Sources and Radiations of the Fermi Bubbles

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} div.type-section h2 { font-size: 20px; line-height: 26px; font-weight: 300; } div.type-section h3 { margin-left: 15px; margin-bottom: 0px; font-weight: 300; } .journal-tabs .tab-title.active a { } </style> <link rel="stylesheet" href="https://pub.mdpi-res.com/assets/css/slick.css?f38b2db10e01b157?1732694169"> <meta name="title" content="Sources and Radiations of the Fermi Bubbles"> <meta name="description" content="Two enigmatic gamma-ray features in the galactic central region, known as Fermi Bubbles (FBs), were found from Fermi-LAT data. An energy release, (e.g., by tidal disruption events in the Galactic Center, GC), generates a cavity with a shock that expands into the local ambient medium of the galactic halo. A decade or so ago, a phenomenological model of the FBs was suggested as a result of routine star disruptions by the supermassive black hole in the GC which might provide enough energy for large-scale structures, like the FBs. In 2020, analytical and numerical models of the FBs as a process of routine tidal disruption of stars near the GC were developed; these disruption events can provide enough cumulative energy to form and maintain large-scale structures like the FBs. The disruption events are expected to be 10−4&sim;10−5yr−1, providing an average power of energy release from the GC into the halo of E˙&sim;3×1041 erg s−1, which is needed to support the FBs. Analysis of the evolution of superbubbles in exponentially stratified disks concluded that the FB envelope would be destroyed by the Rayleigh–Taylor (RT) instabilities at late stages. The shell is composed of swept-up gas of the bubble, whose thickness is much thinner in comparison to the size of the envelope. We assume that hydrodynamic turbulence is excited in the FB envelope by the RT instability. In this case, the universal energy spectrum of turbulence may be developed in the inertial range of wavenumbers of fluctuations (the Kolmogorov–Obukhov spectrum). From our model we suppose the power of the FBs is transformed partly into the energy of hydrodynamic turbulence in the envelope. If so, hydrodynamic turbulence may generate MHD fluctuations, which accelerate cosmic rays there and generate gamma-ray and radio emission from the FBs. We hope that this model may interpret the observed nonthermal emission from the bubbles." > <link rel="image_src" href="https://pub.mdpi-res.com/img/journals/universe-logo.png?8600e93ff98dbf14" > <meta name="dc.title" content="Sources and Radiations of the Fermi Bubbles"> <meta name="dc.creator" content="Vladimir A. Dogiel"> <meta name="dc.creator" content="Chung-Ming Ko"> <meta name="dc.type" content="Article"> <meta name="dc.source" content="Universe 2024, Vol. 10, Page 424"> <meta name="dc.date" content="2024-11-12"> <meta name ="dc.identifier" content="10.3390/universe10110424"> <meta name="dc.publisher" content="Multidisciplinary Digital Publishing Institute"> <meta name="dc.rights" content="http://creativecommons.org/licenses/by/3.0/"> <meta name="dc.format" content="application/pdf" > <meta name="dc.language" content="en" > <meta name="dc.description" content="Two enigmatic gamma-ray features in the galactic central region, known as Fermi Bubbles (FBs), were found from Fermi-LAT data. An energy release, (e.g., by tidal disruption events in the Galactic Center, GC), generates a cavity with a shock that expands into the local ambient medium of the galactic halo. A decade or so ago, a phenomenological model of the FBs was suggested as a result of routine star disruptions by the supermassive black hole in the GC which might provide enough energy for large-scale structures, like the FBs. In 2020, analytical and numerical models of the FBs as a process of routine tidal disruption of stars near the GC were developed; these disruption events can provide enough cumulative energy to form and maintain large-scale structures like the FBs. The disruption events are expected to be 10−4&sim;10−5yr−1, providing an average power of energy release from the GC into the halo of E˙&sim;3×1041 erg s−1, which is needed to support the FBs. Analysis of the evolution of superbubbles in exponentially stratified disks concluded that the FB envelope would be destroyed by the Rayleigh–Taylor (RT) instabilities at late stages. The shell is composed of swept-up gas of the bubble, whose thickness is much thinner in comparison to the size of the envelope. We assume that hydrodynamic turbulence is excited in the FB envelope by the RT instability. In this case, the universal energy spectrum of turbulence may be developed in the inertial range of wavenumbers of fluctuations (the Kolmogorov–Obukhov spectrum). From our model we suppose the power of the FBs is transformed partly into the energy of hydrodynamic turbulence in the envelope. If so, hydrodynamic turbulence may generate MHD fluctuations, which accelerate cosmic rays there and generate gamma-ray and radio emission from the FBs. We hope that this model may interpret the observed nonthermal emission from the bubbles." > <meta name="dc.subject" content="galactic center" > <meta name="dc.subject" content="Fermi bubbles" > <meta name="dc.subject" content="central black hole" > <meta name="dc.subject" content="star disruptions" > <meta name="dc.subject" content="MHD turbulence" > <meta name="dc.subject" content="cosmic rays" > <meta name ="prism.issn" content="2218-1997"> <meta name ="prism.publicationName" content="Universe"> <meta name ="prism.publicationDate" content="2024-11-12"> <meta name ="prism.volume" content="10"> <meta name ="prism.number" content="11"> <meta name ="prism.section" content="Article" > <meta name ="prism.startingPage" content="424" > <meta name="citation_issn" content="2218-1997"> <meta name="citation_journal_title" content="Universe"> <meta name="citation_publisher" content="Multidisciplinary Digital Publishing Institute"> <meta name="citation_title" content="Sources and Radiations of the Fermi Bubbles"> <meta name="citation_publication_date" content="2024/11"> <meta name="citation_online_date" content="2024/11/12"> <meta name="citation_volume" content="10"> <meta name="citation_issue" content="11"> <meta name="citation_firstpage" content="424"> <meta name="citation_author" content="Dogiel, Vladimir A."> <meta name="citation_author" content="Ko, Chung-Ming"> <meta name="citation_doi" content="10.3390/universe10110424"> <meta name="citation_id" content="mdpi-universe10110424"> <meta name="citation_abstract_html_url" content="https://www.mdpi.com/2218-1997/10/11/424"> <meta name="citation_pdf_url" content="https://www.mdpi.com/2218-1997/10/11/424/pdf?version=1732197779"> <link rel="alternate" type="application/pdf" title="PDF Full-Text" href="https://www.mdpi.com/2218-1997/10/11/424/pdf?version=1732197779"> <meta name="fulltext_pdf" content="https://www.mdpi.com/2218-1997/10/11/424/pdf?version=1732197779"> <meta name="citation_fulltext_html_url" content="https://www.mdpi.com/2218-1997/10/11/424/htm"> <link rel="alternate" type="text/html" title="HTML Full-Text" href="https://www.mdpi.com/2218-1997/10/11/424/htm"> <meta name="fulltext_html" content="https://www.mdpi.com/2218-1997/10/11/424/htm"> <link rel="alternate" type="text/xml" title="XML Full-Text" href="https://www.mdpi.com/2218-1997/10/11/424/xml"> <meta name="fulltext_xml" content="https://www.mdpi.com/2218-1997/10/11/424/xml"> <meta name="citation_xml_url" content="https://www.mdpi.com/2218-1997/10/11/424/xml"> <meta name="twitter:card" content="summary" /> <meta name="twitter:site" content="@MDPIOpenAccess" /> <meta name="twitter:image" content="https://pub.mdpi-res.com/img/journals/universe-logo-social.png?8600e93ff98dbf14" /> <meta property="fb:app_id" content="131189377574"/> <meta property="og:site_name" content="MDPI"/> <meta property="og:type" content="article"/> <meta property="og:url" content="https://www.mdpi.com/2218-1997/10/11/424" /> <meta property="og:title" content="Sources and Radiations of the Fermi Bubbles" /> <meta property="og:description" content="Two enigmatic gamma-ray features in the galactic central region, known as Fermi Bubbles (FBs), were found from Fermi-LAT data. An energy release, (e.g., by tidal disruption events in the Galactic Center, GC), generates a cavity with a shock that expands into the local ambient medium of the galactic halo. A decade or so ago, a phenomenological model of the FBs was suggested as a result of routine star disruptions by the supermassive black hole in the GC which might provide enough energy for large-scale structures, like the FBs. In 2020, analytical and numerical models of the FBs as a process of routine tidal disruption of stars near the GC were developed; these disruption events can provide enough cumulative energy to form and maintain large-scale structures like the FBs. The disruption events are expected to be 10−4&sim;10−5yr−1, providing an average power of energy release from the GC into the halo of E˙&sim;3×1041 erg s−1, which is needed to support the FBs. Analysis of the evolution of superbubbles in exponentially stratified disks concluded that the FB envelope would be destroyed by the Rayleigh–Taylor (RT) instabilities at late stages. The shell is composed of swept-up gas of the bubble, whose thickness is much thinner in comparison to the size of the envelope. We assume that hydrodynamic turbulence is excited in the FB envelope by the RT instability. In this case, the universal energy spectrum of turbulence may be developed in the inertial range of wavenumbers of fluctuations (the Kolmogorov–Obukhov spectrum). From our model we suppose the power of the FBs is transformed partly into the energy of hydrodynamic turbulence in the envelope. If so, hydrodynamic turbulence may generate MHD fluctuations, which accelerate cosmic rays there and generate gamma-ray and radio emission from the FBs. We hope that this model may interpret the observed nonthermal emission from the bubbles." /> <meta property="og:image" content="https://pub.mdpi-res.com/universe/universe-10-00424/article_deploy/html/images/universe-10-00424-g001-550.jpg?1732197878" /> <link rel="alternate" type="application/rss+xml" title="MDPI Publishing - Latest articles" href="https://www.mdpi.com/rss"> <meta name="google-site-verification" content="PxTlsg7z2S00aHroktQd57fxygEjMiNHydKn3txhvwY"> <meta name="facebook-domain-verification" content="mcoq8dtq6sb2hf7z29j8w515jjoof7" /> <script id="Cookiebot" data-cfasync="false" src="https://consent.cookiebot.com/uc.js" data-cbid="51491ddd-fe7a-4425-ab39-69c78c55829f" type="text/javascript" async></script>  <script type="text/plain" data-cookieconsent="statistics"> (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start': new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0], j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src= 'https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f); 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class="fa fa-file-text"></i> </div> </div> </div> </div> </div> </div> </div> <article ><div class='html-article-content'> <span itemprop="publisher" content="Multidisciplinary Digital Publishing Institute"></span><span itemprop="url" content="https://www.mdpi.com/2218-1997/10/11/424"></span> <div class="article-icons"><span class="label openaccess" data-dropdown="drop-article-label-openaccess" aria-expanded="false">Open Access</span><span class="label articletype">Article</span></div> <h1 class="title hypothesis_container" itemprop="name"> Sources and Radiations of the Fermi Bubbles </h1> <div class="art-authors hypothesis_container"> by <span class="inlineblock "><div class='profile-card-drop' data-dropdown='profile-card-drop13389305' data-options='is_hover:true, hover_timeout:5000'> Vladimir A. Dogiel</div><div id="profile-card-drop13389305" data-dropdown-content class="f-dropdown content profile-card-content" aria-hidden="true" tabindex="-1"><div class="profile-card__title"><div class="sciprofiles-link" style="display: inline-block"><div class="sciprofiles-link__link"><img class="sciprofiles-link__image" src="/bundles/mdpisciprofileslink/img/unknown-user.png" style="width: auto; height: 16px; border-radius: 50%;"><span class="sciprofiles-link__name">Vladimir A. Dogiel</span></div></div></div><div class="profile-card__buttons" style="margin-bottom: 10px;"><a href="https://sciprofiles.com/profile/3367488?utm_source=mdpi.com&utm_medium=website&utm_campaign=avatar_name" class="button button--color-inversed" target="_blank"> SciProfiles </a><a href="https://scilit.net/scholars?q=Vladimir%20A.%20Dogiel" class="button button--color-inversed" target="_blank"> Scilit </a><a href="https://www.preprints.org/search?search1=Vladimir%20A.%20Dogiel&field1=authors" class="button button--color-inversed" target="_blank"> Preprints.org </a><a href="https://scholar.google.com/scholar?q=Vladimir%20A.%20Dogiel" class="button button--color-inversed" target="_blank" rels="noopener noreferrer"> Google Scholar </a></div></div><sup> 1,*</sup><span style="display: inline; margin-left: 5px;"></span><a class="toEncode emailCaptcha visibility-hidden" data-author-id="13389305" href="/cdn-cgi/l/email-protection#b49bd7dad099d7d3dd9bd89bd1d9d5ddd899c4c6dbc0d1d7c0dddbda97848484d68280848784d08485848c86808584848480d5848c858084d080d585828585"><sup><i class="fa fa-envelope-o"></i></sup></a> and </span><span class="inlineblock "><div class='profile-card-drop' data-dropdown='profile-card-drop13389306' data-options='is_hover:true, hover_timeout:5000'> Chung-Ming Ko</div><div id="profile-card-drop13389306" data-dropdown-content class="f-dropdown content profile-card-content" aria-hidden="true" tabindex="-1"><div class="profile-card__title"><div class="sciprofiles-link" style="display: inline-block"><div class="sciprofiles-link__link"><img class="sciprofiles-link__image" src="/bundles/mdpisciprofileslink/img/unknown-user.png" style="width: auto; height: 16px; border-radius: 50%;"><span class="sciprofiles-link__name">Chung-Ming Ko</span></div></div></div><div class="profile-card__buttons" style="margin-bottom: 10px;"><a href="https://sciprofiles.com/profile/3829786?utm_source=mdpi.com&utm_medium=website&utm_campaign=avatar_name" class="button button--color-inversed" target="_blank"> SciProfiles </a><a href="https://scilit.net/scholars?q=Chung-Ming%20Ko" class="button button--color-inversed" target="_blank"> Scilit </a><a href="https://www.preprints.org/search?search1=Chung-Ming%20Ko&field1=authors" class="button button--color-inversed" target="_blank"> Preprints.org </a><a href="https://scholar.google.com/scholar?q=Chung-Ming%20Ko" class="button button--color-inversed" target="_blank" rels="noopener noreferrer"> Google Scholar </a></div></div><sup> 2,*</sup><span style="display: inline; margin-left: 5px;"></span><a class="toEncode emailCaptcha visibility-hidden" data-author-id="13389306" href="/cdn-cgi/l/email-protection#66490508024b05010f490a49030b070f0a4b161409120305120f090845565656035055565e560554555654575657515757560552025602565657505202565056515750520257515752"><sup><i class="fa fa-envelope-o"></i></sup></a></span> </div> <div class="nrm"></div> <span style="display:block; height:6px;"></span> <div></div> <div style="margin: 5px 0 15px 0;" class="hypothesis_container"> <div class="art-affiliations"> <div class="affiliation "> <div class="affiliation-item"><sup>1</sup></div> <div class="affiliation-name ">I.E.Tamm Theoretical Physics Division of P.N.Lebedev Institute of Physics, Leninskii Pr. 53, 119991 Moscow, Russia</div> </div> <div class="affiliation "> <div class="affiliation-item"><sup>2</sup></div> <div class="affiliation-name ">Institute of Astronomy, Department of Physics and Center for Complex Systems, National Central University, Zhongli Dist., Taoyuan City 320317, Taiwan</div> </div> <div class="affiliation"> <div class="affiliation-item"><sup>*</sup></div> <div class="affiliation-name ">Authors to whom correspondence should be addressed. </div> </div> </div> </div> <div class="bib-identity" style="margin-bottom: 10px;"> <em>Universe</em> <b>2024</b>, <em>10</em>(11), 424; <a href="https://doi.org/10.3390/universe10110424">https://doi.org/10.3390/universe10110424</a> </div> <div class="pubhistory" style="font-weight: bold; padding-bottom: 10px;"> <span style="display: inline-block">Submission received: 11 September 2024</span> / <span style="display: inline-block">Revised: 22 October 2024</span> / <span style="display: inline-block">Accepted: 25 October 2024</span> / <span style="display: inline-block">Published: 12 November 2024</span> </div> <div class="belongsTo" style="margin-bottom: 10px;"> (This article belongs to the Special Issue <a href=" /journal/universe/special_issues/7683EV6AD4 ">Studying Astrophysics with High-Energy Cosmic Particles</a>)<br/> </div> <div class="highlight-box1"> <div class="download"> <a class="button button--color-inversed button--drop-down" data-dropdown="drop-download-1519436" aria-controls="drop-supplementary-1519436" aria-expanded="false"> Download <i class="material-icons">keyboard_arrow_down</i> </a> <div id="drop-download-1519436" class="f-dropdown label__btn__dropdown label__btn__dropdown--button" data-dropdown-content aria-hidden="true" tabindex="-1"> <a class="UD_ArticlePDF" href="/2218-1997/10/11/424/pdf?version=1732197779" data-name="Sources and Radiations of the Fermi Bubbles" data-journal="universe">Download PDF</a> <br/> <a id="js-pdf-with-cover-access-captcha" href="#" data-target="/2218-1997/10/11/424/pdf-with-cover" class="accessCaptcha">Download PDF with Cover</a> <br/> <a id="js-xml-access-captcha" href="#" data-target="/2218-1997/10/11/424/xml" class="accessCaptcha">Download XML</a> <br/> <a href="/2218-1997/10/11/424/epub" id="epub_link">Download Epub</a> <br/> </div> <div class="js-browse-figures" style="display: inline-block;"> <a href="#" class="button button--color-inversed margin-bottom-10 openpopupgallery UI_BrowseArticleFigures" data-target='article-popup' data-counterslink = "https://www.mdpi.com/2218-1997/10/11/424/browse" >Browse Figures</a> </div> <div id="article-popup" class="popupgallery" style="display: inline; line-height: 200%"> <a href="https://pub.mdpi-res.com/universe/universe-10-00424/article_deploy/html/images/universe-10-00424-g001.png?1732197878" title=" <strong>Figure 1</strong><br/> <p>Comparison of the morphology of the gamma-ray bubbles (red) and the X-ray bubbles (cyan) in the direction of the Galactic Center. Figure reproduced from Predehl et al. [<a href="#B1-universe-10-00424" class="html-bibr">1</a>] with permission.</p> "> </a> <a href="https://pub.mdpi-res.com/universe/universe-10-00424/article_deploy/html/images/universe-10-00424-g002.png?1732197879" title=" <strong>Figure 2</strong><br/> <p>X-ray superbubbles in the galaxy NGC 3079. Image from <a href="https://chandra.harvard.edu/photo/2019/ngc3079" target="_blank">https://chandra.harvard.edu/photo/2019/ngc3079</a> (accessed on 10 September 2024). Image credit: X-ray: NASA/CXC/University of Michigan. Optical: NASA/STScI.</p> "> </a> <a href="https://pub.mdpi-res.com/universe/universe-10-00424/article_deploy/html/images/universe-10-00424-g003.png?1732197879" title=" <strong>Figure 3</strong><br/> <p>The observed X-ray light curve of Swift J1644+57 from Swift, XMM-Newton, and Chandra. Figure reproduced from Cheng et al. [<a href="#B43-universe-10-00424" class="html-bibr">43</a>] with permission.</p> "> </a> <a href="https://pub.mdpi-res.com/universe/universe-10-00424/article_deploy/html/images/universe-10-00424-g004.png?1732197881" title=" <strong>Figure 4</strong><br/> <p>A thermonuclear explosion in the terrestrial atmosphere. Image credit: United States Department of Energy. Image from <a href="https://commons.wikimedia.org/wiki/File:Castle_Bravo_nuclear_test_(cropped).jpg" target="_blank">https://commons.wikimedia.org/wiki/File:Castle_Bravo_nuclear_test_(cropped).jpg</a> (accessed on 10 September 2024).</p> "> </a> <a href="https://pub.mdpi-res.com/universe/universe-10-00424/article_deploy/html/images/universe-10-00424-g005.png?1732197881" title=" <strong>Figure 5</strong><br/> <p>Illustration of the double-bubble shock envelope in the halo evolving with time. The gas distribution in the halo in the left panel is exponential and in the right panel follows a power law. Figure adapted from Ko et al. [<a href="#B16-universe-10-00424" class="html-bibr">16</a>] with permission.</p> "> </a> <a href="https://pub.mdpi-res.com/universe/universe-10-00424/article_deploy/html/images/universe-10-00424-g006.png?1732197882" title=" <strong>Figure 6</strong><br/> <p>Temporal variation in the shock velocity of the top of the bubble for the case of exponential halo with <math display="inline"><semantics> <mrow> <mi>H</mi> <mo>=</mo> <mn>0.67</mn> </mrow> </semantics></math> kpc and <math display="inline"><semantics> <mrow> <msub> <mi>n</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>0.03</mn> <msup> <mi>cm</mi> <mrow> <mo>−</mo> <mn>3</mn> </mrow> </msup> </mrow> </semantics></math>. (<b>Left panel</b>): One single input of energy from the GC. (<b>Right panel</b>): Multiple TDEs with different values of power release at the GC. The horizontal dotted line indicates the velocity which is necessary for the shock in order not to stall in the halo, which is three times the sound speed in the halo <math display="inline"><semantics> <mrow> <mn>3</mn> <mo>×</mo> <msup> <mn>10</mn> <mn>7</mn> </msup> </mrow> </semantics></math> cm <math display="inline"><semantics> <mrow> <msup> <mi mathvariant="normal">s</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </semantics></math>. Figure reproduced from Ko et al. [<a href="#B16-universe-10-00424" class="html-bibr">16</a>] with permission.</p> "> </a> <a href="https://pub.mdpi-res.com/universe/universe-10-00424/article_deploy/html/images/universe-10-00424-g007.png?1732197884" title=" <strong>Figure 7</strong><br/> <p>Density distribution of numerical simulations of the FBs in an exponential halo. The two panels in the left column are results of multiple explosions (e.g., TDEs) and in the right column are results of a single huge explosion. In the upper left panel, “Me0.05-3e52erg 18.0 Myr” corresponds to multiple explosions with 0.05 Myr between successive explosions and the energy release by each explosion is <math display="inline"><semantics> <mrow> <mn>3</mn> <mo>×</mo> <msup> <mn>10</mn> <mn>52</mn> </msup> </mrow> </semantics></math> erg, and the simulation ends at 18.0 Myr. In the upper right panel, “1e-1.08e55erg 10.0 Myr” corresponds to a single explosion with an energy release of <math display="inline"><semantics> <mrow> <mn>1.08</mn> <mo>×</mo> <msup> <mn>10</mn> <mn>55</mn> </msup> </mrow> </semantics></math> erg, and the simulation ends at 10.0 Myr. Similar explanation for the lower panels. Lower panel figures reproduced from Ko et al. [<a href="#B16-universe-10-00424" class="html-bibr">16</a>] with permission.</p> "> </a> <a href="https://pub.mdpi-res.com/universe/universe-10-00424/article_deploy/html/images/universe-10-00424-g008.png?1732197886" title=" <strong>Figure 8</strong><br/> <p>Coherent magnetic structure above and below the galactic plane [<a href="#B17-universe-10-00424" class="html-bibr">17</a>]. (<b>a</b>) Polarized synchrotron intensity map at <math display="inline"><semantics> <mrow> <mn>22.8</mn> </mrow> </semantics></math> GHz from WMAP. Green bars show the magnetic field direction. (<b>b</b>) Comparison between the polarized synchrotron emission at <math display="inline"><semantics> <mrow> <mn>22.8</mn> </mrow> </semantics></math> GHz (red) and the X-ray emission at <math display="inline"><semantics> <mrow> <mn>0.6</mn> </mrow> </semantics></math>∼<math display="inline"><semantics> <mrow> <mn>1.0</mn> </mrow> </semantics></math> keV from eROSITA (green). Magnetized ridges are shown in white. Figure reproduced from Zhang et al. [<a href="#B17-universe-10-00424" class="html-bibr">17</a>] with permission.</p> "> </a> <a href="https://pub.mdpi-res.com/universe/universe-10-00424/article_deploy/html/images/universe-10-00424-g009.png?1732197886" title=" <strong>Figure 9</strong><br/> <p>The solid line shows the momentum diffusion coefficient derived for the bubble parameters when the CR absorption is taken into account. The dash-dotted line is the results ignoring the CR absorption. Figure reproduced from Cheng et al. [<a href="#B71-universe-10-00424" class="html-bibr">71</a>] with permission.</p> "> </a> <a href="https://pub.mdpi-res.com/universe/universe-10-00424/article_deploy/html/images/universe-10-00424-g010.png?1732197887" title=" <strong>Figure 10</strong><br/> <p>Spectrum of radio (<b>left</b>) and gamma-ray (<b>right</b>) emission from the FBs (see [<a href="#B71-universe-10-00424" class="html-bibr">71</a>]). The microwave data were taken from Planck Collaboration [<a href="#B3-universe-10-00424" class="html-bibr">3</a>], and the gamma-ray data from Ackermann et al. [<a href="#B72-universe-10-00424" class="html-bibr">72</a>], Ackermann et al. [<a href="#B77-universe-10-00424" class="html-bibr">77</a>]. Figure adapted from Cheng et al. [<a href="#B71-universe-10-00424" class="html-bibr">71</a>] with permission.</p> "> </a> <a href="https://pub.mdpi-res.com/universe/universe-10-00424/article_deploy/html/images/universe-10-00424-g011.png?1732197888" title=" <strong>Figure 11</strong><br/> <p>X-ray emission from the galactic plane whose excess emission is above the equilibrium Maxwellian spectrum. Dash-dotted line is a simple combination of thermal plus nonthermal spectrum. Solid line is the spectrum with the effect of runaway flux. Figure reproduced from Dogiel et al. [<a href="#B96-universe-10-00424" class="html-bibr">96</a>] with permission.</p> "> </a> <a href="https://pub.mdpi-res.com/universe/universe-10-00424/article_deploy/html/images/universe-10-00424-g012.png?1732197888" title=" <strong>Figure 12</strong><br/> <p>The spectrum of electrons accelerated from background plasma (see [<a href="#B100-universe-10-00424" class="html-bibr">100</a>]). The solid line is the density of electrons, <math display="inline"><semantics> <mrow> <mi>f</mi> <mo>(</mo> <mi>p</mi> <mo>)</mo> </mrow> </semantics></math>. The thick solid line is the pure thermal Maxwellian distribution. The dashed line is the power-law approximation of the nonthermal tail. For <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>0</mn> </msub> <mo>&gt;</mo> <msub> <mi>p</mi> <mi>inj</mi> </msub> </mrow> </semantics></math>, overheating is insignificant. Figure adapted from Chernyshov et al. [<a href="#B100-universe-10-00424" class="html-bibr">100</a>] with permission.</p> "> </a> <a href="https://pub.mdpi-res.com/universe/universe-10-00424/article_deploy/html/images/universe-10-00424-g013.png?1732197889" title=" <strong>Figure 13</strong><br/> <p>The spectrum of SNR electrons from the galactic disk that have been re-accelerated in the FBs. The five spectra in the figure correspond to different cases of the model: (1) thick solid line: without re-acceleration, escape, and advection; (2) thick dash-dotted line: without re-acceleration and escape but with advection; (3) thin dash-dotted line: with re-acceleration but without escape from the region and advection; (4) thin dotted line: with re-acceleration and escape from the region but without advection; (5) thin dashed line: with re-acceleration and advection but without escape. The density of electrons needed for the observed gamma-ray flux from the bubbles is shown by the gray region. The electron spectrum of case (5) can reproduce the gamma-ray data from Fermi-LAT and the microwave data from Planck (<a href="#universe-10-00424-f010" class="html-fig">Figure 10</a>). The parameters of case (5) can be found in the main text. For parameters of other cases, the reader is referred to Cheng et al. [<a href="#B101-universe-10-00424" class="html-bibr">101</a>]. Figure reproduced from Cheng et al. [<a href="#B101-universe-10-00424" class="html-bibr">101</a>] with permission.</p> "> </a> <a href="https://pub.mdpi-res.com/universe/universe-10-00424/article_deploy/html/images/universe-10-00424-g014.png?1732197890" title=" <strong>Figure 14</strong><br/> <p>CR spectrum at the Earth as a combination of the contributions from the SNRs in the galactic disk and the stochastic acceleration in the FBs. Figure reproduced from Cheng et al. [<a href="#B116-universe-10-00424" class="html-bibr">116</a>] with permission.</p> "> </a> <a href="https://pub.mdpi-res.com/universe/universe-10-00424/article_deploy/html/images/universe-10-00424-g015.png?1732197891" title=" <strong>Figure 15</strong><br/> <p>A possible multiple-shock structure in the FBs resulting from multiple TDEs at the GC. The figure shows the pressure (<b>left panel</b>) and kinetic energy (<b>right panel</b>) distributions of a numerical simulation of the FBs in an exponential halo. In the panels, “Me0.05-1e53” corresponds to multiple TDEs with 0.05 Myr between successive TDEs and the energy release by each TDE is <math display="inline"><semantics> <msup> <mn>10</mn> <mn>53</mn> </msup> </semantics></math> erg. The simulation ends at 10.0 Myr. The units of the color bars in both panels are <math display="inline"><semantics> <mrow> <mn>1.178</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>8</mn> </mrow> </msup> </mrow> </semantics></math> erg <math display="inline"><semantics> <mrow> <msup> <mi>cm</mi> <mrow> <mo>−</mo> <mn>3</mn> </mrow> </msup> </mrow> </semantics></math>.</p> "> </a> </div> <a class="button button--color-inversed" href="/2218-1997/10/11/424/notes">Versions Notes</a> </div> </div> <div class="responsive-moving-container small hidden" data-id="article-counters" style="margin-top: 15px;"></div> <div class="html-dynamic"> <section> <div class="art-abstract art-abstract-new in-tab hypothesis_container"> <p> <div><section class="html-abstract" id="html-abstract"> <h2 id="html-abstract-title">Abstract</h2><b>:</b> <div class="html-p">Two enigmatic gamma-ray features in the galactic central region, known as Fermi Bubbles (FBs), were found from Fermi-LAT data. An energy release, (e.g., by tidal disruption events in the Galactic Center, GC), generates a cavity with a shock that expands into the local ambient medium of the galactic halo. A decade or so ago, a phenomenological model of the FBs was suggested as a result of routine star disruptions by the supermassive black hole in the GC which might provide enough energy for large-scale structures, like the FBs. In 2020, analytical and numerical models of the FBs as a process of routine tidal disruption of stars near the GC were developed; these disruption events can provide enough cumulative energy to form and maintain large-scale structures like the FBs. The disruption events are expected to be <math display="inline"><semantics> <mrow> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>4</mn> </mrow> </msup> <mo>&sim;</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>5</mn> </mrow> </msup> <msup> <mi>yr</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </semantics></math>, providing an average power of energy release from the GC into the halo of <math display="inline"><semantics> <mrow> <mover accent="true"> <mi mathvariant="script">E</mi> <mo>˙</mo> </mover> <mo>&sim;</mo> <mn>3</mn> <mo>×</mo> <msup> <mn>10</mn> <mn>41</mn> </msup> </mrow> </semantics></math> erg <math display="inline"><semantics> <mrow> <msup> <mi mathvariant="normal">s</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </semantics></math>, which is needed to support the FBs. Analysis of the evolution of superbubbles in exponentially stratified disks concluded that the FB envelope would be destroyed by the Rayleigh–Taylor (RT) instabilities at late stages. The shell is composed of swept-up gas of the bubble, whose thickness is much thinner in comparison to the size of the envelope. We assume that hydrodynamic turbulence is excited in the FB envelope by the RT instability. In this case, the universal energy spectrum of turbulence may be developed in the inertial range of wavenumbers of fluctuations (the Kolmogorov–Obukhov spectrum). From our model we suppose the power of the FBs is transformed partly into the energy of hydrodynamic turbulence in the envelope. If so, hydrodynamic turbulence may generate MHD fluctuations, which accelerate cosmic rays there and generate gamma-ray and radio emission from the FBs. We hope that this model may interpret the observed nonthermal emission from the bubbles.