{"id":84,"date":"2018-11-16T10:01:26","date_gmt":"2018-11-16T10:01:26","guid":{"rendered":"http:\/\/phyp06.epgpbooks.inflibnet.ac.in\/?post_type=chapter&p=84"},"modified":"2022-01-07T05:41:42","modified_gmt":"2022-01-07T05:41:42","slug":"physical-properties-of-stars","status":"publish","type":"chapter","link":"https:\/\/ebooks.inflibnet.ac.in\/phyp06\/chapter\/physical-properties-of-stars\/","title":{"rendered":"Physical Properties of Stars"},"content":{"raw":"
\r\n\r\n \r\n\r\n1.\u00a0 <\/strong>Learning Outcomes<\/strong>\r\n\r\n \r\n

After studying this module, you should be able to<\/p>\r\n\r\n

    \r\n \t
  • grasp the meaning of parallax and demonstrate it<\/li>\r\n \t
  • appreciate why we need a very long baseline for observing stellar parallaxes<\/span><\/li>\r\n \t
  • calculate the distance of a star whose parallax is known<\/span><\/li>\r\n \t
  • understand the importance of stellar parallaxes in astronomy<\/span><\/li>\r\n \t
  • explain the meaning of proper motion of stars<\/span><\/li>\r\n \t
  • deduce the change in equatorial coordinates of a star due to its proper motion<\/span><\/li>\r\n \t
  • derive relationship between parallax and proper motion of stars<\/span><\/li>\r\n<\/ul>\r\n \r\n\r\n2.\u00a0 <\/strong>Introduction<\/strong>\r\n\r\n \r\n

    So far we have managed to fix the direction of a star in terms of two angular coordinates. Refer to the <\/span>horizon system, <\/span>the equatorial system, <\/span>the ecliptic system and the <\/span>galactic system of coordinates in Modules 01, 02 and 03.\u00a0 \u00a0We now embark on the study pf physical characteristics of stars. These characteristics include the luminosity of a star, its effective temperature, its spectra, source of its energy, state of its evolution.\u00a0 Critical to such studies is the astronomical data, such as the distance of a star, its motion in the sky and its brightness. Therefore, in this module we turn our attention first to the distance of stars. Then we shall take up their motion in the sky.<\/span><\/p>\r\n \r\n

    Remember that the coordinates of a star only fix its direction. \u00a0For a complete location of the star in space, we also need its distance. As we shall see below, the distance of a star also fixes its luminosity, spectral class and other characteristics.<\/span><\/p>\r\n \r\n

    Since distances of celestial objects are very large \u2013 even the distance of the Sun is 1.5 \u00d7 1011 m \u2013 special methods and techniques are required to measure them. \u00a0The units required are also special.\u00a0 The distance to the Sun is called Astronomical Unit (AU).\u00a0 Distances of members of the solar system are measured in AU.\u00a0 For stars even AU is too small. Light Year (9.46 \u00d7 1015 m) is a very popular and descriptive unit. However, most suitable unit for astronomers in many respects is a parsec (pc), which we shall define in the next few pages.<\/span><\/p>\r\n \r\n

    3.\u00a0 Stellar Parallaxes<\/span><\/strong><\/p>\r\n \r\n

    The standard method of measuring distances of nearby stars is by observing their parallaxes. \u00a0As you already know, the <\/span>parallax is the apparent shift in the position of an object with respect to the background when seen from two different positions.\u00a0<\/strong>You can realize the effect of parallax if you observe the thumb of your stretched hand first by your one eye and then the other. You will notice that the position of the thumb shifts with respect to the background.<\/span><\/p>\r\n \r\n

    Fig. 6.1 shows how observation of a star from locations A and B changes its direction with respect to the background of distant stars. <\/span>Parallaxes are measured in terms of the angular changes in direction.\u00a0\u00a0<\/strong>However, stellar distances are so large that the angular change in their directions are extremely small, unless the base line AB is long. \u00a0Therefore, we look for a long base line, so that the angles are measurable.<\/span><\/p>\r\n \r\n

    The longest baseline available to observers on the earth\u2019s surface is the diameter of the earth. However, it is too short for the determination of stellar parallaxes.\u00a0 So, observers have to use as base line the diameter of the earth\u2019s orbit round the Sun.\u00a0 This they can do by making observations at intervals of six months,<\/span><\/p>\r\n\r\n<\/div>\r\n

    \r\n\r\n\"\"\r\n

    Fig. 6.1.\u00a0 Seen from point A, the star appears to be in the direction AX.\u00a0 Seen from B, the star appears to be in the direction BY. The change in direction with respect to the background, angle ASB, is a measure of the parallax of star S.<\/p>\r\n \r\n\r\n3.1.\u00a0 Annual Parallax of Stars<\/strong>\r\n\r\n \r\n

    Observations are made from positions E1 and E2 at the opposite ends of the earth\u2019s orbit.\u00a0 \u00a0Half the total change in the angular direction of S, angle p in Fig. 6.2, is called the parallax<\/strong>, or\u00a0annual parallax<\/strong>, of the object S.\u00a0 As said above, these angles are small; they are expressed in seconds of arc.\u00a0 Even for nearest stars, these angles are only a fraction of arc-second. If this angle is one second of arc, the distance of S from the Sun <\/strong>(which is the same thing as the distance from the earth, because the distances between the Sun and the earth is negligible compared with the distance of stars from the Sun) is known as one parsec (written also as pc)<\/strong>.<\/p>\r\n \r\n

    The Sun \u2013 earth distance is called an astronomical unit <\/strong>denoted by AU.\u00a0 It is equal to 1.496 \u00d7 10<\/strong>11 <\/strong>m.\u00a0 It is used as a unit for distances within the solar system. It is now an easy matter to show that<\/p>\r\n\r\n<\/div>\r\n\"\"\r\n

