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</h1> <ol class="breadcrumb"> <li><a href="../../../physics.html"><i class="fa fa-home"></i></a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><a href="../823/astrophysics.html">Astrophysics</a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><span class="gray">AHL Stellar processes</span></li> <span class="pull-right" style="color: #555" title="Suggested study time: 60 minutes"><i class="fa fa-clock-o"></i> 60'</span> </ol> <article id="main-article"> <p><img alt="" src="../../astro/gold-1.jpg" style="float: left; width: 250px; height: 188px;">Having established the<a href="../831/stellar-characteristics.html" title="Stellar characteristics"> characteristics</a> and<a href="../832/stellar-evolution.html" title="Stellar evolution"> evolution</a> of different types of star, here we will dig a little deeper. Under what circumstances will a nebula collapse? What energy changes are happening during fusion beyond the main sequence? What happens to make a star die? And what does all of this have to do with gold ingots?</p> <hr class="hidden-separator"> <div class="panel panel-turquoise panel-has-colored-body"> <div class="panel-heading"> <div> <p>Key Concepts</p> </div> </div> <div class="panel-body"> <div> <div class="panel panel-has-colored-body panel-has-border panel-turquoise"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Jeans criterion</p> </div> </div> <div class="panel-body"> <div> <p>Stars form when a giant molecular cloud collapses. Clouds collapse due to gravitational attraction but this is opposed by the random movement of the particles, so a cloud of given mass will only collapse if the temperature is low and the mass of the particles is high.</p> <p>We can calculate whether a cloud will collapse based on whether its mass exceeds the Jeans criterion:</p> <p style="text-align: center;"><span class="math-tex">\(M_J={3\over 2}{kTR\over Gm}\)</span></p> <ul> <li><span class="math-tex">\(M_J\)</span> is the Jeans criterion, above which an interstellar cloud may collapse (kg)</li> <li><span class="math-tex">\(k\)</span> is the Boltzmann constant (1.38 × 10<sup>-23</sup> JK<sup>-1</sup>)</li> <li><span class="math-tex">\(T\)</span> is absolute temperature (K)</li> <li><span class="math-tex">\(R\)</span> is the radius of the cloud (m)</li> <li><span class="math-tex">\(G\)</span> is the universal gravitational constant (6.67 × 10<sup>−11</sup> m<sup>3</sup> kg<sup>−1</sup> s<sup>−2</sup>)</li> <li><span class="math-tex">\(m\)</span> is the average molecular mass (kg)</li> </ul> <p>The Jeans criterion varies with temperature, radius and composition.</p> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/197763009"></iframe></div> </div> </div> <div class="panel-footer"> <div> <p> </p> </div> </div> </div> <div class="panel panel-has-colored-body panel-has-border panel-turquoise panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Nuclear fusion</p> </div> </div> <div class="panel-body"> <div> <p>Once a giant molecular cloud has collapsed, it forms a protostar. Nucleosynthesis (the production of new elements by combining nuclei) will commence and the star will enter the main sequence if the temperature is sufficient for hydrogren nuclei to overcome electrostatic repulsion due to like charges and fuse to form helium. Fusion releases <a href="../369/nuclear-stability-and-binding-energy.html" title="Nuclear stability and binding energy">binding energy</a> as the strong nuclear force combines the hydrogren nuclei, meaning that a radiation pressure acts outwards, balancing against the inward force of gravity.</p> <p>As the hydrogen starts to run out, the rate of fusion decreases, the radiation pressure is reduced and the star starts to collapse. This can be modelled in Algodoo.</p> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/197777136"></iframe></div> <p>It is stars with the highest mass that spend least time in the main sequence. This is a little contrary to common sense - shouldn't stars with the highest availability of hydrogren nuclei need longer for hydrogen fusion to complete? Instead, let's consider all relevant relationships in turn.