center ltx_border_b" id="S1.T1.28.28.28.6">-73.7233</td> <td class="ltx_td ltx_align_center ltx_border_b" id="S1.T1.28.28.28.7">127.000</td> <td class="ltx_td ltx_align_center ltx_border_b" id="S1.T1.28.28.28.8">128.197</td> <td class="ltx_td ltx_align_center ltx_border_b" id="S1.T1.28.28.28.1"><math alttext="9.315^{\rm\,(d)}" class="ltx_Math" display="inline" id="S1.T1.28.28.28.1.m1.1"><semantics id="S1.T1.28.28.28.1.m1.1a"><msup id="S1.T1.28.28.28.1.m1.1.2" xref="S1.T1.28.28.28.1.m1.1.2.cmml"><mn id="S1.T1.28.28.28.1.m1.1.2.2" xref="S1.T1.28.28.28.1.m1.1.2.2.cmml">9.315</mn><mrow id="S1.T1.28.28.28.1.m1.1.1.1.3" xref="S1.T1.28.28.28.1.m1.1.2.cmml"><mo id="S1.T1.28.28.28.1.m1.1.1.1.3.1" stretchy="false" xref="S1.T1.28.28.28.1.m1.1.2.cmml">(</mo><mi id="S1.T1.28.28.28.1.m1.1.1.1.1" mathvariant="normal" xref="S1.T1.28.28.28.1.m1.1.1.1.1.cmml">d</mi><mo id="S1.T1.28.28.28.1.m1.1.1.1.3.2" stretchy="false" xref="S1.T1.28.28.28.1.m1.1.2.cmml">)</mo></mrow></msup><annotation-xml encoding="MathML-Content" id="S1.T1.28.28.28.1.m1.1b"><apply id="S1.T1.28.28.28.1.m1.1.2.cmml" xref="S1.T1.28.28.28.1.m1.1.2"><csymbol cd="ambiguous" id="S1.T1.28.28.28.1.m1.1.2.1.cmml" xref="S1.T1.28.28.28.1.m1.1.2">superscript</csymbol><cn id="S1.T1.28.28.28.1.m1.1.2.2.cmml" type="float" xref="S1.T1.28.28.28.1.m1.1.2.2">9.315</cn><ci id="S1.T1.28.28.28.1.m1.1.1.1.1.cmml" xref="S1.T1.28.28.28.1.m1.1.1.1.1">d</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.T1.28.28.28.1.m1.1c">9.315^{\rm\,(d)}</annotation><annotation encoding="application/x-llamapun" id="S1.T1.28.28.28.1.m1.1d">9.315 start_POSTSUPERSCRIPT ( roman_d ) end_POSTSUPERSCRIPT</annotation></semantics></math></td> </tr> </tbody> </table> </div> <div class="ltx_flex_break"></div> <div class="ltx_flex_cell ltx_flex_size_1"> <p class="ltx_p ltx_figure_panel" id="S1.T1.33"><span class="ltx_text ltx_font_bold" id="S1.T1.33.1">Notes:</span> (a) Original pulsation period from <cite class="ltx_cite ltx_citemacro_citet">Leavitt & Pickering (<a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#bib.bib21" title="">1912</a>)</cite>. (b) Pulsation period from the OGLE survey <cite class="ltx_cite ltx_citemacro_citep">(Soszyński et al., <a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#bib.bib28" title="">2015</a>)</cite>. (c) Apparent mean magnitude in the optical Wesenheit index, <math alttext="m_{VI}^{W}=I-1.278\,(V-I)" class="ltx_Math" display="inline" id="S1.T1.29.m1.1"><semantics id="S1.T1.29.m1.1a"><mrow id="S1.T1.29.m1.1.1" xref="S1.T1.29.m1.1.1.cmml"><msubsup id="S1.T1.29.m1.1.1.3" xref="S1.T1.29.m1.1.1.3.cmml"><mi id="S1.T1.29.m1.1.1.3.2.2" xref="S1.T1.29.m1.1.1.3.2.2.cmml">m</mi><mrow id="S1.T1.29.m1.1.1.3.2.3" xref="S1.T1.29.m1.1.1.3.2.3.cmml"><mi id="S1.T1.29.m1.1.1.3.2.3.2" xref="S1.T1.29.m1.1.1.3.2.3.2.cmml">V</mi><mo id="S1.T1.29.m1.1.1.3.2.3.1" xref="S1.T1.29.m1.1.1.3.2.3.1.cmml">⁢</mo><mi id="S1.T1.29.m1.1.1.3.2.3.3" xref="S1.T1.29.m1.1.1.3.2.3.3.cmml">I</mi></mrow><mi id="S1.T1.29.m1.1.1.3.3" xref="S1.T1.29.m1.1.1.3.3.cmml">W</mi></msubsup><mo id="S1.T1.29.m1.1.1.2" xref="S1.T1.29.m1.1.1.2.cmml">=</mo><mrow id="S1.T1.29.m1.1.1.1" xref="S1.T1.29.m1.1.1.1.cmml"><mi id="S1.T1.29.m1.1.1.1.3" xref="S1.T1.29.m1.1.1.1.3.cmml">I</mi><mo id="S1.T1.29.m1.1.1.1.2" xref="S1.T1.29.m1.1.1.1.2.cmml">−</mo><mrow id="S1.T1.29.m1.1.1.1.1" xref="S1.T1.29.m1.1.1.1.1.cmml"><mn id="S1.T1.29.m1.1.1.1.1.3" xref="S1.T1.29.m1.1.1.1.1.3.cmml">1.278</mn><mo id="S1.T1.29.m1.1.1.1.1.2" lspace="0.170em" xref="S1.T1.29.m1.1.1.1.1.2.cmml">⁢</mo><mrow id="S1.T1.29.m1.1.1.1.1.1.1" xref="S1.T1.29.m1.1.1.1.1.1.1.1.cmml"><mo id="S1.T1.29.m1.1.1.1.1.1.1.2" stretchy="false" xref="S1.T1.29.m1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S1.T1.29.m1.1.1.1.1.1.1.1" xref="S1.T1.29.m1.1.1.1.1.1.1.1.cmml"><mi id="S1.T1.29.m1.1.1.1.1.1.1.1.2" xref="S1.T1.29.m1.1.1.1.1.1.1.1.2.cmml">V</mi><mo id="S1.T1.29.m1.1.1.1.1.1.1.1.1" xref="S1.T1.29.m1.1.1.1.1.1.1.1.1.cmml">−</mo><mi id="S1.T1.29.m1.1.1.1.1.1.1.1.3" xref="S1.T1.29.m1.1.1.1.1.1.1.1.3.cmml">I</mi></mrow><mo id="S1.T1.29.m1.1.1.1.1.1.1.3" stretchy="false" xref="S1.T1.29.m1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.T1.29.m1.1b"><apply id="S1.T1.29.m1.1.1.cmml" xref="S1.T1.29.m1.1.1"><eq id="S1.T1.29.m1.1.1.2.cmml" xref="S1.T1.29.m1.1.1.2"></eq><apply id="S1.T1.29.m1.1.1.3.cmml" xref="S1.T1.29.m1.1.1.3"><csymbol cd="ambiguous" id="S1.T1.29.m1.1.1.3.1.cmml" xref="S1.T1.29.m1.1.1.3">superscript</csymbol><apply id="S1.T1.29.m1.1.1.3.2.cmml" xref="S1.T1.29.m1.1.1.3"><csymbol cd="ambiguous" id="S1.T1.29.m1.1.1.3.2.1.cmml" xref="S1.T1.29.m1.1.1.3">subscript</csymbol><ci id="S1.T1.29.m1.1.1.3.2.2.cmml" xref="S1.T1.29.m1.1.1.3.2.2">𝑚</ci><apply id="S1.T1.29.m1.1.1.3.2.3.cmml" xref="S1.T1.29.m1.1.1.3.2.3"><times id="S1.T1.29.m1.1.1.3.2.3.1.cmml" xref="S1.T1.29.m1.1.1.3.2.3.1"></times><ci id="S1.T1.29.m1.1.1.3.2.3.2.cmml" xref="S1.T1.29.m1.1.1.3.2.3.2">𝑉</ci><ci id="S1.T1.29.m1.1.1.3.2.3.3.cmml" xref="S1.T1.29.m1.1.1.3.2.3.3">𝐼</ci></apply></apply><ci id="S1.T1.29.m1.1.1.3.3.cmml" xref="S1.T1.29.m1.1.1.3.3">𝑊</ci></apply><apply id="S1.T1.29.m1.1.1.1.cmml" xref="S1.T1.29.m1.1.1.1"><minus id="S1.T1.29.m1.1.1.1.2.cmml" xref="S1.T1.29.m1.1.1.1.2"></minus><ci id="S1.T1.29.m1.1.1.1.3.cmml" xref="S1.T1.29.m1.1.1.1.3">𝐼</ci><apply id="S1.T1.29.m1.1.1.1.1.cmml" xref="S1.T1.29.m1.1.1.1.1"><times id="S1.T1.29.m1.1.1.1.1.2.cmml" xref="S1.T1.29.m1.1.1.1.1.2"></times><cn id="S1.T1.29.m1.1.1.1.1.3.cmml" type="float" xref="S1.T1.29.m1.1.1.1.1.3">1.278</cn><apply id="S1.T1.29.m1.1.1.1.1.1.1.1.cmml" xref="S1.T1.29.m1.1.1.1.1.1.1"><minus id="S1.T1.29.m1.1.1.1.1.1.1.1.1.cmml" xref="S1.T1.29.m1.1.1.1.1.1.1.1.1"></minus><ci id="S1.T1.29.m1.1.1.1.1.1.1.1.2.cmml" xref="S1.T1.29.m1.1.1.1.1.1.1.1.2">𝑉</ci><ci id="S1.T1.29.m1.1.1.1.1.1.1.1.3.cmml" xref="S1.T1.29.m1.1.1.1.1.1.1.1.3">𝐼</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.T1.29.m1.1c">m_{VI}^{W}=I-1.278\,(V-I)</annotation><annotation encoding="application/x-llamapun" id="S1.T1.29.m1.1d">italic_m start_POSTSUBSCRIPT italic_V italic_I end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_W end_POSTSUPERSCRIPT = italic_I - 1.278 ( italic_V - italic_I )</annotation></semantics></math>, built with <math alttext="V" class="ltx_Math" display="inline" id="S1.T1.30.m2.1"><semantics id="S1.T1.30.m2.1a"><mi id="S1.T1.30.m2.1.1" xref="S1.T1.30.m2.1.1.cmml">V</mi><annotation-xml encoding="MathML-Content" id="S1.T1.30.m2.1b"><ci id="S1.T1.30.m2.1.1.cmml" xref="S1.T1.30.m2.1.1">𝑉</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.T1.30.m2.1c">V</annotation><annotation encoding="application/x-llamapun" id="S1.T1.30.m2.1d">italic_V</annotation></semantics></math> and <math alttext="I" class="ltx_Math" display="inline" id="S1.T1.31.m3.1"><semantics id="S1.T1.31.m3.1a"><mi id="S1.T1.31.m3.1.1" xref="S1.T1.31.m3.1.1.cmml">I</mi><annotation-xml encoding="MathML-Content" id="S1.T1.31.m3.1b"><ci id="S1.T1.31.m3.1.1.cmml" xref="S1.T1.31.m3.1.1">𝐼</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.T1.31.m3.1c">I</annotation><annotation encoding="application/x-llamapun" id="S1.T1.31.m3.1d">italic_I</annotation></semantics></math> magnitudes from OGLE <cite class="ltx_cite ltx_citemacro_citep">(Soszyński et al., <a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#bib.bib28" title="">2015</a>)</cite> assuming <math alttext="R_{V}=2.74" class="ltx_Math" display="inline" id="S1.T1.32.m4.1"><semantics id="S1.T1.32.m4.1a"><mrow id="S1.T1.32.m4.1.1" xref="S1.T1.32.m4.1.1.cmml"><msub id="S1.T1.32.m4.1.1.2" xref="S1.T1.32.m4.1.1.2.cmml"><mi id="S1.T1.32.m4.1.1.2.2" xref="S1.T1.32.m4.1.1.2.2.cmml">R</mi><mi id="S1.T1.32.m4.1.1.2.3" xref="S1.T1.32.m4.1.1.2.3.cmml">V</mi></msub><mo id="S1.T1.32.m4.1.1.1" xref="S1.T1.32.m4.1.1.1.cmml">=</mo><mn id="S1.T1.32.m4.1.1.3" xref="S1.T1.32.m4.1.1.3.cmml">2.74</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.T1.32.m4.1b"><apply id="S1.T1.32.m4.1.1.cmml" xref="S1.T1.32.m4.1.1"><eq id="S1.T1.32.m4.1.1.1.cmml" xref="S1.T1.32.m4.1.1.1"></eq><apply id="S1.T1.32.m4.1.1.2.cmml" xref="S1.T1.32.m4.1.1.2"><csymbol cd="ambiguous" id="S1.T1.32.m4.1.1.2.1.cmml" xref="S1.T1.32.m4.1.1.2">subscript</csymbol><ci id="S1.T1.32.m4.1.1.2.2.cmml" xref="S1.T1.32.m4.1.1.2.2">𝑅</ci><ci id="S1.T1.32.m4.1.1.2.3.cmml" xref="S1.T1.32.m4.1.1.2.3">𝑉</ci></apply><cn id="S1.T1.32.m4.1.1.3.cmml" type="float" xref="S1.T1.32.m4.1.1.3">2.74</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.T1.32.m4.1c">R_{V}=2.74</annotation><annotation encoding="application/x-llamapun" id="S1.T1.32.m4.1d">italic_R start_POSTSUBSCRIPT italic_V end_POSTSUBSCRIPT = 2.74</annotation></semantics></math> (see Sect. <a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#S4.SS2" title="4.2 P-L relation with magnitudes from OGLE ‣ 4 Period-Luminosity relation ‣ The Legacy of Henrietta Leavitt: A Re-analysis of the First Cepheid Period-Luminosity Relation"><span class="ltx_text ltx_ref_tag">4.2</span></a>). (d) These <math alttext="V" class="ltx_Math" display="inline" id="S1.T1.33.m5.1"><semantics id="S1.T1.33.m5.1a"><mi id="S1.T1.33.m5.1.1" xref="S1.T1.33.m5.1.1.cmml">V</mi><annotation-xml encoding="MathML-Content" id="S1.T1.33.m5.1b"><ci id="S1.T1.33.m5.1.1.cmml" xref="S1.T1.33.m5.1.1">𝑉</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.T1.33.m5.1c">V</annotation><annotation encoding="application/x-llamapun" id="S1.T1.33.m5.1d">italic_V</annotation></semantics></math>-band mean magnitudes are not available in OGLE so they are taken from <cite class="ltx_cite ltx_citemacro_citet">Henden et al. (<a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#bib.bib13" title="">2015</a>)</cite>. <br class="ltx_break"/> <br class="ltx_break"/></p> </div> </div> </figure> <div class="ltx_pagination ltx_role_newpage"></div> </section> <section class="ltx_section" id="S2"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">2 </span>Cepheid sample</h2> <div class="ltx_para" id="S2.p1"> <p class="ltx_p" id="S2.p1.1">The first step of this analysis is to identify the 25 SMC Cepheids used to derive the original P-L relation from <cite class="ltx_cite ltx_citemacro_cite">Leavitt & Pickering (<a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#bib.bib21" title="">1912</a>)</cite> (see Fig. <a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#S1.F1" title="Figure 1 ‣ 1 Introduction ‣ The Legacy of Henrietta Leavitt: A Re-analysis of the First Cepheid Period-Luminosity Relation"><span class="ltx_text ltx_ref_tag">1</span></a>). The Cepheid names listed in her paper are in the Harvard Variable (HV) system. Their corresponding names in the OGLE survey <cite class="ltx_cite ltx_citemacro_citep">(Soszyński et al., <a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#bib.bib28" title="">2015</a>)</cite> and in the <span class="ltx_text ltx_font_italic" id="S2.p1.1.1">Gaia</span> DR3 catalog can be found using the Simbad<span class="ltx_note ltx_role_footnote" id="footnote1"><sup class="ltx_note_mark">1</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">1</sup><span class="ltx_tag ltx_tag_note">1</span><a class="ltx_ref ltx_href" href="https://simbad.u-strasbg.fr/simbad/" title="">https://simbad.u-strasbg.fr/simbad/</a></span></span></span> database. Their pulsation periods are provided in the OGLE survey and in the <span class="ltx_text ltx_font_italic" id="S2.p1.1.2">Gaia</span> DR3 catalog <cite class="ltx_cite ltx_citemacro_citep">(Gaia Collaboration et al., <a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#bib.bib8" title="">2023</a>)</cite>, they are listed in Table <a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#S1.T1" title="Table 1 ‣ 1 Introduction ‣ The Legacy of Henrietta Leavitt: A Re-analysis of the First Cepheid Period-Luminosity Relation"><span class="ltx_text ltx_ref_tag">1</span></a>.</p> </div> <div class="ltx_para" id="S2.p2"> <p class="ltx_p" id="S2.p2.1">For 16 of Leavitt’s SMC Cepheids, Table VI of <cite class="ltx_cite ltx_citemacro_cite">Leavitt (<a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#bib.bib20" title="">1907</a>)</cite> gives information about the quality of the time-series observations. They were observed between 1889 and 1906 at an average of <math alttext="\sim 41" class="ltx_Math" display="inline" id="S2.p2.1.m1.1"><semantics id="S2.p2.1.m1.1a"><mrow id="S2.p2.1.m1.1.1" xref="S2.p2.1.m1.1.1.cmml"><mi id="S2.p2.1.m1.1.1.2" xref="S2.p2.1.m1.1.1.2.cmml"></mi><mo id="S2.p2.1.m1.1.1.1" xref="S2.p2.1.m1.1.1.1.cmml">∼</mo><mn id="S2.p2.1.m1.1.1.3" xref="S2.p2.1.m1.1.1.3.cmml">41</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.p2.1.m1.1b"><apply id="S2.p2.1.m1.1.1.cmml" xref="S2.p2.1.m1.1.1"><csymbol cd="latexml" id="S2.p2.1.m1.1.1.1.cmml" xref="S2.p2.1.m1.1.1.1">similar-to</csymbol><csymbol cd="latexml" id="S2.p2.1.m1.1.1.2.cmml" xref="S2.p2.1.m1.1.1.2">absent</csymbol><cn id="S2.p2.1.m1.1.1.3.cmml" type="integer" xref="S2.p2.1.m1.1.1.3">41</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p2.1.m1.1c">\sim 41</annotation><annotation encoding="application/x-llamapun" id="S2.p2.1.m1.1d">∼ 41</annotation></semantics></math> times each. All had at least 20 observations, and two had more than 80. While these are several times the number of observations that are typically used to obtain periods for Cepheids for modern cosmic distance ladders <cite class="ltx_cite ltx_citemacro_citep">(Freedman et al., <a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#bib.bib7" title="">2001</a>; Riess et al., <a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#bib.bib25" title="">2022</a>)</cite>, it is important to remember that over such a long baseline, the observational sampling was likely to be quite sparse. In extreme cases, such as that of HV 1436, its 22 epochs of observations covered more than 2,800 pulsational cycles.</p> </div> <div class="ltx_para" id="S2.p3"> <p class="ltx_p" id="S2.p3.4">Figure <a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#S2.F2" title="Figure 2 ‣ 2 Cepheid sample ‣ The Legacy of Henrietta Leavitt: A Re-analysis of the First Cepheid Period-Luminosity Relation"><span class="ltx_text ltx_ref_tag">2</span></a> shows a comparison between the Cepheid pulsation periods measured by <cite class="ltx_cite ltx_citemacro_citet">Leavitt & Pickering (<a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#bib.bib21" title="">1912</a>)</cite> from photographic plates, and those from the OGLE catalog <cite class="ltx_cite ltx_citemacro_citep">(Soszyński et al., <a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#bib.bib28" title="">2015</a>)</cite>. Surprisingly, given the sparsity of Leavitt’s original data, the agreement is very good for most Cepheids. The largest outlier in <math alttext="\log P" class="ltx_Math" display="inline" id="S2.p3.1.m1.1"><semantics id="S2.p3.1.m1.1a"><mrow id="S2.p3.1.m1.1.1" xref="S2.p3.1.m1.1.1.cmml"><mi id="S2.p3.1.m1.1.1.1" xref="S2.p3.1.m1.1.1.1.cmml">log</mi><mo id="S2.p3.1.m1.1.1a" lspace="0.167em" xref="S2.p3.1.m1.1.1.cmml">⁡</mo><mi id="S2.p3.1.m1.1.1.2" xref="S2.p3.1.m1.1.1.2.cmml">P</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.p3.1.m1.1b"><apply id="S2.p3.1.m1.1.1.cmml" xref="S2.p3.1.m1.1.1"><log id="S2.p3.1.m1.1.1.1.cmml" xref="S2.p3.1.m1.1.1.1"></log><ci id="S2.p3.1.m1.1.1.2.cmml" xref="S2.p3.1.m1.1.1.2">𝑃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p3.1.m1.1c">\log P</annotation><annotation encoding="application/x-llamapun" id="S2.p3.1.m1.1d">roman_log italic_P</annotation></semantics></math> is HV 1446 (red point in the top left corner of Fig. <a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#S2.F2" title="Figure 2 ‣ 2 Cepheid sample ‣ The Legacy of Henrietta Leavitt: A Re-analysis of the First Cepheid Period-Luminosity Relation"><span class="ltx_text ltx_ref_tag">2</span></a>), whose 21 observations spanned 2,052 cycles. Even for this variable star, the measured periods were still within <math alttext="\sim 0.1" class="ltx_Math" display="inline" id="S2.p3.2.m2.1"><semantics id="S2.p3.2.m2.1a"><mrow id="S2.p3.2.m2.1.1" xref="S2.p3.2.m2.1.1.cmml"><mi id="S2.p3.2.m2.1.1.2" xref="S2.p3.2.m2.1.1.2.cmml"></mi><mo id="S2.p3.2.m2.1.1.1" xref="S2.p3.2.m2.1.1.1.cmml">∼</mo><mn id="S2.p3.2.m2.1.1.3" xref="S2.p3.2.m2.1.1.3.cmml">0.1</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.p3.2.m2.1b"><apply id="S2.p3.2.m2.1.1.cmml" xref="S2.p3.2.m2.1.1"><csymbol cd="latexml" id="S2.p3.2.m2.1.1.1.cmml" xref="S2.p3.2.m2.1.1.1">similar-to</csymbol><csymbol cd="latexml" id="S2.p3.2.m2.1.1.2.cmml" xref="S2.p3.2.m2.1.1.2">absent</csymbol><cn id="S2.p3.2.m2.1.1.3.cmml" type="float" xref="S2.p3.2.m2.1.1.3">0.1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p3.2.m2.1c">\sim 0.1</annotation><annotation encoding="application/x-llamapun" id="S2.p3.2.m2.1d">∼ 0.1</annotation></semantics></math> days: Leavitt found a period of 1.76 days (<math alttext="\log P=0.246" class="ltx_Math" display="inline" id="S2.p3.3.m3.1"><semantics id="S2.p3.3.m3.1a"><mrow id="S2.p3.3.m3.1.1" xref="S2.p3.3.m3.1.1.cmml"><mrow id="S2.p3.3.m3.1.1.2" xref="S2.p3.3.m3.1.1.2.cmml"><mi id="S2.p3.3.m3.1.1.2.1" xref="S2.p3.3.m3.1.1.2.1.cmml">log</mi><mo id="S2.p3.3.m3.1.1.2a" lspace="0.167em" xref="S2.p3.3.m3.1.1.2.cmml">⁡</mo><mi id="S2.p3.3.m3.1.1.2.2" xref="S2.p3.3.m3.1.1.2.2.cmml">P</mi></mrow><mo id="S2.p3.3.m3.1.1.1" xref="S2.p3.3.m3.1.1.1.cmml">=</mo><mn id="S2.p3.3.m3.1.1.3" xref="S2.p3.3.m3.1.1.3.cmml">0.246</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.p3.3.m3.1b"><apply id="S2.p3.3.m3.1.1.cmml" xref="S2.p3.3.m3.1.1"><eq id="S2.p3.3.m3.1.1.1.cmml" xref="S2.p3.3.m3.1.1.1"></eq><apply id="S2.p3.3.m3.1.1.2.cmml" xref="S2.p3.3.m3.1.1.2"><log id="S2.p3.3.m3.1.1.2.1.cmml" xref="S2.p3.3.m3.1.1.2.1"></log><ci id="S2.p3.3.m3.1.1.2.2.cmml" xref="S2.p3.3.m3.1.1.2.2">𝑃</ci></apply><cn id="S2.p3.3.m3.1.1.3.cmml" type="float" xref="S2.p3.3.m3.1.1.3">0.246</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p3.3.m3.1c">\log P=0.246</annotation><annotation encoding="application/x-llamapun" id="S2.p3.3.m3.1d">roman_log italic_P = 0.246</annotation></semantics></math>) while OGLE reports 1.87 days (<math alttext="\log P=0.272" class="ltx_Math" display="inline" id="S2.p3.4.m4.1"><semantics id="S2.p3.4.m4.1a"><mrow id="S2.p3.4.m4.1.1" xref="S2.p3.4.m4.1.1.cmml"><mrow id="S2.p3.4.m4.1.1.2" xref="S2.p3.4.m4.1.1.2.cmml"><mi id="S2.p3.4.m4.1.1.2.1" xref="S2.p3.4.m4.1.1.2.1.cmml">log</mi><mo id="S2.p3.4.m4.1.1.2a" lspace="0.167em" xref="S2.p3.4.m4.1.1.2.cmml">⁡</mo><mi id="S2.p3.4.m4.1.1.2.2" xref="S2.p3.4.m4.1.1.2.2.cmml">P</mi></mrow><mo id="S2.p3.4.m4.1.1.1" xref="S2.p3.4.m4.1.1.1.cmml">=</mo><mn id="S2.p3.4.m4.1.1.3" xref="S2.p3.4.m4.1.1.3.cmml">0.272</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.p3.4.m4.1b"><apply id="S2.p3.4.m4.1.1.cmml" xref="S2.p3.4.m4.1.1"><eq id="S2.p3.4.m4.1.1.1.cmml" xref="S2.p3.4.m4.1.1.1"></eq><apply id="S2.p3.4.m4.1.1.2.cmml" xref="S2.p3.4.m4.1.1.2"><log id="S2.p3.4.m4.1.1.2.1.cmml" xref="S2.p3.4.m4.1.1.2.1"></log><ci id="S2.p3.4.m4.1.1.2.2.cmml" xref="S2.p3.4.m4.1.1.2.2">𝑃</ci></apply><cn id="S2.p3.4.m4.1.1.3.cmml" type="float" xref="S2.p3.4.m4.1.1.3">0.272</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p3.4.m4.1c">\log P=0.272</annotation><annotation encoding="application/x-llamapun" id="S2.p3.4.m4.1d">roman_log italic_P = 0.272</annotation></semantics></math>). Except for the 6 Cepheids indicated in red and labeled in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#S2.F2" title="Figure 2 ‣ 2 Cepheid sample ‣ The Legacy of Henrietta Leavitt: A Re-analysis of the First Cepheid Period-Luminosity Relation"><span class="ltx_text ltx_ref_tag">2</span></a>, the agreement is excellent (to 0.01 day on average). In the Appendix, we discuss the particular case of BZ Tuc (HV 821) and its various period measurements and potential evolution.</p> </div> <figure class="ltx_figure" id="S2.F2"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="305" id="S2.F2.g1" src="x2.png" width="457"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 2: </span>Residuals between the periods (expressed in <math alttext="\log P" class="ltx_Math" display="inline" id="S2.F2.2.m1.1"><semantics id="S2.F2.2.m1.1b"><mrow id="S2.F2.2.m1.1.1" xref="S2.F2.2.m1.1.1.cmml"><mi id="S2.F2.2.m1.1.1.1" xref="S2.F2.2.m1.1.1.1.cmml">log</mi><mo id="S2.F2.2.m1.1.1b" lspace="0.167em" xref="S2.F2.2.m1.1.1.cmml">⁡</mo><mi id="S2.F2.2.m1.1.1.2" xref="S2.F2.2.m1.1.1.2.cmml">P</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.F2.2.m1.1c"><apply id="S2.F2.2.m1.1.1.cmml" xref="S2.F2.2.m1.1.1"><log id="S2.F2.2.m1.1.1.1.cmml" xref="S2.F2.2.m1.1.1.1"></log><ci id="S2.F2.2.m1.1.1.2.cmml" xref="S2.F2.2.m1.1.1.2">𝑃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.F2.2.m1.1d">\log P</annotation><annotation encoding="application/x-llamapun" id="S2.F2.2.m1.1e">roman_log italic_P</annotation></semantics></math>) from <cite class="ltx_cite ltx_citemacro_citet">Leavitt & Pickering (<a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#bib.bib21" title="">1912</a>)</cite> and from OGLE IV <cite class="ltx_cite ltx_citemacro_citep">(Soszyński et al., <a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#bib.bib28" title="">2015</a>)</cite>.</figcaption> </figure> <div class="ltx_para" id="S2.p4"> <p class="ltx_p" id="S2.p4.1">The paper by <cite class="ltx_cite ltx_citemacro_citet">Leavitt & Pickering (<a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#bib.bib21" title="">1912</a>)</cite> does not provide any classification or pulsation mode for their 25 SMC Cepheids. In fact, at the time, Cepheids were not yet classified between different types and were considered as a unique population. For this reason, and because he was using a mix of classical and Type II variables, Edwin Hubble’s first estimate of the distance to galaxies was overestimated and his initial value for the Hubble constant was around 500 km/s/Mpc. According to the OGLE catalog <cite class="ltx_cite ltx_citemacro_citep">(Soszyński et al., <a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#bib.bib28" title="">2015</a>)</cite>, all Cepheids from <cite class="ltx_cite ltx_citemacro_cite">Leavitt & Pickering (<a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#bib.bib21" title="">1912</a>)</cite> are fundamental mode pulsators, although some of them have very short periods (three Cepheids with <math alttext="\log P<0.25" class="ltx_Math" display="inline" id="S2.p4.1.m1.1"><semantics id="S2.p4.1.m1.1a"><mrow id="S2.p4.1.m1.1.1" xref="S2.p4.1.m1.1.1.cmml"><mrow id="S2.p4.1.m1.1.1.2" xref="S2.p4.1.m1.1.1.2.cmml"><mi id="S2.p4.1.m1.1.1.2.1" xref="S2.p4.1.m1.1.1.2.1.cmml">log</mi><mo id="S2.p4.1.m1.1.1.2a" lspace="0.167em" xref="S2.p4.1.m1.1.1.2.cmml">⁡</mo><mi id="S2.p4.1.m1.1.1.2.2" xref="S2.p4.1.m1.1.1.2.2.cmml">P</mi></mrow><mo id="S2.p4.1.m1.1.1.1" xref="S2.p4.1.m1.1.1.1.cmml"><</mo><mn id="S2.p4.1.m1.1.1.3" xref="S2.p4.1.m1.1.1.3.cmml">0.25</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.p4.1.m1.1b"><apply id="S2.p4.1.m1.1.1.cmml" xref="S2.p4.1.m1.1.1"><lt id="S2.p4.1.m1.1.1.1.cmml" xref="S2.p4.1.m1.1.1.1"></lt><apply id="S2.p4.1.m1.1.1.2.cmml" xref="S2.p4.1.m1.1.1.2"><log id="S2.p4.1.m1.1.1.2.1.cmml" xref="S2.p4.1.m1.1.1.2.1"></log><ci id="S2.p4.1.m1.1.1.2.2.cmml" xref="S2.p4.1.m1.1.1.2.2">𝑃</ci></apply><cn id="S2.p4.1.m1.1.1.3.cmml" type="float" xref="S2.p4.1.m1.1.1.3">0.25</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p4.1.m1.1c">\log P<0.25</annotation><annotation encoding="application/x-llamapun" id="S2.p4.1.m1.1d">roman_log italic_P < 0.25</annotation></semantics></math>) and could be suspected to be first overtone pulsators or to be incorrectly classified as classical Cepheids. Additionally, all 25 SMC Cepheids, even the ones with very short periods, are confirmed fundamental-mode classical Cepheids by the Gaia DR3 reclassification <cite class="ltx_cite ltx_citemacro_citep">(Ripepi et al., <a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#bib.bib26" title="">2023</a>)</cite>. <br class="ltx_break"/></p> </div> <figure class="ltx_figure" id="S2.F3"> <div class="ltx_flex_figure"> <div class="ltx_flex_cell ltx_flex_size_1"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_figure_panel ltx_img_landscape" height="177" id="S2.F3.g1" src="x3.png" width="463"/></div> <div class="ltx_flex_break"></div> <div class="ltx_flex_cell ltx_flex_size_1"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_figure_panel ltx_img_landscape" height="177" id="S2.F3.g2" src="x4.png" width="463"/></div> <div class="ltx_flex_break"></div> <div class="ltx_flex_cell ltx_flex_size_1"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_figure_panel ltx_img_landscape" height="198" id="S2.F3.g3" src="x5.png" width="463"/></div> </div> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 3: </span>Example of Cepheid light curves extracted from Henrietta Leavitt’s notebook <cite class="ltx_cite ltx_citemacro_citep">(Leavitt, <a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#bib.bib19" title="">1905</a>)</cite> in red and from the OGLE IV catalog <cite class="ltx_cite ltx_citemacro_citep">(Soszyński et al., <a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#bib.bib28" title="">2015</a>)</cite> in blue. The light curve shape from OGLE was fitted to Leavitt’s data points.</figcaption> </figure> </section> <section class="ltx_section" id="S3"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">3 </span>Cepheid light curves</h2> <div class="ltx_para" id="S3.p1"> <p class="ltx_p" id="S3.p1.1">At the time of these observations, a standard scale of magnitudes had not yet been developed <cite class="ltx_cite ltx_citemacro_citep">(Pickering, <a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#bib.bib22" title="">1909</a>)</cite>. Thus, all of the measurements in Leavitt’s notebooks and 1912 paper were recorded in <span class="ltx_text ltx_font_italic" id="S3.p1.1.1">provisional</span> magnitudes. However, Leavitt hypothesized that converting the provisional magnitudes of Cepheids to a standard scale could further decrease the scatter in the P-L relation, writing, <span class="ltx_text ltx_font_italic" id="S3.p1.1.2">“It is possible that the deviations from a straight line may become smaller when an absolute scale of magnitudes is used”</span> <cite class="ltx_cite ltx_citemacro_citep">(Leavitt & Pickering, <a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#bib.bib21" title="">1912</a>)</cite>.