    It is important to remember that in Equation (6.1) the angle had to be expressed in radians. Since\u00a0<\/span>the distance is proportional to 1\/?, it is quite clear that smaller the parallax, larger the distance of the object.\u00a0 In fact, if ?is expressed in arc-second, 1\/? gives the distance in parsec. For\u00a0<\/span>example, the parallax of Barnard\u2019s star is 545.6 \u00d7 10-3 arc-second.\u00a0 Therefore, its distance is 1.83\u00a0<\/span>pc.<\/span><\/p>\r\n\r\n

    \r\n\r\n\"\"\r\n\r\n<\/div>\r\n
    \r\n\r\n \r\n

    Fig. 6.2.\u00a0 The diameter of the earth\u2019s orbit round the Sun is the baseline for determining parallaxes of stars. Angle p is called the parallax of the star S.<\/p>\r\n \r\n

    The method of parallaxes serves to measure distances of only nearby stars, say stars at\u00a0distances ~ 100 pc from the Sun (<\/strong>?~?.??\u00a0arc-second)<\/strong>. As parallaxes become smaller, the errors in measurement become comparable to the parallaxes themselves and the uncertainties in measurements become too large to be acceptable.<\/p>\r\n \r\n

    In recent years the European satellite Hipparcos (Hi<\/strong>gh P<\/strong>recision Par<\/strong>allax Co<\/strong>llecting S<\/strong>atellite) <\/strong>has measured parallaxes of a large number of stars which are not accessible from the ground. Moreover, the accuracy achieved by Hipparcos instruments is ~ 0.001 arc-sec compared with .01 arc-sec of that of ground-based facilities.\u00a0 So, whereas the instruments on the ground could measure distances of about 1000 stars, Hipparcos instruments could measure accurately distances of about 100,000 stars within a distance of up to 1000 pc.\u00a0 European Space Agency has launched another satellite Gaia in 2013 whose mission, among other things, is to observe several million\u00a0stars and measure their distances accurately. The accuracy in the measurement of parallax by Gaia is expected to be 10-6 arc-second.<\/span><\/p>\r\n \r\n

    An interesting sidelight: the acronym Hipparcos was chosen probably because it is close to Hipparchus, a Greek Astronomer who lived more than 2000 years ago. Among the more important works of Hipparchus was the measurement of the parallax of the moon and the determination of its distance from the earth. His most important work was the discovery of the phenomenon we now call precession of the equinoxes. He determined quite accurately the inclination of the ecliptic to the equator.<\/em><\/p>\r\n\r\n<\/div>\r\n

    \r\n\r\n \r\n\r\n3.2.\u00a0 Importance of Stellar Parallax<\/strong>\r\n\r\n \r\n

    Why do we lay so much importance on accurate measurement of distances of stars? The fact is that if we wish to know accurately the physical properties of stars and other objects, properties such as luminosity, effective temperature, chemical composition, in order to understand their origin and evolution, the accurate knowledge of their distances is of paramount importance.<\/p>\r\n \r\n

    Historically, the discovery of stellar parallaxes provided another argument against the geocentric universe which held the earth to be at rest and at the centre of the universe. The arguments of Copernicus and Galileo disfavouring the earth as the centre of the universe were not accepted because their critics pointed out that if earth really revolved around the Sun, one must observe the parallactic motion of stars.\u00a0 At that time such an observation was not possible because of lack of suitable instruments.\u00a0 However, the successful observation of stellar parallax in 1853 by Friedrich Bessel<\/strong>, vindicated Copernicus and Galileo and dealt a severe blow to the geocentric universe.<\/p>\r\n \r\n

    For distant objects, in the Galaxy and outside, there are several indirect methods which we shall describe in due course. However, all these methods depend critically on the distances of the\u00a0nearby stars measured by the method of annual parallaxes. In that lies the importance of the accurate measurement of parallaxes by satellites such as Hipparcos and Gaia.<\/span><\/p>\r\n \r\n

    3.3.\u00a0 Complexities of Parallax Measurement<\/span><\/strong><\/p>\r\n \r\n

    The determination of annual parallaxes is a difficult task and only a few observatories specialize in this work. We must remember that only if the star is in the plane of the ecliptic, its parallactic path is a straight line; in all other directions of the star the parallactic path is generally an ellipse, the ellipse being reduced to a circle when the star is in the direction of the ecliptic pole (Fig. 6.3).<\/span><\/p>\r\n\r\n<\/div>\r\n

    \r\n\r\n\"\"\r\n

    Fig. 6.3. Parallactic motion of a star is generally an ellipse. If the star is on a normal to the ecliptic plane, its parallactic motion is a circle. If the star is in the plane of the ecliptic, then its parallactic path is a straight line.<\/p>\r\n \r\n

    It means that the parallactic path of a star is never a simple curve.<\/span><\/p>\r\n \r\n

    Moreover, the parallactic path is always distorted by the star\u2019s own motion relative to the Sun. The change in a star\u2019s direction is called its <\/span>proper motion <\/strong>(see below).\u00a0 Thus, the motion of a star is really a complicated curve.\u00a0 From this complicated curve, the annual parallactic motion of the star has to be extracted.\u00a0 This is a difficult task which requires several photographs of the concerned region of the sky taken at different times of the year.<\/span><\/p>\r\n \r\n

    Table 6.1 lists nearest stars, their equatorial coordinates, parallaxes and distances.<\/span><\/p>\r\n \r\n

    Table 6.1.\u00a0 Parallaxes and Distances of the nearest 20 stars<\/strong><\/p>\r\n\r\n<\/div>\r\n