</p> <p>The mass–luminosity relation shows that luminosity is proportional to mass raised to a high power:</p> <p style="text-align: center;"><span class="math-tex">\(L\propto M^{3.5}\)</span></p> <ul> <li><span class="math-tex">\(L\)</span> is the luminosity of the star (W, or any consistent unit of power due to the proportional relationship)</li> <li><span class="math-tex">\(M\)</span> is the mass of the star (kg, or any consistent unit of mass due to the proportional relationship)</li> <li><span class="math-tex">\(3.5\)</span> is the exponent for main sequence stars</li> </ul> <p>Assuming that luminosity is constant over the lifetime of the main sequence and knowing that luminosity is equivalent to power, we see that luminosity is inversely proportional to time:</p> <p style="text-align: center;"><span class="math-tex">\(L\propto {E\over t_{MS}}\)</span></p> <ul> <li><span class="math-tex">\(E\)</span> is the total binding energy released by fusion (J)</li> <li><span class="math-tex">\(t_{MS}\)</span> is the lifetime of the main sequence (s)</li> </ul> <p>Considering that binding energy is released due to a mass defect:</p> <p style="text-align: center;"><span class="math-tex">\(E\propto M\)</span></p> <ul> <li><span class="math-tex">\(M\)</span> is the mass available for fusion</li> </ul> <p>Therefore, combining these relationships, we see that the lifetime of the main sequence actually decreases with the mass of the star raised to a power:</p> <p style="text-align: center;"><span class="math-tex">\(t_{MS}\propto{1\over M^{2.5}}\)</span></p> </div> </div> <div class="panel-footer"> <div> <p> </p> </div> </div> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-yellow"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Essentials</p> </div> </div> <div class="panel-body"> <div> <div class="panel panel-has-colored-body panel-has-border panel-yellow"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Nucleosynthesis off the main sequence</p> </div> </div> <div class="panel-body"> <div> <p>When a star has fused all available hydrogren, it ceases to be in its main sequence. However, fusion may yet continue depending on the temperature of the core after the collapse due to decreasing radiation pressure.</p> <p>As the core gets hotter and more dense in the helium collapse, the helium nuclei may begin to fuse to beryllium. This requires a higher temperature than for hydrogren fusion, as more energy is required to overcome the electrostatic repulsion between the nuclei. When the rate of fusion decreases once more, the process of star contraction and increased temperature repeats.</p> <p>In large stars with a core of <span class="math-tex">\(M>4 M_\text{sun}\)</span>, cycles of nuclear fusion continue until iron is formed. This is because iron nuclei have the maximum binding energy per nucleon and so further fusion would require more kinetic energy than is released as binding energy. The temperatures required are not sustained.</p> <p>How do we know all of this? Observations of stellar <a href="../826/spectra.html" title="Spectra">spectra</a> show the existence of different elements in stars. Nuclear fusion theory explains these.</p> </div> </div> <div class="panel-footer"> <div> <p> </p> </div> </div> </div> <div class="panel panel-has-colored-body panel-has-border panel-yellow panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Radioactive decay</p> </div> </div> <div class="panel-body"> <div> <p>So where do the heavier elements (like gold!) come from? One mechanism is beta decay. In <a href="../370/radioactivity.html" title="Radioactivity">radioactivity</a> we learned that a surplus of neutrons in a nucleus (e.g. due to absorption of excess neutrons) leads to instability. Energy is released when a neutron is changed into a proton and an electron, and the retaining of the proton means that a new element is formed.</p> <p>Absorption of neutrons can happen in one of the following ways:</p> <ul> <li>Rapid (r) processes in which there is not time to decay between each addition. Resulting nuclei are neutron-rich.</li> <li>Slow (s) processes in which each neutron decays before the next is added. Resulting nuclei are proton-rich. This is how almost all of the heavy elements are created.</li> </ul> </div> </div> <div class="panel-footer"> <div> <p> </p> </div> </div> </div> <div class="panel panel-has-colored-body panel-has-border panel-yellow panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Type Ia and II supernovae</p> </div> </div> <div class="panel-body"> <div> <p>The other route for fusion of larger nuclei occurs if the core becomes so dense that electrons combine with protons to form neutrons. The ensuing sudden collapse of the core leads to an explosion known as a supernova in which there is a rapid increase in temperature and luminosity, as shown in the simulation.</p> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/197781108"></iframe></div> <p>There are two types of supernovae:</p> <ul> <li>Type Ia - an exploding white dwarf that contains all the elements synthesized in the core. Their reliable light curve means that Type Ia supernovae are used as standard candles; their luminosity is known during the decay of nickel and cobalt to iron and so their distance from an observer can be determined.