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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">Stellar quantities</span></li> <span class="pull-right" style="color: #555" title="Suggested study time: 10 minutes"><i class="fa fa-clock-o"></i> 10'</span> </ol> <article id="main-article"> <p><img alt="" src="../../astro/1-distance-sm-1.jpg" style="float: left; width: 250px; height: 167px;">There are two significant measures to get to grips with in astrophysics: distance and brightness.</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-turquoise panel-has-colored-body panel-has-border"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Astronomical distances</p> </div> </div> <div class="panel-body"> <div> <p><img alt="" src="../../astro/1-au.png" style="float: left; width: 200px; height: 200px;">The astronomical unit (AU) is defined as the average distance from the Earth to the Sun. This is 149 597 870 700 m, or the slightly more convenient 150 million <strong>k</strong>m (to 3 sf).</p> <p>A light year (ly) is the distance travelled by light in one year, equivalent to 9.46 × 10<sup>15</sup><strong> </strong>m.</p> <div class="box"> <p>To calculate we need two quantities:</p> <ul> <li>The speed of light, 3 x 10<sup>8</sup> ms<sup>-1</sup>.</li> <li>The number of seconds in one year (365 x 24 x 60 x 60), 3.15 x 10<sup>7</sup> s.</li> </ul> <p>Distance = speed x time</p> </div> </div> </div> <div class="panel-footer"> <div> <p> </p> </div> </div> </div> <div class="panel panel-turquoise panel-has-colored-body panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Parsec</p> </div> </div> <div class="panel-body"> <div> <p><img alt="" src="../../astro/parsec.png" style="float: right; width: 200px; height: 314px;">The parsec (pc) is the largest unit of distance and is defined by the technique of <em>parallax</em>.</p> <ul> </ul> <div class="box"> <p>Parallax is a technique to determine the distance to stars. You might have noticed on car or train journeys that nearby objects appear to move much faster past the window in comparison to the distant landscape.</p> <p>As the Earth orbits the Sun (the process that defines the <em>year</em>), nearby stars move relative to the backdrop of distant stars. Using the mathematical technique of similar triangles, we can observe that the angle subtended by the Earth's movement through 1 AU is equal to that subtended by the near star relative to the distant background.</p> <p><span class="math-tex">\(\tan p \approx p=\)</span>1 AU / distance to near star</p> <p>Rearranging: distance to near star = 1 AU / <span class="math-tex">\(p\)</span></p> </div> <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/358639876"></iframe></div> <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/358639924"></iframe></div> <ul> </ul> <div class="box"> <p>To make things simple the parsec is defined as the distance that subtends an angle of 1 arcsec.</p> <p style="text-align: center;">The distance to a star in parsec =<span class="math-tex">\(1\over \text{the angle in arcsec}\)</span></p> <p>No further calculation is required.</p> </div> <p>One parsec is equal to:</p> <ul> <li>3.09 x 10<sup>16</sup> m</li> <li>6.02 x 10<sup>5</sup> AU</li> <li>3.26 ly</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-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>Luminosity</p> </div> </div> <div class="panel-body"> <div> <p><img alt="" src="../../astro/blackbody.png" style="float: left; width: 375px; height: 300px;"></p> <p>The <em>luminosity </em>of a star is equivalent to its power, the energy released in one second:</p> <p style="text-align: center;"><span class="math-tex">\(L= {\Delta E \over t}\)</span></p> <p>This luminosity</p> <p>is also related to the star's temperature:</p> <div class="box"> <p style="text-align: center;"><span class="math-tex">\(L=\sigma AT^4\)</span></p> <ul> <li><span class="math-tex">\(L\)</span> is the luminosity (W)</li> <li><span class="math-tex">\(\sigma\)</span> is the Stefan-Boltzmann constant (= 5.67 x 10<sup>-8</sup> Wm<sup>-2</sup>K<sup>-4</sup>)</li> <li><span class="math-tex">\(A\)</span> is the surface area of the star (m<sup>2</sup>)</li> <li><span class="math-tex">\(T\)</span> is the <strong>absolute</strong> temperature of the star (K)</li> </ul> </div> <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/358639883"></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-yellow panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Brightness</p> </div> </div> <div class="panel-body"> <div> <p>The <em>apparent brightness</em> of the star at a distance from the centre of the star is equivalent to the intensity, the power (or luminosity) per unit area moved through by the radiation:</p> <div class="box"> <p style="text-align: center;"><span class="math-tex">\(b={L\over {4\pi d^2}}\)</span></p> <ul> <li><span class="math-tex">\(b\)</span> is the apparent brightness (Wm<sup>-</sup><sup>2</sup>)</li> <li><span class="math-tex">\(L\)</span> is the luminosity (W)</li> <li><span class="math-tex">\(d\)</span> is the distance that the star's radiation has travelled from its centre</li> </ul> </div> <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/358639850"></iframe></div> <p>It is important to distinguish between the areas involved:</p> <div class="box"> <ul> <li><span class="math-tex">\(A\)</span> is the surface area of the star</li> <li><span class="math-tex">\(4\pi d^2\)</span> is the area of the sphere that the star's radiation has produced as it expands through space (like the skin of a balloon)</li> </ul> </div> <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/358639898"></iframe></div> <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/358639903"></iframe></div> </div> </div> <div class="panel-footer"> <div> <p> </p> </div> </div> </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><i>Use flashcards to practise your recall.