</div> </section> <div id="html-keywords"> <div class="html-gwd-group"><div id="html-keywords-title">Keywords: </div><a href="/search?q=galactic+center">galactic center</a>; <a href="/search?q=Fermi+bubbles">Fermi bubbles</a>; <a href="/search?q=central+black+hole">central black hole</a>; <a href="/search?q=star+disruptions">star disruptions</a>; <a href="/search?q=MHD+turbulence">MHD turbulence</a>; <a href="/search?q=cosmic+rays">cosmic rays</a></div> <div> </div> </div> </div> </p> </div> </section> </div> <div class="hypothesis_container"> <ul class="menu html-nav" data-prev-node="#html-quick-links-title"> </ul> <div class="html-body"> <section id='sec1-universe-10-00424' type='intro'><h2 data-nested='1'> 1. Introduction: Sources of the Fermi Bubbles</h2><div class='html-p'>In this article, we present our interpretation of the origin of the Fermi Bubbles (FBs). The discussion includes the energy release in the Galactic Center (GC) to the hydrodynamic envelope in the galactic halo, the excitation of MHD turbulence that accelerates cosmic rays (CRs) in the halo, the processes of nonthermal emissions which are observed in X-ray, gamma-ray, and radio ranges, and the high-energy CRs escaping from the FBs to the galactic disk. We present a puzzle of the FB picture, where many fragments are still missing in the mosaic. Many of them are interpreted but not completely understood yet. The goal of the article is to find a way to find a proper solution to this problem.</div><div class='html-p'>The origin of the energy release in the FBs in the GC is still an open question. This kiloparsec-scale structure was interpreted as a manifestation of past activity of the central supermassive black hole (SMBH) Sgr A* in the GC (see <a href="#universe-10-00424-f001" class="html-fig">Figure 1</a>). Observations in the GC showed structures above and below it of gamma-rays, microwaves, and X-rays. The eROSITA [<a href="#B1-universe-10-00424" class="html-bibr">1</a>] found giant bubbles in the X-ray range 0.1∼2.4 keV, extending approximately 14 kiloparsecs above and below the GC. The estimated energy of the bubbles is around <math display='inline'><semantics> <msup> <mn>10</mn> <mn>56</mn> </msup> </semantics></math> erg. The total luminosity in X-rays is about <math display='inline'><semantics> <msup> <mn>10</mn> <mn>39</mn> </msup> </semantics></math> erg <math display='inline'><semantics> <mrow> <msup> <mi mathvariant="normal">s</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </semantics></math>, which could be the result of past activity in the GC. The temperature of the envelope is about <math display='inline'><semantics> <mrow> <mn>0.3</mn> </mrow> </semantics></math> keV, the velocity of the shock is about 340 km <math display='inline'><semantics> <mrow> <msup> <mi mathvariant="normal">s</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </semantics></math> or of Mach number <math display='inline'><semantics> <mrow> <mo>≈</mo> <mn>1.5</mn> </mrow> </semantics></math>, and the energy-release rate of the gas envelope is roughly <math display='inline'><semantics> <msup> <mn>10</mn> <mn>41</mn> </msup> </semantics></math>∼<math display='inline'><semantics> <msup> <mn>10</mn> <mn>42</mn> </msup> </semantics></math> erg <math display='inline'><semantics> <mrow> <msup> <mi mathvariant="normal">s</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </semantics></math>.</div><div class='html-p'>The inner radius of the X-ray shell (about 7 kpc) coincides spatially with the region of GeV gamma rays in the range 1∼100 GeV with the luminosity of <math display='inline'><semantics> <mrow> <msub> <mi>F</mi> <mi>γ</mi> </msub> <mo>≈</mo> <mn>4</mn> <mo>×</mo> <msup> <mn>10</mn> <mn>37</mn> </msup> </mrow> </semantics></math> erg <math display='inline'><semantics> <msup> <mi>s</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </semantics></math> [<a href="#B2-universe-10-00424" class="html-bibr">2</a>]. The bubble structure in the GC was also revealed in the range of microwaves which coincides nicely with that of gamma rays [<a href="#B3-universe-10-00424" class="html-bibr">3</a>]. The flux is in the range 23∼61 GHz, and the luminosity is <math display='inline'><semantics> <mrow> <msub> <mo>Φ</mo> <mi>ν</mi> </msub> <mo>≈</mo> <mn>1</mn> </mrow> </semantics></math>∼<math display='inline'><semantics> <mrow> <mn>5</mn> <mo>×</mo> <msup> <mn>10</mn> <mn>36</mn> </msup> </mrow> </semantics></math> erg <math display='inline'><semantics> <mrow> <msup> <mi mathvariant="normal">s</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </semantics></math>.</div><div class='html-p'>Similar giant structures near the GC were found earlier in the radio range of hundreds of MHz (the North Polar Spur; see [<a href="#B4-universe-10-00424" class="html-bibr">4</a>]) and in 1.5 keV X-ray emission (see [<a href="#B5-universe-10-00424" class="html-bibr">5</a>]). These structures were postulated as bipolar supershells which were produced by starbursts. A shock front was supposed to reach a radius of 10 kpc in the polar regions, which could be consistent with the GC explosions. This model required an energy release of about <math display='inline'><semantics> <msup> <mn>10</mn> <mn>55</mn> </msup> </semantics></math> erg at the GC, and periodic activity on a time scale of 10∼15 Myr.</div><div class='html-p'>Mou et al. [<a href="#B6-universe-10-00424" class="html-bibr">6</a>] suggested that the nature of the North Polar Spur in the GC agreed with the eROSITA bubble of an age of about 20 Myr [<a href="#B1-universe-10-00424" class="html-bibr">1</a>].</div><div class='html-p'>Bubbles have also been discovered in other galaxies (see review by [<a href="#B7-universe-10-00424" class="html-bibr">7</a>]). <a href="#universe-10-00424-f002" class="html-fig">Figure 2</a> shows superbubbles from the galaxy NGC 3079.</div><div class='html-p'>The total energy needed to generate large galactic outflows is assumed to be in the range of up to about <math display='inline'><semantics> <msup> <mn>10</mn> <mn>56</mn> </msup> </semantics></math> erg. This energy release in the GC may be compelling evidence for a huge energetic explosion having occurred in the GC a few (2∼8) million years ago (see, e.g., [<a href="#B5-universe-10-00424" class="html-bibr">5</a>] and references therein). For examples, Nayakshin & Zubovas [<a href="#B8-universe-10-00424" class="html-bibr">8</a>] assumed a capture of a giant molecular cloud of mass ∼<math display='inline'><semantics> <mrow> <msup> <mn>10</mn> <mn>5</mn> </msup> <msub> <mi>M</mi> <mo>⊙</mo> </msub> </mrow> </semantics></math> in the GC about one Myr ago; and Yang et al. [<a href="#B9-universe-10-00424" class="html-bibr">9</a>], Yang et al. [<a href="#B10-universe-10-00424" class="html-bibr">10</a>] suggested a model of FBs as a result of past activity at the GC.</div><div class='html-p'>Alternative models of the bubbles were suggested by, e.g., Cheng et al. [<a href="#B11-universe-10-00424" class="html-bibr">11</a>], Cheng et al. [<a href="#B12-universe-10-00424" class="html-bibr">12</a>], Cheng et al. [<a href="#B13-universe-10-00424" class="html-bibr">13</a>], Zubovas & Nayakshin [<a href="#B14-universe-10-00424" class="html-bibr">14</a>], Mertsch & Sarkar [<a href="#B15-universe-10-00424" class="html-bibr">15</a>], Ko et al. [<a href="#B16-universe-10-00424" class="html-bibr">16</a>], etc. They suggested that the source of energy of the bubbles is sporadic energy releases in the GC by stellar tidal disruption events (TDEs) near the central SMBH (see also other suggestions such as active star formation near the GC, e.g., [<a href="#B17-universe-10-00424" class="html-bibr">17</a>]). The motion of nearby stars orbiting around Sgr A* (the GC) has been observed for more than two decades [<a href="#B18-universe-10-00424" class="html-bibr">18</a>,<a href="#B19-universe-10-00424" class="html-bibr">19</a>,<a href="#B20-universe-10-00424" class="html-bibr">20</a>]. Analysis of the motions gave an estimate of about <math display='inline'><semantics> <mrow> <mn>4.4</mn> <mo>×</mo> <msup> <mn>10</mn> <mn>6</mn> </msup> <mspace width="4pt"/> <msub> <mi>M</mi> <mo>⊙</mo> </msub> </mrow> </semantics></math> for the central SMBH. For an illustration of orbits of stars around Sgr A*, see Figure 16 of Gillessen et al. [<a href="#B20-universe-10-00424" class="html-bibr">20</a>]. For the motion of stars orbiting Sgr A*, the reader is referred to <a href='https://www.eso.org/public/videos/eso1825e' target='_blank' rel="noopener noreferrer">https://www.eso.org/public/videos/eso1825e</a> (accessed on 10 September 2024) (or <a href='https://www.youtube.com/watch?v=TF8THY5spmo' target='_blank' rel="noopener noreferrer">https://www.youtube.com/watch?v=TF8THY5spmo</a> (accessed on 10 September 2024)); and for animation of stellar orbits around Sgr A* to <a href='https://www.eso.org/public/videos/eso1825f' target='_blank' rel="noopener noreferrer">https://www.eso.org/public/videos/eso1825f</a> (accessed on 10 September 2024) (or <a href='https://www.youtube.com/watch?v=wyuj7-XE8RE' target='_blank' rel="noopener noreferrer">https://www.youtube.com/watch?v=wyuj7-XE8RE</a> (accessed on 10 September 2024)), and <a href='https://www.youtube.com/watch?v=tMax0KgyZZU' target='_blank' rel="noopener noreferrer">https://www.youtube.com/watch?v=tMax0KgyZZU</a> (accessed on 10 September 2024).</div><div class='html-p'>A TDE occurs when a star is coming too close to an SMBH (closer than the tidal radius). The classic picture is that the star is disrupted by the tidal force, and after half of the stellar debris is in unbound orbits and the other half in bound orbits, it falls back towards the black hole [<a href="#B21-universe-10-00424" class="html-bibr">21</a>]. The problem is how and how much energy is released into the host galaxy. For instance, how much energy is carried away by the unbound debris, and how much binding energy is released by the bound debris, say through accretion. Theoretically, it is possible to have <math display='inline'><semantics> <mrow> <mn>5</mn> <mo>%</mo> </mrow> </semantics></math> of the rest mass energy of the star being released (for a solar type star this is about <math display='inline'><semantics> <msup> <mn>10</mn> <mn>53</mn> </msup> </semantics></math> erg). However, the results from observations are mixed, from several <math display='inline'><semantics> <msup> <mn>10</mn> <mn>51</mn> </msup> </semantics></math> to <math display='inline'><semantics> <msup> <mn>10</mn> <mn>53</mn> </msup> </semantics></math> erg, It is an active area of research to study TDEs from different perspectives (to name a few, e.g., [<a href="#B22-universe-10-00424" class="html-bibr">22</a>,<a href="#B23-universe-10-00424" class="html-bibr">23</a>,<a href="#B24-universe-10-00424" class="html-bibr">24</a>,<a href="#B25-universe-10-00424" class="html-bibr">25</a>,<a href="#B26-universe-10-00424" class="html-bibr">26</a>,<a href="#B27-universe-10-00424" class="html-bibr">27</a>,<a href="#B28-universe-10-00424" class="html-bibr">28</a>,<a href="#B29-universe-10-00424" class="html-bibr">29</a>,<a href="#B30-universe-10-00424" class="html-bibr">30</a>,<a href="#B31-universe-10-00424" class="html-bibr">31</a>,<a href="#B32-universe-10-00424" class="html-bibr">32</a>]). The reader is also referred to reviews like Dai et al. [<a href="#B33-universe-10-00424" class="html-bibr">33</a>] and Gezari [<a href="#B34-universe-10-00424" class="html-bibr">34</a>]. Another issue is the rate of TDEs at the center of a galaxy with an SMBH. It is conceivable that the rate depends on the type of galaxy and the environment near the black hole. A typical estimation is roughly <math display='inline'><semantics> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>4</mn> </mrow> </msup> </semantics></math>∼<math display='inline'><semantics> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>5</mn> </mrow> </msup> </semantics></math> <math display='inline'><semantics> <mrow> <msup> <mi>yr</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </semantics></math> (e.g., [<a href="#B35-universe-10-00424" class="html-bibr">35</a>,<a href="#B36-universe-10-00424" class="html-bibr">36</a>,<a href="#B37-universe-10-00424" class="html-bibr">37</a>]).</div><div class='html-p'>Ko et al. [<a href="#B16-universe-10-00424" class="html-bibr">16</a>] adopted the TDE model of Dai et al. [<a href="#B23-universe-10-00424" class="html-bibr">23</a>] and obtained the outflow energy of an event as about <math display='inline'><semantics> <msup> <mn>10</mn> <mn>52</mn> </msup> </semantics></math>∼<math display='inline'><semantics> <msup> <mn>10</mn> <mn>53</mn> </msup> </semantics></math> erg (the outflow velocity is about <math display='inline'><semantics> <mrow> <mn>0.1</mn> </mrow> </semantics></math>∼<math display='inline'><semantics> <mrow> <mn>0.3</mn> </mrow> </semantics></math> the speed of light). Together with an event rate of about <math display='inline'><semantics> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>4</mn> </mrow> </msup> </semantics></math> <math display='inline'><semantics> <mrow> <msup> <mi>yr</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </semantics></math>, this provides an average power of about <math display='inline'><semantics> <mrow> <mover accent="true"> <mi mathvariant="script">E</mi> <mo>˙</mo> </mover> <mo>∼</mo> <mn>3</mn> <mo>×</mo> <msup> <mn>10</mn> <mn>41</mn> </msup> </mrow> </semantics></math> erg <math display='inline'><semantics> <mrow> <msup> <mi mathvariant="normal">s</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </semantics></math>, which would be sufficient to power the FBs. Miller & Bregman [<a href="#B38-universe-10-00424" class="html-bibr">38</a>] inferred a bubble expansion rate of 490 km <math display='inline'><semantics> <mrow> <msup> <mi mathvariant="normal">s</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </semantics></math>, an age of 4.3 Myr, and a luminosity <math display='inline'><semantics> <mrow> <mn>2.3</mn> <mo>×</mo> <msup> <mn>10</mn> <mn>42</mn> </msup> </mrow> </semantics></math> erg <math display='inline'><semantics> <mrow> <msup> <mi mathvariant="normal">s</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </semantics></math> (see also the review of FBs in [<a href="#B7-universe-10-00424" class="html-bibr">7</a>]).</div><div class='html-p'>Recently, evidence of the energy release at Sgr A* was interpreted as the result of the latest stellar disruption. Two elongated chimneys of about 150 pc near the GC were found in the X-ray [<a href="#B39-universe-10-00424" class="html-bibr">39</a>,<a href="#B40-universe-10-00424" class="html-bibr">40</a>] and radio [<a href="#B41-universe-10-00424" class="html-bibr">41</a>] ranges. In both cases, the total energy of the chimneys was estimated to be about <math display='inline'><semantics> <msup> <mn>10</mn> <mn>53</mn> </msup> </semantics></math> erg or below, which might be a result of the latest TDE by the central SMBH.</div><div class='html-p'>Similar processes of stellar disruption at the galactic centers of other galaxies have been observed. For example, X-ray transient Swift J164449.3+573451 (also known as GRB 110328A) was detected by Swift in the direction of the constellation Draco with a peak luminosity <math display='inline'><semantics> <msup> <mn>10</mn> <mn>48</mn> </msup> </semantics></math> erg <math display='inline'><semantics> <mrow> <msup> <mi mathvariant="normal">s</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </semantics></math> (see <a href="#universe-10-00424-f003" class="html-fig">Figure 3</a>). Observations showed that the transient originated from the center of a galaxy at cosmological distances involving an SMBH in the galaxy nucleus. It was concluded that Swift J164449.3+573451 most likely originated from the central SMBH, and the X-ray and radio emissions were interpreted as a result of stellar capture by the black hole (see, e.g., [<a href="#B22-universe-10-00424" class="html-bibr">22</a>,<a href="#B32-universe-10-00424" class="html-bibr">32</a>,<a href="#B42-universe-10-00424" class="html-bibr">42</a>]).</div></section><section id='sec2-universe-10-00424' type=''><h2 data-nested='1'> 2. Structure of the Fermi Bubbles</h2><div class='html-p'>A sudden (sporadic) energy release by a TDE in the GC creates a cavity with a shock which expands into the surrounding nonuniform medium of the halo. For example, the gas distribution in the halo above (and below) the galactic plane decays exponentially with a scale height <math display='inline'><semantics> <mrow> <mi>H</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math> kpc and the density at the plane is <math display='inline'><semantics> <mrow> <msub> <mi>n</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>4</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>3</mn> </mrow> </msup> <msup> <mi>cm</mi> <mrow> <mo>−</mo> <mn>3</mn> </mrow> </msup> </mrow> </semantics></math> (see, e.g., [<a href="#B44-universe-10-00424" class="html-bibr">44</a>]). The exponential gas distribution is <div class='html-disp-formula-info' id='FD1-universe-10-00424'> <div class='f'> <math display='block'><semantics> <mrow> <mi mathvariant="script">R</mi> <mrow> <mo>(</mo> <mi>z</mi> <mo>)</mo> </mrow> <mo>=</mo> <mo form="prefix">exp</mo> <mfenced separators="" open="(" close=")"> <mo>−</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mi>z</mi> <mi>H</mi> </mfrac> </mstyle> </mfenced> <mspace width="0.166667em"/> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(1)</label> </div> </div> where <math display='inline'><semantics> <mrow> <mi mathvariant="script">R</mi> <mrow> <mo>(</mo> <mi>z</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>n</mi> <mrow> <mo>(</mo> <mi>z</mi> <mo>)</mo> </mrow> <mo>/</mo> <msub> <mi>n</mi> <mn>0</mn> </msub> </mrow> </semantics></math>. Alternatively, Miller & Bregman [<a href="#B38-universe-10-00424" class="html-bibr">38</a>] suggested a so-called <math display='inline'><semantics> <mi>β</mi> </semantics></math>-model of the gas density profile in the halo from the intensity of absorption lines:<div class='html-disp-formula-info' id='FD2-universe-10-00424'> <div class='f'> <math display='block'><semantics> <mrow> <mi mathvariant="script">R</mi> <mrow> <mo>(</mo> <mi>z</mi> <mo>)</mo> </mrow> <mo>=</mo> <msup> <mfenced separators="" open="(" close=")"> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mi>z</mi> <msub> <mi>z</mi> <mi>c</mi> </msub> </mfrac> </mstyle> </mfenced> <mrow> <mo>−</mo> <mn>3</mn> <mi>β</mi> </mrow> </msup> <mspace width="0.166667em"/> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(2)</label> </div> </div> where <math display='inline'><semantics> <mrow> <msub> <mi>z</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>0.26</mn> </mrow> </semantics></math> kpc and <math display='inline'><semantics> <mrow> <msub> <mi>n</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>0.5</mn> <msup> <mi>cm</mi> <mrow> <mo>−</mo> <mn>3</mn> </mrow> </msup> </mrow> </semantics></math>.</div><div class='html-p'>The formalism of envelope propagation was developed as a solution for a strong explosion (see [<a href="#B45-universe-10-00424" class="html-bibr">45</a>]), and was elaborated by Kompaneets [<a href="#B46-universe-10-00424" class="html-bibr">46</a>] for explosion in a nonuniform atmosphere. The reader is referred to the review of Bisnovatyi-Kogan & Silich [<a href="#B47-universe-10-00424" class="html-bibr">47</a>] and the monograph of Zel’dovich & Raizer [<a href="#B48-universe-10-00424" class="html-bibr">48</a>]. This model was derived for a thermonuclear explosion in the terrestrial atmosphere. <a href="#universe-10-00424-f004" class="html-fig">Figure 4</a> shows an example of a thermonuclear explosion test in the terrestrial atmosphere.</div><div class='html-p'>Kahn [<a href="#B49-universe-10-00424" class="html-bibr">49</a>], Baumgartner & Breitschwerdt [<a href="#B50-universe-10-00424" class="html-bibr">50</a>], Ko et al. [<a href="#B16-universe-10-00424" class="html-bibr">16</a>], and Schulreich & Breitschwerdt [<a href="#B51-universe-10-00424" class="html-bibr">51</a>] developed analytical solutions of a hydrodynamic model for shock wave propagation in nonuniform atmospheres or halos for different energy input rates for single and successive explosions. The shock envelope generated has a double-bubble structure in the halo (see <a href="#universe-10-00424-f005" class="html-fig">Figure 5</a>).</div><div class='html-p'>Following the Kompaneets formalism (see details in [<a href="#B47-universe-10-00424" class="html-bibr">47</a>]), the shock front is described as <div class='html-disp-formula-info' id='FD3-universe-10-00424'> <div class='f'> <math display='block'><semantics> <mrow> <msup> <mfenced separators="" open="(" close=")"> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mo>∂</mo> <mi>r</mi> </mrow> <mrow> <mo>∂</mo> <mi>y</mi> </mrow> </mfrac> </mstyle> </mfenced> <mn>2</mn> </msup> <mo>−</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mn>1</mn> <mrow> <mi mathvariant="script">R</mi> <mo>(</mo> <mi>z</mi> <mo>)</mo> </mrow> </mfrac> </mstyle> <mfenced separators="" open="[" close="]"> <msup> <mfenced separators="" open="(" close=")"> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mo>∂</mo> <mi>r</mi> </mrow> <mrow> <mo>∂</mo> <mi>z</mi> </mrow> </mfrac> </mstyle> </mfenced> <mn>2</mn> </msup> <mo>+</mo> <mn>1</mn> </mfenced> <mo>=</mo> <mn>0</mn> <mspace width="0.166667em"/> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(3)</label> </div> </div> where <math display='inline'><semantics> <mrow> <mi>y</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </semantics></math> is a transformed time (in units of length), <div class='html-disp-formula-info' id='FD4-universe-10-00424'> <div class='f'> <math display='block'><semantics> <mrow> <mi>y</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mn>0</mn> <mi>t</mi> </msubsup> <msqrt> <mrow> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mfenced separators="" open="(" close=")"> <msubsup> <mi>γ</mi> <mrow> <mi mathvariant="normal">g</mi> </mrow> <mn>2</mn> </msubsup> <mo>−</mo> <mn>1</mn> </mfenced> <mn>2</mn> </mfrac> </mstyle> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mn>2</mn> <mi mathvariant="script">E</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mrow> <mn>3</mn> <msub> <mi>ρ</mi> <mn>0</mn> </msub> <mi>V</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </mfrac> </mstyle> <mspace width="0.166667em"/> </mrow> </msqrt> <mspace width="0.166667em"/> <mi>d</mi> <mi>t</mi> <mspace width="0.166667em"/> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(4)</label> </div> </div> and the bubble volume <math display='inline'><semantics> <mrow> <mi>V</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </semantics></math> is <div class='html-disp-formula-info' id='FD5-universe-10-00424'> <div class='f'> <math display='block'><semantics> <mrow> <mi>V</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>π</mi> <msubsup> <mo>∫</mo> <mn>0</mn> <msub> <mi>z</mi> <mi>u</mi> </msub> </msubsup> <msup> <mi>r</mi> <mn>2</mn> </msup> <mrow> <mo>(</mo> <mi>z</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> <mi>d</mi> <mi>z</mi> <mspace width="0.166667em"/> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(5)</label> </div> </div> with <math display='inline'><semantics> <mrow> <mi>r</mi> <mo>(</mo> <mi>z</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </semantics></math> or <math display='inline'><semantics> <mrow> <mi>r</mi> <mo>(</mo> <mi>z</mi> <mo>,</mo> <mi>y</mi> <mo>)</mo> </mrow> </semantics></math> as the bubble radius at the altitude <span class='html-italic'>z</span>, <math display='inline'><semantics> <mrow> <msub> <mi>ρ</mi> <mn>0</mn> </msub> <mo>=</mo> <msub> <mi>n</mi> <mn>0</mn> </msub> <msub> <mi>m</mi> <mi>p</mi> </msub> </mrow> </semantics></math> the mass density corresponding to <math display='inline'><semantics> <msub> <mi>n</mi> <mn>0</mn> </msub> </semantics></math> (the number density at the base <math display='inline'><semantics> <mrow> <mi>z</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>), <math display='inline'><semantics> <mi mathvariant="script">E</mi> </semantics></math> the energy released by the central source into the bubble, and <math display='inline'><semantics> <msub> <mi>γ</mi> <mi mathvariant="normal">g</mi> </msub> </semantics></math> the adiabatic index of the gas.</div><div class='html-p'>For the exponential gas distribution given by Equation (<a href="#FD1-universe-10-00424" class="html-disp-formula">1</a>), the top of the bubble <math display='inline'><semantics> <msub> <mi>z</mi> <mi>u</mi> </msub> </semantics></math> is a function of time <span class='html-italic'>t</span>:<div class='html-disp-formula-info' id='FD6-universe-10-00424'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>z</mi> <mi>u</mi> </msub> <mo>=</mo> <mo>−</mo> <mn>2</mn> <mi>H</mi> <mspace width="0.166667em"/> <mo form="prefix">ln</mo> <mfenced separators="" open="(" close=")"> <mn>1</mn> <mo>−</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mi>y</mi> <mrow> <mn>2</mn> <mi>H</mi> </mrow> </mfrac> </mstyle> </mfenced> <mspace width="0.166667em"/> <mo>.</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(6)</label> </div> </div></div><div class='html-p'>In this model of Baumgartner & Breitschwerdt [<a href="#B50-universe-10-00424" class="html-bibr">50</a>] and Schulreich & Breitschwerdt [<a href="#B51-universe-10-00424" class="html-bibr">51</a>], the velocity at the top of the bubble <math display='inline'><semantics> <msub> <mi>v</mi> <mi>u</mi> </msub> </semantics></math> for the total energy release of <math display='inline'><semantics> <mi mathvariant="script">E</mi> </semantics></math> is <div class='html-disp-formula-info' id='FD7-universe-10-00424'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>v</mi> <mi>u</mi> </msub> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mi>d</mi> <msub> <mi>z</mi> <mi>u</mi> </msub> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mstyle> <mo>=</mo> <mo form="prefix">exp</mo> <mfenced separators="" open="(" close=")"> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <msub> <mi>z</mi> <mi>u</mi> </msub> <mrow> <mn>2</mn> <mi>H</mi> </mrow> </mfrac> </mstyle> </mfenced> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mi>d</mi> <mi>y</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mstyle> <mspace width="0.166667em"/> <mo>.</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(7)</label> </div> </div></div><div class='html-p'>In the early phase, when <math display='inline'><semantics> <mrow> <msub> <mi>z</mi> <mi>u</mi> </msub> <mo>≲</mo> <mi>H</mi> </mrow> </semantics></math>, the expanding cavity can be described by the Sedov solution [<a href="#B45-universe-10-00424" class="html-bibr">45</a>], where the gas density is almost uniform and the velocity of the shock envelope decreases with time. When <math display='inline'><semantics> <mrow> <msub> <mi>z</mi> <mi>u</mi> </msub> <mo>></mo> <mi>H</mi> </mrow> </semantics></math>, the shock propagates in the exponential halo with subsequent acceleration (see [<a href="#B50-universe-10-00424" class="html-bibr">50</a>]).</div><div class='html-p'>The propagation of the shock envelope is derived under the strong shock assumption. In reality, if the velocity of the envelope is below the sound speed <math display='inline'><semantics> <msub> <mi>c</mi> <mi>s</mi> </msub> </semantics></math> of the halo gas, then the shock or the envelope will decay and be absorbed in the halo. On the other hand, if this velocity is higher than the sound speed, the shock will be able to penetrate into the halo and transfer the energy from the initial central source into the exponential halo. The velocity of the top of the bubble is the fastest; Baumgartner & Breitschwerdt [<a href="#B50-universe-10-00424" class="html-bibr">50</a>] defined a condition of shock penetration into the exponential halo: <math display='inline'><semantics> <mrow> <msub> <mi>v</mi> <mi>u</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>y</mi> <mi>acc</mi> </msub> <mo>)</mo> </mrow> <mo>></mo> <mn>3</mn> <msub> <mi>c</mi> <mi>s</mi> </msub> </mrow> </semantics></math>, where <math display='inline'><semantics> <mrow> <msub> <mi>v</mi> <mi>u</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>y</mi> <mi>acc</mi> </msub> <mo>)</mo> </mrow> </mrow> </semantics></math> is the minimum of <math display='inline'><semantics> <msub> <mi>v</mi> <mi>u</mi> </msub> </semantics></math>, which occurs at <math display='inline'><semantics> <mrow> <mi>y</mi> <mo>=</mo> <msub> <mi>y</mi> <mi>acc</mi> </msub> </mrow> </semantics></math>, i.e., <math display='inline'><semantics> <mrow> <msub> <mover accent="true"> <mi>v</mi> <mo>˙</mo> </mover> <mi>u</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>y</mi> <mi>acc</mi> </msub> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mover accent="true"> <mi>z</mi> <mo>¨</mo> </mover> <mi>u</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>y</mi> <mi>acc</mi> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>. The acceleration at the top of the bubble is (see [<a href="#B50-universe-10-00424" class="html-bibr">50</a>]) <div class='html-disp-formula-info' id='FD8-universe-10-00424'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mover accent="true"> <mi>z</mi> <mo>¨</mo> </mover> <mi>u</mi> </msub> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mi>H</mi> <msubsup> <mi>t</mi> <mi>SN</mi> <mn>2</mn> </msubsup> </mfrac> </mstyle> <mspace width="0.166667em"/> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mfenced separators="" open="(" close=")"> <msubsup> <mi>γ</mi> <mrow> <mi mathvariant="normal">g</mi> </mrow> <mn>2</mn> </msubsup> <mo>−</mo> <mn>1</mn> </mfenced> <mrow> <mn>4</mn> <mrow> <mo>(</mo> <mn>1</mn> <mo>−</mo> <mover accent="true"> <mi>y</mi> <mo>˜</mo> </mover> <mo>/</mo> <mn>2</mn> <mo>)</mo> </mrow> <mover accent="true"> <mi>V</mi> <mo>˜</mo> </mover> </mrow> </mfrac> </mstyle> <mfenced separators="" open="[" close="]"> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mn>1</mn> <mfenced separators="" open="(" close=")"> <mn>1</mn> <mo>−</mo> <mover accent="true"> <mi>y</mi> <mo>˜</mo> </mover> <mo>/</mo> <mn>2</mn> </mfenced> </mfrac> </mstyle> <mo>−</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mn>1</mn> <mover accent="true"> <mi>V</mi> <mo>˜</mo> </mover> </mfrac> </mstyle> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mi>d</mi> <mover accent="true"> <mi>V</mi> <mo>˜</mo> </mover> </mrow> <mrow> <mi>d</mi> <mover accent="true"> <mi>y</mi> <mo>˜</mo> </mover> </mrow> </mfrac> </mstyle> </mfenced> <mspace width="0.166667em"/> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(8)</label> </div> </div> where <div class='html-disp-formula-info' id='FD9-universe-10-00424'> <div class='f'> <math display='block'><semantics> <mrow> <mover accent="true"> <mi>y</mi> <mo>˜</mo> </mover> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mi>y</mi> <mi>H</mi> </mfrac> </mstyle> <mspace width="0.166667em"/> <mo>,</mo> <mspace width="1.em"/> <mover accent="true"> <mi>V</mi> <mo>˜</mo> </mover> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mi>V</mi> <msup> <mi>H</mi> <mn>3</mn> </msup> </mfrac> </mstyle> <mspace width="0.166667em"/> <mo>,</mo> <mspace width="1.em"/> <msub> <mi>t</mi> <mi>SN</mi> </msub> <mo>=</mo> <msqrt> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mn>3</mn> <msub> <mi>ρ</mi> <mn>0</mn> </msub> <msup> <mi>H</mi> <mn>5</mn> </msup> </mrow> <mrow> <mn>2</mn> <msub> <mi mathvariant="script">E</mi> <mi>SN</mi> </msub> </mrow> </mfrac> </mstyle> </msqrt> <mspace width="0.166667em"/> <mo>.</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(9)</label> </div> </div></div><div class='html-p'>As an example, in the left panel of <a href="#universe-10-00424-f006" class="html-fig">Figure 6</a> we present the development of the shock velocity at the top of the bubble from a single energy explosion <math display='inline'><semantics> <mi mathvariant="script">E</mi> </semantics></math>. The energy of the explosion occupies more and more volume of the exponential atmosphere, finally approaching infinity in finite time (provided that the velocity is always larger than <math display='inline'><semantics> <mrow> <mn>3</mn> <msub> <mi>c</mi> <mi>s</mi> </msub> </mrow> </semantics></math>).</div><div class='html-p'>For the parameters in the GC, a single star disruption event provides no more than <math display='inline'><semantics> <msup> <mn>10</mn> <mn>52</mn> </msup> </semantics></math>∼<math display='inline'><semantics> <msup> <mn>10</mn> <mn>53</mn> </msup> </semantics></math> erg (see [<a href="#B23-universe-10-00424" class="html-bibr">23</a>,<a href="#B31-universe-10-00424" class="html-bibr">31</a>,<a href="#B33-universe-10-00424" class="html-bibr">33</a>]). This is not enough energy for the FBs or similar structures. An unusually huge single energy release in the past, say exceeding <math display='inline'><semantics> <mrow> <mi mathvariant="script">E</mi> <mo>></mo> <msup> <mn>10</mn> <mn>54</mn> </msup> </mrow> </semantics></math> erg, may explain the origin of the Fermi Bubbles.