</p> </div> <div class="ltx_para" id="S3.p2"> <p class="ltx_p" id="S3.p2.1">Leavitt’s published papers <cite class="ltx_cite ltx_citemacro_citep">(Leavitt, <a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#bib.bib20" title="">1907</a>; Leavitt & Pickering, <a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#bib.bib21" title="">1912</a>)</cite> do not discuss how her Cepheid light curves were constructed. However, her notebooks <cite class="ltx_cite ltx_citemacro_citep">(see Leavitt, <a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#bib.bib19" title="">1905</a>)</cite> contain a handwritten collection of Cepheid measurements in the same regions of the SMC with dozens of photographic plates obtained at Harvard’s Boyden Station located in Arequipa, Peru. Using the measurements reported from pages 4 to 40 of her 1905 notebook, we were able to extract and reconstruct the majority of the original observations for all 25 Cepheids of her sample. These light curves are based on 34 epochs on average, with a minimum and a maximum of 17 and 56 epochs, respectively. We note that additional observations listed in other notebooks might be available but were not included here. These original light curves were then compared with modern and well-sampled <math alttext="V" class="ltx_Math" display="inline" id="S3.p2.1.m1.1"><semantics id="S3.p2.1.m1.1a"><mi id="S3.p2.1.m1.1.1" xref="S3.p2.1.m1.1.1.cmml">V</mi><annotation-xml encoding="MathML-Content" id="S3.p2.1.m1.1b"><ci id="S3.p2.1.m1.1.1.cmml" xref="S3.p2.1.m1.1.1">𝑉</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.1.m1.1c">V</annotation><annotation encoding="application/x-llamapun" id="S3.p2.1.m1.1d">italic_V</annotation></semantics></math>-band light curves from OGLE <cite class="ltx_cite ltx_citemacro_citep">(Soszyński et al., <a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#bib.bib28" title="">2015</a>)</cite>. For 17 of the 25 Cepheids, we were able to determine a clear periodicity and resemblance to light curve shapes of these stars from OGLE. Three Cepheids are displayed in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#S2.F3" title="Figure 3 ‣ 2 Cepheid sample ‣ The Legacy of Henrietta Leavitt: A Re-analysis of the First Cepheid Period-Luminosity Relation"><span class="ltx_text ltx_ref_tag">3</span></a> as an example. They are representative of the 17 conclusive recovered light curves. The similarity between both data sets is remarkable and even clearly shows the Hertzsprung progression, a small local maximum whose phase depends on the period of the Cepheid <cite class="ltx_cite ltx_citemacro_citep">(Hertzsprung, <a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#bib.bib14" title="">1926</a>)</cite>.</p> </div> <div class="ltx_para" id="S3.p3"> <p class="ltx_p" id="S3.p3.8">While the exact wavelength of Leavitt’s measurements is unknown, it is possible to infer the approximate wavelength by examining the amplitude of light curves. Cepheid amplitudes are known to correlate with wavelength, with larger amplitudes in the blue and smaller amplitudes in the infrared. Today, amplitude ratios between different filters have been established with great precision <cite class="ltx_cite ltx_citemacro_citep">(Klagyivik & Szabados, <a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#bib.bib17" title="">2009</a>; Riess et al., <a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#bib.bib24" title="">2020</a>)</cite>. Thus, we use the amplitude ratios between Leavitt’s system and OGLE’s <math alttext="V" class="ltx_Math" display="inline" id="S3.p3.1.m1.1"><semantics id="S3.p3.1.m1.1a"><mi id="S3.p3.1.m1.1.1" xref="S3.p3.1.m1.1.1.cmml">V</mi><annotation-xml encoding="MathML-Content" id="S3.p3.1.m1.1b"><ci id="S3.p3.1.m1.1.1.cmml" xref="S3.p3.1.m1.1.1">𝑉</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p3.1.m1.1c">V</annotation><annotation encoding="application/x-llamapun" id="S3.p3.1.m1.1d">italic_V</annotation></semantics></math>-band light curves to derive the wavelength of her observations. We first fit the well-sampled OGLE light curves with Fourier series. Then, we adopt OGLE’s light curve shapes (which are similar at similar wavelengths) and set the amplitude as a free parameter to fit Leavitt’s light curves. We compare amplitudes in both systems and we find that amplitudes in Leavitt’s provisional scale of magnitudes are 1.21 times larger than in the <math alttext="V" class="ltx_Math" display="inline" id="S3.p3.2.m2.1"><semantics id="S3.p3.2.m2.1a"><mi id="S3.p3.2.m2.1.1" xref="S3.p3.2.m2.1.1.cmml">V</mi><annotation-xml encoding="MathML-Content" id="S3.p3.2.m2.1b"><ci id="S3.p3.2.m2.1.1.cmml" xref="S3.p3.2.m2.1.1">𝑉</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p3.2.m2.1c">V</annotation><annotation encoding="application/x-llamapun" id="S3.p3.2.m2.1d">italic_V</annotation></semantics></math>-band OGLE filter (see Fig. <a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#S3.F4" title="Figure 4 ‣ 3 Cepheid light curves ‣ The Legacy of Henrietta Leavitt: A Re-analysis of the First Cepheid Period-Luminosity Relation"><span class="ltx_text ltx_ref_tag">4</span></a>). Only Cepheids with periods in <math alttext="0.5<\log P<1.5" class="ltx_Math" display="inline" id="S3.p3.3.m3.1"><semantics id="S3.p3.3.m3.1a"><mrow id="S3.p3.3.m3.1.1" xref="S3.p3.3.m3.1.1.cmml"><mn id="S3.p3.3.m3.1.1.2" xref="S3.p3.3.m3.1.1.2.cmml">0.5</mn><mo id="S3.p3.3.m3.1.1.3" xref="S3.p3.3.m3.1.1.3.cmml"><</mo><mrow id="S3.p3.3.m3.1.1.4" xref="S3.p3.3.m3.1.1.4.cmml"><mi id="S3.p3.3.m3.1.1.4.1" xref="S3.p3.3.m3.1.1.4.1.cmml">log</mi><mo id="S3.p3.3.m3.1.1.4a" lspace="0.167em" xref="S3.p3.3.m3.1.1.4.cmml">⁡</mo><mi id="S3.p3.3.m3.1.1.4.2" xref="S3.p3.3.m3.1.1.4.2.cmml">P</mi></mrow><mo id="S3.p3.3.m3.1.1.5" xref="S3.p3.3.m3.1.1.5.cmml"><</mo><mn id="S3.p3.3.m3.1.1.6" xref="S3.p3.3.m3.1.1.6.cmml">1.5</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.p3.3.m3.1b"><apply id="S3.p3.3.m3.1.1.cmml" xref="S3.p3.3.m3.1.1"><and id="S3.p3.3.m3.1.1a.cmml" xref="S3.p3.3.m3.1.1"></and><apply id="S3.p3.3.m3.1.1b.cmml" xref="S3.p3.3.m3.1.1"><lt id="S3.p3.3.m3.1.1.3.cmml" xref="S3.p3.3.m3.1.1.3"></lt><cn id="S3.p3.3.m3.1.1.2.cmml" type="float" xref="S3.p3.3.m3.1.1.2">0.5</cn><apply id="S3.p3.3.m3.1.1.4.cmml" xref="S3.p3.3.m3.1.1.4"><log id="S3.p3.3.m3.1.1.4.1.cmml" xref="S3.p3.3.m3.1.1.4.1"></log><ci id="S3.p3.3.m3.1.1.4.2.cmml" xref="S3.p3.3.m3.1.1.4.2">𝑃</ci></apply></apply><apply id="S3.p3.3.m3.1.1c.cmml" xref="S3.p3.3.m3.1.1"><lt id="S3.p3.3.m3.1.1.5.cmml" xref="S3.p3.3.m3.1.1.5"></lt><share href="https://arxiv.org/html/2502.17438v1#S3.p3.3.m3.1.1.4.cmml" id="S3.p3.3.m3.1.1d.cmml" xref="S3.p3.3.m3.1.1"></share><cn id="S3.p3.3.m3.1.1.6.cmml" type="float" xref="S3.p3.3.m3.1.1.6">1.5</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p3.3.m3.1c">0.5<\log P<1.5</annotation><annotation encoding="application/x-llamapun" id="S3.p3.3.m3.1d">0.5 < roman_log italic_P < 1.5</annotation></semantics></math> were considered in this analysis, in order to exclude faintest and brightest variables which might be affected by the non-linearity of the photographic plates response and possibly crowding (see Sect. <a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#S4.SS1" title="4.1 P-L relation with magnitudes from Leavitt & Pickering (1912) ‣ 4 Period-Luminosity relation ‣ The Legacy of Henrietta Leavitt: A Re-analysis of the First Cepheid Period-Luminosity Relation"><span class="ltx_text ltx_ref_tag">4.1</span></a>). The fact that the amplitude ratio between Leavitt’s filter and the <math alttext="V" class="ltx_Math" display="inline" id="S3.p3.4.m4.1"><semantics id="S3.p3.4.m4.1a"><mi id="S3.p3.4.m4.1.1" xref="S3.p3.4.m4.1.1.cmml">V</mi><annotation-xml encoding="MathML-Content" id="S3.p3.4.m4.1b"><ci id="S3.p3.4.m4.1.1.cmml" xref="S3.p3.4.m4.1.1">𝑉</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p3.4.m4.1c">V</annotation><annotation encoding="application/x-llamapun" id="S3.p3.4.m4.1d">italic_V</annotation></semantics></math>-band was greater than 1 suggests that the effective wavelength of her observations was shorter than the modern <math alttext="V" class="ltx_Math" display="inline" id="S3.p3.5.m5.1"><semantics id="S3.p3.5.m5.1a"><mi id="S3.p3.5.m5.1.1" xref="S3.p3.5.m5.1.1.cmml">V</mi><annotation-xml encoding="MathML-Content" id="S3.p3.5.m5.1b"><ci id="S3.p3.5.m5.1.1.cmml" xref="S3.p3.5.m5.1.1">𝑉</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p3.5.m5.1c">V</annotation><annotation encoding="application/x-llamapun" id="S3.p3.5.m5.1d">italic_V</annotation></semantics></math> band. We can further narrow this range down by comparing with the detailed amplitude ratios analysis from <cite class="ltx_cite ltx_citemacro_citet">Klagyivik & Szabados (<a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#bib.bib17" title="">2009</a>)</cite> which derived that <math alttext="A(B)\sim 1.5~{}A(V)" class="ltx_Math" display="inline" id="S3.p3.6.m6.2"><semantics id="S3.p3.6.m6.2a"><mrow id="S3.p3.6.m6.2.3" xref="S3.p3.6.m6.2.3.cmml"><mrow id="S3.p3.6.m6.2.3.2" xref="S3.p3.6.m6.2.3.2.cmml"><mi id="S3.p3.6.m6.2.3.2.2" xref="S3.p3.6.m6.2.3.2.2.cmml">A</mi><mo id="S3.p3.6.m6.2.3.2.1" xref="S3.p3.6.m6.2.3.2.1.cmml">⁢</mo><mrow id="S3.p3.6.m6.2.3.2.3.2" xref="S3.p3.6.m6.2.3.2.cmml"><mo id="S3.p3.6.m6.2.3.2.3.2.1" stretchy="false" xref="S3.p3.6.m6.2.3.2.cmml">(</mo><mi id="S3.p3.6.m6.1.1" xref="S3.p3.6.m6.1.1.cmml">B</mi><mo id="S3.p3.6.m6.2.3.2.3.2.2" stretchy="false" xref="S3.p3.6.m6.2.3.2.cmml">)</mo></mrow></mrow><mo id="S3.p3.6.m6.2.3.1" xref="S3.p3.6.m6.2.3.1.cmml">∼</mo><mrow id="S3.p3.6.m6.2.3.3" xref="S3.p3.6.m6.2.3.3.cmml"><mn id="S3.p3.6.m6.2.3.3.2" xref="S3.p3.6.m6.2.3.3.2.cmml">1.5</mn><mo id="S3.p3.6.m6.2.3.3.1" lspace="0.330em" xref="S3.p3.6.m6.2.3.3.1.cmml">⁢</mo><mi id="S3.p3.6.m6.2.3.3.3" xref="S3.p3.6.m6.2.3.3.3.cmml">A</mi><mo id="S3.p3.6.m6.2.3.3.1a" xref="S3.p3.6.m6.2.3.3.1.cmml">⁢</mo><mrow id="S3.p3.6.m6.2.3.3.4.2" xref="S3.p3.6.m6.2.3.3.cmml"><mo id="S3.p3.6.m6.2.3.3.4.2.1" stretchy="false" xref="S3.p3.6.m6.2.3.3.cmml">(</mo><mi id="S3.p3.6.m6.2.2" xref="S3.p3.6.m6.2.2.cmml">V</mi><mo id="S3.p3.6.m6.2.3.3.4.2.2" stretchy="false" xref="S3.p3.6.m6.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p3.6.m6.2b"><apply id="S3.p3.6.m6.2.3.cmml" xref="S3.p3.6.m6.2.3"><csymbol cd="latexml" id="S3.p3.6.m6.2.3.1.cmml" xref="S3.p3.6.m6.2.3.1">similar-to</csymbol><apply id="S3.p3.6.m6.2.3.2.cmml" xref="S3.p3.6.m6.2.3.2"><times id="S3.p3.6.m6.2.3.2.1.cmml" xref="S3.p3.6.m6.2.3.2.1"></times><ci id="S3.p3.6.m6.2.3.2.2.cmml" xref="S3.p3.6.m6.2.3.2.2">𝐴</ci><ci id="S3.p3.6.m6.1.1.cmml" xref="S3.p3.6.m6.1.1">𝐵</ci></apply><apply id="S3.p3.6.m6.2.3.3.cmml" xref="S3.p3.6.m6.2.3.3"><times id="S3.p3.6.m6.2.3.3.1.cmml" xref="S3.p3.6.m6.2.3.3.1"></times><cn id="S3.p3.6.m6.2.3.3.2.cmml" type="float" xref="S3.p3.6.m6.2.3.3.2">1.5</cn><ci id="S3.p3.6.m6.2.3.3.3.cmml" xref="S3.p3.6.m6.2.3.3.3">𝐴</ci><ci id="S3.p3.6.m6.2.2.cmml" xref="S3.p3.6.m6.2.2">𝑉</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p3.6.m6.2c">A(B)\sim 1.5~{}A(V)</annotation><annotation encoding="application/x-llamapun" id="S3.p3.6.m6.2d">italic_A ( italic_B ) ∼ 1.5 italic_A ( italic_V )</annotation></semantics></math> with a small dependence on the period. Therefore we conclude that Leavitt’s provisional scale of magnitude is comparable to a blue filter between the <math alttext="B" class="ltx_Math" display="inline" id="S3.p3.7.m7.1"><semantics id="S3.p3.7.m7.1a"><mi id="S3.p3.7.m7.1.1" xref="S3.p3.7.m7.1.1.cmml">B</mi><annotation-xml encoding="MathML-Content" id="S3.p3.7.m7.1b"><ci id="S3.p3.7.m7.1.1.cmml" xref="S3.p3.7.m7.1.1">𝐵</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p3.7.m7.1c">B</annotation><annotation encoding="application/x-llamapun" id="S3.p3.7.m7.1d">italic_B</annotation></semantics></math> and <math alttext="V" class="ltx_Math" display="inline" id="S3.p3.8.m8.1"><semantics id="S3.p3.8.m8.1a"><mi id="S3.p3.8.m8.1.1" xref="S3.p3.8.m8.1.1.cmml">V</mi><annotation-xml encoding="MathML-Content" id="S3.p3.8.m8.1b"><ci id="S3.p3.8.m8.1.1.cmml" xref="S3.p3.8.m8.1.1">𝑉</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p3.8.m8.1c">V</annotation><annotation encoding="application/x-llamapun" id="S3.p3.8.m8.1d">italic_V</annotation></semantics></math> bands. An additional discussion on the wavelength of Leavitt’s observations is presented in Sect. <a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#S4.SS1" title="4.1 P-L relation with magnitudes from Leavitt & Pickering (1912) ‣ 4 Period-Luminosity relation ‣ The Legacy of Henrietta Leavitt: A Re-analysis of the First Cepheid Period-Luminosity Relation"><span class="ltx_text ltx_ref_tag">4.1</span></a>. <br class="ltx_break"/></p> </div> <figure class="ltx_figure" id="S3.F4"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="278" id="S3.F4.g1" src="x6.png" width="463"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 4: </span>Cepheid light curve amplitude ratios between Leavitt’s provisional scale and the OGLE <math alttext="V" class="ltx_Math" display="inline" id="S3.F4.2.m1.1"><semantics id="S3.F4.2.m1.1b"><mi id="S3.F4.2.m1.1.1" xref="S3.F4.2.m1.1.1.cmml">V</mi><annotation-xml encoding="MathML-Content" id="S3.F4.2.m1.1c"><ci id="S3.F4.2.m1.1.1.cmml" xref="S3.F4.2.m1.1.1">𝑉</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.F4.2.m1.1d">V</annotation><annotation encoding="application/x-llamapun" id="S3.F4.2.m1.1e">italic_V</annotation></semantics></math>-filter. </figcaption> </figure> </section> <section class="ltx_section" id="S4"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">4 </span>Period-Luminosity relation</h2> <div class="ltx_para" id="S4.p1"> <p class="ltx_p" id="S4.p1.1">In Sect. <a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#S4.SS1" title="4.1 P-L relation with magnitudes from Leavitt & Pickering (1912) ‣ 4 Period-Luminosity relation ‣ The Legacy of Henrietta Leavitt: A Re-analysis of the First Cepheid Period-Luminosity Relation"><span class="ltx_text ltx_ref_tag">4.1</span></a>, we reproduce the original P-L relation from <cite class="ltx_cite ltx_citemacro_citet">Leavitt & Pickering (<a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#bib.bib21" title="">1912</a>)</cite> by simply using their original magnitudes, but we adopt the pulsation periods from the OGLE survey. We compare the P-L slope to what we expect it to be at the <math alttext="B" class="ltx_Math" display="inline" id="S4.p1.1.m1.1"><semantics id="S4.p1.1.m1.1a"><mi id="S4.p1.1.m1.1.1" xref="S4.p1.1.m1.1.1.cmml">B</mi><annotation-xml encoding="MathML-Content" id="S4.p1.1.m1.1b"><ci id="S4.p1.1.m1.1.1.cmml" xref="S4.p1.1.m1.1.1">𝐵</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.1.m1.1c">B</annotation><annotation encoding="application/x-llamapun" id="S4.p1.1.m1.1d">italic_B</annotation></semantics></math>-band wavelength and attribute differences to the non-linearity of the photographic plate response and to crowding. In Sect. <a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#S4.SS2" title="4.2 P-L relation with magnitudes from OGLE ‣ 4 Period-Luminosity relation ‣ The Legacy of Henrietta Leavitt: A Re-analysis of the First Cepheid Period-Luminosity Relation"><span class="ltx_text ltx_ref_tag">4.2</span></a>, we update Leavitt’s photometry with the most recent light curves from the OGLE survey and we describe the impact of the spatial sampling of SMC Cepheids. <br class="ltx_break"/></p> </div> <figure class="ltx_figure" id="S4.F5"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_square" height="790" id="S4.F5.g1" src="x7.png" width="812"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 5: </span><span class="ltx_text ltx_font_bold" id="S4.F5.21.1">Top</span>: P-L relation obtained in a <span class="ltx_text ltx_font_italic" id="S4.F5.22.2">provisional scale of magnitudes</span> using the mean between the minimum and maximum from Leavitt’s original data <cite class="ltx_cite ltx_citemacro_citep">(Leavitt & Pickering, <a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#bib.bib21" title="">1912</a>)</cite>. The solid black line is the P-L relation fit to the subset of Cepheids within 0.8 deg of the SMC center (red circles). The solid blue line shows the P-L relation fit to Cepheids with <math alttext="\log P>1" class="ltx_Math" display="inline" id="S4.F5.9.m1.1"><semantics id="S4.F5.9.m1.1b"><mrow id="S4.F5.9.m1.1.1" xref="S4.F5.9.m1.1.1.cmml"><mrow id="S4.F5.9.m1.1.1.2" xref="S4.F5.9.m1.1.1.2.cmml"><mi id="S4.F5.9.m1.1.1.2.1" xref="S4.F5.9.m1.1.1.2.1.cmml">log</mi><mo id="S4.F5.9.m1.1.1.2b" lspace="0.167em" xref="S4.F5.9.m1.1.1.2.cmml">⁡</mo><mi id="S4.F5.9.m1.1.1.2.2" xref="S4.F5.9.m1.1.1.2.2.cmml">P</mi></mrow><mo id="S4.F5.9.m1.1.1.1" xref="S4.F5.9.m1.1.1.1.cmml">></mo><mn id="S4.F5.9.m1.1.1.3" xref="S4.F5.9.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.F5.9.m1.1c"><apply id="S4.F5.9.m1.1.1.cmml" xref="S4.F5.9.m1.1.1"><gt id="S4.F5.9.m1.1.1.1.cmml" xref="S4.F5.9.m1.1.1.1"></gt><apply id="S4.F5.9.m1.1.1.2.cmml" xref="S4.F5.9.m1.1.1.2"><log id="S4.F5.9.m1.1.1.2.1.cmml" xref="S4.F5.9.m1.1.1.2.1"></log><ci id="S4.F5.9.m1.1.1.2.2.cmml" xref="S4.F5.9.m1.1.1.2.2">𝑃</ci></apply><cn id="S4.F5.9.m1.1.1.3.cmml" type="integer" xref="S4.F5.9.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F5.9.m1.1d">\log P>1</annotation><annotation encoding="application/x-llamapun" id="S4.F5.9.m1.1e">roman_log italic_P > 1</annotation></semantics></math>, with slope fixed to the modern <math alttext="B" class="ltx_Math" display="inline" id="S4.F5.10.m2.1"><semantics id="S4.F5.10.m2.1b"><mi id="S4.F5.10.m2.1.1" xref="S4.F5.10.m2.1.1.cmml">B</mi><annotation-xml encoding="MathML-Content" id="S4.F5.10.m2.1c"><ci id="S4.F5.10.m2.1.1.cmml" xref="S4.F5.10.m2.1.1">𝐵</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.F5.10.m2.1d">B</annotation><annotation encoding="application/x-llamapun" id="S4.F5.10.m2.1e">italic_B</annotation></semantics></math>-band value of -2.55 (Fig. <a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#S4.F6" title="Figure 6 ‣ 4.1 P-L relation with magnitudes from Leavitt & Pickering (1912) ‣ 4 Period-Luminosity relation ‣ The Legacy of Henrietta Leavitt: A Re-analysis of the First Cepheid Period-Luminosity Relation"><span class="ltx_text ltx_ref_tag">6</span></a>). The dashed-blue line shows the extension of this fit to <math alttext="\log P<1" class="ltx_Math" display="inline" id="S4.F5.11.m3.1"><semantics id="S4.F5.11.m3.1b"><mrow id="S4.F5.11.m3.1.1" xref="S4.F5.11.m3.1.1.cmml"><mrow id="S4.F5.11.m3.1.1.2" xref="S4.F5.11.m3.1.1.2.cmml"><mi id="S4.F5.11.m3.1.1.2.1" xref="S4.F5.11.m3.1.1.2.1.cmml">log</mi><mo id="S4.F5.11.m3.1.1.2b" lspace="0.167em" xref="S4.F5.11.m3.1.1.2.cmml">⁡</mo><mi id="S4.F5.11.m3.1.1.2.2" xref="S4.F5.11.m3.1.1.2.2.cmml">P</mi></mrow><mo id="S4.F5.11.m3.1.1.1" xref="S4.F5.11.m3.1.1.1.cmml"><</mo><mn id="S4.F5.11.m3.1.1.3" xref="S4.F5.11.m3.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.F5.11.m3.1c"><apply id="S4.F5.11.m3.1.1.cmml" xref="S4.F5.11.m3.1.1"><lt id="S4.F5.11.m3.1.1.1.cmml" xref="S4.F5.11.m3.1.1.1"></lt><apply id="S4.F5.11.m3.1.1.2.cmml" xref="S4.F5.11.m3.1.1.2"><log id="S4.F5.11.m3.1.1.2.1.cmml" xref="S4.F5.11.m3.1.1.2.1"></log><ci id="S4.F5.11.m3.1.1.2.2.cmml" xref="S4.F5.11.m3.1.1.2.2">𝑃</ci></apply><cn id="S4.F5.11.m3.1.1.3.cmml" type="integer" xref="S4.F5.11.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F5.11.m3.1d">\log P<1</annotation><annotation encoding="application/x-llamapun" id="S4.F5.11.m3.1e">roman_log italic_P < 1</annotation></semantics></math>. The stamps show five Cepheids of different brightness in this sample, obtained from the digitized photographic plates by the <span class="ltx_text ltx_font_italic" id="S4.F5.23.3">DASCH</span> project. <span class="ltx_text ltx_font_bold" id="S4.F5.24.4">Bottom</span>: PL relation for the same Cepheid sample as above, except that mean magnitudes are expressed in the <math alttext="m_{VI}^{W}" class="ltx_Math" display="inline" id="S4.F5.12.m4.1"><semantics id="S4.F5.12.m4.1b"><msubsup id="S4.F5.12.m4.1.1" xref="S4.F5.12.m4.1.1.cmml"><mi id="S4.F5.12.m4.1.1.2.2" xref="S4.F5.12.m4.1.1.2.2.cmml">m</mi><mrow id="S4.F5.12.m4.1.1.2.3" xref="S4.F5.12.m4.1.1.2.3.cmml"><mi id="S4.F5.12.m4.1.1.2.3.2" xref="S4.F5.12.m4.1.1.2.3.2.cmml">V</mi><mo id="S4.F5.12.m4.1.1.2.3.1" xref="S4.F5.12.m4.1.1.2.3.1.cmml">⁢</mo><mi id="S4.F5.12.m4.1.1.2.3.3" xref="S4.F5.12.m4.1.1.2.3.3.cmml">I</mi></mrow><mi id="S4.F5.12.m4.1.1.3" xref="S4.F5.12.m4.1.1.3.cmml">W</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.F5.12.m4.1c"><apply id="S4.F5.12.m4.1.1.cmml" xref="S4.F5.12.m4.1.1"><csymbol cd="ambiguous" id="S4.F5.12.m4.1.1.1.cmml" xref="S4.F5.12.m4.1.1">superscript</csymbol><apply id="S4.F5.12.m4.1.1.2.cmml" xref="S4.F5.12.m4.1.1"><csymbol cd="ambiguous" id="S4.F5.12.m4.1.1.2.1.cmml" xref="S4.F5.12.m4.1.1">subscript</csymbol><ci id="S4.F5.12.m4.1.1.2.2.cmml" xref="S4.F5.12.m4.1.1.2.2">𝑚</ci><apply id="S4.F5.12.m4.1.1.2.3.cmml" xref="S4.F5.12.m4.1.1.2.3"><times id="S4.F5.12.m4.1.1.2.3.1.cmml" xref="S4.F5.12.m4.1.1.2.3.1"></times><ci id="S4.F5.12.m4.1.1.2.3.2.cmml" xref="S4.F5.12.m4.1.1.2.3.2">𝑉</ci><ci id="S4.F5.12.m4.1.1.2.3.3.cmml" xref="S4.F5.12.m4.1.1.2.3.3">𝐼</ci></apply></apply><ci id="S4.F5.12.m4.1.1.3.cmml" xref="S4.F5.12.m4.1.1.3">𝑊</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F5.12.m4.1d">m_{VI}^{W}</annotation><annotation encoding="application/x-llamapun" id="S4.F5.12.m4.1e">italic_m start_POSTSUBSCRIPT italic_V italic_I end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_W end_POSTSUPERSCRIPT</annotation></semantics></math> Wesenheit index (a combination of <math alttext="V" class="ltx_Math" display="inline" id="S4.F5.13.m5.1"><semantics id="S4.F5.13.m5.1b"><mi id="S4.F5.13.m5.1.1" xref="S4.F5.13.m5.1.1.cmml">V</mi><annotation-xml encoding="MathML-Content" id="S4.F5.13.m5.1c"><ci id="S4.F5.13.m5.1.1.cmml" xref="S4.F5.13.m5.1.1">𝑉</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.F5.13.m5.1d">V</annotation><annotation encoding="application/x-llamapun" id="S4.F5.13.m5.1e">italic_V</annotation></semantics></math> and <math alttext="I" class="ltx_Math" display="inline" id="S4.F5.14.m6.1"><semantics id="S4.F5.14.m6.1b"><mi id="S4.F5.14.m6.1.1" xref="S4.F5.14.m6.1.1.cmml">I</mi><annotation-xml encoding="MathML-Content" id="S4.F5.14.m6.1c"><ci id="S4.F5.14.m6.1.1.cmml" xref="S4.F5.14.m6.1.1">𝐼</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.F5.14.m6.1d">I</annotation><annotation encoding="application/x-llamapun" id="S4.F5.14.m6.1e">italic_I</annotation></semantics></math> filters) and mean magnitudes taken from the OGLE survey. The two longest period Cepheids are shown in orange and are not considered in the fit, as their <math alttext="V" class="ltx_Math" display="inline" id="S4.F5.15.m7.1"><semantics id="S4.F5.15.m7.1b"><mi id="S4.F5.15.m7.1.1" xref="S4.F5.15.m7.1.1.cmml">V</mi><annotation-xml encoding="MathML-Content" id="S4.F5.15.m7.1c"><ci id="S4.F5.15.m7.1.1.cmml" xref="S4.F5.15.m7.1.1">𝑉</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.F5.15.m7.1d">V</annotation><annotation encoding="application/x-llamapun" id="S4.F5.15.m7.1e">italic_V</annotation></semantics></math>-band magnitudes are not available in the OGLE survey. We assumed a reddening parameter of <math alttext="R_{V}=2.74" class="ltx_Math" display="inline" id="S4.F5.16.m8.1"><semantics id="S4.F5.16.m8.1b"><mrow id="S4.F5.16.m8.1.1" xref="S4.F5.16.m8.1.1.cmml"><msub id="S4.F5.16.m8.1.1.2" xref="S4.F5.16.m8.1.1.2.cmml"><mi id="S4.F5.16.m8.1.1.2.2" xref="S4.F5.16.m8.1.1.2.2.cmml">R</mi><mi id="S4.F5.16.m8.1.1.2.3" xref="S4.F5.16.m8.1.1.2.3.cmml">V</mi></msub><mo id="S4.F5.16.m8.1.1.1" xref="S4.F5.16.m8.1.1.1.cmml">=</mo><mn id="S4.F5.16.m8.1.1.3" xref="S4.F5.16.m8.1.1.3.cmml">2.74</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.F5.16.m8.1c"><apply id="S4.F5.16.m8.1.1.cmml" xref="S4.F5.16.m8.1.1"><eq id="S4.F5.16.m8.1.1.1.cmml" xref="S4.F5.16.m8.1.1.1"></eq><apply id="S4.F5.16.m8.1.1.2.cmml" xref="S4.F5.16.m8.1.1.2"><csymbol cd="ambiguous" id="S4.F5.16.m8.1.1.2.1.cmml" xref="S4.F5.16.m8.1.1.2">subscript</csymbol><ci id="S4.F5.16.m8.1.1.2.2.cmml" xref="S4.F5.16.m8.1.1.2.2">𝑅</ci><ci id="S4.F5.16.m8.1.1.2.3.cmml" xref="S4.F5.16.m8.1.1.2.3">𝑉</ci></apply><cn id="S4.F5.16.m8.1.1.3.cmml" type="float" xref="S4.F5.16.m8.1.1.3">2.74</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F5.16.m8.1d">R_{V}=2.74</annotation><annotation encoding="application/x-llamapun" id="S4.F5.16.m8.1e">italic_R start_POSTSUBSCRIPT italic_V end_POSTSUBSCRIPT = 2.74</annotation></semantics></math> and included the geometry correction from <cite class="ltx_cite ltx_citemacro_citet">Graczyk et al. (<a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#bib.bib11" title="">2020</a>)</cite>. The stamps are taken from the OGLE survey and were obtained with the 1.3-m Warsaw telescope at Las Campanas Observatory in Chile. <br class="ltx_break"/></figcaption> </figure> <section class="ltx_subsection" id="S4.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">4.1 </span>P-L relation with magnitudes from Leavitt & Pickering (1912)</h3> <div class="ltx_para" id="S4.SS1.p1"> <p class="ltx_p" id="S4.SS1.p1.1"><cite class="ltx_cite ltx_citemacro_citet">Leavitt & Pickering (<a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#bib.bib21" title="">1912</a>)</cite> only provide the minimum and maximum observed magnitudes for the 25 Cepheids in their Table 1 (see Fig. <a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#S1.F1" title="Figure 1 ‣ 1 Introduction ‣ The Legacy of Henrietta Leavitt: A Re-analysis of the First Cepheid Period-Luminosity Relation"><span class="ltx_text ltx_ref_tag">1</span></a>, left) but do not mention using the full light curves. Therefore, with only the information available in <cite class="ltx_cite ltx_citemacro_citet">Leavitt & Pickering (<a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#bib.bib21" title="">1912</a>)</cite>, we adopt the mean of the minimum and maximum magnitude for each Cepheid and reconstruct their P-L relation in the top panel of Fig. <a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#S4.F5" title="Figure 5 ‣ 4 Period-Luminosity relation ‣ The Legacy of Henrietta Leavitt: A Re-analysis of the First Cepheid Period-Luminosity Relation"><span class="ltx_text ltx_ref_tag">5</span></a>. With these adjustments, the Leavitt law obtained with her original data has a scatter of <math alttext="0.22\,\rm mag" class="ltx_Math" display="inline" id="S4.SS1.p1.1.m1.1"><semantics id="S4.SS1.p1.1.m1.1a"><mrow id="S4.SS1.p1.1.m1.1.1" xref="S4.SS1.p1.1.m1.1.1.cmml"><mn id="S4.SS1.p1.1.m1.1.1.2" xref="S4.SS1.p1.1.m1.1.1.2.cmml">0.22</mn><mo id="S4.SS1.p1.1.m1.1.1.1" lspace="0.170em" xref="S4.SS1.p1.1.m1.1.1.1.cmml">⁢</mo><mi id="S4.SS1.p1.1.m1.1.1.3" xref="S4.SS1.p1.1.m1.1.1.3.cmml">mag</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.1.m1.1b"><apply id="S4.SS1.p1.1.m1.1.1.cmml" xref="S4.SS1.p1.1.m1.1.1"><times id="S4.SS1.p1.1.m1.1.1.1.cmml" xref="S4.SS1.p1.1.m1.1.1.1"></times><cn id="S4.SS1.p1.1.m1.1.1.2.cmml" type="float" xref="S4.SS1.p1.1.m1.1.1.2">0.22</cn><ci id="S4.SS1.p1.1.m1.1.1.3.cmml" xref="S4.SS1.p1.1.m1.1.1.3">mag</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.1.m1.1c">0.22\,\rm mag</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.1.m1.1d">0.22 roman_mag</annotation></semantics></math>. As a comparison, in the modern <span class="ltx_text ltx_font_italic" id="S4.SS1.p1.1.1">Hubble</span> Space Telescope (HST) filters, <cite class="ltx_cite ltx_citemacro_cite">Breuval et al. (<a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#bib.bib4" title="">2024</a>)</cite> find the lowest dispersion to date for the SMC Leavitt law with a scatter of 0.10 mag based on 87 Cepheids in the core of the SMC.