{"id":210,"date":"2018-11-19T06:29:29","date_gmt":"2018-11-19T06:29:29","guid":{"rendered":"http:\/\/phyp06.epgpbooks.inflibnet.ac.in\/?post_type=chapter&#038;p=210"},"modified":"2018-11-19T06:29:29","modified_gmt":"2018-11-19T06:29:29","slug":"masses-of-stars","status":"publish","type":"chapter","link":"https:\/\/ebooks.inflibnet.ac.in\/phyp06\/chapter\/masses-of-stars\/","title":{"rendered":"Masses of Stars"},"content":{"raw":"<div>\r\n\r\n&nbsp;\r\n\r\n<strong>1.\u00a0\u00a0Learning Outcomes<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">After studying this module, you should be able to<\/p>\r\n\r\n<ul>\r\n \t<li style=\"text-align: justify\">relate masses of stars to binary groups<\/li>\r\n \t<li style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">understand that a binary group can be a visual group, spectroscopic group or eclipsing group<\/span><\/li>\r\n \t<li style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">describe the nature of a visual binary group<\/span><\/li>\r\n \t<li style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">calculate the masses of visual binary group using Kepler\u2019s laws<\/span><\/li>\r\n \t<li style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">explain how the masses of both the components can be found if both are visible<\/span><\/li>\r\n \t<li style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">describe how spectroscopic binary stars are recognized from the behaviour of their spectral lines<\/span><\/li>\r\n \t<li style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">recognize the orbital elements of a binary star<\/span><\/li>\r\n \t<li style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">derive relation between the orbital elements and the masses of the binary stars<\/span><\/li>\r\n<\/ul>\r\n<\/div>\r\n&nbsp;\r\n\r\n<strong>2.\u00a0 Introduction<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;text-indent: 1em;font-size: 1em\">So far, we have dealt with the properties of stars like their surface temperature, luminosity and nature of spectra.\u00a0 Based on their spectral properties, stars have been classified in seven major classes.\u00a0 A chance observation of correlation of luminosity with the surface temperature of stars led to the development of H. R. Diagram and luminosity classification of stars. H. R. Diagram and its variant colour-magnitude diagram have been of immense use to astronomers and astrophysicists in learning about the evolution of stars. Several times it has been said in these modules that the mass of a star is an important parameter in determining the rate at which it evolves.\u00a0 It is, therefore, time that we learnt how the masses of stars are actually determined. The next few modules are devoted to this topic.\u00a0 We shall find that not all star masses can be determined directly.\u00a0 Those stars whose masses can be found directly are used to establish the relationship between mass and luminosity, called the <\/span><strong style=\"text-align: initial;text-indent: 1em;font-size: 1em\">mass \u2013 luminosity relation<\/strong><span style=\"text-align: initial;text-indent: 1em;font-size: 1em\">.\u00a0 Masses of most stars are then found indirectly using this relation.<\/span><\/p>\r\n<img class=\"aligncenter wp-image-223 size-full\" src=\"http:\/\/phyp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/97\/2018\/11\/1523530618Module17Text_Page_02-e1542608204743.jpg\" alt=\"\" width=\"2021\" height=\"1254\" \/> <img class=\"aligncenter wp-image-222 size-full\" src=\"http:\/\/phyp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/97\/2018\/11\/1523530618Module17Text_Page_03-e1542608245644.jpg\" alt=\"\" width=\"2054\" height=\"2747\" \/> <img class=\"aligncenter wp-image-221 size-full\" src=\"http:\/\/phyp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/97\/2018\/11\/1523530618Module17Text_Page_04-e1542608274204.jpg\" alt=\"\" width=\"1996\" height=\"2697\" \/> <img class=\"aligncenter wp-image-220 size-full\" src=\"http:\/\/phyp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/97\/2018\/11\/1523530618Module17Text_Page_05-e1542608296687.jpg\" alt=\"\" width=\"1996\" height=\"2788\" \/> <img