</div><div class='html-p'>Alternatively, this huge energy could be supplied by a series of many weaker disruption events with an effective power input of <math display='inline'><semantics> <mrow> <mover accent="true"> <mi mathvariant="script">E</mi> <mo>˙</mo> </mover> <mo>≥</mo> <msup> <mn>10</mn> <mn>40</mn> </msup> </mrow> </semantics></math> erg <math display='inline'><semantics> <mrow> <msup> <mi mathvariant="normal">s</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </semantics></math>; see the right panel of <a href="#universe-10-00424-f006" class="html-fig">Figure 6</a>. This may be interpreted as routine TDEs, each of which produced an energy of <math display='inline'><semantics> <msup> <mn>10</mn> <mn>52</mn> </msup> </semantics></math>∼<math display='inline'><semantics> <msup> <mn>10</mn> <mn>53</mn> </msup> </semantics></math> erg, with the average rate of stellar capture about <math display='inline'><semantics> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>5</mn> </mrow> </msup> </semantics></math>∼<math display='inline'><semantics> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>4</mn> </mrow> </msup> </semantics></math> <math display='inline'><semantics> <mrow> <msup> <mi>yr</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </semantics></math> (see [<a href="#B16-universe-10-00424" class="html-bibr">16</a>]).</div><div class='html-p'>The envelope shell is composed of the swept-up gas of the bubble, and it is much thinner in comparison to the size of the bubble. The shell thickness is defined as <div class='html-disp-formula-info' id='FD10-universe-10-00424'> <div class='f'> <math display='block'><semantics> <mrow> <mi>d</mi> <mrow> <mo>(</mo> <mi>y</mi> <mo>)</mo> </mrow> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <msub> <mi>M</mi> <mi>s</mi> </msub> <mrow> <mo>(</mo> <mi>y</mi> <mo>)</mo> </mrow> </mrow> <mrow> <mn>2</mn> <mi>π</mi> <msub> <mi>ρ</mi> <mrow> <mi>sh</mi> <mn>0</mn> </mrow> </msub> <msubsup> <mo>∫</mo> <mn>0</mn> <mrow> <msub> <mi>z</mi> <mi>u</mi> </msub> <mrow> <mo>(</mo> <mi>y</mi> <mo>)</mo> </mrow> </mrow> </msubsup> <msup> <mi>e</mi> <mrow> <mo>−</mo> <mi>z</mi> <mo>/</mo> <mi>H</mi> </mrow> </msup> <mi>r</mi> <mrow> <mo>(</mo> <mi>z</mi> <mo>,</mo> <mi>y</mi> <mo>)</mo> </mrow> <msqrt> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mfenced separators="" open="(" close=")"> <mfrac> <mrow> <mo>∂</mo> <mi>r</mi> </mrow> <mrow> <mo>∂</mo> <mi>z</mi> </mrow> </mfrac> </mfenced> <mn>2</mn> </msup> <mspace width="0.166667em"/> </mrow> </msqrt> <mi>d</mi> <mi>z</mi> </mrow> </mfrac> </mstyle> <mspace width="0.166667em"/> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(10)</label> </div> </div> where <math display='inline'><semantics> <msub> <mi>M</mi> <mi>s</mi> </msub> </semantics></math> is the total mass of the FB, <div class='html-disp-formula-info' id='FD11-universe-10-00424'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>M</mi> <mi>s</mi> </msub> <mrow> <mo>(</mo> <mi>y</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>π</mi> <msub> <mi>ρ</mi> <mn>0</mn> </msub> <msubsup> <mo>∫</mo> <mn>0</mn> <mrow> <msub> <mi>z</mi> <mi>u</mi> </msub> <mrow> <mo>(</mo> <mi>y</mi> <mo>)</mo> </mrow> </mrow> </msubsup> <msup> <mi>e</mi> <mrow> <mo>−</mo> <mi>z</mi> <mo>/</mo> <mi>H</mi> </mrow> </msup> <msup> <mi>r</mi> <mn>2</mn> </msup> <mrow> <mo>(</mo> <mi>z</mi> <mo>,</mo> <mi>y</mi> <mo>)</mo> </mrow> <mi>d</mi> <mi>z</mi> <mspace width="0.166667em"/> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(11)</label> </div> </div> and <math display='inline'><semantics> <mrow> <msub> <mi>ρ</mi> <mi>sh</mi> </msub> <mo>=</mo> <msub> <mi>ρ</mi> <mrow> <mi>sh</mi> <mn>0</mn> </mrow> </msub> <mspace width="0.166667em"/> <mo form="prefix">exp</mo> <mrow> <mo>(</mo> <mo>−</mo> <mi>z</mi> <mo>/</mo> <mi>H</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> is the density within the shell (see [<a href="#B51-universe-10-00424" class="html-bibr">51</a>]).</div><div class='html-p'><a href="#universe-10-00424-f007" class="html-fig">Figure 7</a> shows some examples of numerical simulations of the FB envelope from sporadic star disruptions or from a single huge explosion.</div></section><section id='sec3-universe-10-00424' type=''><h2 data-nested='1'> 3. Envelope Disruption by Rayleigh–Taylor Instability</h2><div class='html-p'>The interface between a denser fluid supported by a lighter fluid in a gravitational field is susceptible to Rayleigh–Taylor (RT) instability. The amplitude of an infinitesimal perturbation will grow exponentially in the early phase or the linear phase (see, e.g., [<a href="#B52-universe-10-00424" class="html-bibr">52</a>]). The growth time of the instability in the linear phase is <div class='html-disp-formula-info' id='FD12-universe-10-00424'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>τ</mi> <mi>RT</mi> </msub> <mo>=</mo> <msqrt> <mrow> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mi>λ</mi> <mrow> <mn>2</mn> <mi>π</mi> <mi>g</mi> </mrow> </mfrac> </mstyle> <mspace width="0.166667em"/> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mo>(</mo> <msub> <mi>ρ</mi> <mn>2</mn> </msub> <mo>+</mo> <msub> <mi>ρ</mi> <mn>1</mn> </msub> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <msub> <mi>ρ</mi> <mn>2</mn> </msub> <mo>−</mo> <msub> <mi>ρ</mi> <mn>1</mn> </msub> <mo>)</mo> </mrow> </mfrac> </mstyle> <mspace width="0.166667em"/> </mrow> </msqrt> <mspace width="0.166667em"/> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(12)</label> </div> </div> where <math display='inline'><semantics> <mi>λ</mi> </semantics></math> is the wavelength of the perturbation, <span class='html-italic'>g</span> is the gravitational acceleration, and <math display='inline'><semantics> <msub> <mi>ρ</mi> <mn>1</mn> </msub> </semantics></math> and <math display='inline'><semantics> <msub> <mi>ρ</mi> <mn>2</mn> </msub> </semantics></math> are the densities of the lighter and denser fluids, respectively. For an illustration of the time evolution of the RT instability, the reader is referred to Figure 4 of Schulreich & Breitschwerdt [<a href="#B51-universe-10-00424" class="html-bibr">51</a>].</div><div class='html-p'>In the case of a superbubble [<a href="#B50-universe-10-00424" class="html-bibr">50</a>,<a href="#B51-universe-10-00424" class="html-bibr">51</a>], the RT instabilities are excited between the dense shell and the hot interior when the envelope is accelerating into the exponential halo (see Equation (<a href="#FD8-universe-10-00424" class="html-disp-formula">8</a>)). Identifying the gravitational acceleration with the acceleration at the top of the bubble <math display='inline'><semantics> <mrow> <msub> <mover accent="true"> <mi>z</mi> <mo>¨</mo> </mover> <mi>u</mi> </msub> <mrow> <mo>(</mo> <mi>y</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> gives the growth time of the instability (in these coordinates):<div class='html-disp-formula-info' id='FD13-universe-10-00424'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>τ</mi> <mrow> <mi>RT</mi> <mo>,</mo> <msub> <mi>z</mi> <mi>u</mi> </msub> </mrow> </msub> <mrow> <mo>(</mo> <mi>y</mi> <mo>)</mo> </mrow> <mo>=</mo> <msqrt> <mrow> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mi>d</mi> <mo>(</mo> <mi>y</mi> <mo>)</mo> </mrow> <mrow> <mn>2</mn> <mi>π</mi> <msub> <mover accent="true"> <mi>z</mi> <mo>¨</mo> </mover> <mi>u</mi> </msub> <mrow> <mo>(</mo> <mi>y</mi> <mo>)</mo> </mrow> </mrow> </mfrac> </mstyle> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mo>[</mo> <msub> <mi>ρ</mi> <mi>sh</mi> </msub> <mrow> <mo>(</mo> <mi>y</mi> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>ρ</mi> <mi>in</mi> </msub> <mrow> <mo>(</mo> <mi>y</mi> <mo>)</mo> </mrow> <mo>]</mo> </mrow> <mrow> <mo>[</mo> <msub> <mi>ρ</mi> <mi>sh</mi> </msub> <mrow> <mo>(</mo> <mi>y</mi> <mo>)</mo> </mrow> <mo>−</mo> <msub> <mi>ρ</mi> <mi>in</mi> </msub> <mrow> <mo>(</mo> <mi>y</mi> <mo>)</mo> </mrow> <mo>]</mo> </mrow> </mfrac> </mstyle> <mspace width="0.166667em"/> </mrow> </msqrt> <mspace width="0.166667em"/> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(13)</label> </div> </div> where <span class='html-italic'>y</span> is the transformed time (see <a href="#sec2-universe-10-00424" class="html-sec">Section 2</a>). When the wavelength <math display='inline'><semantics> <mi>λ</mi> </semantics></math> of the RT fluctuations is about the envelope shell thickness <span class='html-italic'>d</span>, these instabilities may destroy the bubble; see Figure 13 of Schulreich & Breitschwerdt [<a href="#B51-universe-10-00424" class="html-bibr">51</a>].</div><div class='html-p'>The temporal evolution of the RT instability during the nonlinear regime is obtained by numerically solving the ordinary differential equation for the RT fluctuations of <math display='inline'><semantics> <mi>λ</mi> </semantics></math>, <div class='html-disp-formula-info' id='FD14-universe-10-00424'> <div class='f'> <math display='block'><semantics> <mrow> <mover accent="true"> <mi>λ</mi> <mo>˙</mo> </mover> <mrow> <mo>(</mo> <mi>y</mi> <mo>)</mo> </mrow> <mo>=</mo> <mn>2</mn> <msqrt> <mrow> <mi>α</mi> <msub> <mover accent="true"> <mi>z</mi> <mo>¨</mo> </mover> <mi>u</mi> </msub> <mrow> <mo>(</mo> <mi>y</mi> <mo>)</mo> </mrow> <mi>λ</mi> <mrow> <mo>(</mo> <mi>y</mi> <mo>)</mo> </mrow> <mspace width="0.166667em"/> </mrow> </msqrt> <mspace width="0.166667em"/> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(14)</label> </div> </div> the parameter <math display='inline'><semantics> <mi>α</mi> </semantics></math> is estimated from the initial condition <math display='inline'><semantics> <mrow> <mi>λ</mi> <mrow> <mo>(</mo> <msub> <mi>y</mi> <mn>0</mn> </msub> <mo>)</mo> </mrow> <mo>∼</mo> <mn>0.01</mn> <mi>d</mi> <mrow> <mo>(</mo> <msub> <mi>y</mi> <mn>0</mn> </msub> <mo>)</mo> </mrow> </mrow> </semantics></math> (see [<a href="#B50-universe-10-00424" class="html-bibr">50</a>]).</div></section><section id='sec4-universe-10-00424' type=''><h2 data-nested='1'> 4. Energy and Spectrum of Hydrodynamic Fluctuations</h2><div class='html-p'>From Equation (<a href="#FD14-universe-10-00424" class="html-disp-formula">14</a>), we can estimate the fraction of the total energy of the FBs, <math display='inline'><semantics> <mrow> <mover accent="true"> <mi mathvariant="script">E</mi> <mo>˙</mo> </mover> <mo>∼</mo> <mn>3</mn> <mo>×</mo> <msup> <mn>10</mn> <mn>41</mn> </msup> </mrow> </semantics></math> erg <math display='inline'><semantics> <mrow> <msup> <mi mathvariant="normal">s</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </semantics></math>, that is transformed into the hydrodynamic turbulence in the envelope excited by the RT instabilities there. From Landau & Lifshitz [<a href="#B53-universe-10-00424" class="html-bibr">53</a>], we obtain the rate of energy dissipation in the turbulent flux:<div class='html-disp-formula-info' id='FD15-universe-10-00424'> <div class='f'> <math display='block'><semantics> <mrow> <mi>ε</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <msubsup> <mi>v</mi> <mi>λ</mi> <mn>3</mn> </msubsup> <mi>λ</mi> </mfrac> </mstyle> <mspace width="0.166667em"/> <mo>.</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(15)</label> </div> </div></div><div class='html-p'>Instead of <math display='inline'><semantics> <mi>λ</mi> </semantics></math>, we can introduce a wavenumber <math display='inline'><semantics> <mrow> <mi>k</mi> <mo>=</mo> <mn>2</mn> <mi>π</mi> <mo>/</mo> <mi>λ</mi> </mrow> </semantics></math>. (the Kolmogorov–Obukhov spectrum of turbulence). Then, the kinetic energy spectrum <math display='inline'><semantics> <mrow> <mi>W</mi> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </semantics></math> of the turbulence is <div class='html-disp-formula-info' id='FD16-universe-10-00424'> <div class='f'> <math display='block'><semantics> <mrow> <mi>W</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>∼</mo> <msup> <mi>ε</mi> <mrow> <mn>2</mn> <mo>/</mo> <mn>3</mn> </mrow> </msup> <msup> <mi>k</mi> <mrow> <mo>−</mo> <mn>5</mn> <mo>/</mo> <mn>3</mn> </mrow> </msup> <mspace width="0.166667em"/> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(16)</label> </div> </div> and <div class='html-disp-formula-info' id='FD17-universe-10-00424'> <div class='f'> <math display='block'><semantics> <mrow> <msubsup> <mo>∫</mo> <mi>k</mi> <mo>∞</mo> </msubsup> <mi>W</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mi>d</mi> <mi>k</mi> <mo>∼</mo> <msubsup> <mi>v</mi> <mi>λ</mi> <mn>2</mn> </msubsup> <mspace width="0.166667em"/> <mo>.</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(17)</label> </div> </div></div><div class='html-p'>In this case, a universal energy spectrum <math display='inline'><semantics> <mrow> <mi>W</mi> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </semantics></math> is developed in the inertial range, as in Equation (<a href="#FD16-universe-10-00424" class="html-disp-formula">16</a>), i.e., the Kolmogorov–Obukhov spectrum of turbulence Landau & Lifshitz [<a href="#B53-universe-10-00424" class="html-bibr">53</a>].</div><div class='html-p'>The energy losses of RT, <math display='inline'><semantics> <mrow> <mi>ε</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </semantics></math>, in the envelope, transferred into the turbulence, is <div class='html-disp-formula-info' id='FD18-universe-10-00424'> <div class='f'> <math display='block'><semantics> <mrow> <mi>ε</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <msubsup> <mrow> <mover accent="true"> <mi>λ</mi> <mo>˙</mo> </mover> </mrow> <mn>0</mn> <mn>3</mn> </msubsup> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> <mrow> <msub> <mi>λ</mi> <mn>0</mn> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </mfrac> </mstyle> <mo>=</mo> <mn>8</mn> <msup> <mi>α</mi> <mrow> <mn>3</mn> <mo>/</mo> <mn>2</mn> </mrow> </msup> <msubsup> <mover accent="true"> <mi>z</mi> <mo>¨</mo> </mover> <mi>u</mi> <mrow> <mn>3</mn> <mo>/</mo> <mn>2</mn> </mrow> </msubsup> <msubsup> <mi>λ</mi> <mn>0</mn> <mrow> <mn>1</mn> <mo>/</mo> <mn>2</mn> </mrow> </msubsup> <mspace width="0.166667em"/> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(18)</label> </div> </div> where <math display='inline'><semantics> <mrow> <msub> <mi>λ</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>2</mn> <mi>π</mi> <mo>/</mo> <msub> <mi>k</mi> <mn>0</mn> </msub> </mrow> </semantics></math> is the pumping scale (see below).</div><div class='html-p'>The spectrum of hydrodynamic fluctuations <math display='inline'><semantics> <mrow> <mi>W</mi> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </semantics></math> has a wide inertial interval, where the energy is supplied on the initial scale of <math display='inline'><semantics> <mrow> <msub> <mi>k</mi> <mn>0</mn> </msub> <mo><</mo> <mi>k</mi> </mrow> </semantics></math>, in which only energy transfer along the spectrum is realized. In this energy range, the spectrum at <math display='inline'><semantics> <mrow> <mi>λ</mi> <mo><</mo> <msub> <mi>λ</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>2</mn> <mi>π</mi> <mo>/</mo> <msub> <mi>k</mi> <mn>0</mn> </msub> </mrow> </semantics></math> is determined only by the power of energy pumping at the scale <math display='inline'><semantics> <msub> <mi>λ</mi> <mn>0</mn> </msub> </semantics></math>.</div><div class='html-p'>The growth of this large-scale structure may be understood at each stage in terms of ring vortex pairing, mutual orbiting, and merging, followed by these processes repeating with the just-merged eddies on a larger scale. During this development, ambient material is entrained and intense smaller-scale turbulence is generated in the regions between the vortices, presumably establishing the turbulent cascade to higher wavenumbers, which is eventually dominated by viscosity on the Kolmogorov microscale, <math display='inline'><semantics> <msub> <mi>l</mi> <mi>K</mi> </msub> </semantics></math>. The small-scale of the stretched turbulence-generating regions between the vortices can be associated with the Taylor microscale, <math display='inline'><semantics> <msub> <mi>l</mi> <mi>T</mi> </msub> </semantics></math>.</div><div class='html-p'>The total fluid turbulence energy is given by <math display='inline'><semantics> <mrow> <msub> <mi>E</mi> <mi>t</mi> </msub> <mo>=</mo> <mi>ρ</mi> <msubsup> <mi>v</mi> <mi>t</mi> <mn>2</mn> </msubsup> </mrow> </semantics></math>, where <math display='inline'><semantics> <mi>ρ</mi> </semantics></math> is the fluid density and <math display='inline'><semantics> <msub> <mi>v</mi> <mi>t</mi> </msub> </semantics></math> is the root mean square of the velocity of the turbulence (or simply the velocity of the turbulence) at the largest scale <math display='inline'><semantics> <msub> <mi>λ</mi> <mn>0</mn> </msub> </semantics></math> or <math display='inline'><semantics> <msub> <mi>l</mi> <mn>0</mn> </msub> </semantics></math>. The range is taken to extend from <math display='inline'><semantics> <msub> <mi>l</mi> <mn>0</mn> </msub> </semantics></math> down to the effective damping (or Kolmogorov) scale <math display='inline'><semantics> <msub> <mi>l</mi> <mi>K</mi> </msub> </semantics></math>. The Reynolds number in this case is <div class='html-disp-formula-info' id='FD19-universe-10-00424'> <div class='f'> <math display='block'><semantics> <mrow> <mi>R</mi> <mi>e</mi> <mo>=</mo> <msup> <mfenced separators="" open="(" close=")"> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <msub> <mi>l</mi> <mn>0</mn> </msub> <msub> <mi>l</mi> <mi>K</mi> </msub> </mfrac> </mstyle> </mfenced> <mrow> <mn>4</mn> <mo>/</mo> <mn>3</mn> </mrow> </msup> <mspace width="0.166667em"/> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(19)</label> </div> </div> where the smallest scale, <math display='inline'><semantics> <msub> <mi>l</mi> <mi>K</mi> </msub> </semantics></math>, is defined by the dissipation of turbulence.</div><div class='html-p'>There is no absolute definition of the scale <math display='inline'><semantics> <msub> <mi>l</mi> <mi>T</mi> </msub> </semantics></math> of this transition from the hydrodynamic spectrum to that of MHD (upper limit of MHD turbulence scale). Following Eilek & Hendriksen [<a href="#B54-universe-10-00424" class="html-bibr">54</a>], the Taylor length, <math display='inline'><semantics> <msub> <mi>l</mi> <mi>T</mi> </msub> </semantics></math>, was estimated as <div class='html-disp-formula-info' id='FD20-universe-10-00424'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>l</mi> <mi>T</mi> </msub> <mo>∼</mo> <msub> <mi>l</mi> <mn>0</mn> </msub> <msup> <mfenced separators="" open="(" close=")"> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mn>15</mn> <mrow> <mi>R</mi> <mi>e</mi> </mrow> </mfrac> </mstyle> </mfenced> <mrow> <mn>1</mn> <mo>/</mo> <mn>2</mn> </mrow> </msup> <mspace width="0.166667em"/> <mo>.</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(20)</label> </div> </div></div><div class='html-p'>Therefore, the fluid cascade extends from the Taylor scale <math display='inline'><semantics> <msub> <mi>l</mi> <mi>T</mi> </msub> </semantics></math>, on which the transition from a large-scale ordered turbulence to smaller-scale disordered motion occurs. The cascade proceeds down to the smallest scale, <math display='inline'><semantics> <msub> <mi>l</mi> <mi>K</mi> </msub> </semantics></math>, determined by dissipation:<div class='html-disp-formula-info' id='FD21-universe-10-00424'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>l</mi> <mi>K</mi> </msub> <mo>≈</mo> <msub> <mi>l</mi> <mn>0</mn> </msub> <mspace width="0.166667em"/> <msup> <mrow> <mo>(</mo> <mi>R</mi> <mi>e</mi> <mo>)</mo> </mrow> <mrow> <mo>−</mo> <mn>3</mn> <mo>/</mo> <mn>4</mn> </mrow> </msup> <mspace width="0.166667em"/> <mo>.</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(21)</label> </div> </div></div><div class='html-p'>We adopt the hydrodynamic view that the Kolmogorov equilibrium cascade exists between <math display='inline'><semantics> <msub> <mi>l</mi> <mi>T</mi> </msub> </semantics></math> and <math display='inline'><semantics> <msub> <mi>l</mi> <mi>K</mi> </msub> </semantics></math> as our first approximation to the complex interactions expected in the regime. The relation between (the root mean square of) the turbulence velocity <math display='inline'><semantics> <mrow> <mi>v</mi> <mo>(</mo> <mi>l</mi> <mo>)</mo> </mrow> </semantics></math> and the turbulence scale <span class='html-italic'>l</span> is <div class='html-disp-formula-info' id='FD22-universe-10-00424'> <div class='f'> <math display='block'><semantics> <mrow> <mi>v</mi> <mrow> <mo>(</mo> <mi>l</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>v</mi> <mrow> <mo>(</mo> <msub> <mi>l</mi> <mi>T</mi> </msub> <mo>)</mo> </mrow> <msup> <mfenced separators="" open="(" close=")"> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mi>l</mi> <msub> <mi>l</mi> <mi>T</mi> </msub> </mfrac> </mstyle> </mfenced> <mrow> <mn>1</mn> <mo>/</mo> <mn>3</mn> </mrow> </msup> <mspace width="0.166667em"/> <mo>.</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(22)</label> </div> </div></div></section><section id='sec5-universe-10-00424' type=''><h2 data-nested='1'> 5. Particle Acceleration by Alfvén Fluctuations and the Lighthill Radiation</h2><div class='html-p'>Contrary to shock acceleration, turbulent resonant acceleration does not require strong shocks. The fluid turbulence in the interstellar medium or intercluster medium is another possible origin of particle acceleration, e.g., in galaxy clusters [<a href="#B55-universe-10-00424" class="html-bibr">55</a>], and in radio jets in turbulent mixing regions [<a href="#B56-universe-10-00424" class="html-bibr">56</a>]. Turbulent motions act as a source of waves and are presented as a hierarchy of eddies. The Lighthill mechanism acts as a direct source of energy to the MHD waves over the range of wavenumbers corresponding to the fluid turbulent spectrum. A turbulent eddy has kinetic energy, which it releases when it mixes with its surroundings. Most of this energy is returned to the ambient medium, but a small fraction is transformed into propagating waves (see, e.g., [<a href="#B56-universe-10-00424" class="html-bibr">56</a>,<a href="#B57-universe-10-00424" class="html-bibr">57</a>,<a href="#B58-universe-10-00424" class="html-bibr">58</a>,<a href="#B59-universe-10-00424" class="html-bibr">59</a>,<a href="#B60-universe-10-00424" class="html-bibr">60</a>] and others). A strong coupling between particle energy and turbulent energy spectra can be expected, and the hydrodynamic turbulence in the medium accelerates particles through wave–particle resonance. Alfvén waves are an alternative source of charged-particle acceleration via resonant interaction of MHD waves with relativistic particles.</div><div class='html-p'>In a pioneering paper, Lighthill [<a href="#B61-universe-10-00424" class="html-bibr">61</a>] developed the model of acoustic waves, which are excited by hydrodynamic turbulence in the absence of magnetic fields. This is known as the Lighthill radiation. The radiated power of the waves is roughly the energy density in the turbulent motion, <math display='inline'><semantics> <mi>ε</mi> </semantics></math>, divided by the decay time scale of waves <math display='inline'><semantics> <mi>τ</mi> </semantics></math>, and the compactness of the eddy (which is measured by the ratio of the size of the eddy, <span class='html-italic'>l</span>, to the wavelength, <math display='inline'><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mn>2</mn> <mi>π</mi> <mo>/</mo> <mi>k</mi> </mrow> </semantics></math>).</div><div class='html-p'>Kulsrud [<a href="#B62-universe-10-00424" class="html-bibr">62</a>] showed that this radiation of MHD turbulence is excited if there is an external constant magnetic field. If there is no external magnetic field, the magnetic turbulence generates sound waves only via the Lighthill mechanism. If there is a constant external magnetic field, hydromagnetic waves are generated instead by Alfvén waves, unless the energy density of hydrodynamic turbulence prevails over the energy of magnetic density. The central idea of a coupling between the hydrodynamic eddy cascade and the MHD waves through the process of Lighthill radiation is presented in Kulsrud [<a href="#B62-universe-10-00424" class="html-bibr">62</a>], Parker [<a href="#B63-universe-10-00424" class="html-bibr">63</a>], and Kato [<a href="#B64-universe-10-00424" class="html-bibr">64</a>].</div><div class='html-p'>Kato [<a href="#B64-universe-10-00424" class="html-bibr">64</a>] developed the Lighthill theory of MHD radiation for strong and weak magnetic fields <span class='html-italic'>B</span>, which is characterized by the magnetic Mach number, the ratio of the (root mean square) velocity of the hydrodynamic turbulence <span class='html-italic'>v</span> to the Alfvén speed <math display='inline'><semantics> <mrow> <msub> <mi>v</mi> <mi>A</mi> </msub> <mo>=</mo> <mi>B</mi> <mo>/</mo> <msqrt> <mrow> <mn>4</mn> <mi>π</mi> <mi>ρ</mi> </mrow> </msqrt> </mrow> </semantics></math>:<div class='html-disp-formula-info' id='FD23-universe-10-00424'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>M</mi> <mi>A</mi> </msub> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mi>v</mi> <msub> <mi>v</mi> <mi>A</mi> </msub> </mfrac> </mstyle> <mspace width="0.166667em"/> <mo>.</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(23)</label> </div> </div></div><div class='html-p'>For small Mach numbers (<math display='inline'><semantics> <mrow> <msub> <mi>M</mi> <mi>A</mi> </msub> <mo>≪</mo> <mn>1</mn> </mrow> </semantics></math>), a small fraction of the power is emitted in the form of Alfvén waves, while for large Mach numbers (<math display='inline'><semantics> <mrow> <msub> <mi>M</mi> <mi>A</mi> </msub> <mo>≫</mo> <mn>1</mn> </mrow> </semantics></math>) the power is emitted as sound waves and the radiation of Alfvén waves is insignificant. The turbulence decay time scale is a nonlinear cascade time, which is the eddy turnover time, i.e., the eddy size <span class='html-italic'>l</span> divided by its velocity <span class='html-italic'>v</span>:<div class='html-disp-formula-info' id='FD24-universe-10-00424'> <div class='f'> <math display='block'><semantics> <mrow> <mi>τ</mi> <mo>≈</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mi>l</mi> <mi>v</mi> </mfrac> </mstyle> <mspace width="0.166667em"/> <mo>.</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(24)</label> </div> </div></div><div class='html-p'>For Alfvén waves in a strong magnetic field, and the frequency <math display='inline'><semantics> <mrow> <mi>ω</mi> <mo>=</mo> <mover accent="true"> <mi>k</mi> <mo>¯</mo> </mover> <msub> <mi>v</mi> <mi>A</mi> </msub> </mrow> </semantics></math>, <div class='html-disp-formula-info' id='FD25-universe-10-00424'> <div class='f'> <math display='block'><semantics> <mrow> <mover accent="true"> <mi>k</mi> <mo>¯</mo> </mover> <mi>l</mi> <mo>≈</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mi>v</mi> <msub> <mi>v</mi> <mi>A</mi> </msub> </mfrac> </mstyle> <mspace width="0.166667em"/> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(25)</label> </div> </div> which corresponds to the resonance <math display='inline'><semantics> <mrow> <mi>τ</mi> <mrow> <mo>(</mo> <mi>l</mi> <mo>)</mo> </mrow> <mo>≈</mo> <mn>1</mn> <mo>/</mo> <mi>ω</mi> <mrow> <mo>(</mo> <mover accent="true"> <mi>k</mi> <mo>¯</mo> </mover> <mo>)</mo> </mrow> </mrow> </semantics></math>.</div><div class='html-p'>From radio polarization measurements, Zhang et al. [<a href="#B17-universe-10-00424" class="html-bibr">17</a>] showed that there are large-scale magnetic fields in the Fermi and eROSITA bubbles. <a href="#universe-10-00424-f008" class="html-fig">Figure 8</a> shows several kpc-scale magnetized structures in the bubbles.</div><div class='html-p'>If the turbulent magnetic field dominates the motion (<math display='inline'><semantics> <mrow> <msub> <mi>M</mi> <mi>A</mi> </msub> <mo>≪</mo> <mn>1</mn> </mrow> </semantics></math>), then the power is <div class='html-disp-formula-info' id='FD26-universe-10-00424'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>P</mi> <mi>A</mi> </msub> <mo>∼</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mi>ρ</mi> <msup> <mi>v</mi> <mn>3</mn> </msup> </mrow> <mi>l</mi> </mfrac> </mstyle> <msub> <mi>M</mi> <mi>A</mi> </msub> <mspace width="0.166667em"/> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(26)</label> </div> </div> and a small fraction of the hydrodynamic flux is transformed into Alfvén waves. In the following, we present some details of the radiation of MHD waves in the space medium in the limit of strong external magnetic fields.</div><div class='html-p'>We assume that fluid turbulence is induced by the motion of a smaller cluster in a larger cluster and its energy spectrum is described by a power law (see [<a href="#B54-universe-10-00424" class="html-bibr">54</a>,<a href="#B59-universe-10-00424" class="html-bibr">59</a>]):<div class='html-disp-formula-info' id='FD27-universe-10-00424'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>W</mi> <mi>f</mi> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>=</mo> <msubsup> <mi>W</mi> <mi>f</mi> <mn>0</mn> </msubsup> <msup> <mi>k</mi> <mrow> <mo>−</mo> <mi>m</mi> </mrow> </msup> <mspace width="0.166667em"/> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(27)</label> </div> </div> where <math display='inline'><semantics> <mrow> <mi>k</mi> <mo>=</mo> <mn>2</mn> <mi>π</mi> <mo>/</mo> <mi>l</mi> </mrow> </semantics></math> is the wavenumber corresponding to the scale <span class='html-italic'>l</span>, <math display='inline'><semantics> <mrow> <msub> <mi>W</mi> <mi>f</mi> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mi>δ</mi> <mi>k</mi> </mrow> </semantics></math> is the energy per unit volume in turbulence with wavenumbers between <span class='html-italic'>k</span> and <math display='inline'><semantics> <mrow> <mi>k</mi> <mo>+</mo> <mi>δ</mi> <mi>k</mi> </mrow> </semantics></math>, and <math display='inline'><semantics> <msubsup> <mi>W</mi> <mi>f</mi> <mn>0</mn> </msubsup> </semantics></math> and <span class='html-italic'>m</span> are constants. If one expresses the turbulent spectrum in terms of eddy size, the spectrum is represented by <math display='inline'><semantics> <mrow> <mi>W</mi> <mrow> <mo>(</mo> <mi>l</mi> <mo>)</mo> </mrow> <mo>∝</mo> <msup> <mi>l</mi> <mrow> <mi>m</mi> <mo>−</mo> <mn>2</mn> </mrow> </msup> </mrow> </semantics></math>. The cascade of the fluid turbulence extends from a largest eddy size <math display='inline'><semantics> <mrow> <msub> <mi>l</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>2</mn> <mi>π</mi> <mo>/</mo> <msub> <mi>k</mi> <mn>0</mn> </msub> </mrow> </semantics></math> down to a smallest scale determined by dissipation: <math display='inline'><semantics> <mrow> <msub> <mi>l</mi> <mi>K</mi> </msub> <mo>∼</mo> <msub> <mi>l</mi> <mn>0</mn> </msub> <mi>R</mi> <msup> <mi>e</mi> <mrow> <mo>−</mo> <mn>3</mn> <mo>/</mo> <mn>4</mn> </mrow> </msup> </mrow> </semantics></math> (where <math display='inline'><semantics> <mrow> <mi>R</mi> <mi>e</mi> </mrow> </semantics></math> is the Reynolds number). Since most of the energy of fluid turbulence resides in the largest scale, the total energy density of fluid turbulence is presented in the form <math display='inline'><semantics> <mrow> <msub> <mi>E</mi> <mi>t</mi> </msub> <mspace width="3.33333pt"/> <mo>∼</mo> <mi>ρ</mi> <msubsup> <mi>v</mi> <mi>t</mi> <mn>2</mn> </msubsup> </mrow> </semantics></math>, where <math display='inline'><semantics> <mi>ρ</mi> </semantics></math> is the fluid density and <math display='inline'><semantics> <msub> <mi>v</mi> <mi>t</mi> </msub> </semantics></math> is the turbulent velocity of the largest scale <math display='inline'><semantics> <msub> <mi>l</mi> <mn>0</mn> </msub> </semantics></math>. The normalization <math display='inline'><semantics> <msubsup> <mi>W</mi> <mi>f</mi> <mn>0</mn> </msubsup> </semantics></math> can be derived from the relation <math display='inline'><semantics> <mrow> <msub> <mi>E</mi> <mi>t</mi> </msub> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow> <msub> <mi>k</mi> <mn>0</mn> </msub> </mrow> <msub> <mi>k</mi> <mi>T</mi> </msub> </msubsup> <msub> <mi>W</mi> <mi>f</mi> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mi>d</mi> <mi>k</mi> </mrow> </semantics></math>, <div class='html-disp-formula-info' id='FD28-universe-10-00424'> <div class='f'> <math display='block'><semantics> <mrow> <msubsup> <mi>W</mi> <mi>f</mi> <mn>0</mn> </msubsup> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <msub> <mi>E</mi> <mi>t</mi> </msub> <mi>R</mi> </mfrac> </mstyle> <msubsup> <mi>k</mi> <mi>T</mi> <mrow> <mo>(</mo> <mi>m</mi> <mo>−</mo> <mn>1</mn> <mo>)</mo> </mrow> </msubsup> <mspace width="0.166667em"/> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(28)</label> </div> </div> where <div class='html-disp-formula-info' id='FD29-universe-10-00424'> <div class='f'> <math display='block'><semantics> <mtable displaystyle="true"> <mtr> <mtd columnalign="right"> <mspace width="1.em"/> </mtd> <mtd> <mi>R</mi> </mtd> <mtd columnalign="left"> <mrow> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mn>1</mn> <mrow> <mo>(</mo> <mi>m</mi> <mo>−</mo> <mn>1</mn> <mo>)</mo> </mrow> </mfrac> </mstyle> <mfenced separators="" open="[" close="]"> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <msub> <mi>k</mi> <mn>0</mn> </msub> <msub> <mi>W</mi> <mi>f</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>k</mi> <mn>0</mn> </msub> <mo>)</mo> </mrow> </mrow> <mrow> <msub> <mi>k</mi> <mi>T</mi> </msub> <msub> <mi>W</mi> <mi>f</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>k</mi> <mi>T</mi> </msub> <mo>)</mo> </mrow> </mrow> </mfrac> </mstyle> <mo>−</mo> <mn>1</mn> </mfenced> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="right"> <mspace width="1.em"/> </mtd> <mtd> <mspace width="1.em"/> </mtd> <mtd columnalign="left"> <mrow> <mo>≈</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mn>1</mn> <mrow> <mo>(</mo> <mi>m</mi> <mo>−</mo> <mn>1</mn> <mo>)</mo> </mrow> </mfrac> </mstyle> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <msub> <mi>k</mi> <mn>0</mn> </msub> <msub> <mi>W</mi> <mi>f</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>k</mi> <mn>0</mn> </msub> <mo>)</mo> </mrow> </mrow> <mrow> <msub> <mi>k</mi> <mi>T</mi> </msub> <msub> <mi>W</mi> <mi>f</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>k</mi> <mi>T</mi> </msub> <mo>)</mo> </mrow> </mrow> </mfrac> </mstyle> <mspace width="0.166667em"/> <mo>.