</p> </div> <div class="ltx_para" id="S4.SS1.p2"> <p class="ltx_p" id="S4.SS1.p2.7">We also investigate the impact of the spatial distribution of Leavitt’s Cepheid sample in the SMC. <cite class="ltx_cite ltx_citemacro_citet">Breuval et al. (<a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#bib.bib2" title="">2022</a>, <a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#bib.bib4" title="">2024</a>)</cite> show that the elongated shape of the SMC produces additional scatter in the Leavitt law. For this reason, it is recommended to limit the Cepheid sample to a narrow region around the SMC center. Progressively reducing the sample to <math alttext="R=1^{\circ}" class="ltx_Math" display="inline" id="S4.SS1.p2.1.m1.1"><semantics id="S4.SS1.p2.1.m1.1a"><mrow id="S4.SS1.p2.1.m1.1.1" xref="S4.SS1.p2.1.m1.1.1.cmml"><mi id="S4.SS1.p2.1.m1.1.1.2" xref="S4.SS1.p2.1.m1.1.1.2.cmml">R</mi><mo id="S4.SS1.p2.1.m1.1.1.1" xref="S4.SS1.p2.1.m1.1.1.1.cmml">=</mo><msup id="S4.SS1.p2.1.m1.1.1.3" xref="S4.SS1.p2.1.m1.1.1.3.cmml"><mn id="S4.SS1.p2.1.m1.1.1.3.2" xref="S4.SS1.p2.1.m1.1.1.3.2.cmml">1</mn><mo id="S4.SS1.p2.1.m1.1.1.3.3" xref="S4.SS1.p2.1.m1.1.1.3.3.cmml">∘</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.1.m1.1b"><apply id="S4.SS1.p2.1.m1.1.1.cmml" xref="S4.SS1.p2.1.m1.1.1"><eq id="S4.SS1.p2.1.m1.1.1.1.cmml" xref="S4.SS1.p2.1.m1.1.1.1"></eq><ci id="S4.SS1.p2.1.m1.1.1.2.cmml" xref="S4.SS1.p2.1.m1.1.1.2">𝑅</ci><apply id="S4.SS1.p2.1.m1.1.1.3.cmml" xref="S4.SS1.p2.1.m1.1.1.3"><csymbol cd="ambiguous" id="S4.SS1.p2.1.m1.1.1.3.1.cmml" xref="S4.SS1.p2.1.m1.1.1.3">superscript</csymbol><cn id="S4.SS1.p2.1.m1.1.1.3.2.cmml" type="integer" xref="S4.SS1.p2.1.m1.1.1.3.2">1</cn><compose id="S4.SS1.p2.1.m1.1.1.3.3.cmml" xref="S4.SS1.p2.1.m1.1.1.3.3"></compose></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.1.m1.1c">R=1^{\circ}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.1.m1.1d">italic_R = 1 start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT</annotation></semantics></math>, <math alttext="0.9^{\circ}" class="ltx_Math" display="inline" id="S4.SS1.p2.2.m2.1"><semantics id="S4.SS1.p2.2.m2.1a"><msup id="S4.SS1.p2.2.m2.1.1" xref="S4.SS1.p2.2.m2.1.1.cmml"><mn id="S4.SS1.p2.2.m2.1.1.2" xref="S4.SS1.p2.2.m2.1.1.2.cmml">0.9</mn><mo id="S4.SS1.p2.2.m2.1.1.3" xref="S4.SS1.p2.2.m2.1.1.3.cmml">∘</mo></msup><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.2.m2.1b"><apply id="S4.SS1.p2.2.m2.1.1.cmml" xref="S4.SS1.p2.2.m2.1.1"><csymbol cd="ambiguous" id="S4.SS1.p2.2.m2.1.1.1.cmml" xref="S4.SS1.p2.2.m2.1.1">superscript</csymbol><cn id="S4.SS1.p2.2.m2.1.1.2.cmml" type="float" xref="S4.SS1.p2.2.m2.1.1.2">0.9</cn><compose id="S4.SS1.p2.2.m2.1.1.3.cmml" xref="S4.SS1.p2.2.m2.1.1.3"></compose></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.2.m2.1c">0.9^{\circ}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.2.m2.1d">0.9 start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="0.8^{\circ}" class="ltx_Math" display="inline" id="S4.SS1.p2.3.m3.1"><semantics id="S4.SS1.p2.3.m3.1a"><msup id="S4.SS1.p2.3.m3.1.1" xref="S4.SS1.p2.3.m3.1.1.cmml"><mn id="S4.SS1.p2.3.m3.1.1.2" xref="S4.SS1.p2.3.m3.1.1.2.cmml">0.8</mn><mo id="S4.SS1.p2.3.m3.1.1.3" xref="S4.SS1.p2.3.m3.1.1.3.cmml">∘</mo></msup><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.3.m3.1b"><apply id="S4.SS1.p2.3.m3.1.1.cmml" xref="S4.SS1.p2.3.m3.1.1"><csymbol cd="ambiguous" id="S4.SS1.p2.3.m3.1.1.1.cmml" xref="S4.SS1.p2.3.m3.1.1">superscript</csymbol><cn id="S4.SS1.p2.3.m3.1.1.2.cmml" type="float" xref="S4.SS1.p2.3.m3.1.1.2">0.8</cn><compose id="S4.SS1.p2.3.m3.1.1.3.cmml" xref="S4.SS1.p2.3.m3.1.1.3"></compose></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.3.m3.1c">0.8^{\circ}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.3.m3.1d">0.8 start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT</annotation></semantics></math> decreases the scatter of the Leavitt law to <math alttext="0.20\,\rm mag" class="ltx_Math" display="inline" id="S4.SS1.p2.4.m4.1"><semantics id="S4.SS1.p2.4.m4.1a"><mrow id="S4.SS1.p2.4.m4.1.1" xref="S4.SS1.p2.4.m4.1.1.cmml"><mn id="S4.SS1.p2.4.m4.1.1.2" xref="S4.SS1.p2.4.m4.1.1.2.cmml">0.20</mn><mo id="S4.SS1.p2.4.m4.1.1.1" lspace="0.170em" xref="S4.SS1.p2.4.m4.1.1.1.cmml">⁢</mo><mi id="S4.SS1.p2.4.m4.1.1.3" xref="S4.SS1.p2.4.m4.1.1.3.cmml">mag</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.4.m4.1b"><apply id="S4.SS1.p2.4.m4.1.1.cmml" xref="S4.SS1.p2.4.m4.1.1"><times id="S4.SS1.p2.4.m4.1.1.1.cmml" xref="S4.SS1.p2.4.m4.1.1.1"></times><cn id="S4.SS1.p2.4.m4.1.1.2.cmml" type="float" xref="S4.SS1.p2.4.m4.1.1.2">0.20</cn><ci id="S4.SS1.p2.4.m4.1.1.3.cmml" xref="S4.SS1.p2.4.m4.1.1.3">mag</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.4.m4.1c">0.20\,\rm mag</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.4.m4.1d">0.20 roman_mag</annotation></semantics></math> (N = 23), <math alttext="0.18\,\rm mag" class="ltx_Math" display="inline" id="S4.SS1.p2.5.m5.1"><semantics id="S4.SS1.p2.5.m5.1a"><mrow id="S4.SS1.p2.5.m5.1.1" xref="S4.SS1.p2.5.m5.1.1.cmml"><mn id="S4.SS1.p2.5.m5.1.1.2" xref="S4.SS1.p2.5.m5.1.1.2.cmml">0.18</mn><mo id="S4.SS1.p2.5.m5.1.1.1" lspace="0.170em" xref="S4.SS1.p2.5.m5.1.1.1.cmml">⁢</mo><mi id="S4.SS1.p2.5.m5.1.1.3" xref="S4.SS1.p2.5.m5.1.1.3.cmml">mag</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.5.m5.1b"><apply id="S4.SS1.p2.5.m5.1.1.cmml" xref="S4.SS1.p2.5.m5.1.1"><times id="S4.SS1.p2.5.m5.1.1.1.cmml" xref="S4.SS1.p2.5.m5.1.1.1"></times><cn id="S4.SS1.p2.5.m5.1.1.2.cmml" type="float" xref="S4.SS1.p2.5.m5.1.1.2">0.18</cn><ci id="S4.SS1.p2.5.m5.1.1.3.cmml" xref="S4.SS1.p2.5.m5.1.1.3">mag</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.5.m5.1c">0.18\,\rm mag</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.5.m5.1d">0.18 roman_mag</annotation></semantics></math> (N = 22), and <math alttext="0.17\,\rm mag" class="ltx_Math" display="inline" id="S4.SS1.p2.6.m6.1"><semantics id="S4.SS1.p2.6.m6.1a"><mrow id="S4.SS1.p2.6.m6.1.1" xref="S4.SS1.p2.6.m6.1.1.cmml"><mn id="S4.SS1.p2.6.m6.1.1.2" xref="S4.SS1.p2.6.m6.1.1.2.cmml">0.17</mn><mo id="S4.SS1.p2.6.m6.1.1.1" lspace="0.170em" xref="S4.SS1.p2.6.m6.1.1.1.cmml">⁢</mo><mi id="S4.SS1.p2.6.m6.1.1.3" xref="S4.SS1.p2.6.m6.1.1.3.cmml">mag</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.6.m6.1b"><apply id="S4.SS1.p2.6.m6.1.1.cmml" xref="S4.SS1.p2.6.m6.1.1"><times id="S4.SS1.p2.6.m6.1.1.1.cmml" xref="S4.SS1.p2.6.m6.1.1.1"></times><cn id="S4.SS1.p2.6.m6.1.1.2.cmml" type="float" xref="S4.SS1.p2.6.m6.1.1.2">0.17</cn><ci id="S4.SS1.p2.6.m6.1.1.3.cmml" xref="S4.SS1.p2.6.m6.1.1.3">mag</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.6.m6.1c">0.17\,\rm mag</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.6.m6.1d">0.17 roman_mag</annotation></semantics></math> (N = 18) respectively. Limiting the sample to an even narrower region would discard more than half of Leavitt’s sample, therefore, in the following we assume that the ideal subsample for this study would be a region of radius <math alttext="0.8^{\circ}" class="ltx_Math" display="inline" id="S4.SS1.p2.7.m7.1"><semantics id="S4.SS1.p2.7.m7.1a"><msup id="S4.SS1.p2.7.m7.1.1" xref="S4.SS1.p2.7.m7.1.1.cmml"><mn id="S4.SS1.p2.7.m7.1.1.2" xref="S4.SS1.p2.7.m7.1.1.2.cmml">0.8</mn><mo id="S4.SS1.p2.7.m7.1.1.3" xref="S4.SS1.p2.7.m7.1.1.3.cmml">∘</mo></msup><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.7.m7.1b"><apply id="S4.SS1.p2.7.m7.1.1.cmml" xref="S4.SS1.p2.7.m7.1.1"><csymbol cd="ambiguous" id="S4.SS1.p2.7.m7.1.1.1.cmml" xref="S4.SS1.p2.7.m7.1.1">superscript</csymbol><cn id="S4.SS1.p2.7.m7.1.1.2.cmml" type="float" xref="S4.SS1.p2.7.m7.1.1.2">0.8</cn><compose id="S4.SS1.p2.7.m7.1.1.3.cmml" xref="S4.SS1.p2.7.m7.1.1.3"></compose></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.7.m7.1c">0.8^{\circ}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.7.m7.1d">0.8 start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT</annotation></semantics></math> around the SMC center and use this subset (red circles) for the P-L relation fit (shown in black) in Figure <a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#S4.F5" title="Figure 5 ‣ 4 Period-Luminosity relation ‣ The Legacy of Henrietta Leavitt: A Re-analysis of the First Cepheid Period-Luminosity Relation"><span class="ltx_text ltx_ref_tag">5</span></a>. This test confirms that Cepheids located outside of the SMC core increase the P-L scatter, and shows that after excluding the latter, Leavitt’s P-L scatter is comparable to that obtained with modern data. The P-L scatter for different regions around the SMC center is listed in the upper part of Table <a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#S4.T2" title="Table 2 ‣ 4.1 P-L relation with magnitudes from Leavitt & Pickering (1912) ‣ 4 Period-Luminosity relation ‣ The Legacy of Henrietta Leavitt: A Re-analysis of the First Cepheid Period-Luminosity Relation"><span class="ltx_text ltx_ref_tag">2</span></a>.</p> </div> <div class="ltx_para" id="S4.SS1.p3"> <p class="ltx_p" id="S4.SS1.p3.7">The color response of Harvard photographic plates, including those used by Leavitt for her discovery, is comparable to the Johnson <math alttext="B" class="ltx_Math" display="inline" id="S4.SS1.p3.1.m1.1"><semantics id="S4.SS1.p3.1.m1.1a"><mi id="S4.SS1.p3.1.m1.1.1" xref="S4.SS1.p3.1.m1.1.1.cmml">B</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.1.m1.1b"><ci id="S4.SS1.p3.1.m1.1.1.cmml" xref="S4.SS1.p3.1.m1.1.1">𝐵</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.1.m1.1c">B</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.1.m1.1d">italic_B</annotation></semantics></math> filter <cite class="ltx_cite ltx_citemacro_citep">(Tang et al., <a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#bib.bib29" title="">2013</a>)</cite>. This can be verified empirically by comparing Leavitt’s measurements with modern data. For example, in Sect. <a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#S3" title="3 Cepheid light curves ‣ The Legacy of Henrietta Leavitt: A Re-analysis of the First Cepheid Period-Luminosity Relation"><span class="ltx_text ltx_ref_tag">3</span></a>, the amplitude ratios between Cepheid light curves from Leavitt’s notebook and from the OGLE survey point towards a wavelength between the <math alttext="B" class="ltx_Math" display="inline" id="S4.SS1.p3.2.m2.1"><semantics id="S4.SS1.p3.2.m2.1a"><mi id="S4.SS1.p3.2.m2.1.1" xref="S4.SS1.p3.2.m2.1.1.cmml">B</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.2.m2.1b"><ci id="S4.SS1.p3.2.m2.1.1.cmml" xref="S4.SS1.p3.2.m2.1.1">𝐵</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.2.m2.1c">B</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.2.m2.1d">italic_B</annotation></semantics></math>-band and the <math alttext="V" class="ltx_Math" display="inline" id="S4.SS1.p3.3.m3.1"><semantics id="S4.SS1.p3.3.m3.1a"><mi id="S4.SS1.p3.3.m3.1.1" xref="S4.SS1.p3.3.m3.1.1.cmml">V</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.3.m3.1b"><ci id="S4.SS1.p3.3.m3.1.1.cmml" xref="S4.SS1.p3.3.m3.1.1">𝑉</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.3.m3.1c">V</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.3.m3.1d">italic_V</annotation></semantics></math>-band, around <math alttext="\lambda\sim 0.50\,\mu" class="ltx_Math" display="inline" id="S4.SS1.p3.4.m4.1"><semantics id="S4.SS1.p3.4.m4.1a"><mrow id="S4.SS1.p3.4.m4.1.1" xref="S4.SS1.p3.4.m4.1.1.cmml"><mi id="S4.SS1.p3.4.m4.1.1.2" xref="S4.SS1.p3.4.m4.1.1.2.cmml">λ</mi><mo id="S4.SS1.p3.4.m4.1.1.1" xref="S4.SS1.p3.4.m4.1.1.1.cmml">∼</mo><mrow id="S4.SS1.p3.4.m4.1.1.3" xref="S4.SS1.p3.4.m4.1.1.3.cmml"><mn id="S4.SS1.p3.4.m4.1.1.3.2" xref="S4.SS1.p3.4.m4.1.1.3.2.cmml">0.50</mn><mo id="S4.SS1.p3.4.m4.1.1.3.1" lspace="0.170em" xref="S4.SS1.p3.4.m4.1.1.3.1.cmml">⁢</mo><mi id="S4.SS1.p3.4.m4.1.1.3.3" xref="S4.SS1.p3.4.m4.1.1.3.3.cmml">μ</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.4.m4.1b"><apply id="S4.SS1.p3.4.m4.1.1.cmml" xref="S4.SS1.p3.4.m4.1.1"><csymbol cd="latexml" id="S4.SS1.p3.4.m4.1.1.1.cmml" xref="S4.SS1.p3.4.m4.1.1.1">similar-to</csymbol><ci id="S4.SS1.p3.4.m4.1.1.2.cmml" xref="S4.SS1.p3.4.m4.1.1.2">𝜆</ci><apply id="S4.SS1.p3.4.m4.1.1.3.cmml" xref="S4.SS1.p3.4.m4.1.1.3"><times id="S4.SS1.p3.4.m4.1.1.3.1.cmml" xref="S4.SS1.p3.4.m4.1.1.3.1"></times><cn id="S4.SS1.p3.4.m4.1.1.3.2.cmml" type="float" xref="S4.SS1.p3.4.m4.1.1.3.2">0.50</cn><ci id="S4.SS1.p3.4.m4.1.1.3.3.cmml" xref="S4.SS1.p3.4.m4.1.1.3.3">𝜇</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.4.m4.1c">\lambda\sim 0.50\,\mu</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.4.m4.1d">italic_λ ∼ 0.50 italic_μ</annotation></semantics></math>m. In principle and assuming accurate magnitude measurements over a wide dynamic range of 5 magnitudes, we might infer the wavelength of the photographic plates by measuring the P-L slope, which is strongly correlated with the wavelength at which it is measured, with steeper slopes in the infrared and shallower slopes in the optical. For a plate response near the <math alttext="B" class="ltx_Math" display="inline" id="S4.SS1.p3.5.m5.1"><semantics id="S4.SS1.p3.5.m5.1a"><mi id="S4.SS1.p3.5.m5.1.1" xref="S4.SS1.p3.5.m5.1.1.cmml">B</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.5.m5.1b"><ci id="S4.SS1.p3.5.m5.1.1.cmml" xref="S4.SS1.p3.5.m5.1.1">𝐵</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.5.m5.1c">B</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.5.m5.1d">italic_B</annotation></semantics></math>-band, Leavitt’s P-L slope is expected around <math alttext="-2.55" class="ltx_Math" display="inline" id="S4.SS1.p3.6.m6.1"><semantics id="S4.SS1.p3.6.m6.1a"><mrow id="S4.SS1.p3.6.m6.1.1" xref="S4.SS1.p3.6.m6.1.1.cmml"><mo id="S4.SS1.p3.6.m6.1.1a" xref="S4.SS1.p3.6.m6.1.1.cmml">−</mo><mn id="S4.SS1.p3.6.m6.1.1.2" xref="S4.SS1.p3.6.m6.1.1.2.cmml">2.55</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.6.m6.1b"><apply id="S4.SS1.p3.6.m6.1.1.cmml" xref="S4.SS1.p3.6.m6.1.1"><minus id="S4.SS1.p3.6.m6.1.1.1.cmml" xref="S4.SS1.p3.6.m6.1.1"></minus><cn id="S4.SS1.p3.6.m6.1.1.2.cmml" type="float" xref="S4.SS1.p3.6.m6.1.1.2">2.55</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.6.m6.1c">-2.55</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.6.m6.1d">- 2.55</annotation></semantics></math> mag/dex (see Fig. <a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#S4.F6" title="Figure 6 ‣ 4.1 P-L relation with magnitudes from Leavitt & Pickering (1912) ‣ 4 Period-Luminosity relation ‣ The Legacy of Henrietta Leavitt: A Re-analysis of the First Cepheid Period-Luminosity Relation"><span class="ltx_text ltx_ref_tag">6</span></a>). In practice, the observed slope of Leavitt’s original P-L relation (Fig. <a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#S4.F5" title="Figure 5 ‣ 4 Period-Luminosity relation ‣ The Legacy of Henrietta Leavitt: A Re-analysis of the First Cepheid Period-Luminosity Relation"><span class="ltx_text ltx_ref_tag">5</span></a>, upper panel, solid black line) is much shallower, with <math alttext="-2.03\pm 0.11" class="ltx_Math" display="inline" id="S4.SS1.p3.7.m7.1"><semantics id="S4.SS1.p3.7.m7.1a"><mrow id="S4.SS1.p3.7.m7.1.1" xref="S4.SS1.p3.7.m7.1.1.cmml"><mrow id="S4.SS1.p3.7.m7.1.1.2" xref="S4.SS1.p3.7.m7.1.1.2.cmml"><mo id="S4.SS1.p3.7.m7.1.1.2a" xref="S4.SS1.p3.7.m7.1.1.2.cmml">−</mo><mn id="S4.SS1.p3.7.m7.1.1.2.2" xref="S4.SS1.p3.7.m7.1.1.2.2.cmml">2.03</mn></mrow><mo id="S4.SS1.p3.7.m7.1.1.1" xref="S4.SS1.p3.7.m7.1.1.1.cmml">±</mo><mn id="S4.SS1.p3.7.m7.1.1.3" xref="S4.SS1.p3.7.m7.1.1.3.cmml">0.11</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.7.m7.1b"><apply id="S4.SS1.p3.7.m7.1.1.cmml" xref="S4.SS1.p3.7.m7.1.1"><csymbol cd="latexml" id="S4.SS1.p3.7.m7.1.1.1.cmml" xref="S4.SS1.p3.7.m7.1.1.1">plus-or-minus</csymbol><apply id="S4.SS1.p3.7.m7.1.1.2.cmml" xref="S4.SS1.p3.7.m7.1.1.2"><minus id="S4.SS1.p3.7.m7.1.1.2.1.cmml" xref="S4.SS1.p3.7.m7.1.1.2"></minus><cn id="S4.SS1.p3.7.m7.1.1.2.2.cmml" type="float" xref="S4.SS1.p3.7.m7.1.1.2.2">2.03</cn></apply><cn id="S4.SS1.p3.7.m7.1.1.3.cmml" type="float" xref="S4.SS1.p3.7.m7.1.1.3">0.11</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.7.m7.1c">-2.03\pm 0.11</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.7.m7.1d">- 2.03 ± 0.11</annotation></semantics></math> mag/dex. It is very likely that the shallow P-L slope obtained by Leavitt is a consequence of two systematic errors, 1) the well-known non-linearity of the photographic plates over a wide dynamic range and 2) environmental crowding, exacerbated by low resolution.</p> </div> <figure class="ltx_table" id="S4.T2"> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_table">Table 2: </span>Successive improvements in the scatter of the P-L relation from Leavitt’s Cepheid sample. The columns are: (1) the radius <math alttext="R" class="ltx_Math" display="inline" id="S4.T2.5.m1.1"><semantics id="S4.T2.5.m1.1b"><mi id="S4.T2.5.m1.1.1" xref="S4.T2.5.m1.1.1.cmml">R</mi><annotation-xml encoding="MathML-Content" id="S4.T2.5.m1.1c"><ci id="S4.T2.5.m1.1.1.cmml" xref="S4.T2.5.m1.1.1">𝑅</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.T2.5.m1.1d">R</annotation><annotation encoding="application/x-llamapun" id="S4.T2.5.m1.1e">italic_R</annotation></semantics></math> of the region around the SMC center, (2) the number <math alttext="N" class="ltx_Math" display="inline" id="S4.T2.6.m2.1"><semantics id="S4.T2.6.m2.1b"><mi id="S4.T2.6.m2.1.1" xref="S4.T2.6.m2.1.1.cmml">N</mi><annotation-xml encoding="MathML-Content" id="S4.T2.6.m2.1c"><ci id="S4.T2.6.m2.1.1.cmml" xref="S4.T2.6.m2.1.1">𝑁</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.T2.6.m2.1d">N</annotation><annotation encoding="application/x-llamapun" id="S4.T2.6.m2.1e">italic_N</annotation></semantics></math> of Cepheids, (3) the inclusion or not of geometry corrections, (4) the value of <math alttext="R_{V}" class="ltx_Math" display="inline" id="S4.T2.7.m3.1"><semantics id="S4.T2.7.m3.1b"><msub id="S4.T2.7.m3.1.1" xref="S4.T2.7.m3.1.1.cmml"><mi id="S4.T2.7.m3.1.1.2" xref="S4.T2.7.m3.1.1.2.cmml">R</mi><mi id="S4.T2.7.m3.1.1.3" xref="S4.T2.7.m3.1.1.3.cmml">V</mi></msub><annotation-xml encoding="MathML-Content" id="S4.T2.7.m3.1c"><apply id="S4.T2.7.m3.1.1.cmml" xref="S4.T2.7.m3.1.1"><csymbol cd="ambiguous" id="S4.T2.7.m3.1.1.1.cmml" xref="S4.T2.7.m3.1.1">subscript</csymbol><ci id="S4.T2.7.m3.1.1.2.cmml" xref="S4.T2.7.m3.1.1.2">𝑅</ci><ci id="S4.T2.7.m3.1.1.3.cmml" xref="S4.T2.7.m3.1.1.3">𝑉</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T2.7.m3.1d">R_{V}</annotation><annotation encoding="application/x-llamapun" id="S4.T2.7.m3.1e">italic_R start_POSTSUBSCRIPT italic_V end_POSTSUBSCRIPT</annotation></semantics></math> and (5) the P-L scatter <math alttext="\sigma" class="ltx_Math" display="inline" id="S4.T2.8.m4.1"><semantics id="S4.T2.8.m4.1b"><mi id="S4.T2.8.m4.1.1" xref="S4.T2.8.m4.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S4.T2.8.m4.1c"><ci id="S4.T2.8.m4.1.1.cmml" xref="S4.T2.8.m4.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.T2.8.m4.1d">\sigma</annotation><annotation encoding="application/x-llamapun" id="S4.T2.8.m4.1e">italic_σ</annotation></semantics></math> in mag. <br class="ltx_break"/></figcaption> <table class="ltx_tabular ltx_guessed_headers ltx_align_middle" id="S4.T2.19"> <tbody class="ltx_tbody"> <tr class="ltx_tr" id="S4.T2.12.4"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_tt" id="S4.T2.9.1.1"><math alttext="R" class="ltx_Math" display="inline" id="S4.T2.9.1.1.m1.1"><semantics id="S4.T2.9.1.1.m1.1a"><mi id="S4.T2.9.1.1.m1.1.1" xref="S4.T2.9.1.1.m1.1.1.cmml">R</mi><annotation-xml encoding="MathML-Content" id="S4.T2.9.1.1.m1.1b"><ci id="S4.T2.9.1.1.m1.1.1.cmml" xref="S4.T2.9.1.1.m1.1.1">𝑅</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.T2.9.1.1.m1.1c">R</annotation><annotation encoding="application/x-llamapun" id="S4.T2.9.1.1.m1.1d">italic_R</annotation></semantics></math></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_tt" id="S4.T2.10.2.2"><math alttext="N" class="ltx_Math" display="inline" id="S4.T2.10.2.2.m1.1"><semantics id="S4.T2.10.2.2.m1.1a"><mi id="S4.T2.10.2.2.m1.1.1" xref="S4.T2.10.2.2.m1.1.1.cmml">N</mi><annotation-xml encoding="MathML-Content" id="S4.T2.10.2.2.m1.1b"><ci id="S4.T2.10.2.2.m1.1.1.cmml" xref="S4.T2.10.2.2.m1.1.1">𝑁</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.T2.10.2.2.m1.1c">N</annotation><annotation encoding="application/x-llamapun" id="S4.T2.10.2.2.m1.1d">italic_N</annotation></semantics></math></th> <td class="ltx_td ltx_align_center ltx_border_tt" id="S4.T2.12.4.5">geo. corr.</td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S4.T2.11.3.3"><math alttext="R_{V}" class="ltx_Math" display="inline" id="S4.T2.11.3.3.m1.1"><semantics id="S4.T2.11.3.3.m1.1a"><msub id="S4.T2.11.3.3.m1.1.1" xref="S4.T2.11.3.3.m1.1.1.cmml"><mi id="S4.T2.11.3.3.m1.1.1.2" xref="S4.T2.11.3.3.m1.1.1.2.cmml">R</mi><mi id="S4.T2.11.3.3.m1.1.1.3" xref="S4.T2.11.3.3.m1.1.1.3.cmml">V</mi></msub><annotation-xml encoding="MathML-Content" id="S4.T2.11.3.3.m1.1b"><apply id="S4.T2.11.3.3.m1.1.1.cmml" xref="S4.T2.11.3.3.m1.1.1"><csymbol cd="ambiguous" id="S4.T2.11.3.3.m1.1.1.1.cmml" xref="S4.T2.11.3.3.m1.1.1">subscript</csymbol><ci id="S4.T2.11.3.3.m1.1.1.2.cmml" xref="S4.T2.11.3.3.m1.1.1.2">𝑅</ci><ci id="S4.T2.11.3.3.m1.1.1.3.cmml" xref="S4.T2.11.3.3.m1.1.1.3">𝑉</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T2.11.3.3.m1.1c">R_{V}</annotation><annotation encoding="application/x-llamapun" id="S4.T2.11.3.3.m1.1d">italic_R start_POSTSUBSCRIPT italic_V end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S4.T2.12.4.4"><math alttext="\sigma" class="ltx_Math" display="inline" id="S4.T2.12.4.4.m1.1"><semantics id="S4.T2.12.4.4.m1.1a"><mi id="S4.T2.12.4.4.m1.1.1" xref="S4.T2.12.4.4.m1.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S4.T2.12.4.4.m1.1b"><ci id="S4.T2.12.4.4.m1.1.1.cmml" xref="S4.T2.12.4.4.m1.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.T2.12.4.4.m1.