class=\"aligncenter wp-image-219 size-full\" src=\"http:\/\/phyp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/97\/2018\/11\/1523530618Module17Text_Page_06-e1542608324589.jpg\" alt=\"\" width=\"2087\" height=\"2788\" \/> <img class=\"aligncenter wp-image-218 size-full\" src=\"http:\/\/phyp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/97\/2018\/11\/1523530618Module17Text_Page_07-e1542608344785.jpg\" alt=\"\" width=\"1988\" height=\"2442\" \/> <img class=\"aligncenter wp-image-217 size-full\" src=\"http:\/\/phyp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/97\/2018\/11\/1523530618Module17Text_Page_08-e1542608368466.jpg\" alt=\"\" width=\"2161\" height=\"2697\" \/> <img class=\"aligncenter wp-image-216 size-full\" src=\"http:\/\/phyp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/97\/2018\/11\/1523530618Module17Text_Page_09-e1542608395606.jpg\" alt=\"\" width=\"2029\" height=\"2549\" \/> <img class=\"aligncenter wp-image-215 size-full\" src=\"http:\/\/phyp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/97\/2018\/11\/1523530618Module17Text_Page_10-e1542608437796.jpg\" alt=\"\" width=\"2070\" height=\"2730\" \/> <img class=\"aligncenter wp-image-214 size-full\" src=\"http:\/\/phyp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/97\/2018\/11\/1523530618Module17Text_Page_11-e1542608459384.jpg\" alt=\"\" width=\"2013\" height=\"2524\" \/> <img class=\"aligncenter wp-image-213 size-full\" src=\"http:\/\/phyp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/97\/2018\/11\/1523530618Module17Text_Page_12-e1542608478597.jpg\" alt=\"\" width=\"1963\" height=\"2582\" \/> <img class=\"aligncenter wp-image-212 size-full\" src=\"http:\/\/phyp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/97\/2018\/11\/1523530618Module17Text_Page_13-e1542608507841.jpg\" alt=\"\" width=\"2013\" height=\"2631\" \/> <img class=\"aligncenter wp-image-211 size-full\" src=\"http:\/\/phyp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/97\/2018\/11\/1523530618Module17Text_Page_14-e1542608534453.jpg\" alt=\"\" width=\"1988\" height=\"2211\" \/>\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n<strong><span style=\"text-align: initial;font-size: 1em\">5.\u00a0 Summary<\/span><\/strong>\r\n<div>\r\n<ul>\r\n \t<li style=\"text-align: justify\">Mass of a star is an important parameter which determines the rate at which it evolves.<\/li>\r\n \t<li style=\"text-align: justify\">Most stars exist in binary groups and groups of three or more stars.<\/li>\r\n \t<li style=\"text-align: justify\">Application of Kepler\u2019s laws to the binary groups is the source of stellar masses.<\/li>\r\n \t<li style=\"text-align: justify\">Binary groups can be visual, spectroscopic or eclipsing binary groups.<\/li>\r\n \t<li style=\"text-align: justify\">Visual binaries are accessible to observation with telescopes.<\/li>\r\n \t<li style=\"text-align: justify\">Spectroscopic binaries reveal themselves through an oscillating set of spectral lines.<\/li>\r\n \t<li style=\"text-align: justify\">Eclipsing binaries are recognized when one component eclipses the other and gets eclipsed in turn.<\/li>\r\n \t<li style=\"text-align: justify\">The more massive component of the group is called the primary, while the less massive is called the companion.<\/li>\r\n \t<li style=\"text-align: justify\">Since the centre of mass (or barycenter) of a binary system moves in a straight line in the sky, the two components move around this moving centre and execute serpentine motions in the sky.<\/li>\r\n \t<li style=\"text-align: justify\">The orbit of a binary system is an ellipse. The plane of the ellipse is generally inclined to the plane of the sky. The orbit projected on the plane of the sky is also an ellipse, but the primary in not at the focus.<\/li>\r\n \t<li style=\"text-align: justify\">Kepler\u2019s third law can be applied directly to the visible system. If both the components are visible, mass of individual member of the group can be found.<\/li>\r\n \t<li style=\"text-align: justify\">By studying the behaviour of oscillating lines of a spectroscopic binary system, we can plot the velocity curve of the binary system.