</mo> </mrow> </mtd> </mtr> </mtable> </semantics></math> </div> <div class='l'> <label >(29)</label> </div> </div>Here <math display='inline'><semantics> <mrow> <msub> <mi>k</mi> <mi>T</mi> </msub> <mo>=</mo> <mn>2</mn> <mi>π</mi> <mo>/</mo> <msub> <mi>l</mi> <mi>T</mi> </msub> </mrow> </semantics></math> and <math display='inline'><semantics> <msub> <mi>l</mi> <mi>T</mi> </msub> </semantics></math> is the wavelength below which Alfvén waves are driven.</div><div class='html-p'>A fluid eddy of size <span class='html-italic'>l</span> has a velocity <div class='html-disp-formula-info' id='FD30-universe-10-00424'> <div class='f'> <math display='block'><semantics> <mtable displaystyle="true"> <mtr> <mtd/> <mtd> <mrow> <mi>v</mi> <mo>(</mo> <mi>l</mi> <mo>)</mo> </mrow> </mtd> <mtd columnalign="left"> <mrow> <mo>≈</mo> <msup> <mfenced separators="" open="[" close="]"> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mi>l</mi> <msub> <mi>W</mi> <mi>f</mi> </msub> <mrow> <mo>(</mo> <mi>l</mi> <mo>)</mo> </mrow> </mrow> <mi>ρ</mi> </mfrac> </mstyle> </mfenced> <mrow> <mn>1</mn> <mo>/</mo> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mfenced separators="" open="[" close="]"> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mi>k</mi> <msub> <mi>W</mi> <mi>f</mi> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </mrow> <mi>ρ</mi> </mfrac> </mstyle> </mfenced> <mrow> <mn>1</mn> <mo>/</mo> <mn>2</mn> </mrow> </msup> </mrow> </mtd> </mtr> <mtr> <mtd/> <mtd/> <mtd columnalign="left"> <mrow> <mo>=</mo> <msup> <mfenced separators="" open="(" close=")"> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <msub> <mi>E</mi> <mi>t</mi> </msub> <mrow> <mi>ρ</mi> <mi>R</mi> </mrow> </mfrac> </mstyle> </mfenced> <mrow> <mn>1</mn> <mo>/</mo> <mn>2</mn> </mrow> </msup> <msup> <mfenced separators="" open="(" close=")"> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mi>k</mi> <msub> <mi>k</mi> <mi>T</mi> </msub> </mfrac> </mstyle> </mfenced> <mrow> <mo>(</mo> <mn>1</mn> <mo>−</mo> <mi>m</mi> <mo>)</mo> <mo>/</mo> <mn>2</mn> </mrow> </msup> <mspace width="0.166667em"/> <mo>.</mo> </mrow> </mtd> </mtr> </mtable> </semantics></math> </div> <div class='l'> <label >(30)</label> </div> </div></div><div class='html-p'>Turbulence on a scale <span class='html-italic'>k</span> will radiate Alfvén waves at the wavenumber <div class='html-disp-formula-info' id='FD31-universe-10-00424'> <div class='f'> <math display='block'><semantics> <mrow> <mover accent="true"> <mi>k</mi> <mo>¯</mo> </mover> <mo>=</mo> <mfenced separators="" open="[" close="]"> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mi>v</mi> <mo>(</mo> <mi>l</mi> <mo>)</mo> </mrow> <msub> <mi>v</mi> <mi>A</mi> </msub> </mfrac> </mstyle> </mfenced> <mi>k</mi> <mspace width="0.166667em"/> <mo>.</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(31)</label> </div> </div></div><div class='html-p'>Here, we recall that <span class='html-italic'>k</span> is the wavenumber of hydrodynamic turbulence and <math display='inline'><semantics> <mover accent="true"> <mi>k</mi> <mo>¯</mo> </mover> </semantics></math> is the wavenumber of the Alfvén waves.</div><div class='html-p'>Let <math display='inline'><semantics> <mrow> <mi>v</mi> <mo>[</mo> <mi>k</mi> <mrow> <mo>(</mo> <mover accent="true"> <mi>k</mi> <mo>¯</mo> </mover> <mo>)</mo> </mrow> <mo>]</mo> </mrow> </semantics></math> be the fluid velocity on the fluid scale <math display='inline'><semantics> <mrow> <mi>k</mi> <mo>(</mo> <mover accent="true"> <mi>k</mi> <mo>¯</mo> </mover> <mo>)</mo> </mrow> </semantics></math> that drives Alfvén waves of wavenumber <math display='inline'><semantics> <mover accent="true"> <mi>k</mi> <mo>¯</mo> </mover> </semantics></math>. From Equations (<a href="#FD30-universe-10-00424" class="html-disp-formula">30</a>) and (<a href="#FD31-universe-10-00424" class="html-disp-formula">31</a>), we obtain <div class='html-disp-formula-info' id='FD32-universe-10-00424'> <div class='f'> <math display='block'><semantics> <mrow> <mi>v</mi> <mrow> <mo>[</mo> <mi>k</mi> <mrow> <mo>(</mo> <mover accent="true"> <mi>k</mi> <mo>¯</mo> </mover> <mo>)</mo> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <msub> <mi>v</mi> <mi>A</mi> </msub> <msup> <mfenced separators="" open="(" close=")"> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <msub> <mi>E</mi> <mi>t</mi> </msub> <mrow> <mi>ρ</mi> <msubsup> <mi>v</mi> <mi>A</mi> <mn>2</mn> </msubsup> <mi>R</mi> </mrow> </mfrac> </mstyle> </mfenced> <mrow> <mn>1</mn> <mo>/</mo> <mo>(</mo> <mn>3</mn> <mo>−</mo> <mi>m</mi> <mo>)</mo> </mrow> </msup> <msup> <mfenced separators="" open="(" close=")"> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mover accent="true"> <mi>k</mi> <mo>¯</mo> </mover> <msub> <mi>k</mi> <mi>T</mi> </msub> </mfrac> </mstyle> </mfenced> <mrow> <mo>(</mo> <mn>1</mn> <mo>−</mo> <mi>m</mi> <mo>)</mo> <mo>/</mo> <mo>(</mo> <mn>3</mn> <mo>−</mo> <mi>m</mi> <mo>)</mo> </mrow> </msup> <mspace width="0.166667em"/> <mo>.</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(32)</label> </div> </div></div><div class='html-p'>Assume that the energy going into Alfvén waves at wavenumber <math display='inline'><semantics> <mover accent="true"> <mi>k</mi> <mo>¯</mo> </mover> </semantics></math> has an energy flux <div class='html-disp-formula-info' id='FD33-universe-10-00424'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>I</mi> <mi>A</mi> </msub> <mrow> <mo>(</mo> <mover accent="true"> <mi>k</mi> <mo>¯</mo> </mover> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>I</mi> <mn>0</mn> </msub> <msup> <mrow> <mo>(</mo> <mover accent="true"> <mi>k</mi> <mo>¯</mo> </mover> <mo>/</mo> <msub> <mi>k</mi> <mi>T</mi> </msub> <mo>)</mo> </mrow> <mrow> <mo>−</mo> <msub> <mi>s</mi> <mi>t</mi> </msub> </mrow> </msup> <mspace width="0.166667em"/> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(33)</label> </div> </div> where <math display='inline'><semantics> <mrow> <msub> <mi>I</mi> <mi>A</mi> </msub> <mrow> <mo>(</mo> <mover accent="true"> <mi>k</mi> <mo>¯</mo> </mover> <mo>)</mo> </mrow> <mi>δ</mi> <mover accent="true"> <mi>k</mi> <mo>¯</mo> </mover> </mrow> </semantics></math> is the power per unit volume going into Alfvén waves with wavenumbers in the range <math display='inline'><semantics> <mrow> <mover accent="true"> <mi>k</mi> <mo>¯</mo> </mover> <mo>→</mo> <mover accent="true"> <mi>k</mi> <mo>¯</mo> </mover> <mo>+</mo> <mi>δ</mi> <mover accent="true"> <mi>k</mi> <mo>¯</mo> </mover> </mrow> </semantics></math> and <math display='inline'><semantics> <msub> <mi>I</mi> <mn>0</mn> </msub> </semantics></math> and <math display='inline'><semantics> <msub> <mi>s</mi> <mi>t</mi> </msub> </semantics></math> are constants. In this case, the power per unit volume going into the Alfvén mode from fluid turbulence is <div class='html-disp-formula-info' id='FD34-universe-10-00424'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>P</mi> <mi>A</mi> </msub> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow> <mover accent="true"> <mi>k</mi> <mo>¯</mo> </mover> </mrow> <msub> <mover accent="true"> <mi>k</mi> <mo>¯</mo> </mover> <mi>max</mi> </msub> </msubsup> <msub> <mi>I</mi> <mi>A</mi> </msub> <mrow> <mo>(</mo> <mover accent="true"> <mi>k</mi> <mo>¯</mo> </mover> <mo>)</mo> </mrow> <mi>d</mi> <mover accent="true"> <mi>k</mi> <mo>¯</mo> </mover> <mo>≈</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <msub> <mi>I</mi> <mn>0</mn> </msub> <msub> <mi>k</mi> <mi>T</mi> </msub> </mrow> <mrow> <mo>(</mo> <msub> <mi>s</mi> <mi>t</mi> </msub> <mo>−</mo> <mn>1</mn> <mo>)</mo> </mrow> </mfrac> </mstyle> <msup> <mfenced separators="" open="(" close=")"> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mover accent="true"> <mi>k</mi> <mo>¯</mo> </mover> <msub> <mi>k</mi> <mi>T</mi> </msub> </mfrac> </mstyle> </mfenced> <mrow> <mo>(</mo> <mn>1</mn> <mo>−</mo> <msub> <mi>s</mi> <mi>t</mi> </msub> <mo>)</mo> </mrow> </msup> <mspace width="0.166667em"/> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(34)</label> </div> </div> where <math display='inline'><semantics> <mrow> <mover accent="true"> <mi>k</mi> <mo>¯</mo> </mover> <mo>≪</mo> <msub> <mover accent="true"> <mi>k</mi> <mo>¯</mo> </mover> <mi>max</mi> </msub> </mrow> </semantics></math> and <math display='inline'><semantics> <mrow> <msub> <mi>s</mi> <mi>t</mi> </msub> <mo>></mo> <mn>1</mn> </mrow> </semantics></math>.</div><div class='html-p'>On the other hand, according to the Lighthill theory, <math display='inline'><semantics> <msub> <mi>P</mi> <mi>A</mi> </msub> </semantics></math> is given by Equation (<a href="#FD26-universe-10-00424" class="html-disp-formula">26</a>):<div class='html-disp-formula-info' id='FD35-universe-10-00424'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>P</mi> <mi>A</mi> </msub> <mo>=</mo> <msub> <mi>η</mi> <mi>A</mi> </msub> <mfenced separators="" open="[" close="]"> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mi>v</mi> <mo>(</mo> <mi>l</mi> <mo>)</mo> </mrow> <msub> <mi>v</mi> <mi>A</mi> </msub> </mfrac> </mstyle> </mfenced> <mspace width="0.166667em"/> <mi>ρ</mi> <msup> <mi>v</mi> <mn>3</mn> </msup> <mrow> <mo>(</mo> <mi>l</mi> <mo>)</mo> </mrow> <mi>k</mi> <mspace width="0.166667em"/> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(35)</label> </div> </div> where <math display='inline'><semantics> <msub> <mi>η</mi> <mi>A</mi> </msub> </semantics></math> is an efficiency factor of order unity [<a href="#B54-universe-10-00424" class="html-bibr">54</a>,<a href="#B56-universe-10-00424" class="html-bibr">56</a>,<a href="#B64-universe-10-00424" class="html-bibr">64</a>]. By comparing Equations (<a href="#FD34-universe-10-00424" class="html-disp-formula">34</a>) and (<a href="#FD35-universe-10-00424" class="html-disp-formula">35</a>), and using Equations (<a href="#FD31-universe-10-00424" class="html-disp-formula">31</a>) and (<a href="#FD32-universe-10-00424" class="html-disp-formula">32</a>), we obtain (see [<a href="#B54-universe-10-00424" class="html-bibr">54</a>,<a href="#B59-universe-10-00424" class="html-bibr">59</a>]) <div class='html-disp-formula-info' id='FD36-universe-10-00424'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>s</mi> <mi>t</mi> </msub> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mn>3</mn> <mo>(</mo> <mi>m</mi> <mo>−</mo> <mn>1</mn> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mn>3</mn> <mo>−</mo> <mi>m</mi> <mo>)</mo> </mrow> </mfrac> </mstyle> <mspace width="0.166667em"/> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(36)</label> </div> </div><div class='html-disp-formula-info' id='FD37-universe-10-00424'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>I</mi> <mn>0</mn> </msub> <mo>=</mo> <msub> <mi>η</mi> <mi>A</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>s</mi> <mi>t</mi> </msub> <mo>−</mo> <mn>1</mn> <mo>)</mo> </mrow> <mi>ρ</mi> <msubsup> <mi>v</mi> <mi>A</mi> <mn>3</mn> </msubsup> <msup> <mfenced separators="" open="(" close=")"> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <msub> <mi>E</mi> <mi>t</mi> </msub> <mrow> <mi>ρ</mi> <msubsup> <mi>v</mi> <mi>A</mi> <mn>2</mn> </msubsup> <mi>R</mi> </mrow> </mfrac> </mstyle> </mfenced> <mrow> <mn>3</mn> <mo>/</mo> <mo>(</mo> <mn>3</mn> <mo>−</mo> <mi>m</mi> <mo>)</mo> </mrow> </msup> <mspace width="0.166667em"/> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(37)</label> </div> </div></div><div class='html-p'>The power radiated in the form of Alfvén waves <math display='inline'><semantics> <msub> <mi>P</mi> <mi>A</mi> </msub> </semantics></math> is dominated by small <math display='inline'><semantics> <mover accent="true"> <mi>k</mi> <mo>¯</mo> </mover> </semantics></math> and the smallest is <math display='inline'><semantics> <msub> <mover accent="true"> <mi>k</mi> <mo>¯</mo> </mover> <mi>T</mi> </msub> </semantics></math>. With Equation (<a href="#FD30-universe-10-00424" class="html-disp-formula">30</a>) and <math display='inline'><semantics> <mrow> <msub> <mover accent="true"> <mi>k</mi> <mo>¯</mo> </mover> <mi>T</mi> </msub> <mo>=</mo> <mrow> <mo>[</mo> <mi>v</mi> <mrow> <mo>(</mo> <msub> <mi>l</mi> <mi>T</mi> </msub> <mo>)</mo> </mrow> <mo>/</mo> <msub> <mi>v</mi> <mi>A</mi> </msub> <mo>]</mo> </mrow> <msub> <mi>k</mi> <mi>T</mi> </msub> </mrow> </semantics></math>, the total power in the form of Alfv’en waves is approximately <div class='html-disp-formula-info' id='FD38-universe-10-00424'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>P</mi> <mi>A</mi> </msub> <mo>=</mo> <msub> <mi>η</mi> <mi>A</mi> </msub> <msup> <mfenced separators="" open="(" close=")"> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <msub> <mi>E</mi> <mi>t</mi> </msub> <mrow> <mi>ρ</mi> <msubsup> <mi>v</mi> <mi>A</mi> <mn>2</mn> </msubsup> <mi>R</mi> </mrow> </mfrac> </mstyle> </mfenced> <mn>2</mn> </msup> <mi>ρ</mi> <msubsup> <mi>v</mi> <mi>A</mi> <mn>3</mn> </msubsup> <msub> <mi>k</mi> <mi>T</mi> </msub> <mspace width="0.166667em"/> <mo>.</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(38)</label> </div> </div></div><div class='html-p'>For Kolmogorov turbulence, the spectral index is <math display='inline'><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>5</mn> <mo>/</mo> <mn>3</mn> </mrow> </semantics></math> (see Equation (<a href="#FD27-universe-10-00424" class="html-disp-formula">27</a>)), and the spectral index of the Alfvén wave flux is <math display='inline'><semantics> <mrow> <msub> <mi>s</mi> <mi>t</mi> </msub> <mo>=</mo> <mn>3</mn> <mo>/</mo> <mn>2</mn> </mrow> </semantics></math> (Equation (<a href="#FD33-universe-10-00424" class="html-disp-formula">33</a>)).</div></section><section id='sec6-universe-10-00424' type=''><h2 data-nested='1'> 6. Spectrum of MHD Turbulence in the Fermi Bubble Envelope</h2><div class='html-p'>The evolution of the spectrum of the Alfvén waves, <math display='inline'><semantics> <mrow> <msub> <mi>W</mi> <mi>k</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>, is described by the equation of nonlinear diffusion presented in, e.g., Brunetti et al. [<a href="#B55-universe-10-00424" class="html-bibr">55</a>] and Brunetti & Blasi [<a href="#B65-universe-10-00424" class="html-bibr">65</a>]:<div class='html-disp-formula-info' id='FD39-universe-10-00424'> <div class='f'> <math display='block'><semantics> <mrow> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>W</mi> <mi>k</mi> </msub> <mrow> <mo>(</mo> <mover accent="true"> <mi>k</mi> <mo>¯</mo> </mover> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> <mrow> <mo>∂</mo> <mi>t</mi> </mrow> </mfrac> </mstyle> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mo>∂</mo> <mrow> <mo>∂</mo> <mover accent="true"> <mi>k</mi> <mo>¯</mo> </mover> </mrow> </mfrac> </mstyle> <mfenced separators="" open="[" close="]"> <msub> <mi>D</mi> <mrow> <mi>k</mi> <mi>k</mi> </mrow> </msub> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>W</mi> <mi>k</mi> </msub> <mrow> <mo>(</mo> <mover accent="true"> <mi>k</mi> <mo>¯</mo> </mover> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> <mrow> <mo>∂</mo> <mover accent="true"> <mi>k</mi> <mo>¯</mo> </mover> </mrow> </mfrac> </mstyle> </mfenced> <mo>−</mo> <mo>Γ</mo> <mrow> <mo>(</mo> <mover accent="true"> <mi>k</mi> <mo>¯</mo> </mover> <mo>)</mo> </mrow> <msub> <mi>W</mi> <mi>k</mi> </msub> <mrow> <mo>(</mo> <mover accent="true"> <mi>k</mi> <mo>¯</mo> </mover> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>I</mi> <mi>A</mi> </msub> <mrow> <mo>(</mo> <mover accent="true"> <mi>k</mi> <mo>¯</mo> </mover> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> <mspace width="0.166667em"/> <mo>.</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(39)</label> </div> </div></div><div class='html-p'>The first term on the right-hand side of Equation (<a href="#FD39-universe-10-00424" class="html-disp-formula">39</a>) describes the nonlinear MHD wave–wave cascade. The diffusion coefficient for the Kolmogorov and the Iroshnikov–Kraichnan spectra is (e.g., [<a href="#B66-universe-10-00424" class="html-bibr">66</a>]) <div class='html-disp-formula-info' id='FD40-universe-10-00424'> <div class='f'> <math display='block'><semantics> <mtable displaystyle="true"> <mtr> <mtd columnalign="right"> <msub> <mi>D</mi> <mrow> <mi>k</mi> <mi>k</mi> </mrow> </msub> </mtd> <mtd> <mo>=</mo> </mtd> <mtd columnalign="left"> <mrow> <msub> <mi>v</mi> <mi>A</mi> </msub> <mfenced separators="" open="{" close=""> <mtable> <mtr> <mtd columnalign="left"> <mrow> <mstyle scriptlevel="0" displaystyle="true"> <mrow> <msup> <mover accent="true"> <mi>k</mi> <mo>¯</mo> </mover> <mrow> <mn>7</mn> <mo>/</mo> <mn>2</mn> </mrow> </msup> <msup> <mfenced separators="" open="[" close="]"> <mfrac> <mrow> <msub> <mi>W</mi> <mi>k</mi> </msub> <mrow> <mo>(</mo> <mover accent="true"> <mi>k</mi> <mo>¯</mo> </mover> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> <mrow> <mn>2</mn> <msub> <mi>W</mi> <mi>B</mi> </msub> </mrow> </mfrac> </mfenced> <mrow> <mn>1</mn> <mo>/</mo> <mn>2</mn> </mrow> </msup> </mrow> </mstyle> <mspace width="0.166667em"/> <mo>,</mo> <mspace width="4pt"/> <mi>Kolmogorov</mi> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="left"> <mrow> <mstyle scriptlevel="0" displaystyle="true"> <mrow> <msup> <mover accent="true"> <mi>k</mi> <mo>¯</mo> </mover> <mn>4</mn> </msup> <mfenced separators="" open="[" close="]"> <mfrac> <mrow> <msub> <mi>W</mi> <mi>k</mi> </msub> <mrow> <mo>(</mo> <mover accent="true"> <mi>k</mi> <mo>¯</mo> </mover> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> <mrow> <mn>2</mn> <msub> <mi>W</mi> <mi>B</mi> </msub> </mrow> </mfrac> </mfenced> </mrow> </mstyle> <mspace width="0.166667em"/> <mo>,</mo> <mspace width="4pt"/> <mrow> <mi>Iroshnikov</mi> <mtext>-</mtext> <mi>Kraichnan</mi> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced> </mrow> </mtd> </mtr> </mtable> </semantics></math> </div> <div class='l'> <label >(40)</label> </div> </div> where <math display='inline'><semantics> <mrow> <msub> <mi>W</mi> <mi>B</mi> </msub> <mo>=</mo> <msubsup> <mi>B</mi> <mn>0</mn> <mn>2</mn> </msubsup> <mo>/</mo> <mn>8</mn> <mi>π</mi> </mrow> </semantics></math>.</div><div class='html-p'>The second term on the right-hand side of Equation (<a href="#FD39-universe-10-00424" class="html-disp-formula">39</a>) describes the damping of MHD waves by collisions of relativistic and thermal particles in the interstellar or intercluster medium (see [<a href="#B67-universe-10-00424" class="html-bibr">67</a>]):<div class='html-disp-formula-info' id='FD41-universe-10-00424'> <div class='f'> <math display='block'><semantics> <mtable displaystyle="true"> <mtr> <mtd/> <mtd> <msub> <mo>Γ</mo> <mi>k</mi> </msub> </mtd> <mtd columnalign="left"> <mrow> <mo>≃</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mn>4</mn> <msup> <mi>π</mi> <mn>3</mn> </msup> <msup> <mi>e</mi> <mn>2</mn> </msup> <msubsup> <mi>v</mi> <mi>A</mi> <mn>2</mn> </msubsup> </mrow> <mrow> <mover accent="true"> <mi>k</mi> <mo>¯</mo> </mover> <msup> <mi>c</mi> <mn>2</mn> </msup> </mrow> </mfrac> </mstyle> <msubsup> <mo>∫</mo> <mrow> <msub> <mi>p</mi> <mi>min</mi> </msub> </mrow> <msub> <mi>p</mi> <mi>max</mi> </msub> </msubsup> <msup> <mi>p</mi> <mn>2</mn> </msup> <mrow> <mo>(</mo> <mn>1</mn> <mo>−</mo> <msub> <mi>μ</mi> <mi>α</mi> </msub> <mo>)</mo> </mrow> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mo>∂</mo> <mi>F</mi> <mo>(</mo> <mi>p</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> <mrow> <mo>∂</mo> <mi>p</mi> </mrow> </mfrac> </mstyle> <mi>d</mi> <mi>p</mi> </mrow> </mtd> </mtr> <mtr> <mtd/> <mtd/> <mtd columnalign="left"> <mrow> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <msup> <mi>π</mi> <mn>2</mn> </msup> <msup> <mi>e</mi> <mn>2</mn> </msup> <msubsup> <mi>v</mi> <mi>A</mi> <mn>2</mn> </msubsup> </mrow> <mrow> <mover accent="true"> <mi>k</mi> <mo>¯</mo> </mover> <msup> <mi>c</mi> <mn>2</mn> </msup> </mrow> </mfrac> </mstyle> <msubsup> <mo>∫</mo> <mrow> <msub> <mi>p</mi> <mi>min</mi> </msub> </mrow> <msub> <mi>p</mi> <mi>max</mi> </msub> </msubsup> <mrow> <mo>(</mo> <mn>1</mn> <mo>−</mo> <msub> <mi>μ</mi> <mi>α</mi> </msub> <mo>)</mo> </mrow> <mfenced separators="" open="[" close="]"> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mo>∂</mo> <mi>N</mi> <mo>(</mo> <mi>p</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> <mrow> <mo>∂</mo> <mi>p</mi> </mrow> </mfrac> </mstyle> <mo>−</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mn>2</mn> <mi>N</mi> <mo>(</mo> <mi>p</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> <mi>p</mi> </mfrac> </mstyle> </mfenced> <mi>d</mi> <mi>p</mi> <mspace width="0.166667em"/> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </semantics></math> </div> <div class='l'> <label >(41)</label> </div> </div> where <math display='inline'><semantics> <mrow> <mi>F</mi> <mo>(</mo> <mi>p</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </semantics></math> is the particle distribution function and <math display='inline'><semantics> <mrow> <mi>N</mi> <mrow> <mo>(</mo> <mi>p</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mn>4</mn> <mi>π</mi> <msup> <mi>p</mi> <mn>2</mn> </msup> <mi>F</mi> <mrow> <mo>(</mo> <mi>p</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>, and <div class='html-disp-formula-info' id='FD42-universe-10-00424'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>μ</mi> <mi>α</mi> </msub> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <msub> <mi>v</mi> <mi>A</mi> </msub> <mi>c</mi> </mfrac> </mstyle> <mo>±</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mi>m</mi> <mo>Ω</mo> </mrow> <mrow> <mi>p</mi> <mover accent="true"> <mi>k</mi> <mo>¯</mo> </mover> </mrow> </mfrac> </mstyle> <mspace width="0.166667em"/> <mo>.</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(42)</label> </div> </div>Here, the upper and lower signs are for negatively and positively charged particles, respectively.</div><div class='html-p'>The third term on the right-hand side of Equation (<a href="#FD39-universe-10-00424" class="html-disp-formula">39</a>), <math display='inline'><semantics> <msub> <mi>I</mi> <mi>A</mi> </msub> </semantics></math>, describes the injection of Alfvén waves by the fluid turbulence through the Lighthill mechanism.</div><div class='html-p'>The time scale of the damping with the thermal pool is considerably shorter than the cascade time scale for <math display='inline'><semantics> <mrow> <mover accent="true"> <mi>k</mi> <mo>¯</mo> </mover> <mo>/</mo> <msub> <mover accent="true"> <mi>k</mi> <mo>¯</mo> </mover> <mi>max</mi> </msub> <mo>≫</mo> <mn>0.1</mn> </mrow> </semantics></math>. Thus, a break or a cutoff in the wave spectrum is expected at large wavenumbers.</div><div class='html-p'>In the following, we present an alternative formulation of the steady-state equation for <math display='inline'><semantics> <mrow> <msub> <mi>W</mi> <mi>k</mi> </msub> <mrow> <mo>(</mo> <mover accent="true"> <mi>k</mi> <mo>¯</mo> </mover> <mo>)</mo> </mrow> </mrow> </semantics></math>. For simplicity, it is described in a compact form (see [<a href="#B68-universe-10-00424" class="html-bibr">68</a>,<a href="#B69-universe-10-00424" class="html-bibr">69</a>] and references therein):<div class='html-disp-formula-info' id='FD43-universe-10-00424'> <div class='f'> <math display='block'><semantics> <mrow> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mo>∂</mo> <mrow> <mo>∂</mo> <mover accent="true"> <mi>k</mi> <mo>¯</mo> </mover> </mrow> </mfrac> </mstyle> <mfenced separators="" open="(" close=")"> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mover accent="true"> <mi>k</mi> <mo>¯</mo> </mover> <msub> <mi>W</mi> <mi>k</mi> </msub> </mrow> <msub> <mi>T</mi> <mi>NL</mi> </msub> </mfrac> </mstyle> </mfenced> <mo>=</mo> <mn>2</mn> <msub> <mo>Γ</mo> <mi>CR</mi> </msub> <msub> <mi>W</mi> <mi>k</mi> </msub> <mo>+</mo> <msub> <mi>I</mi> <mi>A</mi> </msub> <mrow> <mo>(</mo> <mover accent="true"> <mi>k</mi> <mo>¯</mo> </mover> <mo>)</mo> </mrow> <mspace width="0.166667em"/> <mo>.</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(43)</label> </div> </div>Here, the source of MHD fluctuations, <math display='inline'><semantics> <mrow> <msub> <mi>I</mi> <mi>A</mi> </msub> <mrow> <mo>(</mo> <mover accent="true"> <mi>k</mi> <mo>¯</mo> </mover> <mo>)</mo> </mrow> </mrow> </semantics></math>, is given by Equation (<a href="#FD33-universe-10-00424" class="html-disp-formula">33</a>).</div><div class='html-p'>The rate of damping of MHD waves by cosmic rays, <math display='inline'><semantics> <mrow> <msub> <mo>Γ</mo> <mi>CR</mi> </msub> <mrow> <mo>(</mo> <mover accent="true"> <mi>k</mi> <mo>¯</mo> </mover> <mo>)</mo> </mrow> </mrow> </semantics></math>, is (see, e.g., [<a href="#B70-universe-10-00424" class="html-bibr">70</a>]) <div class='html-disp-formula-info' id='FD44-universe-10-00424'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mo>Γ</mo> <mi>CR</mi> </msub> <mrow> <mo>(</mo> <mover accent="true"> <mi>k</mi> <mo>¯</mo> </mover> <mo>)</mo> </mrow> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mi>π</mi> <msup> <mi>Z</mi> <mn>2</mn> </msup> <msup> <mi>e</mi> <mn>2</mn> </msup> <msubsup> <mi>v</mi> <mi>A</mi> <mn>2</mn> </msubsup> </mrow> <mrow> <mn>2</mn> <mover accent="true"> <mi>k</mi> <mo>¯</mo> </mover> <msup> <mi>c</mi> <mn>2</mn> </msup> </mrow> </mfrac> </mstyle> <msubsup> <mo>∫</mo> <mrow> <msub> <mi>p</mi> <mi>res</mi> </msub> <mrow> <mo>(</mo> <mover accent="true"> <mi>k</mi> <mo>¯</mo> </mover> <mo>)</mo> </mrow> </mrow> <mo>∞</mo> </msubsup> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mi>d</mi> <mi>p</mi> </mrow> <mi>p</mi> </mfrac> </mstyle> <mi>F</mi> <mrow> <mo>(</mo> <mi>p</mi> <mo>)</mo> </mrow> <mspace width="0.166667em"/> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(44)</label> </div> </div> where <math display='inline'><semantics> <mrow> <mi>F</mi> <mo>(</mo> <mi>p</mi> <mo>)</mo> </mrow> </semantics></math> is the CR distribution, <span class='html-italic'>p</span> is the particle momentum, and <math display='inline'><semantics> <mrow> <msub> <mi>p</mi> <mi>res</mi> </msub> <mrow> <mo>(</mo> <mover accent="true"> <mi>k</mi> <mo>¯</mo> </mover> <mo>)</mo> </mrow> <mo>=</mo> <mi>Z</mi> <mi>e</mi> <mi>B</mi> <mo>/</mo> <mi>c</mi> <mover accent="true"> <mi>k</mi> <mo>¯</mo> </mover> </mrow> </semantics></math>.</div><div class='html-p'>In the form of the Iroshnikov–Kraichnan spectrum, the term on the left-hand side of Equation (<a href="#FD43-universe-10-00424" class="html-disp-formula">43</a>) is <div class='html-disp-formula-info' id='FD45-universe-10-00424'> <div class='f'> <math display='block'><semantics> <mrow> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mo>∂</mo> <mrow> <mo>∂</mo> <mover accent="true"> <mi>k</mi> <mo>¯</mo> </mover> </mrow> </mfrac> </mstyle> <mfenced separators="" open="(" close=")"> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mover accent="true"> <mi>k</mi> <mo>¯</mo> </mover> <msub> <mi>W</mi> <mi>k</mi> </msub> </mrow> <msub> <mi>T</mi> <mi>NL</mi> </msub> </mfrac> </mstyle> </mfenced> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mover accent="true"> <mi>k</mi> <mo>¯</mo> </mover> </mrow> </mfrac> </mstyle> <mfenced separators="" open="[" close="]"> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mi>C</mi> <mfenced separators="" open="(" close=")"> <msup> <mover accent="true"> <mi>k</mi> <mo>¯</mo> </mover> <mn>3</mn> </msup> <msup> <mi>W</mi> <mn>2</mn> </msup> <mrow> <mo>(</mo> <mover accent="true"> <mi>k</mi> <mo>¯</mo> </mover> <mo>)</mo> </mrow> </mfenced> </mrow> <mrow> <mi>ρ</mi> <msub> <mi>v</mi> <mi>A</mi> </msub> </mrow> </mfrac> </mstyle> </mfenced> <mspace width="0.166667em"/> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(45)</label> </div> </div> where the interaction is <div class='html-disp-formula-info' id='FD46-universe-10-00424'> <div class='f'> <math display='block'><semantics> <mrow> <msubsup> <mi>T</mi> <mrow> <mi>NL</mi> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> <mrow> <mo>(</mo> <mover accent="true"> <mi>k</mi> <mo>¯</mo> </mover> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>C</mi> <mi>NL</mi> </msub> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <msup> <mover accent="true"> <mi>k</mi> <mo>¯</mo> </mover> <mn>2</mn> </msup> <msub> <mi>W</mi> <mi>k</mi> </msub> <mrow> <mo>(</mo> <mover accent="true"> <mi>k</mi> <mo>¯</mo> </mover> <mo>)</mo> </mrow> </mrow> <mrow> <msub> <mi>m</mi> <mi mathvariant="normal">i</mi> </msub> <msub> <mi>n</mi> <mi mathvariant="normal">i</mi> </msub> <msub> <mi>v</mi> <mi mathvariant="normal">A</mi> </msub> </mrow> </mfrac> </mstyle> <mspace width="0.166667em"/> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(46)</label> </div> </div> and the constant <math display='inline'><semantics> <mrow> <msub> <mi>C</mi> <mi>NL</mi> </msub> <mo>∼</mo> <mn>1</mn> </mrow> </semantics></math>.</div><div class='html-p'>If the magnetic field fluctuations are injected by an external source at the scale <math display='inline'><semantics> <mrow> <mi>L</mi> <mo>=</mo> <mn>1</mn> <mo>/</mo> <msub> <mover accent="true"> <mi>k</mi> <mo>¯</mo> </mover> <mi>L</mi> </msub> </mrow> </semantics></math>, Equation (<a href="#FD43-universe-10-00424" class="html-disp-formula">43</a>) can be simplified to <div class='html-disp-formula-info' id='FD47-universe-10-00424'> <div class='f'> <math display='block'><semantics> <mrow> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mo>∂</mo> <mrow> <mo>∂</mo> <mover accent="true"> <mi>k</mi> <mo>¯</mo> </mover> </mrow> </mfrac> </mstyle> <mfenced separators="" open="(" close=")"> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mover accent="true"> <mi>k</mi> <mo>¯</mo> </mover> <msub> <mi>W</mi> <mi>k</mi> </msub> </mrow> <msub> <mi>T</mi> <mi>NL</mi> </msub> </mfrac> </mstyle> </mfenced> <mo>=</mo> <mn>2</mn> <msub> <mo>Γ</mo> <mi>CR</mi> </msub> <msub> <mi>W</mi> <mi>k</mi> </msub> <mo>+</mo> <mo>Φ</mo> <mspace width="0.166667em"/> <mi>δ</mi> <mrow> <mo>(</mo> <mover accent="true"> <mi>k</mi> <mo>¯</mo> </mover> <mo>−</mo> <msub> <mover accent="true"> <mi>k</mi> <mo>¯</mo> </mover> <mi>L</mi> </msub> <mo>)</mo> </mrow> <mspace width="0.166667em"/> <mo>.</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(47)</label> </div> </div></div><div class='html-p'>This result has been applied to the spectrum of MHD turbulence in the FB envelope for <math display='inline'><semantics> <mrow> <mover accent="true"> <mi>k</mi> <mo>¯</mo> </mover> <mo>></mo> <msub> <mover accent="true"> <mi>k</mi> <mo>¯</mo> </mover> <mn>0</mn> </msub> </mrow> </semantics></math> (see [<a href="#B71-universe-10-00424" class="html-bibr">71</a>]). The solution of Equation (<a href="#FD47-universe-10-00424" class="html-disp-formula">47</a>) is given by <div class='html-disp-formula-info' id='FD48-universe-10-00424'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>W</mi> <mi>k</mi> </msub> <mrow> <mo>(</mo> <mover accent="true"> <mi>k</mi> <mo>¯</mo> </mover> <mo>)</mo> </mrow> <mo>=</mo> <msup> <mfenced separators="" open="(" close=")"> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <msub> <mover accent="true"> <mi>k</mi> <mo>¯</mo> </mover> <mn>0</mn> </msub> <mover accent="true"> <mi>k</mi> <mo>¯</mo> </mover> </mfrac> </mstyle> </mfenced> <mrow> <mn>3</mn> <mo>/</mo> <mn>2</mn> </mrow> </msup> <msub> <mi>W</mi> <mi>k</mi> </msub> <mrow> <mo>(</mo> <msub> <mover accent="true"> <mi>k</mi> <mo>¯</mo> </mover> <mn>0</mn> </msub> <mo>)</mo> </mrow> <mo>−</mo> <mspace width="0.166667em"/> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <msup> <mi>Z</mi> <mn>2</mn> </msup> <msup> <mi>e</mi> <mn>2</mn> </msup> <msup> <mi>B</mi> <mn>2</mn> </msup> <msub> <mi>v</mi> <mi>A</mi> </msub> </mrow> <mrow> <mn>8</mn> <mi>C</mi> <msup> <mi>c</mi> <mn>2</mn> </msup> <msup> <mover accent="true"> <mi>k</mi> <mo>¯</mo> </mover> <mrow> <mn>3</mn> <mo>/</mo> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mstyle> <msubsup> <mo>∫</mo> <mrow> <msub> <mover accent="true"> <mi>k</mi> <mo>¯</mo> </mover> <mn>0</mn> </msub> </mrow> <mover accent="true"> <mi>k</mi> <mo>¯</mo> </mover> </msubsup> <msubsup> <mover accent="true"> <mi>k</mi> <mo>¯</mo> </mover> <mi>L</mi> <mrow> <mo>−</mo> <mn>5</mn> <mo>/</mo> <mn>2</mn> </mrow> </msubsup> <mspace width="0.166667em"/> <mi>d</mi> <mover accent="true"> <mi>k</mi> <mo>¯</mo> </mover> <msubsup> <mo>∫</mo> <mrow> <msub> <mi>p</mi> <mi>res</mi> </msub> <mrow> <mo>(</mo> <msub> <mover accent="true"> <mi>k</mi> <mo>¯</mo> </mover> <mi>L</mi> </msub> <mo>)</mo> </mrow> </mrow> <mo>∞</mo> </msubsup> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mi>F</mi> <mo>(</mo> <mi>p</mi> <mo>)</mo> <mi>d</mi> <mi>p</mi> </mrow> <mi>p</mi> </mfrac> </mstyle> <mspace width="0.166667em"/> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(48)</label> </div> </div> where <math display='inline'><semantics> <mrow> <msub> <mi>W</mi> <mi>k</mi> </msub> <mrow> <mo>(</mo> <msub> <mover accent="true"> <mi>k</mi> <mo>¯</mo> </mover> <mi>L</mi> </msub> <mo>)</mo> </mrow> <mo>=</mo> <msubsup> <mover accent="true"> <mi>k</mi> <mo>¯</mo> </mover> <mi>L</mi> <mrow> <mo>−</mo> <mn>3</mn> <mo>/</mo> <mn>2</mn> </mrow> </msubsup> <msqrt> <mrow> <mi>ρ</mi> <msub> <mi>v</mi> <mi>A</mi> </msub> <mo>Φ</mo> <mo>/</mo> <mi>C</mi> <mspace width="0.166667em"/> </mrow> </msqrt> </mrow> </semantics></math>, where <math display='inline'><semantics> <mo>Φ</mo> </semantics></math> describes the source injection at <math display='inline'><semantics> <msub> <mover accent="true"> <mi>k</mi> <mo>¯</mo> </mover> <mi>L</mi> </msub> </semantics></math>.