{"id":66,"date":"2018-11-16T07:25:52","date_gmt":"2018-11-16T07:25:52","guid":{"rendered":"http:\/\/phyp06.epgpbooks.inflibnet.ac.in\/?post_type=chapter&p=66"},"modified":"2018-11-16T09:43:26","modified_gmt":"2018-11-16T09:43:26","slug":"measurement-of-time","status":"publish","type":"chapter","link":"https:\/\/ebooks.inflibnet.ac.in\/phyp06\/chapter\/measurement-of-time\/","title":{"rendered":"Measurement of Time"},"content":{"raw":"
\r\n\r\n \r\n\r\n1.\u00a0 <\/strong>Learning Outcomes<\/strong>\r\n\r\n \r\n\r\nAfter studying this module, you should be able to\r\n
    \r\n \t
  • define time<\/li>\r\n \t
  • grasp the fact that time measurement is tied to the rotation of the earth<\/span><\/li>\r\n \t
  • appreciate that the solar day is not a constant quantity<\/span><\/li>\r\n \t
  • explain what factors affect the length of the solar day<\/span><\/li>\r\n \t
  • understand the need to define a fictitious object called the mean Sun from which mean solar time is derived<\/span><\/li>\r\n \t
  • deduce from the Equation of Time the difference between the mean solar time and the apparent solar time<\/span><\/li>\r\n \t
  • define sidereal time and calculate the difference between a mean solar day and a sidereal day<\/span><\/li>\r\n \t
  • reason out the need to define Universal Time and Standard and Zone Times of countries<\/span><\/li>\r\n<\/ul>\r\n<\/div>\r\n
    \r\n\r\n \r\n\r\n2.\u00a0\u00a0<\/strong>Introduction<\/strong>\r\n\r\n \r\n

    The concept of time is basic to the whole of science, indeed, to whole of knowledge. We intuitively know what time and its passage mean. For a formal definition, consider a point moving relative to a frame of reference.\u00a0 With each position of the point we can associate a number, which defines time.\u00a0 The passage of time is measured in terms of the motion of the point relative to the frame of reference.<\/p>\r\n \r\n

    The motion of the earth relative to the Sun and stars forms the basis of the various systems of measurement of time.\u00a0 It must be remembered that all our watches, clocks and other time measuring devices are tied to the daily and annual motion of the earth relative to the Sun and stars.\u00a0 We discuss below how a measure of time is derived from these motions of the earth.<\/p>\r\n \r\n\r\n3.\u00a0 Solar Time<\/strong>\r\n\r\n \r\n

    In the last module, we discussed the diurnal motion of stars caused by the rotation of the earth on<\/p>\r\n\"\"\r\n\r\n<\/div>\r\n

    \r\n\r\n \r\n

    Fig. 5.1. In addition to its annual motion in the plane of the ecliptic, the Sun has also diurnal motion.\u00a0 Its diurnal circles on 21 June and 23 December are shown.<\/span><\/p>\r\n \r\n

    its axis. Like other stars, the Sun, too, has diurnal motion. Because of the annual motion of the Sun, however, its diurnal circle changes every day. \u00a0Fig. 5.1 shows the diurnal circles of the Sun on the days of summer and winter solstices. \u00a0For other days, the diurnal circles of the Sun are located in between these two extremes.\u00a0 From 21 June to 23 December, the diurnal circles of the Sun migrate southwards, while from 23 December to 21 June, the diurnal circle drift northwards along with the position of the Sun on the ecliptic.\u00a0 For six months, its diurnal circles are north of the equator and during the remaining six months, the diurnal circles are south of the equator. \u00a0In northern hemisphere, the Sun is high in the sky in the summer and low in the sky in the winter (Fig. 5.2); it is summer there.\u00a0 It is exactly the opposite in the southern hemisphere. Fig. 5.3 depicts the causations of seasons on the earth.<\/span><\/p>\r\n\r\n<\/div>\r\n

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

    Fig. 5.2.\u00a0 In the northern hemisphere the Sun is high in June and low in December.<\/span><\/p>\r\n \r\n

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

    \r\n\r\n \r\n

    Fig. 5.3. Cause of the seasons on the earth is the motion of the Sun in an orbit inclined to the equator, or equivalently, the inclination of the earth\u2019s axis of rotation to the plane of the ecliptic.<\/p>\r\n \r\n

    During its daily motion, when the Sun is exactly overhead, it is local noon<\/strong>. At that time the Sun is on the local meridian. Recall that the local meridian or the observer\u2019s meridian includes the zenith along with the celestial poles. \u00a0Since zenith is the point directly overhead the observer, the Sun appears overhead when on the local meridian. The interval between two successive passages of the Sun over the local meridian of a place is called a solar day<\/strong>.\u00a0 However, the length of the solar day is not constant throughout the year. Therefore, the solar day and its smaller divisions are not suited for accurate time measurement.<\/p>\r\n \r\n\r\n3.1.\u00a0 Reasons for Variation of Solar Day \u2013 Effect of Elliptic Orbit and Obliquity of Ecliptic<\/strong>\r\n\r\n \r\n\r\nBut why is the solar day not constant? \u00a0There are two reasons:\r\n\r\n \r\n