<\/li>\r\n \t<li style=\"text-align: justify\">Analyses of velocity curves leads to the determination of the orbital elements of the system, which lead to the determination of stellar masses.<\/li>\r\n \t<li style=\"text-align: justify\">Since the inclination of the orbit to the plane of the sky is not generally known, we can place only a lower limit on the combined mass of the two stars.<\/li>\r\n \t<li style=\"text-align: justify\">The masses of only a small fraction of stars can be found by analyzing binary systems. These masses can, however, be used to establish a mass-luminosity relation, which can be used for finding masses of most stars.<\/li>\r\n<\/ul>\r\n<\/div>\r\n&nbsp;\r\n\r\n<strong><span style=\"font-size: 1em\">Know More<\/span><\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"font-size: 1em\">The basic source of the material on binary stars is\u00a0<\/span><span style=\"font-size: 1em\">Textbook of Astronomy and Astrophysics with\u00a0<\/span><span style=\"font-size: 1em\">Elements of Cosmology <\/span><span style=\"font-size: 1em\">by V. B. Bhatia. This material has been supplemented by the following\u00a0<\/span><span style=\"font-size: 1em\">web sources:<\/span><\/p>\r\n\r\n<ul>\r\n \t<li style=\"text-align: justify\"><span style=\"font-size: 1em\">https:\/\/en.wikipedia.org\/wiki\/Visual_binary<\/span><\/li>\r\n \t<li style=\"text-align: justify\"><span style=\"font-size: 1em\">http:\/\/www.ast.cam.ac.uk\/~pettini\/STARS\/Lecture04.pdf<\/span><\/li>\r\n \t<li style=\"text-align: justify\"><span style=\"font-size: 1em\">http:\/\/www.atnf.csiro.au\/outreach\/education\/senior\/astrophysics\/binary_types.html<\/span><\/li>\r\n \t<li style=\"text-align: justify\"><span style=\"font-size: 1em\">http:\/\/burro.astr.<\/span><span style=\"font-size: 1em\">cwru.edu\/Academics\/Astr221\/StarProp\/speclipse.html<\/span><\/li>\r\n \t<li style=\"text-align: justify\"><span style=\"font-size: 1em\">http:\/\/www.astro.caltech.edu\/~george\/ay20\/Ay20<\/span><span style=\"font-size: 1em\">-<\/span><span style=\"font-size: 1em\">Lec4x.pdf<\/span><\/li>\r\n \t<li style=\"text-align: justify\"><span style=\"font-size: 1em\">https:\/\/en.wikipedia.org\/wiki\/Orbital_elements<\/span><\/li>\r\n \t<li style=\"text-align: justify\"><span style=\"font-size: 1em\">https:\/\/com<\/span><span style=\"font-size: 1em\">mons.wikimedia.org\/wiki\/File:WZ_Cephei_binary.png<\/span><\/li>\r\n \t<li style=\"text-align: justify\"><span style=\"font-size: 1em\">https:\/\/commons.wikimedia.org\/wiki\/File:Eta_carinae_orbit.png<\/span><\/li>\r\n \t<li style=\"text-align: justify\"><span style=\"font-size: 1em\">https:\/\/en.wikipedia.org\/wiki\/Alpha_Centauri#\/media\/File:Alpha,_Beta_and_Proxima_Centauri_<\/span><span style=\"font-size: 1em\">(1).jpg<\/span><\/li>\r\n \t<li style=\"text-align: justify\"><span style=\"font-size: 1em\">http:\/\/spiff.rit.edu<\/span><span style=\"font-size: 1em\">\/classes\/phys440\/lectures\/reduced\/reduced.html<\/span><\/li>\r\n<\/ul>","rendered":"<div>\n<p>&nbsp;<\/p>\n<p><strong>1.\u00a0\u00a0Learning Outcomes<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">After studying this module, you should be able to<\/p>\n<ul>\n<li style=\"text-align: justify\">relate masses of stars to binary groups<\/li>\n<li style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">understand that a binary group can be a visual group, spectroscopic group or eclipsing group<\/span><\/li>\n<li style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">describe the nature of a visual binary group<\/span><\/li>\n<li style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">calculate the masses of visual binary group using Kepler\u2019s laws<\/span><\/li>\n<li style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">explain how the masses of both the components can be found if both are visible<\/span><\/li>\n<li style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">describe how spectroscopic binary stars are recognized from the behaviour of their spectral lines<\/span><\/li>\n<li style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">recognize the orbital elements of a binary star<\/span><\/li>\n<li style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">derive relation between the orbital elements and the masses of the binary stars<\/span><\/li>\n<\/ul>\n<\/div>\n<p>&nbsp;<\/p>\n<p><strong>2.