</i></p> <div class="tib-flashcard"><a class="show-flashcards btn btn-success btn-xs-block btn-block " data-levels="1" data-mode="Normal" data-topics="833" data-subject-id="6" data-n-flashcards="6" style="text-align:center">Show flashcards</a></div><hr> <p><i>Use quizzes to practise application of theory.</i> </p> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#35352104"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="35352104"> <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%;"> Stellar quantities <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="6-308-825" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p>A telescope is lined up with a star on two nights 6 months apart. The situation is represented by the diagram below.</p><p style="text-align: center;"><img alt="" height="152" src="../../astro/screenshot-2019-08-08-at-08.04.43.png" width="376"></p><p>Distance <span class="math-tex">\(X\)</span> is:</p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span>2 pc</span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span>2 AU</span></label> </p><p><label class="radio"> <input type="radio"> <span>1 AU</span></label> </p><p><label class="radio"> <input type="radio"> <span>1 pc</span></label> </p></div><div class="q-explanation"><p>The circle on the left hand side of the diagram represents the Earth orbitting the yellow Sun. The diameter is twice the distance between the Earth and the Sun (which is 1 astronomical unit (AU)).</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 telescope is lined up with a star on two nights 6 months apart. The situation is represented by the diagram below.</p><p style="text-align: center;"><img alt="" height="152" src="../../astro/screenshot-2019-08-08-at-08.04.43.png" width="376"></p><p>If the distance from Earth to the star is 4 pc then the angle <span class="math-tex">\(A\)</span> is:</p></div><div class="q-answer"><p><label class="radio"> <input class="c" type="radio"> <span>0.25 <span class="scayt-misspell-word" data-scayt-word="arcsec" data-wsc-lang="en_US">arcsec</span></span></label> </p><p><label class="radio"> <input type="radio"> <span>4 <span class="scayt-misspell-word" data-scayt-word="arcsec" data-wsc-lang="en_US">arcsec</span></span></label> </p><p><label class="radio"> <input type="radio"> <span>0.5 <span class="scayt-misspell-word" data-scayt-word="arcsec" data-wsc-lang="en_US">arcsec</span></span></label> </p><p><label class="radio"> <input type="radio"> <span>2 <span class="scayt-misspell-word" data-scayt-word="arcsec" data-wsc-lang="en_US">arcsec</span></span></label> </p></div><div class="q-explanation"><p><span class="math-tex">\(\text{distance in pc} ={1\over \text{angle in arcsec}}\)</span></p><p><span class="math-tex">\(\Rightarrow \text{angle in arcsec} ={1\over \text{distance in pc}}\)</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>A telescope is lined up with a star on two nights 6 months apart. The situation is represented by the diagram below.</p><p style="text-align: center;"><img alt="" height="152" src="../../astro/screenshot-2019-08-08-at-08.04.43.png" width="376"></p><p>If the angle <span class="math-tex">\(A\)</span> is 0.1 <span class="scayt-misspell-word" data-scayt-word="arcsec" data-wsc-lang="en_US">arcsec,</span> the distance from Earth to the star is:</p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span>0.1 pc</span></label> </p><p><label class="radio"> <input type="radio"> <span>2 pc</span></label> </p><p><label class="radio"> <input type="radio"> <span>5 pc</span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span>10 pc</span></label> </p></div><div class="q-explanation"><p><span class="math-tex">\(\text{distance in pc} ={1\over \text{angle in arcsec}}\)</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>This photograph shows the stars in the constellation Orion.</p><p style="text-align: center;"><img alt="" height="280" src="../../astro/orion_spinelli_c1.jpg" width="328"></p><p>From this image alone we can deduce that the stars are of different...</p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span>distance from Earth</span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span>brightness</span></label> </p><p><label class="radio"> <input type="radio"> <span>luminosity</span></label> </p><p><label class="radio"> <input type="radio"> <span>mass</span></label> </p></div><div class="q-explanation"><p>In fact, all of these quantities are different. However, the others can't be deduced from the photograph (e.g. one could be more luminous but further away).</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>This photograph shows the stars in the constellation Orion.