    \r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
    No.<\/strong><\/td>\r\nName of the Star<\/strong><\/td>\r\nRight Ascension<\/strong><\/td>\r\nDeclination<\/strong><\/td>\r\nParallax (milliarcsec)<\/strong><\/td>\r\nDistance (pc)<\/strong><\/td>\r\n<\/tr>\r\n
    1<\/td>\r\nProxima Centauri<\/td>\r\n14h 29m 43.0s<\/td>\r\n\u221262\u00b0 40\u2032 46\u2033<\/td>\r\n768.87<\/td>\r\n1.30<\/td>\r\n<\/tr>\r\n
    2<\/td>\r\n\u03b1 Centauri A<\/td>\r\n14h 39m 36.5s<\/td>\r\n\u221260\u00b0 50\u2032 02\u2033<\/td>\r\n747.23<\/td>\r\n1.33<\/td>\r\n<\/tr>\r\n
    3<\/td>\r\n\u03b1 Centauri B<\/td>\r\n14h 39m 35.1s<\/td>\r\n\u221260\u00b0 50\u2032 14\u2033<\/td>\r\n747.23<\/td>\r\n1.33<\/td>\r\n<\/tr>\r\n
    4<\/td>\r\nBarnard\u2019s Star<\/td>\r\n17h 57m 48.5s<\/td>\r\n+04\u00b0 41\u2032 36\u2033<\/td>\r\n546.98<\/td>\r\n1.83<\/td>\r\n<\/tr>\r\n
    5<\/td>\r\nWolf 359<\/td>\r\n10h 56m 29.2s<\/td>\r\n+07\u00b0 00\u2032 53\u2033<\/td>\r\n419.10<\/td>\r\n2.38<\/td>\r\n<\/tr>\r\n
    6<\/td>\r\nLalande 21185<\/td>\r\n11h 03m 20.2s<\/td>\r\n+35\u00b0 58\u2032 12\u2033<\/td>\r\n393.42<\/td>\r\n2.54<\/td>\r\n<\/tr>\r\n
    7<\/td>\r\nSirius A<\/td>\r\n06h 45m 08.9s<\/td>\r\n\u221216\u00b0 42\u2032 58\u2033<\/td>\r\n380.02<\/td>\r\n2.63<\/td>\r\n<\/tr>\r\n
    8<\/td>\r\nSirius B<\/td>\r\n06h 45m 08.9s<\/td>\r\n\u221216\u00b0 42\u2032 58\u2033<\/td>\r\n380.02<\/td>\r\n2.63<\/td>\r\n<\/tr>\r\n
    9<\/td>\r\nLuyten 726-8 A<\/td>\r\n01h 39m 01.3s<\/td>\r\n\u221217\u00b0 57\u2032 01\u2033<\/td>\r\n373.70<\/td>\r\n2.67<\/td>\r\n<\/tr>\r\n
    10<\/td>\r\nLuyten 726-8 B<\/td>\r\n01h 39m 01.3s<\/td>\r\n\u221217\u00b0 57\u2032 01\u2033<\/td>\r\n373.70<\/td>\r\n2.67<\/td>\r\n<\/tr>\r\n
    11<\/td>\r\nRoss 154<\/td>\r\n18h 49m 49.4s<\/td>\r\n\u221223\u00b0 50\u2032 10\u2033<\/td>\r\n336.90<\/td>\r\n2.97<\/td>\r\n<\/tr>\r\n
    12<\/td>\r\nRoss 248<\/td>\r\n23h 41m 54.7s<\/td>\r\n+44\u00b0 10\u2032 30\u2033<\/td>\r\n316.00<\/td>\r\n3.16<\/td>\r\n<\/tr>\r\n
    13<\/td>\r\nEpsilon Eridani<\/td>\r\n03h 32m 55.8s<\/td>\r\n\u221209\u00b0 27\u2032 30\u2033<\/td>\r\n309.99<\/td>\r\n3.23<\/td>\r\n<\/tr>\r\n
    14<\/td>\r\nLacaille 9352<\/td>\r\n23h 05m 52.0s<\/td>\r\n\u221235\u00b0 51\u2032 11\u2033<\/td>\r\n303.64<\/td>\r\n3.29<\/td>\r\n<\/tr>\r\n
    15<\/td>\r\nRoss 128<\/td>\r\n11h 47m 44.4s<\/td>\r\n+00\u00b0 48\u2032 16\u2033<\/td>\r\n298.72<\/td>\r\n3.35<\/td>\r\n<\/tr>\r\n
    16<\/td>\r\nEZ Aquarii A<\/td>\r\n22h 38m 33.4s<\/td>\r\n\u221215\u00b0 17\u2032 57\u2033<\/td>\r\n289.50<\/td>\r\n3.45<\/td>\r\n<\/tr>\r\n
    17<\/td>\r\nEZ Aquarii B<\/td>\r\n22h 38m 33.4s<\/td>\r\n\u221215\u00b0 17\u2032 57\u2033<\/td>\r\n289.50<\/td>\r\n3.45<\/td>\r\n<\/tr>\r\n
    18<\/td>\r\nEZ Aquarii C<\/td>\r\n22h 38m 33.4s<\/td>\r\n\u221215\u00b0 17\u2032 57\u2033<\/td>\r\n289.50<\/td>\r\n3.45<\/td>\r\n<\/tr>\r\n
    19<\/td>\r\nProcyon A<\/td>\r\n07h 39m 18.1s<\/td>\r\n+05\u00b0 13\u2032 30\u2033<\/td>\r\n286.05<\/td>\r\n3.50<\/td>\r\n<\/tr>\r\n
    20<\/td>\r\nProcyon B<\/td>\r\n07h 39m 18.1s<\/td>\r\n+05\u00b0 13\u2032 30\u2033<\/td>\r\n286.05<\/td>\r\n3.50<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n
    \r\n\r\n \r\n\r\nNotes for Table 6.1:\r\n\r\n1. Equatorial coordinates are for the epoch J2000.0\r\n\r\n2. Many of these stars belong to double or triple systems.\r\n\r\n3. Based on the list of nearest stars and brown dwarfs given in:\r\n\r\nhttps:\/\/en.wikipedia.org\/wiki\/List_of_nearest_stars_and_brown_dwarfs\r\n\r\n \r\n\r\n4. Proper Motion of Stars<\/strong>\r\n\r\n \r\n