\u00a0 Introduction<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;text-indent: 1em;font-size: 1em\">So far, we have dealt with the properties of stars like their surface temperature, luminosity and nature of spectra.\u00a0 Based on their spectral properties, stars have been classified in seven major classes.\u00a0 A chance observation of correlation of luminosity with the surface temperature of stars led to the development of H. R. Diagram and luminosity classification of stars. H. R. Diagram and its variant colour-magnitude diagram have been of immense use to astronomers and astrophysicists in learning about the evolution of stars. Several times it has been said in these modules that the mass of a star is an important parameter in determining the rate at which it evolves.\u00a0 It is, therefore, time that we learnt how the masses of stars are actually determined. The next few modules are devoted to this topic.\u00a0 We shall find that not all star masses can be determined directly.\u00a0 Those stars whose masses can be found directly are used to establish the relationship between mass and luminosity, called the <\/span><strong style=\"text-align: initial;text-indent: 1em;font-size: 1em\">mass \u2013 luminosity relation<\/strong><span style=\"text-align: initial;text-indent: 1em;font-size: 1em\">.\u00a0 Masses of most stars are then found indirectly using this relation.<\/span><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-223 size-full\" src=\"http:\/\/phyp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/97\/2018\/11\/1523530618Module17Text_Page_02-e1542608204743.jpg\" alt=\"\" width=\"2021\" height=\"1254\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/11\/1523530618Module17Text_Page_02-e1542608204743.jpg 2021w, https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/11\/1523530618Module17Text_Page_02-e1542608204743-300x186.jpg 300w, https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/11\/1523530618Module17Text_Page_02-e1542608204743-768x477.jpg 768w, https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/11\/1523530618Module17Text_Page_02-e1542608204743-1024x635.jpg 1024w, https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/11\/1523530618Module17Text_Page_02-e1542608204743-65x40.jpg 65w, https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/11\/1523530618Module17Text_Page_02-e1542608204743-225x140.jpg 225w, https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/11\/1523530618Module17Text_Page_02-e1542608204743-350x217.jpg 350w\" sizes=\"auto, (max-width: 2021px) 100vw, 2021px\" \/> <img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-222 size-full\" src=\"http:\/\/phyp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/97\/2018\/11\/1523530618Module17Text_Page_03-e1542608245644.jpg\" alt=\"\" width=\"2054\" height=\"2747\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/11\/1523530618Module17Text_Page_03-e1542608245644.jpg 2054w, https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/11\/1523530618Module17Text_Page_03-e1542608245644-224x300.jpg 224w, https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/11\/1523530618Module17Text_Page_03-e1542608245644-768x1027.jpg 768w, https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/11\/1523530618Module17Text_Page_03-e1542608245644-766x1024.jpg 766w, https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/11\/1523530618Module17Text_Page_03-e1542608245644-65x87.jpg 65w, 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src=\"http:\/\/phyp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/97\/2018\/11\/1523530618Module17Text_Page_05-e1542608296687.jpg\" alt=\"\" width=\"1996\" height=\"2788\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/11\/1523530618Module17Text_Page_05-e1542608296687.jpg 