{"id":415,"date":"2018-12-04T12:02:33","date_gmt":"2018-12-04T12:02:33","guid":{"rendered":"http:\/\/phyp06.epgpbooks.inflibnet.ac.in\/?post_type=chapter&#038;p=415"},"modified":"2018-12-04T12:36:23","modified_gmt":"2018-12-04T12:36:23","slug":"physical-properties-of-stars-2","status":"publish","type":"chapter","link":"https:\/\/ebooks.inflibnet.ac.in\/phyp06\/chapter\/physical-properties-of-stars-2\/","title":{"rendered":"Physical Properties of Stars"},"content":{"raw":"<div>\r\n<div><span style=\"float: right\"><a href=\"https:\/\/youtu.be\/NA\" target=\"_blank\" rel=\"noopener\"><img src=\"http:\/\/epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/2018\/11\/download.png\" alt=\"epgp books\" width=\"75px\" height=\"75px;\" \/><\/a>\r\n<\/span><\/div>\r\n&nbsp;\r\n\r\n<strong>1.\u00a0 <\/strong><strong>Learning Outcomes<\/strong>\r\n\r\n&nbsp;\r\n\r\nAfter studying this module, you should be able to\r\n<ul>\r\n \t<li>Understand the concept of space velocities of stars<\/li>\r\n \t<li><span style=\"text-align: initial;font-size: 1em\">Explain the need for a Local Standard of Rest (LSR)<\/span><\/li>\r\n \t<li><span style=\"text-align: initial;font-size: 1em\">Grasp how LSR is defined<\/span><\/li>\r\n \t<li><span style=\"text-align: initial;font-size: 1em\">Account for the motion of the Sun with respect to the LSR<\/span><\/li>\r\n \t<li><span style=\"text-align: initial;font-size: 1em\">Describe the terms Solar Apex and Antapex<\/span><\/li>\r\n \t<li><span style=\"text-align: initial;font-size: 1em\">Understand the notion of peculiar velocities of Star<\/span><\/li>\r\n \t<li><span style=\"text-align: initial;font-size: 1em\">Account for the long term change in the shape of stellar constellations<\/span><\/li>\r\n \t<li><span style=\"text-align: initial;font-size: 1em\">Grasp how the distances of stellar clusters are measured using their space velocities<\/span><\/li>\r\n<\/ul>\r\n&nbsp;\r\n\r\n<strong>2.\u00a0 <\/strong><strong>Introduction<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">In the last module we introduced the proper motion of stars. \u00a0It must be borne in mind that the stars appear fixed because of their large distances; actually they are in constant motion. \u00a0The\u00a0<span style=\"text-align: initial;font-size: 1em\">proper motion is the motion transverse to the line of sight of the star.\u00a0 The proper motion is such a small angular displacement of stars that in most cases it amounts to a mere arc - second, or even smaller, in a year. That is why we need to compare photographs of the star field containing the star separated by decades.\u00a0 The proper motion of stars being so small, it is not possible to measure it for stars in other galaxies.<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">It must be remembered that the proper motion depends on the distance of the star from the Sun. Two stars having the same displacement normal to the line of sight in a given time will have different proper motions depending on their distances (Fig. 7.1). \u00a0<\/span><strong style=\"text-align: initial;font-size: 1em\">However, smaller proper motion does not necessarily imply a larger distance<\/strong><span style=\"text-align: initial;font-size: 1em\">. An important effect of the proper motion is that it changes slowly the coordinates of a star. Therefore, whenever we state the coordinates of a star, we also state the epoch when the coordinates were measured.<\/span><\/p>\r\n\r\n<\/div>\r\n<div>\r\n\r\n<img class=\"aligncenter size-full wp-image-428\" src=\"http:\/\/phyp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/97\/2018\/12\/6.28.1.png\" alt=\"\" width=\"579\" height=\"145\" \/>\r\n<p style=\"text-align: center\"><span style=\"text-align: initial;font-size: 1em\">Fig. 7.1. Transverse displacement of two stars, S1 and S2 in a given time are equal. However, their proper motions ?1 and ?2 are not equal because of unequal distances.<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">If the star is attended by an invisible companion, say a satellite, then the proper motion shows a periodicity. In fact, this an important method by which the presence of extrasolar planets is detected.<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">Combined with the velocity along the line of sight of the star, the radial velocity, the proper motion gives the space velocity of the star, which we discuss in this module.<\/span><\/p>\r\n\r\n<\/div>\r\n<div>\r\n\r\n&nbsp;\r\n\r\n<strong>3.\u00a0 Space Velocity of Stars<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"font-size: 1em;text-align: initial\">The product of the proper motion of a star with its distance from the Sun gives its transverse velocity ??. If we express ??in km\/s, the distance ?in pc and the proper motion in \u02dd\/yr, then we get the following relation:<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: center\"><span style=\"text-align: initial;font-size: 1em\">??= 4.74?<em>d<\/em>= 4.74?\/?km\/s,\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 (7.1)<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">where ?is the parallax of the star in arc-second. This is a particularly useful relation, for it <\/span><span style=\"text-align: initial;font-size: 1em\">enables us to find stellar distances when the annual parallaxes cannot be directly <\/span><span style=\"text-align: initial;font-size: 1em\">measured. If we wait long enough, ?can be accurately determined and then a knowledge <\/span><span style=\"text-align: initial;font-size: 1em\">of ??give space velocity. Application of this relation to stellar clusters will be discussed <\/span><span style=\"text-align: initial;font-size: 1em\">later.