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<script type="text/javascript" src="/_static/js/bundle-playback.js?v=HxkREWBo" charset="utf-8"></script> <script type="text/javascript" src="/_static/js/wombat.js?v=txqj7nKC" charset="utf-8"></script> <script>window.RufflePlayer=window.RufflePlayer||{};window.RufflePlayer.config={"autoplay":"on","unmuteOverlay":"hidden"};</script> <script type="text/javascript" src="/_static/js/ruffle/ruffle.js"></script> <script type="text/javascript"> __wm.init("https://web.archive.org/web"); __wm.wombat("http://zebu.uoregon.edu:80/~imamura/208/feb6/mass.html","20061214065335","https://web.archive.org/","web","/_static/", "1166079215"); </script> <link rel="stylesheet" type="text/css" href="/_static/css/banner-styles.css?v=S1zqJCYt" /> <link rel="stylesheet" type="text/css" href="/_static/css/iconochive.css?v=3PDvdIFv" />  <title>Mass-Luminosity Relationship</title> <h1>Mass-Luminosity Relationship</h1> <p> The <i>Mass-Luminosity relationship</i>, interestingly enough, can be roughly dervied without specifying how stars generate energy! It turns out that the form of the <i>Mass-Luminosity relation</i> depends upon how energy is moved around inside of a star, i.e., it depends upon how easily energy can flow from the core of the star to the surface of the star. To get a feel for why this comes about, we will perform a <i>rough and ready</i> analysis of the energy flow between regions of differing temperatures. <p> <ul> <li>Region 1 has T1 and Region 2 has T2 where T1 > T2 and so <p> energy flows from Region 1 ===> Region 2 <p> <li>The hotter (more energetic) region 1 <i>mixes</i> with the lower temperature (lower energy) material in region 2 and heats it. This energy <i>mixing</i> leads to the transport of energy. <p> <li>The rate at which energy moves (is mixed) is <ul> <li><a href="mix.xbm">Flux ~ - c x <i>l</i> x <i>d</i>E/<i>d</i>R</a> </ul> <p> Now note that: <ul> <li>Flux ~ L/R**2 <li>l = distance traveled before interacting ~ 1/(opacity x density) <li>Radiation energy ~ T**4 and so dE/DR ~ T(central)**4/R <p> We assumed that the star only produces energy near its center from where the energy then simply works its way outward to the surface of the star. <p> </ul> <p> <li>Substituting these things into the Flux relation leads to <p> <ul> <li>Flux ~ L/R**2 ~ T**4/(opacity x density x R) ~ (M**4/R**4) x (R**2/(opacity x M)) <p> So that we have <p> L ~ M**3/opacity !!!! <p> <li>That is, depending upon the opacity, how strongly the photons interact with matter, we have a <i>Mass-Luminosity relation</i> which is not too bad when compared to the <a href="ml.gif">empirical one</a>. <p> We arrived at this relation without having to specify how the star produced energy, we only required that the star produce energy the bulk of its luminosity in its core.