1996w, https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/11\/1523530618Module17Text_Page_05-e1542608296687-215x300.jpg 215w, https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/11\/1523530618Module17Text_Page_05-e1542608296687-768x1073.jpg 768w, https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/11\/1523530618Module17Text_Page_05-e1542608296687-733x1024.jpg 733w, https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/11\/1523530618Module17Text_Page_05-e1542608296687-65x91.jpg 65w, 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https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/11\/1523530618Module17Text_Page_13-e1542608507841-350x457.jpg 350w\" sizes=\"auto, (max-width: 2013px) 100vw, 2013px\" \/> <img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-211 size-full\" src=\"http:\/\/phyp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/97\/2018\/11\/1523530618Module17Text_Page_14-e1542608534453.jpg\" alt=\"\" width=\"1988\" height=\"2211\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/11\/1523530618Module17Text_Page_14-e1542608534453.jpg 1988w, https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/11\/1523530618Module17Text_Page_14-e1542608534453-270x300.jpg 270w, https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/11\/1523530618Module17Text_Page_14-e1542608534453-768x854.jpg 768w, https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/11\/1523530618Module17Text_Page_14-e1542608534453-921x1024.jpg 921w, https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/11\/1523530618Module17Text_Page_14-e1542608534453-65x72.jpg 65w, https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/11\/1523530618Module17Text_Page_14-e1542608534453-225x250.jpg 225w, https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/11\/1523530618Module17Text_Page_14-e1542608534453-350x389.jpg 350w\" sizes=\"auto, (max-width: 1988px) 100vw, 1988px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p><strong><span style=\"text-align: initial;font-size: 1em\">5.\u00a0 Summary<\/span><\/strong><\/p>\n<div>\n<ul>\n<li style=\"text-align: justify\">Mass of a star is an important parameter which determines the rate at which it evolves.<\/li>\n<li style=\"text-align: justify\">Most stars exist in binary groups and groups of three or more stars.<\/li>\n<li style=\"text-align: justify\">Application of Kepler\u2019s laws to the binary groups is the source of stellar masses.<\/li>\n<li style=\"text-align: justify\">Binary groups can be visual, spectroscopic or eclipsing binary groups.<\/li>\n<li style=\"text-align: justify\">Visual binaries are accessible to observation with telescopes.<\/li>\n<li style=\"text-align: justify\">Spectroscopic binaries reveal themselves through an oscillating set of spectral lines.<\/li>\n<li style=\"text-align: justify\">Eclipsing binaries are recognized when one component eclipses the other and gets eclipsed in turn.<\/li>\n<li style=\"text-align: justify\">The more massive component of the group is called the primary, while the less massive is called the companion.<\/li>\n<li style=\"text-align: justify\">Since the centre of mass (or barycenter) of a binary system moves in a straight line in the sky, the two components move around this moving centre and execute serpentine motions in the sky.<\/li>\n<li style=\"text-align: justify\">The orbit of a binary system is an ellipse. The plane of the ellipse is generally inclined to the plane of the sky. The orbit projected on the plane of the sky is also an ellipse, but the primary in not at the focus.<\/li>\n<li style=\"text-align: justify\">Kepler\u2019s third law can be applied directly to the visible system. If both the components are visible, mass of individual member of the group can be found.<\/li>\n<li style=\"text-align: justify\">By studying the behaviour of oscillating lines of a spectroscopic binary system, we can plot the velocity curve of the binary system.<\/li>\n<li style=\"text-align: justify\">Analyses of velocity curves leads to the determination of the orbital elements of the system, which lead to the determination of stellar masses.