<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">The velocity of a star in the line of sight is called its <\/span><strong style=\"text-align: initial;font-size: 1em\">radial velocity, <\/strong><span style=\"text-align: initial;font-size: 1em\">denoted by ??. It is\u00a0<\/span><span style=\"text-align: initial;font-size: 1em\">inferred from the Doppler shifts of the lines in the star\u2019s spectrum.\u00a0 Combining the radial\u00a0<\/span><span style=\"text-align: initial;font-size: 1em\">velocity with the transverse velocity, we get the <\/span><strong style=\"text-align: initial;font-size: 1em\">space velocity <\/strong><span style=\"text-align: initial;font-size: 1em\">of the star (Fig. 7.2).\u00a0 Typical space velocities of stars range from 20 to 100 km\/s.<\/span><\/p>\r\n\r\n<\/div>\r\n<div>\r\n\r\n<img class=\"aligncenter size-full wp-image-427\" src=\"http:\/\/phyp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/97\/2018\/12\/6.28.2.png\" alt=\"\" width=\"763\" height=\"367\" \/>\r\n<p style=\"text-align: center\">Fig. 7.2.\u00a0 Radial velocity, transverse velocity, space velocity and proper motion of a star.<\/p>\r\n&nbsp;\r\n\r\nClearly,\r\n\r\n<img class=\"aligncenter wp-image-426\" src=\"http:\/\/phyp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/97\/2018\/12\/6.28.2.1.png\" alt=\"\" width=\"566\" height=\"62\" \/>\r\n\r\n<strong><span style=\"font-size: 1em;text-align: initial\">4.\u00a0 Peculiar Velocity of Stars<\/span><\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">So far we have described the motion of a <\/span><strong style=\"text-align: initial;font-size: 1em\">star relative to the Sun<\/strong><span style=\"text-align: initial;font-size: 1em\">. But it is quite conceivable that the Sun itself may be moving in some direction. This, indeed, is the case. We, therefore, define a <\/span><strong style=\"text-align: initial;font-size: 1em\">Local Standard of Rest (LSR), <\/strong><span style=\"text-align: initial;font-size: 1em\">a sort of origin<\/span><strong style=\"text-align: initial;font-size: 1em\">, <\/strong><span style=\"text-align: initial;font-size: 1em\">with respect to which we state the motion of the Sun and stars in its neighbourhood.<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><strong><span style=\"text-align: initial;font-size: 1em\">4.1.\u00a0 Local Standard of Rest (LSR)<\/span><\/strong><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">The <\/span><strong style=\"text-align: initial;font-size: 1em\">solar neighbourhood<\/strong><span style=\"text-align: initial;font-size: 1em\">, is a region roughly of radius ~ 100 pc centred on the Sun.\u00a0 These stars are supposed to share the motion of the Sun round the centre of the Galaxy, so that their motion due to <\/span><strong style=\"text-align: initial;font-size: 1em\">differential rotation of the Galaxy <\/strong><span style=\"text-align: initial;font-size: 1em\">does not become conspicuous. (Differential rotation implies that the various subsystems in the Galaxy rotate with their own characteristic speed round the centre of the Galaxy.) We have seen earlier that the Sun is situated at a<\/span><\/p>\r\n<p style=\"text-align: justify\"><strong style=\"text-align: initial;font-size: 1em\">\u00a0<\/strong><\/p>\r\n\r\n<\/div>\r\n<div><img class=\"aligncenter size-full wp-image-425\" src=\"http:\/\/phyp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/97\/2018\/12\/6.28.3.png\" alt=\"\" width=\"828\" height=\"505\" \/><\/div>\r\n<div>\r\n<p style=\"text-align: center\">Fig. 7.3.\u00a0 Local Standard of Rest orbits the galactic centre in a circular orbit at the distance of the Sun and completes one revolution in the same time as the Sun.<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">distance of about 8.5 kpc from the centre of the Galaxy.\u00a0 The Sun and its neighbourhood move round the centre of the Galaxy, completing one revolution in about 250 million years, the orbital velocity being ~ 220 km\/s.\u00a0 The orbit of the Sun round the galactic centre is elliptic. Therefore, <\/span>we define the Local Standard of Rest a point which orbits the galactic centre in a circular orbit at the galacto-centric distance of the Sun and completing one revolution in the same time as the Sun <span style=\"text-align: initial;font-size: 1em\">(Fig. 7.3).<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">Now, if we plot all the velocity vectors of stars in the solar neighbourhood, we find that they all converge towards the Sun and in the general direction of the constellation Columba with a resultant velocity of about 20 km\/s (Fig. 7.4).\u00a0 This is obviously a reflection of the Sun\u2019s own<\/span><\/p>\r\n\r\n<\/div>\r\n<div>\r\n\r\n<img class=\"aligncenter size-full wp-image-424\" src=\"http:\/\/phyp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/97\/2018\/12\/6.28.4.png\" alt=\"\" width=\"672\" height=\"563\" \/>\r\n<p style=\"text-align: center\">Fig. 7.4.\u00a0 A sketch of the velocity vectors of stars in the solar neighbourhood. All these vectors converge towards the Sun.<\/p>\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">motion in the opposite direction, that is, in the direction of the star Vega in the constellation of Lyra with a velocity of 20 km\/s.