    (1) The speed of the earth is not uniform in its elliptic orbit round the Sun (or equivalently, the speed of the apparent motion of the Sun round the earth). Kepler\u2019s laws say that the\u00a0<\/strong>earth revolves round the Sun in an elliptic orbit with the Sun at one of the foci, and the area carved out by the line joining the Sun and the earth per unit time is constant.\u00a0 <\/strong>The consequence is that the speed of the earth in its orbit is not constant: it is maximum at <\/span>perihelion <\/strong>(the point nearest to the Sun) and minimum at <\/span>aphelion <\/strong>(point farthest from the Sun).\u00a0 This implies that the difference between the solar day and the more constant <\/span>sidereal day <\/strong>(more about it later) keeps varying throughout the year<\/span>.<\/strong><\/p>\r\n\r\n<\/div>\r\n

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

    Fig. 5.4. According to the second law of Kepler the area bounded by green lines is equal to the area bounded by red lines.\u00a0 These areas are carved out in unit time. \u00a0The law implies that the orbital speed of the earth is higher at perihelion than at aphelion.<\/p>\r\n \r\n

    (2) <\/strong>The plane of the Sun\u2019s annual motion, the ecliptic, is inclined to the equatorial plane, so that the hour angle of the Sun does not advance at a uniform rate. \u00a0Remember that the hour angle of any object, such as the Sun, is measured from the observer\u2019s meridian along the equator.\u00a0 Even if the Sun were to move uniformly along the ecliptic, the obliquity of the ecliptic will cause equal intervals along the ecliptic to have unequal projections on the equator (Fig. 5.5). T<\/em>herefore, the Sun does not progress uniformly along the equator.\u00a0 <\/em>This causes a variation in the length of the day. The effect of obliquity is more pronounced near the solstices than near the equinoxes.<\/p>\r\n\r\n<\/div>\r\n\"\"\r\n

    \r\n\r\n \r\n

    Fig. 5.5.\u00a0 Segments of equal length along the ecliptic do not have equal projections on the equator. \u00a0So, while the Sun progresses uniformly along the ecliptic, it does not progress along the equator.<\/p>\r\n \r\n\r\n3.2.\u00a0 Mean Solar Time<\/strong>\r\n\r\n \r\n

    To eliminate these two effects, a fictitious body, the mean Sun<\/strong>, is defined whose apparent motion round the earth takes place (i) at a constant speed (circular path) <\/strong>and (ii) the plane of whose orbit coincides with the plane of the equator <\/strong>(Fig. 5.6).\u00a0 The time derived from the mean Sun is called the mean solar time<\/strong>, as against the true, or apparent, solar time, <\/strong>derived from the actual Sun.\u00a0 Our clocks and watches run on mean solar time.\u00a0 To bring the time given by a sundial (apparent solar time) must be corrected to bring it in line with the time on our watches, that is, the mean solar time.\u00a0 The difference between the mean solar time and the apparent solar time is called the equation of time.<\/strong><\/p>\r\n\r\n<\/div>\r\n

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

    Fig. 5.6. While the true Sun orbits in the ecliptic plane the fictitious mean Sun moves in the plane of the equator in a circular orbit. Both take the same time in completing one orbit.<\/p>\r\n \r\n\r\n3.3.\u00a0 Equation of Time<\/strong>\r\n\r\n \r\n\r\nEquation of time is given by\r\n\r\n?= Hour Angle of actual Sun \u2212 Hour Angle of mean Sun\r\n\r\n \r\n\r\n= HA\u0298 \u2212 HAMS.\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 (5.1)\r\n\r\n \r\n\r\nThe Equation of Time is shown in Fig. 5.7.\r\n\r\n\"\"\r\n\r\n<\/div>\r\n

    \r\n\r\n \r\n

    Fig. 5.7.\u00a0 The equation of time is shown by the red line.\u00a0 The two factors contributing to it, eccentricity of the earth\u2019s orbit and the obliquity of the ecliptic plane to the equatorial plane,\u00a0<\/span>are shown by broken lines. <\/span>\u03d2 <\/strong>and ? are the vernal and autumn equinoxes. Points A and P\u00a0<\/span>denote aphelion (the point where the Sun \u2013 earth distance is longest) and perihelion\u00a0<\/span>(the point where the Sun \u2013 earth distance is shortest). The red line shows the number\u00a0<\/span>of minutes by which the apparent solar time (sundial time) differs from the mean solar\u00a0<\/span>time. On 1 Jan, for example, the apparent solar time is 4 min behind the mean solar time.<\/span><\/p>\r\n \r\n

    That means that 4 min must be added to the time given by the sundial to bring it in ?\u00a0<\/span>agreement with the mean solar time. (Based on the figure from Wikipedia)<\/span><\/p>\r\n \r\n

    4.\u00a0 Sidereal Time<\/span><\/strong><\/p>\r\n \r\n

    Sidereal time is the time measured with respect to the distant, and therefore apparently fixed, stars. We have seen above that the stars appear to move round the earth in diurnal circles parallel to the equator.\u00a0 Therefore, they appear on the observer\u2019s meridian at some instant and\u00a0<\/span>then progress steadily, increasing steadily their hour angle.\u00a0 Instead of any star, we use the Vernal equinox as the reference point.\u00a0 So, <\/span>the sidereal time is defined as the hour angle of the Vernal equinox.<\/strong><\/p>\r\n \r\n

