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src="https://www.gstatic.com/_/translate_http/_/js/k=translate_http.tr.en_GB.HWThTSqfXy8.O/am=DAY/d=1/exm=corsproxy,phishing_protection/ed=1/rs=AN8SPforCdnxlZDJUkR1p8RUr9HyDr68Rg/m=navigationui" data-environment="prod" data-proxy-url="https://web-archive-org.translate.goog" data-proxy-full-url="https://web-archive-org.translate.goog/web/20161123021114/https://briankoberlein.com/2013/10/14/mutual-attraction/?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" data-source-url="https://web.archive.org/web/20161123021114/https://briankoberlein.com/2013/10/14/mutual-attraction/" data-source-language="auto" data-target-language="en" data-display-language="en-GB" data-detected-source-language="en" data-is-source-untranslated="false" data-source-untranslated-url="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://web.archive.org/web/20161123021114/https://briankoberlein.com/2013/10/14/mutual-attraction/&anno=2" data-client="tr"></script> <script>__wm.rw(0);</script> <div id="wm-ipp-base" lang="en" style="display:none;direction:ltr;"> <div id="wm-ipp" style="position:fixed;left:0;top:0;right:0;"> <div id="donato" style="position:relative;width:100%;"> <div id="donato-base"><iframe id="donato-if" src="https://archive.org/includes/donate.php?as_page=1&platform=wb&referer=https%3A//web.archive.org/web/20161123021114/https%3A//briankoberlein.com/2013/10/14/mutual-attraction/" scrolling="no" frameborder="0" style="width:100%; height:100%"> </iframe> </div> </div> <div id="wm-ipp-inside"> <div id="wm-toolbar" style="position:relative;display:flex;flex-flow:row nowrap;justify-content:space-between;"> <div id="wm-logo" style="/*width:110px;*/padding-top:12px;"><a href="https://web-archive-org.translate.goog/web/?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Wayback Machine home page"><img src="/_static/images/toolbar/wayback-toolbar-logo-200.png" srcset="https://web-archive-org.translate.goog/_static/images/toolbar/wayback-toolbar-logo-100.png?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB,https://web-archive-org.translate.goog/_static/images/toolbar/wayback-toolbar-logo-150.png?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB 1.5x,https://web-archive-org.translate.goog/_static/images/toolbar/wayback-toolbar-logo-200.png?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB 2x" alt="Wayback Machine" style="width:100px" border="0"></a> </div> <div class="c" style="display:flex;flex-flow:column nowrap;justify-content:space-between;flex:1;"> <form class="u" style="display:flex;flex-direction:row;flex-wrap:nowrap;" target="_top" method="get" action="/web/submit" name="wmtb" id="wmtb"> <input type="text" name="url" id="wmtbURL" value="https://briankoberlein.com/2013/10/14/mutual-attraction/" onfocus="this.focus();this.select();" style="flex:1;"><input type="hidden" name="type" value="replay"><input type="hidden" name="date" value="20161123021114"><input type="submit" value="Go"> </form> <div style="display:flex;flex-flow:row nowrap;align-items:flex-end;"> <div class="s" id="wm-nav-captures" style="flex:1;"><a class="t" href="https://web-archive-org.translate.goog/web/20161123021114*/https://briankoberlein.com/2013/10/14/mutual-attraction/?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="See a list of every capture for this URL">10 captures</a> <div class="r" title="Timespan for captures of this URL"> 16 Mar 2015 - 06 Dec 2022 </div> </div> <div class="k"><a href="https://web-archive-org.translate.goog/web/20161123021114/https://briankoberlein.com/2013/10/14/mutual-attraction/?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" id="wm-graph-anchor"> <div id="wm-ipp-sparkline" title="Explore captures for this URL" style="position: relative"> <canvas id="wm-sparkline-canvas" width="725" height="27" border="0"></canvas> </div></a> </div> </div> </div> <div class="n"> <table> <tbody> <tr class="m"> <td class="b" nowrap><a href="https://web-archive-org.translate.goog/web/20160827013844/https://briankoberlein.com/2013/10/14/mutual-attraction/?