<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><strong><span style=\"text-align: initial;font-size: 1em\">4.2.\u00a0 Solar Apex<\/span><\/strong><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">The direction in which the Sun is moving with respect to the LSR is called <\/span><strong style=\"text-align: initial;font-size: 1em\">Solar Apex. <\/strong><span style=\"text-align: initial;font-size: 1em\">The direction diametrically opposite to that of Apex in which the stars in the solar neighbourhood are moving is called <\/span><strong style=\"text-align: initial;font-size: 1em\">Antapex. <\/strong><span style=\"text-align: initial;font-size: 1em\">As stated above, <\/span>Apex is in the direction of star Vega and Antapex is in the direction of the constellation Columba<strong style=\"text-align: initial;font-size: 1em\">.<\/strong><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">Consider Fig. 7.5.\u00a0 The radial velocity vector of all the stars in the solar neighbourhood add to zero. Their space velocities all point in the direction of Antapex. Equivalently, the Sun motion with respect to the LSR is in the direction of the Solar Apex.<\/span><\/p>\r\n\r\n<\/div>\r\n<div>\r\n\r\n<img class=\"aligncenter size-full wp-image-423\" src=\"http:\/\/phyp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/97\/2018\/12\/6.28.5.png\" alt=\"\" width=\"684\" height=\"415\" \/>\r\n<p style=\"text-align: center\">Fig. 7.5.\u00a0 Velocities of stars surrounding the Sun. Red arrows indicate radial velocities, which all add to zero.\u00a0 The black arrows are the space velocities of star pointing in the direction of Antapex. \u00a0This is due to the Sun\u2019s own motion in the direction of Apex at a speed of ~ 20 km\/s.<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">Space velocities corrected for the Sun\u2019s own motion are called the <\/span><strong style=\"text-align: initial;font-size: 1em\">peculiar velocities<\/strong><span style=\"text-align: initial;font-size: 1em\">.\u00a0 It is found that the peculiar velocities of stars are essentially random.\u00a0 Note that the net gravitational field due to all the stars surrounding a given star is zero. Therefore, being under no force, the stars move in random directions like the particles of a perfect gas. The typical magnitude of the peculiar velocities is such that in a time interval of ~ 105 year the stars will switch their nearest neighbours. Thus, in a period of about a million years the present constellations will be thoroughly shuffled (Fig. 7.6).<\/span><\/p>\r\n\r\n<\/div>\r\n<div>\r\n\r\n<img class=\"aligncenter size-full wp-image-422\" src=\"http:\/\/phyp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/97\/2018\/12\/6.28.6.png\" alt=\"\" width=\"330\" height=\"574\" \/>\r\n<p style=\"text-align: center\">Fig. 7.6.\u00a0 Changing shape of the familiar constellation due to the random motion of stars.<\/p>\r\n&nbsp;\r\n\r\n<strong><span style=\"font-size: 1em;text-align: initial\">5.\u00a0 Distances of Star Clusters<\/span><\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">In the case of some star clusters it is found that the space velocity vectors of their individual members converge in a certain direction (Fig. 7.7).\u00a0 This could only mean that the cluster as a\u00a0<\/span><span style=\"text-align: initial;font-size: 1em\">whole is moving in that direction. Let \ufffd\u20d7 \u00a0be the resultant velocity of the stars of the cluster.<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">From Fig. 7.8 it is obvious that the parallax of the cluster is given by ?<\/span><\/p>\r\n\r\n<\/div>\r\n<div>\r\n\r\n<img class=\"aligncenter wp-image-421\" src=\"http:\/\/phyp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/97\/2018\/12\/6.28.7.png\" alt=\"\" width=\"693\" height=\"632\" \/>\r\n<p style=\"text-align: center\">Fig. 7.7. The space velocity vectors of the members of a stellar cluster converge towards the point of convergence.<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"font-size: 1em;text-align: initial\">using Equation (7.1). Here ?is the angle between the line of sight and the direction in which the velocity vectors of the members of the cluster converge. Thus, a knowledge of ?and ? leads to the distances of these clusters. This is the moving cluster method for finding cluster distances.<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">The differential rotation of the Galaxy (to be discussed later) has a systematic effect on the proper motions. It will be shown that for a star at galactic longitude ??\ufffd, the transverse velocity produced by the galactic rotation is given by<\/span><\/p>\r\n\r\n<\/div>\r\n<div>\r\n\r\n<img class=\"aligncenter size-full wp-image-420\" src=\"http:\/\/phyp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/97\/2018\/12\/6.28.8.png\" alt=\"\" width=\"727\" height=\"546\" \/>\r\n\r\n<\/div>\r\n<div>\r\n<p style=\"text-align: center\"><span style=\"text-align: initial;font-size: 1em\">Fig. 7.8.\u00a0 The velocity vectors of the stars of a cluster converge in a direction\u00a0<\/span><span style=\"text-align: initial;font-size: 1em\">which makes an angle ?with the line of sight<\/span><strong style=\"text-align: initial;font-size: 1em\">.<\/strong><\/p>\r\n&nbsp;\r\n\r\n<strong style=\"text-align: initial;font-size: 1em\">6.