    It must be noted, however, that vernal equinox itself is not a fixed point in the sky. It undergoes drifts due to <\/span>precessional <\/strong>and <\/span>nutational <\/strong>motions of the earth.\u00a0 Since Vernal equinox is affected<\/span><\/p>\r\n \r\n

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

    \r\n\r\n \r\n

    Fig. 5.7. The earth precesses around a line perpendicular to the ecliptic.<\/p>\r\n \r\n

    this effect is also called the precession of equinoxes<\/strong>.\u00a0 You must have learnt precession and nutation as part of your course in classical mechanics.\u00a0 Since the Vernal equinox drifts\u00a0irregularly, we define a <\/span>mean Vernal equinox, <\/strong>much like the mean Sun<\/span>, <\/strong>which is supposed to move uniformly along the celestial equator.<\/span><\/p>\r\n\r\n<\/div>\r\n

    \r\n\r\n \r\n\r\n4.1.\u00a0 Precession and Nutation of the Earth<\/strong>\r\n\r\n \r\n

    We know that the earth is not a perfect sphere.\u00a0 It has a slight bulge at the equator. This makes it behave like a spinning top.\u00a0 The gravitation field of the Sun and the moon and, to much lesser extent, the planets causes the earth to precess<\/strong>. The axis of the earth charts out a cone about a line perpendicular to the plane of the ecliptic (Fig. 5.7). The direction of the precession is<\/p>\r\n\r\n<\/div>\r\n

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

    Fig. 5.8. Due to precession, Polaris will no longer be our pole star after a few thousand years. \u00a0Around 14000 CE star Vega will replace Polaris as our pole star. About 4500 years ago, the time the Egyptian pyramids were built, Thuban was our pole star.<\/p>\r\n \r\n

    clockwise if we look down on the earth from the north. The time period of precession is about 26000 years. This, so-called <\/span>astronomical precession<\/strong>, has very important consequences.<\/span><\/p>\r\n \r\n

    (i) The axis of the earth does not point in a fixed direction in space but wanders among stars (Fig. 5.8).\u00a0 At present the axis points towards the star Polaris, which at present is our Pole Star.\u00a0 In a few thousand years, Polaris will no longer be our Pole Star.<\/span><\/p>\r\n \r\n

    (ii) The Vernal equinox, or the First Point of Aires, shifts continuously eastwards. \u00a0That is why precession of the earth is also called the <\/span>precession of equinoxes<\/strong>. While precessing, the earth also undergoes a wobbling motion with a period of about 19 years, called <\/span>nutation.<\/strong><\/p>\r\n\r\n<\/div>\r\n

    \r\n\r\n \r\n

    We know that the gravitational forces of the sun and the moon vary across the earth, giving rise to tides. Since the sources of these tidal forces are changing their location continuously with respect to each other as well as with respect to the earth, they produce a wobbling motion in the axis of the earth. This wobbling motion has multiple periods, the principal period being 18.6 years. This motion is called nutation.<\/p>\r\n \r\n

    (iii) The continuous drift of the Vernal equinox due to precession causes a slow apparent drift in the zodiacal constellations.\u00a0 Two thousand years ago, the Vernal equinox occurred when the Sun entered the constellation of Aires, that is why Vernal equinox is also called the First Point of Aries.\u00a0 At present, however, the vernal equinox occurs at the time the Sun is in the constellation Pisces (Fig. 5.9).\u00a0 The result is that the zodiacal signs are no longer in step with the constellations from which they derive their names.<\/p>\r\n\"\"\r\n

    Fig. 5.9.\u00a0 Vernal equinox at present occurs when the Sun is in the constellation Pisces.\u00a0 Two thousand years ago this event occurred when the Sun was in Aries.\u00a0 This change is the result of precession of equinoxes.<\/span><\/p>\r\n\r\n<\/div>\r\n

    \r\n\r\n \r\n\r\n4.2.\u00a0 Drift of Vernal Equinox and its Effect on Equatorial Coordinates<\/strong>\r\n\r\n \r\n

    Since the sidereal time is the hour angle of the Vernal equinox, one effect of the drift of Vernal equinox is that sidereal days and years are not uniform in length. To overcome this non- uniformity, we have to define a mean Vernal equinox.\u00a0 The rate of eastward drift of the mean equinox is about 50 arc-sec per year. Actually, this number is the first term in a series in time which contains higher order terms.\u00a0 The drift of Vernal equinox causes the tropical year <\/strong>to be slightly shorter than sidereal year.<\/p>\r\n \r\n

    Tropical year is the same as solar year, which is the average interval of time between the two successive returns of the Sun to the same point in the sky, for example the First Point of Aires.<\/p>\r\n \r\n

    The drift of Vernal equinox also results in a continuous change in the equatorial coordinates of astronomical objects<\/strong>. Recall that the equatorial system has the Vernal equinox\u00a0as the reference point.\u00a0 So, while stating the right ascension and declination of objects we have also to state the relevant epoch.<\/span><\/p>\r\n\"\"\r\n