</div><div class='html-p'>The coefficient of momentum diffusion of CRs is (see [<a href="#B70-universe-10-00424" class="html-bibr">70</a>]) <div class='html-disp-formula-info' id='FD49-universe-10-00424'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>D</mi> <mi>p</mi> </msub> <mrow> <mo>(</mo> <mi>p</mi> <mo>)</mo> </mrow> <mo>=</mo> <msup> <mi>p</mi> <mn>2</mn> </msup> <mi>κ</mi> <mrow> <mo>(</mo> <mi>p</mi> <mo>)</mo> </mrow> <mspace width="0.166667em"/> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(49)</label> </div> </div> where <div class='html-disp-formula-info' id='FD50-universe-10-00424'> <div class='f'> <math display='block'><semantics> <mrow> <mi>κ</mi> <mrow> <mo>(</mo> <mi>p</mi> <mo>)</mo> </mrow> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mn>3</mn> <msubsup> <mover accent="true"> <mi>k</mi> <mo>¯</mo> </mover> <mrow> <mi>res</mi> </mrow> <mn>2</mn> </msubsup> <msub> <mi>W</mi> <mi>k</mi> </msub> <mrow> <mo>(</mo> <msub> <mover accent="true"> <mi>k</mi> <mo>¯</mo> </mover> <mi>res</mi> </msub> <mo>)</mo> </mrow> </mrow> <mrow> <mi>ρ</mi> <mi>v</mi> </mrow> </mfrac> </mstyle> <mspace width="0.166667em"/> <mo>.</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(50)</label> </div> </div>Here, <math display='inline'><semantics> <mrow> <msub> <mover accent="true"> <mi>k</mi> <mo>¯</mo> </mover> <mi>res</mi> </msub> <mo>=</mo> <mn>1</mn> <mo>/</mo> <msub> <mi>r</mi> <mi>L</mi> </msub> <mo>=</mo> <mi>Z</mi> <mi>e</mi> <mi>B</mi> <mo>/</mo> <mi>p</mi> <mi>c</mi> </mrow> </semantics></math>, <math display='inline'><semantics> <msub> <mi>r</mi> <mi>L</mi> </msub> </semantics></math> is the particle Larmor radius, and <span class='html-italic'>B</span> is the magnetic field strength.</div><div class='html-p'>The momentum diffusion coefficient <math display='inline'><semantics> <msub> <mi>D</mi> <mi>p</mi> </msub> </semantics></math> for the bubble parameters is shown in <a href="#universe-10-00424-f009" class="html-fig">Figure 9</a> (solid line). For comparison, the dash-dotted line is the diffusion coefficient for the Kraichnan spectrum of turbulence without CR absorption.</div><div class='html-p'>As shown in <a href="#universe-10-00424-f009" class="html-fig">Figure 9</a>, the wave damping by cosmic rays can terminate the cascade for relatively small CR momenta <span class='html-italic'>p</span>. Brunetti et al. [<a href="#B55-universe-10-00424" class="html-bibr">55</a>] showed that the time of damping was considerably shorter than the cascade time for large wavenumbers. They also concluded that the damping rate for protons largely dominates that for electrons. These protons can exhibit a resonance with the relativistic electrons, which may be important for their acceleration.</div><div class='html-p'>From Equation (<a href="#FD48-universe-10-00424" class="html-disp-formula">48</a>), the distribution function of CR electrons, <math display='inline'><semantics> <mrow> <mi>F</mi> <mo>(</mo> <mi>p</mi> <mo>)</mo> </mrow> </semantics></math>, can be estimated from the observed gamma-ray [<a href="#B2-universe-10-00424" class="html-bibr">2</a>,<a href="#B72-universe-10-00424" class="html-bibr">72</a>] and microwave [<a href="#B3-universe-10-00424" class="html-bibr">3</a>] emissions for the expected parameter of the FB envelope of hydrodynamic turbulence.</div><div class='html-p'>If the effect of CR damping is insignificant and <math display='inline'><semantics> <mrow> <mn>2</mn> <msub> <mo>Γ</mo> <mi>CR</mi> </msub> <msub> <mi>W</mi> <mi>k</mi> </msub> </mrow> </semantics></math> can be ignored in Equation (<a href="#FD43-universe-10-00424" class="html-disp-formula">43</a>), then Equation (<a href="#FD43-universe-10-00424" class="html-disp-formula">43</a>) is simply the balance of wave cascading and MHD excitation by the hydrodynamic turbulence:<div class='html-disp-formula-info' id='FD51-universe-10-00424'> <div class='f'> <math display='block'><semantics> <mrow> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mo>∂</mo> <mrow> <mo>∂</mo> <mover accent="true"> <mi>k</mi> <mo>¯</mo> </mover> </mrow> </mfrac> </mstyle> <mfenced separators="" open="(" close=")"> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mover accent="true"> <mi>k</mi> <mo>¯</mo> </mover> <msub> <mi>W</mi> <mi>k</mi> </msub> </mrow> <msub> <mi>T</mi> <mi>NL</mi> </msub> </mfrac> </mstyle> </mfenced> <mo>≈</mo> <msub> <mi>I</mi> <mi>A</mi> </msub> <mrow> <mo>(</mo> <mover accent="true"> <mi>k</mi> <mo>¯</mo> </mover> <mo>)</mo> </mrow> <mspace width="0.166667em"/> <mo>.</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(51)</label> </div> </div></div><div class='html-p'>With Equation (<a href="#FD33-universe-10-00424" class="html-disp-formula">33</a>), the spectrum of MHD fluctuations <math display='inline'><semantics> <msub> <mi>W</mi> <mi>k</mi> </msub> </semantics></math> is <div class='html-disp-formula-info' id='FD52-universe-10-00424'> <div class='f'> <math display='block'><semantics> <mrow> <msubsup> <mi>W</mi> <mi>k</mi> <mn>2</mn> </msubsup> <mrow> <mo>(</mo> <mover accent="true"> <mi>k</mi> <mo>¯</mo> </mover> <mo>)</mo> </mrow> <mo>≈</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mi>ρ</mi> <msub> <mi>v</mi> <mi>A</mi> </msub> </mrow> <mrow> <msub> <mi>C</mi> <mi>NL</mi> </msub> <msup> <mover accent="true"> <mi>k</mi> <mo>¯</mo> </mover> <mn>3</mn> </msup> </mrow> </mfrac> </mstyle> <mfenced separators="" open="[" close="]"> <msub> <mi mathvariant="script">P</mi> <mn>0</mn> </msub> <mo>−</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <msub> <mi>I</mi> <mn>0</mn> </msub> <msub> <mi>k</mi> <mn>0</mn> </msub> </mrow> <mrow> <mo>(</mo> <msub> <mi>s</mi> <mi>t</mi> </msub> <mo>−</mo> <mn>1</mn> <mo>)</mo> </mrow> </mfrac> </mstyle> <msup> <mfenced separators="" open="(" close=")"> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <msub> <mi>k</mi> <mn>0</mn> </msub> <mover accent="true"> <mi>k</mi> <mo>¯</mo> </mover> </mfrac> </mstyle> </mfenced> <mrow> <mo>(</mo> <msub> <mi>s</mi> <mi>t</mi> </msub> <mo>−</mo> <mn>1</mn> <mo>)</mo> </mrow> </msup> </mfenced> <mspace width="0.166667em"/> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(52)</label> </div> </div> where <div class='html-disp-formula-info' id='FD53-universe-10-00424'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi mathvariant="script">P</mi> <mn>0</mn> </msub> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <msub> <mi>I</mi> <mn>0</mn> </msub> <msub> <mi>k</mi> <mn>0</mn> </msub> </mrow> <mrow> <mo>(</mo> <msub> <mi>s</mi> <mi>t</mi> </msub> <mo>−</mo> <mn>1</mn> <mo>)</mo> </mrow> </mfrac> </mstyle> <mo>+</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <msub> <mi>C</mi> <mi>NL</mi> </msub> <mrow> <mi>ρ</mi> <msub> <mi>v</mi> <mi>A</mi> </msub> </mrow> </mfrac> </mstyle> <msubsup> <mi>k</mi> <mn>0</mn> <mn>3</mn> </msubsup> <msubsup> <mi>W</mi> <mi>k</mi> <mn>2</mn> </msubsup> <mrow> <mo>(</mo> <msub> <mi>k</mi> <mn>0</mn> </msub> <mo>)</mo> </mrow> <mspace width="0.166667em"/> <mo>.</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(53)</label> </div> </div></div><div class='html-p'>A cutoff in the spectrum of the waves <math display='inline'><semantics> <msub> <mi>W</mi> <mi>k</mi> </msub> </semantics></math> can be estimated from the balance between damping and the cascade at large wavenumbers.</div><div class='html-p'>Unlike the classical GALPROP code with accepted arbitrary parameters of the kinetic diffusion (e.g., [<a href="#B73-universe-10-00424" class="html-bibr">73</a>,<a href="#B74-universe-10-00424" class="html-bibr">74</a>,<a href="#B75-universe-10-00424" class="html-bibr">75</a>]), we derive the coefficients of the kinetic equation for CRs in the FB envelope from Equation (<a href="#FD47-universe-10-00424" class="html-disp-formula">47</a>) for the MHD turbulence:<div class='html-disp-formula-info' id='FD54-universe-10-00424'> <div class='f'> <math display='block'><semantics> <mrow> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mo>∂</mo> <mi>F</mi> <mo>(</mo> <mi>p</mi> <mo>)</mo> </mrow> <mrow> <mo>∂</mo> <mi>t</mi> </mrow> </mfrac> </mstyle> <mo>+</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mi>F</mi> <mo>(</mo> <mi>p</mi> <mo>)</mo> </mrow> <msub> <mi>τ</mi> <mi>esc</mi> </msub> </mfrac> </mstyle> <mo>−</mo> <mi>Q</mi> <mrow> <mo>(</mo> <mi>p</mi> <mo>,</mo> <mi>z</mi> <mo>)</mo> </mrow> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mn>1</mn> <msup> <mi>p</mi> <mn>2</mn> </msup> </mfrac> </mstyle> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mo>∂</mo> <mrow> <mo>∂</mo> <mi>p</mi> </mrow> </mfrac> </mstyle> <msup> <mi>p</mi> <mn>2</mn> </msup> <mfenced separators="" open="[" close="]"> <msub> <mi>D</mi> <mi>p</mi> </msub> <mrow> <mo>(</mo> <mi>p</mi> <mo>)</mo> </mrow> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mo>∂</mo> <mi>F</mi> <mo>(</mo> <mi>p</mi> <mo>)</mo> </mrow> <mrow> <mo>∂</mo> <mi>p</mi> </mrow> </mfrac> </mstyle> <mo>−</mo> <mfenced separators="" open="(" close=")"> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mi>d</mi> <mi>p</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mstyle> </mfenced> <mi>F</mi> <mrow> <mo>(</mo> <mi>p</mi> <mo>)</mo> </mrow> </mfenced> <mo>+</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mo>∂</mo> <mrow> <mo>∂</mo> <mi>z</mi> </mrow> </mfrac> </mstyle> <msub> <mi>D</mi> <mrow> <mi>z</mi> <mi>z</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>p</mi> <mo>)</mo> </mrow> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mo>∂</mo> <mi>F</mi> <mo>(</mo> <mi>p</mi> <mo>)</mo> </mrow> <mrow> <mo>∂</mo> <mi>z</mi> </mrow> </mfrac> </mstyle> <mspace width="0.166667em"/> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(54)</label> </div> </div> where <math display='inline'><semantics> <mrow> <mo>(</mo> <mi>d</mi> <mi>p</mi> <mo>/</mo> <mi>d</mi> <mi>t</mi> <mo>)</mo> </mrow> </semantics></math> (>0) is the rate of continuous energy losses, <math display='inline'><semantics> <msub> <mi>τ</mi> <mi>esc</mi> </msub> </semantics></math> is catastrophic CR losses or the characteristic time of CR escape from the envelope, <math display='inline'><semantics> <mrow> <mi>Q</mi> <mo>(</mo> <mi>p</mi> <mo>,</mo> <mi>z</mi> <mo>)</mo> </mrow> </semantics></math> is the internal sources of CRs, <math display='inline'><semantics> <msub> <mi>D</mi> <mrow> <mi>z</mi> <mi>z</mi> </mrow> </msub> </semantics></math> is the coefficient of spatial diffusion, <div class='html-disp-formula-info' id='FD55-universe-10-00424'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>D</mi> <mrow> <mi>z</mi> <mi>z</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>p</mi> <mo>)</mo> </mrow> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mi>v</mi> <msup> <mi>B</mi> <mn>2</mn> </msup> </mrow> <mrow> <mn>6</mn> <msup> <mi>π</mi> <mn>2</mn> </msup> <msup> <mi>k</mi> <mn>2</mn> </msup> <msub> <mi>W</mi> <mi>k</mi> </msub> <mrow> <mo>(</mo> <mover accent="true"> <mi>k</mi> <mo>¯</mo> </mover> <mo>)</mo> </mrow> </mrow> </mfrac> </mstyle> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mn>2</mn> <mi>ρ</mi> <msubsup> <mi>v</mi> <mi>A</mi> <mn>2</mn> </msubsup> <mi>v</mi> </mrow> <mrow> <mn>3</mn> <mi>π</mi> <msup> <mover accent="true"> <mi>k</mi> <mo>¯</mo> </mover> <mn>2</mn> </msup> <mi>W</mi> <mrow> <mo>(</mo> <mover accent="true"> <mi>k</mi> <mo>¯</mo> </mover> <mo>)</mo> </mrow> </mrow> </mfrac> </mstyle> <mspace width="0.166667em"/> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(55)</label> </div> </div> and <math display='inline'><semantics> <msub> <mi>D</mi> <mi>p</mi> </msub> </semantics></math> is the coefficient of momentum diffusion and is described by Equations (<a href="#FD49-universe-10-00424" class="html-disp-formula">49</a>) and (<a href="#FD50-universe-10-00424" class="html-disp-formula">50</a>). Here, <math display='inline'><semantics> <mrow> <mover accent="true"> <mi>k</mi> <mo>¯</mo> </mover> <mo>=</mo> <mn>1</mn> <mo>/</mo> <msub> <mi>r</mi> <mi>L</mi> </msub> <mo>=</mo> <mi>Z</mi> <mi>e</mi> <mi>B</mi> <mo>/</mo> <mi>p</mi> <mi>c</mi> </mrow> </semantics></math> (cf. Equations (<a href="#FD49-universe-10-00424" class="html-disp-formula">49</a>) and (<a href="#FD50-universe-10-00424" class="html-disp-formula">50</a>)).</div><div class='html-p'>Our goal is to derive the spectrum of CRs from a combination of kinetic MHD/CR equations and to estimate the proper and correct coefficients of Equation (<a href="#FD54-universe-10-00424" class="html-disp-formula">54</a>). However, there is still a gap between the correct coefficients of the kinetic equations and some rough estimations of the spatial and momentum diffusion from the observed gamma-ray and microwave emissions from the FBs. At present we are unable to derive reliable numerical values for these coefficients, and try to estimate these parameters roughly from the data, ignoring the equation for the origin of MHD turbulence needed for CR scattering and propagation. These parameters of the spatial and momentum diffusion coefficients have been roughly or arbitrarily estimated, e.g., by weak random waves of a hydromagnetic turbulence (see [<a href="#B15-universe-10-00424" class="html-bibr">15</a>]), by a supersonic turbulence (see [<a href="#B76-universe-10-00424" class="html-bibr">76</a>]), or by simple estimations of electron acceleration from shocks of the FBs (see [<a href="#B13-universe-10-00424" class="html-bibr">13</a>]), etc.</div><div class='html-p'>In the following, we describe how to roughly estimate the parameters of the spatial and momentum diffusion coefficients from the observed gamma-ray and microwave emissions from the FBs.</div></section><section id='sec7-universe-10-00424' type=''><h2 data-nested='1'> 7. Leptonic and Hadronic Origins of the Radiation from the Fermi Bubbles</h2><div class='html-p'>The origin of CRs in the envelope of the giant bubbles is still an open question. The structure of the bubbles is complicated. It is seen in thermal X-rays as an outer envelope with the following parameters: <div class='html-disp-formula-info' id=''> <div class='f'> <math display='block'><semantics> <mtable displaystyle="true"> <mtr> <mtd/> <mtd/> <mtd columnalign="left"> <mrow> <mi>Power</mi> <mspace width="4.pt"/> <mi>of</mi> <mspace width="4.pt"/> <mi>hydrodynamic</mi> <mspace width="4.pt"/> <mrow> <mi>turbulence</mi> <mtext>:</mtext> </mrow> <mspace width="4pt"/> <mo>∼</mo> <msup> <mn>10</mn> <mn>39</mn> </msup> <mspace width="4pt"/> <mrow> <mi>erg</mi> <msup> <mi mathvariant="normal">s</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> <mo>;</mo> </mrow> </mtd> </mtr> <mtr> <mtd/> <mtd/> <mtd columnalign="left"> <mrow> <mi>Thickness</mi> <mspace width="4.pt"/> <mi>of</mi> <mspace width="4.pt"/> <mi>the</mi> <mspace width="4.pt"/> <mrow> <mi>envelope</mi> <mtext>:</mtext> </mrow> <mspace width="4pt"/> <mo>∼</mo> <mn>100</mn> <mspace width="4pt"/> <mi>pc</mi> <mo>;</mo> </mrow> </mtd> </mtr> <mtr> <mtd/> <mtd/> <mtd columnalign="left"> <mrow> <mi>Scale</mi> <mspace width="4.pt"/> <mi>of</mi> <mspace width="4.pt"/> <mi>eROSITA</mi> <mspace width="4.pt"/> <mrow> <mi>bubbles</mi> <mo>;</mo> </mrow> <mspace width="4pt"/> <mo>∼</mo> <mn>14</mn> <mspace width="4pt"/> <mi>kpc</mi> <mo>;</mo> </mrow> </mtd> </mtr> <mtr> <mtd/> <mtd/> <mtd columnalign="left"> <mrow> <mi>Magnetic</mi> <mspace width="4.pt"/> <mrow> <mi>field</mi> <mo>;</mo> </mrow> <mspace width="4pt"/> <mo>∼</mo> <mn>8</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>6</mn> </mrow> </msup> <mspace width="4pt"/> <mi mathvariant="normal">G</mi> <mo>;</mo> </mrow> </mtd> </mtr> <mtr> <mtd/> <mtd/> <mtd columnalign="left"> <mrow> <mi>Gas</mi> <mspace width="4.pt"/> <mrow> <mi>density</mi> <mtext>:</mtext> </mrow> <mspace width="4pt"/> <mo>∼</mo> <mn>4</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>3</mn> </mrow> </msup> <mspace width="4pt"/> <msup> <mi>cm</mi> <mrow> <mo>−</mo> <mn>3</mn> </mrow> </msup> <mo>;</mo> </mrow> </mtd> </mtr> <mtr> <mtd/> <mtd/> <mtd columnalign="left"> <mrow> <mrow> <mi>Alfvén</mi> </mrow> <mspace width="4.pt"/> <mrow> <mi>velocity</mi> <mtext>:</mtext> </mrow> <mspace width="4pt"/> <mo>∼</mo> <mn>3</mn> <mo>×</mo> <msup> <mn>10</mn> <mn>7</mn> </msup> <mspace width="4pt"/> <mi>cm</mi> <msup> <mi mathvariant="normal">s</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>.</mo> </mrow> </mtd> </mtr> </mtable> </semantics></math> </div> <div class='l'> </div> </div></div><div class='html-p'>An inner envelope of size ∼8 kpc is seen in the nonthermal gamma-ray and microwave emissions (see <a href="#universe-10-00424-f010" class="html-fig">Figure 10</a>). The microwave emission is evidently produced by the synchrotron losses of relativistic electrons, while the origin of the gamma rays is not clear.</div><div class='html-p'>The total gamma-ray luminosity of the bubbles between 100 MeV and 500 GeV is <math display='inline'><semantics> <mrow> <msub> <mi>F</mi> <mi>γ</mi> </msub> <mo>≃</mo> <mn>4.4</mn> <mo>×</mo> <msup> <mn>10</mn> <mn>37</mn> </msup> </mrow> </semantics></math> erg <math display='inline'><semantics> <mrow> <msup> <mi mathvariant="normal">s</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </semantics></math> (see [<a href="#B72-universe-10-00424" class="html-bibr">72</a>]). The spectrum can be described by a power law, <math display='inline'><semantics> <mrow> <mi>d</mi> <msub> <mi>F</mi> <mi>γ</mi> </msub> <mo>/</mo> <mi>d</mi> <msub> <mi>E</mi> <mi>γ</mi> </msub> <mo>∝</mo> <msubsup> <mi>E</mi> <mi>γ</mi> <mrow> <mo>−</mo> <mn>1.87</mn> </mrow> </msubsup> </mrow> </semantics></math>, with a cutoff <math display='inline'><semantics> <mrow> <msub> <mi>E</mi> <mi>cut</mi> </msub> <mo>≃</mo> <mn>113</mn> </mrow> </semantics></math> GeV.</div><section id='sec7dot1-universe-10-00424' type=''><h4 class='html-italic' data-nested='2'> 7.1. Origin of Gamma-Ray Emission from the Fermi Bubbles</h4><div class='html-p'>The spectrum of gamma rays can be fitted by either a leptonic or hadronic model.</div><ul class='html-bullet'><li><div class='html-p'>Leptonic model: The rate of gamma-ray production by relativistic electrons interacting with low-energy interstellar photons is (see [<a href="#B72-universe-10-00424" class="html-bibr">72</a>]) <div class='html-disp-formula-info' id='FD56-universe-10-00424'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>ε</mi> <mi>IC</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>E</mi> <mi>γ</mi> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mi>c</mi> <munder> <mo>∑</mo> <mi>i</mi> </munder> <msub> <mi>n</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>ϵ</mi> <mi>ph</mi> </msub> <mo>)</mo> </mrow> <mo>∫</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mi>d</mi> <msub> <mi>σ</mi> <mi>IC</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>E</mi> <mi>γ</mi> </msub> <mo>,</mo> <msub> <mi>E</mi> <mi>e</mi> </msub> <mo>,</mo> <msub> <mi>ϵ</mi> <mi>ph</mi> </msub> <mo>)</mo> </mrow> </mrow> <mrow> <mi>d</mi> <msub> <mi>E</mi> <mi>γ</mi> </msub> </mrow> </mfrac> </mstyle> <msub> <mi>N</mi> <mi>e</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>E</mi> <mi>e</mi> </msub> <mo>)</mo> </mrow> <mi>d</mi> <msub> <mi>E</mi> <mi>e</mi> </msub> <mspace width="0.166667em"/> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(56)</label> </div> </div> where <math display='inline'><semantics> <msub> <mi>σ</mi> <mi>IC</mi> </msub> </semantics></math> is the inverse Compton (IC) cross-section (see, e.g., [<a href="#B78-universe-10-00424" class="html-bibr">78</a>]). The parameters of the CR electron spectrum were derived from the observed gamma-ray emission from the envelope: <math display='inline'><semantics> <mrow> <msub> <mi>N</mi> <mi>e</mi> </msub> <mo>∝</mo> <msubsup> <mi>E</mi> <mi>e</mi> <mrow> <mo>−</mo> <mn>2.17</mn> </mrow> </msubsup> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msub> <mi>E</mi> <mi>cut</mi> </msub> <mo>∼</mo> <mn>1.25</mn> </mrow> </semantics></math> TeV. The required total energy in electrons above 1 GeV is <math display='inline'><semantics> <mrow> <msub> <mi mathvariant="script">E</mi> <mi>e</mi> </msub> <mo>∼</mo> <msup> <mn>10</mn> <mn>52</mn> </msup> </mrow> </semantics></math> erg.</div></li><li><div class='html-p'>Hadronic model: Gamma rays can be produced by proton–proton (p-p) collisions. For calculations of the emission from p-p collision, Ackermann et al. [<a href="#B72-universe-10-00424" class="html-bibr">72</a>] used the p-p cross-section from Kamae et al. [<a href="#B79-universe-10-00424" class="html-bibr">79</a>]. The rate of gamma-ray production by p-p collisions is <div class='html-disp-formula-info' id='FD57-universe-10-00424'> <div class='f'> <math display='block'><semantics> <mrow> <mspace width="-0.166667em"/> <mspace width="-0.166667em"/> <msub> <mi>ε</mi> <mi>pp</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>E</mi> <mi>γ</mi> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mi>c</mi> <mo>∫</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mi>d</mi> <msub> <mi>σ</mi> <mi>pp</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>E</mi> <mi>γ</mi> </msub> <mo>,</mo> <msub> <mi>E</mi> <mi>p</mi> </msub> <mo>)</mo> </mrow> </mrow> <mrow> <mi>d</mi> <msub> <mi>E</mi> <mi>γ</mi> </msub> </mrow> </mfrac> </mstyle> <msub> <mi>n</mi> <mi>H</mi> </msub> <msub> <mi>N</mi> <mi>p</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>E</mi> <mi>p</mi> </msub> <mo>)</mo> </mrow> <mi>d</mi> <msub> <mi>E</mi> <mi>p</mi> </msub> <mspace width="0.166667em"/> <mo>.</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(57)</label> </div> </div></div><div class='html-p'>The required spectrum of CR protons is expressed as <math display='inline'><semantics> <mrow> <mi>d</mi> <msub> <mi>N</mi> <mi>p</mi> </msub> <mo>/</mo> <mi>d</mi> <msub> <mi>E</mi> <mi>p</mi> </msub> <mo>∝</mo> <msubsup> <mi>E</mi> <mi>p</mi> <mrow> <mo>−</mo> <mn>2.13</mn> </mrow> </msubsup> <mo form="prefix">exp</mo> <mrow> <mo>(</mo> <mo>−</mo> <msub> <mi>E</mi> <mi>p</mi> </msub> <mo>/</mo> <msub> <mi>E</mi> <mi>cut</mi> </msub> <mo>)</mo> </mrow> </mrow> </semantics></math>, where <math display='inline'><semantics> <mrow> <msub> <mi>E</mi> <mi>cut</mi> </msub> <mo>∼</mo> <mn>14</mn> </mrow> </semantics></math> TeV. The total required energy in CR protons above 1 GeV is <math display='inline'><semantics> <mrow> <msub> <mi mathvariant="script">E</mi> <mi>p</mi> </msub> <mo>∼</mo> <mn>3.5</mn> <mo>×</mo> <msup> <mn>10</mn> <mn>55</mn> </msup> </mrow> </semantics></math> erg for <math display='inline'><semantics> <mrow> <msub> <mi>n</mi> <mi>H</mi> </msub> <mo>=</mo> <mn>0.01</mn> <msup> <mi>cm</mi> <mrow> <mo>−</mo> <mn>3</mn> </mrow> </msup> </mrow> </semantics></math>.</div></li></ul><div class='html-p'>In principle, we can interpret the gamma-ray emission (the right panel of <a href="#universe-10-00424-f010" class="html-fig">Figure 10</a>) by both leptonic (IC) and hadronic (p-p) models for the correspondingly derived parameters of CRs. The question is whether the observed microwave spectrum from the FBs is also compatible with the leptonic or hadronic model.</div></section><section id='sec7dot2-universe-10-00424' type=''><h4 class='html-italic' data-nested='2'> 7.2. Microwave Origin in Cosmic-Ray Electron Model</h4><div class='html-p'>The origin of the microwave radiation (see left panel of <a href="#universe-10-00424-f010" class="html-fig">Figure 10</a>) was analyzed for the leptonic and hadronic models (see [<a href="#B72-universe-10-00424" class="html-bibr">72</a>]), with the goal of fitting the gamma-ray and microwave observations within the same model.</div><div class='html-p'>The electrons in the IC scenario should also produce the observed WMAP and Planck microwave spectrum and flux. Their properties can be derived from the observed density of gamma rays produced by relativistic electrons that interact with the low-energy interstellar photons in the IC scenario for a magnetic field in the FBs in the range <math display='inline'><semantics> <mn>5</mn> </semantics></math> μG to <math display='inline'><semantics> <mn>20</mn> </semantics></math> μG (see [<a href="#B72-universe-10-00424" class="html-bibr">72</a>]). The best-fit magnetic field is about <math display='inline'><semantics> <mn>8.4</mn> </semantics></math> μG. The synchrotron flux of the FBs from these electrons can be estimated as (see [<a href="#B80-universe-10-00424" class="html-bibr">80</a>,<a href="#B81-universe-10-00424" class="html-bibr">81</a>]) <div class='html-disp-formula-info' id='FD58-universe-10-00424'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mo>Φ</mo> <mi>ν</mi> </msub> <mo>≃</mo> <mn>4</mn> <mi>π</mi> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <msqrt> <mn>3</mn> </msqrt> <msup> <mi>e</mi> <mn>3</mn> </msup> </mrow> <mrow> <msub> <mi>m</mi> <mi>e</mi> </msub> <msup> <mi>c</mi> <mn>2</mn> </msup> </mrow> </mfrac> </mstyle> <msubsup> <mo>∫</mo> <mrow> <msub> <mi>r</mi> <mrow> <mi>m</mi> <mi>i</mi> <mi>n</mi> </mrow> </msub> </mrow> <msub> <mi>r</mi> <mn>0</mn> </msub> </msubsup> <mi>B</mi> <mrow> <mo>(</mo> <mi>r</mi> <mo>)</mo> </mrow> <msup> <mi>r</mi> <mn>2</mn> </msup> <mspace width="0.166667em"/> <mi>d</mi> <mi>r</mi> <mspace width="0.166667em"/> <mo>∫</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mi>ν</mi> <msub> <mi>ν</mi> <mi>c</mi> </msub> </mfrac> </mstyle> <msub> <mi>N</mi> <mi>e</mi> </msub> <mrow> <mo>(</mo> <mi>E</mi> <mo>)</mo> </mrow> <mspace width="0.166667em"/> <mi>d</mi> <mi>E</mi> <mspace width="0.166667em"/> <msubsup> <mo>∫</mo> <mrow> <mi>ν</mi> <mo>/</mo> <msub> <mi>ν</mi> <mi>c</mi> </msub> </mrow> <mo>∞</mo> </msubsup> <msub> <mi>K</mi> <mrow> <mn>5</mn> <mo>/</mo> <mn>3</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>η</mi> <mo>)</mo> </mrow> <mspace width="0.166667em"/> <mi>d</mi> <mi>η</mi> <mspace width="0.166667em"/> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(58)</label> </div> </div> where <math display='inline'><semantics> <mrow> <msub> <mi>K</mi> <mi>μ</mi> </msub> <mrow> <mo>(</mo> <mi>η</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> is the McDonald function, and <div class='html-disp-formula-info' id='FD59-universe-10-00424'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>ν</mi> <mi>c</mi> </msub> <mrow> <mo>(</mo> <mi>r</mi> <mo>,</mo> <mi>E</mi> <mo>)</mo> </mrow> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mn>3</mn> <mi>e</mi> <mi>B</mi> <mo>(</mo> <mi>r</mi> <mo>)</mo> </mrow> <mrow> <mn>4</mn> <mi>π</mi> <msub> <mi>m</mi> <mi>e</mi> </msub> <mi>c</mi> </mrow> </mfrac> </mstyle> <msup> <mfenced separators="" open="(" close=")"> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mi>E</mi> <mrow> <msub> <mi>m</mi> <mi>e</mi> </msub> <msup> <mi>c</mi> <mn>2</mn> </msup> </mrow> </mfrac> </mstyle> </mfenced> <mn>2</mn> </msup> <mspace width="0.166667em"/> <mo>.</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(59)</label> </div> </div></div><div class='html-p'><math display='inline'><semantics> <mrow> <msub> <mo>Φ</mo> <mi>ν</mi> </msub> <mrow> <mo>(</mo> <mi>ν</mi> <mrow> <mo>(</mo> <msub> <mi>E</mi> <mi>e</mi> </msub> <mo>)</mo> </mrow> <mo>)</mo> </mrow> </mrow> </semantics></math> can be estimated from the density of relativistic electrons <math display='inline'><semantics> <mrow> <msub> <mi>N</mi> <mi>e</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>E</mi> <mi>e</mi> </msub> <mo>)</mo> </mrow> </mrow> </semantics></math> in the FB envelope as <div class='html-disp-formula-info' id='FD60-universe-10-00424'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>N</mi> <mi>e</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>E</mi> <mi>e</mi> </msub> <mo>)</mo> </mrow> <mo>≃</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mi>χ</mi> <msub> <mi>V</mi> <mn>0</mn> </msub> </mfrac> </mstyle> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <msub> <mo>Φ</mo> <mi>ν</mi> </msub> <mrow> <mo>(</mo> <mi>ν</mi> <mrow> <mo>(</mo> <msub> <mi>E</mi> <mi>e</mi> </msub> <mo>)</mo> </mrow> <mo>)</mo> </mrow> </mrow> <msub> <mi>E</mi> <mi>e</mi> </msub> </mfrac> </mstyle> <mspace width="0.166667em"/> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(60)</label> </div> </div> where the parameter <math display='inline'><semantics> <mi>χ</mi> </semantics></math> is <div class='html-disp-formula-info' id='FD61-universe-10-00424'> <div class='f'> <math display='block'><semantics> <mrow> <mi>χ</mi> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mn>3</mn> <msubsup> <mi>m</mi> <mi>e</mi> <mn>3</mn> </msubsup> <msup> <mi>c</mi> <mn>5</mn> </msup> <msub> <mi>ν</mi> <mn>0</mn> </msub> </mrow> <mrow> <msup> <mi>e</mi> <mn>4</mn> </msup> <msubsup> <mi>B</mi> <mn>0</mn> <mn>2</mn> </msubsup> </mrow> </mfrac> </mstyle> <mspace width="0.166667em"/> <mo>.</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(61)</label> </div> </div></div><div class='html-p'>Although the main contribution to the IC signal comes from electrons at energies <math display='inline'><semantics> <mrow> <mo>></mo> <mn>100</mn> </mrow> </semantics></math> GeV, while the main contribution to the Planck frequencies comes from electrons between 10∼30 GeV, they match each other in the IC model.</div></section><section id='sec7dot3-universe-10-00424' type=''><h4 class='html-italic' data-nested='2'> 7.3. Microwave Origin in Cosmic-Ray Proton Model</h4><div class='html-p'>In the pure hadronic model (see [<a href="#B72-universe-10-00424" class="html-bibr">72</a>,<a href="#B82-universe-10-00424" class="html-bibr">82</a>]), the FB synchrotron emission is produced by secondary electrons from collisions of primary protons.