\u00a0 Summary<\/strong>\r\n\r\n<\/div>\r\n<div>\r\n<ul>\r\n \t<li style=\"text-align: justify\">Proper motion gives the transverse velocity, which together with the radial velocity (found by using Doppler Effect) gives the space velocity of a star.<\/li>\r\n \t<li style=\"text-align: justify\">The Sun\u2019s own motion is defined in terms of the Local Standard of Rest (LSR).<\/li>\r\n \t<li style=\"text-align: justify\">LSR is a point which orbits the centre of the Galaxy in a circular orbit at the galacto-centric distance of the Sun and completes one revolution in the same time as the Sun.<\/li>\r\n \t<li style=\"text-align: justify\">The convergence of velocity vectors of stars in the solar neighbourhood shows that the Sun is moving in a certain direction with respect to the LSR with a velocity of about 20 km\/s.<\/li>\r\n \t<li style=\"text-align: justify\">The point towards which the Sun is moving in space is called the solar apex.<\/li>\r\n \t<li style=\"text-align: justify\">The velocity of a star transverse to the line of sight is called the transverse velocity.\u00a0It is obtained by multiplying the proper motion by the distance of the star.<\/li>\r\n \t<li style=\"text-align: justify\">Velocity along the line of sight is called the radial velocity.<\/li>\r\n \t<li style=\"text-align: justify\">Transverse and radial velocities together give the space velocity of a star.<\/li>\r\n \t<li style=\"text-align: justify\">Space velocity of stars corrected for the Sun\u2019s motion is called the peculiar velocity.<\/li>\r\n \t<li style=\"text-align: justify\">Peculiar velocities of stars are purely random. In a time of the order of 100,000 years all the constellations will be thoroughly shuffled.<\/li>\r\n \t<li style=\"text-align: justify\">Stars in some clusters show that their velocity vectors converge in a certain direction. This observation gives the space velocity of the cluster.<\/li>\r\n \t<li style=\"text-align: justify\">Armed with the space velocities of clusters and their radial velocities, we can determine their distances.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<table style=\"height: 22px\" width=\"716\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 613.063px\"><strong>You can view video on Physical Properties of Stars<\/strong><\/td>\r\n<td style=\"width: 74.0625px\"><a href=\"https:\/\/www.youtube.com\/watch?v=NA&amp;feature=youtu.be\" target=\"_blank\" rel=\"noopener\"><img class=\"alignnone wp-image-120\" src=\"http:\/\/epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/2018\/11\/download.png\" alt=\"\" width=\"36\" height=\"36\" \/><\/a><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n&nbsp;\r\n\r\n<strong>The following are the references for more-material on the topics covered in this module:<\/strong>\r\n<ul>\r\n \t<li>http:\/\/www.cosmos.esa.int\/web\/hipparcos<\/li>\r\n \t<li>http:\/\/hyperphysics.phy-astr.gsu.edu\/hbase\/astro\/para.html<\/li>\r\n \t<li>http:\/\/cseligman.com\/text\/stars\/propermotion.htm<\/li>\r\n \t<li>https:\/\/inspirehep.net\/record\/818899\/plots<\/li>\r\n \t<li>https:\/\/medium.com\/starts-with-a-bang\/how-fast-are-we-moving-through-space-985bf470378d#.gvhngkinf<\/li>\r\n<\/ul>","rendered":"<div>\n<div><span style=\"float: right\"><a href=\"https:\/\/youtu.be\/NA\" target=\"_blank\" rel=\"noopener\"><img decoding=\"async\" src=\"http:\/\/epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/2018\/11\/download.png\" alt=\"epgp books\" width=\"75px\" height=\"75px;\" \/><\/a><br \/>\n<\/span><\/div>\n<p>&nbsp;<\/p>\n<p><strong>1.\u00a0 <\/strong><strong>Learning Outcomes<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p>After studying this module, you should be able to<\/p>\n<ul>\n<li>Understand the concept of space velocities of stars<\/li>\n<li><span style=\"text-align: initial;font-size: 1em\">Explain the need for a Local Standard of Rest (LSR)<\/span><\/li>\n<li><span style=\"text-align: initial;font-size: 1em\">Grasp how LSR is defined<\/span><\/li>\n<li><span style=\"text-align: initial;font-size: 1em\">Account for the motion of the Sun with respect to the LSR<\/span><\/li>\n<li><span style=\"text-align: initial;font-size: 1em\">Describe the terms Solar Apex and Antapex<\/span><\/li>\n<li><span style=\"text-align: initial;font-size: 1em\">Understand the notion of peculiar velocities of Star<\/span><\/li>\n<li><span style=\"text-align: initial;font-size: 1em\">Account for the long term change in the shape of stellar constellations<\/span><\/li>\n<li><span style=\"text-align: initial;font-size: 1em\">Grasp how the distances of stellar clusters are measured using their space velocities<\/span><\/li>\n<\/ul>\n<p>&nbsp;<\/p>\n<p><strong>2.\u00a0 <\/strong><strong>Introduction<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">In the last module we introduced the proper motion of stars. \u00a0It must be borne in mind that the stars appear fixed because of their large distances; actually they are in constant motion. \u00a0The\u00a0<span style=\"text-align: initial;font-size: 1em\">proper motion is the motion transverse to the line of sight of the star.\u00a0 The proper motion is such a small angular displacement of stars that in most cases it amounts to a mere arc &#8211; second, or even smaller, in a year. That is why we need to compare photographs of the star field containing the star separated by decades.\u00a0 The proper motion of stars being so small, it is not possible to measure it for stars in other galaxies.