    Fig. 5.10. Drift of Vernal equinox due to the precession of the earth\u00a0<\/span>causing changes in the equatorial coordinates.<\/span><\/p>\r\n\r\n<\/div>\r\n

    \r\n\r\n \r\n\r\n4.3. Comparison of Sidereal Day and Solar Day<\/strong>\r\n\r\n \r\n

    The time interval between the two successive passages of the mean Vernal equinox over the observer\u2019s meridian defines a sidereal day<\/strong>. A sidereal day is slightly shorter than the solar day, as the following argument shows.<\/p>\r\n \r\n

    Consider a point X on the earth\u2019s surface (Fig. 5.11) at the time of the Vernal equinox.\u00a0 In situation ?\ufffd both the Sun and the Vernal equinox (or the distant stars) are overhead. \u00a0After one rotation of the earth on its axis, at ?\ufffd, the Vernal equinox, being very distant, is again overhead of X. So, as reckoned by distant stars, one day has been completed. This day is called the\u00a0sidereal day. <\/strong>However, <\/strong>the Sun will be overhead of X, and one solar day <\/strong>will be completed,\u00a0only when X moves to Y, that is, the earth undergoes an additional rotation through the angle\u00a0<\/span>XO\u00b4Y. \u00a0<\/span>This means that the solar day is longer than the sidereal day by the time required to cover the angle XO<\/strong>\u00b4<\/span>Y<\/strong>.\u00a0\u00a0 Since the earth rotates through 360\u00b0 in a day, angle XO\u00b4Y (covered in<\/span><\/p>\r\n\r\n<\/div>\r\n

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

    one day) is roughly one degree and the time taken by the earth to cover this angle is about 4 minutes. Thus, the sidereal day is shorter than the solar day by 4 minutes, or to be more exact, by 3 min 56 s.<\/strong><\/p>\r\n \r\n\r\n5.\u00a0 Local Mean Time<\/strong>\r\n\r\n \r\n

    The sidereal time is not very convenient for everyday life.\u00a0 The sidereal noon, for example, occurs at night for part of the year.\u00a0 A different system of time, the Universal Time <\/strong>(UT<\/strong>), has therefore been created for civil purposes.\u00a0 The UT is based on the mean Sun.\u00a0 We have already stated that the difference between the apparent solar time and the mean solar time, that is, the Equation of Time<\/strong>, can be precisely calculated, and hence the hour angle of the mean Sun\u00a0<\/strong>(<\/span>HAMS<\/strong>) can be worked out.\u00a0 When the mean Sun is on the local meridian of a place, it is <\/span>mean noon <\/strong>at that place.\u00a0 The time of the day is, however, reckoned from the midnight, and so the time of the day, styled as <\/span>local mean time <\/strong>(<\/span>LMT<\/strong>), is given by the following equation:<\/span><\/p>\r\n\r\n<\/div>\r\n

    \r\n\r\n \r\n\r\nLMT = HAMS + 12 h.\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 (5.2)\r\n\r\n \r\n\r\n6.\u00a0 The Universal Time<\/strong>\r\n\r\n \r\n\r\nTo correlate the local time at various places on the earth, it has been agreed to adopt the Greenwich local time as the reference time. \u00a0The universal time (UT) is then defined as follows:\r\n\r\n \r\n\r\nUT = HAMS at Greenwich + 12 h.\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 (5.3)\r\n\r\n \r\n\r\nThe relation then between the local mean time at any place and the universal time is\r\n\r\n \r\n\r\nUT = LMT \u2213 longitude of the place in hours,\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 (5.4)\r\n\r\n \r\n

    the minus sign being for places east of Greenwich and the plus sign for places west of Greenwich. For example, at a place at longitude 15\u00b0 East, the local mean time is ahead of the universal time by one hour. By the same token, the local mean time at a place at 15\u00b0 West is one hour behind the universal time. In order to avoid continuous setting of clocks as people travel from one place to another, Standard Times and Zone Times, spanning whole countries or their major parts, have been adopted. The Indian standard Time (IST) is the local mean time of a place (near Allahabad) whose longitude is 82.5\u00b0 East.<\/p>\r\n \r\n