    Like everything else in the universe, the stars are also always in motion. \u00a0They appear fixed because they are very far off and the change in their direction is so small that it cannot be appreciated by the naked eye. The motion of a star in the direction perpendicular to the line of sight is called its proper motion <\/strong>(Fig. 6.4).\u00a0 It is denoted by ? and is measured in arc-second per year (\u02dd\/yr).<\/p>\r\n\"\"\r\n

    Fig. 6.4. The line of sight from the observer O to the star is OS.\u00a0 The motion perpendicular to the line of sight, measured in angle, is the proper motion of the star.<\/p>\r\n \r\n

    4.1.\u00a0 Proper Motion in Declination and Right Ascension<\/span><\/strong><\/p>\r\n \r\n

    It is obvious that the motion of a star will bring about a change in its coordinates. In Fig.\u00a0<\/span>6.5, the star moves from position X to position X'. \u00a0It now lies on a different meridian and\u00a0<\/span>not at the same arc length from the equator as before. \u00a0Its new declination is ?+ ??, the\u00a0<\/span>change in its declination being ??, often denoted by ??. Similarly the change in its right\u00a0<\/span>ascension is ??denoted by ??. \u00a0This nomenclature implies that these changes have been\u00a0<\/span>brought about by the proper motion of the star. It is usual to express the proper motion in\u00a0<\/span>terms of ??and ??, the changes in the coordinates of the star:<\/span><\/p>\r\n\r\n<\/div>\r\n

    \r\n\r\n\"\"\r\n

    Fig. 6.5.\u00a0 Star X moves to X\u00b4 in one year due to proper motion ?.\u00a0 Its right ascension changes by ??= ??while its declination changes by ??= ??.<\/span><\/p>\r\n \r\n

    ?= \u221a\u00af??2<\/sup> + ??2<\/sup> .\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\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 (6.2)<\/span><\/p>\r\n\r\n<\/div>\r\n

    \r\n\r\n \r\n\r\n4.2. Determination of Proper Motion<\/strong>\r\n\r\n \r\n

    The proper motion of a star is cumulative.\u00a0 Therefore, it can be determined by comparing the photographs of the region of the sky containing the star with the background of distant objects which do not change their positions. Since proper motion is a small quantity, the photographs are separated by decades. \u00a0It is important that the position measurements at two epochs be referred to the same equator and equinoxes so that changes in coordinates caused by the earth\u2019s precession are avoided.<\/p>\r\n\"\"\r\n

    Fig. 6.6. (a) and (b) are the photograph several decades apart of the region of the sky which contains the star whose proper motion is to be<\/p>\r\n \r\n

    Figures 6.6 (a) and (b) show, schematically, two photographs taken a few decades apart.<\/span><\/p>\r\n \r\n

    Superposition of these photographs shows that the star X has moved to X\u00b4.\u00a0 Knowing the displacement XX\u00b4 (after reduction by the appropriate photographic plate factors) and the lapse of time between the two photographs, ?can be calculated. It is obvious that the accuracy in the measurement of ?can be increased by selecting photographs separated by a long interval; however, much of this advantage is offset by the errors in the position measurement at earlier epochs. Measurement of the proper motion of a star is not as simple as it might appear; it is obtained by analyzing the complicated path that the star describes in the sky due to the combination of the proper motion and the parallactic motion. Hipparcos has also measured the proper motion of a large number of stars.<\/span><\/p>\r\n\"\"\r\n

    Fig. 6.7.\u00a0 Hipparcos also measured the proper motion of stars. The figure is taken from the introduction to <\/span>The Millennium Star Atlas, and was produced by Dennis di Cicco for Sky Publishing Corporation. Note that proper motion of the star stands out clearly as a movement of the star against the background stars with time. Dennis di Cicco's observations were so accurate that the effect of the parallax (the distance) of the stars is also evident. It is seen as the \"wavy\" motion of the star (the individual observations are shown as black circles with the relevant observation date) about its linear motion (shown as the straight line dissecting the figure); this wavy motion has a period of one year, corresponding to the Earth's orbital motion around the Sun. (<\/span>http:\/\/www.cosmos.esa.int\/web\/hipparcos\/high-proper-motion-stars)<\/span><\/p>\r\n \r\n

    The largest proper motion known is that of the Barnard\u2019s star, which is 10.3\u02dd yr-1. Among the stars visible with naked eye, the largest proper motion belongs to 61 Cygni (5.22\u00b4 yr-1). For pictures of the Barnard\u2019s star with the stellar background about 50 years apart, and animation of the same pictures showing movement of the star against the same background, visit <\/span>http:\/\/cseligman.com\/text\/stars\/propermotion.htm<\/p>\r\n\r\n<\/div>\r\n

    \r\n\r\n \r\n\r\n5. Summary<\/span><\/strong>\r\n\r\n<\/div>\r\n
      \r\n \t
    • Stellar distances are essential for fixing physical properties of stars, such as their luminosities.<\/li>\r\n \t
    • Distances of nearby stars are determined by the method of parallax.<\/li>\r\n \t
    • Observations are made from the two ends of the earth\u2019s orbit round the Sun.<\/li>\r\n \t
    • Half the angular displacement in direction observed from these points is called the annual parallax of a star.<\/li>\r\n \t
    • If the annual parallax is one arc-second, the distance is said to be one parsec.<\/li>\r\n \t
    • Generally, the reciprocal of parallax in arc-second is the distance in parsec.<\/li>\r\n \t
    • The motion of a star perpendicular to its line of sight is called the proper motion of the star.<\/li>\r\n \t
    • Proper motion changes the equatorial coordinates of the star. Therefore, proper motion is given in terms of its component changes in right ascension and declination.<\/li>\r\n \t
    • Proper motion is a small quantity. It is measured in arc-second\/year, or milli arc second\/year.<\/li>\r\n<\/ul>\r\n \r\n\r\nReferences<\/strong>\r\n
        \r\n \t
      • https:\/\/en.wikipedia.org\/wiki\/Parallax<\/li>\r\n \t
      • http:\/\/hyperphysics.phy-astr.gsu.edu\/hbase\/astro\/para.html<\/li>\r\n \t
      • https:\/\/en.wikipedia.org\/wiki\/Gaia_%28spacecraft%29<\/li>\r\n \t
      • http:\/\/www.ast.cam.ac.uk\/~mjp\/calc_parallax.html<\/li>\r\n \t
      • http:\/\/www.scientus.org\/Copernicus-Stellar-Parallax.html<\/li>\r\n \t
      • http:\/\/physics.unm.edu\/101lab\/lab4\/lab4_C.html(For animation)<\/li>\r\n<\/ul>","rendered":"
        \n

