{"id":180607,"date":"2025-01-07T18:00:15","date_gmt":"2025-01-07T18:00:15","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=180607"},"modified":"2025-01-07T18:00:18","modified_gmt":"2025-01-07T18:00:18","slug":"if-you-were-to-record-the-spectrum-of-saturn-as-well-as-the-a-ring","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/01\/07\/if-you-were-to-record-the-spectrum-of-saturn-as-well-as-the-a-ring\/","title":{"rendered":"If you were to record the spectrum of Saturn as well as the A ring"},"content":{"rendered":"\n<p>If you were to record the spectrum of Saturn as well as the A ring, you would find light from one edge of the rings redshifted and light from the other edge blueshifted. If you observed a spectral line at a wavelength of 480 nm, what difference in wavelength (in nm) should you expect between the opposite edges of the rings? (Hints: The radius of the edge of the A ring is 136,500 km. Use the formula for orbital velocity, Vc =\u00c3\u201a\u00c2 \u00c3\u201a\u00c2 GM r<br>If you were to record the spectrum of Saturn as well as the A ring, you would find light from one edge of the rings redshifted and light from the other edge blueshifted. If you observed a spectral line at a wavelength of 480 nm, what difference in wavelength (in nm) should you expect between the opposite edges of the rings? (Hints: The radius of the edge of the A ring is 136,500 km. Use the formula for orbital velocity,<\/p>\n\n\n\n<p>Vc =\u00c3\u201a\u00c2<br>\u00c3\u201a\u00c2<br>GM<br>r<br>and the formula for Doppler shift,<\/p>\n\n\n\n<p>Vr<br>c<br>\u00c3\u201a\u00c2 =\u00c3\u201a\u00c2<br>\u00c3\u017d\u00e2\u20ac\ufffd?<br>?0<br>.<br>\u00c3\u201a\u00c2<\/p>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-6-color\">The correct answer and explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p>To solve the problem, we need to calculate the Doppler shift difference in the observed wavelength caused by the orbital motion of the A ring. Here are the steps:<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Step 1: Orbital velocity of the A ring (Vc)<\/h3>\n\n\n\n<p>The formula for orbital velocity is: Vc=GMrV_c = \\sqrt{\\frac{GM}{r}}<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>GG (gravitational constant): 6.674\u00d710\u221211\u2009m3\u2009kg\u22121\u2009s\u221226.674 \\times 10^{-11} \\, \\text{m}^3 \\, \\text{kg}^{-1} \\, \\text{s}^{-2}<\/li>\n\n\n\n<li>MM (mass of Saturn): 5.683\u00d71026\u2009kg5.683 \\times 10^{26} \\, \\text{kg}<\/li>\n\n\n\n<li>rr (radius of the A ring): 136,500\u2009km=1.365\u00d7108\u2009m136,500 \\, \\text{km} = 1.365 \\times 10^8 \\, \\text{m}<\/li>\n<\/ul>\n\n\n\n<p>Vc=(6.674\u00d710\u221211)(5.683\u00d71026)1.365\u00d7108\u224819,700\u2009m\/sV_c = \\sqrt{\\frac{(6.674 \\times 10^{-11})(5.683 \\times 10^{26})}{1.365 \\times 10^8}} \\approx 19,700 \\, \\text{m\/s}<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Step 2: Doppler shift formula<\/h3>\n\n\n\n<p>The Doppler shift formula relates the velocity of the source to the observed wavelength change: \u0394\u03bb\u03bb0=Vrc\\frac{\\Delta \\lambda}{\\lambda_0} = \\frac{V_r}{c}<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u03bb0\\lambda_0: original wavelength (480\u2009nm480 \\, \\text{nm})<\/li>\n\n\n\n<li>\u0394\u03bb\\Delta \\lambda: wavelength shift<\/li>\n\n\n\n<li>VrV_r: radial velocity (orbital velocity VcV_c)<\/li>\n\n\n\n<li>cc: speed of light (3.0\u00d7108\u2009m\/s3.0 \\times 10^8 \\, \\text{m\/s})<\/li>\n<\/ul>\n\n\n\n<p>Substitute values for one edge of the ring: \u0394\u03bb=\u03bb0Vcc=480\u00d719,7003.0\u00d7108\u22480.0315\u2009nm\\Delta \\lambda = \\lambda_0 \\frac{V_c}{c} = 480 \\times \\frac{19,700}{3.0 \\times 10^8} \\approx 0.0315 \\, \\text{nm}<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Step 3: Total wavelength difference<\/h3>\n\n\n\n<p>The total difference between the redshifted and blueshifted edges is twice this value: \u0394\u03bbtotal=2\u00d70.0315\u22480.063\u2009nm\\Delta \\lambda_\\text{total} = 2 \\times 0.0315 \\approx 0.063 \\, \\text{nm}<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Final Answer:<\/h3>\n\n\n\n<p>The difference in wavelength between the opposite edges of Saturn&#8217;s A ring is <strong>0.063 nm<\/strong>.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Explanation:<\/h3>\n\n\n\n<p>This phenomenon occurs because one side of the ring moves toward the observer (blueshift), while the other moves away (redshift). The orbital velocity VcV_c determines the magnitude of this effect. By calculating the velocity using Saturn\u2019s mass and the radius of the ring, we find how much the wavelength changes. The total Doppler shift combines the contributions from both edges, giving the final wavelength difference of 0.063 nm. This is a direct result of the relative motion of the ring particles and demonstrates the application of both orbital mechanics and the Doppler effect in astrophysical observations.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>If you were to record the spectrum of Saturn as well as the A ring, you would find light from one edge of the rings redshifted and light from the other edge blueshifted. If you observed a spectral line at a wavelength of 480 nm, what difference in wavelength (in nm) should you expect between [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-180607","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/180607","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=180607"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/180607\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=180607"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=180607"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=180607"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}