    7.\u00a0 Summary<\/span><\/strong><\/p>\r\n\r\n<\/div>\r\n

      \r\n \t
    • Interval between two successive passages of the Sun over the observer\u2019s meridian defines a solar day.<\/li>\r\n \t
    • Length of the solar day is not constant.<\/li>\r\n \t
    • The two reasons are (i) variation in the speed of the earth in its elliptic path round the Sun, and (ii) obliquity of the plane of the ecliptic to the celestial equator.<\/li>\r\n \t
    • To overcome these problems, a mean Sun is imagined, travelling along the equator at a uniform speed.<\/li>\r\n \t
    • Mean solar time is defined by the mean Sun. Our clocks and watches keep the mean solar time.<\/li>\r\n \t
    • The difference between the mean solar time and the apparent solar time is given by the Equation of Time.<\/li>\r\n \t
    • Sidereal time is defined as the interval between two successive passages of the vernal equinox over the observer\u2019s meridian.<\/li>\r\n \t
    • Vernal equinox is not a fixed point in the sky; it drifts irregularly due to precession and nutation of the earth.<\/li>\r\n \t
    • Precession of the earth is caused by the inclination of its axis of rotation to the plane of the ecliptic.<\/li>\r\n \t
    • Precession has three major effects: (i) it makes the sidereal year shorter than the tropical year, (ii) it changes the equatorial coordinates of celestial objects, and (iii) it causes a slow apparent drift in the zodiacal constellations.<\/li>\r\n \t
    • Sidereal day is shorter than the solar day by four minutes.<\/li>\r\n \t
    • Every day a star rises four minutes earlier than the previous day. Similarly, a star sets four minutes earlier than the previous day.<\/li>\r\n \t
    • Early rising and setting of stars than the previous day causes the sky to change slowly every night. The effect is apparent over a time of a few weeks.<\/li>\r\n \t
    • Local Mean Time (LMT) is defined as hour angle of the mean Sun + 12 hours.<\/li>\r\n \t
    • LMT of various places is coordinated by the LMT of Greenwich, which is called the Universal Time.<\/li>\r\n \t
    • Standard and Zone times are defined with respect to the Universal Time.<\/li>\r\n<\/ul>\r\n \r\n\r\nThe material covered in this module and related material can be found in the following websites:\r\n
        \r\n \t
      • https:\/\/en.wikipedia.org\/wiki\/Sidereal_time<\/li>\r\n \t
      • http:\/\/www.astro.cornell.edu\/academics\/courses\/astro201\/sidereal.htm<\/li>\r\n \t
      • http:\/\/astro.wsu.edu\/worthey\/astro\/html\/lec-precession.html<\/li>\r\n \t
      • http:\/\/davidpratt.info\/pole1.htm<\/li>\r\n \t
      • http:\/\/www.heavens-above.com\/SkyChart.aspx?lat=0&lng=0&loc=Unspecified&alt=0&tz=UCT<\/li>\r\n \t
      • http:\/\/www.thetropicalevents.com\/<\/li>\r\n \t
      • https:\/\/en.wikipedia.org\/wiki\/Axial_precession<\/li>\r\n \t
      • http:\/\/www.universetoday.com\/113328\/having-fun-with-the-equation-of-time\/<\/li>\r\n \t
      • https:\/\/en.wikipedia.org\/wiki\/Equation_of_time<\/li>\r\n \t
      • http:\/\/www.intmath.com\/blog\/mathematics\/the-equation-of-time-5039<\/li>\r\n<\/ul>\r\n \r\n\r\nMost books on classical mechanics discuss phenomena of precession in the context of tops. It is directly applicable to the earth.","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
        • define time<\/li>\n
        • grasp the fact that time measurement is tied to the rotation of the earth<\/span><\/li>\n
        • appreciate that the solar day is not a constant quantity<\/span><\/li>\n
        • explain what factors affect the length of the solar day<\/span><\/li>\n
        • understand the need to define a fictitious object called the mean Sun from which mean solar time is derived<\/span><\/li>\n
        • deduce from the Equation of Time the difference between the mean solar time and the apparent solar time<\/span><\/li>\n
        • define sidereal time and calculate the difference between a mean solar day and a sidereal day<\/span><\/li>\n
        • reason out the need to define Universal Time and Standard and Zone Times of countries<\/span><\/li>\n<\/ul>\n<\/div>\n
          \n

           <\/p>\n

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

           <\/p>\n

          The concept of time is basic to the whole of science, indeed, to whole of knowledge. We intuitively know what time and its passage mean. For a formal definition, consider a point moving relative to a frame of reference.\u00a0 With each position of the point we can associate a number, which defines time.\u00a0 The passage of time is measured in terms of the motion of the point relative to the frame of reference.<\/p>\n

           <\/p>\n

          The motion of the earth relative to the Sun and stars forms the basis of the various systems of measurement of time.\u00a0 It must be remembered that all our watches, clocks and other time measuring devices are tied to the daily and annual motion of the earth relative to the Sun and stars.\u00a0 We discuss below how a measure of time is derived from these motions of the earth.<\/p>\n

           <\/p>\n

          3.\u00a0 Solar Time<\/strong><\/p>\n

           <\/p>\n

          In the last module, we discussed the diurnal motion of stars caused by the rotation of the earth on<\/p>\n

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

          \n

           <\/p>\n

          Fig. 5.1. In addition to its annual motion in the plane of the ecliptic, the Sun has also diurnal motion.\u00a0 Its diurnal circles on 21 June and 23 December are shown.<\/span><\/p>\n