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S4.T2.12.4.4.m1.1d">italic_σ</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S4.T2.19.12.1"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row" id="S4.T2.19.12.1.1">(deg)</th> <th class="ltx_td ltx_th ltx_th_row" id="S4.T2.19.12.1.2"></th> <td class="ltx_td" id="S4.T2.19.12.1.3"></td> <td class="ltx_td" id="S4.T2.19.12.1.4"></td> <td class="ltx_td ltx_align_center" id="S4.T2.19.12.1.5">(mag)</td> </tr> <tr class="ltx_tr" id="S4.T2.19.13.2"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_t" colspan="5" id="S4.T2.19.13.2.1">Leavitt’s provisional magnitude scale</th> </tr> <tr class="ltx_tr" id="S4.T2.19.14.3"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_t" id="S4.T2.19.14.3.1">Full</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_t" id="S4.T2.19.14.3.2">25</th> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T2.19.14.3.3">no</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T2.19.14.3.4">–</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T2.19.14.3.5"><span class="ltx_text ltx_font_bold" id="S4.T2.19.14.3.5.1">0.224</span></td> </tr> <tr class="ltx_tr" id="S4.T2.13.5"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row" id="S4.T2.13.5.1"><math alttext="R<1.0^{\circ}" class="ltx_Math" display="inline" id="S4.T2.13.5.1.m1.1"><semantics id="S4.T2.13.5.1.m1.1a"><mrow id="S4.T2.13.5.1.m1.1.1" xref="S4.T2.13.5.1.m1.1.1.cmml"><mi id="S4.T2.13.5.1.m1.1.1.2" xref="S4.T2.13.5.1.m1.1.1.2.cmml">R</mi><mo id="S4.T2.13.5.1.m1.1.1.1" xref="S4.T2.13.5.1.m1.1.1.1.cmml"><</mo><msup id="S4.T2.13.5.1.m1.1.1.3" xref="S4.T2.13.5.1.m1.1.1.3.cmml"><mn id="S4.T2.13.5.1.m1.1.1.3.2" xref="S4.T2.13.5.1.m1.1.1.3.2.cmml">1.0</mn><mo id="S4.T2.13.5.1.m1.1.1.3.3" xref="S4.T2.13.5.1.m1.1.1.3.3.cmml">∘</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.T2.13.5.1.m1.1b"><apply id="S4.T2.13.5.1.m1.1.1.cmml" xref="S4.T2.13.5.1.m1.1.1"><lt id="S4.T2.13.5.1.m1.1.1.1.cmml" xref="S4.T2.13.5.1.m1.1.1.1"></lt><ci id="S4.T2.13.5.1.m1.1.1.2.cmml" xref="S4.T2.13.5.1.m1.1.1.2">𝑅</ci><apply id="S4.T2.13.5.1.m1.1.1.3.cmml" xref="S4.T2.13.5.1.m1.1.1.3"><csymbol cd="ambiguous" id="S4.T2.13.5.1.m1.1.1.3.1.cmml" xref="S4.T2.13.5.1.m1.1.1.3">superscript</csymbol><cn id="S4.T2.13.5.1.m1.1.1.3.2.cmml" type="float" xref="S4.T2.13.5.1.m1.1.1.3.2">1.0</cn><compose id="S4.T2.13.5.1.m1.1.1.3.3.cmml" xref="S4.T2.13.5.1.m1.1.1.3.3"></compose></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T2.13.5.1.m1.1c">R<1.0^{\circ}</annotation><annotation encoding="application/x-llamapun" id="S4.T2.13.5.1.m1.1d">italic_R < 1.0 start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT</annotation></semantics></math></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_row" id="S4.T2.13.5.2">23</th> <td class="ltx_td ltx_align_center" id="S4.T2.13.5.3">no</td> <td class="ltx_td ltx_align_center" id="S4.T2.13.5.4">–</td> <td class="ltx_td ltx_align_center" id="S4.T2.13.5.5"><span class="ltx_text ltx_font_bold" id="S4.T2.13.5.5.1">0.197</span></td> </tr> <tr class="ltx_tr" id="S4.T2.14.6"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row" id="S4.T2.14.6.1"><math alttext="R<0.9^{\circ}" class="ltx_Math" display="inline" id="S4.T2.14.6.1.m1.1"><semantics id="S4.T2.14.6.1.m1.1a"><mrow id="S4.T2.14.6.1.m1.1.1" xref="S4.T2.14.6.1.m1.1.1.cmml"><mi id="S4.T2.14.6.1.m1.1.1.2" xref="S4.T2.14.6.1.m1.1.1.2.cmml">R</mi><mo id="S4.T2.14.6.1.m1.1.1.1" xref="S4.T2.14.6.1.m1.1.1.1.cmml"><</mo><msup id="S4.T2.14.6.1.m1.1.1.3" xref="S4.T2.14.6.1.m1.1.1.3.cmml"><mn id="S4.T2.14.6.1.m1.1.1.3.2" xref="S4.T2.14.6.1.m1.1.1.3.2.cmml">0.9</mn><mo id="S4.T2.14.6.1.m1.1.1.3.3" xref="S4.T2.14.6.1.m1.1.1.3.3.cmml">∘</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.T2.14.6.1.m1.1b"><apply id="S4.T2.14.6.1.m1.1.1.cmml" xref="S4.T2.14.6.1.m1.1.1"><lt id="S4.T2.14.6.1.m1.1.1.1.cmml" xref="S4.T2.14.6.1.m1.1.1.1"></lt><ci id="S4.T2.14.6.1.m1.1.1.2.cmml" xref="S4.T2.14.6.1.m1.1.1.2">𝑅</ci><apply id="S4.T2.14.6.1.m1.1.1.3.cmml" xref="S4.T2.14.6.1.m1.1.1.3"><csymbol cd="ambiguous" id="S4.T2.14.6.1.m1.1.1.3.1.cmml" xref="S4.T2.14.6.1.m1.1.1.3">superscript</csymbol><cn id="S4.T2.14.6.1.m1.1.1.3.2.cmml" type="float" xref="S4.T2.14.6.1.m1.1.1.3.2">0.9</cn><compose id="S4.T2.14.6.1.m1.1.1.3.3.cmml" xref="S4.T2.14.6.1.m1.1.1.3.3"></compose></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T2.14.6.1.m1.1c">R<0.9^{\circ}</annotation><annotation encoding="application/x-llamapun" id="S4.T2.14.6.1.m1.1d">italic_R < 0.9 start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT</annotation></semantics></math></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_row" id="S4.T2.14.6.2">22</th> <td class="ltx_td ltx_align_center" id="S4.T2.14.6.3">no</td> <td class="ltx_td ltx_align_center" id="S4.T2.14.6.4">–</td> <td class="ltx_td ltx_align_center" id="S4.T2.14.6.5"><span class="ltx_text ltx_font_bold" id="S4.T2.14.6.5.1">0.179</span></td> </tr> <tr class="ltx_tr" id="S4.T2.15.7"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row" id="S4.T2.15.7.1"><math alttext="R<0.8^{\circ}" class="ltx_Math" display="inline" id="S4.T2.15.7.1.m1.1"><semantics id="S4.T2.15.7.1.m1.1a"><mrow id="S4.T2.15.7.1.m1.1.1" xref="S4.T2.15.7.1.m1.1.1.cmml"><mi id="S4.T2.15.7.1.m1.1.1.2" xref="S4.T2.15.7.1.m1.1.1.2.cmml">R</mi><mo id="S4.T2.15.7.1.m1.1.1.1" xref="S4.T2.15.7.1.m1.1.1.1.cmml"><</mo><msup id="S4.T2.15.7.1.m1.1.1.3" xref="S4.T2.15.7.1.m1.1.1.3.cmml"><mn id="S4.T2.15.7.1.m1.1.1.3.2" xref="S4.T2.15.7.1.m1.1.1.3.2.cmml">0.8</mn><mo id="S4.T2.15.7.1.m1.1.1.3.3" xref="S4.T2.15.7.1.m1.1.1.3.3.cmml">∘</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.T2.15.7.1.m1.1b"><apply id="S4.T2.15.7.1.m1.1.1.cmml" xref="S4.T2.15.7.1.m1.1.1"><lt id="S4.T2.15.7.1.m1.1.1.1.cmml" xref="S4.T2.15.7.1.m1.1.1.1"></lt><ci id="S4.T2.15.7.1.m1.1.1.2.cmml" xref="S4.T2.15.7.1.m1.1.1.2">𝑅</ci><apply id="S4.T2.15.7.1.m1.1.1.3.cmml" xref="S4.T2.15.7.1.m1.1.1.3"><csymbol cd="ambiguous" id="S4.T2.15.7.1.m1.1.1.3.1.cmml" xref="S4.T2.15.7.1.m1.1.1.3">superscript</csymbol><cn id="S4.T2.15.7.1.m1.1.1.3.2.cmml" type="float" xref="S4.T2.15.7.1.m1.1.1.3.2">0.8</cn><compose id="S4.T2.15.7.1.m1.1.1.3.3.cmml" xref="S4.T2.15.7.1.m1.1.1.3.3"></compose></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T2.15.7.1.m1.1c">R<0.8^{\circ}</annotation><annotation encoding="application/x-llamapun" id="S4.T2.15.7.1.m1.1d">italic_R < 0.8 start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT</annotation></semantics></math></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_row" id="S4.T2.15.7.2">18</th> <td class="ltx_td ltx_align_center" id="S4.T2.15.7.3">no</td> <td class="ltx_td ltx_align_center" id="S4.T2.15.7.4">–</td> <td class="ltx_td ltx_align_center" id="S4.T2.15.7.5"><span class="ltx_text ltx_font_bold" id="S4.T2.15.7.5.1">0.172</span></td> </tr> <tr class="ltx_tr" id="S4.T2.16.8"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_t" colspan="5" id="S4.T2.16.8.1">OGLE mean magnitudes (<math alttext="W_{VI}" class="ltx_Math" display="inline" id="S4.T2.16.8.1.m1.1"><semantics id="S4.T2.16.8.1.m1.1a"><msub id="S4.T2.16.8.1.m1.1.1" xref="S4.T2.16.8.1.m1.1.1.cmml"><mi id="S4.T2.16.8.1.m1.1.1.2" xref="S4.T2.16.8.1.m1.1.1.2.cmml">W</mi><mrow id="S4.T2.16.8.1.m1.1.1.3" xref="S4.T2.16.8.1.m1.1.1.3.cmml"><mi id="S4.T2.16.8.1.m1.1.1.3.2" xref="S4.T2.16.8.1.m1.1.1.3.2.cmml">V</mi><mo id="S4.T2.16.8.1.m1.1.1.3.1" xref="S4.T2.16.8.1.m1.1.1.3.1.cmml">⁢</mo><mi id="S4.T2.16.8.1.m1.1.1.3.3" xref="S4.T2.16.8.1.m1.1.1.3.3.cmml">I</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S4.T2.16.8.1.m1.1b"><apply id="S4.T2.16.8.1.m1.1.1.cmml" xref="S4.T2.16.8.1.m1.1.1"><csymbol cd="ambiguous" id="S4.T2.16.8.1.m1.1.1.1.cmml" xref="S4.T2.16.8.1.m1.1.1">subscript</csymbol><ci id="S4.T2.16.8.1.m1.1.1.2.cmml" xref="S4.T2.16.8.1.m1.1.1.2">𝑊</ci><apply id="S4.T2.16.8.1.m1.1.1.3.cmml" xref="S4.T2.16.8.1.m1.1.1.3"><times id="S4.T2.16.8.1.m1.1.1.3.1.cmml" xref="S4.T2.16.8.1.m1.1.1.3.1"></times><ci id="S4.T2.16.8.1.m1.1.1.3.2.cmml" xref="S4.T2.16.8.1.m1.1.1.3.2">𝑉</ci><ci id="S4.T2.16.8.1.m1.1.1.3.3.cmml" xref="S4.T2.16.8.1.m1.1.1.3.3">𝐼</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T2.16.8.1.m1.1c">W_{VI}</annotation><annotation encoding="application/x-llamapun" id="S4.T2.16.8.1.m1.1d">italic_W start_POSTSUBSCRIPT italic_V italic_I end_POSTSUBSCRIPT</annotation></semantics></math>)</th> </tr> <tr class="ltx_tr" id="S4.T2.19.15.4"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_t" id="S4.T2.19.15.4.1">Full</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_t" id="S4.T2.19.15.4.2">23</th> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T2.19.15.4.3">no</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T2.19.15.4.4">3.3</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T2.19.15.4.5"><span class="ltx_text ltx_font_bold" id="S4.T2.19.15.4.5.1">0.137</span></td> </tr> <tr class="ltx_tr" id="S4.T2.19.16.5"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row" id="S4.T2.19.16.5.1">Full</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_row" id="S4.T2.19.16.5.2">23</th> <td class="ltx_td ltx_align_center" id="S4.T2.19.16.5.3">yes</td> <td class="ltx_td ltx_align_center" id="S4.T2.19.16.5.4">3.3</td> <td class="ltx_td ltx_align_center" id="S4.T2.19.16.5.5"><span class="ltx_text ltx_font_bold" id="S4.T2.19.16.5.5.1">0.126</span></td> </tr> <tr class="ltx_tr" id="S4.T2.17.9"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row" id="S4.T2.17.9.1"><math alttext="R<1.0^{\circ}" class="ltx_Math" display="inline" id="S4.T2.17.9.1.m1.1"><semantics id="S4.T2.17.9.1.m1.1a"><mrow id="S4.T2.17.9.1.m1.1.1" xref="S4.T2.17.9.1.m1.1.1.cmml"><mi id="S4.T2.17.9.1.m1.1.1.2" xref="S4.T2.17.9.1.m1.1.1.2.cmml">R</mi><mo id="S4.T2.17.9.1.m1.1.1.1" xref="S4.T2.17.9.1.m1.1.1.1.cmml"><</mo><msup id="S4.T2.17.9.1.m1.1.1.3" xref="S4.T2.17.9.1.m1.1.1.3.cmml"><mn id="S4.T2.17.9.1.m1.1.1.3.2" xref="S4.T2.17.9.1.m1.1.1.3.2.cmml">1.0</mn><mo id="S4.T2.17.9.1.m1.1.1.3.3" xref="S4.T2.17.9.1.m1.1.1.3.3.cmml">∘</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.T2.17.9.1.m1.1b"><apply id="S4.T2.17.9.1.m1.1.1.cmml" xref="S4.T2.17.9.1.m1.1.1"><lt id="S4.T2.17.9.1.m1.1.1.1.cmml" xref="S4.T2.17.9.1.m1.1.1.1"></lt><ci id="S4.T2.17.9.1.m1.1.1.2.cmml" xref="S4.T2.17.9.1.m1.1.1.2">𝑅</ci><apply id="S4.T2.17.9.1.m1.1.1.3.cmml" xref="S4.T2.17.9.1.m1.1.1.3"><csymbol cd="ambiguous" id="S4.T2.17.9.1.m1.1.1.3.1.cmml" xref="S4.T2.17.9.1.m1.1.1.3">superscript</csymbol><cn id="S4.T2.17.9.1.m1.1.1.3.2.cmml" type="float" xref="S4.T2.17.9.1.m1.1.1.3.2">1.0</cn><compose id="S4.T2.17.9.1.m1.1.1.3.3.cmml" xref="S4.T2.17.9.1.m1.1.1.3.3"></compose></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T2.17.9.1.m1.1c">R<1.0^{\circ}</annotation><annotation encoding="application/x-llamapun" id="S4.T2.17.9.1.m1.1d">italic_R < 1.0 start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT</annotation></semantics></math></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_row" id="S4.T2.17.9.2">21</th> <td class="ltx_td ltx_align_center" id="S4.T2.17.9.3">yes</td> <td class="ltx_td ltx_align_center" id="S4.T2.17.9.4">3.3</td> <td class="ltx_td ltx_align_center" id="S4.T2.17.9.5"><span class="ltx_text ltx_font_bold" id="S4.T2.17.9.5.1">0.118</span></td> </tr> <tr class="ltx_tr" id="S4.T2.18.10"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row" id="S4.T2.18.10.1"><math alttext="R<0.8^{\circ}" class="ltx_Math" display="inline" id="S4.T2.18.10.1.m1.1"><semantics id="S4.T2.18.10.1.m1.1a"><mrow id="S4.T2.18.10.1.m1.1.1" xref="S4.T2.18.10.1.m1.1.1.cmml"><mi id="S4.T2.18.10.1.m1.1.1.2" xref="S4.T2.18.10.1.m1.1.1.2.cmml">R</mi><mo id="S4.T2.18.10.1.m1.1.1.1" xref="S4.T2.18.10.1.m1.1.1.1.cmml"><</mo><msup id="S4.T2.18.10.1.m1.1.1.3" xref="S4.T2.18.10.1.m1.1.1.3.cmml"><mn id="S4.T2.18.10.1.m1.1.1.3.2" xref="S4.T2.18.10.1.m1.1.1.3.2.cmml">0.8</mn><mo id="S4.T2.18.10.1.m1.1.1.3.3" xref="S4.T2.18.10.1.m1.1.1.3.3.cmml">∘</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.T2.18.10.1.m1.1b"><apply id="S4.T2.18.10.1.m1.1.1.cmml" xref="S4.T2.18.10.1.m1.1.1"><lt id="S4.T2.18.10.1.m1.1.1.1.cmml" xref="S4.T2.18.10.1.m1.1.1.1"></lt><ci id="S4.T2.18.10.1.m1.1.1.2.cmml" xref="S4.T2.18.10.1.m1.1.1.2">𝑅</ci><apply id="S4.T2.18.10.1.m1.1.1.3.cmml" xref="S4.T2.18.10.1.m1.1.1.3"><csymbol cd="ambiguous" id="S4.T2.18.10.1.m1.1.1.3.1.cmml" xref="S4.T2.18.10.1.m1.1.1.3">superscript</csymbol><cn id="S4.T2.18.10.1.m1.1.1.3.2.cmml" type="float" xref="S4.T2.18.10.1.m1.1.1.3.2">0.8</cn><compose id="S4.T2.18.10.1.m1.1.1.3.3.cmml" xref="S4.T2.18.10.1.m1.1.1.3.3"></compose></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T2.18.10.1.m1.1c">R<0.8^{\circ}</annotation><annotation encoding="application/x-llamapun" id="S4.T2.18.10.1.m1.1d">italic_R < 0.8 start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT</annotation></semantics></math></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_row" id="S4.T2.18.10.2">17</th> <td class="ltx_td ltx_align_center" id="S4.T2.18.10.3">yes</td> <td class="ltx_td ltx_align_center" id="S4.T2.18.10.4">3.3</td> <td class="ltx_td ltx_align_center" id="S4.T2.18.10.5"><span class="ltx_text ltx_font_bold" id="S4.T2.18.10.5.1">0.112</span></td> </tr> <tr class="ltx_tr" id="S4.T2.19.11"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_b" id="S4.T2.19.11.1"><math alttext="R<0.8^{\circ}" class="ltx_Math" display="inline" id="S4.T2.19.11.1.m1.1"><semantics id="S4.T2.19.11.1.m1.1a"><mrow id="S4.T2.19.11.1.m1.1.1" xref="S4.T2.19.11.1.m1.1.1.cmml"><mi id="S4.T2.19.11.1.m1.1.1.2" xref="S4.T2.19.11.1.m1.1.1.2.cmml">R</mi><mo id="S4.T2.19.11.1.m1.1.1.1" xref="S4.T2.19.11.1.m1.1.1.1.cmml"><</mo><msup id="S4.T2.19.11.1.m1.1.1.3" xref="S4.T2.19.11.1.m1.1.1.3.cmml"><mn id="S4.T2.19.11.1.m1.1.1.3.2" xref="S4.T2.19.11.1.m1.1.1.3.2.cmml">0.8</mn><mo id="S4.T2.19.11.1.m1.1.1.3.3" xref="S4.T2.19.11.1.m1.1.1.3.3.cmml">∘</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.T2.19.11.1.m1.1b"><apply id="S4.T2.19.11.1.m1.1.1.cmml" xref="S4.T2.19.11.1.m1.1.1"><lt id="S4.T2.19.11.1.m1.1.1.1.cmml" xref="S4.T2.19.11.1.m1.1.1.1"></lt><ci id="S4.T2.19.11.1.m1.1.1.2.cmml" xref="S4.T2.19.11.1.m1.1.1.2">𝑅</ci><apply id="S4.T2.19.11.1.m1.1.1.3.cmml" xref="S4.T2.19.11.1.m1.1.1.3"><csymbol cd="ambiguous" id="S4.T2.19.11.1.m1.1.1.3.1.cmml" xref="S4.T2.19.11.1.m1.1.1.3">superscript</csymbol><cn id="S4.T2.19.11.1.m1.1.1.3.2.cmml" type="float" xref="S4.T2.19.11.1.m1.1.1.3.2">0.8</cn><compose id="S4.T2.19.11.1.m1.1.1.3.3.cmml" xref="S4.T2.19.11.1.m1.1.1.3.3"></compose></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T2.19.11.1.m1.1c">R<0.8^{\circ}</annotation><annotation encoding="application/x-llamapun" id="S4.T2.19.11.1.m1.1d">italic_R < 0.8 start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT</annotation></semantics></math></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_b" id="S4.T2.19.11.2">17</th> <td class="ltx_td ltx_align_center ltx_border_b" id="S4.T2.19.11.3">yes</td> <td class="ltx_td ltx_align_center ltx_border_b" id="S4.T2.19.11.4">2.74</td> <td class="ltx_td ltx_align_center ltx_border_b" id="S4.T2.19.11.5"><span class="ltx_text ltx_font_bold" id="S4.T2.19.11.5.1">0.105</span></td> </tr> </tbody> </table> </figure> <figure class="ltx_figure" id="S4.F6"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="359" id="S4.F6.g1" src="x8.png" width="449"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 6: </span>P-L slope for the SMC relations as a function of wavelength, labeled with the filters. Values in <span class="ltx_text ltx_font_italic" id="S4.F6.7.1">Gaia</span>, <span class="ltx_text ltx_font_italic" id="S4.F6.8.2">Spitzer</span> and ground-based filters are from <cite class="ltx_cite ltx_citemacro_citet">Breuval et al. (<a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#bib.bib2" title="">2022</a>)</cite>, and values in HST filters are from <cite class="ltx_cite ltx_citemacro_citet">Breuval et al. (<a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#bib.bib4" title="">2024</a>)</cite>. The red square shows the observed P-L slope in Leavitt’s provisional scale of magnitude (<math alttext="-2.033\pm 0.098" class="ltx_Math" display="inline" id="S4.F6.3.m1.1"><semantics id="S4.F6.3.m1.1b"><mrow id="S4.F6.3.m1.1.1" xref="S4.F6.3.m1.1.1.cmml"><mrow id="S4.F6.3.m1.1.1.2" xref="S4.F6.3.m1.1.1.2.cmml"><mo id="S4.F6.3.m1.1.1.2b" xref="S4.F6.3.m1.1.1.2.cmml">−</mo><mn id="S4.F6.3.m1.1.1.2.2" xref="S4.F6.3.m1.1.1.2.2.cmml">2.033</mn></mrow><mo id="S4.F6.3.m1.1.1.1" xref="S4.F6.3.m1.1.1.1.cmml">±</mo><mn id="S4.F6.3.m1.1.1.3" xref="S4.F6.3.m1.1.1.3.cmml">0.098</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.F6.3.m1.1c"><apply id="S4.F6.3.m1.1.1.cmml" xref="S4.F6.3.m1.1.1"><csymbol cd="latexml" id="S4.F6.3.m1.1.1.1.cmml" xref="S4.F6.3.m1.1.1.1">plus-or-minus</csymbol><apply id="S4.F6.3.m1.1.1.2.cmml" xref="S4.F6.3.m1.1.1.2"><minus id="S4.F6.3.m1.1.1.2.1.cmml" xref="S4.F6.3.m1.1.1.2"></minus><cn id="S4.F6.3.m1.1.1.2.2.cmml" type="float" xref="S4.F6.3.m1.1.1.2.2">2.033</cn></apply><cn id="S4.F6.3.m1.1.1.3.cmml" type="float" xref="S4.F6.3.m1.1.1.3">0.098</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F6.3.m1.1d">-2.033\pm 0.098</annotation><annotation encoding="application/x-llamapun" id="S4.F6.3.m1.1e">- 2.033 ± 0.098</annotation></semantics></math> mag/dex) at the position of the expected wavelength of photographic plate observations (0.5 <math alttext="\mu" class="ltx_Math" display="inline" id="S4.F6.4.m2.1"><semantics id="S4.F6.4.m2.1b"><mi id="S4.F6.4.m2.1.1" xref="S4.F6.4.m2.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S4.F6.4.m2.1c"><ci id="S4.F6.4.m2.1.1.cmml" xref="S4.F6.4.m2.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.F6.4.m2.1d">\mu</annotation><annotation encoding="application/x-llamapun" id="S4.F6.4.m2.1e">italic_μ</annotation></semantics></math>m). <br class="ltx_break"/></figcaption> </figure> <figure class="ltx_table" id="S4.T3"> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_table">Table 3: </span>Leavitt law (<math alttext="m_{VI}^{W}=\alpha\log P+\beta" class="ltx_Math" display="inline" id="S4.T3.4.m1.1"><semantics id="S4.T3.4.m1.1b"><mrow id="S4.T3.4.m1.1.1" xref="S4.T3.4.m1.1.1.cmml"><msubsup id="S4.T3.4.m1.1.1.2" xref="S4.T3.4.m1.1.1.2.cmml"><mi id="S4.T3.4.m1.1.1.2.2.2" xref="S4.T3.4.m1.1.1.2.2.2.cmml">m</mi><mrow id="S4.T3.4.m1.1.1.2.2.3" xref="S4.T3.4.m1.1.1.2.2.3.cmml"><mi id="S4.T3.4.m1.1.1.2.2.3.2" xref="S4.T3.4.m1.1.1.2.2.3.2.cmml">V</mi><mo id="S4.T3.4.m1.1.1.2.2.3.1" xref="S4.T3.4.m1.1.1.2.2.3.1.cmml">⁢</mo><mi id="S4.T3.4.m1.1.1.2.2.3.3" xref="S4.T3.4.m1.1.1.2.2.3.3.cmml">I</mi></mrow><mi id="S4.T3.4.m1.1.1.2.3" xref="S4.T3.4.m1.1.1.2.3.cmml">W</mi></msubsup><mo id="S4.T3.4.m1.1.1.1" xref="S4.T3.4.m1.1.1.1.cmml">=</mo><mrow id="S4.T3.4.m1.1.1.3" xref="S4.T3.4.m1.1.1.3.cmml"><mrow id="S4.T3.4.m1.1.1.3.2" xref="S4.T3.4.m1.1.1.3.2.cmml"><mi id="S4.T3.4.m1.1.1.3.2.2" xref="S4.T3.4.m1.1.1.3.2.2.cmml">α</mi><mo id="S4.T3.4.m1.1.1.3.2.1" lspace="0.167em" xref="S4.T3.4.m1.1.1.3.2.1.cmml">⁢</mo><mrow id="S4.T3.4.m1.1.1.3.2.3" xref="S4.T3.4.m1.1.1.3.2.3.cmml"><mi id="S4.T3.4.m1.1.1.3.2.3.1" xref="S4.T3.4.m1.1.1.3.2.3.1.cmml">log</mi><mo id="S4.T3.4.m1.1.1.3.2.3b" lspace="0.167em" xref="S4.T3.4.m1.1.1.3.2.3.cmml">⁡</mo><mi id="S4.T3.4.m1.1.1.3.2.3.2" xref="S4.T3.4.m1.1.1.3.2.3.2.cmml">P</mi></mrow></mrow><mo id="S4.T3.4.m1.1.1.3.1" xref="S4.T3.4.m1.1.1.3.1.cmml">+</mo><mi id="S4.T3.4.m1.1.1.3.3" xref="S4.T3.4.m1.1.1.3.3.cmml">β</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.T3.4.m1.1c"><apply id="S4.T3.4.m1.1.1.cmml" xref="S4.T3.4.m1.1.1"><eq id="S4.T3.4.m1.1.1.1.cmml" xref="S4.T3.4.m1.1.1.1"></eq><apply id="S4.T3.4.m1.1.1.2.cmml" xref="S4.T3.4.m1.1.1.2"><csymbol cd="ambiguous" id="S4.T3.4.m1.1.1.2.1.cmml" xref="S4.T3.4.m1.1.1.2">superscript</csymbol><apply id="S4.T3.4.m1.1.1.2.2.cmml" xref="S4.T3.4.m1.1.1.2"><csymbol cd="ambiguous" id="S4.T3.4.m1.1.1.2.2.1.cmml" xref="S4.T3.4.m1.1.1.2">subscript</csymbol><ci id="S4.T3.4.m1.1.1.2.2.2.cmml" xref="S4.T3.4.m1.1.1.2.2.2">𝑚</ci><apply id="S4.T3.4.m1.1.1.2.2.3.cmml" xref="S4.T3.4.m1.1.1.2.2.3"><times id="S4.T3.4.m1.1.1.2.2.3.1.cmml" xref="S4.T3.4.m1.1.1.2.2.3.1"></times><ci id="S4.T3.4.m1.1.1.2.2.3.2.cmml" xref="S4.T3.4.m1.1.1.2.2.3.2">𝑉</ci><ci id="S4.T3.4.m1.1.1.2.2.3.3.cmml" xref="S4.T3.4.m1.1.1.2.2.3.3">𝐼</ci></apply></apply><ci id="S4.T3.4.m1.1.1.2.3.cmml" xref="S4.T3.4.m1.1.1.2.3">𝑊</ci></apply><apply id="S4.T3.4.m1.1.1.3.cmml" xref="S4.T3.4.m1.1.1.3"><plus id="S4.T3.4.m1.1.1.3.1.cmml" xref="S4.T3.4.m1.1.1.3.1"></plus><apply id="S4.T3.4.m1.1.1.3.2.cmml" xref="S4.T3.4.m1.1.1.3.2"><times id="S4.T3.4.m1.1.1.3.2.1.cmml" xref="S4.T3.4.m1.1.1.3.2.1"></times><ci id="S4.T3.4.m1.1.1.3.2.2.cmml" xref="S4.T3.4.m1.1.1.3.2.2">𝛼</ci><apply id="S4.T3.4.m1.1.1.3.2.3.cmml" xref="S4.T3.4.m1.1.1.3.2.3"><log id="S4.T3.4.m1.1.1.3.2.3.1.cmml" xref="S4.T3.4.m1.1.1.3.2.3.1"></log><ci id="S4.T3.4.m1.1.1.3.2.3.2.cmml" xref="S4.T3.4.m1.1.1.3.2.3.2">𝑃</ci></apply></apply><ci id="S4.T3.4.m1.1.1.3.3.cmml" xref="S4.T3.4.m1.1.1.3.3">𝛽</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T3.4.m1.1d">m_{VI}^{W}=\alpha\log P+\beta</annotation><annotation encoding="application/x-llamapun" id="S4.T3.4.m1.1e">italic_m start_POSTSUBSCRIPT italic_V italic_I end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_W end_POSTSUPERSCRIPT = italic_α roman_log italic_P + italic_β</annotation></semantics></math>) obtained with Leavitt’s Cepheid sample for different <math alttext="R_{V}" class="ltx_Math" display="inline" id="S4.T3.5.m2.1"><semantics id="S4.T3.5.m2.1b"><msub id="S4.T3.5.m2.1.1" xref="S4.T3.5.m2.1.1.cmml"><mi id="S4.T3.5.m2.1.1.2" xref="S4.T3.5.m2.1.1.2.cmml">R</mi><mi id="S4.T3.5.m2.1.1.3" xref="S4.T3.5.m2.1.1.3.cmml">V</mi></msub><annotation-xml encoding="MathML-Content" id="S4.T3.5.m2.1c"><apply id="S4.T3.5.m2.1.1.cmml" xref="S4.T3.5.m2.1.1"><csymbol cd="ambiguous" id="S4.T3.5.m2.1.1.1.cmml" xref="S4.T3.5.m2.1.1">subscript</csymbol><ci id="S4.T3.5.m2.1.1.2.cmml" xref="S4.T3.5.m2.1.1.2">𝑅</ci><ci id="S4.T3.5.m2.1.1.3.cmml" xref="S4.T3.5.m2.1.1.3">𝑉</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T3.5.m2.1d">R_{V}</annotation><annotation encoding="application/x-llamapun" id="S4.T3.5.m2.1e">italic_R start_POSTSUBSCRIPT italic_V end_POSTSUBSCRIPT</annotation></semantics></math> values, with periods and magnitudes taken from the OGLE catalog, after excluding the two longest period Cepheids which do not have <math alttext="V" class="ltx_Math" display="inline" id="S4.T3.6.m3.1"><semantics id="S4.T3.6.m3.1b"><mi id="S4.T3.6.m3.1.1" xref="S4.T3.6.m3.1.1.cmml">V</mi><annotation-xml encoding="MathML-Content" id="S4.T3.6.m3.1c"><ci id="S4.T3.6.m3.1.1.cmml" xref="S4.T3.6.m3.1.1">𝑉</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.T3.6.m3.1d">V</annotation><annotation encoding="application/x-llamapun" id="S4.T3.6.m3.1e">italic_V</annotation></semantics></math>-band photometry in OGLE.</figcaption><div class="ltx_flex_figure"> <div class="ltx_flex_cell ltx_flex_size_1"> <table class="ltx_tabular ltx_figure_panel ltx_guessed_headers ltx_align_middle" id="S4.T3.24"> <thead class="ltx_thead"> <tr class="ltx_tr" id="S4.T3.16.10"> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" id="S4.T3.7.1.1"><math alttext="\alpha" class="ltx_Math" display="inline" id="S4.T3.7.1.1.m1.1"><semantics id="S4.T3.7.1.1.m1.1a"><mi id="S4.T3.7.1.1.m1.1.1" xref="S4.T3.7.1.1.m1.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S4.T3.7.1.1.m1.1b"><ci id="S4.T3.7.1.1.m1.1.1.cmml" xref="S4.T3.7.1.1.m1.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.T3.7.1.1.m1.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S4.T3.7.1.1.m1.1d">italic_α</annotation></semantics></math></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" id="S4.T3.8.2.2"><math alttext="\beta" class="ltx_Math" display="inline" id="S4.T3.8.2.2.m1.1"><semantics id="S4.T3.8.2.2.m1.1a"><mi id="S4.T3.8.2.2.m1.1.1" xref="S4.T3.8.2.2.m1.1.1.cmml">β</mi><annotation-xml encoding="MathML-Content" id="S4.T3.8.2.2.m1.1b"><ci id="S4.T3.8.2.2.m1.1.1.cmml" xref="S4.T3.8.2.2.m1.1.1">𝛽</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.T3.8.2.2.m1.1c">\beta</annotation><annotation encoding="application/x-llamapun" id="S4.T3.8.2.2.m1.1d">italic_β</annotation></semantics></math></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_r ltx_border_tt" id="S4.T3.9.3.3"><math alttext="\sigma" class="ltx_Math" display="inline" id="S4.T3.9.3.3.m1.1"><semantics id="S4.T3.9.3.3.m1.1a"><mi id="S4.T3.9.3.3.m1.1.1" xref="S4.T3.9.3.3.m1.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S4.T3.9.3.3.m1.1b"><ci id="S4.T3.9.3.3.m1.1.1.cmml" xref="S4.T3.9.3.3.m1.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.T3.9.3.3.m1.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S4.T3.9.3.3.m1.1d">italic_σ</annotation></semantics></math></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" id="S4.T3.10.4.4"><math alttext="\alpha" class="ltx_Math" display="inline" id="S4.T3.10.4.4.m1.1"><semantics id="S4.T3.10.4.4.m1.1a"><mi id="S4.T3.10.4.4.m1.1.1" xref="S4.T3.10.4.4.m1.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S4.T3.10.4.4.m1.1b"><ci id="S4.T3.10.4.4.m1.1.1.cmml" xref="S4.T3.10.4.4.m1.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.T3.10.4.4.m1.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S4.T3.10.4.4.m1.1d">italic_α</annotation></semantics></math></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" id="S4.T3.11.5.5"><math alttext="\beta" class="ltx_Math" display="inline" id="S4.T3.11.5.5.m1.1"><semantics id="S4.T3.11.5.5.m1.1a"><mi id="S4.T3.11.5.5.m1.1.1" xref="S4.T3.11.5.5.m1.1.1.cmml">β</mi><annotation-xml encoding="MathML-Content" id="S4.T3.11.5.5.m1.1b"><ci id="S4.T3.11.5.5.m1.1.1.cmml" xref="S4.T3.11.5.5.m1.1.1">𝛽</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.T3.11.5.5.m1.1c">\beta</annotation><annotation encoding="application/x-llamapun" id="S4.T3.11.5.5.m1.1d">italic_β</annotation></semantics></math></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_r ltx_border_tt" id="S4.T3.12.6.6"><math alttext="\sigma" class="ltx_Math" display="inline" id="S4.T3.12.6.6.m1.1"><semantics id="S4.T3.12.6.6.m1.1a"><mi id="S4.T3.12.6.6.m1.1.1" xref="S4.T3.12.6.6.m1.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S4.T3.12.6.6.m1.1b"><ci id="S4.T3.12.6.6.m1.1.1.cmml" xref="S4.T3.12.6.6.m1.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.T3.12.6.6.m1.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S4.T3.12.6.6.m1.1d">italic_σ</annotation></semantics></math></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" id="S4.T3.13.7.7"><math alttext="\alpha" class="ltx_Math" display="inline" id="S4.T3.13.7.7.m1.1"><semantics id="S4.T3.13.7.7.m1.1a"><mi id="S4.T3.13.7.7.m1.1.1" xref="S4.T3.13.7.7.m1.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S4.T3.13.7.7.m1.1b"><ci id="S4.T3.13.7.7.m1.1.1.cmml" xref="S4.T3.13.7.7.m1.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.T3.13.7.7.m1.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S4.T3.13.7.7.m1.1d">italic_α</annotation></semantics></math></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" id="S4.T3.14.8.8"><math alttext="\beta" class="ltx_Math" display="inline" id="S4.T3.14.8.8.m1.1"><semantics id="S4.T3.14.8.8.m1.1a"><mi id="S4.T3.14.8.8.m1.1.1" xref="S4.T3.14.8.8.m1.1.1.cmml">β</mi><annotation-xml encoding="MathML-Content" id="S4.T3.14.8.8.m1.1b"><ci id="S4.T3.14.8.8.m1.1.1.cmml" xref="S4.T3.14.8.8.m1.1.1">𝛽</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.T3.14.8.8.m1.1c">\beta</annotation><annotation encoding="application/x-llamapun" id="S4.T3.14.8.8.m1.1d">italic_β</annotation></semantics></math></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_r ltx_border_tt" id="S4.T3.15.9.9"><math alttext="\sigma" class="ltx_Math" display="inline" id="S4.T3.15.9.9.m1.1"><semantics id="S4.T3.15.9.9.m1.1a"><mi id="S4.T3.15.9.9.m1.1.1" xref="S4.T3.15.9.9.m1.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S4.T3.15.9.9.m1.1b"><ci id="S4.T3.15.9.9.m1.1.1.cmml" xref="S4.T3.15.9.9.m1.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.T3.15.9.9.m1.