</li> </ul> <p style="text-align: center;"><img alt="" src="../../astro/light-curve.png" style="width: 250px; height: 179px;"></p> <ul> <li>Type II (as shown above) - composed mainly of hydrogen since this is what makes up the outer layers of the star. The remaining core is a neutron star.</li> </ul> </div> </div> <div class="panel-footer"> <div> <p> </p> </div> </div> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-green"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Test Yourself</p> </div> </div> <div class="panel-body"> <div> <p><em>Use flashcards to practise your recall.</em></p> <div class="tib-flashcard"><a class="show-flashcards btn btn-success btn-xs-block btn-block " data-levels="3" data-mode="Normal" data-topics="1071" data-subject-id="6" data-n-flashcards="13" style="text-align:center">Show flashcards</a></div><hr> <p><em>Use quizzes to practise application of theory.</em></p> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#d568b511"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="d568b511"> <div class="modal-dialog" style="width: 95vw; max-width: 960px"> <div class="modal-content"> <div class="modal-header slide-quiz-title"> <h4 class="modal-title" style="width: 100%;"> AHL Stellar processes <strong class="q-number pull-right"> <span class="counter">1</span>/<span class="total">1</span> </strong> </h4> </div> <div class="modal-body p-xs-3"> <div class="slide-quiz" data-stats="6-412-1153" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p>A computer simulation of a giant molecular cloud is set so that it doesn't collapse. It could collapse if:</p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span>the radius was increased</span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span>the velocity of the particles was reduced</span></label> </p><p><label class="radio"> <input type="radio"> <span>the number of particles was reduced</span></label> </p><p><label class="radio"> <input type="radio"> <span>the mass of the particles was reduced</span></label> </p></div><div class="q-explanation"><p>For collapse the mass of the cloud must exceed <span class="math-tex">\(M_J={3\over 2}{kTR\over Gm}\)</span></p><p>Reducing velocity of the particles means that the temperature is lower, but the mass of the cloud remains the same.</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>A and B are two giant molecular clouds of equal mass and radius. A collapses B doesn't.</p><p>The ratio <span class="math-tex">\(T_Am_B\over T_Bm_A\)</span> could be:</p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span>1.5</span></label> </p><p><label class="radio"> <input type="radio"> <span>-1</span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span>0.5</span></label> </p><p><label class="radio"> <input type="radio"> <span>1</span></label> </p></div><div class="q-explanation"><p>For collapse the mass of the cloud must exceed <span class="math-tex">\(M_J={3\over 2}{kTR\over Gm}\)</span></p><p>The mass of each cloud is the same so <span class="math-tex">\(M_J\)</span> for A must be lower than B:</p><p><span class="math-tex">\({{T_A\over m_A}\over {T_B\over m_B}}={T_Am_B\over T_Bm_A}\)</span><sub> </sub></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>From observations of the absorption spectrum of a star, the outer layers are seen to contain iron. The mass of the core of this star could be...</p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(M_\text{sun}\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(3M_\text{sun}\)</span></span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span><span class="math-tex">\(5M_\text{sun}\)</span></span></label></p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(0.5M_\text{sun}\)</span></span></label> </p></div><div class="q-explanation"><p>Fusion of iron takes place in stars with core mass <span class="math-tex">\(> 4M_\text{sun}\)</span>.</p><p>Note that the presence of iron in the outer layers implies that the core contains iron since the layers mix.</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The r and s processes refer to...</p></div><div class="q-answer"><p><label class="radio"> <input class="c" type="radio"> <span>neutron absorption</span></label> </p><p><label class="radio"> <input type="radio"> <span>proton absorption</span></label> </p><p><label class="radio"> <input type="radio"> <span>alpha decay</span></label> </p><p><label class="radio"> <input type="radio"> <span>the fusion of hydrogen</span></label> </p></div><div class="q-explanation"><p>The rapid and slow absorption of neutrons, which lead to different outcomes from beta decay.