</p><p style="text-align: center;"><img alt="" height="277" src="../../astro/orionlabel.png" width="325"></p><p>Alnilam has a higher luminosity than Betelgeuse. From the photo we can deduce that Alnilam is:</p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span>smaller</span></label></p><p><label class="radio"> <input type="radio"> <span>colder</span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span>further from the Earth</span></label> </p><p><label class="radio"> <input type="radio"> <span>heavier</span></label></p></div><div class="q-explanation"><p>Alnilam is less bright to observers here at Earth and so must be further away. The other options are irrelevant - only distance could affect the observed brightness for a fixed luminosity.</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>This photograph shows the stars in the constellation Orion.</p><p style="text-align: center;"><img alt="" height="277" src="../../astro/orionlabel.png" width="325"></p><p>Betelgeuse and Rigel are a similar distance from the Earth, they also have similar...</p></div><div class="q-answer"><p><label class="radio"> <input class="c" type="radio"> <span>luminosity</span></label> </p><p><label class="radio"> <input type="radio"> <span>mass</span></label> </p><p><label class="radio"> <input type="radio"> <span>temperature</span></label></p><p><label class="radio"> <input type="radio"> <span>size</span></label></p></div><div class="q-explanation"><p>They have a similar brightness to observers in the same location. Therefore, they must have similar luminosity.</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>Star X has the same radius as the Sun and temperature 12 000 K.</p><p>The ratio <span class="math-tex">\(\text{luminosity}_\text{X}\over \text{luminosity}_\text{sun}\)</span> is equal to:</p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span>4</span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span>16</span></label> </p><p><label class="radio"> <input type="radio"> <span>2</span></label> </p><p><label class="radio"> <input type="radio"> <span>8</span></label> </p></div><div class="q-explanation"><p>The surface temperature of the Sun is approximately 6000 K. HINT: It's good to learn this!</p><p><span class="math-tex">\(\Rightarrow T_\text{X}=2T_\text{sun}\)</span></p><p><span class="math-tex">\(L \propto T^4\)</span> so <span class="math-tex">\({L_\text{X}\over L_\text{sun}}=2^4\)</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>Star Y has twice the temperature of the Sun but half the radius.</p><p>The ratio <span class="math-tex">\(\text{luminosity}_\text{Y}\over \text{luminosity} _\text{sun}\)</span> is equal to:</p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span>8</span></label> </p><p><label class="radio"> <input type="radio"> <span>2</span></label> </p><p><label class="radio"> <input type="radio"> <span>16</span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span>4</span></label> </p></div><div class="q-explanation"><p><span class="math-tex">\(L = σAT^4\)</span> and <span class="math-tex">\(A = 4πr^2\)</span></p><p><span class="math-tex">\(\Rightarrow L\propto r^2T^4\)</span></p><p><span class="math-tex">\({L_\text{Y}\over L_\text{sun}}=({1\over2})^2\times 2^4\)</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>Star Z has the same luminosity as the Sun but half the radius.</p><p>The ratio <span class="math-tex">\(T_\text{Z}\over T_\text{sun}\)</span> =</p></div><div class="q-answer"><p><label class="radio"> <input class="c" type="radio"> <span><span class="math-tex">\(\sqrt 2\)</span></span></label></p><p><label class="radio"> <input type="radio"> <span>4</span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(2\)</span></span></label></p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(1\over \sqrt 2\)</span></span></label> </p></div><div class="q-explanation"><p><span class="math-tex">\(L = σAT^4\)</span> and <span class="math-tex">\(A = 4πr^2\)</span></p><p><span class="math-tex">\(T^4\propto{1\over r^2}\)</span> so <span class="math-tex">\(T\propto{1\over \sqrt r}\)</span></p><p>Since the radius has halved, the temperature is increased by a factor <span class="math-tex">\(\sqrt 2\)</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>The units of the Stefan-Boltzmann constant are:</p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span>Wm<sup>2</sup>K<sup>-4</sup></span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span>Wm<sup>-2</sup>K<sup>-4</sup></span></label> </p><p><label class="radio"> <input type="radio"> <span>Wm<sup>2</sup>K<sup>4</sup></span></label> </p><p><label class="radio"> <input type="radio"> <span>Wm<sup>-2</sup>K<sup>4</sup></span></label> </p></div><div class="q-explanation"><p><span class="math-tex">\(L = σAT^4\)</span></p><p><span class="math-tex">\(σ ={L\over AT^4}\)</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> </div> <div class="modal-footer slide-quiz-actions"> <div class=""> <div class="pull-left pull-xs-none mb-xs-3"> <button class="btn btn-default d-xs-none btn-prev"> <i class="fa fa-arrow-left"></i> Prev </button> </div> <div class="pull-right pull-xs-none"> <button class="btn btn-success btn-xs-block text-xs-center btn-results" style="display: none"> <i class="fa fa-bar-chart"></i> Check Results </button> 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