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">It must be remembered that the proper motion depends on the distance of the star from the Sun. Two stars having the same displacement normal to the line of sight in a given time will have different proper motions depending on their distances (Fig. 7.1). \u00a0<\/span><strong style=\"text-align: initial;font-size: 1em\">However, smaller proper motion does not necessarily imply a larger distance<\/strong><span style=\"text-align: initial;font-size: 1em\">. An important effect of the proper motion is that it changes slowly the coordinates of a star. Therefore, whenever we state the coordinates of a star, we also state the epoch when the coordinates were measured.<\/span><\/p>\n<\/div>\n<div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-428\" src=\"http:\/\/phyp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/97\/2018\/12\/6.28.1.png\" alt=\"\" width=\"579\" height=\"145\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/12\/6.28.1.png 579w, https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/12\/6.28.1-300x75.png 300w, https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/12\/6.28.1-65x16.png 65w, https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/12\/6.28.1-225x56.png 225w, https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/12\/6.28.1-350x88.png 350w\" sizes=\"auto, (max-width: 579px) 100vw, 579px\" \/><\/p>\n<p style=\"text-align: center\"><span style=\"text-align: initial;font-size: 1em\">Fig. 7.1. Transverse displacement of two stars, S1 and S2 in a given time are equal. However, their proper motions ?1 and ?2 are not equal because of unequal distances.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">If the star is attended by an invisible companion, say a satellite, then the proper motion shows a periodicity. In fact, this an important method by which the presence of extrasolar planets is detected.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">Combined with the velocity along the line of sight of the star, the radial velocity, the proper motion gives the space velocity of the star, which we discuss in this module.<\/span><\/p>\n<\/div>\n<div>\n<p>&nbsp;<\/p>\n<p><strong>3.\u00a0 Space Velocity of Stars<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"font-size: 1em;text-align: initial\">The product of the proper motion of a star with its distance from the Sun gives its transverse velocity ??. If we express ??in km\/s, the distance ?in pc and the proper motion in \u02dd\/yr, then we get the following relation:<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: center\"><span style=\"text-align: initial;font-size: 1em\">??= 4.74?<em>d<\/em>= 4.74?\/?km\/s,\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 (7.1)<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">where ?is the parallax of the star in arc-second. This is a particularly useful relation, for it <\/span><span style=\"text-align: initial;font-size: 1em\">enables us to find stellar distances when the annual parallaxes cannot be directly <\/span><span style=\"text-align: initial;font-size: 1em\">measured. If we wait long enough, ?can be accurately determined and then a knowledge <\/span><span style=\"text-align: initial;font-size: 1em\">of ??give space velocity. Application of this relation to stellar clusters will be discussed <\/span><span style=\"text-align: initial;font-size: 1em\">later.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">The velocity of a star in the line of sight is called its <\/span><strong style=\"text-align: initial;font-size: 1em\">radial velocity, <\/strong><span style=\"text-align: initial;font-size: 1em\">denoted by ??. It is\u00a0<\/span><span style=\"text-align: initial;font-size: 1em\">inferred from the Doppler shifts of the lines in the star\u2019s spectrum.\u00a0 Combining the radial\u00a0<\/span><span style=\"text-align: initial;font-size: 1em\">velocity with the transverse velocity, we get the <\/span><strong style=\"text-align: initial;font-size: 1em\">space velocity <\/strong><span style=\"text-align: initial;font-size: 1em\">of the star (Fig. 7.2).\u00a0 Typical space velocities of stars range from 20 to 100 km\/s.<\/span><\/p>\n<\/div>\n<div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-427\" src=\"http:\/\/phyp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/97\/2018\/12\/6.28.2.png\" alt=\"\" width=\"763\" height=\"367\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/12\/6.28.2.png 763w, https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/12\/6.28.2-300x144.png 300w, https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/12\/6.28.2-65x31.png 65w, https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/12\/6.28.2-225x108.png 225w, https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/12\/6.28.2-350x168.png 350w\" sizes=\"auto, (max-width: 763px) 100vw, 763px\" \/><\/p>\n<p style=\"text-align: center\">Fig. 7.2.\u00a0 Radial velocity, transverse velocity, space velocity and proper motion of a star.<\/p>\n<p>&nbsp;<\/p>\n<p>Clearly,<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-426\" src=\"http:\/\/phyp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/97\/2018\/12\/6.28.2.1.png\" alt=\"\" width=\"566\" height=\"62\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/12\/6.28.2.1.png 748w, https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/12\/6.28.2.1-300x33.png 300w, https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/12\/6.28.2.1-65x7.png 65w, https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/12\/6.28.2.1-225x25.png 225w, https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/12\/6.28.2.1-350x38.png 350w\" sizes=\"auto, (max-width: 566px) 100vw, 566px\" \/><\/p>\n<p><strong><span style=\"font-size: 1em;text-align: initial\">4.