<\/li>\n<li style=\"text-align: justify\">Since the inclination of the orbit to the plane of the sky is not generally known, we can place only a lower limit on the combined mass of the two stars.<\/li>\n<li style=\"text-align: justify\">The masses of only a small fraction of stars can be found by analyzing binary systems. These masses can, however, be used to establish a mass-luminosity relation, which can be used for finding masses of most stars.<\/li>\n<\/ul>\n<\/div>\n<p>&nbsp;<\/p>\n<p><strong><span style=\"font-size: 1em\">Know More<\/span><\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"font-size: 1em\">The basic source of the material on binary stars is\u00a0<\/span><span style=\"font-size: 1em\">Textbook of Astronomy and Astrophysics with\u00a0<\/span><span style=\"font-size: 1em\">Elements of Cosmology <\/span><span style=\"font-size: 1em\">by V. B. Bhatia. This material has been supplemented by the following\u00a0<\/span><span style=\"font-size: 1em\">web sources:<\/span><\/p>\n<ul>\n<li style=\"text-align: justify\"><span style=\"font-size: 1em\">https:\/\/en.wikipedia.org\/wiki\/Visual_binary<\/span><\/li>\n<li style=\"text-align: justify\"><span style=\"font-size: 1em\">http:\/\/www.ast.cam.ac.uk\/~pettini\/STARS\/Lecture04.pdf<\/span><\/li>\n<li style=\"text-align: justify\"><span style=\"font-size: 1em\">http:\/\/www.atnf.csiro.au\/outreach\/education\/senior\/astrophysics\/binary_types.html<\/span><\/li>\n<li style=\"text-align: justify\"><span style=\"font-size: 1em\">http:\/\/burro.astr.<\/span><span style=\"font-size: 1em\">cwru.edu\/Academics\/Astr221\/StarProp\/speclipse.html<\/span><\/li>\n<li style=\"text-align: justify\"><span style=\"font-size: 1em\">http:\/\/www.astro.caltech.edu\/~george\/ay20\/Ay20<\/span><span style=\"font-size: 1em\">&#8211;<\/span><span style=\"font-size: 1em\">Lec4x.pdf<\/span><\/li>\n<li style=\"text-align: justify\"><span style=\"font-size: 1em\">https:\/\/en.wikipedia.org\/wiki\/Orbital_elements<\/span><\/li>\n<li style=\"text-align: justify\"><span style=\"font-size: 1em\">https:\/\/com<\/span><span style=\"font-size: 1em\">mons.wikimedia.org\/wiki\/File:WZ_Cephei_binary.png<\/span><\/li>\n<li style=\"text-align: justify\"><span style=\"font-size: 1em\">https:\/\/commons.wikimedia.org\/wiki\/File:Eta_carinae_orbit.png<\/span><\/li>\n<li style=\"text-align: justify\"><span style=\"font-size: 1em\">https:\/\/en.wikipedia.org\/wiki\/Alpha_Centauri#\/media\/File:Alpha,_Beta_and_Proxima_Centauri_<\/span><span style=\"font-size: 1em\">(1).jpg<\/span><\/li>\n<li style=\"text-align: justify\"><span style=\"font-size: 1em\">http:\/\/spiff.rit.edu<\/span><span style=\"font-size: 1em\">\/classes\/phys440\/lectures\/reduced\/reduced.html<\/span><\/li>\n<\/ul>\n","protected":false},"author":4,"menu_order":12,"template":"","meta":{"_acf_changed":false,"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":["prof-v-b-bhatia"],"pb_section_license":""},"chapter-type":[],"contributor":[58],"license":[],"class_list":["post-210","chapter","type-chapter","status-publish","hentry","contributor-prof-v-b-bhatia"],"part":3,"_links":{"self":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-json\/pressbooks\/v2\/chapters\/210","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-json\/wp\/v2\/users\/4"}],"version-history":[{"count":1,"href":"https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-json\/pressbooks\/v2\/chapters\/210\/revisions"}],"predecessor-version":[{"id":224,"href":"https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-json\/pressbooks\/v2\/chapters\/210\/revisions\/224"}],"part":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-json\/pressbooks\/v2\/parts\/3"}],"metadata":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-json\/pressbooks\/v2\/chapters\/210\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-json\/wp\/v2\/media?parent=210"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-json\/pressbooks\/v2\/chapter-type?post=210"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-json\/wp\/v2\/contributor?post=210"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-json\/wp\/v2\/license?post=210"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}