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="27 Aug 2016"><strong>Aug</strong></a></td> <td class="c" id="displayMonthEl" title="You are here: 02:11:14 Nov 23, 2016">NOV</td> <td class="f" nowrap><a href="https://web-archive-org.translate.goog/web/20170220221618/https://briankoberlein.com/2013/10/14/mutual-attraction/?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="20 Feb 2017"><strong>Feb</strong></a></td> </tr> <tr class="d"> <td class="b" nowrap><a href="https://web-archive-org.translate.goog/web/20160827013844/https://briankoberlein.com/2013/10/14/mutual-attraction/?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="01:38:44 Aug 27, 2016"><img src="/_static/images/toolbar/wm_tb_prv_on.png" alt="Previous capture" width="14" height="16" border="0"></a></td> <td class="c" id="displayDayEl" style="width:34px;font-size:22px;white-space:nowrap;" title="You are here: 02:11:14 Nov 23, 2016">23</td> <td class="f" nowrap><a href="https://web-archive-org.translate.goog/web/20170220221618/https://briankoberlein.com/2013/10/14/mutual-attraction/?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="22:16:18 Feb 20, 2017"><img src="/_static/images/toolbar/wm_tb_nxt_on.png" alt="Next capture" width="14" height="16" border="0"></a></td> </tr> <tr class="y"> <td class="b" nowrap><a href="https://web-archive-org.translate.goog/web/20150907031357/https://briankoberlein.com/2013/10/14/mutual-attraction/?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="07 Sep 2015"><strong>2015</strong></a></td> <td class="c" id="displayYearEl" title="You are here: 02:11:14 Nov 23, 2016">2016</td> <td class="f" nowrap><a href="https://web-archive-org.translate.goog/web/20180908062859/https://briankoberlein.com/2013/10/14/mutual-attraction/?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="08 Sep 2018"><strong>2018</strong></a></td> </tr> </tbody> </table> </div> <div class="r" style="display:flex;flex-flow:column nowrap;align-items:flex-end;justify-content:space-between;"> <div id="wm-btns" style="text-align:right;height:23px;"><span class="xxs"> <div id="wm-save-snapshot-success"> success </div> <div id="wm-save-snapshot-fail"> fail </div><a id="wm-save-snapshot-open" href="https://web-archive-org.translate.goog/web/20161123021114/https://briankoberlein.com/2013/10/14/mutual-attraction/?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#" title="Share via My Web Archive"> <span class="iconochive-web"></span> </a> <a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://archive.org/account/login.php" title="Sign In" id="wm-sign-in"> <span class="iconochive-person"></span> </a> <span id="wm-save-snapshot-in-progress" class="iconochive-web"></span> </span> <a class="xxs" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=http://faq.web.archive.org/" title="Get some help using the Wayback Machine" style="top:-6px;"><span class="iconochive-question" style="color:rgb(87,186,244);font-size:160%;"></span></a> <a id="wm-tb-close" href="https://web-archive-org.translate.goog/web/20161123021114/https://briankoberlein.com/2013/10/14/mutual-attraction/?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#close" style="top:-2px;" title="Close the toolbar"><span class="iconochive-remove-circle" style="color:#888888;font-size:240%;"></span></a> </div> <div id="wm-share" class="xxs"><a href="https://web-archive-org.translate.goog/web/20161123021114/http://web.archive.org/screenshot/https://briankoberlein.com/2013/10/14/mutual-attraction/?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" id="wm-screenshot" title="screenshot"> <span class="wm-icon-screen-shot"></span> </a> <a href="https://web-archive-org.translate.goog/web/20161123021114/https://briankoberlein.com/2013/10/14/mutual-attraction/?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#" id="wm-video" title="video"> <span class="iconochive-movies"></span> </a> <a id="wm-share-facebook" href="https://web-archive-org.translate.goog/web/20161123021114/https://briankoberlein.com/2013/10/14/mutual-attraction/?