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S4.T3.15.9.9.m1.1d">italic_σ</annotation></semantics></math></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" id="S4.T3.16.10.11">sample</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" id="S4.T3.16.10.10"><math alttext="N" class="ltx_Math" display="inline" id="S4.T3.16.10.10.m1.1"><semantics id="S4.T3.16.10.10.m1.1a"><mi id="S4.T3.16.10.10.m1.1.1" xref="S4.T3.16.10.10.m1.1.1.cmml">N</mi><annotation-xml encoding="MathML-Content" id="S4.T3.16.10.10.m1.1b"><ci id="S4.T3.16.10.10.m1.1.1.cmml" xref="S4.T3.16.10.10.m1.1.1">𝑁</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.T3.16.10.10.m1.1c">N</annotation><annotation encoding="application/x-llamapun" id="S4.T3.16.10.10.m1.1d">italic_N</annotation></semantics></math></th> </tr> <tr class="ltx_tr" id="S4.T3.19.13"> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_t" colspan="3" id="S4.T3.17.11.1"><math alttext="R_{V}=3.3" class="ltx_Math" display="inline" id="S4.T3.17.11.1.m1.1"><semantics id="S4.T3.17.11.1.m1.1a"><mrow id="S4.T3.17.11.1.m1.1.1" xref="S4.T3.17.11.1.m1.1.1.cmml"><msub id="S4.T3.17.11.1.m1.1.1.2" xref="S4.T3.17.11.1.m1.1.1.2.cmml"><mi id="S4.T3.17.11.1.m1.1.1.2.2" xref="S4.T3.17.11.1.m1.1.1.2.2.cmml">R</mi><mi id="S4.T3.17.11.1.m1.1.1.2.3" xref="S4.T3.17.11.1.m1.1.1.2.3.cmml">V</mi></msub><mo id="S4.T3.17.11.1.m1.1.1.1" xref="S4.T3.17.11.1.m1.1.1.1.cmml">=</mo><mn id="S4.T3.17.11.1.m1.1.1.3" xref="S4.T3.17.11.1.m1.1.1.3.cmml">3.3</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T3.17.11.1.m1.1b"><apply id="S4.T3.17.11.1.m1.1.1.cmml" xref="S4.T3.17.11.1.m1.1.1"><eq id="S4.T3.17.11.1.m1.1.1.1.cmml" xref="S4.T3.17.11.1.m1.1.1.1"></eq><apply id="S4.T3.17.11.1.m1.1.1.2.cmml" xref="S4.T3.17.11.1.m1.1.1.2"><csymbol cd="ambiguous" id="S4.T3.17.11.1.m1.1.1.2.1.cmml" xref="S4.T3.17.11.1.m1.1.1.2">subscript</csymbol><ci id="S4.T3.17.11.1.m1.1.1.2.2.cmml" xref="S4.T3.17.11.1.m1.1.1.2.2">𝑅</ci><ci id="S4.T3.17.11.1.m1.1.1.2.3.cmml" xref="S4.T3.17.11.1.m1.1.1.2.3">𝑉</ci></apply><cn id="S4.T3.17.11.1.m1.1.1.3.cmml" type="float" xref="S4.T3.17.11.1.m1.1.1.3">3.3</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T3.17.11.1.m1.1c">R_{V}=3.3</annotation><annotation encoding="application/x-llamapun" id="S4.T3.17.11.1.m1.1d">italic_R start_POSTSUBSCRIPT italic_V end_POSTSUBSCRIPT = 3.3</annotation></semantics></math></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_t" colspan="3" id="S4.T3.18.12.2"><math alttext="R_{V}=3.1" class="ltx_Math" display="inline" id="S4.T3.18.12.2.m1.1"><semantics id="S4.T3.18.12.2.m1.1a"><mrow id="S4.T3.18.12.2.m1.1.1" xref="S4.T3.18.12.2.m1.1.1.cmml"><msub id="S4.T3.18.12.2.m1.1.1.2" xref="S4.T3.18.12.2.m1.1.1.2.cmml"><mi id="S4.T3.18.12.2.m1.1.1.2.2" xref="S4.T3.18.12.2.m1.1.1.2.2.cmml">R</mi><mi id="S4.T3.18.12.2.m1.1.1.2.3" xref="S4.T3.18.12.2.m1.1.1.2.3.cmml">V</mi></msub><mo id="S4.T3.18.12.2.m1.1.1.1" xref="S4.T3.18.12.2.m1.1.1.1.cmml">=</mo><mn id="S4.T3.18.12.2.m1.1.1.3" xref="S4.T3.18.12.2.m1.1.1.3.cmml">3.1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T3.18.12.2.m1.1b"><apply id="S4.T3.18.12.2.m1.1.1.cmml" xref="S4.T3.18.12.2.m1.1.1"><eq id="S4.T3.18.12.2.m1.1.1.1.cmml" xref="S4.T3.18.12.2.m1.1.1.1"></eq><apply id="S4.T3.18.12.2.m1.1.1.2.cmml" xref="S4.T3.18.12.2.m1.1.1.2"><csymbol cd="ambiguous" id="S4.T3.18.12.2.m1.1.1.2.1.cmml" xref="S4.T3.18.12.2.m1.1.1.2">subscript</csymbol><ci id="S4.T3.18.12.2.m1.1.1.2.2.cmml" xref="S4.T3.18.12.2.m1.1.1.2.2">𝑅</ci><ci id="S4.T3.18.12.2.m1.1.1.2.3.cmml" xref="S4.T3.18.12.2.m1.1.1.2.3">𝑉</ci></apply><cn id="S4.T3.18.12.2.m1.1.1.3.cmml" type="float" xref="S4.T3.18.12.2.m1.1.1.3">3.1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T3.18.12.2.m1.1c">R_{V}=3.1</annotation><annotation encoding="application/x-llamapun" id="S4.T3.18.12.2.m1.1d">italic_R start_POSTSUBSCRIPT italic_V end_POSTSUBSCRIPT = 3.1</annotation></semantics></math></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_t" colspan="3" id="S4.T3.19.13.3"><math alttext="R_{V}=2.74" class="ltx_Math" display="inline" id="S4.T3.19.13.3.m1.1"><semantics id="S4.T3.19.13.3.m1.1a"><mrow id="S4.T3.19.13.3.m1.1.1" xref="S4.T3.19.13.3.m1.1.1.cmml"><msub id="S4.T3.19.13.3.m1.1.1.2" xref="S4.T3.19.13.3.m1.1.1.2.cmml"><mi id="S4.T3.19.13.3.m1.1.1.2.2" xref="S4.T3.19.13.3.m1.1.1.2.2.cmml">R</mi><mi id="S4.T3.19.13.3.m1.1.1.2.3" xref="S4.T3.19.13.3.m1.1.1.2.3.cmml">V</mi></msub><mo id="S4.T3.19.13.3.m1.1.1.1" xref="S4.T3.19.13.3.m1.1.1.1.cmml">=</mo><mn id="S4.T3.19.13.3.m1.1.1.3" xref="S4.T3.19.13.3.m1.1.1.3.cmml">2.74</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T3.19.13.3.m1.1b"><apply id="S4.T3.19.13.3.m1.1.1.cmml" xref="S4.T3.19.13.3.m1.1.1"><eq id="S4.T3.19.13.3.m1.1.1.1.cmml" xref="S4.T3.19.13.3.m1.1.1.1"></eq><apply id="S4.T3.19.13.3.m1.1.1.2.cmml" xref="S4.T3.19.13.3.m1.1.1.2"><csymbol cd="ambiguous" id="S4.T3.19.13.3.m1.1.1.2.1.cmml" xref="S4.T3.19.13.3.m1.1.1.2">subscript</csymbol><ci id="S4.T3.19.13.3.m1.1.1.2.2.cmml" xref="S4.T3.19.13.3.m1.1.1.2.2">𝑅</ci><ci id="S4.T3.19.13.3.m1.1.1.2.3.cmml" xref="S4.T3.19.13.3.m1.1.1.2.3">𝑉</ci></apply><cn id="S4.T3.19.13.3.m1.1.1.3.cmml" type="float" xref="S4.T3.19.13.3.m1.1.1.3">2.74</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T3.19.13.3.m1.1c">R_{V}=2.74</annotation><annotation encoding="application/x-llamapun" id="S4.T3.19.13.3.m1.1d">italic_R start_POSTSUBSCRIPT italic_V end_POSTSUBSCRIPT = 2.74</annotation></semantics></math></th> <th class="ltx_td ltx_th ltx_th_column ltx_border_t" id="S4.T3.19.13.4"></th> <th class="ltx_td ltx_th ltx_th_column ltx_border_t" id="S4.T3.19.13.5"></th> </tr> </thead> <tbody class="ltx_tbody"> <tr class="ltx_tr" id="S4.T3.20.14"> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T3.20.14.1"><math alttext="-3.459" class="ltx_Math" display="inline" id="S4.T3.20.14.1.m1.1"><semantics id="S4.T3.20.14.1.m1.1a"><mrow id="S4.T3.20.14.1.m1.1.1" xref="S4.T3.20.14.1.m1.1.1.cmml"><mo id="S4.T3.20.14.1.m1.1.1a" xref="S4.T3.20.14.1.m1.1.1.cmml">−</mo><mn id="S4.T3.20.14.1.m1.1.1.2" xref="S4.T3.20.14.1.m1.1.1.2.cmml">3.459</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T3.20.14.1.m1.1b"><apply id="S4.T3.20.14.1.m1.1.1.cmml" xref="S4.T3.20.14.1.m1.1.1"><minus id="S4.T3.20.14.1.m1.1.1.1.cmml" xref="S4.T3.20.14.1.m1.1.1"></minus><cn id="S4.T3.20.14.1.m1.1.1.2.cmml" type="float" xref="S4.T3.20.14.1.m1.1.1.2">3.459</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T3.20.14.1.m1.1c">-3.459</annotation><annotation encoding="application/x-llamapun" id="S4.T3.20.14.1.m1.1d">- 3.459</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T3.20.14.2">16.548</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T3.20.14.3">0.126</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T3.20.14.4">-3.441</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T3.20.14.5">16.575</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T3.20.14.6">0.125</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T3.20.14.7">-3.407</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T3.20.14.8">16.626</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T3.20.14.9">0.123</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T3.20.14.10">full</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T3.20.14.11">23</td> </tr> <tr class="ltx_tr" id="S4.T3.22.16"> <td class="ltx_td ltx_align_center" id="S4.T3.21.15.1"><math alttext="-3.526" class="ltx_Math" display="inline" id="S4.T3.21.15.1.m1.1"><semantics id="S4.T3.21.15.1.m1.1a"><mrow id="S4.T3.21.15.1.m1.1.1" xref="S4.T3.21.15.1.m1.1.1.cmml"><mo id="S4.T3.21.15.1.m1.1.1a" xref="S4.T3.21.15.1.m1.1.1.cmml">−</mo><mn id="S4.T3.21.15.1.m1.1.1.2" xref="S4.T3.21.15.1.m1.1.1.2.cmml">3.526</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T3.21.15.1.m1.1b"><apply id="S4.T3.21.15.1.m1.1.1.cmml" xref="S4.T3.21.15.1.m1.1.1"><minus id="S4.T3.21.15.1.m1.1.1.1.cmml" xref="S4.T3.21.15.1.m1.1.1"></minus><cn id="S4.T3.21.15.1.m1.1.1.2.cmml" type="float" xref="S4.T3.21.15.1.m1.1.1.2">3.526</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T3.21.15.1.m1.1c">-3.526</annotation><annotation encoding="application/x-llamapun" id="S4.T3.21.15.1.m1.1d">- 3.526</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S4.T3.22.16.3">16.617</td> <td class="ltx_td ltx_align_center ltx_border_r" id="S4.T3.22.16.4">0.118</td> <td class="ltx_td ltx_align_center" id="S4.T3.22.16.5">-3.509</td> <td class="ltx_td ltx_align_center" id="S4.T3.22.16.6">16.645</td> <td class="ltx_td ltx_align_center ltx_border_r" id="S4.T3.22.16.7">0.116</td> <td class="ltx_td ltx_align_center" id="S4.T3.22.16.8">-3.478</td> <td class="ltx_td ltx_align_center" id="S4.T3.22.16.9">16.698</td> <td class="ltx_td ltx_align_center ltx_border_r" id="S4.T3.22.16.10">0.114</td> <td class="ltx_td ltx_align_center" id="S4.T3.22.16.2"><math alttext="R<1.0^{\circ}" class="ltx_Math" display="inline" id="S4.T3.22.16.2.m1.1"><semantics id="S4.T3.22.16.2.m1.1a"><mrow id="S4.T3.22.16.2.m1.1.1" xref="S4.T3.22.16.2.m1.1.1.cmml"><mi id="S4.T3.22.16.2.m1.1.1.2" xref="S4.T3.22.16.2.m1.1.1.2.cmml">R</mi><mo id="S4.T3.22.16.2.m1.1.1.1" xref="S4.T3.22.16.2.m1.1.1.1.cmml"><</mo><msup id="S4.T3.22.16.2.m1.1.1.3" xref="S4.T3.22.16.2.m1.1.1.3.cmml"><mn id="S4.T3.22.16.2.m1.1.1.3.2" xref="S4.T3.22.16.2.m1.1.1.3.2.cmml">1.0</mn><mo id="S4.T3.22.16.2.m1.1.1.3.3" xref="S4.T3.22.16.2.m1.1.1.3.3.cmml">∘</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.T3.22.16.2.m1.1b"><apply id="S4.T3.22.16.2.m1.1.1.cmml" xref="S4.T3.22.16.2.m1.1.1"><lt id="S4.T3.22.16.2.m1.1.1.1.cmml" xref="S4.T3.22.16.2.m1.1.1.1"></lt><ci id="S4.T3.22.16.2.m1.1.1.2.cmml" xref="S4.T3.22.16.2.m1.1.1.2">𝑅</ci><apply id="S4.T3.22.16.2.m1.1.1.3.cmml" xref="S4.T3.22.16.2.m1.1.1.3"><csymbol cd="ambiguous" id="S4.T3.22.16.2.m1.1.1.3.1.cmml" xref="S4.T3.22.16.2.m1.1.1.3">superscript</csymbol><cn id="S4.T3.22.16.2.m1.1.1.3.2.cmml" type="float" xref="S4.T3.22.16.2.m1.1.1.3.2">1.0</cn><compose id="S4.T3.22.16.2.m1.1.1.3.3.cmml" xref="S4.T3.22.16.2.m1.1.1.3.3"></compose></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T3.22.16.2.m1.1c">R<1.0^{\circ}</annotation><annotation encoding="application/x-llamapun" id="S4.T3.22.16.2.m1.1d">italic_R < 1.0 start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S4.T3.22.16.11">21</td> </tr> <tr class="ltx_tr" id="S4.T3.24.18"> <td class="ltx_td ltx_align_center ltx_border_b" id="S4.T3.23.17.1"><math alttext="-3.529" class="ltx_Math" display="inline" id="S4.T3.23.17.1.m1.1"><semantics id="S4.T3.23.17.1.m1.1a"><mrow id="S4.T3.23.17.1.m1.1.1" xref="S4.T3.23.17.1.m1.1.1.cmml"><mo id="S4.T3.23.17.1.m1.1.1a" xref="S4.T3.23.17.1.m1.1.1.cmml">−</mo><mn id="S4.T3.23.17.1.m1.1.1.2" xref="S4.T3.23.17.1.m1.1.1.2.cmml">3.529</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T3.23.17.1.m1.1b"><apply id="S4.T3.23.17.1.m1.1.1.cmml" xref="S4.T3.23.17.1.m1.1.1"><minus id="S4.T3.23.17.1.m1.1.1.1.cmml" xref="S4.T3.23.17.1.m1.1.1"></minus><cn id="S4.T3.23.17.1.m1.1.1.2.cmml" type="float" xref="S4.T3.23.17.1.m1.1.1.2">3.529</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T3.23.17.1.m1.1c">-3.529</annotation><annotation encoding="application/x-llamapun" id="S4.T3.23.17.1.m1.1d">- 3.529</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_b" id="S4.T3.24.18.3">16.697</td> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_r" id="S4.T3.24.18.4">0.112</td> <td class="ltx_td ltx_align_center ltx_border_b" id="S4.T3.24.18.5">-3.510</td> <td class="ltx_td ltx_align_center ltx_border_b" id="S4.T3.24.18.6">16.624</td> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_r" id="S4.T3.24.18.7">0.109</td> <td class="ltx_td ltx_align_center ltx_border_b" id="S4.T3.24.18.8">-3.475</td> <td class="ltx_td ltx_align_center ltx_border_b" id="S4.T3.24.18.9">16.674</td> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_r" id="S4.T3.24.18.10">0.105</td> <td class="ltx_td ltx_align_center ltx_border_b" id="S4.T3.24.18.2"><math alttext="R<0.8^{\circ}" class="ltx_Math" display="inline" id="S4.T3.24.18.2.m1.1"><semantics id="S4.T3.24.18.2.m1.1a"><mrow id="S4.T3.24.18.2.m1.1.1" xref="S4.T3.24.18.2.m1.1.1.cmml"><mi id="S4.T3.24.18.2.m1.1.1.2" xref="S4.T3.24.18.2.m1.1.1.2.cmml">R</mi><mo id="S4.T3.24.18.2.m1.1.1.1" xref="S4.T3.24.18.2.m1.1.1.1.cmml"><</mo><msup id="S4.T3.24.18.2.m1.1.1.3" xref="S4.T3.24.18.2.m1.1.1.3.cmml"><mn id="S4.T3.24.18.2.m1.1.1.3.2" xref="S4.T3.24.18.2.m1.1.1.3.2.cmml">0.8</mn><mo id="S4.T3.24.18.2.m1.1.1.3.3" xref="S4.T3.24.18.2.m1.1.1.3.3.cmml">∘</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.T3.24.18.2.m1.1b"><apply id="S4.T3.24.18.2.m1.1.1.cmml" xref="S4.T3.24.18.2.m1.1.1"><lt id="S4.T3.24.18.2.m1.1.1.1.cmml" xref="S4.T3.24.18.2.m1.1.1.1"></lt><ci id="S4.T3.24.18.2.m1.1.1.2.cmml" xref="S4.T3.24.18.2.m1.1.1.2">𝑅</ci><apply id="S4.T3.24.18.2.m1.1.1.3.cmml" xref="S4.T3.24.18.2.m1.1.1.3"><csymbol cd="ambiguous" id="S4.T3.24.18.2.m1.1.1.3.1.cmml" xref="S4.T3.24.18.2.m1.1.1.3">superscript</csymbol><cn id="S4.T3.24.18.2.m1.1.1.3.2.cmml" type="float" xref="S4.T3.24.18.2.m1.1.1.3.2">0.8</cn><compose id="S4.T3.24.18.2.m1.1.1.3.3.cmml" xref="S4.T3.24.18.2.m1.1.1.3.3"></compose></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T3.24.18.2.m1.1c">R<0.8^{\circ}</annotation><annotation encoding="application/x-llamapun" id="S4.T3.24.18.2.m1.1d">italic_R < 0.8 start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_b" id="S4.T3.24.18.11">17</td> </tr> </tbody> </table> </div> <div class="ltx_flex_break"></div> <div class="ltx_flex_cell ltx_flex_size_1"> <p class="ltx_p ltx_figure_panel ltx_align_left" id="S4.T3.27.3"><span class="ltx_text ltx_font_bold" id="S4.T3.27.3.1">Notes:</span> In the modern literature, the P-L slope in the SMC and in the <math alttext="m_{VI}^{W}" class="ltx_Math" display="inline" id="S4.T3.25.1.m1.1"><semantics id="S4.T3.25.1.m1.1a"><msubsup id="S4.T3.25.1.m1.1.1" xref="S4.T3.25.1.m1.1.1.cmml"><mi id="S4.T3.25.1.m1.1.1.2.2" xref="S4.T3.25.1.m1.1.1.2.2.cmml">m</mi><mrow id="S4.T3.25.1.m1.1.1.2.3" xref="S4.T3.25.1.m1.1.1.2.3.cmml"><mi id="S4.T3.25.1.m1.1.1.2.3.2" xref="S4.T3.25.1.m1.1.1.2.3.2.cmml">V</mi><mo id="S4.T3.25.1.m1.1.1.2.3.1" xref="S4.T3.25.1.m1.1.1.2.3.1.cmml">⁢</mo><mi id="S4.T3.25.1.m1.1.1.2.3.3" xref="S4.T3.25.1.m1.1.1.2.3.3.cmml">I</mi></mrow><mi id="S4.T3.25.1.m1.1.1.3" xref="S4.T3.25.1.m1.1.1.3.cmml">W</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.T3.25.1.m1.1b"><apply id="S4.T3.25.1.m1.1.1.cmml" xref="S4.T3.25.1.m1.1.1"><csymbol cd="ambiguous" id="S4.T3.25.1.m1.1.1.1.cmml" xref="S4.T3.25.1.m1.1.1">superscript</csymbol><apply id="S4.T3.25.1.m1.1.1.2.cmml" xref="S4.T3.25.1.m1.1.1"><csymbol cd="ambiguous" id="S4.T3.25.1.m1.1.1.2.1.cmml" xref="S4.T3.25.1.m1.1.1">subscript</csymbol><ci id="S4.T3.25.1.m1.1.1.2.2.cmml" xref="S4.T3.25.1.m1.1.1.2.2">𝑚</ci><apply id="S4.T3.25.1.m1.1.1.2.3.cmml" xref="S4.T3.25.1.m1.1.1.2.3"><times id="S4.T3.25.1.m1.1.1.2.3.1.cmml" xref="S4.T3.25.1.m1.1.1.2.3.1"></times><ci id="S4.T3.25.1.m1.1.1.2.3.2.cmml" xref="S4.T3.25.1.m1.1.1.2.3.2">𝑉</ci><ci id="S4.T3.25.1.m1.1.1.2.3.3.cmml" xref="S4.T3.25.1.m1.1.1.2.3.3">𝐼</ci></apply></apply><ci id="S4.T3.25.1.m1.1.1.3.cmml" xref="S4.T3.25.1.m1.1.1.3">𝑊</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T3.25.1.m1.1c">m_{VI}^{W}</annotation><annotation encoding="application/x-llamapun" id="S4.T3.25.1.m1.1d">italic_m start_POSTSUBSCRIPT italic_V italic_I end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_W end_POSTSUPERSCRIPT</annotation></semantics></math> index is found between <math alttext="-3.32" class="ltx_Math" display="inline" id="S4.T3.26.2.m2.1"><semantics id="S4.T3.26.2.m2.1a"><mrow id="S4.T3.26.2.m2.1.1" xref="S4.T3.26.2.m2.1.1.cmml"><mo id="S4.T3.26.2.m2.1.1a" xref="S4.T3.26.2.m2.1.1.cmml">−</mo><mn id="S4.T3.26.2.m2.1.1.2" xref="S4.T3.26.2.m2.1.1.2.cmml">3.32</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T3.26.2.m2.1b"><apply id="S4.T3.26.2.m2.1.1.cmml" xref="S4.T3.26.2.m2.1.1"><minus id="S4.T3.26.2.m2.1.1.1.cmml" xref="S4.T3.26.2.m2.1.1"></minus><cn id="S4.T3.26.2.m2.1.1.2.cmml" type="float" xref="S4.T3.26.2.m2.1.1.2">3.32</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T3.26.2.m2.1c">-3.32</annotation><annotation encoding="application/x-llamapun" id="S4.T3.26.2.m2.1d">- 3.32</annotation></semantics></math> <cite class="ltx_cite ltx_citemacro_citep">(Breuval et al., <a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#bib.bib2" title="">2022</a>)</cite> and <math alttext="-3.46" class="ltx_Math" display="inline" id="S4.T3.27.3.m3.1"><semantics id="S4.T3.27.3.m3.1a"><mrow id="S4.T3.27.3.m3.1.1" xref="S4.T3.27.3.m3.1.1.cmml"><mo id="S4.T3.27.3.m3.1.1a" xref="S4.T3.27.3.m3.1.1.cmml">−</mo><mn id="S4.T3.27.3.m3.1.1.2" xref="S4.T3.27.3.m3.1.1.2.cmml">3.46</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T3.27.3.m3.1b"><apply id="S4.T3.27.3.m3.1.1.cmml" xref="S4.T3.27.3.m3.1.1"><minus id="S4.T3.27.3.m3.1.1.1.cmml" xref="S4.T3.27.3.m3.1.1"></minus><cn id="S4.T3.27.3.m3.1.1.2.cmml" type="float" xref="S4.T3.27.3.m3.1.1.2">3.46</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T3.27.3.m3.1c">-3.46</annotation><annotation encoding="application/x-llamapun" id="S4.T3.27.3.m3.1d">- 3.46</annotation></semantics></math> mag/dex <cite class="ltx_cite ltx_citemacro_citep">(Soszyński et al., <a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#bib.bib28" title="">2015</a>)</cite>. The steeper slope found by <cite class="ltx_cite ltx_citemacro_cite">Soszyński et al. (<a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#bib.bib28" title="">2015</a>)</cite> is mostly due to their use of short period Cepheids and a different coefficient for the Wesenheit index.</p> </div> </div> </figure> <div class="ltx_para" id="S4.SS1.p4"> <p class="ltx_p" id="S4.SS1.p4.3">The sensitivity of a photographic plate is described by a <span class="ltx_text ltx_font_italic" id="S4.SS1.p4.3.1">characteristic curve</span> (or Hurter-Driffield curve), which is commonly described as function of <math alttext="\log" class="ltx_Math" display="inline" id="S4.SS1.p4.1.m1.1"><semantics id="S4.SS1.p4.1.m1.1a"><mi id="S4.SS1.p4.1.m1.1.1" xref="S4.SS1.p4.1.m1.1.1.cmml">log</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p4.1.m1.1b"><log id="S4.SS1.p4.1.m1.1.1.cmml" xref="S4.SS1.p4.1.m1.1.1"></log></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p4.1.m1.1c">\log</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p4.1.m1.1d">roman_log</annotation></semantics></math>(Exposure) versus <math alttext="\log" class="ltx_Math" display="inline" id="S4.SS1.p4.2.m2.1"><semantics id="S4.SS1.p4.2.m2.1a"><mi id="S4.SS1.p4.2.m2.1.1" xref="S4.SS1.p4.2.m2.1.1.cmml">log</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p4.2.m2.1b"><log id="S4.SS1.p4.2.m2.1.1.cmml" xref="S4.SS1.p4.2.m2.1.1"></log></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p4.2.m2.1c">\log</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p4.2.m2.1d">roman_log</annotation></semantics></math>(Density). Modern texts often represent Exposure as <math alttext="H" class="ltx_Math" display="inline" id="S4.SS1.p4.3.m3.1"><semantics id="S4.SS1.p4.3.m3.1a"><mi id="S4.SS1.p4.3.m3.1.1" xref="S4.SS1.p4.3.m3.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p4.3.m3.1b"><ci id="S4.SS1.p4.3.m3.1.1.cmml" xref="S4.SS1.p4.3.m3.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p4.3.m3.1c">H</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p4.3.m3.1d">italic_H</annotation></semantics></math>, and it is defined as,</p> <table class="ltx_equation ltx_eqn_table" id="S4.E1"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="H=t\times I" class="ltx_Math" display="block" id="S4.E1.m1.1"><semantics id="S4.E1.m1.1a"><mrow id="S4.E1.m1.1.1" xref="S4.E1.m1.1.1.cmml"><mi id="S4.E1.m1.1.1.2" xref="S4.E1.m1.1.1.2.cmml">H</mi><mo id="S4.E1.m1.1.1.1" xref="S4.E1.m1.1.1.1.cmml">=</mo><mrow id="S4.E1.m1.1.1.3" xref="S4.E1.m1.1.1.3.cmml"><mi id="S4.E1.m1.1.1.3.2" xref="S4.E1.m1.1.1.3.2.cmml">t</mi><mo id="S4.E1.m1.1.1.3.1" lspace="0.222em" rspace="0.222em" xref="S4.E1.m1.1.1.3.1.cmml">×</mo><mi id="S4.E1.m1.1.1.3.3" xref="S4.E1.m1.1.1.3.3.cmml">I</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.E1.m1.1b"><apply id="S4.E1.m1.1.1.cmml" xref="S4.E1.m1.1.1"><eq id="S4.E1.m1.1.1.1.cmml" xref="S4.E1.m1.1.1.1"></eq><ci id="S4.E1.m1.1.1.2.cmml" xref="S4.E1.m1.1.1.2">𝐻</ci><apply id="S4.E1.m1.1.1.3.cmml" xref="S4.E1.m1.1.1.3"><times id="S4.E1.m1.1.1.3.1.cmml" xref="S4.E1.m1.1.1.3.1"></times><ci id="S4.E1.m1.1.1.3.2.cmml" xref="S4.E1.m1.1.1.3.2">𝑡</ci><ci id="S4.E1.m1.1.1.3.3.cmml" xref="S4.E1.m1.1.1.3.3">𝐼</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E1.m1.1c">H=t\times I</annotation><annotation encoding="application/x-llamapun" id="S4.E1.m1.1d">italic_H = italic_t × italic_I</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(1)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS1.p4.15">where <math alttext="t" class="ltx_Math" display="inline" id="S4.SS1.p4.4.m1.1"><semantics id="S4.SS1.p4.4.m1.1a"><mi id="S4.SS1.p4.4.m1.1.1" xref="S4.SS1.p4.4.m1.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p4.4.m1.1b"><ci id="S4.SS1.p4.4.m1.1.1.cmml" xref="S4.SS1.p4.4.m1.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p4.4.m1.1c">t</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p4.4.m1.1d">italic_t</annotation></semantics></math> is the exposure time and <math alttext="I" class="ltx_Math" display="inline" id="S4.SS1.p4.5.m2.1"><semantics id="S4.SS1.p4.5.m2.1a"><mi id="S4.SS1.p4.5.m2.1.1" xref="S4.SS1.p4.5.m2.1.1.cmml">I</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p4.5.m2.1b"><ci id="S4.SS1.p4.5.m2.1.1.cmml" xref="S4.SS1.p4.5.m2.1.1">𝐼</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p4.5.m2.1c">I</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p4.5.m2.1d">italic_I</annotation></semantics></math> is the intensity of the incident light. Density (<math alttext="D" class="ltx_Math" display="inline" id="S4.SS1.p4.6.m3.1"><semantics id="S4.SS1.p4.6.m3.1a"><mi id="S4.SS1.p4.6.m3.1.1" xref="S4.SS1.p4.6.m3.1.1.cmml">D</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p4.6.m3.1b"><ci id="S4.SS1.p4.6.m3.1.1.cmml" xref="S4.SS1.p4.6.m3.1.1">𝐷</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p4.6.m3.1c">D</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p4.6.m3.1d">italic_D</annotation></semantics></math>), on the other hand, describes the optical density of the photographic plate as a result of the development process. In <math alttext="\log-\log" class="ltx_Math" display="inline" id="S4.SS1.p4.7.m4.1"><semantics id="S4.SS1.p4.7.m4.1a"><mrow id="S4.SS1.p4.7.m4.1.1" xref="S4.SS1.p4.7.m4.1.1.cmml"><mi id="S4.SS1.p4.7.m4.1.1.2" xref="S4.SS1.p4.7.m4.1.1.2.cmml">log</mi><mo id="S4.SS1.p4.7.m4.1.1.1" lspace="0em" xref="S4.SS1.p4.7.m4.1.1.1.cmml">−</mo><mi id="S4.SS1.p4.7.m4.1.1.3" xref="S4.SS1.p4.7.m4.1.1.3.cmml">log</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p4.7.m4.1b"><apply id="S4.SS1.p4.7.m4.1.1.cmml" xref="S4.SS1.p4.7.m4.1.1"><minus id="S4.SS1.p4.7.m4.1.1.1.cmml" xref="S4.SS1.p4.7.m4.1.1.1"></minus><log id="S4.SS1.p4.7.m4.1.1.2.cmml" xref="S4.SS1.p4.7.m4.1.1.2"></log><log id="S4.SS1.p4.7.m4.1.1.3.cmml" xref="S4.SS1.p4.7.m4.1.1.3"></log></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p4.7.m4.1c">\log-\log</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p4.7.m4.1d">roman_log - roman_log</annotation></semantics></math> space, the characteristic curve generally has an “S” shape. In the “toe region” where there is low exposure, the photographic plate response is nonlinear and relatively flat. The mid-exposure region has a predictable, linear relationship between <math alttext="D" class="ltx_Math" display="inline" id="S4.SS1.p4.8.m5.1"><semantics id="S4.SS1.p4.8.m5.1a"><mi id="S4.SS1.p4.8.m5.1.1" xref="S4.SS1.p4.8.m5.1.1.cmml">D</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p4.8.m5.1b"><ci id="S4.SS1.p4.8.m5.1.1.cmml" xref="S4.SS1.p4.8.m5.1.1">𝐷</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p4.8.m5.1c">D</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p4.8.m5.1d">italic_D</annotation></semantics></math> and <math alttext="H" class="ltx_Math" display="inline" id="S4.SS1.p4.9.m6.1"><semantics id="S4.SS1.p4.9.m6.1a"><mi id="S4.SS1.p4.9.m6.1.1" xref="S4.SS1.p4.9.m6.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p4.9.m6.1b"><ci id="S4.SS1.p4.9.m6.1.1.cmml" xref="S4.SS1.p4.9.m6.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p4.9.m6.1c">H</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p4.9.m6.1d">italic_H</annotation></semantics></math>. The high exposure region is known as the shoulder – here saturation causes the characteristic curve to be flat again. The procedures and emulsions used will affect the shape of the characteristic curve. As explained in <cite class="ltx_cite ltx_citemacro_cite">Laycock et al. (<a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#bib.bib18" title="">2010</a>)</cite>, the photographic plate collection is a heterogeneous dataset and there is scant information about the procedures, emulsions, or developers. Thus, the characteristic curve is likely to be different for each plate, though we can assume that the “S” shape will be present in the sensitivity. It is beyond the scope of this work to derive a new photometric calibration for each plate used here, and also orthogonal to our main goal of specifically re-examining Leavitt’s own work. Empirically, these nonlinearities can have the effect of compressing the observed magnitude range, e.g., from a true <math alttext="\Delta m" class="ltx_Math" display="inline" id="S4.SS1.p4.10.m7.1"><semantics id="S4.SS1.p4.10.m7.1a"><mrow id="S4.SS1.p4.10.m7.1.1" xref="S4.SS1.p4.10.m7.1.1.cmml"><mi id="S4.SS1.p4.10.m7.1.1.2" mathvariant="normal" xref="S4.SS1.p4.10.m7.1.1.2.cmml">Δ</mi><mo id="S4.SS1.p4.10.m7.1.1.1" xref="S4.SS1.p4.10.m7.1.1.1.cmml">⁢</mo><mi id="S4.SS1.p4.10.m7.1.1.3" xref="S4.SS1.p4.10.m7.1.1.3.cmml">m</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p4.10.m7.1b"><apply id="S4.SS1.p4.10.m7.1.1.cmml" xref="S4.SS1.p4.10.m7.1.1"><times id="S4.SS1.p4.10.m7.1.1.1.cmml" xref="S4.SS1.p4.10.m7.1.1.1"></times><ci id="S4.SS1.p4.10.m7.1.1.2.cmml" xref="S4.SS1.p4.10.m7.1.1.2">Δ</ci><ci id="S4.SS1.p4.10.m7.1.1.3.cmml" xref="S4.SS1.p4.10.m7.1.1.3">𝑚</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p4.10.m7.1c">\Delta m</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p4.10.m7.1d">roman_Δ italic_m</annotation></semantics></math> from faintest to brightest of <math alttext="\sim" class="ltx_Math" display="inline" id="S4.SS1.p4.11.m8.1"><semantics id="S4.SS1.p4.11.m8.1a"><mo id="S4.SS1.p4.11.m8.1.1" xref="S4.SS1.p4.11.m8.1.1.cmml">∼</mo><annotation-xml encoding="MathML-Content" id="S4.SS1.p4.11.m8.1b"><csymbol cd="latexml" id="S4.SS1.p4.11.m8.1.1.cmml" xref="S4.SS1.p4.11.m8.1.1">similar-to</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p4.11.m8.1c">\sim</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p4.11.m8.1d">∼</annotation></semantics></math>5 to a measured <math alttext="\Delta m\sim" class="ltx_Math" display="inline" id="S4.SS1.p4.12.m9.1"><semantics id="S4.SS1.p4.12.m9.1a"><mrow id="S4.SS1.p4.12.m9.1.1" xref="S4.SS1.p4.12.m9.1.1.cmml"><mrow id="S4.SS1.p4.12.m9.1.1.2" xref="S4.SS1.p4.12.m9.1.1.2.cmml"><mi id="S4.SS1.p4.12.m9.1.1.2.2" mathvariant="normal" xref="S4.SS1.p4.12.m9.1.1.2.2.cmml">Δ</mi><mo id="S4.SS1.p4.12.m9.1.1.2.1" xref="S4.SS1.p4.12.m9.1.1.2.1.cmml">⁢</mo><mi id="S4.SS1.p4.12.m9.1.1.2.3" xref="S4.SS1.p4.12.m9.1.1.2.3.cmml">m</mi></mrow><mo id="S4.SS1.p4.12.m9.1.1.1" xref="S4.SS1.p4.12.m9.1.1.1.cmml">∼</mo><mi id="S4.SS1.p4.12.m9.1.1.3" xref="S4.SS1.p4.12.m9.1.1.3.cmml"></mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p4.12.m9.1b"><apply id="S4.SS1.p4.12.m9.1.1.cmml" xref="S4.SS1.p4.12.m9.1.1"><csymbol cd="latexml" id="S4.SS1.p4.12.m9.1.1.1.cmml" xref="S4.SS1.p4.12.m9.1.1.1">similar-to</csymbol><apply id="S4.SS1.p4.12.m9.1.1.2.cmml" xref="S4.SS1.p4.12.m9.1.1.2"><times id="S4.SS1.p4.12.m9.1.1.2.1.cmml" xref="S4.SS1.p4.12.m9.1.1.2.1"></times><ci id="S4.SS1.p4.12.m9.1.1.2.2.cmml" xref="S4.SS1.p4.12.m9.1.1.2.2">Δ</ci><ci id="S4.SS1.p4.12.m9.1.1.2.3.cmml" xref="S4.SS1.p4.12.m9.1.1.2.3">𝑚</ci></apply><csymbol cd="latexml" id="S4.SS1.p4.12.m9.1.1.3.cmml" xref="S4.SS1.p4.12.m9.1.1.3">absent</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p4.12.m9.1c">\Delta m\sim</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p4.12.m9.1d">roman_Δ italic_m ∼</annotation></semantics></math> 4 <cite class="ltx_cite ltx_citemacro_citep">(Laycock et al., <a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#bib.bib18" title="">2010</a>; Tang et al., <a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#bib.bib29" title="">2013</a>)</cite>. This compressed magnitude range over the same period range could be responsible for the relatively flatter slope in Leavitt’s