\u00a0 Peculiar Velocity of Stars<\/span><\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">So far we have described the motion of a <\/span><strong style=\"text-align: initial;font-size: 1em\">star relative to the Sun<\/strong><span style=\"text-align: initial;font-size: 1em\">. But it is quite conceivable that the Sun itself may be moving in some direction. This, indeed, is the case. We, therefore, define a <\/span><strong style=\"text-align: initial;font-size: 1em\">Local Standard of Rest (LSR), <\/strong><span style=\"text-align: initial;font-size: 1em\">a sort of origin<\/span><strong style=\"text-align: initial;font-size: 1em\">, <\/strong><span style=\"text-align: initial;font-size: 1em\">with respect to which we state the motion of the Sun and stars in its neighbourhood.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><strong><span style=\"text-align: initial;font-size: 1em\">4.1.\u00a0 Local Standard of Rest (LSR)<\/span><\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">The <\/span><strong style=\"text-align: initial;font-size: 1em\">solar neighbourhood<\/strong><span style=\"text-align: initial;font-size: 1em\">, is a region roughly of radius ~ 100 pc centred on the Sun.\u00a0 These stars are supposed to share the motion of the Sun round the centre of the Galaxy, so that their motion due to <\/span><strong style=\"text-align: initial;font-size: 1em\">differential rotation of the Galaxy <\/strong><span style=\"text-align: initial;font-size: 1em\">does not become conspicuous. (Differential rotation implies that the various subsystems in the Galaxy rotate with their own characteristic speed round the centre of the Galaxy.) We have seen earlier that the Sun is situated at a<\/span><\/p>\n<p style=\"text-align: justify\"><strong style=\"text-align: initial;font-size: 1em\">\u00a0<\/strong><\/p>\n<\/div>\n<div><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-425\" src=\"http:\/\/phyp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/97\/2018\/12\/6.28.3.png\" alt=\"\" width=\"828\" height=\"505\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/12\/6.28.3.png 828w, https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/12\/6.28.3-300x183.png 300w, https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/12\/6.28.3-768x468.png 768w, https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/12\/6.28.3-65x40.png 65w, https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/12\/6.28.3-225x137.png 225w, https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/12\/6.28.3-350x213.png 350w\" sizes=\"auto, (max-width: 828px) 100vw, 828px\" \/><\/div>\n<div>\n<p style=\"text-align: center\">Fig. 7.3.\u00a0 Local Standard of Rest orbits the galactic centre in a circular orbit at the distance of the Sun and completes one revolution in the same time as the Sun.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">distance of about 8.5 kpc from the centre of the Galaxy.\u00a0 The Sun and its neighbourhood move round the centre of the Galaxy, completing one revolution in about 250 million years, the orbital velocity being ~ 220 km\/s.\u00a0 The orbit of the Sun round the galactic centre is elliptic. Therefore, <\/span>we define the Local Standard of Rest a point which orbits the galactic centre in a circular orbit at the galacto-centric distance of the Sun and completing one revolution in the same time as the Sun <span style=\"text-align: initial;font-size: 1em\">(Fig. 7.3).<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">Now, if we plot all the velocity vectors of stars in the solar neighbourhood, we find that they all converge towards the Sun and in the general direction of the constellation Columba with a resultant velocity of about 20 km\/s (Fig. 7.4).\u00a0 This is obviously a reflection of the Sun\u2019s own<\/span><\/p>\n<\/div>\n<div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-424\" src=\"http:\/\/phyp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/97\/2018\/12\/6.28.4.png\" alt=\"\" width=\"672\" height=\"563\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/12\/6.28.4.png 672w, https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/12\/6.28.4-300x251.png 300w, https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/12\/6.28.4-65x54.png 65w, https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/12\/6.28.4-225x189.png 225w, https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/12\/6.28.4-350x293.png 350w\" sizes=\"auto, (max-width: 672px) 100vw, 672px\" \/><\/p>\n<p style=\"text-align: center\">Fig. 7.4.\u00a0 A sketch of the velocity vectors of stars in the solar neighbourhood. All these vectors converge towards the Sun.<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">motion in the opposite direction, that is, in the direction of the star Vega in the constellation of Lyra with a velocity of 20 km\/s.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><strong><span style=\"text-align: initial;font-size: 1em\">4.2.