</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Neutron degeneracy is a consequence of:</p></div><div class="q-answer"><p><label class="radio"> <input class="c" type="radio"> <span>The Pauli exclusion principle</span></label> </p><p><label class="radio"> <input type="radio"> <span>Einstein's mass-energy equivalence</span></label> </p><p><label class="radio"> <input type="radio"> <span>General relativity</span></label> </p><p><label class="radio"> <input type="radio"> <span>Heisenberg's uncertainty principle</span></label> </p></div><div class="q-explanation"><p>The Pauli exclusion principle applies to all particles with spin <span class="math-tex">\(1\over 2\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Type Ia and type II supernovae can be distinguished by their:</p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span>lifetime</span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span>spectra</span></label> </p><p><label class="radio"> <input type="radio"> <span>luminosity</span></label> </p><p><label class="radio"> <input type="radio"> <span>brightness</span></label> </p></div><div class="q-explanation"><p>Type I are exploding white dwarfs so contain all the elements of the core.</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Which of the following star masses is the smallest that could form a black hole?</p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(10M_\text{sun}\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(M_\text{sun}\)</span></span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span><span class="math-tex">\(20M_\text{sun}\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(3M_\text{sun}\)</span></span></label> </p></div><div class="q-explanation"><p>The core must be <span class="math-tex">\(>3M_\text{sun}\)</span> but the whole star is <span class="math-tex">\(>20M_\text{sun}\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Type Ia supernovae can be used as standard candles because they all have the same:</p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span>temperature</span></label> </p><p><label class="radio"> <input type="radio"> <span>absorption spectra</span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span>luminosity</span></label> </p><p><label class="radio"> <input type="radio"> <span>brightness</span></label> </p></div><div class="q-explanation"><p>They all have the same peak luminosity so the distance from an observer can be calculated from their brightness.</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The brightness of a type Ia supernova known to be 100 Mpc from Earth is <span class="math-tex">\(b\)</span>.</p><p>How far from the Earth is a type Ia supernova with brightness <span class="math-tex">\(b\over2\)</span>?</p></div><div class="q-answer"><p><label class="radio"> <input class="c" type="radio"> <span>141 Mpc</span></label> </p><p><label class="radio"> <input type="radio"> <span>50 Mpc</span></label> </p><p><label class="radio"> <input type="radio"> <span>400 Mpc</span></label> </p><p><label class="radio"> <input type="radio"> <span>200 Mpc</span></label> </p></div><div class="q-explanation"><p>The luminosities of both supernovae is the same so <span class="math-tex">\(b\propto{1\over r^2}\)</span> and <span class="math-tex">\(r\propto {1\over b^2}\)</span></p><p>If <span class="math-tex">\(b\times {1\over 2}\)</span>, <span class="math-tex">\(\Rightarrow r\times \sqrt2\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Which of the following statements is correct about a star of <span class="math-tex">\(10M_\text{sun}\)</span>?</p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span>It will be a main sequence star for more time than the Sun because it has a greater luminosity</span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span>It will be a main sequence star for less time than the Sun because it has a greater luminosity</span></label> </p><p><label class="radio"> <input type="radio"> <span>It will be a main sequence star for more time than the Sun because it has a lower luminosity</span></label> </p><p><label class="radio"> <input type="radio"> <span>It will be a main sequence star for less time than the Sun because it has a lower luminosity</span></label> </p></div><div class="q-explanation"><p>The lifetime of a main sequence star is inversely proportional to <span class="math-tex">\(M^{2.5}\)</span> and luminosity is proportional to <span class="math-tex">\(M^{3.5}\)</span>. 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