results compared to modern measurements. However, if we limit the magnitude range of the Cepheids to the subset that fall within the linear regime of the photographic plate, we expect that we will see better agreement in the measurement of the slope, even in Leavitt’s provisional scale of magnitude. To test this hypothesis, we excluded faint and bright Cepheids outside of the range <math alttext="0.5<\log P" class="ltx_Math" display="inline" id="S4.SS1.p4.13.m10.1"><semantics id="S4.SS1.p4.13.m10.1a"><mrow id="S4.SS1.p4.13.m10.1.1" xref="S4.SS1.p4.13.m10.1.1.cmml"><mn id="S4.SS1.p4.13.m10.1.1.2" xref="S4.SS1.p4.13.m10.1.1.2.cmml">0.5</mn><mo id="S4.SS1.p4.13.m10.1.1.1" xref="S4.SS1.p4.13.m10.1.1.1.cmml"><</mo><mrow id="S4.SS1.p4.13.m10.1.1.3" xref="S4.SS1.p4.13.m10.1.1.3.cmml"><mi id="S4.SS1.p4.13.m10.1.1.3.1" xref="S4.SS1.p4.13.m10.1.1.3.1.cmml">log</mi><mo id="S4.SS1.p4.13.m10.1.1.3a" lspace="0.167em" xref="S4.SS1.p4.13.m10.1.1.3.cmml">⁡</mo><mi id="S4.SS1.p4.13.m10.1.1.3.2" xref="S4.SS1.p4.13.m10.1.1.3.2.cmml">P</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p4.13.m10.1b"><apply id="S4.SS1.p4.13.m10.1.1.cmml" xref="S4.SS1.p4.13.m10.1.1"><lt id="S4.SS1.p4.13.m10.1.1.1.cmml" xref="S4.SS1.p4.13.m10.1.1.1"></lt><cn id="S4.SS1.p4.13.m10.1.1.2.cmml" type="float" xref="S4.SS1.p4.13.m10.1.1.2">0.5</cn><apply id="S4.SS1.p4.13.m10.1.1.3.cmml" xref="S4.SS1.p4.13.m10.1.1.3"><log id="S4.SS1.p4.13.m10.1.1.3.1.cmml" xref="S4.SS1.p4.13.m10.1.1.3.1"></log><ci id="S4.SS1.p4.13.m10.1.1.3.2.cmml" xref="S4.SS1.p4.13.m10.1.1.3.2">𝑃</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p4.13.m10.1c">0.5<\log P</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p4.13.m10.1d">0.5 < roman_log italic_P</annotation></semantics></math> (days) <math alttext="<1.5" class="ltx_Math" display="inline" id="S4.SS1.p4.14.m11.1"><semantics id="S4.SS1.p4.14.m11.1a"><mrow id="S4.SS1.p4.14.m11.1.1" xref="S4.SS1.p4.14.m11.1.1.cmml"><mi id="S4.SS1.p4.14.m11.1.1.2" xref="S4.SS1.p4.14.m11.1.1.2.cmml"></mi><mo id="S4.SS1.p4.14.m11.1.1.1" xref="S4.SS1.p4.14.m11.1.1.1.cmml"><</mo><mn id="S4.SS1.p4.14.m11.1.1.3" xref="S4.SS1.p4.14.m11.1.1.3.cmml">1.5</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p4.14.m11.1b"><apply id="S4.SS1.p4.14.m11.1.1.cmml" xref="S4.SS1.p4.14.m11.1.1"><lt id="S4.SS1.p4.14.m11.1.1.1.cmml" xref="S4.SS1.p4.14.m11.1.1.1"></lt><csymbol cd="latexml" id="S4.SS1.p4.14.m11.1.1.2.cmml" xref="S4.SS1.p4.14.m11.1.1.2">absent</csymbol><cn id="S4.SS1.p4.14.m11.1.1.3.cmml" type="float" xref="S4.SS1.p4.14.m11.1.1.3">1.5</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p4.14.m11.1c"><1.5</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p4.14.m11.1d">< 1.5</annotation></semantics></math> which are most likely to be affected by this non-linearity: the resulting slope measured in Leavitt’s P-L relation becomes steeper, <math alttext="-2.19\pm 0.11" class="ltx_Math" display="inline" id="S4.SS1.p4.15.m12.1"><semantics id="S4.SS1.p4.15.m12.1a"><mrow id="S4.SS1.p4.15.m12.1.1" xref="S4.SS1.p4.15.m12.1.1.cmml"><mrow id="S4.SS1.p4.15.m12.1.1.2" xref="S4.SS1.p4.15.m12.1.1.2.cmml"><mo id="S4.SS1.p4.15.m12.1.1.2a" xref="S4.SS1.p4.15.m12.1.1.2.cmml">−</mo><mn id="S4.SS1.p4.15.m12.1.1.2.2" xref="S4.SS1.p4.15.m12.1.1.2.2.cmml">2.19</mn></mrow><mo id="S4.SS1.p4.15.m12.1.1.1" xref="S4.SS1.p4.15.m12.1.1.1.cmml">±</mo><mn id="S4.SS1.p4.15.m12.1.1.3" xref="S4.SS1.p4.15.m12.1.1.3.cmml">0.11</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p4.15.m12.1b"><apply id="S4.SS1.p4.15.m12.1.1.cmml" xref="S4.SS1.p4.15.m12.1.1"><csymbol cd="latexml" id="S4.SS1.p4.15.m12.1.1.1.cmml" xref="S4.SS1.p4.15.m12.1.1.1">plus-or-minus</csymbol><apply id="S4.SS1.p4.15.m12.1.1.2.cmml" xref="S4.SS1.p4.15.m12.1.1.2"><minus id="S4.SS1.p4.15.m12.1.1.2.1.cmml" xref="S4.SS1.p4.15.m12.1.1.2"></minus><cn id="S4.SS1.p4.15.m12.1.1.2.2.cmml" type="float" xref="S4.SS1.p4.15.m12.1.1.2.2">2.19</cn></apply><cn id="S4.SS1.p4.15.m12.1.1.3.cmml" type="float" xref="S4.SS1.p4.15.m12.1.1.3">0.11</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p4.15.m12.1c">-2.19\pm 0.11</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p4.15.m12.1d">- 2.19 ± 0.11</annotation></semantics></math> mag/dex, and in better agreement with the expected value.</p> </div> <div class="ltx_para" id="S4.SS1.p5"> <p class="ltx_p" id="S4.SS1.p5.4">This “toe” and “shoulder” effects can potentially also be seen in the amplitude ratios shown in Figure <a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#S3.F4" title="Figure 4 ‣ 3 Cepheid light curves ‣ The Legacy of Henrietta Leavitt: A Re-analysis of the First Cepheid Period-Luminosity Relation"><span class="ltx_text ltx_ref_tag">4</span></a>. For the faintest Cepheids with <math alttext="\log P\lesssim 0.9" class="ltx_Math" display="inline" id="S4.SS1.p5.1.m1.1"><semantics id="S4.SS1.p5.1.m1.1a"><mrow id="S4.SS1.p5.1.m1.1.1" xref="S4.SS1.p5.1.m1.1.1.cmml"><mrow id="S4.SS1.p5.1.m1.1.1.2" xref="S4.SS1.p5.1.m1.1.1.2.cmml"><mi id="S4.SS1.p5.1.m1.1.1.2.1" xref="S4.SS1.p5.1.m1.1.1.2.1.cmml">log</mi><mo id="S4.SS1.p5.1.m1.1.1.2a" lspace="0.167em" xref="S4.SS1.p5.1.m1.1.1.2.cmml">⁡</mo><mi id="S4.SS1.p5.1.m1.1.1.2.2" xref="S4.SS1.p5.1.m1.1.1.2.2.cmml">P</mi></mrow><mo id="S4.SS1.p5.1.m1.1.1.1" xref="S4.SS1.p5.1.m1.1.1.1.cmml">≲</mo><mn id="S4.SS1.p5.1.m1.1.1.3" xref="S4.SS1.p5.1.m1.1.1.3.cmml">0.9</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p5.1.m1.1b"><apply id="S4.SS1.p5.1.m1.1.1.cmml" xref="S4.SS1.p5.1.m1.1.1"><csymbol cd="latexml" id="S4.SS1.p5.1.m1.1.1.1.cmml" xref="S4.SS1.p5.1.m1.1.1.1">less-than-or-similar-to</csymbol><apply id="S4.SS1.p5.1.m1.1.1.2.cmml" xref="S4.SS1.p5.1.m1.1.1.2"><log id="S4.SS1.p5.1.m1.1.1.2.1.cmml" xref="S4.SS1.p5.1.m1.1.1.2.1"></log><ci id="S4.SS1.p5.1.m1.1.1.2.2.cmml" xref="S4.SS1.p5.1.m1.1.1.2.2">𝑃</ci></apply><cn id="S4.SS1.p5.1.m1.1.1.3.cmml" type="float" xref="S4.SS1.p5.1.m1.1.1.3">0.9</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p5.1.m1.1c">\log P\lesssim 0.9</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p5.1.m1.1d">roman_log italic_P ≲ 0.9</annotation></semantics></math>, the amplitude ratios all fall below the mean of 1.2. Greater than <math alttext="\log P\gtrsim 0.9" class="ltx_Math" display="inline" id="S4.SS1.p5.2.m2.1"><semantics id="S4.SS1.p5.2.m2.1a"><mrow id="S4.SS1.p5.2.m2.1.1" xref="S4.SS1.p5.2.m2.1.1.cmml"><mrow id="S4.SS1.p5.2.m2.1.1.2" xref="S4.SS1.p5.2.m2.1.1.2.cmml"><mi id="S4.SS1.p5.2.m2.1.1.2.1" xref="S4.SS1.p5.2.m2.1.1.2.1.cmml">log</mi><mo id="S4.SS1.p5.2.m2.1.1.2a" lspace="0.167em" xref="S4.SS1.p5.2.m2.1.1.2.cmml">⁡</mo><mi id="S4.SS1.p5.2.m2.1.1.2.2" xref="S4.SS1.p5.2.m2.1.1.2.2.cmml">P</mi></mrow><mo id="S4.SS1.p5.2.m2.1.1.1" xref="S4.SS1.p5.2.m2.1.1.1.cmml">≳</mo><mn id="S4.SS1.p5.2.m2.1.1.3" xref="S4.SS1.p5.2.m2.1.1.3.cmml">0.9</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p5.2.m2.1b"><apply id="S4.SS1.p5.2.m2.1.1.cmml" xref="S4.SS1.p5.2.m2.1.1"><csymbol cd="latexml" id="S4.SS1.p5.2.m2.1.1.1.cmml" xref="S4.SS1.p5.2.m2.1.1.1">greater-than-or-equivalent-to</csymbol><apply id="S4.SS1.p5.2.m2.1.1.2.cmml" xref="S4.SS1.p5.2.m2.1.1.2"><log id="S4.SS1.p5.2.m2.1.1.2.1.cmml" xref="S4.SS1.p5.2.m2.1.1.2.1"></log><ci id="S4.SS1.p5.2.m2.1.1.2.2.cmml" xref="S4.SS1.p5.2.m2.1.1.2.2">𝑃</ci></apply><cn id="S4.SS1.p5.2.m2.1.1.3.cmml" type="float" xref="S4.SS1.p5.2.m2.1.1.3">0.9</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p5.2.m2.1c">\log P\gtrsim 0.9</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p5.2.m2.1d">roman_log italic_P ≳ 0.9</annotation></semantics></math>, the amplitude ratios are all larger than the mean. It is possible that the shortest period Cepheids are either not detected at the minima in their pulsational phases, or fall in the regime of the toe effect at their minima, which would lead to a compressed amplitude. We illustrate this potential effect in the top panel of Figure <a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#S4.F5" title="Figure 5 ‣ 4 Period-Luminosity relation ‣ The Legacy of Henrietta Leavitt: A Re-analysis of the First Cepheid Period-Luminosity Relation"><span class="ltx_text ltx_ref_tag">5</span></a>: we fit the P-L relation (solid blue line) with the expected slope based on modern observations to all of the Cepheids with <math alttext="\log P>1" class="ltx_Math" display="inline" id="S4.SS1.p5.3.m3.1"><semantics id="S4.SS1.p5.3.m3.1a"><mrow id="S4.SS1.p5.3.m3.1.1" xref="S4.SS1.p5.3.m3.1.1.cmml"><mrow id="S4.SS1.p5.3.m3.1.1.2" xref="S4.SS1.p5.3.m3.1.1.2.cmml"><mi id="S4.SS1.p5.3.m3.1.1.2.1" xref="S4.SS1.p5.3.m3.1.1.2.1.cmml">log</mi><mo id="S4.SS1.p5.3.m3.1.1.2a" lspace="0.167em" xref="S4.SS1.p5.3.m3.1.1.2.cmml">⁡</mo><mi id="S4.SS1.p5.3.m3.1.1.2.2" xref="S4.SS1.p5.3.m3.1.1.2.2.cmml">P</mi></mrow><mo id="S4.SS1.p5.3.m3.1.1.1" xref="S4.SS1.p5.3.m3.1.1.1.cmml">></mo><mn id="S4.SS1.p5.3.m3.1.1.3" xref="S4.SS1.p5.3.m3.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p5.3.m3.1b"><apply id="S4.SS1.p5.3.m3.1.1.cmml" xref="S4.SS1.p5.3.m3.1.1"><gt id="S4.SS1.p5.3.m3.1.1.1.cmml" xref="S4.SS1.p5.3.m3.1.1.1"></gt><apply id="S4.SS1.p5.3.m3.1.1.2.cmml" xref="S4.SS1.p5.3.m3.1.1.2"><log id="S4.SS1.p5.3.m3.1.1.2.1.cmml" xref="S4.SS1.p5.3.m3.1.1.2.1"></log><ci id="S4.SS1.p5.3.m3.1.1.2.2.cmml" xref="S4.SS1.p5.3.m3.1.1.2.2">𝑃</ci></apply><cn id="S4.SS1.p5.3.m3.1.1.3.cmml" type="integer" xref="S4.SS1.p5.3.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p5.3.m3.1c">\log P>1</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p5.3.m3.1d">roman_log italic_P > 1</annotation></semantics></math>. Even using only Leavitt’s provisional scale of magnitudes, points qualitatively appear to fit the modern slope quite well. When we extend the relation to <math alttext="\log P<1" class="ltx_Math" display="inline" id="S4.SS1.p5.4.m4.1"><semantics id="S4.SS1.p5.4.m4.1a"><mrow id="S4.SS1.p5.4.m4.1.1" xref="S4.SS1.p5.4.m4.1.1.cmml"><mrow id="S4.SS1.p5.4.m4.1.1.2" xref="S4.SS1.p5.4.m4.1.1.2.cmml"><mi id="S4.SS1.p5.4.m4.1.1.2.1" xref="S4.SS1.p5.4.m4.1.1.2.1.cmml">log</mi><mo id="S4.SS1.p5.4.m4.1.1.2a" lspace="0.167em" xref="S4.SS1.p5.4.m4.1.1.2.cmml">⁡</mo><mi id="S4.SS1.p5.4.m4.1.1.2.2" xref="S4.SS1.p5.4.m4.1.1.2.2.cmml">P</mi></mrow><mo id="S4.SS1.p5.4.m4.1.1.1" xref="S4.SS1.p5.4.m4.1.1.1.cmml"><</mo><mn id="S4.SS1.p5.4.m4.1.1.3" xref="S4.SS1.p5.4.m4.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p5.4.m4.1b"><apply id="S4.SS1.p5.4.m4.1.1.cmml" xref="S4.SS1.p5.4.m4.1.1"><lt id="S4.SS1.p5.4.m4.1.1.1.cmml" xref="S4.SS1.p5.4.m4.1.1.1"></lt><apply id="S4.SS1.p5.4.m4.1.1.2.cmml" xref="S4.SS1.p5.4.m4.1.1.2"><log id="S4.SS1.p5.4.m4.1.1.2.1.cmml" xref="S4.SS1.p5.4.m4.1.1.2.1"></log><ci id="S4.SS1.p5.4.m4.1.1.2.2.cmml" xref="S4.SS1.p5.4.m4.1.1.2.2">𝑃</ci></apply><cn id="S4.SS1.p5.4.m4.1.1.3.cmml" type="integer" xref="S4.SS1.p5.4.m4.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p5.4.m4.1c">\log P<1</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p5.4.m4.1d">roman_log italic_P < 1</annotation></semantics></math> (dashed blue line) we see that the shorter period end appears to be flattened compared to the modern version. Together, these observations suggest that at least some of the differences in the steepness of the slope compared to modern measurements can be attributed to the nonlinearity of the photographic plate response.</p> </div> <div class="ltx_para" id="S4.SS1.p6"> <p class="ltx_p" id="S4.SS1.p6.2">In addition to the non-linear response of photographic plates, the possibility of a higher level of crowding for fainter Cepheids could also explain the shallow slope observed by Leavitt. Indeed, short period Cepheids are barely visible on photographic plates and the presence of nearby stars around the Cepheids is very clear on Fig. <a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#S4.F5" title="Figure 5 ‣ 4 Period-Luminosity relation ‣ The Legacy of Henrietta Leavitt: A Re-analysis of the First Cepheid Period-Luminosity Relation"><span class="ltx_text ltx_ref_tag">5</span></a>. To test this hypothesis, we performed aperture photometry on OGLE images (A. Udalski, private communication) of Leavitt’s Cepheids using different aperture sizes. We find that contamination by nearby stars is about five times larger at <math alttext="\log P<0.25" class="ltx_Math" display="inline" id="S4.SS1.p6.1.m1.1"><semantics id="S4.SS1.p6.1.m1.1a"><mrow id="S4.SS1.p6.1.m1.1.1" xref="S4.SS1.p6.1.m1.1.1.cmml"><mrow id="S4.SS1.p6.1.m1.1.1.2" xref="S4.SS1.p6.1.m1.1.1.2.cmml"><mi id="S4.SS1.p6.1.m1.1.1.2.1" xref="S4.SS1.p6.1.m1.1.1.2.1.cmml">log</mi><mo id="S4.SS1.p6.1.m1.1.1.2a" lspace="0.167em" xref="S4.SS1.p6.1.m1.1.1.2.cmml">⁡</mo><mi id="S4.SS1.p6.1.m1.1.1.2.2" xref="S4.SS1.p6.1.m1.1.1.2.2.cmml">P</mi></mrow><mo id="S4.SS1.p6.1.m1.1.1.1" xref="S4.SS1.p6.1.m1.1.1.1.cmml"><</mo><mn id="S4.SS1.p6.1.m1.1.1.3" xref="S4.SS1.p6.1.m1.1.1.3.cmml">0.25</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p6.1.m1.1b"><apply id="S4.SS1.p6.1.m1.1.1.cmml" xref="S4.SS1.p6.1.m1.1.1"><lt id="S4.SS1.p6.1.m1.1.1.1.cmml" xref="S4.SS1.p6.1.m1.1.1.1"></lt><apply id="S4.SS1.p6.1.m1.1.1.2.cmml" xref="S4.SS1.p6.1.m1.1.1.2"><log id="S4.SS1.p6.1.m1.1.1.2.1.cmml" xref="S4.SS1.p6.1.m1.1.1.2.1"></log><ci id="S4.SS1.p6.1.m1.1.1.2.2.cmml" xref="S4.SS1.p6.1.m1.1.1.2.2">𝑃</ci></apply><cn id="S4.SS1.p6.1.m1.1.1.3.cmml" type="float" xref="S4.SS1.p6.1.m1.1.1.3">0.25</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p6.1.m1.1c">\log P<0.25</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p6.1.m1.1d">roman_log italic_P < 0.25</annotation></semantics></math> than at <math alttext="\log P>1.30" class="ltx_Math" display="inline" id="S4.SS1.p6.2.m2.1"><semantics id="S4.SS1.p6.2.m2.1a"><mrow id="S4.SS1.p6.2.m2.1.1" xref="S4.SS1.p6.2.m2.1.1.cmml"><mrow id="S4.SS1.p6.2.m2.1.1.2" xref="S4.SS1.p6.2.m2.1.1.2.cmml"><mi id="S4.SS1.p6.2.m2.1.1.2.1" xref="S4.SS1.p6.2.m2.1.1.2.1.cmml">log</mi><mo id="S4.SS1.p6.2.m2.1.1.2a" lspace="0.167em" xref="S4.SS1.p6.2.m2.1.1.2.cmml">⁡</mo><mi id="S4.SS1.p6.2.m2.1.1.2.2" xref="S4.SS1.p6.2.m2.1.1.2.2.cmml">P</mi></mrow><mo id="S4.SS1.p6.2.m2.1.1.1" xref="S4.SS1.p6.2.m2.1.1.1.cmml">></mo><mn id="S4.SS1.p6.2.m2.1.1.3" xref="S4.SS1.p6.2.m2.1.1.3.cmml">1.30</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p6.2.m2.1b"><apply id="S4.SS1.p6.2.m2.1.1.cmml" xref="S4.SS1.p6.2.m2.1.1"><gt id="S4.SS1.p6.2.m2.1.1.1.cmml" xref="S4.SS1.p6.2.m2.1.1.1"></gt><apply id="S4.SS1.p6.2.m2.1.1.2.cmml" xref="S4.SS1.p6.2.m2.1.1.2"><log id="S4.SS1.p6.2.m2.1.1.2.1.cmml" xref="S4.SS1.p6.2.m2.1.1.2.1"></log><ci id="S4.SS1.p6.2.m2.1.1.2.2.cmml" xref="S4.SS1.p6.2.m2.1.1.2.2">𝑃</ci></apply><cn id="S4.SS1.p6.2.m2.1.1.3.cmml" type="float" xref="S4.SS1.p6.2.m2.1.1.3">1.30</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p6.2.m2.1c">\log P>1.30</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p6.2.m2.1d">roman_log italic_P > 1.30</annotation></semantics></math>. These short-period Cepheids were likely measured brighter than they actually are, resulting in a shallower slope at the faint end of the P-L relation. <br class="ltx_break"/></p> </div> <figure class="ltx_figure" id="S4.F7"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="485" id="S4.F7.g1" src="x9.png" width="988"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 7: </span><span class="ltx_text ltx_font_bold" id="S4.F7.3.1">Left</span>: Photographic plate of the Small Magellanic Cloud analyzed by Leavitt for her study of variable stars, which led to her discovery <cite class="ltx_cite ltx_citemacro_citep">(Leavitt & Pickering, <a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#bib.bib21" title="">1912</a>)</cite>. <span class="ltx_text ltx_font_bold" id="S4.F7.4.2">Right</span>: Same region as on the left, but from DSS2-IR. The light green circles are the SMC Cepheids that were observed by HST/WFC3 in <cite class="ltx_cite ltx_citemacro_cite">Breuval et al. (<a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#bib.bib4" title="">2024</a>)</cite>. The magenta squares are the 25 Cepheids from Leavitt’s study. The orientation and scale of the two maps are identical. <br class="ltx_break"/></figcaption> </figure> </section> <section class="ltx_subsection" id="S4.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">4.2 </span>P-L relation with magnitudes from OGLE</h3> <div class="ltx_para" id="S4.SS2.p1"> <p class="ltx_p" id="S4.SS2.p1.11">In this section, we adopt the <math alttext="V" class="ltx_Math" display="inline" id="S4.SS2.p1.1.m1.1"><semantics id="S4.SS2.p1.1.m1.1a"><mi id="S4.SS2.p1.1.m1.1.1" xref="S4.SS2.p1.1.m1.1.1.cmml">V</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.1.m1.1b"><ci id="S4.SS2.p1.1.m1.1.1.cmml" xref="S4.SS2.p1.1.m1.1.1">𝑉</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.1.m1.1c">V</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.1.m1.1d">italic_V</annotation></semantics></math> and <math alttext="I" class="ltx_Math" display="inline" id="S4.SS2.p1.2.m2.1"><semantics id="S4.SS2.p1.2.m2.1a"><mi id="S4.SS2.p1.2.m2.1.1" xref="S4.SS2.p1.2.m2.1.1.cmml">I</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.2.m2.1b"><ci id="S4.SS2.p1.2.m2.1.1.cmml" xref="S4.SS2.p1.2.m2.1.1">𝐼</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.2.m2.1c">I</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.2.m2.1d">italic_I</annotation></semantics></math> mean magnitudes from the OGLE IV survey <cite class="ltx_cite ltx_citemacro_citep">(Soszyński et al., <a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#bib.bib28" title="">2015</a>)</cite> for the same sample of 25 SMC Cepheids used by Leavitt. For the two longest period Cepheids, the <math alttext="V" class="ltx_Math" display="inline" id="S4.SS2.p1.3.m3.1"><semantics id="S4.SS2.p1.3.m3.1a"><mi id="S4.SS2.p1.3.m3.1.1" xref="S4.SS2.p1.3.m3.1.1.cmml">V</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.3.m3.1b"><ci id="S4.SS2.p1.3.m3.1.1.cmml" xref="S4.SS2.p1.3.m3.1.1">𝑉</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.3.m3.1c">V</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.3.m3.1d">italic_V</annotation></semantics></math>-band mean magnitudes are taken from <cite class="ltx_cite ltx_citemacro_citet">Henden et al. (<a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#bib.bib13" title="">2015</a>)</cite> as they are not available in OGLE IV, they were not considered in the P-L fit. To mitigate the effect of dust<span class="ltx_note ltx_role_footnote" id="footnote2"><sup class="ltx_note_mark">2</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">2</sup><span class="ltx_tag ltx_tag_note">2</span> <math alttext="E(B-V)\sim 0.05" class="ltx_Math" display="inline" id="footnote2.m1.1"><semantics id="footnote2.m1.1b"><mrow id="footnote2.m1.1.1" xref="footnote2.m1.1.1.cmml"><mrow id="footnote2.m1.1.1.1" xref="footnote2.m1.1.1.1.cmml"><mi id="footnote2.m1.1.1.1.3" xref="footnote2.m1.1.1.1.3.cmml">E</mi><mo id="footnote2.m1.1.1.1.2" xref="footnote2.m1.1.1.1.2.cmml">⁢</mo><mrow id="footnote2.m1.1.1.1.1.1" xref="footnote2.m1.1.1.1.1.1.1.cmml"><mo id="footnote2.m1.1.1.1.1.1.2" stretchy="false" xref="footnote2.m1.1.1.1.1.1.1.cmml">(</mo><mrow id="footnote2.m1.1.1.1.1.1.1" xref="footnote2.m1.1.1.1.1.1.1.cmml"><mi id="footnote2.m1.1.1.1.1.1.1.2" xref="footnote2.m1.1.1.1.1.1.1.2.cmml">B</mi><mo id="footnote2.m1.1.1.1.1.1.1.1" xref="footnote2.m1.1.1.1.1.1.1.1.cmml">−</mo><mi id="footnote2.m1.1.1.1.1.1.1.3" xref="footnote2.m1.1.1.1.1.1.1.3.cmml">V</mi></mrow><mo id="footnote2.m1.1.1.1.1.1.3" stretchy="false" xref="footnote2.m1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="footnote2.m1.1.1.2" xref="footnote2.m1.1.1.2.cmml">∼</mo><mn id="footnote2.m1.1.1.3" xref="footnote2.m1.1.1.3.cmml">0.05</mn></mrow><annotation-xml encoding="MathML-Content" id="footnote2.m1.1c"><apply id="footnote2.m1.1.1.cmml" xref="footnote2.m1.1.1"><csymbol cd="latexml" id="footnote2.m1.1.1.2.cmml" xref="footnote2.m1.1.1.2">similar-to</csymbol><apply id="footnote2.m1.1.1.1.cmml" xref="footnote2.m1.1.1.1"><times id="footnote2.m1.1.1.1.2.cmml" xref="footnote2.m1.1.1.1.2"></times><ci id="footnote2.m1.1.1.1.3.cmml" xref="footnote2.m1.1.1.1.3">𝐸</ci><apply id="footnote2.m1.1.1.1.1.1.1.cmml" xref="footnote2.m1.1.1.1.1.1"><minus id="footnote2.m1.1.1.1.1.1.1.1.cmml" xref="footnote2.m1.1.1.1.1.1.1.1"></minus><ci id="footnote2.m1.1.1.1.1.1.1.2.cmml" xref="footnote2.m1.1.1.1.1.1.1.2">𝐵</ci><ci id="footnote2.m1.1.1.1.1.1.1.3.cmml" xref="footnote2.m1.1.1.1.1.1.1.3">𝑉</ci></apply></apply><cn id="footnote2.m1.1.1.3.cmml" type="float" xref="footnote2.m1.1.1.3">0.05</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote2.m1.1d">E(B-V)\sim 0.05</annotation><annotation encoding="application/x-llamapun" id="footnote2.m1.1e">italic_E ( italic_B - italic_V ) ∼ 0.05</annotation></semantics></math> mag in the SMC <cite class="ltx_cite ltx_citemacro_citep">(Skowron et al., <a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#bib.bib27" title="">2021</a>)</cite>.</span></span></span>, we derive a reddening-free Wesenheit index defined as <math alttext="m_{VI}^{W}=I-R\,(V-I)" class="ltx_Math" display="inline" id="S4.SS2.p1.4.m4.1"><semantics id="S4.SS2.p1.4.m4.1a"><mrow id="S4.SS2.p1.4.m4.1.1" xref="S4.SS2.p1.4.m4.1.1.cmml"><msubsup id="S4.SS2.p1.4.m4.1.1.3" xref="S4.SS2.p1.4.m4.1.1.3.cmml"><mi id="S4.SS2.p1.4.m4.1.1.3.2.2" xref="S4.SS2.p1.4.m4.1.1.3.2.2.cmml">m</mi><mrow id="S4.SS2.p1.4.m4.1.1.3.2.3" xref="S4.SS2.p1.4.m4.1.1.3.2.3.cmml"><mi id="S4.SS2.p1.4.m4.1.1.3.2.3.2" xref="S4.SS2.p1.4.m4.1.1.3.2.3.2.cmml">V</mi><mo id="S4.SS2.p1.4.m4.1.1.3.2.3.1" xref="S4.SS2.p1.4.m4.1.1.3.2.3.1.cmml">⁢</mo><mi id="S4.SS2.p1.4.m4.1.1.3.2.3.3" xref="S4.SS2.p1.4.m4.1.1.3.2.3.3.cmml">I</mi></mrow><mi id="S4.SS2.p1.4.m4.1.1.3.3" xref="S4.SS2.p1.4.m4.1.1.3.3.cmml">W</mi></msubsup><mo id="S4.SS2.p1.4.m4.1.1.2" xref="S4.SS2.p1.4.m4.1.1.2.cmml">=</mo><mrow id="S4.SS2.p1.4.m4.1.1.1" xref="S4.SS2.p1.4.m4.1.1.1.cmml"><mi id="S4.SS2.p1.4.m4.1.1.1.3" xref="S4.SS2.p1.4.m4.1.1.1.3.cmml">I</mi><mo id="S4.SS2.p1.4.m4.1.1.1.2" xref="S4.SS2.p1.4.m4.1.1.1.2.cmml">−</mo><mrow id="S4.SS2.p1.4.m4.1.1.1.1" xref="S4.SS2.p1.4.m4.1.1.1.1.cmml"><mi id="S4.SS2.p1.4.m4.1.1.1.1.3" xref="S4.SS2.p1.4.m4.1.1.1.1.3.cmml">R</mi><mo id="S4.SS2.p1.4.m4.1.1.1.1.2" lspace="0.170em" xref="S4.SS2.p1.4.m4.1.1.1.1.2.cmml">⁢</mo><mrow id="S4.SS2.p1.4.m4.1.1.1.1.1.1" xref="S4.SS2.p1.4.m4.1.1.1.1.1.1.1.cmml"><mo id="S4.SS2.p1.4.m4.1.1.1.1.1.1.2" stretchy="false" xref="S4.SS2.p1.4.m4.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S4.SS2.p1.4.m4.1.1.1.1.1.1.1" xref="S4.SS2.p1.4.m4.1.1.1.1.1.1.1.cmml"><mi id="S4.SS2.p1.4.m4.1.1.1.1.1.1.1.2" xref="S4.SS2.p1.4.m4.1.1.1.1.1.1.1.2.cmml">V</mi><mo id="S4.SS2.p1.4.m4.1.1.1.1.1.1.1.1" xref="S4.SS2.p1.4.m4.1.1.1.1.1.1.1.1.cmml">−</mo><mi id="S4.SS2.p1.4.m4.1.1.1.1.1.1.1.3" xref="S4.SS2.p1.4.m4.1.1.1.1.1.1.1.3.cmml">I</mi></mrow><mo id="S4.SS2.p1.4.m4.1.1.1.1.1.1.3" stretchy="false" xref="S4.SS2.p1.4.m4.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.4.m4.1b"><apply id="S4.SS2.p1.4.m4.1.1.cmml" xref="S4.SS2.p1.4.m4.1.1"><eq id="S4.SS2.p1.4.m4.1.1.2.cmml" xref="S4.SS2.p1.4.m4.1.1.2"></eq><apply id="S4.SS2.p1.4.m4.1.1.3.cmml" xref="S4.SS2.p1.4.m4.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.p1.4.m4.1.1.3.1.cmml" xref="S4.SS2.p1.4.m4.1.1.3">superscript</csymbol><apply id="S4.SS2.p1.4.m4.1.1.3.2.cmml" xref="S4.SS2.p1.4.m4.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.p1.4.m4.1.1.3.2.1.cmml" xref="S4.SS2.p1.4.m4.1.1.3">subscript</csymbol><ci id="S4.SS2.p1.4.m4.1.1.3.2.2.cmml" xref="S4.SS2.p1.4.m4.1.1.3.2.2">𝑚</ci><apply id="S4.SS2.p1.4.m4.1.1.3.2.3.cmml" xref="S4.SS2.p1.4.m4.1.1.3.2.3"><times id="S4.SS2.p1.4.m4.1.1.3.2.3.1.cmml" xref="S4.SS2.p1.4.m4.1.1.3.2.3.1"></times><ci id="S4.SS2.p1.4.m4.1.1.3.2.3.2.cmml" xref="S4.SS2.p1.4.m4.1.1.3.2.3.2">𝑉</ci><ci id="S4.SS2.p1.4.m4.1.1.3.2.3.3.cmml" xref="S4.SS2.p1.4.m4.1.1.3.2.3.3">𝐼</ci></apply></apply><ci id="S4.SS2.p1.4.m4.1.1.3.3.cmml" xref="S4.SS2.p1.4.m4.1.1.3.3">𝑊</ci></apply><apply id="S4.SS2.p1.4.m4.1.1.1.cmml" xref="S4.SS2.p1.4.m4.1.1.1"><minus id="S4.SS2.p1.4.m4.1.1.1.2.cmml" xref="S4.SS2.p1.4.m4.1.1.1.2"></minus><ci id="S4.SS2.p1.4.m4.1.1.1.3.cmml" xref="S4.SS2.p1.4.m4.1.1.1.3">𝐼</ci><apply id="S4.SS2.p1.4.m4.1.1.1.1.cmml" xref="S4.SS2.p1.4.m4.1.1.1.1"><times id="S4.SS2.p1.4.m4.1.1.1.1.2.cmml" xref="S4.SS2.p1.4.m4.1.1.1.1.2"></times><ci id="S4.SS2.p1.4.m4.1.1.1.1.3.cmml" xref="S4.SS2.p1.4.m4.1.1.1.1.3">𝑅</ci><apply id="S4.SS2.p1.4.m4.1.1.1.1.1.1.1.cmml" xref="S4.SS2.p1.4.m4.1.1.1.1.1.1"><minus id="S4.SS2.p1.4.m4.1.1.1.1.1.1.1.1.cmml" xref="S4.SS2.p1.4.m4.1.1.1.1.1.1.1.1"></minus><ci id="S4.SS2.p1.4.m4.1.1.1.1.1.1.1.2.cmml" xref="S4.SS2.p1.4.m4.1.1.1.1.1.1.1.2">𝑉</ci><ci id="S4.SS2.p1.4.m4.1.1.1.1.1.1.1.3.cmml" xref="S4.SS2.p1.4.m4.1.1.1.1.1.1.1.3">𝐼</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.4.m4.1c">m_{VI}^{W}=I-R\,(V-I)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.4.m4.1d">italic_m start_POSTSUBSCRIPT italic_V italic_I end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_W end_POSTSUPERSCRIPT = italic_I - italic_R ( italic_V - italic_I )</annotation></semantics></math>. Assuming the reddening law from <cite class="ltx_cite ltx_citemacro_citet">Fitzpatrick (<a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#bib.bib6" title="">1999</a>)</cite> and <math alttext="R_{V}=3.1" class="ltx_Math" display="inline" id="S4.SS2.p1.5.m5.1"><semantics id="S4.SS2.p1.5.m5.1a"><mrow id="S4.SS2.p1.5.m5.1.1" xref="S4.SS2.p1.5.m5.1.1.cmml"><msub id="S4.SS2.p1.5.m5.1.1.2" xref="S4.SS2.p1.5.m5.1.1.2.cmml"><mi id="S4.SS2.p1.5.m5.1.1.2.2" xref="S4.SS2.p1.5.m5.1.1.2.2.cmml">R</mi><mi id="S4.SS2.p1.5.m5.1.1.2.3" xref="S4.SS2.p1.5.m5.1.1.2.3.cmml">V</mi></msub><mo id="S4.SS2.p1.5.m5.1.1.1" xref="S4.SS2.p1.5.m5.1.1.1.cmml">=</mo><mn id="S4.SS2.p1.5.m5.1.1.3" xref="S4.SS2.p1.5.m5.1.1.3.cmml">3.1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.5.m5.1b"><apply id="S4.SS2.p1.5.m5.1.1.cmml" xref="S4.SS2.p1.5.m5.1.1"><eq id="S4.SS2.p1.5.m5.1.1.1.cmml" xref="S4.SS2.p1.5.m5.1.1.1"></eq><apply id="S4.SS2.p1.5.m5.1.1.2.cmml" xref="S4.SS2.p1.5.m5.