\u00a0 Solar Apex<\/span><\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">The direction in which the Sun is moving with respect to the LSR is called <\/span><strong style=\"text-align: initial;font-size: 1em\">Solar Apex. <\/strong><span style=\"text-align: initial;font-size: 1em\">The direction diametrically opposite to that of Apex in which the stars in the solar neighbourhood are moving is called <\/span><strong style=\"text-align: initial;font-size: 1em\">Antapex. <\/strong><span style=\"text-align: initial;font-size: 1em\">As stated above, <\/span>Apex is in the direction of star Vega and Antapex is in the direction of the constellation Columba<strong style=\"text-align: initial;font-size: 1em\">.<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">Consider Fig. 7.5.\u00a0 The radial velocity vector of all the stars in the solar neighbourhood add to zero. Their space velocities all point in the direction of Antapex. Equivalently, the Sun motion with respect to the LSR is in the direction of the Solar Apex.<\/span><\/p>\n<\/div>\n<div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-423\" src=\"http:\/\/phyp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/97\/2018\/12\/6.28.5.png\" alt=\"\" width=\"684\" height=\"415\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/12\/6.28.5.png 684w, https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/12\/6.28.5-300x182.png 300w, https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/12\/6.28.5-65x39.png 65w, https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/12\/6.28.5-225x137.png 225w, https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/12\/6.28.5-350x212.png 350w\" sizes=\"auto, (max-width: 684px) 100vw, 684px\" \/><\/p>\n<p style=\"text-align: center\">Fig. 7.5.\u00a0 Velocities of stars surrounding the Sun. Red arrows indicate radial velocities, which all add to zero.\u00a0 The black arrows are the space velocities of star pointing in the direction of Antapex. \u00a0This is due to the Sun\u2019s own motion in the direction of Apex at a speed of ~ 20 km\/s.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">Space velocities corrected for the Sun\u2019s own motion are called the <\/span><strong style=\"text-align: initial;font-size: 1em\">peculiar velocities<\/strong><span style=\"text-align: initial;font-size: 1em\">.\u00a0 It is found that the peculiar velocities of stars are essentially random.\u00a0 Note that the net gravitational field due to all the stars surrounding a given star is zero. Therefore, being under no force, the stars move in random directions like the particles of a perfect gas. The typical magnitude of the peculiar velocities is such that in a time interval of ~ 105 year the stars will switch their nearest neighbours. Thus, in a period of about a million years the present constellations will be thoroughly shuffled (Fig. 7.6).<\/span><\/p>\n<\/div>\n<div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-422\" src=\"http:\/\/phyp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/97\/2018\/12\/6.28.6.png\" alt=\"\" width=\"330\" height=\"574\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/12\/6.28.6.png 330w, https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/12\/6.28.6-172x300.png 172w, https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/12\/6.28.6-65x113.png 65w, https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/12\/6.28.6-225x391.png 225w\" sizes=\"auto, (max-width: 330px) 100vw, 330px\" \/><\/p>\n<p style=\"text-align: center\">Fig. 7.6.\u00a0 Changing shape of the familiar constellation due to the random motion of stars.<\/p>\n<p>&nbsp;<\/p>\n<p><strong><span style=\"font-size: 1em;text-align: initial\">5.\u00a0 Distances of Star Clusters<\/span><\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">In the case of some star clusters it is found that the space velocity vectors of their individual members converge in a certain direction (Fig. 7.7).\u00a0 This could only mean that the cluster as a\u00a0<\/span><span style=\"text-align: initial;font-size: 1em\">whole is moving in that direction. Let \ufffd\u20d7 \u00a0be the resultant velocity of the stars of the cluster.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">From Fig. 7.8 it is obvious that the parallax of the cluster is given by ?<\/span><\/p>\n<\/div>\n<div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-421\" src=\"http:\/\/phyp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/97\/2018\/12\/6.28.7.png\" alt=\"\" width=\"693\" height=\"632\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/12\/6.28.7.png 572w, https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/12\/6.28.7-300x274.png 300w, https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/12\/6.28.7-65x59.png 65w, https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/12\/6.28.7-225x205.png 225w, https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/12\/6.28.7-350x319.png 350w\" sizes=\"auto, (max-width: 693px) 100vw, 693px\" \/><\/p>\n<p style=\"text-align: center\">Fig. 7.7. The space velocity vectors of the members of a stellar cluster converge towards the point of convergence.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"font-size: 1em;text-align: initial\">using Equation (7.1). Here ?is the angle between the line of sight and the direction in which the velocity vectors of the members of the cluster converge. Thus, a knowledge of ?and ? leads to the distances of these clusters. This is the moving cluster method for finding cluster distances.