         <\/p>\n

        1.\u00a0 <\/strong>Learning Outcomes<\/strong><\/p>\n

         <\/p>\n

        After studying this module, you should be able to<\/p>\n

          \n
        • grasp the meaning of parallax and demonstrate it<\/li>\n
        • appreciate why we need a very long baseline for observing stellar parallaxes<\/span><\/li>\n
        • calculate the distance of a star whose parallax is known<\/span><\/li>\n
        • understand the importance of stellar parallaxes in astronomy<\/span><\/li>\n
        • explain the meaning of proper motion of stars<\/span><\/li>\n
        • deduce the change in equatorial coordinates of a star due to its proper motion<\/span><\/li>\n
        • derive relationship between parallax and proper motion of stars<\/span><\/li>\n<\/ul>\n

           <\/p>\n

          2.\u00a0 <\/strong>Introduction<\/strong><\/p>\n

           <\/p>\n

          So far we have managed to fix the direction of a star in terms of two angular coordinates. Refer to the <\/span>horizon system, <\/span>the equatorial system, <\/span>the ecliptic system and the <\/span>galactic system of coordinates in Modules 01, 02 and 03.\u00a0 \u00a0We now embark on the study pf physical characteristics of stars. These characteristics include the luminosity of a star, its effective temperature, its spectra, source of its energy, state of its evolution.\u00a0 Critical to such studies is the astronomical data, such as the distance of a star, its motion in the sky and its brightness. Therefore, in this module we turn our attention first to the distance of stars. Then we shall take up their motion in the sky.<\/span><\/p>\n

           <\/p>\n

          Remember that the coordinates of a star only fix its direction. \u00a0For a complete location of the star in space, we also need its distance. As we shall see below, the distance of a star also fixes its luminosity, spectral class and other characteristics.<\/span><\/p>\n

           <\/p>\n

          Since distances of celestial objects are very large \u2013 even the distance of the Sun is 1.5 \u00d7 1011 m \u2013 special methods and techniques are required to measure them. \u00a0The units required are also special.\u00a0 The distance to the Sun is called Astronomical Unit (AU).\u00a0 Distances of members of the solar system are measured in AU.\u00a0 For stars even AU is too small. Light Year (9.46 \u00d7 1015 m) is a very popular and descriptive unit. However, most suitable unit for astronomers in many respects is a parsec (pc), which we shall define in the next few pages.<\/span><\/p>\n

           <\/p>\n

          3.\u00a0 Stellar Parallaxes<\/span><\/strong><\/p>\n

           <\/p>\n

          The standard method of measuring distances of nearby stars is by observing their parallaxes. \u00a0As you already know, the <\/span>parallax is the apparent shift in the position of an object with respect to the background when seen from two different positions.\u00a0<\/strong>You can realize the effect of parallax if you observe the thumb of your stretched hand first by your one eye and then the other. You will notice that the position of the thumb shifts with respect to the background.<\/span><\/p>\n

           <\/p>\n

          Fig. 6.1 shows how observation of a star from locations A and B changes its direction with respect to the background of distant stars. <\/span>Parallaxes are measured in terms of the angular changes in direction.\u00a0\u00a0<\/strong>However, stellar distances are so large that the angular change in their directions are extremely small, unless the base line AB is long. \u00a0Therefore, we look for a long base line, so that the angles are measurable.<\/span><\/p>\n

           <\/p>\n

          The longest baseline available to observers on the earth\u2019s surface is the diameter of the earth. However, it is too short for the determination of stellar parallaxes.\u00a0 So, observers have to use as base line the diameter of the earth\u2019s orbit round the Sun.\u00a0 This they can do by making observations at intervals of six months,<\/span><\/p>\n<\/div>\n

          \n

          \"\"<\/p>\n

          Fig. 6.1.\u00a0 Seen from point A, the star appears to be in the direction AX.\u00a0 Seen from B, the star appears to be in the direction BY. The change in direction with respect to the background, angle ASB, is a measure of the parallax of star S.<\/p>\n

           <\/p>\n

          3.1.\u00a0 Annual Parallax of Stars<\/strong><\/p>\n

           <\/p>\n

          Observations are made from positions E1 and E2 at the opposite ends of the earth\u2019s orbit.\u00a0 \u00a0Half the total change in the angular direction of S, angle p in Fig. 6.2, is called the parallax<\/strong>, or\u00a0annual parallax<\/strong>, of the object S.\u00a0 As said above, these angles are small; they are expressed in seconds of arc.\u00a0 Even for nearest stars, these angles are only a fraction of arc-second. If this angle is one second of arc, the distance of S from the Sun <\/strong>(which is the same thing as the distance from the earth, because the distances between the Sun and the earth is negligible compared with the distance of stars from the Sun) is known as one parsec (written also as pc)<\/strong>.<\/p>\n

           <\/p>\n

          The Sun \u2013 earth distance is called an astronomical unit <\/strong>denoted by AU.\u00a0 It is equal to 1.496 \u00d7 10<\/strong>11 <\/strong>m.\u00a0 It is used as a unit for distances within the solar system. It is now an easy matter to show that<\/p>\n<\/div>\n

          \"\"<\/p>\n

          It is important to remember that in Equation (6.1) the angle had to be expressed in radians. Since\u00a0<\/span>the distance is proportional to 1\/?, it is quite clear that smaller the parallax, larger the distance of the object.\u00a0 In fact, if ?is expressed in arc-second, 1\/? gives the distance in parsec. For\u00a0<\/span>example, the parallax of Barnard\u2019s star is 545.6 \u00d7 10-3 arc-second.\u00a0 Therefore, its distance is 1.83\u00a0<\/span>pc.<\/span><\/p>\n