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.p1.5.m5.1.1.2.1.cmml" xref="S4.SS2.p1.5.m5.1.1.2">subscript</csymbol><ci id="S4.SS2.p1.5.m5.1.1.2.2.cmml" xref="S4.SS2.p1.5.m5.1.1.2.2">𝑅</ci><ci id="S4.SS2.p1.5.m5.1.1.2.3.cmml" xref="S4.SS2.p1.5.m5.1.1.2.3">𝑉</ci></apply><cn id="S4.SS2.p1.5.m5.1.1.3.cmml" type="float" xref="S4.SS2.p1.5.m5.1.1.3">3.1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.5.m5.1c">R_{V}=3.1</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.5.m5.1d">italic_R start_POSTSUBSCRIPT italic_V end_POSTSUBSCRIPT = 3.1</annotation></semantics></math> (as usually adopted in the Milky Way) yields <math alttext="R=1.387" class="ltx_Math" display="inline" id="S4.SS2.p1.6.m6.1"><semantics id="S4.SS2.p1.6.m6.1a"><mrow id="S4.SS2.p1.6.m6.1.1" xref="S4.SS2.p1.6.m6.1.1.cmml"><mi id="S4.SS2.p1.6.m6.1.1.2" xref="S4.SS2.p1.6.m6.1.1.2.cmml">R</mi><mo id="S4.SS2.p1.6.m6.1.1.1" xref="S4.SS2.p1.6.m6.1.1.1.cmml">=</mo><mn id="S4.SS2.p1.6.m6.1.1.3" xref="S4.SS2.p1.6.m6.1.1.3.cmml">1.387</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.6.m6.1b"><apply id="S4.SS2.p1.6.m6.1.1.cmml" xref="S4.SS2.p1.6.m6.1.1"><eq id="S4.SS2.p1.6.m6.1.1.1.cmml" xref="S4.SS2.p1.6.m6.1.1.1"></eq><ci id="S4.SS2.p1.6.m6.1.1.2.cmml" xref="S4.SS2.p1.6.m6.1.1.2">𝑅</ci><cn id="S4.SS2.p1.6.m6.1.1.3.cmml" type="float" xref="S4.SS2.p1.6.m6.1.1.3">1.387</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.6.m6.1c">R=1.387</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.6.m6.1d">italic_R = 1.387</annotation></semantics></math>. However, <cite class="ltx_cite ltx_citemacro_citet">Gordon et al. (<a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#bib.bib10" title="">2003</a>)</cite> reported a value of <math alttext="R_{V}=2.74" class="ltx_Math" display="inline" id="S4.SS2.p1.7.m7.1"><semantics id="S4.SS2.p1.7.m7.1a"><mrow id="S4.SS2.p1.7.m7.1.1" xref="S4.SS2.p1.7.m7.1.1.cmml"><msub id="S4.SS2.p1.7.m7.1.1.2" xref="S4.SS2.p1.7.m7.1.1.2.cmml"><mi id="S4.SS2.p1.7.m7.1.1.2.2" xref="S4.SS2.p1.7.m7.1.1.2.2.cmml">R</mi><mi id="S4.SS2.p1.7.m7.1.1.2.3" xref="S4.SS2.p1.7.m7.1.1.2.3.cmml">V</mi></msub><mo id="S4.SS2.p1.7.m7.1.1.1" xref="S4.SS2.p1.7.m7.1.1.1.cmml">=</mo><mn id="S4.SS2.p1.7.m7.1.1.3" xref="S4.SS2.p1.7.m7.1.1.3.cmml">2.74</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.7.m7.1b"><apply id="S4.SS2.p1.7.m7.1.1.cmml" xref="S4.SS2.p1.7.m7.1.1"><eq id="S4.SS2.p1.7.m7.1.1.1.cmml" xref="S4.SS2.p1.7.m7.1.1.1"></eq><apply id="S4.SS2.p1.7.m7.1.1.2.cmml" xref="S4.SS2.p1.7.m7.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.p1.7.m7.1.1.2.1.cmml" xref="S4.SS2.p1.7.m7.1.1.2">subscript</csymbol><ci id="S4.SS2.p1.7.m7.1.1.2.2.cmml" xref="S4.SS2.p1.7.m7.1.1.2.2">𝑅</ci><ci id="S4.SS2.p1.7.m7.1.1.2.3.cmml" xref="S4.SS2.p1.7.m7.1.1.2.3">𝑉</ci></apply><cn id="S4.SS2.p1.7.m7.1.1.3.cmml" type="float" xref="S4.SS2.p1.7.m7.1.1.3">2.74</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.7.m7.1c">R_{V}=2.74</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.7.m7.1d">italic_R start_POSTSUBSCRIPT italic_V end_POSTSUBSCRIPT = 2.74</annotation></semantics></math> in the SMC, which gives <math alttext="R=1.278" class="ltx_Math" display="inline" id="S4.SS2.p1.8.m8.1"><semantics id="S4.SS2.p1.8.m8.1a"><mrow id="S4.SS2.p1.8.m8.1.1" xref="S4.SS2.p1.8.m8.1.1.cmml"><mi id="S4.SS2.p1.8.m8.1.1.2" xref="S4.SS2.p1.8.m8.1.1.2.cmml">R</mi><mo id="S4.SS2.p1.8.m8.1.1.1" xref="S4.SS2.p1.8.m8.1.1.1.cmml">=</mo><mn id="S4.SS2.p1.8.m8.1.1.3" xref="S4.SS2.p1.8.m8.1.1.3.cmml">1.278</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.8.m8.1b"><apply id="S4.SS2.p1.8.m8.1.1.cmml" xref="S4.SS2.p1.8.m8.1.1"><eq id="S4.SS2.p1.8.m8.1.1.1.cmml" xref="S4.SS2.p1.8.m8.1.1.1"></eq><ci id="S4.SS2.p1.8.m8.1.1.2.cmml" xref="S4.SS2.p1.8.m8.1.1.2">𝑅</ci><cn id="S4.SS2.p1.8.m8.1.1.3.cmml" type="float" xref="S4.SS2.p1.8.m8.1.1.3">1.278</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.8.m8.1c">R=1.278</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.8.m8.1d">italic_R = 1.278</annotation></semantics></math>. Finally, the value adopted in the SH0ES distance ladder <cite class="ltx_cite ltx_citemacro_citep">(Riess et al., <a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#bib.bib25" title="">2022</a>)</cite> is <math alttext="R_{V}=3.3" class="ltx_Math" display="inline" id="S4.SS2.p1.9.m9.1"><semantics id="S4.SS2.p1.9.m9.1a"><mrow id="S4.SS2.p1.9.m9.1.1" xref="S4.SS2.p1.9.m9.1.1.cmml"><msub id="S4.SS2.p1.9.m9.1.1.2" xref="S4.SS2.p1.9.m9.1.1.2.cmml"><mi id="S4.SS2.p1.9.m9.1.1.2.2" xref="S4.SS2.p1.9.m9.1.1.2.2.cmml">R</mi><mi id="S4.SS2.p1.9.m9.1.1.2.3" xref="S4.SS2.p1.9.m9.1.1.2.3.cmml">V</mi></msub><mo id="S4.SS2.p1.9.m9.1.1.1" xref="S4.SS2.p1.9.m9.1.1.1.cmml">=</mo><mn id="S4.SS2.p1.9.m9.1.1.3" xref="S4.SS2.p1.9.m9.1.1.3.cmml">3.3</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.9.m9.1b"><apply id="S4.SS2.p1.9.m9.1.1.cmml" xref="S4.SS2.p1.9.m9.1.1"><eq id="S4.SS2.p1.9.m9.1.1.1.cmml" xref="S4.SS2.p1.9.m9.1.1.1"></eq><apply id="S4.SS2.p1.9.m9.1.1.2.cmml" xref="S4.SS2.p1.9.m9.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.p1.9.m9.1.1.2.1.cmml" xref="S4.SS2.p1.9.m9.1.1.2">subscript</csymbol><ci id="S4.SS2.p1.9.m9.1.1.2.2.cmml" xref="S4.SS2.p1.9.m9.1.1.2.2">𝑅</ci><ci id="S4.SS2.p1.9.m9.1.1.2.3.cmml" xref="S4.SS2.p1.9.m9.1.1.2.3">𝑉</ci></apply><cn id="S4.SS2.p1.9.m9.1.1.3.cmml" type="float" xref="S4.SS2.p1.9.m9.1.1.3">3.3</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.9.m9.1c">R_{V}=3.3</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.9.m9.1d">italic_R start_POSTSUBSCRIPT italic_V end_POSTSUBSCRIPT = 3.3</annotation></semantics></math>, which gives <math alttext="R=1.445" class="ltx_Math" display="inline" id="S4.SS2.p1.10.m10.1"><semantics id="S4.SS2.p1.10.m10.1a"><mrow id="S4.SS2.p1.10.m10.1.1" xref="S4.SS2.p1.10.m10.1.1.cmml"><mi id="S4.SS2.p1.10.m10.1.1.2" xref="S4.SS2.p1.10.m10.1.1.2.cmml">R</mi><mo id="S4.SS2.p1.10.m10.1.1.1" xref="S4.SS2.p1.10.m10.1.1.1.cmml">=</mo><mn id="S4.SS2.p1.10.m10.1.1.3" xref="S4.SS2.p1.10.m10.1.1.3.cmml">1.445</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.10.m10.1b"><apply id="S4.SS2.p1.10.m10.1.1.cmml" xref="S4.SS2.p1.10.m10.1.1"><eq id="S4.SS2.p1.10.m10.1.1.1.cmml" xref="S4.SS2.p1.10.m10.1.1.1"></eq><ci id="S4.SS2.p1.10.m10.1.1.2.cmml" xref="S4.SS2.p1.10.m10.1.1.2">𝑅</ci><cn id="S4.SS2.p1.10.m10.1.1.3.cmml" type="float" xref="S4.SS2.p1.10.m10.1.1.3">1.445</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.10.m10.1c">R=1.445</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.10.m10.1d">italic_R = 1.445</annotation></semantics></math>. The following results are given for all three <math alttext="R_{V}" class="ltx_Math" display="inline" id="S4.SS2.p1.11.m11.1"><semantics id="S4.SS2.p1.11.m11.1a"><msub id="S4.SS2.p1.11.m11.1.1" xref="S4.SS2.p1.11.m11.1.1.cmml"><mi id="S4.SS2.p1.11.m11.1.1.2" xref="S4.SS2.p1.11.m11.1.1.2.cmml">R</mi><mi id="S4.SS2.p1.11.m11.1.1.3" xref="S4.SS2.p1.11.m11.1.1.3.cmml">V</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.11.m11.1b"><apply id="S4.SS2.p1.11.m11.1.1.cmml" xref="S4.SS2.p1.11.m11.1.1"><csymbol cd="ambiguous" id="S4.SS2.p1.11.m11.1.1.1.cmml" xref="S4.SS2.p1.11.m11.1.1">subscript</csymbol><ci id="S4.SS2.p1.11.m11.1.1.2.cmml" xref="S4.SS2.p1.11.m11.1.1.2">𝑅</ci><ci id="S4.SS2.p1.11.m11.1.1.3.cmml" xref="S4.SS2.p1.11.m11.1.1.3">𝑉</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.11.m11.1c">R_{V}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.11.m11.1d">italic_R start_POSTSUBSCRIPT italic_V end_POSTSUBSCRIPT</annotation></semantics></math> values in Table <a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#S4.T3" title="Table 3 ‣ 4.1 P-L relation with magnitudes from Leavitt & Pickering (1912) ‣ 4 Period-Luminosity relation ‣ The Legacy of Henrietta Leavitt: A Re-analysis of the First Cepheid Period-Luminosity Relation"><span class="ltx_text ltx_ref_tag">3</span></a>.</p> </div> <div class="ltx_para" id="S4.SS2.p2"> <p class="ltx_p" id="S4.SS2.p2.6">Similarly to Sect. <a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#S4.SS1" title="4.1 P-L relation with magnitudes from Leavitt & Pickering (1912) ‣ 4 Period-Luminosity relation ‣ The Legacy of Henrietta Leavitt: A Re-analysis of the First Cepheid Period-Luminosity Relation"><span class="ltx_text ltx_ref_tag">4.1</span></a>, using only Cepheids within <math alttext="0.8^{\circ}" class="ltx_Math" display="inline" id="S4.SS2.p2.1.m1.1"><semantics id="S4.SS2.p2.1.m1.1a"><msup id="S4.SS2.p2.1.m1.1.1" xref="S4.SS2.p2.1.m1.1.1.cmml"><mn id="S4.SS2.p2.1.m1.1.1.2" xref="S4.SS2.p2.1.m1.1.1.2.cmml">0.8</mn><mo id="S4.SS2.p2.1.m1.1.1.3" xref="S4.SS2.p2.1.m1.1.1.3.cmml">∘</mo></msup><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.1.m1.1b"><apply id="S4.SS2.p2.1.m1.1.1.cmml" xref="S4.SS2.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S4.SS2.p2.1.m1.1.1.1.cmml" xref="S4.SS2.p2.1.m1.1.1">superscript</csymbol><cn id="S4.SS2.p2.1.m1.1.1.2.cmml" type="float" xref="S4.SS2.p2.1.m1.1.1.2">0.8</cn><compose id="S4.SS2.p2.1.m1.1.1.3.cmml" xref="S4.SS2.p2.1.m1.1.1.3"></compose></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.1.m1.1c">0.8^{\circ}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.1.m1.1d">0.8 start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT</annotation></semantics></math> from the SMC center reduces the P-L scatter from 0.13 to 0.11 mag. <cite class="ltx_cite ltx_citemacro_citet">Breuval et al. (<a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#bib.bib3" title="">2021</a>, <a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#bib.bib4" title="">2024</a>)</cite> show that, additionally to excluding Cepheids located far from the SMC core, adopting a geometric model of the SMC to correct for residual depth effects further reduces the scatter of the Leavitt law: we apply the SMC model from <cite class="ltx_cite ltx_citemacro_cite">Graczyk et al. (<a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#bib.bib11" title="">2020</a>)</cite> so that each Cepheid is considered at its own individual distance depending on its position, contrary to previous studies <cite class="ltx_cite ltx_citemacro_citep">(e.g. Soszyński et al., <a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#bib.bib28" title="">2015</a>; Wielgórski et al., <a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#bib.bib31" title="">2017</a>; Gieren et al., <a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#bib.bib9" title="">2018</a>)</cite> which assumed the same distance for all SMC Cepheids. The Leavitt law obtained for the three values of <math alttext="R_{V}" class="ltx_Math" display="inline" id="S4.SS2.p2.2.m2.1"><semantics id="S4.SS2.p2.2.m2.1a"><msub id="S4.SS2.p2.2.m2.1.1" xref="S4.SS2.p2.2.m2.1.1.cmml"><mi id="S4.SS2.p2.2.m2.1.1.2" xref="S4.SS2.p2.2.m2.1.1.2.cmml">R</mi><mi id="S4.SS2.p2.2.m2.1.1.3" xref="S4.SS2.p2.2.m2.1.1.3.cmml">V</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.2.m2.1b"><apply id="S4.SS2.p2.2.m2.1.1.cmml" xref="S4.SS2.p2.2.m2.1.1"><csymbol cd="ambiguous" id="S4.SS2.p2.2.m2.1.1.1.cmml" xref="S4.SS2.p2.2.m2.1.1">subscript</csymbol><ci id="S4.SS2.p2.2.m2.1.1.2.cmml" xref="S4.SS2.p2.2.m2.1.1.2">𝑅</ci><ci id="S4.SS2.p2.2.m2.1.1.3.cmml" xref="S4.SS2.p2.2.m2.1.1.3">𝑉</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.2.m2.1c">R_{V}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.2.m2.1d">italic_R start_POSTSUBSCRIPT italic_V end_POSTSUBSCRIPT</annotation></semantics></math> and for different regions around the SMC center are listed in Table <a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#S4.T3" title="Table 3 ‣ 4.1 P-L relation with magnitudes from Leavitt & Pickering (1912) ‣ 4 Period-Luminosity relation ‣ The Legacy of Henrietta Leavitt: A Re-analysis of the First Cepheid Period-Luminosity Relation"><span class="ltx_text ltx_ref_tag">3</span></a>. Adopting <math alttext="R_{V}=2.74" class="ltx_Math" display="inline" id="S4.SS2.p2.3.m3.1"><semantics id="S4.SS2.p2.3.m3.1a"><mrow id="S4.SS2.p2.3.m3.1.1" xref="S4.SS2.p2.3.m3.1.1.cmml"><msub id="S4.SS2.p2.3.m3.1.1.2" xref="S4.SS2.p2.3.m3.1.1.2.cmml"><mi id="S4.SS2.p2.3.m3.1.1.2.2" xref="S4.SS2.p2.3.m3.1.1.2.2.cmml">R</mi><mi id="S4.SS2.p2.3.m3.1.1.2.3" xref="S4.SS2.p2.3.m3.1.1.2.3.cmml">V</mi></msub><mo id="S4.SS2.p2.3.m3.1.1.1" xref="S4.SS2.p2.3.m3.1.1.1.cmml">=</mo><mn id="S4.SS2.p2.3.m3.1.1.3" xref="S4.SS2.p2.3.m3.1.1.3.cmml">2.74</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.3.m3.1b"><apply id="S4.SS2.p2.3.m3.1.1.cmml" xref="S4.SS2.p2.3.m3.1.1"><eq id="S4.SS2.p2.3.m3.1.1.1.cmml" xref="S4.SS2.p2.3.m3.1.1.1"></eq><apply id="S4.SS2.p2.3.m3.1.1.2.cmml" xref="S4.SS2.p2.3.m3.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.p2.3.m3.1.1.2.1.cmml" xref="S4.SS2.p2.3.m3.1.1.2">subscript</csymbol><ci id="S4.SS2.p2.3.m3.1.1.2.2.cmml" xref="S4.SS2.p2.3.m3.1.1.2.2">𝑅</ci><ci id="S4.SS2.p2.3.m3.1.1.2.3.cmml" xref="S4.SS2.p2.3.m3.1.1.2.3">𝑉</ci></apply><cn id="S4.SS2.p2.3.m3.1.1.3.cmml" type="float" xref="S4.SS2.p2.3.m3.1.1.3">2.74</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.3.m3.1c">R_{V}=2.74</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.3.m3.1d">italic_R start_POSTSUBSCRIPT italic_V end_POSTSUBSCRIPT = 2.74</annotation></semantics></math> <cite class="ltx_cite ltx_citemacro_citep">(Gordon et al., <a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#bib.bib10" title="">2003</a>)</cite> slightly reduces the P-L scatter compared with using <math alttext="R_{V}=3.1" class="ltx_Math" display="inline" id="S4.SS2.p2.4.m4.1"><semantics id="S4.SS2.p2.4.m4.1a"><mrow id="S4.SS2.p2.4.m4.1.1" xref="S4.SS2.p2.4.m4.1.1.cmml"><msub id="S4.SS2.p2.4.m4.1.1.2" xref="S4.SS2.p2.4.m4.1.1.2.cmml"><mi id="S4.SS2.p2.4.m4.1.1.2.2" xref="S4.SS2.p2.4.m4.1.1.2.2.cmml">R</mi><mi id="S4.SS2.p2.4.m4.1.1.2.3" xref="S4.SS2.p2.4.m4.1.1.2.3.cmml">V</mi></msub><mo id="S4.SS2.p2.4.m4.1.1.1" xref="S4.SS2.p2.4.m4.1.1.1.cmml">=</mo><mn id="S4.SS2.p2.4.m4.1.1.3" xref="S4.SS2.p2.4.m4.1.1.3.cmml">3.1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.4.m4.1b"><apply id="S4.SS2.p2.4.m4.1.1.cmml" xref="S4.SS2.p2.4.m4.1.1"><eq id="S4.SS2.p2.4.m4.1.1.1.cmml" xref="S4.SS2.p2.4.m4.1.1.1"></eq><apply id="S4.SS2.p2.4.m4.1.1.2.cmml" xref="S4.SS2.p2.4.m4.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.p2.4.m4.1.1.2.1.cmml" xref="S4.SS2.p2.4.m4.1.1.2">subscript</csymbol><ci id="S4.SS2.p2.4.m4.1.1.2.2.cmml" xref="S4.SS2.p2.4.m4.1.1.2.2">𝑅</ci><ci id="S4.SS2.p2.4.m4.1.1.2.3.cmml" xref="S4.SS2.p2.4.m4.1.1.2.3">𝑉</ci></apply><cn id="S4.SS2.p2.4.m4.1.1.3.cmml" type="float" xref="S4.SS2.p2.4.m4.1.1.3">3.1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.4.m4.1c">R_{V}=3.1</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.4.m4.1d">italic_R start_POSTSUBSCRIPT italic_V end_POSTSUBSCRIPT = 3.1</annotation></semantics></math> or <math alttext="R_{V}=3.3" class="ltx_Math" display="inline" id="S4.SS2.p2.5.m5.1"><semantics id="S4.SS2.p2.5.m5.1a"><mrow id="S4.SS2.p2.5.m5.1.1" xref="S4.SS2.p2.5.m5.1.1.cmml"><msub id="S4.SS2.p2.5.m5.1.1.2" xref="S4.SS2.p2.5.m5.1.1.2.cmml"><mi id="S4.SS2.p2.5.m5.1.1.2.2" xref="S4.SS2.p2.5.m5.1.1.2.2.cmml">R</mi><mi id="S4.SS2.p2.5.m5.1.1.2.3" xref="S4.SS2.p2.5.m5.1.1.2.3.cmml">V</mi></msub><mo id="S4.SS2.p2.5.m5.1.1.1" xref="S4.SS2.p2.5.m5.1.1.1.cmml">=</mo><mn id="S4.SS2.p2.5.m5.1.1.3" xref="S4.SS2.p2.5.m5.1.1.3.cmml">3.3</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.5.m5.1b"><apply id="S4.SS2.p2.5.m5.1.1.cmml" xref="S4.SS2.p2.5.m5.1.1"><eq id="S4.SS2.p2.5.m5.1.1.1.cmml" xref="S4.SS2.p2.5.m5.1.1.1"></eq><apply id="S4.SS2.p2.5.m5.1.1.2.cmml" xref="S4.SS2.p2.5.m5.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.p2.5.m5.1.1.2.1.cmml" xref="S4.SS2.p2.5.m5.1.1.2">subscript</csymbol><ci id="S4.SS2.p2.5.m5.1.1.2.2.cmml" xref="S4.SS2.p2.5.m5.1.1.2.2">𝑅</ci><ci id="S4.SS2.p2.5.m5.1.1.2.3.cmml" xref="S4.SS2.p2.5.m5.1.1.2.3">𝑉</ci></apply><cn id="S4.SS2.p2.5.m5.1.1.3.cmml" type="float" xref="S4.SS2.p2.5.m5.1.1.3">3.3</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.5.m5.1c">R_{V}=3.3</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.5.m5.1d">italic_R start_POSTSUBSCRIPT italic_V end_POSTSUBSCRIPT = 3.3</annotation></semantics></math>, but only represents a minor improvement (see lower part of Table <a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#S4.T2" title="Table 2 ‣ 4.1 P-L relation with magnitudes from Leavitt & Pickering (1912) ‣ 4 Period-Luminosity relation ‣ The Legacy of Henrietta Leavitt: A Re-analysis of the First Cepheid Period-Luminosity Relation"><span class="ltx_text ltx_ref_tag">2</span></a> for the successive improvements in the P-L scatter). Our final P-L relation based on Leavitt’s sample and modern photometry is shown in the bottom panel of Fig. <a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#S4.F5" title="Figure 5 ‣ 4 Period-Luminosity relation ‣ The Legacy of Henrietta Leavitt: A Re-analysis of the First Cepheid Period-Luminosity Relation"><span class="ltx_text ltx_ref_tag">5</span></a>, with periods and magnitudes from OGLE, <math alttext="R_{V}=2.74" class="ltx_Math" display="inline" id="S4.SS2.p2.6.m6.1"><semantics id="S4.SS2.p2.6.m6.1a"><mrow id="S4.SS2.p2.6.m6.1.1" xref="S4.SS2.p2.6.m6.1.1.cmml"><msub id="S4.SS2.p2.6.m6.1.1.2" xref="S4.SS2.p2.6.m6.1.1.2.cmml"><mi id="S4.SS2.p2.6.m6.1.1.2.2" xref="S4.SS2.p2.6.m6.1.1.2.2.cmml">R</mi><mi id="S4.SS2.p2.6.m6.1.1.2.3" xref="S4.SS2.p2.6.m6.1.1.2.3.cmml">V</mi></msub><mo id="S4.SS2.p2.6.m6.1.1.1" xref="S4.SS2.p2.6.m6.1.1.1.cmml">=</mo><mn id="S4.SS2.p2.6.m6.1.1.3" xref="S4.SS2.p2.6.m6.1.1.3.cmml">2.74</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.6.m6.1b"><apply id="S4.SS2.p2.6.m6.1.1.cmml" xref="S4.SS2.p2.6.m6.1.1"><eq id="S4.SS2.p2.6.m6.1.1.1.cmml" xref="S4.SS2.p2.6.m6.1.1.1"></eq><apply id="S4.SS2.p2.6.m6.1.1.2.cmml" xref="S4.SS2.p2.6.m6.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.p2.6.m6.1.1.2.1.cmml" xref="S4.SS2.p2.6.m6.1.1.2">subscript</csymbol><ci id="S4.SS2.p2.6.m6.1.1.2.2.cmml" xref="S4.SS2.p2.6.m6.1.1.2.2">𝑅</ci><ci id="S4.SS2.p2.6.m6.1.1.2.3.cmml" xref="S4.SS2.p2.6.m6.1.1.2.3">𝑉</ci></apply><cn id="S4.SS2.p2.6.m6.1.1.3.cmml" type="float" xref="S4.SS2.p2.6.m6.1.1.3">2.74</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.6.m6.1c">R_{V}=2.74</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.6.m6.1d">italic_R start_POSTSUBSCRIPT italic_V end_POSTSUBSCRIPT = 2.74</annotation></semantics></math> and geometry corrections from <cite class="ltx_cite ltx_citemacro_cite">Graczyk et al. (<a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#bib.bib11" title="">2020</a>)</cite>’s planar model of the SMC based on eclisping binaries.</p> </div> <div class="ltx_para" id="S4.SS2.p3"> <p class="ltx_p" id="S4.SS2.p3.1">Below each P-L relation shown in Figure <a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#S4.F5" title="Figure 5 ‣ 4 Period-Luminosity relation ‣ The Legacy of Henrietta Leavitt: A Re-analysis of the First Cepheid Period-Luminosity Relation"><span class="ltx_text ltx_ref_tag">5</span></a>, we have also shown cutouts of the field surrounding five of the Cepheids from the SMC sample (HV 1446, HV 1400, HV 1351, HV 823, HV 821) at a range of periods. The images are taken from both the digitized plates from the <span class="ltx_text ltx_font_italic" id="S4.SS2.p3.1.1">DASCH</span> project <cite class="ltx_cite ltx_citemacro_citep">(Grindlay et al., <a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#bib.bib12" title="">2012</a>)</cite> and also from the OGLE survey to give a sense of the crowding in the field around each representative star. The photographic plate cutouts were extracted using the <span class="ltx_text ltx_font_typewriter" id="S4.SS2.p3.1.2">daschlab</span> Python package from <span class="ltx_text ltx_font_italic" id="S4.SS2.p3.1.3">DASCH</span> Data Release 7 <cite class="ltx_cite ltx_citemacro_citep">(Williams, <a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#bib.bib33" title="">2025</a>)</cite>. The software is available on Github<span class="ltx_note ltx_role_footnote" id="footnote3"><sup class="ltx_note_mark">3</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">3</sup><span class="ltx_tag ltx_tag_note">3</span><span class="ltx_text ltx_font_typewriter" id="footnote3.1">daschlab</span> codebase: <a class="ltx_ref ltx_url ltx_font_typewriter" href="https://github.com/pkgw/daschlab" title="">https://github.com/pkgw/daschlab</a></span></span></span> and is archived in Zenodo <cite class="ltx_cite ltx_citemacro_citep">(Williams, <a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#bib.bib32" title="">2024</a>)</cite>. <br class="ltx_break"/></p> </div> </section> </section> <section class="ltx_section" id="S5"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">5 </span>Discussion</h2> <div class="ltx_para" id="S5.p1"> <p class="ltx_p" id="S5.p1.1">This paper aims to highlight the quality of the work performed by Henrietta Leavitt in <cite class="ltx_cite ltx_citemacro_cite">Leavitt & Pickering (<a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#bib.bib21" title="">1912</a>)</cite> and the significant improvements of our knowledge, both in astronomy in general and on the Cepheid Leavitt law, since her pioneering discovery of the Leavitt law. The decrease by a factor of two in dispersion between the first P-L relation derived by Leavitt and the present reanalysis with OGLE data (Table <a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#S4.T2" title="Table 2 ‣ 4.1 P-L relation with magnitudes from Leavitt & Pickering (1912) ‣ 4 Period-Luminosity relation ‣ The Legacy of Henrietta Leavitt: A Re-analysis of the First Cepheid Period-Luminosity Relation"><span class="ltx_text ltx_ref_tag">2</span></a>) can be explained by the tremendous advancement of photometry over the last century, in particular the linearity of modern detectors compared to photographic plates and the availability of fully covered light curves in standard photometric systems for a large number of Cepheids, as compared with visual inspection of photographic plates performed by Leavitt. <span class="ltx_text" id="S5.p1.1.1" style="color:#000000;">As Leavitt herself noted, “<span class="ltx_text ltx_font_italic" id="S5.p1.1.1.1">The measurement and discussion of these objects present problems of unusual difficulty, on account of the large area covered by the two regions, the extremely crowded distribution of the stars contained in them, the faintness of the variables, and the shortness of their periods</span>” <cite class="ltx_cite ltx_citemacro_citep">(Leavitt & Pickering, <a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#bib.bib21" title="">1912</a>)</cite>.</span></p> </div> <div class="ltx_para" id="S5.p2"> <p class="ltx_p" id="S5.p2.1"><span class="ltx_text" id="S5.p2.1.1" style="color:#000000;">In addition to these technological advances, developments in our astrophysical knowledge have also contributed to the reduction in the P-L scatter. These include the relatively recent use of the reddening-free Wesenheit index and an improved understanding of the geometry of the SMC and its depth effects <cite class="ltx_cite ltx_citemacro_citep">(Breuval et al., <a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#bib.bib2" title="">2022</a>)</cite>.</span></p> </div> <div class="ltx_para" id="S5.p3"> <p class="ltx_p" id="S5.p3.2">With the <span class="ltx_text ltx_font_italic" id="S5.p3.2.1">Hubble</span> Space Telescope program GO-17097 (PI: A. Riess, Cycle 30), <cite class="ltx_cite ltx_citemacro_cite">Breuval et al. (<a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#bib.bib4" title="">2024</a>)</cite> provide consistent WFC3 photometry for 88 SMC Cepheids and improve the local measurement of the Hubble constant by including the SMC as the fourth anchor of the SH0ES distance ladder. This sample focuses on long period Cepheids (<math alttext="P>6" class="ltx_Math" display="inline" id="S5.p3.1.m1.1"><semantics id="S5.p3.1.m1.1a"><mrow id="S5.p3.1.m1.1.1" xref="S5.p3.1.m1.1.1.cmml"><mi id="S5.p3.1.m1.1.1.2" xref="S5.p3.1.m1.1.1.2.cmml">P</mi><mo id="S5.p3.1.m1.1.1.1" xref="S5.p3.1.m1.1.1.1.cmml">></mo><mn id="S5.p3.1.m1.1.1.3" xref="S5.p3.1.m1.1.1.3.cmml">6</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.p3.1.m1.1b"><apply id="S5.p3.1.m1.1.1.cmml" xref="S5.p3.1.m1.1.1"><gt id="S5.p3.1.m1.1.1.1.cmml" xref="S5.p3.1.m1.1.1.1"></gt><ci id="S5.p3.1.m1.1.1.2.cmml" xref="S5.p3.1.m1.1.1.2">𝑃</ci><cn id="S5.p3.1.m1.1.1.3.cmml" type="integer" xref="S5.p3.1.m1.1.1.3">6</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p3.1.m1.1c">P>6</annotation><annotation encoding="application/x-llamapun" id="S5.p3.1.m1.1d">italic_P > 6</annotation></semantics></math> days) in the inner region of the SMC (<math alttext="R<0.6^{\circ}" class="ltx_Math" display="inline" id="S5.p3.2.m2.1"><semantics id="S5.p3.2.m2.1a"><mrow id="S5.p3.2.m2.1.1" xref="S5.p3.2.m2.1.1.cmml"><mi id="S5.p3.2.m2.1.1.2" xref="S5.p3.2.m2.1.1.2.cmml">R</mi><mo id="S5.p3.2.m2.1.1.1" xref="S5.p3.2.m2.1.1.1.cmml"><</mo><msup id="S5.p3.2.m2.1.1.3" xref="S5.p3.2.m2.1.1.3.cmml"><mn id="S5.p3.2.m2.1.1.3.2" xref="S5.p3.2.m2.1.1.3.2.cmml">0.6</mn><mo id="S5.p3.2.m2.1.1.3.3" xref="S5.p3.2.m2.1.1.3.3.cmml">∘</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S5.p3.2.m2.1b"><apply id="S5.p3.2.m2.1.1.cmml" xref="S5.p3.2.m2.1.1"><lt id="S5.p3.2.m2.1.1.1.cmml" xref="S5.p3.2.m2.1.1.1"></lt><ci id="S5.p3.2.m2.1.1.2.cmml" xref="S5.p3.2.m2.1.1.2">𝑅</ci><apply id="S5.p3.2.m2.1.1.3.cmml" xref="S5.p3.2.m2.1.1.3"><csymbol cd="ambiguous" id="S5.p3.2.m2.1.1.3.1.cmml" xref="S5.p3.2.m2.1.1.3">superscript</csymbol><cn id="S5.p3.2.m2.1.1.3.2.cmml" type="float" xref="S5.p3.2.m2.1.1.3.2">0.6</cn><compose id="S5.p3.2.m2.1.1.3.3.cmml" xref="S5.p3.2.m2.1.1.3.3"></compose></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p3.2.m2.1c">R<0.6^{\circ}</annotation><annotation encoding="application/x-llamapun" id="S5.p3.2.m2.1d">italic_R < 0.6 start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT</annotation></semantics></math>), and unfortunately none of Leavitt’s variables are included in this sample (see Fig. <a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#S4.F7" title="Figure 7 ‣ 4.1 P-L relation with magnitudes from Leavitt & Pickering (1912) ‣ 4 Period-Luminosity relation ‣ The Legacy of Henrietta Leavitt: A Re-analysis of the First Cepheid Period-Luminosity Relation"><span class="ltx_text ltx_ref_tag">7</span></a>).</p> </div> <div class="ltx_para" id="S5.p4"> <p class="ltx_p" id="S5.p4.1">To place the quality of Leavitt’s work in a historical context, the first Cepheid Leavitt law can be compared to Hubble’s first velocity-distance diagram. Because Hubble’s measurements were affected by serious errors due to confusion between different types of Cepheids, modern distances to Hubble’s galaxies are seven times larger than his original values. As a result, his first estimate of the expansion rate of the universe, later called the Hubble constant, was about 500 km/s/Mpc, almost seven times larger than currently measured. Nevertheless, all his distances were underestimated by about the same factor, so his conclusion about the expanding universe was still valid. On the other hand, Leavitt’s first measurements of Cepheid light curves and periods were already extremely accurate and agree very well with modern data. The 25 Cepheids used by Leavitt are still listed in widely used catalogs such as OGLE and <span class="ltx_text ltx_font_italic" id="S5.p4.1.1">Gaia</span>, and are all classified as fundamental mode pulsators.