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">The differential rotation of the Galaxy (to be discussed later) has a systematic effect on the proper motions. It will be shown that for a star at galactic longitude ??\ufffd, the transverse velocity produced by the galactic rotation is given by<\/span><\/p>\n<\/div>\n<div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-420\" src=\"http:\/\/phyp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/97\/2018\/12\/6.28.8.png\" alt=\"\" width=\"727\" height=\"546\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/12\/6.28.8.png 727w, https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/12\/6.28.8-300x225.png 300w, https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/12\/6.28.8-65x49.png 65w, https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/12\/6.28.8-225x169.png 225w, https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/12\/6.28.8-350x263.png 350w\" sizes=\"auto, (max-width: 727px) 100vw, 727px\" \/><\/p>\n<\/div>\n<div>\n<p style=\"text-align: center\"><span style=\"text-align: initial;font-size: 1em\">Fig. 7.8.\u00a0 The velocity vectors of the stars of a cluster converge in a direction\u00a0<\/span><span style=\"text-align: initial;font-size: 1em\">which makes an angle ?with the line of sight<\/span><strong style=\"text-align: initial;font-size: 1em\">.<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p><strong style=\"text-align: initial;font-size: 1em\">6.\u00a0 Summary<\/strong><\/p>\n<\/div>\n<div>\n<ul>\n<li style=\"text-align: justify\">Proper motion gives the transverse velocity, which together with the radial velocity (found by using Doppler Effect) gives the space velocity of a star.<\/li>\n<li style=\"text-align: justify\">The Sun\u2019s own motion is defined in terms of the Local Standard of Rest (LSR).<\/li>\n<li style=\"text-align: justify\">LSR is a point which orbits the centre of the Galaxy in a circular orbit at the galacto-centric distance of the Sun and completes one revolution in the same time as the Sun.<\/li>\n<li style=\"text-align: justify\">The convergence of velocity vectors of stars in the solar neighbourhood shows that the Sun is moving in a certain direction with respect to the LSR with a velocity of about 20 km\/s.<\/li>\n<li style=\"text-align: justify\">The point towards which the Sun is moving in space is called the solar apex.<\/li>\n<li style=\"text-align: justify\">The velocity of a star transverse to the line of sight is called the transverse velocity.\u00a0It is obtained by multiplying the proper motion by the distance of the star.<\/li>\n<li style=\"text-align: justify\">Velocity along the line of sight is called the radial velocity.<\/li>\n<li style=\"text-align: justify\">Transverse and radial velocities together give the space velocity of a star.<\/li>\n<li style=\"text-align: justify\">Space velocity of stars corrected for the Sun\u2019s motion is called the peculiar velocity.<\/li>\n<li style=\"text-align: justify\">Peculiar velocities of stars are purely random. In a time of the order of 100,000 years all the constellations will be thoroughly shuffled.<\/li>\n<li style=\"text-align: justify\">Stars in some clusters show that their velocity vectors converge in a certain direction. This observation gives the space velocity of the cluster.<\/li>\n<li style=\"text-align: justify\">Armed with the space velocities of clusters and their radial velocities, we can determine their distances.<\/li>\n<\/ul>\n<\/div>\n<table style=\"height: 22px; width: 716px;\">\n<tbody>\n<tr>\n<td style=\"width: 613.063px\"><strong>You can view video on Physical Properties of Stars<\/strong><\/td>\n<td style=\"width: 74.0625px\"><a href=\"https:\/\/www.youtube.com\/watch?v=NA&amp;feature=youtu.be\" target=\"_blank\" rel=\"noopener\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-120\" src=\"http:\/\/epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/2018\/11\/download.png\" alt=\"\" width=\"36\" height=\"36\" \/><\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<p><strong>The following are the references for more-material on the topics covered in this module:<\/strong><\/p>\n<ul>\n<li>http:\/\/www.cosmos.esa.int\/web\/hipparcos<\/li>\n<li>http:\/\/hyperphysics.phy-astr.gsu.edu\/hbase\/astro\/para.html<\/li>\n<li>http:\/\/cseligman.com\/text\/stars\/propermotion.htm<\/li>\n<li>https:\/\/inspirehep.net\/record\/818899\/plots<\/li>\n<li>https:\/\/medium.com\/starts-with-a-bang\/how-fast-are-we-moving-through-space-985bf470378d#.gvhngkinf<\/li>\n<\/ul>\n","protected":false},"author":4,"menu_order":28,"template":"","meta":{"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-415","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\/415","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":5,"href":"https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-json\/pressbooks\/v2\/chapters\/415\/revisions"}],"predecessor-version":[{"id":429,"href":"https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-json\/pressbooks\/v2\/chapters\/415\/revisions\/429"}],"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\/415\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-json\/wp\/v2\/media?parent=415"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-json\/pressbooks\/v2\/chapter-type?post=415"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-json\/wp\/v2\/contributor?post=415"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-json\/wp\/v2\/license?post=415"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}