          \n

          \"\"<\/p>\n<\/div>\n

          \n

           <\/p>\n

          Fig. 6.2.\u00a0 The diameter of the earth\u2019s orbit round the Sun is the baseline for determining parallaxes of stars. Angle p is called the parallax of the star S.<\/p>\n

           <\/p>\n

          The method of parallaxes serves to measure distances of only nearby stars, say stars at\u00a0distances ~ 100 pc from the Sun (<\/strong>?~?.??\u00a0arc-second)<\/strong>. As parallaxes become smaller, the errors in measurement become comparable to the parallaxes themselves and the uncertainties in measurements become too large to be acceptable.<\/p>\n

           <\/p>\n

          In recent years the European satellite Hipparcos (Hi<\/strong>gh P<\/strong>recision Par<\/strong>allax Co<\/strong>llecting S<\/strong>atellite) <\/strong>has measured parallaxes of a large number of stars which are not accessible from the ground. Moreover, the accuracy achieved by Hipparcos instruments is ~ 0.001 arc-sec compared with .01 arc-sec of that of ground-based facilities.\u00a0 So, whereas the instruments on the ground could measure distances of about 1000 stars, Hipparcos instruments could measure accurately distances of about 100,000 stars within a distance of up to 1000 pc.\u00a0 European Space Agency has launched another satellite Gaia in 2013 whose mission, among other things, is to observe several million\u00a0stars and measure their distances accurately. The accuracy in the measurement of parallax by Gaia is expected to be 10-6 arc-second.<\/span><\/p>\n

           <\/p>\n

          An interesting sidelight: the acronym Hipparcos was chosen probably because it is close to Hipparchus, a Greek Astronomer who lived more than 2000 years ago. Among the more important works of Hipparchus was the measurement of the parallax of the moon and the determination of its distance from the earth. His most important work was the discovery of the phenomenon we now call precession of the equinoxes. He determined quite accurately the inclination of the ecliptic to the equator.<\/em><\/p>\n<\/div>\n

          \n

           <\/p>\n

          3.2.\u00a0 Importance of Stellar Parallax<\/strong><\/p>\n

           <\/p>\n

          Why do we lay so much importance on accurate measurement of distances of stars? The fact is that if we wish to know accurately the physical properties of stars and other objects, properties such as luminosity, effective temperature, chemical composition, in order to understand their origin and evolution, the accurate knowledge of their distances is of paramount importance.<\/p>\n

           <\/p>\n

          Historically, the discovery of stellar parallaxes provided another argument against the geocentric universe which held the earth to be at rest and at the centre of the universe. The arguments of Copernicus and Galileo disfavouring the earth as the centre of the universe were not accepted because their critics pointed out that if earth really revolved around the Sun, one must observe the parallactic motion of stars.\u00a0 At that time such an observation was not possible because of lack of suitable instruments.\u00a0 However, the successful observation of stellar parallax in 1853 by Friedrich Bessel<\/strong>, vindicated Copernicus and Galileo and dealt a severe blow to the geocentric universe.<\/p>\n

           <\/p>\n

          For distant objects, in the Galaxy and outside, there are several indirect methods which we shall describe in due course. However, all these methods depend critically on the distances of the\u00a0nearby stars measured by the method of annual parallaxes. In that lies the importance of the accurate measurement of parallaxes by satellites such as Hipparcos and Gaia.<\/span><\/p>\n

           <\/p>\n

          3.3.\u00a0 Complexities of Parallax Measurement<\/span><\/strong><\/p>\n

           <\/p>\n

          The determination of annual parallaxes is a difficult task and only a few observatories specialize in this work. We must remember that only if the star is in the plane of the ecliptic, its parallactic path is a straight line; in all other directions of the star the parallactic path is generally an ellipse, the ellipse being reduced to a circle when the star is in the direction of the ecliptic pole (Fig. 6.3).<\/span><\/p>\n<\/div>\n

          \n

          \"\"<\/p>\n

          Fig. 6.3. Parallactic motion of a star is generally an ellipse. If the star is on a normal to the ecliptic plane, its parallactic motion is a circle. If the star is in the plane of the ecliptic, then its parallactic path is a straight line.<\/p>\n

           <\/p>\n

          It means that the parallactic path of a star is never a simple curve.<\/span><\/p>\n

           <\/p>\n

          Moreover, the parallactic path is always distorted by the star\u2019s own motion relative to the Sun. The change in a star\u2019s direction is called its <\/span>proper motion <\/strong>(see below).\u00a0 Thus, the motion of a star is really a complicated curve.\u00a0 From this complicated curve, the annual parallactic motion of the star has to be extracted.\u00a0 This is a difficult task which requires several photographs of the concerned region of the sky taken at different times of the year.<\/span><\/p>\n

           <\/p>\n

          Table 6.1 lists nearest stars, their equatorial coordinates, parallaxes and distances.<\/span><\/p>\n

           <\/p>\n

          Table 6.1.\u00a0 Parallaxes and Distances of the nearest 20 stars<\/strong><\/p>\n<\/div>\n