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#" data-url="https://web.archive.org/web/20161123021114/https://briankoberlein.com/2013/10/14/mutual-attraction/" title="Share on Facebook" style="margin-right:5px;" target="_blank"><span class="iconochive-facebook" style="color:#3b5998;font-size:160%;"></span></a> <a id="wm-share-twitter" href="https://web-archive-org.translate.goog/web/20161123021114/https://briankoberlein.com/2013/10/14/mutual-attraction/?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#" data-url="https://web.archive.org/web/20161123021114/https://briankoberlein.com/2013/10/14/mutual-attraction/" title="Share on Twitter" style="margin-right:5px;" target="_blank"><span class="iconochive-twitter" style="color:#1dcaff;font-size:160%;"></span></a> </div> <div style="padding-right:2px;text-align:right;white-space:nowrap;"><a id="wm-expand" class="wm-btn wm-closed" href="https://web-archive-org.translate.goog/web/20161123021114/https://briankoberlein.com/2013/10/14/mutual-attraction/?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#expand"><span id="wm-expand-icon" class="iconochive-down-solid"></span> <span class="xxs" style="font-size:80%;">About this capture</span></a> </div> </div> </div> <div id="wm-capinfo" style="border-top:1px solid #777;display:none; overflow: hidden"> <div id="wm-capinfo-notice" source="api"></div> <div id="wm-capinfo-collected-by"> <div style="background-color:#666;color:#fff;font-weight:bold;text-align:center"> COLLECTED BY </div> <div style="padding:3px;position:relative" id="wm-collected-by-content"> <div style="display:inline-block;vertical-align:top;width:50%;"><span class="c-logo" style="background-image:url(https://archive.org/services/img/webwidecrawl);"></span> Organization: <a style="color:#33f;" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://archive.org/details/webwidecrawl" target="_new"><span class="wm-title">Internet Archive</span></a> <div style="max-height:75px;overflow:hidden;position:relative;"> <div style="position:absolute;top:0;left:0;width:100%;height:75px;background:linear-gradient(to bottom,rgba(255,255,255,0) 0%,rgba(255,255,255,0) 90%,rgba(255,255,255,255) 100%);"></div> The Internet Archive discovers and captures web pages through many different web crawls. At any given time several distinct crawls are running, some for months, and some every day or longer. View the web archive through the <a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=http://archive.org/web/web.php">Wayback Machine</a>. </div> </div> <div style="display:inline-block;vertical-align:top;width:49%;"><span class="c-logo" style="background-image:url(https://archive.org/services/img/wide00015)"></span> <div> Collection: <a style="color:#33f;" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://archive.org/details/wide00015" target="_new"><span class="wm-title">Wide Crawl Number 15: Started Oct 1st, 2016 - Ended May 8th, 2017</span></a> </div> <div style="max-height:75px;overflow:hidden;position:relative;"> <div style="position:absolute;top:0;left:0;width:100%;height:75px;background:linear-gradient(to bottom,rgba(255,255,255,0) 0%,rgba(255,255,255,0) 90%,rgba(255,255,255,255) 100%);"></div> Web wide crawl. </div> </div> </div> </div> <div id="wm-capinfo-timestamps"> <div style="background-color:#666;color:#fff;font-weight:bold;text-align:center" title="Timestamps for the elements of this page"> TIMESTAMPS </div> <div> <div id="wm-capresources" style="margin:0 5px 5px 5px;max-height:250px;overflow-y:scroll !important"></div> <div id="wm-capresources-loading" style="text-align:left;margin:0 20px 5px 5px;display:none"> <img src="/_static/images/loading.gif" alt="loading"> </div> </div> </div> </div> </div> </div> </div> <div id="wm-ipp-print"> The Wayback Machine - https://web.archive.org/web/20161123021114/https://briankoberlein.com/2013/10/14/mutual-attraction/ </div> <script type="text/javascript">//<![CDATA[ __wm.bt(725,27,25,2,"web","https://briankoberlein.com/2013/10/14/mutual-attraction/","20161123021114",1996,"/_static/",["/_static/css/banner-styles.css?v=S1zqJCYt","/_static/css/iconochive.css?v=3PDvdIFv"], false); __wm.rw(1); //]]></script> <div id="top" class="site"> <header class="masthead masthead-inline" role="banner"> <div 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href="https://web-archive-org.translate.goog/web/20161123021114/https://briankoberlein.com/2013/10/15/mass-transit/?