</div><div class='html-p'>The approximated equations for the spectrum of secondary electrons produced by p-p and knock-on (KO) collisions (<math display='inline'><semantics> <mrow> <mi>p</mi> <mi>e</mi> </mrow> </semantics></math> and <math display='inline'><semantics> <mrow> <mi>e</mi> <mi>e</mi> </mrow> </semantics></math>), can be written as (see, e.g., [<a href="#B83-universe-10-00424" class="html-bibr">83</a>,<a href="#B84-universe-10-00424" class="html-bibr">84</a>]) <div class='html-disp-formula-info' id='FD62-universe-10-00424'> <div class='f'> <math display='block'><semantics> <mtable displaystyle="true"> <mtr> <mtd/> <mtd/> <mtd columnalign="left"> <mrow> <msub> <mi>N</mi> <mi>se</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>E</mi> <mi>e</mi> </msub> <mo>,</mo> <mi>r</mi> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>τ</mi> <mi>e</mi> </msub> <msubsup> <mo>∫</mo> <mrow> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <msub> <mi>m</mi> <mi>p</mi> </msub> <msub> <mi>m</mi> <mi>e</mi> </msub> </mfrac> </mstyle> <msub> <mi>E</mi> <mi>e</mi> </msub> </mrow> <mo>∞</mo> </msubsup> <msub> <mi>N</mi> <mi>p</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>E</mi> <mi>p</mi> </msub> <mo>,</mo> <mi>r</mi> <mo>)</mo> </mrow> <msub> <mi>n</mi> <mi>H</mi> </msub> <msub> <mi>v</mi> <mi>p</mi> </msub> <mspace width="0.166667em"/> <mi>d</mi> <msub> <mi>σ</mi> <mi>pp</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>E</mi> <mi>p</mi> </msub> <mo>)</mo> </mrow> <mspace width="0.166667em"/> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </semantics></math> </div> <div class='l'> <label >(62)</label> </div> </div><div class='html-disp-formula-info' id='FD63-universe-10-00424'> <div class='f'> <math display='block'><semantics> <mtable displaystyle="true"> <mtr> <mtd/> <mtd/> <mtd columnalign="left"> <mrow> <msub> <mi>N</mi> <mi>ss</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>E</mi> <mi>e</mi> </msub> <mo>,</mo> <mi>r</mi> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>τ</mi> <mi>e</mi> </msub> <msubsup> <mo>∫</mo> <mrow> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <msub> <mi>m</mi> <mi>p</mi> </msub> <msub> <mi>m</mi> <mi>e</mi> </msub> </mfrac> </mstyle> <msub> <mi>E</mi> <mi>e</mi> </msub> </mrow> <mo>∞</mo> </msubsup> <msub> <mi>N</mi> <mi>p</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>E</mi> <mi>p</mi> </msub> <mo>,</mo> <mi>r</mi> <mo>)</mo> </mrow> <msub> <mi>v</mi> <mi>p</mi> </msub> <msub> <mi>n</mi> <mi>H</mi> </msub> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mi>d</mi> <msub> <mi>σ</mi> <mi>KO</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>E</mi> <mi>p</mi> </msub> <mo>,</mo> <msup> <mi>E</mi> <mo>′</mo> </msup> <mo>)</mo> </mrow> </mrow> <mrow> <mi>d</mi> <msup> <mi>E</mi> <mo>′</mo> </msup> </mrow> </mfrac> </mstyle> <mspace width="0.166667em"/> <mi>d</mi> <msub> <mi>E</mi> <mi>p</mi> </msub> <mspace width="0.166667em"/> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </semantics></math> </div> <div class='l'> <label >(63)</label> </div> </div> where <math display='inline'><semantics> <msub> <mi>τ</mi> <mi>e</mi> </msub> </semantics></math> is an integral over the rate of electron energy losses:<div class='html-disp-formula-info' id='FD64-universe-10-00424'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>τ</mi> <mi>e</mi> </msub> <mo>∼</mo> <msubsup> <mo>∫</mo> <mrow> <msub> <mi>E</mi> <mi>e</mi> </msub> </mrow> <msub> <mi>E</mi> <mi>max</mi> </msub> </msubsup> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mi>d</mi> <msub> <mi>E</mi> <mi>e</mi> </msub> </mrow> <msub> <mrow> <mo>(</mo> <mi>d</mi> <msub> <mi>E</mi> <mi>e</mi> </msub> <mo>/</mo> <mi>d</mi> <mi>t</mi> <mo>)</mo> </mrow> <mi>i</mi> </msub> </mfrac> </mstyle> <mspace width="0.166667em"/> <mo>.</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(64)</label> </div> </div>Here <math display='inline'><semantics> <msub> <mrow> <mo>(</mo> <mi>d</mi> <msub> <mi>E</mi> <mi>e</mi> </msub> <mo>/</mo> <mi>d</mi> <mi>t</mi> <mo>)</mo> </mrow> <mi>i</mi> </msub> </semantics></math> can be determined by bremsstrahlung, ionization, synchrotron, or particle escape.</div><div class='html-p'>The synchrotron emission of secondary electrons from the FB envelope can be calculated from Equation (<a href="#FD58-universe-10-00424" class="html-disp-formula">58</a>).</div><div class='html-p'>The pure hadronic model is unable to reproduce both gamma-ray and radio fluxes from the FBs at the same time. The problem is that the secondary electrons and positrons in the hadronic scenario produce synchrotron radiation with a spectrum that is too soft compared to the microwave haze spectrum, whereas the overall normalization of the synchrotron radiation from the secondary particles is at least a factor of three to four smaller than the microwave level that a hadronic model requires (see [<a href="#B82-universe-10-00424" class="html-bibr">82</a>]).</div><div class='html-p'>Thus, we conclude that a purely hadronic origin of the nonthermal emission (gamma and radio) from the FBs is problematic.</div></section></section><section id='sec8-universe-10-00424' type=''><h2 data-nested='1'> 8. Number of Relativistic Electrons in the Fermi Bubble Envelope</h2><div class='html-p'>The origin of relativistic electrons in the FB envelope is an open question. It was assumed that the FB envelopes might be bounded by a shock with the velocity <math display='inline'><semantics> <mrow> <msub> <mi>v</mi> <mi>sh</mi> </msub> <mo>∼</mo> <msup> <mn>10</mn> <mn>8</mn> </msup> </mrow> </semantics></math> cm <math display='inline'><semantics> <mrow> <msup> <mi mathvariant="normal">s</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </semantics></math> (see, e.g., [<a href="#B13-universe-10-00424" class="html-bibr">13</a>,<a href="#B85-universe-10-00424" class="html-bibr">85</a>,<a href="#B86-universe-10-00424" class="html-bibr">86</a>]). It was proposed that these CRs were accelerated at the shock by the standard mechanism of shock acceleration, with the CR spectrum <math display='inline'><semantics> <mrow> <mi>Q</mi> <mrow> <mo>(</mo> <msub> <mi>E</mi> <mi>e</mi> </msub> <mo>)</mo> </mrow> <mo>∝</mo> <msubsup> <mi>E</mi> <mi>e</mi> <mrow> <mo>−</mo> <mn>2</mn> </mrow> </msubsup> </mrow> </semantics></math> (see, e.g., [<a href="#B87-universe-10-00424" class="html-bibr">87</a>,<a href="#B88-universe-10-00424" class="html-bibr">88</a>,<a href="#B89-universe-10-00424" class="html-bibr">89</a>,<a href="#B90-universe-10-00424" class="html-bibr">90</a>]). However, eROSITA (see [<a href="#B1-universe-10-00424" class="html-bibr">1</a>]) found that the velocity of the shock with which X-ray giant bubbles propagate is about 340 km <math display='inline'><semantics> <mrow> <msup> <mi mathvariant="normal">s</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </semantics></math> and its Mach number is only ≈1.5, which does not correspond to an effective CR shock acceleration. Thus, CRs are not accelerated by a shock near the outer shell of eROSITA. Therefore, CRs should be produced by in situ stochastic acceleration by MHD turbulence <math display='inline'><semantics> <msub> <mi>W</mi> <mi>k</mi> </msub> </semantics></math> near the inner bubble surface (see <a href="#sec6-universe-10-00424" class="html-sec">Section 6</a> the function <math display='inline'><semantics> <msub> <mi>W</mi> <mi>k</mi> </msub> </semantics></math>).</div><div class='html-p'>In the following, we focus on the in situ stochastic (Fermi) acceleration of CRs by a hydromagnetic/supersonic turbulence.</div><section id='sec8dot1-universe-10-00424' type=''><h4 class='html-italic' data-nested='2'> 8.1. Electrons Accelerated from Background Plasma in the Fermi Bubbles</h4><div class='html-p'>In the model of Cheng et al. [<a href="#B71-universe-10-00424" class="html-bibr">71</a>], CR electrons can be directly accelerated from a background plasma. They suggested that the acceleration from the background plasma is able to explain the origin of the nonthermal particles responsible for producing the observed fluxes of radio and gamma-ray emissions from the bubbles.</div><div class='html-p'>The kinetic equation for the distribution function of electrons, <math display='inline'><semantics> <mrow> <mi>F</mi> <mo>(</mo> <mi>p</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </semantics></math>, in the case of in situ acceleration is described as <div class='html-disp-formula-info' id='FD65-universe-10-00424'> <div class='f'> <math display='block'><semantics> <mrow> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mo>∂</mo> <mi>F</mi> <mo>(</mo> <mi>p</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> <mrow> <mo>∂</mo> <mi>t</mi> </mrow> </mfrac> </mstyle> <mo>+</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mi>F</mi> <mo>(</mo> <mi>p</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> <msub> <mi>τ</mi> <mi>esc</mi> </msub> </mfrac> </mstyle> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mn>1</mn> <msup> <mi>p</mi> <mn>2</mn> </msup> </mfrac> </mstyle> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mo>∂</mo> <mrow> <mo>∂</mo> <mi>p</mi> </mrow> </mfrac> </mstyle> <msup> <mi>p</mi> <mn>2</mn> </msup> <mfenced separators="" open="[" close="]"> <mo>−</mo> <msub> <mfenced separators="" open="(" close=")"> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mi>d</mi> <mi>p</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mstyle> </mfenced> <mi mathvariant="normal">C</mi> </msub> <mi>F</mi> <mrow> <mo>(</mo> <mi>p</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <mfenced separators="" open="{" close="}"> <msub> <mi>D</mi> <mi mathvariant="normal">C</mi> </msub> <mrow> <mo>(</mo> <mi>p</mi> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>D</mi> <mi mathvariant="normal">F</mi> </msub> <mrow> <mo>(</mo> <mi>p</mi> <mo>)</mo> </mrow> </mfenced> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mo>∂</mo> <mi>F</mi> <mo>(</mo> <mi>p</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> <mrow> <mo>∂</mo> <mi>p</mi> </mrow> </mfrac> </mstyle> </mfenced> <mspace width="0.166667em"/> <mo>.</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(65)</label> </div> </div>The distribution function includes the thermal and nonthermal components of the particle distribution. The coefficient <math display='inline'><semantics> <msub> <mrow> <mo>(</mo> <mi>d</mi> <mi>p</mi> <mo>/</mo> <mi>d</mi> <mi>t</mi> <mo>)</mo> </mrow> <mi mathvariant="normal">C</mi> </msub> </semantics></math> describes the particle ionization/Coulomb energy losses. <math display='inline'><semantics> <mrow> <msub> <mi>D</mi> <mi mathvariant="normal">C</mi> </msub> <mrow> <mo>(</mo> <mi>p</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> describes the diffusion in the momentum space due to Coulomb collisions (for details, see [<a href="#B91-universe-10-00424" class="html-bibr">91</a>]). The stochastic (Fermi) acceleration is described as diffusion in the momentum space with the diffusion coefficient <math display='inline'><semantics> <mrow> <msub> <mi>D</mi> <mi mathvariant="normal">F</mi> </msub> <mrow> <mo>(</mo> <mi>p</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>. <math display='inline'><semantics> <msub> <mi>τ</mi> <mi>esc</mi> </msub> </semantics></math> is the lifetime of particles in the region of acceleration, e.g., due to escape from the region.</div><div class='html-p'>In an ionized plasma, the equilibrium (Maxwellian) spectrum of background charged particles is formed by Coulomb collisions. For the case of CR acceleration, there is a boundary, <math display='inline'><semantics> <mrow> <mi>E</mi> <mo>=</mo> <msub> <mi>E</mi> <mi>inj</mi> </msub> </mrow> </semantics></math>, between the equilibrium Maxwellian distribution and a power-law nonthermal spectrum of accelerated particles:<div class='html-disp-formula-info' id='FD66-universe-10-00424'> <div class='f'> <math display='block'><semantics> <mrow> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mi>d</mi> <mi>E</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mstyle> <mo>=</mo> <msub> <mi>α</mi> <mn>0</mn> </msub> <mi>E</mi> <mo>−</mo> <msub> <mi>ν</mi> <mn>0</mn> </msub> <mi>E</mi> <msup> <mfenced separators="" open="(" close=")"> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mi>k</mi> <mi>T</mi> </mrow> <mi>E</mi> </mfrac> </mstyle> </mfenced> <mrow> <mn>3</mn> <mo>/</mo> <mn>2</mn> </mrow> </msup> <mspace width="0.166667em"/> <mo>.</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(66)</label> </div> </div>The energy of injection is <div class='html-disp-formula-info' id='FD67-universe-10-00424'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>E</mi> <mi>inj</mi> </msub> <mo>∼</mo> <mi>k</mi> <mi>T</mi> <msup> <mfenced separators="" open="(" close=")"> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <msub> <mi>ν</mi> <mn>0</mn> </msub> <msub> <mi>α</mi> <mn>0</mn> </msub> </mfrac> </mstyle> </mfenced> <mrow> <mn>2</mn> <mo>/</mo> <mn>3</mn> </mrow> </msup> <mspace width="0.166667em"/> <mo>.</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(67)</label> </div> </div>Here, the parameters <math display='inline'><semantics> <msub> <mi>α</mi> <mn>0</mn> </msub> </semantics></math> and <math display='inline'><semantics> <msub> <mi>ν</mi> <mn>0</mn> </msub> </semantics></math> are the acceleration and the ionization loss (by Coulomb collisions), respectively.</div><div class='html-p'>Making use of Equation (<a href="#FD65-universe-10-00424" class="html-disp-formula">65</a>), Gurevich [<a href="#B92-universe-10-00424" class="html-bibr">92</a>] studied the process of the formal connection between the equilibrium Maxwellian distribution of background particles and a power-law non-equilibrium spectrum of accelerated particles.</div><div class='html-p'>For slow time variations, the runaway flux of particles from the region of thermal Maxwellian distribution into the acceleration range is (see, e.g., [<a href="#B93-universe-10-00424" class="html-bibr">93</a>]) <div class='html-disp-formula-info' id='FD68-universe-10-00424'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>S</mi> <mi>ζ</mi> </msub> <mo>=</mo> <msub> <mi>S</mi> <mn>0</mn> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <msqrt> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mn>2</mn> <mi>π</mi> </mfrac> </mstyle> </msqrt> <msubsup> <mo>∫</mo> <mn>0</mn> <mi>ζ</mi> </msubsup> <msup> <mi>ζ</mi> <mn>2</mn> </msup> <mo form="prefix">exp</mo> <mfenced separators="" open="(" close=")"> <mo>−</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <msup> <mi>ζ</mi> <mn>2</mn> </msup> <mn>2</mn> </mfrac> </mstyle> </mfenced> <mi>d</mi> <mi>ζ</mi> <mspace width="0.166667em"/> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(68)</label> </div> </div> where <math display='inline'><semantics> <mrow> <mi>ζ</mi> <mo>=</mo> <mi>p</mi> <mo>/</mo> <msub> <mi>p</mi> <mn>0</mn> </msub> </mrow> </semantics></math> is the normalized momentum, and <div class='html-disp-formula-info' id='FD69-universe-10-00424'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>S</mi> <mn>0</mn> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <msqrt> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mn>2</mn> <mi>π</mi> </mfrac> </mstyle> </msqrt> <mi>n</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo form="prefix">exp</mo> <mfenced separators="" open="(" close=")"> <mo>−</mo> <msubsup> <mo>∫</mo> <mn>0</mn> <mo>∞</mo> </msubsup> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mi>ζ</mi> <mspace width="0.166667em"/> <mi>d</mi> <mi>ζ</mi> </mrow> <mrow> <mo>(</mo> <mn>1</mn> <mo>+</mo> <msub> <mi>α</mi> <mn>0</mn> </msub> <msup> <mi>ζ</mi> <mn>5</mn> </msup> <mo>/</mo> <msub> <mi>ν</mi> <mn>0</mn> </msub> <mo>)</mo> </mrow> </mfrac> </mstyle> </mfenced> <mspace width="0.166667em"/> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(69)</label> </div> </div> and <math display='inline'><semantics> <mrow> <mi>n</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </semantics></math> is the slow variation in the gas density.</div><div class='html-p'>In a non-equilibrium case, this process forms an escape flux of runaway particles for energy <math display='inline'><semantics> <mrow> <mi>E</mi> <mo>></mo> <msub> <mi>E</mi> <mi>inj</mi> </msub> </mrow> </semantics></math>. On the other hand, this process also forms an excess density at energies <math display='inline'><semantics> <mrow> <mi>E</mi> <mo>≲</mo> <msub> <mi>E</mi> <mi>inj</mi> </msub> </mrow> </semantics></math>, which distorts the thermal Maxwellian distribution. This can be interpreted as a “second” effective temperature, higher than the gas equilibrium temperature.</div><div class='html-p'>This model of Gurevich [<a href="#B92-universe-10-00424" class="html-bibr">92</a>] has been applied to the processes of particle acceleration from background plasma in galaxy clusters (see [<a href="#B93-universe-10-00424" class="html-bibr">93</a>,<a href="#B94-universe-10-00424" class="html-bibr">94</a>,<a href="#B95-universe-10-00424" class="html-bibr">95</a>]), in the GC (see [<a href="#B71-universe-10-00424" class="html-bibr">71</a>]), and in the galactic disk [<a href="#B96-universe-10-00424" class="html-bibr">96</a>], in which excesses above thermal particles in the X-ray range is expected.</div><div class='html-p'>For example, the spectrum of X-ray emission from the galactic plane can be described as a multi-temperature emission. Dogiel et al. [<a href="#B96-universe-10-00424" class="html-bibr">96</a>] interpreted this X-ray emission as the flux of runaway particles from background gas as a common effect of Coulomb collision (ionization losses) and stochastic acceleration (see <a href="#universe-10-00424-f011" class="html-fig">Figure 11</a>). When compared with the spectrum of a simple combination of thermal and nonthermal gas, the spectrum with runaway flux is larger, in particular, in the transition range between the thermal and nonthermal parts.</div><div class='html-p'>The model of particle excess from background gas proposed for CRs by Dogiel [<a href="#B93-universe-10-00424" class="html-bibr">93</a>] was challenged by Petrosian [<a href="#B97-universe-10-00424" class="html-bibr">97</a>], Wolfe & Melia [<a href="#B98-universe-10-00424" class="html-bibr">98</a>], and Petrosian & East [<a href="#B99-universe-10-00424" class="html-bibr">99</a>]. The problem was that the stochastic acceleration of the accelerated particles from a background plasma would (over) heat the plasma, because their energy would be quickly dumped into the thermal plasma by ionization losses. The energy gained by the particles is distributed to the whole plasma on a time scale much shorter than that of the acceleration process itself. As a result of the relative inefficiency of bremsstrahlung for cooling the accelerated electrons, this tail is quickly dumped into the thermal body of the background plasma (plasma overheating without a prominent tail of accelerated particles). This effect completely prevents the formation of nonthermal spectra from background plasma.</div><div class='html-p'>However, Chernyshov et al. [<a href="#B100-universe-10-00424" class="html-bibr">100</a>] showed that the effect of overheating depends on the parameters of acceleration. It is insignificant if the stochastic acceleration is effective. This model depends on a value of <math display='inline'><semantics> <msub> <mi>p</mi> <mi>inj</mi> </msub> </semantics></math> and a free parameter of stochastic acceleration <math display='inline'><semantics> <msub> <mi>p</mi> <mn>0</mn> </msub> </semantics></math> in the form of <div class='html-disp-formula-info' id='FD70-universe-10-00424'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>D</mi> <mi mathvariant="normal">F</mi> </msub> <mrow> <mo>(</mo> <mi>p</mi> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>D</mi> <mn>0</mn> </msub> <mspace width="0.166667em"/> <msup> <mi>p</mi> <mi>ς</mi> </msup> <mi>θ</mi> <mrow> <mo>(</mo> <mi>p</mi> <mo>−</mo> <msub> <mi>p</mi> <mn>0</mn> </msub> <mo>)</mo> </mrow> <mspace width="0.166667em"/> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(70)</label> </div> </div> where <math display='inline'><semantics> <msub> <mi>D</mi> <mn>0</mn> </msub> </semantics></math> and <math display='inline'><semantics> <mi>ς</mi> </semantics></math> are constants. In general, <math display='inline'><semantics> <msub> <mi>p</mi> <mi>inj</mi> </msub> </semantics></math> is determined by <math display='inline'><semantics> <mrow> <msub> <mi>D</mi> <mi mathvariant="normal">F</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>p</mi> <mi>inj</mi> </msub> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mrow> <mo>[</mo> <mi>p</mi> <msub> <mrow> <mo>(</mo> <mi>d</mi> <mi>p</mi> <mo>/</mo> <mi>d</mi> <mi>t</mi> <mo>)</mo> </mrow> <mi mathvariant="normal">C</mi> </msub> <mo>]</mo> </mrow> <msub> <mi>p</mi> <mi>inj</mi> </msub> </msub> </mrow> </semantics></math>. In this model, the injection momentum is given by <div class='html-disp-formula-info' id='FD71-universe-10-00424'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>D</mi> <mn>0</mn> </msub> <mspace width="0.166667em"/> <msubsup> <mi>p</mi> <mrow> <mi>inj</mi> </mrow> <mrow> <mo>(</mo> <mi>ς</mi> <mo>−</mo> <mn>1</mn> <mo>)</mo> </mrow> </msubsup> <mo>=</mo> <msub> <mfenced separators="" open="[" close="]"> <msub> <mfenced separators="" open="(" close=")"> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mi>d</mi> <mi>p</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mstyle> </mfenced> <mi mathvariant="normal">C</mi> </msub> </mfenced> <msub> <mi>p</mi> <mi>inj</mi> </msub> </msub> <mspace width="0.166667em"/> <mo>.</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(71)</label> </div> </div></div><div class='html-p'>For a high value of the acceleration momentum <math display='inline'><semantics> <msub> <mi>p</mi> <mn>0</mn> </msub> </semantics></math>, the runaway flux of thermal particles cools the plasma down from the very beginning. In spite of energy supply by external sources, the plasma temperature drops (analogous to Maxwell demon). The acceleration generates a prominent tail of accelerated particles but the “excess”, that was expected in Gurevich [<a href="#B92-universe-10-00424" class="html-bibr">92</a>], is not produced in the range around <math display='inline'><semantics> <msub> <mi>p</mi> <mi>inj</mi> </msub> </semantics></math>; see <a href="#universe-10-00424-f012" class="html-fig">Figure 12</a>. For <math display='inline'><semantics> <mrow> <msub> <mi>p</mi> <mn>0</mn> </msub> <mo>></mo> <msub> <mi>p</mi> <mi>inj</mi> </msub> </mrow> </semantics></math>, plasma overheating is insignificant and stochastic acceleration works well. For <math display='inline'><semantics> <mrow> <msub> <mi>p</mi> <mn>0</mn> </msub> <mo><</mo> <msub> <mi>p</mi> <mi>inj</mi> </msub> </mrow> </semantics></math>, plasma overheating is significant and stochastic acceleration is inhibited.</div><div class='html-p'>In any case, numerical calculations showed that the permitted parameters of the FB model are strongly restricted, and the model is unable to explain the observed fluxes of radio and gamma-ray emissions from the bubbles (see [<a href="#B71-universe-10-00424" class="html-bibr">71</a>]).</div></section><section id='sec8dot2-universe-10-00424' type=''><h4 class='html-italic' data-nested='2'> 8.2. Cosmic-Ray Electrons Re-Accelerated in the Fermi Bubbles</h4><div class='html-p'>CR electrons can be generated by sources in the galactic disk (e.g., supernova remnant shocks, SNR shocks). Due to synchrotron, inverse Compton, and adiabatic losses in the halo, Cheng et al. [<a href="#B101-universe-10-00424" class="html-bibr">101</a>] deemed that these CR electrons with energies above several GeV are unable to reach the height of the FB envelope (which is about 8∼10 kpc). With appropriate parameters in the FB envelope, these electrons can be re-accelerated in situ up to an energy of about <math display='inline'><semantics> <msup> <mn>10</mn> <mn>12</mn> </msup> </semantics></math> eV, which is needed to reproduce the observed radio and gamma-ray emissions from the FBs and supply the required power.</div><div class='html-p'>The steady-state kinetic equation for the relativistic electrons in FBs can be described in the form (see [<a href="#B101-universe-10-00424" class="html-bibr">101</a>]) <div class='html-disp-formula-info' id='FD72-universe-10-00424'> <div class='f'> <math display='block'><semantics> <mrow> <mi>Q</mi> <mrow> <mo>(</mo> <mi>p</mi> <mo>)</mo> </mrow> <mi>δ</mi> <mrow> <mo>(</mo> <mi>z</mi> <mo>)</mo> </mrow> <mo>=</mo> <mo>−</mo> <mo>∇</mo> <mo>·</mo> <mfenced separators="" open="[" close="]"> <msub> <mi>D</mi> <mi>s</mi> </msub> <mo>∇</mo> <mi>F</mi> <mo>−</mo> <mi mathvariant="bold">v</mi> <mi>F</mi> </mfenced> <mo>+</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mn>1</mn> <msup> <mi>p</mi> <mn>2</mn> </msup> </mfrac> </mstyle> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mo>∂</mo> <mrow> <mo>∂</mo> <mi>p</mi> </mrow> </mfrac> </mstyle> <msup> <mi>p</mi> <mn>2</mn> </msup> <mfenced separators="" open="[" close="]"> <mfenced separators="" open="(" close=")"> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mi>d</mi> <mi>p</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mstyle> <mo>−</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mo>∇</mo> <mo>·</mo> <mi mathvariant="bold">v</mi> </mrow> <mn>3</mn> </mfrac> </mstyle> <mi>p</mi> </mfenced> <mi>F</mi> <mo>−</mo> <msub> <mi>D</mi> <mi>p</mi> </msub> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mo>∂</mo> <mi>F</mi> </mrow> <mrow> <mo>∂</mo> <mi>p</mi> </mrow> </mfrac> </mstyle> </mfenced> <mspace width="0.166667em"/> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(72)</label> </div> </div> where <math display='inline'><semantics> <mrow> <mi>F</mi> <mo>(</mo> <mi>p</mi> <mo>,</mo> <mi>r</mi> <mo>,</mo> <mi>z</mi> <mo>)</mo> </mrow> </semantics></math> is the particle distribution function, <math display='inline'><semantics> <mrow> <mi>Q</mi> <mrow> <mo>(</mo> <mi>p</mi> <mo>,</mo> <mi>r</mi> <mo>)</mo> </mrow> <mo>∝</mo> <msup> <mi>p</mi> <mrow> <mo>−</mo> <mi>γ</mi> </mrow> </msup> </mrow> </semantics></math> is the source function of electrons in the galactic disk (<math display='inline'><semantics> <mrow> <mi>z</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>), <math display='inline'><semantics> <mrow> <mi>d</mi> <mi>p</mi> <mo>/</mo> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mi>μ</mi> <msup> <mi>E</mi> <mn>2</mn> </msup> </mrow> </semantics></math> is the rate of synchrotron and inverse Compton energy losses, <math display='inline'><semantics> <mrow> <msub> <mi>D</mi> <mi>s</mi> </msub> <mrow> <mo>(</mo> <mi>p</mi> <mo>,</mo> <mi>r</mi> <mo>,</mo> <mi>z</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> is the coefficient of spatial diffusion, <math display='inline'><semantics> <mrow> <msub> <mi>D</mi> <mi>p</mi> </msub> <mrow> <mo>(</mo> <mi>p</mi> <mo>,</mo> <mi>r</mi> <mo>,</mo> <mi>z</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>κ</mi> <msup> <mi>p</mi> <mn>2</mn> </msup> </mrow> </semantics></math> is the coefficient of momentum diffusion (coefficient of the Fermi re-acceleration), and <math display='inline'><semantics> <mrow> <mi mathvariant="bold">v</mi> <mrow> <mo>(</mo> <mi>r</mi> <mo>,</mo> <mi>z</mi> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>v</mi> <mi>z</mi> </msub> <msub> <mover accent="true"> <mi>e</mi> <mo>^</mo> </mover> <mi>z</mi> </msub> <mo>=</mo> <mn>3</mn> <msup> <mi>v</mi> <mo>′</mo> </msup> <mi>z</mi> <msub> <mover accent="true"> <mi>e</mi> <mo>^</mo> </mover> <mi>z</mi> </msub> </mrow> </semantics></math> (<math display='inline'><semantics> <mrow> <msup> <mi>v</mi> <mo>′</mo> </msup> <mo>=</mo> <mi>d</mi> <msub> <mi>v</mi> <mi>z</mi> </msub> <mo>/</mo> <mi>d</mi> <mi>z</mi> </mrow> </semantics></math>) is the wind velocity of advection in the halo in the <span class='html-italic'>z</span> direction. The effect of the wind advection leads to adiabatic losses of CRs, <math display='inline'><semantics> <mrow> <mi>d</mi> <mi>p</mi> <mo>/</mo> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mo>−</mo> <mi>p</mi> <mo>∇</mo> <mo>·</mo> <mi mathvariant="bold">v</mi> <mo>/</mo> <mn>3</mn> </mrow> </semantics></math>. Consequently, the spectrum of electrons in the FBs in the acceleration region is harder than the case without advection.</div><div class='html-p'>Cheng et al. [<a href="#B101-universe-10-00424" class="html-bibr">101</a>] showed that the gamma-ray and radio emissions of the re-accelerated electrons nicely reproduced the Fermi-LAT and Planck data points for the parameters: the spatial diffusion coefficient <math display='inline'><semantics> <mrow> <msub> <mi>D</mi> <mi>s</mi> </msub> <mo>=</mo> <msup> <mn>10</mn> <mn>29</mn> </msup> <msup> <mi>cm</mi> <mn>2</mn> </msup> <msup> <mi mathvariant="normal">s</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </semantics></math>, the energy loss rate <math display='inline'><semantics> <mrow> <mi>ν</mi> <mo>=</mo> <mn>2</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>16</mn> </mrow> </msup> <msup> <mi mathvariant="normal">s</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msup> <mi>GeV</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </semantics></math>, the velocity gradient of the advection in the halo <math display='inline'><semantics> <mrow> <msup> <mi>v</mi> <mo>′</mo> </msup> <mo>=</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>15</mn> </mrow> </msup> <msup> <mi mathvariant="normal">s</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </semantics></math>, the magnetic field strength is <math display='inline'><semantics> <mrow> <mi>B</mi> <mo>=</mo> <mn>3</mn> </mrow> </semantics></math> μG, the thickness of the re-acceleration region (say, the FB envelope) is about <math display='inline'><semantics> <mrow> <mo>Δ</mo> <msub> <mi>r</mi> <mi>FB</mi> </msub> <mo>=</mo> <mn>60</mn> </mrow> </semantics></math> pc, and the parameter of re-acceleration in the FBs is <math display='inline'><semantics> <mrow> <mi>κ</mi> <mo>=</mo> <mn>2</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>14</mn> </mrow> </msup> <msup> <mi mathvariant="normal">s</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </semantics></math>.</div><div class='html-p'>The total power <math display='inline'><semantics> <mover accent="true"> <mi mathvariant="script">E</mi> <mo>˙</mo> </mover> </semantics></math> needed to be supplied by sources of Fermi re-acceleration in the FBs to produce high-energy electrons is <div class='html-disp-formula-info' id='FD73-universe-10-00424'> <div class='f'> <math display='block'><semantics> <mrow> <mover accent="true"> <mi mathvariant="script">E</mi> <mo>˙</mo> </mover> <mo>=</mo> <mo>−</mo> <msubsup> <mo>∫</mo> <mn>0</mn> <mo>∞</mo> </msubsup> <mn>4</mn> <mi>π</mi> <mi>E</mi> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mo>∂</mo> <mrow> <mo>∂</mo> <mi>p</mi> </mrow> </mfrac> </mstyle> <mfenced separators="" open="(" close=")"> <msup> <mi>p</mi> <mn>2</mn> </msup> <msub> <mi>D</mi> <mi>p</mi> </msub> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mo>∂</mo> <mi>F</mi> </mrow> <mrow> <mo>∂</mo> <mi>p</mi> </mrow> </mfrac> </mstyle> </mfenced> <mi>d</mi> <mi>p</mi> <mspace width="0.166667em"/> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(73)</label> </div> </div> where <span class='html-italic'>p</span> and <span class='html-italic'>E</span> are the particle momentum and particle kinetic energy, respectively.</div><div class='html-p'>To define the spectrum of accelerated electrons we estimated the number of GeV electrons that can reach an altitude of several kpc when the effect of advection <math display='inline'><semantics> <mrow> <msub> <mi>v</mi> <mi>z</mi> </msub> <mrow> <mo>(</mo> <mi>z</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> is essential in the galactic halo (see [<a href="#B102-universe-10-00424" class="html-bibr">102</a>,<a href="#B103-universe-10-00424" class="html-bibr">103</a>,<a href="#B104-universe-10-00424" class="html-bibr">104</a>,<a href="#B105-universe-10-00424" class="html-bibr">105</a>]). The spectrum of re-accelerated SNR electrons in the FBs is shown in <a href="#universe-10-00424-f013" class="html-fig">Figure 13</a> (cf. [<a href="#B101-universe-10-00424" class="html-bibr">101</a>]). In the figure, the thick solid line is the spectrum of CR electrons from their sources in the galactic disk (see, e.g., [<a href="#B70-universe-10-00424" class="html-bibr">70</a>]). When re-acceleration (stochastic acceleration) and adiabatic losses are taken into account, the spectrum of electrons (thin dashed line) becomes harder than that of the spectrum emitted by sources (thick solid line), but softer than the spectrum of pure re-acceleration (thin dash-dotted line). The spectrum for a velocity gradient <math display='inline'><semantics> <mrow> <msup> <mi>v</mi> <mo>′</mo> </msup> <mo>=</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>15</mn> </mrow> </msup> <msup> <mi mathvariant="normal">s</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </semantics></math> (see, e.g., [<a href="#B103-universe-10-00424" class="html-bibr">103</a>,<a href="#B105-universe-10-00424" class="html-bibr">105</a>]) is consistent with the observed one (dashed line in <a href="#universe-10-00424-f013" class="html-fig">Figure 13</a>).</div><div class='html-p'>The model of re-acceleration within the envelope coincides nicely with the observed microwave and gamma-ray emissions, shown in the left and right panels of <a href="#universe-10-00424-f010" class="html-fig">Figure 10</a>, respectively.</div><div class='html-p'>In the phenomenological model, the power, <math display='inline'><semantics> <mover accent="true"> <mi mathvariant="script">E</mi> <mo>˙</mo> </mover> </semantics></math>, is estimated numerically from the observed FB gamma-ray and microwave fluxes, and is about <math display='inline'><semantics> <mrow> <mover accent="true"> <mi mathvariant="script">E</mi> <mo>˙</mo> </mover> <mo>∼</mo> <mn>2</mn> <mo>×</mo> <msup> <mn>10</mn> <mn>38</mn> </msup> </mrow> </semantics></math> erg <math display='inline'><semantics> <mrow> <msup> <mi mathvariant="normal">s</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </semantics></math>. The density of high-energy electrons needed for the observed gamma-ray flux is shown by the shaded gray region in <a href="#universe-10-00424-f013" class="html-fig">Figure 13</a>.</div><div class='html-p'>With appropriate parameters, these electrons can be re-accelerated up to an energy of <math display='inline'><semantics> <msup> <mn>10</mn> <mn>12</mn> </msup> </semantics></math> eV, which explains the origin of the observed gamma-ray and radio emissions from the FBs in this model. However, although the model gamma-ray spectrum is consistent with the Fermi results, the model radio spectrum in the pure diffusion model is steeper than that observed by WMAP and Planck.