           <\/p>\n

          its axis. Like other stars, the Sun, too, has diurnal motion. Because of the annual motion of the Sun, however, its diurnal circle changes every day. \u00a0Fig. 5.1 shows the diurnal circles of the Sun on the days of summer and winter solstices. \u00a0For other days, the diurnal circles of the Sun are located in between these two extremes.\u00a0 From 21 June to 23 December, the diurnal circles of the Sun migrate southwards, while from 23 December to 21 June, the diurnal circle drift northwards along with the position of the Sun on the ecliptic.\u00a0 For six months, its diurnal circles are north of the equator and during the remaining six months, the diurnal circles are south of the equator. \u00a0In northern hemisphere, the Sun is high in the sky in the summer and low in the sky in the winter (Fig. 5.2); it is summer there.\u00a0 It is exactly the opposite in the southern hemisphere. Fig. 5.3 depicts the causations of seasons on the earth.<\/span><\/p>\n<\/div>\n

          \n

          \"\"<\/p>\n

          Fig. 5.2.\u00a0 In the northern hemisphere the Sun is high in June and low in December.<\/span><\/p>\n

           <\/p>\n

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

          \n

           <\/p>\n

          Fig. 5.3. Cause of the seasons on the earth is the motion of the Sun in an orbit inclined to the equator, or equivalently, the inclination of the earth\u2019s axis of rotation to the plane of the ecliptic.<\/p>\n

           <\/p>\n

          During its daily motion, when the Sun is exactly overhead, it is local noon<\/strong>. At that time the Sun is on the local meridian. Recall that the local meridian or the observer\u2019s meridian includes the zenith along with the celestial poles. \u00a0Since zenith is the point directly overhead the observer, the Sun appears overhead when on the local meridian. The interval between two successive passages of the Sun over the local meridian of a place is called a solar day<\/strong>.\u00a0 However, the length of the solar day is not constant throughout the year. Therefore, the solar day and its smaller divisions are not suited for accurate time measurement.<\/p>\n

           <\/p>\n

          3.1.\u00a0 Reasons for Variation of Solar Day \u2013 Effect of Elliptic Orbit and Obliquity of Ecliptic<\/strong><\/p>\n

           <\/p>\n

          But why is the solar day not constant? \u00a0There are two reasons:<\/p>\n

           <\/p>\n

          (1) The speed of the earth is not uniform in its elliptic orbit round the Sun (or equivalently, the speed of the apparent motion of the Sun round the earth). Kepler\u2019s laws say that the\u00a0<\/strong>earth revolves round the Sun in an elliptic orbit with the Sun at one of the foci, and the area carved out by the line joining the Sun and the earth per unit time is constant.\u00a0 <\/strong>The consequence is that the speed of the earth in its orbit is not constant: it is maximum at <\/span>perihelion <\/strong>(the point nearest to the Sun) and minimum at <\/span>aphelion <\/strong>(point farthest from the Sun).\u00a0 This implies that the difference between the solar day and the more constant <\/span>sidereal day <\/strong>(more about it later) keeps varying throughout the year<\/span>.<\/strong><\/p>\n<\/div>\n

          \n

          \"\"<\/p>\n

          Fig. 5.4. According to the second law of Kepler the area bounded by green lines is equal to the area bounded by red lines.\u00a0 These areas are carved out in unit time. \u00a0The law implies that the orbital speed of the earth is higher at perihelion than at aphelion.<\/p>\n

           <\/p>\n

          (2) <\/strong>The plane of the Sun\u2019s annual motion, the ecliptic, is inclined to the equatorial plane, so that the hour angle of the Sun does not advance at a uniform rate. \u00a0Remember that the hour angle of any object, such as the Sun, is measured from the observer\u2019s meridian along the equator.\u00a0 Even if the Sun were to move uniformly along the ecliptic, the obliquity of the ecliptic will cause equal intervals along the ecliptic to have unequal projections on the equator (Fig. 5.5). T<\/em>herefore, the Sun does not progress uniformly along the equator.\u00a0 <\/em>This causes a variation in the length of the day. The effect of obliquity is more pronounced near the solstices than near the equinoxes.<\/p>\n<\/div>\n

          \"\"<\/p>\n

          \n

           <\/p>\n

          Fig. 5.5.\u00a0 Segments of equal length along the ecliptic do not have equal projections on the equator. \u00a0So, while the Sun progresses uniformly along the ecliptic, it does not progress along the equator.<\/p>\n

           <\/p>\n

          3.2.\u00a0 Mean Solar Time<\/strong><\/p>\n

           <\/p>\n

          To eliminate these two effects, a fictitious body, the mean Sun<\/strong>, is defined whose apparent motion round the earth takes place (i) at a constant speed (circular path) <\/strong>and (ii) the plane of whose orbit coincides with the plane of the equator <\/strong>(Fig. 5.6).\u00a0 The time derived from the mean Sun is called the mean solar time<\/strong>, as against the true, or apparent, solar time, <\/strong>derived from the actual Sun.\u00a0 Our clocks and watches run on mean solar time.\u00a0 To bring the time given by a sundial (apparent solar time) must be corrected to bring it in line with the time on our watches, that is, the mean solar time.\u00a0 The difference between the mean solar time and the apparent solar time is called the equation of time.<\/strong><\/p>\n<\/div>\n

          \n

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

          \n

           <\/p>\n

          Fig. 5.6. While the true Sun orbits in the ecliptic plane the fictitious mean Sun moves in the plane of the equator in a circular orbit. Both take the same time in completing one orbit.<\/p>\n