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="" class="prev"> <i class="x-icon-chevron-left" data-x-icon=""></i> </a> <a href="https://web-archive-org.translate.goog/web/20161123021114/https://briankoberlein.com/2013/10/14/aether-or/?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="" class="next"> <i class="x-icon-chevron-right" data-x-icon=""></i> </a> </div> <h1 class="entry-title">Mutual Attraction</h1> <p class="p-meta"><span>In <a href="https://web-archive-org.translate.goog/web/20161123021114/https://briankoberlein.com/category/astronomy-2/?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="View all posts in: “Astronomy”"> Astronomy</a> by Brian Koberlein</span><span><time class="entry-date" datetime="2013-10-14T15:00:20+00:00">14 October 2013</time></span><span><a href="https://web-archive-org.translate.goog/web/20161123021114/https://briankoberlein.com/2013/10/14/mutual-attraction/?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#respond" title="Leave a comment on: “Mutual Attraction”" class="meta-comments">0 Comments</a></span></p> </header> <div class="entry-content content"> <p>If you look at the night sky with a small telescope you will notice that some stars are actually double stars.  Some of these, such as Mizar and Alcor in the big dipper, are simply two separate stars that happen to be in the same direction. Most, however, are binary stars that orbit each other.</p> <p>The first known binary stars were observed in the 1800s, but it wasn’t until the 1900s with the introduction of the filar micrometer that decently accurate measurements could be made.  This device allowed you to center your telescope on the primary star, and then measure the position of the secondary star relative to the primary.  By taking measurements over time (sometimes years or decades) you can start to see the companion star trace its path.  In the figure below, I’ve plotted observations of the star Xi Ursae Majoris.</p> <p><a href="https://web-archive-org.translate.goog/web/20161123021114/https://2ai9u93bg0gn4e99nu2g8mbj-wpengine.netdna-ssl.com/wp-content/uploads/2013/10/binary.png?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB"><img class="aligncenter size-full wp-image-414" alt="binary" src="https://web.archive.org/web/20161123021114im_/https://2ai9u93bg0gn4e99nu2g8mbj-wpengine.netdna-ssl.com/wp-content/uploads/2013/10/binary.png" width="288" height="360"></a></p> <p>The good news is that the motion of a binary companion is an ellipse, so with a bit of mathematics you can fit your observation to an elliptical path.  The bad news is that the positions we observe are within the plane of our field of view.  The actual orbit is usually tilted relative to us, so we have to do more math to determine its actual orbit.  In times past, this had to be done by hand and was really laborious. With modern computers, we can fit the data and calculate orbits pretty easily.</p> <p>Of course, if you know the orbit of the companion star, you can also calculate their masses.  Knowing their separation distance and orbital period, you can calculate their masses via Kepler’s laws.  Once we’ve determined their masses, we can look at other properties such as their brightness and temperature to see how they correlate to mass.</p> <p>Turns out you can learn a lot from stars by looking at their mutual attraction.</p> <div id="whp-modern-related-posts-413" class="whp-modern-related-posts"> <h3 class="mrp-title">Related Posts</h3> <div class="owl-carousel mrpc-carousel"> <div class="mrpc-item"> <span class="mrpc-overlay"></span><img width="300" height="300" class="attachment-medium size-medium wp-post-image" alt="Credit: Keck/UCLA" 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