</div><div class='html-p'>If adiabatic losses due to plasma outflows from the galactic central regions are taken into account, we expect that the spectrum of electrons in the acceleration region will be harder than the one without advection. Our calculations with divergent outflows show that the gamma-ray and radio emissions of the re-accelerated electrons nicely reproduce the Fermi-LAT and Planck data (see <a href="#universe-10-00424-f010" class="html-fig">Figure 10</a>).</div><div class='html-p'>In essence, both gamma-ray and microwave observations can be explained by only one source of high-energy electrons. The basic idea is summarized as follows. CR electrons from SNRs in the galactic disk are re-accelerated in the FB via supersonic turbulence (or multiple shocks). The resulting spectrum is hard, and in the high-energy region the density of electrons exceeds that required to produce the observed gamma-ray emission. With adiabatic loss by the divergent flow, the density reduces but the spectrum is still hard enough to produce the observed microwave emission. Inevitably, some delicate balance or fine-tuning of parameters is needed.</div></section></section><section id='sec9-universe-10-00424' type=''><h2 data-nested='1'> 9. Cosmic-Ray Protons Escaping from the Fermi Bubbles into the Galaxy</h2><div class='html-p'>The fundamental question of the sources of CRs in the galaxy is still open. We present a number of models which may interpret the origin of CRs escaping from the FBs into the galactic disk. The effect can be observed from the spectrum of CRs near Earth.</div><div class='html-p'>As described in Berezinsky et al. [<a href="#B70-universe-10-00424" class="html-bibr">70</a>], the classical model of CR origin in the galaxy is that CRs are generated by SNRs in the galactic disk with energies below <math display='inline'><semantics> <msup> <mn>10</mn> <mn>15</mn> </msup> </semantics></math> eV. They escape into the galactic halo with an effective spatial diffusion coefficient about <math display='inline'><semantics> <mrow> <msub> <mi>D</mi> <mi mathvariant="normal">G</mi> </msub> <mo>=</mo> <msub> <mi>D</mi> <mo>*</mo> </msub> <msup> <mrow> <mo>(</mo> <mi>E</mi> <mo>/</mo> <mn>4</mn> <mspace width="0.166667em"/> <mi>GeV</mi> <mo>)</mo> </mrow> <mrow> <mn>0.6</mn> </mrow> </msup> </mrow> </semantics></math> (<math display='inline'><semantics> <mrow> <msub> <mi>D</mi> <mo>*</mo> </msub> <mo>=</mo> <mn>6.2</mn> <mo>×</mo> <msup> <mn>10</mn> <mn>28</mn> </msup> <msup> <mi>cm</mi> <mn>2</mn> </msup> <msup> <mi mathvariant="normal">s</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </semantics></math>), which is estimated from the observed chemical composition of CRs (for a modern nonlinear model of CR, see [<a href="#B106-universe-10-00424" class="html-bibr">106</a>] and references therein). Plane shock acceleration produces a CR spectrum <math display='inline'><semantics> <mrow> <mi>F</mi> <mrow> <mo>(</mo> <mi>E</mi> <mo>)</mo> </mrow> <mo>∝</mo> <msup> <mi>E</mi> <mrow> <mo>−</mo> <mn>2</mn> </mrow> </msup> </mrow> </semantics></math> (see, e.g., [<a href="#B87-universe-10-00424" class="html-bibr">87</a>,<a href="#B88-universe-10-00424" class="html-bibr">88</a>,<a href="#B89-universe-10-00424" class="html-bibr">89</a>,<a href="#B90-universe-10-00424" class="html-bibr">90</a>]). The actual CR spectrum observed outside the acceleration region is a result of the process of acceleration in the region together with the process of particle leakage or escape from the region. Furthermore, the maximum energy that can be attained by CR particles depends on the size and/or lifetime of the acceleration region. Suppose <math display='inline'><semantics> <msub> <mi>v</mi> <mi>sh</mi> </msub> </semantics></math> is the shock velocity and <math display='inline'><semantics> <msub> <mi>D</mi> <mi>sh</mi> </msub> </semantics></math> is the spatial diffusion coefficient in the shock vicinity. The minimum value of <math display='inline'><semantics> <msub> <mi>D</mi> <mi>sh</mi> </msub> </semantics></math> follows Bohm’s limit, <math display='inline'><semantics> <mrow> <msub> <mi>D</mi> <mi>sh</mi> </msub> <mo>≈</mo> <mi>u</mi> <msub> <mi>r</mi> <mi>L</mi> </msub> <mo>/</mo> <mn>3</mn> </mrow> </semantics></math> (where <math display='inline'><semantics> <mrow> <mi>u</mi> <mo>≈</mo> <mi>c</mi> </mrow> </semantics></math> is the speed of CR particles). The maximum energy is constrained by the size of the acceleration region becoming smaller than the diffusion length scale of the particles <math display='inline'><semantics> <mrow> <msub> <mi>l</mi> <mi>D</mi> </msub> <mo>∼</mo> <msub> <mi>D</mi> <mi>sh</mi> </msub> <mo>/</mo> <msub> <mi>v</mi> <mi>sh</mi> </msub> </mrow> </semantics></math>, and/or the lifetime of the region being smaller than the acceleration time scale of the particles <math display='inline'><semantics> <mrow> <msub> <mi>τ</mi> <mi>acc</mi> </msub> <mo>∼</mo> <msub> <mi>D</mi> <mi>sh</mi> </msub> <mo>/</mo> <msubsup> <mi>v</mi> <mrow> <mi>sh</mi> </mrow> <mn>2</mn> </msubsup> </mrow> </semantics></math>.</div><div class='html-p'>The CR spectrum (above 1 GeV) observed from Earth can be described by a broken power law; see the classic spectrum in Swordy [<a href="#B107-universe-10-00424" class="html-bibr">107</a>] (or <a href="#universe-10-00424-f014" class="html-fig">Figure 14</a>, which shows the part above <math display='inline'><semantics> <msup> <mn>10</mn> <mn>13</mn> </msup> </semantics></math> eV). The power-law index is <math display='inline'><semantics> <mrow> <mo>−</mo> <mn>2.7</mn> </mrow> </semantics></math> between <math display='inline'><semantics> <msup> <mn>10</mn> <mn>9</mn> </msup> </semantics></math> and <math display='inline'><semantics> <msup> <mn>10</mn> <mn>15</mn> </msup> </semantics></math> eV, and <math display='inline'><semantics> <mrow> <mo>−</mo> <mn>3.1</mn> </mrow> </semantics></math> between <math display='inline'><semantics> <msup> <mn>10</mn> <mn>15</mn> </msup> </semantics></math> and several <math display='inline'><semantics> <msup> <mn>10</mn> <mn>18</mn> </msup> </semantics></math> eV. An energy around <math display='inline'><semantics> <msup> <mn>10</mn> <mn>15</mn> </msup> </semantics></math> eV is called the ‘knee’, where the spectrum changes from a harder one to a softer one. The spectrum beyond several <math display='inline'><semantics> <msup> <mn>10</mn> <mn>18</mn> </msup> </semantics></math> eV becomes harder again, and this region is called the ‘ankle’. The apparent cutoff at somewhat less than <math display='inline'><semantics> <msup> <mn>10</mn> <mn>20</mn> </msup> </semantics></math> eV is commonly attributed to the Greisen–Zatsepin–Kuzmin limit due to the interaction of ultrahigh-energy CRs with the cosmic microwave background [<a href="#B108-universe-10-00424" class="html-bibr">108</a>,<a href="#B109-universe-10-00424" class="html-bibr">109</a>]. At energies smaller than 1 GeV, the spectrum is heavily affected by solar modulation and activity of the Sun. Although the spectrum is a broken power law, the ‘joints’ (say the ‘knee’ and the ‘ankle’) are smooth. The origins of different parts of the spectrum should be somehow related (e.g., [<a href="#B110-universe-10-00424" class="html-bibr">110</a>]).</div><div class='html-p'>SNR shocks are believed to be the source of CRs (e.g., [<a href="#B70-universe-10-00424" class="html-bibr">70</a>]). They can produce the spectral index <math display='inline'><semantics> <mrow> <mo>−</mo> <mn>2.7</mn> </mrow> </semantics></math> reasonably well. With the magnetic field around the SNR comparable to the general interstellar field, Lagage & Cesarsky [<a href="#B111-universe-10-00424" class="html-bibr">111</a>] and Berezhko & Völk [<a href="#B112-universe-10-00424" class="html-bibr">112</a>] estimated that the maximum energy of protons from SNRs is about <math display='inline'><semantics> <msup> <mn>10</mn> <mn>13</mn> </msup> </semantics></math>∼<math display='inline'><semantics> <msup> <mn>10</mn> <mn>14</mn> </msup> </semantics></math> eV. However, with instabilities caused by cosmic ray streaming (e.g., non-resonant hybrid instability), Bell [<a href="#B113-universe-10-00424" class="html-bibr">113</a>] and Bykov et al. [<a href="#B114-universe-10-00424" class="html-bibr">114</a>] found that the fluctuated magnetic field can be orders of magnitude larger at the SNR shock (and so <math display='inline'><semantics> <mrow> <msub> <mi>D</mi> <mi>sh</mi> </msub> <mo>≪</mo> <msub> <mi>D</mi> <mi mathvariant="normal">G</mi> </msub> </mrow> </semantics></math>). The confinement time of CRs is longer and they can be accelerated up to <math display='inline'><semantics> <msup> <mn>10</mn> <mn>15</mn> </msup> </semantics></math> eV (see also [<a href="#B115-universe-10-00424" class="html-bibr">115</a>]).</div><div class="html-fig-wrap" id="universe-10-00424-f014"> <div class='html-fig_img'> <div class="html-figpopup html-figpopup-link" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/11/424/display" href="#fig_body_display_universe-10-00424-f014"> <img data-large="/universe/universe-10-00424/article_deploy/html/images/universe-10-00424-g014.png" data-original="/universe/universe-10-00424/article_deploy/html/images/universe-10-00424-g014.png" alt="Universe 10 00424 g014" data-lsrc="/universe/universe-10-00424/article_deploy/html/images/universe-10-00424-g014-550.jpg" /> <a class="html-expand html-figpopup" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/11/424/display" href="#fig_body_display_universe-10-00424-f014"></a> </div> </div> <div class="html-fig_description"> <b>Figure 14.</b> CR spectrum at the Earth as a combination of the contributions from the SNRs in the galactic disk and the stochastic acceleration in the FBs. Figure reproduced from Cheng et al. [<a href="#B116-universe-10-00424" class="html-bibr">116</a>] with permission.  </div> </div> <div class="html-fig_show mfp-hide" id="fig_body_display_universe-10-00424-f014"> <div class="html-caption"> <b>Figure 14.</b> CR spectrum at the Earth as a combination of the contributions from the SNRs in the galactic disk and the stochastic acceleration in the FBs. Figure reproduced from Cheng et al. [<a href="#B116-universe-10-00424" class="html-bibr">116</a>] with permission.</div> <div class="html-img"><img data-large="/universe/universe-10-00424/article_deploy/html/images/universe-10-00424-g014.png" data-original="/universe/universe-10-00424/article_deploy/html/images/universe-10-00424-g014.png" alt="Universe 10 00424 g014" data-lsrc="/universe/universe-10-00424/article_deploy/html/images/universe-10-00424-g014.png" /></div> </div> <div class='html-p'>The SNR framework is able to describe the spectrum in the range <math display='inline'><semantics> <mrow> <mi>E</mi> <mo><</mo> <msup> <mn>10</mn> <mn>15</mn> </msup> </mrow> </semantics></math> eV (see <a href="#universe-10-00424-f014" class="html-fig">Figure 14</a>). However, for <math display='inline'><semantics> <mrow> <mi>E</mi> <mo>></mo> <msup> <mn>10</mn> <mn>15</mn> </msup> </mrow> </semantics></math> eV, we need some other ideas (see, e.g., [<a href="#B76-universe-10-00424" class="html-bibr">76</a>]). Larger and longer lifetime shocks are required, such as superbubbles (like FBs) and strong galactic winds (e.g., [<a href="#B85-universe-10-00424" class="html-bibr">85</a>,<a href="#B86-universe-10-00424" class="html-bibr">86</a>,<a href="#B116-universe-10-00424" class="html-bibr">116</a>,<a href="#B117-universe-10-00424" class="html-bibr">117</a>,<a href="#B118-universe-10-00424" class="html-bibr">118</a>,<a href="#B119-universe-10-00424" class="html-bibr">119</a>]). In the following, we analyze the spectrum of CRs accelerated in the FBs and discuss whether the bubble’s contribution may explain the ‘knee’ steepening. The idea is based on acceleration by multiple shocks. In a multiple-shock system, two length scales are important: (1) the average separation between two shocks <math display='inline'><semantics> <mrow> <msub> <mi>l</mi> <mi>sh</mi> </msub> <mo>∼</mo> <msub> <mi>v</mi> <mi>sh</mi> </msub> <msub> <mi>τ</mi> <mi mathvariant="normal">c</mi> </msub> </mrow> </semantics></math> (where <math display='inline'><semantics> <msub> <mi>τ</mi> <mi mathvariant="normal">c</mi> </msub> </semantics></math> is the average time between the creation of two consecutive shocks), and (2) the diffusion length scale at the shock <math display='inline'><semantics> <msub> <mi>l</mi> <mi>D</mi> </msub> </semantics></math> (∼<math display='inline'><semantics> <mrow> <msub> <mi>D</mi> <mi>sh</mi> </msub> <mo>/</mo> <msub> <mi>v</mi> <mi>sh</mi> </msub> </mrow> </semantics></math>). We focus on the regime <math display='inline'><semantics> <mrow> <msub> <mi>l</mi> <mi>sh</mi> </msub> <mo>≪</mo> <msub> <mi>l</mi> <mi>D</mi> </msub> </mrow> </semantics></math> (the supersonic turbulence regime), for energies beyond the ‘knee’ (<math display='inline'><semantics> <mrow> <mi>E</mi> <mo>></mo> <msup> <mn>10</mn> <mn>15</mn> </msup> </mrow> </semantics></math> eV).</div><div class='html-p'>An illustration of a possible multiple-shock structure in the FBs is shown in <a href="#universe-10-00424-f015" class="html-fig">Figure 15</a>.</div><section id='EscapeofCosmicRayProtonsReAcceleratedbySupersonicTurbulenceInsidetheFermiBubbles' type=''><h4 class='html-italic' data-nested='2'> Escape of Cosmic-Ray Protons Re-Accelerated by Supersonic Turbulence Inside the Fermi Bubbles</h4><div class='html-p'>Cheng et al. [<a href="#B116-universe-10-00424" class="html-bibr">116</a>] suggested an alternative model of CRs in the FBs (see also [<a href="#B11-universe-10-00424" class="html-bibr">11</a>,<a href="#B12-universe-10-00424" class="html-bibr">12</a>,<a href="#B120-universe-10-00424" class="html-bibr">120</a>,<a href="#B121-universe-10-00424" class="html-bibr">121</a>,<a href="#B122-universe-10-00424" class="html-bibr">122</a>]). They assumed that up to several hundred TDEs might have occurred in the past 10 Myr. This would have generated a series of shocks propagating through the central part of the galactic halo, which would produce relativistic CRs via multiple-shock acceleration. The average separation between two shocks is then <div class='html-disp-formula-info' id='FD74-universe-10-00424'> <div class='f'> <math display='block'><semantics> <mtable displaystyle="true"> <mtr> <mtd/> <mtd> <msub> <mi>l</mi> <mi>sh</mi> </msub> </mtd> <mtd columnalign="left"> <mrow> <mo>=</mo> <msub> <mi>v</mi> <mi>sh</mi> </msub> <msub> <mi>τ</mi> <mi>cap</mi> </msub> </mrow> </mtd> </mtr> <mtr> <mtd/> <mtd/> <mtd columnalign="left"> <mrow> <mo>≃</mo> <mn>30</mn> <mfenced separators="" open="(" close=")"> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <msub> <mi>τ</mi> <mi>cap</mi> </msub> <mrow> <mn>3</mn> <mo>×</mo> <msup> <mn>10</mn> <mn>4</mn> </msup> <mspace width="0.166667em"/> <mi>yr</mi> </mrow> </mfrac> </mstyle> </mfenced> <mfenced separators="" open="(" close=")"> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <msub> <mi>v</mi> <mi>sh</mi> </msub> <mrow> <msup> <mn>10</mn> <mn>8</mn> </msup> <mspace width="0.166667em"/> <msup> <mrow> <mi>cm</mi> <mspace width="0.166667em"/> <mi mathvariant="normal">s</mi> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </mfrac> </mstyle> </mfenced> <mspace width="4pt"/> <mi>pc</mi> <mspace width="0.166667em"/> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </semantics></math> </div> <div class='l'> <label >(74)</label> </div> </div> where <math display='inline'><semantics> <msub> <mi>τ</mi> <mi>cap</mi> </msub> </semantics></math> is the average time between two stellar captures by the SMBH.</div><div class='html-p'>We applied the model of Bykov & Fleishman [<a href="#B123-universe-10-00424" class="html-bibr">123</a>] and Bykov & Toptygin [<a href="#B76-universe-10-00424" class="html-bibr">76</a>] for CR acceleration by multiple shocks in the FBs. Under the conditions of supersonic turbulence (multiple-shock structure) the regime of acceleration is characterized by <math display='inline'><semantics> <mrow> <msub> <mi>l</mi> <mi>sh</mi> </msub> <mo>≪</mo> <msub> <mi>l</mi> <mi mathvariant="normal">D</mi> </msub> </mrow> </semantics></math>.</div><div class='html-p'>The steady state kinetic equation in axisymmetric geometry is <div class='html-disp-formula-info' id='FD75-universe-10-00424'> <div class='f'> <math display='block'><semantics> <mrow> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mo>∂</mo> <mrow> <mo>∂</mo> <mi>z</mi> </mrow> </mfrac> </mstyle> <mfenced separators="" open="[" close="]"> <msub> <mi>D</mi> <mi>s</mi> </msub> <mrow> <mo>(</mo> <mi>p</mi> <mo>,</mo> <mi>r</mi> <mo>)</mo> </mrow> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mo>∂</mo> <mi>F</mi> </mrow> <mrow> <mo>∂</mo> <mi>z</mi> </mrow> </mfrac> </mstyle> </mfenced> <mo>+</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mn>1</mn> <mi>r</mi> </mfrac> </mstyle> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mo>∂</mo> <mrow> <mo>∂</mo> <mi>r</mi> </mrow> </mfrac> </mstyle> <mfenced separators="" open="[" close="]"> <msub> <mi>D</mi> <mi>s</mi> </msub> <mrow> <mo>(</mo> <mi>p</mi> <mo>,</mo> <mi>r</mi> <mo>)</mo> </mrow> <mspace width="0.166667em"/> <mi>r</mi> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mo>∂</mo> <mi>F</mi> </mrow> <mrow> <mo>∂</mo> <mi>r</mi> </mrow> </mfrac> </mstyle> </mfenced> <mo>+</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mn>1</mn> <msup> <mi>p</mi> <mn>2</mn> </msup> </mfrac> </mstyle> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mo>∂</mo> <mrow> <mo>∂</mo> <mi>p</mi> </mrow> </mfrac> </mstyle> <mfenced separators="" open="[" close="]"> <msub> <mi>D</mi> <mi>p</mi> </msub> <mrow> <mo>(</mo> <mi>p</mi> <mo>,</mo> <mi>r</mi> <mo>)</mo> </mrow> <msup> <mi>p</mi> <mn>2</mn> </msup> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mo>∂</mo> <mi>F</mi> </mrow> <mrow> <mo>∂</mo> <mi>p</mi> </mrow> </mfrac> </mstyle> </mfenced> <mo>=</mo> <mo>−</mo> <mi>Q</mi> <mrow> <mo>(</mo> <mi>p</mi> <mo>,</mo> <mi>r</mi> <mo>,</mo> <mi>z</mi> <mo>)</mo> </mrow> <mspace width="0.166667em"/> <mo>.</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(75)</label> </div> </div>The spatial diffusion coefficient inside and outside the bubble is <div class='html-disp-formula-info' id='FD76-universe-10-00424'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>D</mi> <mi>s</mi> </msub> <mrow> <mo>(</mo> <mi>p</mi> <mo>,</mo> <mi>r</mi> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>D</mi> <mi>B</mi> </msub> <mi>θ</mi> <mrow> <mo>(</mo> <msub> <mi>r</mi> <mi mathvariant="normal">B</mi> </msub> <mo>−</mo> <mi>r</mi> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>D</mi> <mi mathvariant="normal">G</mi> </msub> <mi>θ</mi> <mrow> <mo>(</mo> <mi>r</mi> <mo>−</mo> <msub> <mi>r</mi> <mi mathvariant="normal">B</mi> </msub> <mo>)</mo> </mrow> <mspace width="0.166667em"/> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(76)</label> </div> </div> where <math display='inline'><semantics> <msub> <mi>r</mi> <mi mathvariant="normal">B</mi> </msub> </semantics></math> is the radius of the bubbles. Inside the bubbles, as a result of interaction with supersonic turbulence, <math display='inline'><semantics> <mrow> <msub> <mi>D</mi> <mi>s</mi> </msub> <mo>=</mo> <msub> <mi>D</mi> <mi mathvariant="normal">B</mi> </msub> <mo>≈</mo> <mi>u</mi> <msub> <mi>l</mi> <mi>sh</mi> </msub> <mo>/</mo> <mn>3</mn> <mo>≈</mo> <mi>c</mi> <msub> <mi>l</mi> <mi>sh</mi> </msub> <mo>/</mo> <mn>3</mn> </mrow> </semantics></math>. Outside the bubbles, <math display='inline'><semantics> <mrow> <msub> <mi>D</mi> <mi>s</mi> </msub> <mo>=</mo> <msub> <mi>D</mi> <mi mathvariant="normal">G</mi> </msub> </mrow> </semantics></math>, which is the average diffusion coefficient in the galaxy, i.e., the one described in Berezinsky et al. [<a href="#B70-universe-10-00424" class="html-bibr">70</a>]. The momentum diffusion coefficient is nonzero inside the bubbles only:<div class='html-disp-formula-info' id='FD77-universe-10-00424'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>D</mi> <mi>p</mi> </msub> <mrow> <mo>(</mo> <mi>p</mi> <mo>,</mo> <mi>r</mi> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>κ</mi> <mi mathvariant="normal">B</mi> </msub> <msup> <mi>p</mi> <mn>2</mn> </msup> <mi>θ</mi> <mrow> <mo>(</mo> <msub> <mi>r</mi> <mi mathvariant="normal">B</mi> </msub> <mo>−</mo> <mi>r</mi> <mo>)</mo> </mrow> <mspace width="0.166667em"/> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(77)</label> </div> </div> and <math display='inline'><semantics> <mrow> <msub> <mi>κ</mi> <mi mathvariant="normal">B</mi> </msub> <mo>∼</mo> <msubsup> <mi>v</mi> <mrow> <mi>sh</mi> </mrow> <mn>2</mn> </msubsup> <mo>/</mo> <msub> <mi>D</mi> <mi mathvariant="normal">B</mi> </msub> </mrow> </semantics></math>. The parameters in the FBs can be estimated numerically from the observed CR spectrum.</div><div class='html-p'>The method of separation variables as in Bulanov et al. [<a href="#B124-universe-10-00424" class="html-bibr">124</a>] and Bulanov & Dogel [<a href="#B125-universe-10-00424" class="html-bibr">125</a>] can be used to solve Equation (<a href="#FD75-universe-10-00424" class="html-disp-formula">75</a>) for the spectrum of CRs generated by SNRs with the standard model of CR propagation and escape in the galactic halo (see, e.g., [<a href="#B70-universe-10-00424" class="html-bibr">70</a>,<a href="#B80-universe-10-00424" class="html-bibr">80</a>,<a href="#B126-universe-10-00424" class="html-bibr">126</a>]). This model describes the observed CR spectrum near Earth in the range below the ‘knee’, <math display='inline'><semantics> <msup> <mn>10</mn> <mn>15</mn> </msup> </semantics></math> eV (see <a href="#universe-10-00424-f014" class="html-fig">Figure 14</a>).</div><div class='html-p'>Cheng et al. [<a href="#B116-universe-10-00424" class="html-bibr">116</a>] interpreted the CR spectrum near Earth in the energy range from <math display='inline'><semantics> <msup> <mn>10</mn> <mn>15</mn> </msup> </semantics></math> eV to a few <math display='inline'><semantics> <msup> <mn>10</mn> <mn>18</mn> </msup> </semantics></math> eV (from the ‘knee’ to the ‘ankle’) as a combined result of acceleration in the FB and escape. The acceleration is provided by the supersonic turbulence in the FB (see [<a href="#B76-universe-10-00424" class="html-bibr">76</a>]). Under the set of parameters in Cheng et al. [<a href="#B116-universe-10-00424" class="html-bibr">116</a>], the ratio of the escape time to the acceleration time is about 1.9.</div><div class='html-p'>In order to derive the spectrum near Earth, Cheng et al. [<a href="#B116-universe-10-00424" class="html-bibr">116</a>] matched the solutions for the spectra inside the FB and outside (in the halo) at <math display='inline'><semantics> <mrow> <mi>r</mi> <mo>=</mo> <msub> <mi>r</mi> <mi mathvariant="normal">B</mi> </msub> </mrow> </semantics></math> (see Equations (<a href="#FD75-universe-10-00424" class="html-disp-formula">75</a>) and (<a href="#FD76-universe-10-00424" class="html-disp-formula">76</a>)). Basically, the model can reproduce the CR spectrum from <math display='inline'><semantics> <msup> <mn>10</mn> <mn>13</mn> </msup> </semantics></math> eV to several <math display='inline'><semantics> <msup> <mn>10</mn> <mn>18</mn> </msup> </semantics></math> eV. However, we should point out that the numerical result described here [<a href="#B116-universe-10-00424" class="html-bibr">116</a>] has free parameters and some physics have been ignored (e.g., adiabatic loss), and the result might not be very solid. It is imperative to perform further investigations.</div><div class='html-p'>A brief summary of the idea is as follows. CRs from SNRs (<<math display='inline'><semantics> <msup> <mn>10</mn> <mn>15</mn> </msup> </semantics></math> eV) are re-accelerated in the FB by multiple shocks or supersonic turbulence. As the shocks are larger and live longer in the FB, CRs can be accelerated to much higher energies (up to <math display='inline'><semantics> <msup> <mn>10</mn> <mn>19</mn> </msup> </semantics></math>). Acceleration by multiple shocks inside the bubbles gives a harder spectrum, hence the contribution of FB to CRs within the ‘knee’ (<<math display='inline'><semantics> <msup> <mn>10</mn> <mn>15</mn> </msup> </semantics></math> eV is subordinate to SNR). The high-energy CRs escaping from the bubbles constitute the sole source of CRs in the range between the ‘knee’ and the ‘ankle’ (<math display='inline'><semantics> <msup> <mn>10</mn> <mn>15</mn> </msup> </semantics></math> to several <math display='inline'><semantics> <msup> <mn>10</mn> <mn>18</mn> </msup> </semantics></math> eV) observed at Earth. In the model, some fine-tuning of parameters is inevitable.</div></section></section><section id='sec10-universe-10-00424' type=''><h2 data-nested='1'> 10. Summary</h2><div class='html-p'>Here, we present a brief summary of our perspective of the Fermi Bubbles at the Galactic Center.</div><ul class='html-bullet'><li><div class='html-p'>The key point of the bubbles is a huge energy release of <math display='inline'><semantics> <msup> <mn>10</mn> <mn>55</mn> </msup> </semantics></math>∼<math display='inline'><semantics> <msup> <mn>10</mn> <mn>56</mn> </msup> </semantics></math> erg in the Galactic Center, whose origin is still unknown.</div></li><li><div class='html-p'>We assume that the energy source of the bubbles could be a routine tidal disruption of stars near the central supermassive black hole. Each disruption of a star releases a total energy about <math display='inline'><semantics> <msup> <mn>10</mn> <mn>52</mn> </msup> </semantics></math>∼<math display='inline'><semantics> <msup> <mn>10</mn> <mn>53</mn> </msup> </semantics></math> erg. For a typical rate of stellar capture (once every <math display='inline'><semantics> <msup> <mn>10</mn> <mn>4</mn> </msup> </semantics></math> years), this can provide a luminosity ≳<math display='inline'><semantics> <msup> <mn>10</mn> <mn>41</mn> </msup> </semantics></math> erg <math display='inline'><semantics> <mrow> <msup> <mi mathvariant="normal">s</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </semantics></math> from the Galactic Center. These processes of stellar tidal disruption events can be directly observed in some external galaxies.</div></li><li><div class='html-p'>Hydrodynamic models can describe the envelope of bubble propagation in the galactic halo where the gas distribution is nonuniform. The distribution is commonly characterized by an exponential or a power-law function. The observed shape of the Fermi bubbles seems to suggest an exponential halo. If the velocity of the top of the envelope prevails over the sound velocity in the halo, then the envelope may reach the size of 10 kpc.</div></li><li><div class='html-p'>The surface of the top of the envelope propagates with acceleration in the halo. As a result, Rayleigh–Taylor instabilities are developed and they will destroy the bubble envelope at the top. We expect excitation of hydrodynamic instabilities and generation of hydrodynamic turbulence there.</div></li><li><div class='html-p'>Turbulent motions act as a source of waves, which are manifested as a hierarchy of eddies, and act as a direct source of energy to the MHD waves (via the Lighthill mechanism). For small Mach numbers, a small fraction of the power radiated by the turbulent motion is Afvén waves.</div></li><li><div class='html-p'>The coefficients of the spatial and momentum diffusion of the system of nonlinear kinetic equations of the cosmic-ray distribution function are derived from the spectrum of MHD waves. These coefficients were calculated analytically, but we were unable to estimate the numerical values for the bubbles because of the lack of available observations on the wave spectrum.</div></li><li><div class='html-p'>We roughly estimated the spatial and momentum diffusion of cosmic rays in the envelope from the data of gamma-ray and microwave radiations from the Fermi bubbles.</div></li><li><div class='html-p'>We concluded that the observed gamma-ray and microwave radiations from the envelope of the Fermi Bubbles are generated by cosmic-ray electrons only. The contribution of cosmic-ray protons can be neglected.</div></li><li><div class='html-p'>We prefer the model that GeV cosmic-ray electrons from supernova remnants in the galactic disk are re-accelerated in situ in the bubbles to TeV energy range. With the help of a divergent flow, this can reproduce the data of both gamma-ray and microwave observations.</div></li><li><div class='html-p'>On the other hand, high-energy cosmic ray protons can escape the bubbles and reach the Earth. Cosmic ray protons from supernova remnants can be accelerated in the bubbles by supersonic turbulence to higher energies. We found that the escaped high-energy protons that arrive the Earth can reproduce the spectrum and flux of cosmic rays in the range <math display='inline'><semantics> <msup> <mn>10</mn> <mn>15</mn> </msup> </semantics></math>∼<math display='inline'><semantics> <msup> <mn>10</mn> <mn>18</mn> </msup> </semantics></math> eV (from the ‘knee’ to the ‘ankle’), observed near Earth.</div></li></ul></section> </div> <div class="html-back"> <section class='html-notes'><h2 >Author Contributions</h2><div class='html-p'>Conceptualization, V.A.D.; formal analysis, V.A.D. and C.M.K.; writing—original draft preparation, V.A.D. and C.M.K.; writing—review and editing, C.M.K. All authors have read and agreed to the published version of the manuscript.</div></section><section class='html-notes'><h2>Funding</h2><div class='html-p'>C.M.K. is supported in part by the Taiwan National Science and Technology Council grant NSTC 113-2112-M-008-001.</div></section><section class='html-notes'><h2 >Data Availability Statement</h2><div class='html-p'>No new data were created or analyzed in this study. Data sharing is not applicable to this article.</div></section><section id='html-ack' class='html-ack'><h2 >Acknowledgments</h2><div class='html-p'>First of all, let us thank colleagues who participated as co-authors of publications, mentioned in the list of references. It was a great joy for us to collaborate with them, and a significant contribution for this review we got from their collaboration. All of them we thank very much and keep memory about any of you. We are grateful for the nice atmosphere where we spent time participating in seminars and private talks with our colleagues from: P.N. Lebedev Institute of Physics (Russia), National Central University (Taiwan), The University of Hong Kong (Hong Kong), Max-Planck-Institut für Extraterrestrische Physik (Germany), Institute of Space and Astronautical Science (Japan), University of Bristol (UK), etc. We are grateful to the International Space Science Institute (ISSI(Bern) and ISSI-BJ (Beijing)) which organized several workshops, whose informal and warm atmosphere helped us to understand some problems on Fermi Bubbles origin.</div></section><section class='html-notes'><h2 >Conflicts of Interest</h2><div class='html-p'>The authors declare no conflicts of interest.</div></section><section id='html-references_list'><h2>References</h2><ol class='html-xxx'><li id='B1-universe-10-00424' class='html-x' data-content='1.'>Predehl, P.; Sunyaev, R.A.; Becker, W.; Brunner, H.; Burenin, R.; Bykov, A.; Cherepashchuk, A.; Chugai, N.; Churazov, E.; Doroshenko, V.; et al. 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Figure reproduced from Predehl et al. [<a href="#B1-universe-10-00424" class="html-bibr">1</a>] with permission.  </div> </div> <div class="html-fig_show mfp-hide" id="fig_body_display_universe-10-00424-f001"> <div class="html-caption"> <b>Figure 1.</b> Comparison of the morphology of the gamma-ray bubbles (red) and the X-ray bubbles (cyan) in the direction of the Galactic Center. Figure reproduced from Predehl et al. [<a href="#B1-universe-10-00424" class="html-bibr">1</a>] with permission.</div> <div class="html-img"><img data-large="/universe/universe-10-00424/article_deploy/html/images/universe-10-00424-g001.png" data-original="/universe/universe-10-00424/article_deploy/html/images/universe-10-00424-g001.png" alt="Universe 10 00424 g001" data-lsrc="/universe/universe-10-00424/article_deploy/html/images/universe-10-00424-g001.png" /></div> </div> <div class="html-fig-wrap" id="universe-10-00424-f002"> <div class='html-fig_img'> <div class="html-figpopup html-figpopup-link" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/11/424/display" href="#fig_body_display_universe-10-00424-f002"> <img data-large="/universe/universe-10-00424/article_deploy/html/images/universe-10-00424-g002.png" data-original="/universe/universe-10-00424/article_deploy/html/images/universe-10-00424-g002.png" alt="Universe 10 00424 g002" data-lsrc="/universe/universe-10-00424/article_deploy/html/images/universe-10-00424-g002-550.jpg" /> <a class="html-expand html-figpopup" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/11/424/display" href="#fig_body_display_universe-10-00424-f002"></a> </div> </div> <div class="html-fig_description"> <b>Figure 2.