          \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
          No.<\/strong><\/td>\nName of the Star<\/strong><\/td>\nRight Ascension<\/strong><\/td>\nDeclination<\/strong><\/td>\nParallax (milliarcsec)<\/strong><\/td>\nDistance (pc)<\/strong><\/td>\n<\/tr>\n
          1<\/td>\nProxima Centauri<\/td>\n14h 29m 43.0s<\/td>\n\u221262\u00b0 40\u2032 46\u2033<\/td>\n768.87<\/td>\n1.30<\/td>\n<\/tr>\n
          2<\/td>\n\u03b1 Centauri A<\/td>\n14h 39m 36.5s<\/td>\n\u221260\u00b0 50\u2032 02\u2033<\/td>\n747.23<\/td>\n1.33<\/td>\n<\/tr>\n
          3<\/td>\n\u03b1 Centauri B<\/td>\n14h 39m 35.1s<\/td>\n\u221260\u00b0 50\u2032 14\u2033<\/td>\n747.23<\/td>\n1.33<\/td>\n<\/tr>\n
          4<\/td>\nBarnard\u2019s Star<\/td>\n17h 57m 48.5s<\/td>\n+04\u00b0 41\u2032 36\u2033<\/td>\n546.98<\/td>\n1.83<\/td>\n<\/tr>\n
          5<\/td>\nWolf 359<\/td>\n10h 56m 29.2s<\/td>\n+07\u00b0 00\u2032 53\u2033<\/td>\n419.10<\/td>\n2.38<\/td>\n<\/tr>\n
          6<\/td>\nLalande 21185<\/td>\n11h 03m 20.2s<\/td>\n+35\u00b0 58\u2032 12\u2033<\/td>\n393.42<\/td>\n2.54<\/td>\n<\/tr>\n
          7<\/td>\nSirius A<\/td>\n06h 45m 08.9s<\/td>\n\u221216\u00b0 42\u2032 58\u2033<\/td>\n380.02<\/td>\n2.63<\/td>\n<\/tr>\n
          8<\/td>\nSirius B<\/td>\n06h 45m 08.9s<\/td>\n\u221216\u00b0 42\u2032 58\u2033<\/td>\n380.02<\/td>\n2.63<\/td>\n<\/tr>\n
          9<\/td>\nLuyten 726-8 A<\/td>\n01h 39m 01.3s<\/td>\n\u221217\u00b0 57\u2032 01\u2033<\/td>\n373.70<\/td>\n2.67<\/td>\n<\/tr>\n
          10<\/td>\nLuyten 726-8 B<\/td>\n01h 39m 01.3s<\/td>\n\u221217\u00b0 57\u2032 01\u2033<\/td>\n373.70<\/td>\n2.67<\/td>\n<\/tr>\n
          11<\/td>\nRoss 154<\/td>\n18h 49m 49.4s<\/td>\n\u221223\u00b0 50\u2032 10\u2033<\/td>\n336.90<\/td>\n2.97<\/td>\n<\/tr>\n
          12<\/td>\nRoss 248<\/td>\n23h 41m 54.7s<\/td>\n+44\u00b0 10\u2032 30\u2033<\/td>\n316.00<\/td>\n3.16<\/td>\n<\/tr>\n
          13<\/td>\nEpsilon Eridani<\/td>\n03h 32m 55.8s<\/td>\n\u221209\u00b0 27\u2032 30\u2033<\/td>\n309.99<\/td>\n3.23<\/td>\n<\/tr>\n
          14<\/td>\nLacaille 9352<\/td>\n23h 05m 52.0s<\/td>\n\u221235\u00b0 51\u2032 11\u2033<\/td>\n303.64<\/td>\n3.29<\/td>\n<\/tr>\n
          15<\/td>\nRoss 128<\/td>\n11h 47m 44.4s<\/td>\n+00\u00b0 48\u2032 16\u2033<\/td>\n298.72<\/td>\n3.35<\/td>\n<\/tr>\n
          16<\/td>\nEZ Aquarii A<\/td>\n22h 38m 33.4s<\/td>\n\u221215\u00b0 17\u2032 57\u2033<\/td>\n289.50<\/td>\n3.45<\/td>\n<\/tr>\n
          17<\/td>\nEZ Aquarii B<\/td>\n22h 38m 33.4s<\/td>\n\u221215\u00b0 17\u2032 57\u2033<\/td>\n289.50<\/td>\n3.45<\/td>\n<\/tr>\n
          18<\/td>\nEZ Aquarii C<\/td>\n22h 38m 33.4s<\/td>\n\u221215\u00b0 17\u2032 57\u2033<\/td>\n289.50<\/td>\n3.45<\/td>\n<\/tr>\n
          19<\/td>\nProcyon A<\/td>\n07h 39m 18.1s<\/td>\n+05\u00b0 13\u2032 30\u2033<\/td>\n286.05<\/td>\n3.50<\/td>\n<\/tr>\n
          20<\/td>\nProcyon B<\/td>\n07h 39m 18.1s<\/td>\n+05\u00b0 13\u2032 30\u2033<\/td>\n286.05<\/td>\n3.50<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n
          \n

           <\/p>\n

          Notes for Table 6.1:<\/p>\n

          1. Equatorial coordinates are for the epoch J2000.0<\/p>\n

          2. Many of these stars belong to double or triple systems.<\/p>\n

          3. Based on the list of nearest stars and brown dwarfs given in:<\/p>\n

          https:\/\/en.wikipedia.org\/wiki\/List_of_nearest_stars_and_brown_dwarfs<\/p>\n

           <\/p>\n

          4. Proper Motion of Stars<\/strong><\/p>\n

           <\/p>\n

          Like everything else in the universe, the stars are also always in motion. \u00a0They appear fixed because they are very far off and the change in their direction is so small that it cannot be appreciated by the naked eye. The motion of a star in the direction perpendicular to the line of sight is called its proper motion <\/strong>(Fig. 6.4).\u00a0 It is denoted by ? and is measured in arc-second per year (\u02dd\/yr).<\/p>\n

          \"\"<\/p>\n

          Fig. 6.4. The line of sight from the observer O to the star is OS.\u00a0 The motion perpendicular to the line of sight, measured in angle, is the proper motion of the star.<\/p>\n

           <\/p>\n

          4.1.\u00a0 Proper Motion in Declination and Right Ascension<\/span><\/strong><\/p>\n