           <\/p>\n

          3.3.\u00a0 Equation of Time<\/strong><\/p>\n

           <\/p>\n

          Equation of time is given by<\/p>\n

          ?= Hour Angle of actual Sun \u2212 Hour Angle of mean Sun<\/p>\n

           <\/p>\n

          = HA\u0298 \u2212 HAMS.\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 (5.1)<\/p>\n

           <\/p>\n

          The Equation of Time is shown in Fig. 5.7.<\/p>\n

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

          \n

           <\/p>\n

          Fig. 5.7.\u00a0 The equation of time is shown by the red line.\u00a0 The two factors contributing to it, eccentricity of the earth\u2019s orbit and the obliquity of the ecliptic plane to the equatorial plane,\u00a0<\/span>are shown by broken lines. <\/span>\u03d2 <\/strong>and ? are the vernal and autumn equinoxes. Points A and P\u00a0<\/span>denote aphelion (the point where the Sun \u2013 earth distance is longest) and perihelion\u00a0<\/span>(the point where the Sun \u2013 earth distance is shortest). The red line shows the number\u00a0<\/span>of minutes by which the apparent solar time (sundial time) differs from the mean solar\u00a0<\/span>time. On 1 Jan, for example, the apparent solar time is 4 min behind the mean solar time.<\/span><\/p>\n

           <\/p>\n

          That means that 4 min must be added to the time given by the sundial to bring it in ?\u00a0<\/span>agreement with the mean solar time. (Based on the figure from Wikipedia)<\/span><\/p>\n

           <\/p>\n

          4.\u00a0 Sidereal Time<\/span><\/strong><\/p>\n

           <\/p>\n

          Sidereal time is the time measured with respect to the distant, and therefore apparently fixed, stars. We have seen above that the stars appear to move round the earth in diurnal circles parallel to the equator.\u00a0 Therefore, they appear on the observer\u2019s meridian at some instant and\u00a0<\/span>then progress steadily, increasing steadily their hour angle.\u00a0 Instead of any star, we use the Vernal equinox as the reference point.\u00a0 So, <\/span>the sidereal time is defined as the hour angle of the Vernal equinox.<\/strong><\/p>\n

           <\/p>\n

          It must be noted, however, that vernal equinox itself is not a fixed point in the sky. It undergoes drifts due to <\/span>precessional <\/strong>and <\/span>nutational <\/strong>motions of the earth.\u00a0 Since Vernal equinox is affected<\/span><\/p>\n

           <\/p>\n

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

          \n

           <\/p>\n

          Fig. 5.7. The earth precesses around a line perpendicular to the ecliptic.<\/p>\n

           <\/p>\n

          this effect is also called the precession of equinoxes<\/strong>.\u00a0 You must have learnt precession and nutation as part of your course in classical mechanics.\u00a0 Since the Vernal equinox drifts\u00a0irregularly, we define a <\/span>mean Vernal equinox, <\/strong>much like the mean Sun<\/span>, <\/strong>which is supposed to move uniformly along the celestial equator.<\/span><\/p>\n<\/div>\n

          \n

           <\/p>\n

          4.1.\u00a0 Precession and Nutation of the Earth<\/strong><\/p>\n

           <\/p>\n

          We know that the earth is not a perfect sphere.\u00a0 It has a slight bulge at the equator. This makes it behave like a spinning top.\u00a0 The gravitation field of the Sun and the moon and, to much lesser extent, the planets causes the earth to precess<\/strong>. The axis of the earth charts out a cone about a line perpendicular to the plane of the ecliptic (Fig. 5.7). The direction of the precession is<\/p>\n<\/div>\n

          \n

          \"\"<\/p>\n

          Fig. 5.8. Due to precession, Polaris will no longer be our pole star after a few thousand years. \u00a0Around 14000 CE star Vega will replace Polaris as our pole star. About 4500 years ago, the time the Egyptian pyramids were built, Thuban was our pole star.<\/p>\n

           <\/p>\n

          clockwise if we look down on the earth from the north. The time period of precession is about 26000 years. This, so-called <\/span>astronomical precession<\/strong>, has very important consequences.<\/span><\/p>\n

           <\/p>\n

          (i) The axis of the earth does not point in a fixed direction in space but wanders among stars (Fig. 5.8).\u00a0 At present the axis points towards the star Polaris, which at present is our Pole Star.\u00a0 In a few thousand years, Polaris will no longer be our Pole Star.<\/span><\/p>\n

           <\/p>\n

          (ii) The Vernal equinox, or the First Point of Aires, shifts continuously eastwards. \u00a0That is why precession of the earth is also called the <\/span>precession of equinoxes<\/strong>. While precessing, the earth also undergoes a wobbling motion with a period of about 19 years, called <\/span>nutation.<\/strong><\/p>\n<\/div>\n

          \n

           <\/p>\n

          We know that the gravitational forces of the sun and the moon vary across the earth, giving rise to tides. Since the sources of these tidal forces are changing their location continuously with respect to each other as well as with respect to the earth, they produce a wobbling motion in the axis of the earth. This wobbling motion has multiple periods, the principal period being 18.6 years. This motion is called nutation.<\/p>\n