</b> X-ray superbubbles in the galaxy NGC 3079. Image from <a href='https://chandra.harvard.edu/photo/2019/ngc3079' target='_blank' rel="noopener noreferrer">https://chandra.harvard.edu/photo/2019/ngc3079</a> (accessed on 10 September 2024). Image credit: X-ray: NASA/CXC/University of Michigan. Optical: NASA/STScI.  </div> </div> <div class="html-fig_show mfp-hide" id="fig_body_display_universe-10-00424-f002"> <div class="html-caption"> <b>Figure 2.</b> X-ray superbubbles in the galaxy NGC 3079. Image from <a href='https://chandra.harvard.edu/photo/2019/ngc3079' target='_blank' rel="noopener noreferrer">https://chandra.harvard.edu/photo/2019/ngc3079</a> (accessed on 10 September 2024). Image credit: X-ray: NASA/CXC/University of Michigan. Optical: NASA/STScI.</div> <div class="html-img"><img data-large="/universe/universe-10-00424/article_deploy/html/images/universe-10-00424-g002.png" data-original="/universe/universe-10-00424/article_deploy/html/images/universe-10-00424-g002.png" alt="Universe 10 00424 g002" data-lsrc="/universe/universe-10-00424/article_deploy/html/images/universe-10-00424-g002.png" /></div> </div> <div class="html-fig-wrap" id="universe-10-00424-f003"> <div class='html-fig_img'> <div class="html-figpopup html-figpopup-link" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/11/424/display" href="#fig_body_display_universe-10-00424-f003"> <img data-large="/universe/universe-10-00424/article_deploy/html/images/universe-10-00424-g003.png" data-original="/universe/universe-10-00424/article_deploy/html/images/universe-10-00424-g003.png" alt="Universe 10 00424 g003" data-lsrc="/universe/universe-10-00424/article_deploy/html/images/universe-10-00424-g003-550.jpg" /> <a class="html-expand html-figpopup" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/11/424/display" href="#fig_body_display_universe-10-00424-f003"></a> </div> </div> <div class="html-fig_description"> <b>Figure 3.</b> The observed X-ray light curve of Swift J1644+57 from Swift, XMM-Newton, and Chandra. Figure reproduced from Cheng et al. [<a href="#B43-universe-10-00424" class="html-bibr">43</a>] with permission.  </div> </div> <div class="html-fig_show mfp-hide" id="fig_body_display_universe-10-00424-f003"> <div class="html-caption"> <b>Figure 3.</b> The observed X-ray light curve of Swift J1644+57 from Swift, XMM-Newton, and Chandra. Figure reproduced from Cheng et al. [<a href="#B43-universe-10-00424" class="html-bibr">43</a>] with permission.</div> <div class="html-img"><img data-large="/universe/universe-10-00424/article_deploy/html/images/universe-10-00424-g003.png" data-original="/universe/universe-10-00424/article_deploy/html/images/universe-10-00424-g003.png" alt="Universe 10 00424 g003" data-lsrc="/universe/universe-10-00424/article_deploy/html/images/universe-10-00424-g003.png" /></div> </div> <div class="html-fig-wrap" id="universe-10-00424-f004"> <div class='html-fig_img'> <div class="html-figpopup html-figpopup-link" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/11/424/display" href="#fig_body_display_universe-10-00424-f004"> <img data-large="/universe/universe-10-00424/article_deploy/html/images/universe-10-00424-g004.png" data-original="/universe/universe-10-00424/article_deploy/html/images/universe-10-00424-g004.png" alt="Universe 10 00424 g004" data-lsrc="/universe/universe-10-00424/article_deploy/html/images/universe-10-00424-g004-550.jpg" /> <a class="html-expand html-figpopup" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/11/424/display" href="#fig_body_display_universe-10-00424-f004"></a> </div> </div> <div class="html-fig_description"> <b>Figure 4.</b> A thermonuclear explosion in the terrestrial atmosphere. Image credit: United States Department of Energy. Image from <a href='https://commons.wikimedia.org/wiki/File:Castle_Bravo_nuclear_test_(cropped).jpg' target='_blank' rel="noopener noreferrer">https://commons.wikimedia.org/wiki/File:Castle_Bravo_nuclear_test_(cropped).jpg</a> (accessed on 10 September 2024).  </div> </div> <div class="html-fig_show mfp-hide" id="fig_body_display_universe-10-00424-f004"> <div class="html-caption"> <b>Figure 4.</b> A thermonuclear explosion in the terrestrial atmosphere. Image credit: United States Department of Energy. Image from <a href='https://commons.wikimedia.org/wiki/File:Castle_Bravo_nuclear_test_(cropped).jpg' target='_blank' rel="noopener noreferrer">https://commons.wikimedia.org/wiki/File:Castle_Bravo_nuclear_test_(cropped).jpg</a> (accessed on 10 September 2024).</div> <div class="html-img"><img data-large="/universe/universe-10-00424/article_deploy/html/images/universe-10-00424-g004.png" data-original="/universe/universe-10-00424/article_deploy/html/images/universe-10-00424-g004.png" alt="Universe 10 00424 g004" data-lsrc="/universe/universe-10-00424/article_deploy/html/images/universe-10-00424-g004.png" /></div> </div> <div class="html-fig-wrap" id="universe-10-00424-f005"> <div class='html-fig_img'> <div class="html-figpopup html-figpopup-link" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/11/424/display" href="#fig_body_display_universe-10-00424-f005"> <img data-large="/universe/universe-10-00424/article_deploy/html/images/universe-10-00424-g005.png" data-original="/universe/universe-10-00424/article_deploy/html/images/universe-10-00424-g005.png" alt="Universe 10 00424 g005" data-lsrc="/universe/universe-10-00424/article_deploy/html/images/universe-10-00424-g005-550.jpg" /> <a class="html-expand html-figpopup" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/11/424/display" href="#fig_body_display_universe-10-00424-f005"></a> </div> </div> <div class="html-fig_description"> <b>Figure 5.</b> Illustration of the double-bubble shock envelope in the halo evolving with time. The gas distribution in the halo in the left panel is exponential and in the right panel follows a power law. Figure adapted from Ko et al. [<a href="#B16-universe-10-00424" class="html-bibr">16</a>] with permission.  </div> </div> <div class="html-fig_show mfp-hide" id="fig_body_display_universe-10-00424-f005"> <div class="html-caption"> <b>Figure 5.</b> Illustration of the double-bubble shock envelope in the halo evolving with time. The gas distribution in the halo in the left panel is exponential and in the right panel follows a power law. Figure adapted from Ko et al. [<a href="#B16-universe-10-00424" class="html-bibr">16</a>] with permission.</div> <div class="html-img"><img data-large="/universe/universe-10-00424/article_deploy/html/images/universe-10-00424-g005.png" data-original="/universe/universe-10-00424/article_deploy/html/images/universe-10-00424-g005.png" alt="Universe 10 00424 g005" data-lsrc="/universe/universe-10-00424/article_deploy/html/images/universe-10-00424-g005.png" /></div> </div> <div class="html-fig-wrap" id="universe-10-00424-f006"> <div class='html-fig_img'> <div class="html-figpopup html-figpopup-link" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/11/424/display" href="#fig_body_display_universe-10-00424-f006"> <img data-large="/universe/universe-10-00424/article_deploy/html/images/universe-10-00424-g006.png" data-original="/universe/universe-10-00424/article_deploy/html/images/universe-10-00424-g006.png" alt="Universe 10 00424 g006" data-lsrc="/universe/universe-10-00424/article_deploy/html/images/universe-10-00424-g006-550.jpg" /> <a class="html-expand html-figpopup" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/11/424/display" href="#fig_body_display_universe-10-00424-f006"></a> </div> </div> <div class="html-fig_description"> <b>Figure 6.</b> Temporal variation in the shock velocity of the top of the bubble for the case of exponential halo with <math display='inline'><semantics> <mrow> <mi>H</mi> <mo>=</mo> <mn>0.67</mn> </mrow> </semantics></math> kpc and <math display='inline'><semantics> <mrow> <msub> <mi>n</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>0.03</mn> <msup> <mi>cm</mi> <mrow> <mo>−</mo> <mn>3</mn> </mrow> </msup> </mrow> </semantics></math>. (<b>Left panel</b>): One single input of energy from the GC. (<b>Right panel</b>): Multiple TDEs with different values of power release at the GC. The horizontal dotted line indicates the velocity which is necessary for the shock in order not to stall in the halo, which is three times the sound speed in the halo <math display='inline'><semantics> <mrow> <mn>3</mn> <mo>×</mo> <msup> <mn>10</mn> <mn>7</mn> </msup> </mrow> </semantics></math> cm <math display='inline'><semantics> <mrow> <msup> <mi mathvariant="normal">s</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </semantics></math>. Figure reproduced from Ko et al. [<a href="#B16-universe-10-00424" class="html-bibr">16</a>] with permission.  </div> </div> <div class="html-fig_show mfp-hide" id="fig_body_display_universe-10-00424-f006"> <div class="html-caption"> <b>Figure 6.</b> Temporal variation in the shock velocity of the top of the bubble for the case of exponential halo with <math display='inline'><semantics> <mrow> <mi>H</mi> <mo>=</mo> <mn>0.67</mn> </mrow> </semantics></math> kpc and <math display='inline'><semantics> <mrow> <msub> <mi>n</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>0.03</mn> <msup> <mi>cm</mi> <mrow> <mo>−</mo> <mn>3</mn> </mrow> </msup> </mrow> </semantics></math>. (<b>Left panel</b>): One single input of energy from the GC. (<b>Right panel</b>): Multiple TDEs with different values of power release at the GC. The horizontal dotted line indicates the velocity which is necessary for the shock in order not to stall in the halo, which is three times the sound speed in the halo <math display='inline'><semantics> <mrow> <mn>3</mn> <mo>×</mo> <msup> <mn>10</mn> <mn>7</mn> </msup> </mrow> </semantics></math> cm <math display='inline'><semantics> <mrow> <msup> <mi mathvariant="normal">s</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </semantics></math>. Figure reproduced from Ko et al. [<a href="#B16-universe-10-00424" class="html-bibr">16</a>] with permission.</div> <div class="html-img"><img data-large="/universe/universe-10-00424/article_deploy/html/images/universe-10-00424-g006.png" data-original="/universe/universe-10-00424/article_deploy/html/images/universe-10-00424-g006.png" alt="Universe 10 00424 g006" data-lsrc="/universe/universe-10-00424/article_deploy/html/images/universe-10-00424-g006.png" /></div> </div> <div class="html-fig-wrap" id="universe-10-00424-f007"> <div class='html-fig_img'> <div class="html-figpopup html-figpopup-link" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/11/424/display" href="#fig_body_display_universe-10-00424-f007"> <img data-large="/universe/universe-10-00424/article_deploy/html/images/universe-10-00424-g007.png" data-original="/universe/universe-10-00424/article_deploy/html/images/universe-10-00424-g007.png" alt="Universe 10 00424 g007" data-lsrc="/universe/universe-10-00424/article_deploy/html/images/universe-10-00424-g007-550.jpg" /> <a class="html-expand html-figpopup" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/11/424/display" href="#fig_body_display_universe-10-00424-f007"></a> </div> </div> <div class="html-fig_description"> <b>Figure 7.</b> Density distribution of numerical simulations of the FBs in an exponential halo. The two panels in the left column are results of multiple explosions (e.g., TDEs) and in the right column are results of a single huge explosion. In the upper left panel, “Me0.05-3e52erg 18.0 Myr” corresponds to multiple explosions with 0.05 Myr between successive explosions and the energy release by each explosion is <math display='inline'><semantics> <mrow> <mn>3</mn> <mo>×</mo> <msup> <mn>10</mn> <mn>52</mn> </msup> </mrow> </semantics></math> erg, and the simulation ends at 18.0 Myr. In the upper right panel, “1e-1.08e55erg 10.0 Myr” corresponds to a single explosion with an energy release of <math display='inline'><semantics> <mrow> <mn>1.08</mn> <mo>×</mo> <msup> <mn>10</mn> <mn>55</mn> </msup> </mrow> </semantics></math> erg, and the simulation ends at 10.0 Myr. Similar explanation for the lower panels. Lower panel figures reproduced from Ko et al. [<a href="#B16-universe-10-00424" class="html-bibr">16</a>] with permission.  </div> </div> <div class="html-fig_show mfp-hide" id="fig_body_display_universe-10-00424-f007"> <div class="html-caption"> <b>Figure 7.</b> Density distribution of numerical simulations of the FBs in an exponential halo. The two panels in the left column are results of multiple explosions (e.g., TDEs) and in the right column are results of a single huge explosion. In the upper left panel, “Me0.05-3e52erg 18.0 Myr” corresponds to multiple explosions with 0.05 Myr between successive explosions and the energy release by each explosion is <math display='inline'><semantics> <mrow> <mn>3</mn> <mo>×</mo> <msup> <mn>10</mn> <mn>52</mn> </msup> </mrow> </semantics></math> erg, and the simulation ends at 18.0 Myr. In the upper right panel, “1e-1.08e55erg 10.0 Myr” corresponds to a single explosion with an energy release of <math display='inline'><semantics> <mrow> <mn>1.08</mn> <mo>×</mo> <msup> <mn>10</mn> <mn>55</mn> </msup> </mrow> </semantics></math> erg, and the simulation ends at 10.0 Myr. Similar explanation for the lower panels. Lower panel figures reproduced from Ko et al. [<a href="#B16-universe-10-00424" class="html-bibr">16</a>] with permission.</div> <div class="html-img"><img data-large="/universe/universe-10-00424/article_deploy/html/images/universe-10-00424-g007.png" data-original="/universe/universe-10-00424/article_deploy/html/images/universe-10-00424-g007.png" alt="Universe 10 00424 g007" data-lsrc="/universe/universe-10-00424/article_deploy/html/images/universe-10-00424-g007.png" /></div> </div> <div class="html-fig-wrap" id="universe-10-00424-f008"> <div class='html-fig_img'> <div class="html-figpopup html-figpopup-link" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/11/424/display" href="#fig_body_display_universe-10-00424-f008"> <img data-large="/universe/universe-10-00424/article_deploy/html/images/universe-10-00424-g008.png" data-original="/universe/universe-10-00424/article_deploy/html/images/universe-10-00424-g008.png" alt="Universe 10 00424 g008" data-lsrc="/universe/universe-10-00424/article_deploy/html/images/universe-10-00424-g008-550.jpg" /> <a class="html-expand html-figpopup" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/11/424/display" href="#fig_body_display_universe-10-00424-f008"></a> </div> </div> <div class="html-fig_description"> <b>Figure 8.</b> Coherent magnetic structure above and below the galactic plane [<a href="#B17-universe-10-00424" class="html-bibr">17</a>]. (<b>a</b>) Polarized synchrotron intensity map at <math display='inline'><semantics> <mrow> <mn>22.8</mn> </mrow> </semantics></math> GHz from WMAP. Green bars show the magnetic field direction. (<b>b</b>) Comparison between the polarized synchrotron emission at <math display='inline'><semantics> <mrow> <mn>22.8</mn> </mrow> </semantics></math> GHz (red) and the X-ray emission at <math display='inline'><semantics> <mrow> <mn>0.6</mn> </mrow> </semantics></math>∼<math display='inline'><semantics> <mrow> <mn>1.0</mn> </mrow> </semantics></math> keV from eROSITA (green). Magnetized ridges are shown in white. Figure reproduced from Zhang et al. [<a href="#B17-universe-10-00424" class="html-bibr">17</a>] with permission.  </div> </div> <div class="html-fig_show mfp-hide" id="fig_body_display_universe-10-00424-f008"> <div class="html-caption"> <b>Figure 8.</b> Coherent magnetic structure above and below the galactic plane [<a href="#B17-universe-10-00424" class="html-bibr">17</a>]. (<b>a</b>) Polarized synchrotron intensity map at <math display='inline'><semantics> <mrow> <mn>22.8</mn> </mrow> </semantics></math> GHz from WMAP. Green bars show the magnetic field direction. (<b>b</b>) Comparison between the polarized synchrotron emission at <math display='inline'><semantics> <mrow> <mn>22.8</mn> </mrow> </semantics></math> GHz (red) and the X-ray emission at <math display='inline'><semantics> <mrow> <mn>0.6</mn> </mrow> </semantics></math>∼<math display='inline'><semantics> <mrow> <mn>1.0</mn> </mrow> </semantics></math> keV from eROSITA (green). Magnetized ridges are shown in white. Figure reproduced from Zhang et al. [<a href="#B17-universe-10-00424" class="html-bibr">17</a>] with permission.</div> <div class="html-img"><img data-large="/universe/universe-10-00424/article_deploy/html/images/universe-10-00424-g008.png" data-original="/universe/universe-10-00424/article_deploy/html/images/universe-10-00424-g008.png" alt="Universe 10 00424 g008" data-lsrc="/universe/universe-10-00424/article_deploy/html/images/universe-10-00424-g008.png" /></div> </div> <div class="html-fig-wrap" id="universe-10-00424-f009"> <div class='html-fig_img'> <div class="html-figpopup html-figpopup-link" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/11/424/display" href="#fig_body_display_universe-10-00424-f009"> <img data-large="/universe/universe-10-00424/article_deploy/html/images/universe-10-00424-g009.png" data-original="/universe/universe-10-00424/article_deploy/html/images/universe-10-00424-g009.png" alt="Universe 10 00424 g009" data-lsrc="/universe/universe-10-00424/article_deploy/html/images/universe-10-00424-g009-550.jpg" /> <a class="html-expand html-figpopup" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/11/424/display" href="#fig_body_display_universe-10-00424-f009"></a> </div> </div> <div class="html-fig_description"> <b>Figure 9.</b> The solid line shows the momentum diffusion coefficient derived for the bubble parameters when the CR absorption is taken into account. The dash-dotted line is the results ignoring the CR absorption. Figure reproduced from Cheng et al. [<a href="#B71-universe-10-00424" class="html-bibr">71</a>] with permission.  </div> </div> <div class="html-fig_show mfp-hide" id="fig_body_display_universe-10-00424-f009"> <div class="html-caption"> <b>Figure 9.</b> The solid line shows the momentum diffusion coefficient derived for the bubble parameters when the CR absorption is taken into account. The dash-dotted line is the results ignoring the CR absorption. Figure reproduced from Cheng et al. [<a href="#B71-universe-10-00424" class="html-bibr">71</a>] with permission.</div> <div class="html-img"><img data-large="/universe/universe-10-00424/article_deploy/html/images/universe-10-00424-g009.png" data-original="/universe/universe-10-00424/article_deploy/html/images/universe-10-00424-g009.png" alt="Universe 10 00424 g009" data-lsrc="/universe/universe-10-00424/article_deploy/html/images/universe-10-00424-g009.png" /></div> </div> <div class="html-fig-wrap" id="universe-10-00424-f010"> <div class='html-fig_img'> <div class="html-figpopup html-figpopup-link" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/11/424/display" href="#fig_body_display_universe-10-00424-f010"> <img data-large="/universe/universe-10-00424/article_deploy/html/images/universe-10-00424-g010.png" data-original="/universe/universe-10-00424/article_deploy/html/images/universe-10-00424-g010.png" alt="Universe 10 00424 g010" data-lsrc="/universe/universe-10-00424/article_deploy/html/images/universe-10-00424-g010-550.jpg" /> <a class="html-expand html-figpopup" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/11/424/display" href="#fig_body_display_universe-10-00424-f010"></a> </div> </div> <div class="html-fig_description"> <b>Figure 10.</b> Spectrum of radio (<b>left</b>) and gamma-ray (<b>right</b>) emission from the FBs (see [<a href="#B71-universe-10-00424" class="html-bibr">71</a>]). The microwave data were taken from Planck Collaboration [<a href="#B3-universe-10-00424" class="html-bibr">3</a>], and the gamma-ray data from Ackermann et al. [<a href="#B72-universe-10-00424" class="html-bibr">72</a>], Ackermann et al. [<a href="#B77-universe-10-00424" class="html-bibr">77</a>]. Figure adapted from Cheng et al. [<a href="#B71-universe-10-00424" class="html-bibr">71</a>] with permission.  </div> </div> <div class="html-fig_show mfp-hide" id="fig_body_display_universe-10-00424-f010"> <div class="html-caption"> <b>Figure 10.</b> Spectrum of radio (<b>left</b>) and gamma-ray (<b>right</b>) emission from the FBs (see [<a href="#B71-universe-10-00424" class="html-bibr">71</a>]). The microwave data were taken from Planck Collaboration [<a href="#B3-universe-10-00424" class="html-bibr">3</a>], and the gamma-ray data from Ackermann et al. [<a href="#B72-universe-10-00424" class="html-bibr">72</a>], Ackermann et al. [<a href="#B77-universe-10-00424" class="html-bibr">77</a>]. Figure adapted from Cheng et al. [<a href="#B71-universe-10-00424" class="html-bibr">71</a>] with permission.</div> <div class="html-img"><img data-large="/universe/universe-10-00424/article_deploy/html/images/universe-10-00424-g010.png" data-original="/universe/universe-10-00424/article_deploy/html/images/universe-10-00424-g010.png" alt="Universe 10 00424 g010" data-lsrc="/universe/universe-10-00424/article_deploy/html/images/universe-10-00424-g010.png" /></div> </div> <div class="html-fig-wrap" id="universe-10-00424-f011"> <div class='html-fig_img'> <div class="html-figpopup html-figpopup-link" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/11/424/display" href="#fig_body_display_universe-10-00424-f011"> <img data-large="/universe/universe-10-00424/article_deploy/html/images/universe-10-00424-g011.png" data-original="/universe/universe-10-00424/article_deploy/html/images/universe-10-00424-g011.png" alt="Universe 10 00424 g011" data-lsrc="/universe/universe-10-00424/article_deploy/html/images/universe-10-00424-g011-550.jpg" /> <a class="html-expand html-figpopup" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/11/424/display" href="#fig_body_display_universe-10-00424-f011"></a> </div> </div> <div class="html-fig_description"> <b>Figure 11.</b> X-ray emission from the galactic plane whose excess emission is above the equilibrium Maxwellian spectrum. Dash-dotted line is a simple combination of thermal plus nonthermal spectrum. Solid line is the spectrum with the effect of runaway flux. Figure reproduced from Dogiel et al. [<a href="#B96-universe-10-00424" class="html-bibr">96</a>] with permission.  </div> </div> <div class="html-fig_show mfp-hide" id="fig_body_display_universe-10-00424-f011"> <div class="html-caption"> <b>Figure 11.</b> X-ray emission from the galactic plane whose excess emission is above the equilibrium Maxwellian spectrum. Dash-dotted line is a simple combination of thermal plus nonthermal spectrum. Solid line is the spectrum with the effect of runaway flux. Figure reproduced from Dogiel et al. [<a href="#B96-universe-10-00424" class="html-bibr">96</a>] with permission.</div> <div class="html-img"><img data-large="/universe/universe-10-00424/article_deploy/html/images/universe-10-00424-g011.png" data-original="/universe/universe-10-00424/article_deploy/html/images/universe-10-00424-g011.png" alt="Universe 10 00424 g011" data-lsrc="/universe/universe-10-00424/article_deploy/html/images/universe-10-00424-g011.png" /></div> </div> <div class="html-fig-wrap" id="universe-10-00424-f012"> <div class='html-fig_img'> <div class="html-figpopup html-figpopup-link" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/11/424/display" href="#fig_body_display_universe-10-00424-f012"> <img data-large="/universe/universe-10-00424/article_deploy/html/images/universe-10-00424-g012.png" data-original="/universe/universe-10-00424/article_deploy/html/images/universe-10-00424-g012.png" alt="Universe 10 00424 g012" data-lsrc="/universe/universe-10-00424/article_deploy/html/images/universe-10-00424-g012-550.jpg" /> <a class="html-expand html-figpopup" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/11/424/display" href="#fig_body_display_universe-10-00424-f012"></a> </div> </div> <div class="html-fig_description"> <b>Figure 12.</b> The spectrum of electrons accelerated from background plasma (see [<a href="#B100-universe-10-00424" class="html-bibr">100</a>]). The solid line is the density of electrons, <math display='inline'><semantics> <mrow> <mi>f</mi> <mo>(</mo> <mi>p</mi> <mo>)</mo> </mrow> </semantics></math>. The thick solid line is the pure thermal Maxwellian distribution. The dashed line is the power-law approximation of the nonthermal tail. For <math display='inline'><semantics> <mrow> <msub> <mi>p</mi> <mn>0</mn> </msub> <mo>></mo> <msub> <mi>p</mi> <mi>inj</mi> </msub> </mrow> </semantics></math>, overheating is insignificant. Figure adapted from Chernyshov et al. [<a href="#B100-universe-10-00424" class="html-bibr">100</a>] with permission.  </div> </div> <div class="html-fig_show mfp-hide" id="fig_body_display_universe-10-00424-f012"> <div class="html-caption"> <b>Figure 12.</b> The spectrum of electrons accelerated from background plasma (see [<a href="#B100-universe-10-00424" class="html-bibr">100</a>]). The solid line is the density of electrons, <math display='inline'><semantics> <mrow> <mi>f</mi> <mo>(</mo> <mi>p</mi> <mo>)</mo> </mrow> </semantics></math>. The thick solid line is the pure thermal Maxwellian distribution. The dashed line is the power-law approximation of the nonthermal tail. For <math display='inline'><semantics> <mrow> <msub> <mi>p</mi> <mn>0</mn> </msub> <mo>></mo> <msub> <mi>p</mi> <mi>inj</mi> </msub> </mrow> </semantics></math>, overheating is insignificant. Figure adapted from Chernyshov et al. [<a href="#B100-universe-10-00424" class="html-bibr">100</a>] with permission.</div> <div class="html-img"><img data-large="/universe/universe-10-00424/article_deploy/html/images/universe-10-00424-g012.png" data-original="/universe/universe-10-00424/article_deploy/html/images/universe-10-00424-g012.png" alt="Universe 10 00424 g012" data-lsrc="/universe/universe-10-00424/article_deploy/html/images/universe-10-00424-g012.png" /></div> </div> <div class="html-fig-wrap" id="universe-10-00424-f013"> <div class='html-fig_img'> <div class="html-figpopup html-figpopup-link" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/11/424/display" href="#fig_body_display_universe-10-00424-f013"> <img data-large="/universe/universe-10-00424/article_deploy/html/images/universe-10-00424-g013.png" data-original="/universe/universe-10-00424/article_deploy/html/images/universe-10-00424-g013.png" alt="Universe 10 00424 g013" data-lsrc="/universe/universe-10-00424/article_deploy/html/images/universe-10-00424-g013-550.jpg" /> <a class="html-expand html-figpopup" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/11/424/display" href="#fig_body_display_universe-10-00424-f013"></a> </div> </div> <div class="html-fig_description"> <b>Figure 13.</b> The spectrum of SNR electrons from the galactic disk that have been re-accelerated in the FBs. The five spectra in the figure correspond to different cases of the model: (1) thick solid line: without re-acceleration, escape, and advection; (2) thick dash-dotted line: without re-acceleration and escape but with advection; (3) thin dash-dotted line: with re-acceleration but without escape from the region and advection; (4) thin dotted line: with re-acceleration and escape from the region but without advection; (5) thin dashed line: with re-acceleration and advection but without escape. The density of electrons needed for the observed gamma-ray flux from the bubbles is shown by the gray region. The electron spectrum of case (5) can reproduce the gamma-ray data from Fermi-LAT and the microwave data from Planck (<a href="#universe-10-00424-f010" class="html-fig">Figure 10</a>). The parameters of case (5) can be found in the main text. For parameters of other cases, the reader is referred to Cheng et al. [<a href="#B101-universe-10-00424" class="html-bibr">101</a>]. Figure reproduced from Cheng et al. [<a href="#B101-universe-10-00424" class="html-bibr">101</a>] with permission.  </div> </div> <div class="html-fig_show mfp-hide" id="fig_body_display_universe-10-00424-f013"> <div class="html-caption"> <b>Figure 13.</b> The spectrum of SNR electrons from the galactic disk that have been re-accelerated in the FBs. The five spectra in the figure correspond to different cases of the model: (1) thick solid line: without re-acceleration, escape, and advection; (2) thick dash-dotted line: without re-acceleration and escape but with advection; (3) thin dash-dotted line: with re-acceleration but without escape from the region and advection; (4) thin dotted line: with re-acceleration and escape from the region but without advection; (5) thin dashed line: with re-acceleration and advection but without escape. The density of electrons needed for the observed gamma-ray flux from the bubbles is shown by the gray region. The electron spectrum of case (5) can reproduce the gamma-ray data from Fermi-LAT and the microwave data from Planck (<a href="#universe-10-00424-f010" class="html-fig">Figure 10</a>). The parameters of case (5) can be found in the main text. For parameters of other cases, the reader is referred to Cheng et al. [<a href="#B101-universe-10-00424" class="html-bibr">101</a>]. Figure reproduced from Cheng et al. [<a href="#B101-universe-10-00424" class="html-bibr">101</a>] with permission.</div> <div class="html-img"><img data-large="/universe/universe-10-00424/article_deploy/html/images/universe-10-00424-g013.png" data-original="/universe/universe-10-00424/article_deploy/html/images/universe-10-00424-g013.png" alt="Universe 10 00424 g013" data-lsrc="/universe/universe-10-00424/article_deploy/html/images/universe-10-00424-g013.png" /></div> </div> <div class="html-fig-wrap" id="universe-10-00424-f015"> <div class='html-fig_img'> <div class="html-figpopup html-figpopup-link" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/11/424/display" href="#fig_body_display_universe-10-00424-f015"> <img data-large="/universe/universe-10-00424/article_deploy/html/images/universe-10-00424-g015.png" data-original="/universe/universe-10-00424/article_deploy/html/images/universe-10-00424-g015.png" alt="Universe 10 00424 g015" data-lsrc="/universe/universe-10-00424/article_deploy/html/images/universe-10-00424-g015-550.jpg" /> <a class="html-expand html-figpopup" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/11/424/display" href="#fig_body_display_universe-10-00424-f015"></a> </div> </div> <div class="html-fig_description"> <b>Figure 15.</b> A possible multiple-shock structure in the FBs resulting from multiple TDEs at the GC. The figure shows the pressure (<b>left panel</b>) and kinetic energy (<b>right panel</b>) distributions of a numerical simulation of the FBs in an exponential halo. In the panels, “Me0.05-1e53” corresponds to multiple TDEs with 0.05 Myr between successive TDEs and the energy release by each TDE is <math display='inline'><semantics> <msup> <mn>10</mn> <mn>53</mn> </msup> </semantics></math> erg. The simulation ends at 10.0 Myr. The units of the color bars in both panels are <math display='inline'><semantics> <mrow> <mn>1.178</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>8</mn> </mrow> </msup> </mrow> </semantics></math> erg <math display='inline'><semantics> <mrow> <msup> <mi>cm</mi> <mrow> <mo>−</mo> <mn>3</mn> </mrow> </msup> </mrow> </semantics></math>.  </div> </div> <div class="html-fig_show mfp-hide" id="fig_body_display_universe-10-00424-f015"> <div class="html-caption"> <b>Figure 15.</b> A possible multiple-shock structure in the FBs resulting from multiple TDEs at the GC. The figure shows the pressure (<b>left panel</b>) and kinetic energy (<b>right panel</b>) distributions of a numerical simulation of the FBs in an exponential halo. In the panels, “Me0.05-1e53” corresponds to multiple TDEs with 0.05 Myr between successive TDEs and the energy release by each TDE is <math display='inline'><semantics> <msup> <mn>10</mn> <mn>53</mn> </msup> </semantics></math> erg. The simulation ends at 10.0 Myr. The units of the color bars in both panels are <math display='inline'><semantics> <mrow> <mn>1.178</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>8</mn> </mrow> </msup> </mrow> </semantics></math> erg <math display='inline'><semantics> <mrow> <msup> <mi>cm</mi> <mrow> <mo>−</mo> <mn>3</mn> </mrow> </msup> </mrow> </semantics></math>.</div> <div class="html-img"><img data-large="/universe/universe-10-00424/article_deploy/html/images/universe-10-00424-g015.png" data-original="/universe/universe-10-00424/article_deploy/html/images/universe-10-00424-g015.png" alt="Universe 10 00424 g015" data-lsrc="/universe/universe-10-00424/article_deploy/html/images/universe-10-00424-g015.png" /></div> </div> </section><section class='html-fn_group'><table><tr id=''><td></td><td><div class='html-p'><b>Disclaimer/Publisher’s Note:</b> The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). 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Sources and Radiations of the Fermi Bubbles