           <\/p>\n

          It is obvious that the motion of a star will bring about a change in its coordinates. In Fig.\u00a0<\/span>6.5, the star moves from position X to position X’. \u00a0It now lies on a different meridian and\u00a0<\/span>not at the same arc length from the equator as before. \u00a0Its new declination is ?+ ??, the\u00a0<\/span>change in its declination being ??, often denoted by ??. Similarly the change in its right\u00a0<\/span>ascension is ??denoted by ??. \u00a0This nomenclature implies that these changes have been\u00a0<\/span>brought about by the proper motion of the star. It is usual to express the proper motion in\u00a0<\/span>terms of ??and ??, the changes in the coordinates of the star:<\/span><\/p>\n<\/div>\n

          \n

          \"\"<\/p>\n

          Fig. 6.5.\u00a0 Star X moves to X\u00b4 in one year due to proper motion ?.\u00a0 Its right ascension changes by ??= ??while its declination changes by ??= ??.<\/span><\/p>\n

           <\/p>\n

          ?= \u221a\u00af??2<\/sup> + ??2<\/sup> .\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\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 (6.2)<\/span><\/p>\n<\/div>\n

          \n

           <\/p>\n

          4.2. Determination of Proper Motion<\/strong><\/p>\n

           <\/p>\n

          The proper motion of a star is cumulative.\u00a0 Therefore, it can be determined by comparing the photographs of the region of the sky containing the star with the background of distant objects which do not change their positions. Since proper motion is a small quantity, the photographs are separated by decades. \u00a0It is important that the position measurements at two epochs be referred to the same equator and equinoxes so that changes in coordinates caused by the earth\u2019s precession are avoided.<\/p>\n

          \"\"<\/p>\n

          Fig. 6.6. (a) and (b) are the photograph several decades apart of the region of the sky which contains the star whose proper motion is to be<\/p>\n

           <\/p>\n

          Figures 6.6 (a) and (b) show, schematically, two photographs taken a few decades apart.<\/span><\/p>\n

           <\/p>\n

          Superposition of these photographs shows that the star X has moved to X\u00b4.\u00a0 Knowing the displacement XX\u00b4 (after reduction by the appropriate photographic plate factors) and the lapse of time between the two photographs, ?can be calculated. It is obvious that the accuracy in the measurement of ?can be increased by selecting photographs separated by a long interval; however, much of this advantage is offset by the errors in the position measurement at earlier epochs. Measurement of the proper motion of a star is not as simple as it might appear; it is obtained by analyzing the complicated path that the star describes in the sky due to the combination of the proper motion and the parallactic motion. Hipparcos has also measured the proper motion of a large number of stars.<\/span><\/p>\n

          \"\"<\/p>\n

          Fig. 6.7.\u00a0 Hipparcos also measured the proper motion of stars. The figure is taken from the introduction to <\/span>The Millennium Star Atlas, and was produced by Dennis di Cicco for Sky Publishing Corporation. Note that proper motion of the star stands out clearly as a movement of the star against the background stars with time. Dennis di Cicco’s observations were so accurate that the effect of the parallax (the distance) of the stars is also evident. It is seen as the “wavy” motion of the star (the individual observations are shown as black circles with the relevant observation date) about its linear motion (shown as the straight line dissecting the figure); this wavy motion has a period of one year, corresponding to the Earth’s orbital motion around the Sun. (<\/span>http:\/\/www.cosmos.esa.int\/web\/hipparcos\/high-proper-motion-stars)<\/span><\/p>\n

           <\/p>\n

          The largest proper motion known is that of the Barnard\u2019s star, which is 10.3\u02dd yr-1. Among the stars visible with naked eye, the largest proper motion belongs to 61 Cygni (5.22\u00b4 yr-1). For pictures of the Barnard\u2019s star with the stellar background about 50 years apart, and animation of the same pictures showing movement of the star against the same background, visit <\/span>http:\/\/cseligman.com\/text\/stars\/propermotion.htm<\/p>\n<\/div>\n

          \n

           <\/p>\n

          5. Summary<\/span><\/strong><\/p>\n<\/div>\n

            \n
          • Stellar distances are essential for fixing physical properties of stars, such as their luminosities.<\/li>\n
          • Distances of nearby stars are determined by the method of parallax.<\/li>\n
          • Observations are made from the two ends of the earth\u2019s orbit round the Sun.<\/li>\n
          • Half the angular displacement in direction observed from these points is called the annual parallax of a star.<\/li>\n
          • If the annual parallax is one arc-second, the distance is said to be one parsec.<\/li>\n
          • Generally, the reciprocal of parallax in arc-second is the distance in parsec.<\/li>\n
          • The motion of a star perpendicular to its line of sight is called the proper motion of the star.<\/li>\n
          • Proper motion changes the equatorial coordinates of the star. Therefore, proper motion is given in terms of its component changes in right ascension and declination.<\/li>\n
          • Proper motion is a small quantity. It is measured in arc-second\/year, or milli arc second\/year.<\/li>\n<\/ul>\n

             <\/p>\n

            References<\/strong><\/p>\n

              \n
            • https:\/\/en.wikipedia.org\/wiki\/Parallax<\/li>\n
            • http:\/\/hyperphysics.phy-astr.gsu.edu\/hbase\/astro\/para.html<\/li>\n
            • https:\/\/en.wikipedia.org\/wiki\/Gaia_%28spacecraft%29<\/li>\n
            • http:\/\/www.ast.cam.ac.uk\/~mjp\/calc_parallax.html<\/li>\n
            • http:\/\/www.scientus.org\/Copernicus-Stellar-Parallax.html<\/li>\n
            • http:\/\/physics.unm.edu\/101lab\/lab4\/lab4_C.html(For animation)<\/li>\n<\/ul>\n","protected":false},"author":4,"menu_order":5,"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-84","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\/84","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":4,"href":"https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-json\/pressbooks\/v2\/chapters\/84\/revisions"}],"predecessor-version":[{"id":468,"href":"https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-json\/pressbooks\/v2\/chapters\/84\/revisions\/468"}],"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\/84\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-json\/wp\/v2\/media?parent=84"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-json\/pressbooks\/v2\/chapter-type?post=84"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-json\/wp\/v2\/contributor?post=84"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-json\/wp\/v2\/license?post=84"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}