</p> </div> <div class="ltx_para" id="S5.p5"> <p class="ltx_p" id="S5.p5.1">Leavitt’s discovery had a tremendous impact on our understanding of the universe. By allowing astronomers to determine distances to objects too distant for parallax, the Leavitt law opened the path to the discovery of the extragalactic universe in the 1920s, and subsequently, the expansion of the universe and Hubble’s law. The latest measurements of the Hubble Constant using Cepheids have achieved a 1.2% precision <cite class="ltx_cite ltx_citemacro_citep">(Breuval et al., <a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#bib.bib4" title="">2024</a>; Riess et al., <a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#bib.bib25" title="">2022</a>)</cite> and play an integral role in characterizing the current Hubble tension. A hundred and thirteen years after the publication of her paper, and thanks to her major breakthrough, Cepheid distance measurements and Leavitt’s law remain at the forefront of observational cosmology. <br class="ltx_break"/></p> </div> </section> <section class="ltx_section" id="Sx1"> <h2 class="ltx_title ltx_title_section">Acknowledgements</h2> <div class="ltx_para" id="Sx1.p1"> <p class="ltx_p" id="Sx1.p1.1">We would like to thank the members of Project PHaEDRA, Wolbach Library, and the Smithsonian Institution Transcription Center for helpful discussions and providing historical context and transcriptions of Leavitt’s notebooks. In particular, we thank Katie Frey, Riley Rhiannon, Giancarlo Romeo, Emily R. Cain, and the Smithsonian Institution Transcription Center volunteers that transcribed the digitized notebooks. We would also like to thank Peter K. G. Williams of the <span class="ltx_text ltx_font_italic" id="Sx1.p1.1.1">DASCH</span> Project for providing the digitized photographic plates of the SMC and for his helpful discussions regarding the nonlinearities in photographic plate magnitudes and in extracting the images from the photographic plates. This material is based on work supported by the National Science Foundation Astronomy & Astrophysics Postdoctoral Fellowship under Grant No. 2401770. We are grateful to Igor Soszyński and Andrzej Udalski for providing a sample of images from the OGLE survey for Leavitt’s Cepheid sample. This work has made use of data provided by Digital Access to a Sky Century @ Harvard (<span class="ltx_text ltx_font_italic" id="Sx1.p1.1.2">DASCH</span>), which has been partially supported by NSF grants AST-0407380, AST-0909073, and AST-1313370. Work on <span class="ltx_text ltx_font_italic" id="Sx1.p1.1.3">DASCH</span> Data Release 7 received support from the Smithsonian American Women’s History Initiative Pool. <br class="ltx_break"/></p> </div> </section> <section class="ltx_bibliography" id="bib"> <h2 class="ltx_title ltx_title_bibliography">References</h2> <ul class="ltx_biblist"> <li class="ltx_bibitem" id="bib.bib1"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Berdnikov & Turner (1997)</span> <span class="ltx_bibblock"> Berdnikov, L. N., & Turner, D. G. 1997, Information Bulletin on Variable Stars, 4437, 1 </span> </li> <li class="ltx_bibitem" id="bib.bib2"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Breuval et al. (2022)</span> <span class="ltx_bibblock"> Breuval, L., Riess, A. G., Kervella, P., Anderson, R. I., & Romaniello, M. 2022, ApJ, 939, 89 </span> </li> <li class="ltx_bibitem" id="bib.bib3"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Breuval et al. (2021)</span> <span class="ltx_bibblock"> Breuval, L., Kervella, P., Wielgórski, P., et al. 2021, ApJ, 913, 38 </span> </li> <li class="ltx_bibitem" id="bib.bib4"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Breuval et al. (2024)</span> <span class="ltx_bibblock"> Breuval, L., Riess, A. G., Casertano, S., et al. 2024, ApJ, 973, 30 </span> </li> <li class="ltx_bibitem" id="bib.bib5"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Eddington (1917)</span> <span class="ltx_bibblock"> Eddington, A. S. 1917, The Observatory, 40, 290 </span> </li> <li class="ltx_bibitem" id="bib.bib6"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Fitzpatrick (1999)</span> <span class="ltx_bibblock"> Fitzpatrick, E. L. 1999, PASP, 111, 63 </span> </li> <li class="ltx_bibitem" id="bib.bib7"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Freedman et al. (2001)</span> <span class="ltx_bibblock"> Freedman, W. L., Madore, B. F., Gibson, B. K., et al. 2001, ApJ, 553, 47 </span> </li> <li class="ltx_bibitem" id="bib.bib8"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Gaia Collaboration et al. (2023)</span> <span class="ltx_bibblock"> Gaia Collaboration, Vallenari, A., Brown, A. G. A., et al. 2023, A&A, 674, A1 </span> </li> <li class="ltx_bibitem" id="bib.bib9"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Gieren et al. (2018)</span> <span class="ltx_bibblock"> Gieren, W., Storm, J., Konorski, P., et al. 2018, A&A, 620, A99 </span> </li> <li class="ltx_bibitem" id="bib.bib10"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Gordon et al. (2003)</span> <span class="ltx_bibblock"> Gordon, K. D., Clayton, G. C., Misselt, K. A., Landolt, A. U., & Wolff, M. J. 2003, ApJ, 594, 279 </span> </li> <li class="ltx_bibitem" id="bib.bib11"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Graczyk et al. (2020)</span> <span class="ltx_bibblock"> Graczyk, D., Pietrzyński, G., Thompson, I. B., et al. 2020, ApJ, 904, 13 </span> </li> <li class="ltx_bibitem" id="bib.bib12"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Grindlay et al. (2012)</span> <span class="ltx_bibblock"> Grindlay, J., Tang, S., Los, E., & Servillat, M. 2012, in IAU Symposium, Vol. 285, New Horizons in Time Domain Astronomy, ed. E. Griffin, R. Hanisch, & R. Seaman, 29–34 </span> </li> <li class="ltx_bibitem" id="bib.bib13"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Henden et al. (2015)</span> <span class="ltx_bibblock"> Henden, A. A., Levine, S., Terrell, D., & Welch, D. L. 2015, in American Astronomical Society Meeting Abstracts, Vol. 225, American Astronomical Society Meeting Abstracts #225, 336.16 </span> </li> <li class="ltx_bibitem" id="bib.bib14"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Hertzsprung (1926)</span> <span class="ltx_bibblock"> Hertzsprung, E. 1926, Bull. Astron. Inst. Netherlands, 3, 115 </span> </li> <li class="ltx_bibitem" id="bib.bib15"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Hubble (1929)</span> <span class="ltx_bibblock"> Hubble, E. 1929, Proceedings of the National Academy of Science, 15, 168 </span> </li> <li class="ltx_bibitem" id="bib.bib16"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Kirshner (2003)</span> <span class="ltx_bibblock"> Kirshner, R. P. 2003, Proceedings of the National Academy of Science, 101, 8 </span> </li> <li class="ltx_bibitem" id="bib.bib17"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Klagyivik & Szabados (2009)</span> <span class="ltx_bibblock"> Klagyivik, P., & Szabados, L. 2009, A&A, 504, 959 </span> </li> <li class="ltx_bibitem" id="bib.bib18"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Laycock et al. (2010)</span> <span class="ltx_bibblock"> Laycock, S., Tang, S., Grindlay, J., et al. 2010, AJ, 140, 1062 </span> </li> <li class="ltx_bibitem" id="bib.bib19"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Leavitt (1905)</span> <span class="ltx_bibblock"> Leavitt, H. S. 1905, Project PHAEDRA: Preserving Harvard’s Early Data and Research in Astronomy (https://library.cfa.harvard.edu/project-phaedra). Harvard College Observatory observations, 2526 </span> </li> <li class="ltx_bibitem" id="bib.bib20"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Leavitt (1907)</span> <span class="ltx_bibblock"> —. 1907, Annals of Harvard College Observatory, 60, 87 </span> </li> <li class="ltx_bibitem" id="bib.bib21"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Leavitt & Pickering (1912)</span> <span class="ltx_bibblock"> Leavitt, H. S., & Pickering, E. C. 1912, Harvard College Observatory Circular, 173, 1 </span> </li> <li class="ltx_bibitem" id="bib.bib22"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Pickering (1909)</span> <span class="ltx_bibblock"> Pickering, E. C. 1909, Astronomische Nachrichten, 181, 7 </span> </li> <li class="ltx_bibitem" id="bib.bib23"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Pojmanski (1997)</span> <span class="ltx_bibblock"> Pojmanski, G. 1997, Acta Astron., 47, 467 </span> </li> <li class="ltx_bibitem" id="bib.bib24"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Riess et al. (2020)</span> <span class="ltx_bibblock"> Riess, A. G., Yuan, W., Casertano, S., Macri, L. M., & Scolnic, D. 2020, ApJ, 896, L43 </span> </li> <li class="ltx_bibitem" id="bib.bib25"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Riess et al. (2022)</span> <span class="ltx_bibblock"> Riess, A. G., Yuan, W., Macri, L. M., et al. 2022, ApJ, 934, L7 </span> </li> <li class="ltx_bibitem" id="bib.bib26"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Ripepi et al. (2023)</span> <span class="ltx_bibblock"> Ripepi, V., Clementini, G., Molinaro, R., et al. 2023, A&A, 674, A17 </span> </li> <li class="ltx_bibitem" id="bib.bib27"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Skowron et al. (2021)</span> <span class="ltx_bibblock"> Skowron, D. M., Skowron, J., Udalski, A., et al. 2021, ApJS, 252, 23 </span> </li> <li class="ltx_bibitem" id="bib.bib28"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Soszyński et al. (2015)</span> <span class="ltx_bibblock"> Soszyński, I., Udalski, A., Szymański, M. K., et al. 2015, Acta Astron., 65, 297 </span> </li> <li class="ltx_bibitem" id="bib.bib29"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Tang et al. (2013)</span> <span class="ltx_bibblock"> Tang, S., Grindlay, J., Los, E., & Servillat, M. 2013, PASP, 125, 857 </span> </li> <li class="ltx_bibitem" id="bib.bib30"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Trahin et al. (2021)</span> <span class="ltx_bibblock"> Trahin, B., Breuval, L., Kervella, P., et al. 2021, A&A, 656, A102 </span> </li> <li class="ltx_bibitem" id="bib.bib31"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Wielgórski et al. (2017)</span> <span class="ltx_bibblock"> Wielgórski, P., Pietrzyński, G., Gieren, W., et al. 2017, ApJ, 842, 116 </span> </li> <li class="ltx_bibitem" id="bib.bib32"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Williams (2024)</span> <span class="ltx_bibblock"> Williams, P. K. G. 2024, daschlab 1.0.0, 1.0.0, Zenodo, doi: <a class="ltx_ref ltx_href" href="http://doi.org/10.5281/zenodo.14574817" title=""><span class="ltx_ref ltx_nolink">10.5281/zenodo.14574817</span></a> </span> </li> <li class="ltx_bibitem" id="bib.bib33"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Williams (2025)</span> <span class="ltx_bibblock"> Williams, P. K. G. 2025, arXiv e-prints, arXiv:2501.12977 </span> </li> </ul> </section> <section class="ltx_appendix" id="A1"> <h2 class="ltx_title ltx_title_appendix"> <span class="ltx_tag ltx_tag_appendix">Appendix A </span>Note On the Periodicity of BZ Tuc</h2> <div class="ltx_para" id="A1.p1"> <p class="ltx_p" id="A1.p1.2">One variable of particular interest in Leavitt’s sample is BZ Tuc (HV 821), which showed one of the largest discrepancies in period when comparing the result obtained by modern authors and Leavitt. This is also the Cepheid with the longest period in Leavitt’s 1912 work. Depending on the catalog, BZ Tuc has a period of 127 days <cite class="ltx_cite ltx_citemacro_citep">(Leavitt & Pickering, <a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#bib.bib21" title="">1912</a>)</cite>, 128.13428 days <cite class="ltx_cite ltx_citemacro_citep">(Gaia Collaboration et al., <a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#bib.bib8" title="">2023</a>)</cite> or 128.197 days <cite class="ltx_cite ltx_citemacro_citep">(Soszyński et al., <a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#bib.bib28" title="">2015</a>)</cite>, respectively <math alttext="\log P=" class="ltx_Math" display="inline" id="A1.p1.1.m1.1"><semantics id="A1.p1.1.m1.1a"><mrow id="A1.p1.1.m1.1.1" xref="A1.p1.1.m1.1.1.cmml"><mrow id="A1.p1.1.m1.1.1.2" xref="A1.p1.1.m1.1.1.2.cmml"><mi id="A1.p1.1.m1.1.1.2.1" xref="A1.p1.1.m1.1.1.2.1.cmml">log</mi><mo id="A1.p1.1.m1.1.1.2a" lspace="0.167em" xref="A1.p1.1.m1.1.1.2.cmml">⁡</mo><mi id="A1.p1.1.m1.1.1.2.2" xref="A1.p1.1.m1.1.1.2.2.cmml">P</mi></mrow><mo id="A1.p1.1.m1.1.1.1" xref="A1.p1.1.m1.1.1.1.cmml">=</mo><mi id="A1.p1.1.m1.1.1.3" xref="A1.p1.1.m1.1.1.3.cmml"></mi></mrow><annotation-xml encoding="MathML-Content" id="A1.p1.1.m1.1b"><apply id="A1.p1.1.m1.1.1.cmml" xref="A1.p1.1.m1.1.1"><eq id="A1.p1.1.m1.1.1.1.cmml" xref="A1.p1.1.m1.1.1.1"></eq><apply id="A1.p1.1.m1.1.1.2.cmml" xref="A1.p1.1.m1.1.1.2"><log id="A1.p1.1.m1.1.1.2.1.cmml" xref="A1.p1.1.m1.1.1.2.1"></log><ci id="A1.p1.1.m1.1.1.2.2.cmml" xref="A1.p1.1.m1.1.1.2.2">𝑃</ci></apply><csymbol cd="latexml" id="A1.p1.1.m1.1.1.3.cmml" xref="A1.p1.1.m1.1.1.3">absent</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.p1.1.m1.1c">\log P=</annotation><annotation encoding="application/x-llamapun" id="A1.p1.1.m1.1d">roman_log italic_P =</annotation></semantics></math> 2.1038, 2.1077, and 2.1079. Compared to other Cepheids in Leavitt’s catalogue, it is remarkably well-sampled, with 89 observations over the course of 45 cycles. Therefore, it is surprising that it is one of the Cepheids with the largest difference in <math alttext="\log P" class="ltx_Math" display="inline" id="A1.p1.2.m2.1"><semantics id="A1.p1.2.m2.1a"><mrow id="A1.p1.2.m2.1.1" xref="A1.p1.2.m2.1.1.cmml"><mi id="A1.p1.2.m2.1.1.1" xref="A1.p1.2.m2.1.1.1.cmml">log</mi><mo id="A1.p1.2.m2.1.1a" lspace="0.167em" xref="A1.p1.2.m2.1.1.cmml">⁡</mo><mi id="A1.p1.2.m2.1.1.2" xref="A1.p1.2.m2.1.1.2.cmml">P</mi></mrow><annotation-xml encoding="MathML-Content" id="A1.p1.2.m2.1b"><apply id="A1.p1.2.m2.1.1.cmml" xref="A1.p1.2.m2.1.1"><log id="A1.p1.2.m2.1.1.1.cmml" xref="A1.p1.2.m2.1.1.1"></log><ci id="A1.p1.2.m2.1.1.2.cmml" xref="A1.p1.2.m2.1.1.2">𝑃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.p1.2.m2.1c">\log P</annotation><annotation encoding="application/x-llamapun" id="A1.p1.2.m2.1d">roman_log italic_P</annotation></semantics></math> with modern estimates.</p> </div> <div class="ltx_para" id="A1.p2"> <p class="ltx_p" id="A1.p2.1">In order to investigate this period discrepancy, we collected ASAS light curves <cite class="ltx_cite ltx_citemacro_citep">(Pojmanski, <a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#bib.bib23" title="">1997</a>)</cite> covering about 10 years of observations, from 1998 to 2009. We folded these by the various literature periods and found that Leavitt’s pulsation period minimizes the dispersion in phase slightly better than the OGLE period does during the span of the ASAS observations. We also determine a new period that minimizes the phase dispersion for this particular set of data (<math alttext="P=" class="ltx_Math" display="inline" id="A1.p2.1.m1.1"><semantics id="A1.p2.1.m1.1a"><mrow id="A1.p2.1.m1.1.1" xref="A1.p2.1.m1.1.1.cmml"><mi id="A1.p2.1.m1.1.1.2" xref="A1.p2.1.m1.1.1.2.cmml">P</mi><mo id="A1.p2.1.m1.1.1.1" xref="A1.p2.1.m1.1.1.1.cmml">=</mo><mi id="A1.p2.1.m1.1.1.3" xref="A1.p2.1.m1.1.1.3.cmml"></mi></mrow><annotation-xml encoding="MathML-Content" id="A1.p2.1.m1.1b"><apply id="A1.p2.1.m1.1.1.cmml" xref="A1.p2.1.m1.1.1"><eq id="A1.p2.1.m1.1.1.1.cmml" xref="A1.p2.1.m1.1.1.1"></eq><ci id="A1.p2.1.m1.1.1.2.cmml" xref="A1.p2.1.m1.1.1.2">𝑃</ci><csymbol cd="latexml" id="A1.p2.1.m1.1.1.3.cmml" xref="A1.p2.1.m1.1.1.3">absent</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.p2.1.m1.1c">P=</annotation><annotation encoding="application/x-llamapun" id="A1.p2.1.m1.1d">italic_P =</annotation></semantics></math> 127.445 days). However, even with this period, the folded light curve still shows significant scatter from cycle-to-cycle (see Fig. <a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#A1.F8" title="Figure 8 ‣ Appendix A Note On the Periodicity of BZ Tuc ‣ The Legacy of Henrietta Leavitt: A Re-analysis of the First Cepheid Period-Luminosity Relation"><span class="ltx_text ltx_ref_tag">8</span></a>, left), which can be indicative of a period change.</p> </div> <div class="ltx_para" id="A1.p3"> <p class="ltx_p" id="A1.p3.1">Using these observations, we constructed an O-C diagram for this Cepheid (Fig. <a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#A1.F8" title="Figure 8 ‣ Appendix A Note On the Periodicity of BZ Tuc ‣ The Legacy of Henrietta Leavitt: A Re-analysis of the First Cepheid Period-Luminosity Relation"><span class="ltx_text ltx_ref_tag">8</span></a>, right). Rather than a measurement error by Leavitt, the variations in the O-C diagram of HV 821 suggest that these different periods may be explained by an unstable period caused by a <span class="ltx_text ltx_font_italic" id="A1.p3.1.1">physical</span> period change, which is frequently observed in long-period Cepheids <cite class="ltx_cite ltx_citemacro_citep">(see Trahin et al., <a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#bib.bib30" title="">2021</a>, Table D)</cite>. Long-period Cepheids are expected to have more massive progenitors and to evolve more rapidly across the instability strip. The period instability of HV 821 was also briefly noted by <cite class="ltx_cite ltx_citemacro_citet">Berdnikov & Turner (<a class="ltx_ref" href="https://arxiv.org/html/2502.17438v1#bib.bib1" title="">1997</a>)</cite> who also speculated on its classification. Despite this, BZ Tuc has been classified by as a classical Cepheid in most modern catalogs.</p> </div> <figure class="ltx_figure" id="A1.F8"> <div class="ltx_flex_figure"> <div class="ltx_flex_cell ltx_flex_size_2"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_figure_panel ltx_img_landscape" height="271" id="A1.F8.g1" src="x10.png" width="407"/></div> <div class="ltx_flex_cell ltx_flex_size_2"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_figure_panel ltx_img_landscape" height="271" id="A1.F8.g2" src="x11.png" width="407"/></div> </div> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 8: </span><span class="ltx_text ltx_font_italic" id="A1.F8.9.1">Left:</span> Folded light curve for HV 821 (BZ Tuc) assuming a period of 127.445 days, which we find minimizes the phase dispersion during the span of the observations. Points are colored chronologically from blue (earlier) to red (later). <span class="ltx_text ltx_font_italic" id="A1.F8.10.2">Right:</span> O-C diagram of HV 821 (BZ Tuc) constructed using ASAS time series collected over a baseline of 10 years. O-C is expressed as a fraction of the fit period. Uncertainties are estimated as <math alttext="\sim\sigma_{phs}/\sqrt{N_{obs}}" class="ltx_Math" display="inline" id="A1.F8.4.m1.1"><semantics id="A1.F8.4.m1.1b"><mrow id="A1.F8.4.m1.1.1" xref="A1.F8.4.m1.1.1.cmml"><mi id="A1.F8.4.m1.1.1.2" xref="A1.F8.4.m1.1.1.2.cmml"></mi><mo id="A1.F8.4.m1.1.1.1" xref="A1.F8.4.m1.1.1.1.cmml">∼</mo><mrow id="A1.F8.4.m1.1.1.3" xref="A1.F8.4.m1.1.1.3.cmml"><msub id="A1.F8.4.m1.1.1.3.2" xref="A1.F8.4.m1.1.1.3.2.cmml"><mi id="A1.F8.4.m1.1.1.3.2.2" xref="A1.F8.4.m1.1.1.3.2.2.cmml">σ</mi><mrow id="A1.F8.4.m1.1.1.3.2.3" xref="A1.F8.4.m1.1.1.3.2.3.cmml"><mi id="A1.F8.4.m1.1.1.3.2.3.2" xref="A1.F8.4.m1.1.1.3.2.3.2.cmml">p</mi><mo id="A1.F8.4.m1.1.1.3.2.3.1" xref="A1.F8.4.m1.1.1.3.2.3.1.cmml">⁢</mo><mi id="A1.F8.4.m1.1.1.3.2.3.3" xref="A1.F8.4.m1.1.1.3.2.3.3.cmml">h</mi><mo id="A1.F8.4.m1.1.1.3.2.3.1b" xref="A1.F8.4.m1.1.1.3.2.3.1.cmml">⁢</mo><mi id="A1.F8.4.m1.1.1.3.2.3.4" xref="A1.F8.4.m1.1.1.3.2.3.4.cmml">s</mi></mrow></msub><mo id="A1.F8.4.m1.1.1.3.1" xref="A1.F8.4.m1.1.1.3.1.cmml">/</mo><msqrt id="A1.F8.4.m1.1.1.3.3" xref="A1.F8.4.m1.1.1.3.3.cmml"><msub id="A1.F8.4.m1.1.1.3.3.2" xref="A1.F8.4.m1.1.1.3.3.2.cmml"><mi id="A1.F8.4.m1.1.1.3.3.2.2" xref="A1.F8.4.m1.1.1.3.3.2.2.cmml">N</mi><mrow id="A1.F8.4.m1.1.1.3.3.2.3" xref="A1.F8.4.m1.1.1.3.3.2.3.cmml"><mi id="A1.F8.4.m1.1.1.3.3.2.3.2" xref="A1.F8.4.m1.1.1.3.3.2.3.2.cmml">o</mi><mo id="A1.F8.4.m1.1.1.3.3.2.3.1" xref="A1.F8.4.m1.1.1.3.3.2.3.1.cmml">⁢</mo><mi id="A1.F8.4.m1.1.1.3.3.2.3.3" xref="A1.F8.4.m1.1.1.3.3.2.3.3.cmml">b</mi><mo id="A1.F8.4.m1.1.1.3.3.2.3.1b" xref="A1.F8.4.m1.1.1.3.3.2.3.1.cmml">⁢</mo><mi id="A1.F8.4.m1.1.1.3.3.2.3.4" xref="A1.F8.4.m1.1.1.3.3.2.3.4.cmml">s</mi></mrow></msub></msqrt></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.F8.4.m1.1c"><apply id="A1.F8.4.m1.1.1.cmml" xref="A1.F8.4.m1.1.1"><csymbol cd="latexml" id="A1.F8.4.m1.1.1.1.cmml" xref="A1.F8.4.m1.1.1.1">similar-to</csymbol><csymbol cd="latexml" id="A1.F8.4.m1.1.1.2.cmml" xref="A1.F8.4.m1.1.1.2">absent</csymbol><apply id="A1.F8.4.m1.1.1.3.cmml" xref="A1.F8.4.m1.1.1.3"><divide id="A1.F8.4.m1.1.1.3.1.cmml" xref="A1.F8.4.m1.1.1.3.1"></divide><apply id="A1.F8.4.m1.1.1.3.2.cmml" xref="A1.F8.4.m1.1.1.3.2"><csymbol cd="ambiguous" id="A1.F8.4.m1.1.1.3.2.1.cmml" xref="A1.F8.4.m1.1.1.3.2">subscript</csymbol><ci id="A1.F8.4.m1.1.1.3.2.2.cmml" xref="A1.F8.4.m1.1.1.3.2.2">𝜎</ci><apply id="A1.F8.4.m1.1.1.3.2.3.cmml" xref="A1.F8.4.m1.1.1.3.2.3"><times id="A1.F8.4.m1.1.1.3.2.3.1.cmml" xref="A1.F8.4.m1.1.1.3.2.3.1"></times><ci id="A1.F8.4.m1.1.1.3.2.3.2.cmml" xref="A1.F8.4.m1.1.1.3.2.3.2">𝑝</ci><ci id="A1.F8.4.m1.1.1.3.2.3.3.cmml" xref="A1.F8.4.m1.1.1.3.2.3.3">ℎ</ci><ci id="A1.F8.4.m1.1.1.3.2.3.4.cmml" xref="A1.F8.4.m1.1.1.3.2.3.4">𝑠</ci></apply></apply><apply id="A1.F8.4.m1.1.1.3.3.cmml" xref="A1.F8.4.m1.1.1.3.3"><root id="A1.F8.4.m1.1.1.3.3a.cmml" xref="A1.F8.4.m1.1.1.3.3"></root><apply id="A1.F8.4.m1.1.1.3.3.2.cmml" xref="A1.F8.4.m1.1.1.3.3.2"><csymbol cd="ambiguous" id="A1.F8.4.m1.1.1.3.3.2.1.cmml" xref="A1.F8.4.m1.1.1.3.3.2">subscript</csymbol><ci id="A1.F8.4.m1.1.1.3.3.2.2.cmml" xref="A1.F8.4.m1.1.1.3.3.2.2">𝑁</ci><apply id="A1.F8.4.m1.1.1.3.3.2.3.cmml" xref="A1.F8.4.m1.1.1.3.3.2.3"><times id="A1.F8.4.m1.1.1.3.3.2.3.1.cmml" xref="A1.F8.4.m1.1.1.3.3.2.3.1"></times><ci id="A1.F8.4.m1.1.1.3.3.2.3.2.cmml" xref="A1.F8.4.m1.1.1.3.3.2.3.2">𝑜</ci><ci id="A1.F8.4.m1.1.1.3.3.2.3.3.cmml" xref="A1.F8.4.m1.1.1.3.3.2.3.3">𝑏</ci><ci id="A1.F8.4.m1.1.1.3.3.2.3.4.cmml" xref="A1.F8.4.m1.1.1.3.3.2.3.4">𝑠</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.F8.4.m1.1d">\sim\sigma_{phs}/\sqrt{N_{obs}}</annotation><annotation encoding="application/x-llamapun" id="A1.F8.4.m1.1e">∼ italic_σ start_POSTSUBSCRIPT italic_p italic_h italic_s end_POSTSUBSCRIPT / square-root start_ARG italic_N start_POSTSUBSCRIPT italic_o italic_b italic_s end_POSTSUBSCRIPT end_ARG</annotation></semantics></math> where <math alttext="\sigma_{phs}" class="ltx_Math" display="inline" id="A1.F8.5.m2.1"><semantics id="A1.F8.5.m2.1b"><msub id="A1.F8.5.m2.1.1" xref="A1.F8.5.m2.1.1.cmml"><mi id="A1.F8.5.m2.1.1.2" xref="A1.F8.5.m2.1.1.2.cmml">σ</mi><mrow id="A1.F8.5.m2.1.1.3" xref="A1.F8.5.m2.1.1.3.cmml"><mi id="A1.F8.5.m2.1.1.3.2" xref="A1.F8.5.m2.1.1.3.2.cmml">p</mi><mo id="A1.F8.5.m2.1.1.3.1" xref="A1.F8.5.m2.1.1.3.1.cmml">⁢</mo><mi id="A1.F8.5.m2.1.1.3.3" xref="A1.F8.5.m2.1.1.3.3.cmml">h</mi><mo id="A1.F8.5.m2.1.1.3.1b" xref="A1.F8.5.m2.1.1.3.1.cmml">⁢</mo><mi id="A1.F8.5.m2.1.1.3.4" xref="A1.F8.5.m2.1.1.3.4.cmml">s</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="A1.F8.5.m2.1c"><apply id="A1.F8.5.m2.1.1.cmml" xref="A1.F8.5.m2.1.1"><csymbol cd="ambiguous" id="A1.F8.5.m2.1.1.1.cmml" xref="A1.F8.5.m2.1.1">subscript</csymbol><ci id="A1.F8.5.m2.1.1.2.cmml" xref="A1.F8.5.m2.1.1.2">𝜎</ci><apply id="A1.F8.5.m2.1.1.3.cmml" xref="A1.F8.5.m2.1.1.3"><times id="A1.F8.5.m2.1.1.3.1.cmml" xref="A1.F8.5.m2.1.1.3.1"></times><ci id="A1.F8.5.m2.1.1.3.2.cmml" xref="A1.F8.5.m2.1.1.3.2">𝑝</ci><ci id="A1.F8.5.m2.1.1.3.3.cmml" xref="A1.F8.5.m2.1.1.3.3">ℎ</ci><ci id="A1.F8.5.m2.1.1.3.4.cmml" xref="A1.F8.5.m2.1.1.3.4">𝑠</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.F8.5.m2.1d">\sigma_{phs}</annotation><annotation encoding="application/x-llamapun" id="A1.F8.5.m2.1e">italic_σ start_POSTSUBSCRIPT italic_p italic_h italic_s end_POSTSUBSCRIPT</annotation></semantics></math> is the phase uncertainty <math alttext="N_{obs}" class="ltx_Math" display="inline" id="A1.F8.6.m3.1"><semantics id="A1.F8.6.m3.1b"><msub id="A1.F8.6.m3.1.1" xref="A1.F8.6.m3.1.1.cmml"><mi id="A1.F8.6.m3.1.1.2" xref="A1.F8.6.m3.1.1.2.cmml">N</mi><mrow id="A1.F8.6.m3.1.1.3" xref="A1.F8.6.m3.1.1.3.cmml"><mi id="A1.F8.6.m3.1.1.3.2" xref="A1.F8.6.m3.1.1.3.2.cmml">o</mi><mo id="A1.F8.6.m3.1.1.3.1" xref="A1.F8.6.m3.1.1.3.1.cmml">⁢</mo><mi id="A1.F8.6.m3.1.1.3.3" xref="A1.F8.6.m3.1.1.3.3.cmml">b</mi><mo id="A1.F8.6.m3.1.1.3.1b" xref="A1.F8.6.m3.1.1.3.1.cmml">⁢</mo><mi id="A1.F8.6.m3.1.1.3.4" xref="A1.F8.6.m3.1.1.3.4.cmml">s</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="A1.F8.6.m3.1c"><apply id="A1.F8.6.m3.1.1.cmml" xref="A1.F8.6.m3.1.1"><csymbol cd="ambiguous" id="A1.F8.6.m3.1.1.1.cmml" xref="A1.F8.6.m3.1.1">subscript</csymbol><ci id="A1.F8.6.m3.1.1.2.cmml" xref="A1.F8.6.m3.1.1.2">𝑁</ci><apply id="A1.F8.6.m3.1.1.3.cmml" xref="A1.F8.6.m3.1.1.3"><times id="A1.F8.6.m3.1.1.3.1.cmml" xref="A1.F8.6.m3.1.1.3.1"></times><ci id="A1.F8.6.m3.1.1.3.2.cmml" xref="A1.F8.6.m3.1.1.3.2">𝑜</ci><ci id="A1.F8.6.m3.1.1.3.3.cmml" xref="A1.F8.6.m3.1.1.3.3">𝑏</ci><ci id="A1.F8.6.m3.1.1.3.4.cmml" xref="A1.F8.6.m3.1.1.3.4">𝑠</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.F8.6.m3.1d">N_{obs}</annotation><annotation encoding="application/x-llamapun" id="A1.F8.6.m3.1e">italic_N start_POSTSUBSCRIPT italic_o italic_b italic_s end_POSTSUBSCRIPT</annotation></semantics></math> is the number of observations per cycle. Only cycles observed near peak are included.</figcaption> </figure> </section> </article> </div> <footer class="ltx_page_footer"> <div class="ltx_page_logo">Generated on Mon Feb 24 18:19:28 2025 by <a class="ltx_LaTeXML_logo" href="http://dlmf.nist.gov/LaTeXML/"><span style="letter-spacing:-0.2em; margin-right:0.1em;">L<span class="ltx_font_smallcaps" style="position:relative; bottom:2.2pt;">a</span>T<span class="ltx_font_smallcaps" style="font-size:120%;position:relative; bottom:-0.2ex;">e</span></span><span style="font-size:90%; position:relative; bottom:-0.2ex;">XML</span><img alt="Mascot Sammy" src="data:image/png;base64,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"/></a> </div></footer> </div> </body> </html>