           <\/p>\n

          (iii) The continuous drift of the Vernal equinox due to precession causes a slow apparent drift in the zodiacal constellations.\u00a0 Two thousand years ago, the Vernal equinox occurred when the Sun entered the constellation of Aires, that is why Vernal equinox is also called the First Point of Aries.\u00a0 At present, however, the vernal equinox occurs at the time the Sun is in the constellation Pisces (Fig. 5.9).\u00a0 The result is that the zodiacal signs are no longer in step with the constellations from which they derive their names.<\/p>\n

          \"\"<\/p>\n

          Fig. 5.9.\u00a0 Vernal equinox at present occurs when the Sun is in the constellation Pisces.\u00a0 Two thousand years ago this event occurred when the Sun was in Aries.\u00a0 This change is the result of precession of equinoxes.<\/span><\/p>\n<\/div>\n

          \n

           <\/p>\n

          4.2.\u00a0 Drift of Vernal Equinox and its Effect on Equatorial Coordinates<\/strong><\/p>\n

           <\/p>\n

          Since the sidereal time is the hour angle of the Vernal equinox, one effect of the drift of Vernal equinox is that sidereal days and years are not uniform in length. To overcome this non- uniformity, we have to define a mean Vernal equinox.\u00a0 The rate of eastward drift of the mean equinox is about 50 arc-sec per year. Actually, this number is the first term in a series in time which contains higher order terms.\u00a0 The drift of Vernal equinox causes the tropical year <\/strong>to be slightly shorter than sidereal year.<\/p>\n

           <\/p>\n

          Tropical year is the same as solar year, which is the average interval of time between the two successive returns of the Sun to the same point in the sky, for example the First Point of Aires.<\/p>\n

           <\/p>\n

          The drift of Vernal equinox also results in a continuous change in the equatorial coordinates of astronomical objects<\/strong>. Recall that the equatorial system has the Vernal equinox\u00a0as the reference point.\u00a0 So, while stating the right ascension and declination of objects we have also to state the relevant epoch.<\/span><\/p>\n

          \"\"<\/p>\n

          Fig. 5.10. Drift of Vernal equinox due to the precession of the earth\u00a0<\/span>causing changes in the equatorial coordinates.<\/span><\/p>\n<\/div>\n

          \n

           <\/p>\n

          4.3. Comparison of Sidereal Day and Solar Day<\/strong><\/p>\n

           <\/p>\n

          The time interval between the two successive passages of the mean Vernal equinox over the observer\u2019s meridian defines a sidereal day<\/strong>. A sidereal day is slightly shorter than the solar day, as the following argument shows.<\/p>\n

           <\/p>\n

          Consider a point X on the earth\u2019s surface (Fig. 5.11) at the time of the Vernal equinox.\u00a0 In situation ?\ufffd both the Sun and the Vernal equinox (or the distant stars) are overhead. \u00a0After one rotation of the earth on its axis, at ?\ufffd, the Vernal equinox, being very distant, is again overhead of X. So, as reckoned by distant stars, one day has been completed. This day is called the\u00a0sidereal day. <\/strong>However, <\/strong>the Sun will be overhead of X, and one solar day <\/strong>will be completed,\u00a0only when X moves to Y, that is, the earth undergoes an additional rotation through the angle\u00a0<\/span>XO\u00b4Y. \u00a0<\/span>This means that the solar day is longer than the sidereal day by the time required to cover the angle XO<\/strong>\u00b4<\/span>Y<\/strong>.\u00a0\u00a0 Since the earth rotates through 360\u00b0 in a day, angle XO\u00b4Y (covered in<\/span><\/p>\n<\/div>\n

          \n

          \"\"<\/p>\n

          one day) is roughly one degree and the time taken by the earth to cover this angle is about 4 minutes. Thus, the sidereal day is shorter than the solar day by 4 minutes, or to be more exact, by 3 min 56 s.<\/strong><\/p>\n

           <\/p>\n

          5.\u00a0 Local Mean Time<\/strong><\/p>\n

           <\/p>\n

          The sidereal time is not very convenient for everyday life.\u00a0 The sidereal noon, for example, occurs at night for part of the year.\u00a0 A different system of time, the Universal Time <\/strong>(UT<\/strong>), has therefore been created for civil purposes.\u00a0 The UT is based on the mean Sun.\u00a0 We have already stated that the difference between the apparent solar time and the mean solar time, that is, the Equation of Time<\/strong>, can be precisely calculated, and hence the hour angle of the mean Sun\u00a0<\/strong>(<\/span>HAMS<\/strong>) can be worked out.\u00a0 When the mean Sun is on the local meridian of a place, it is <\/span>mean noon <\/strong>at that place.\u00a0 The time of the day is, however, reckoned from the midnight, and so the time of the day, styled as <\/span>local mean time <\/strong>(<\/span>LMT<\/strong>), is given by the following equation:<\/span><\/p>\n<\/div>\n

          \n

           <\/p>\n

          LMT = HAMS + 12 h.\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 (5.2)<\/p>\n

           <\/p>\n

          6.\u00a0 The Universal Time<\/strong><\/p>\n

           <\/p>\n

          To correlate the local time at various places on the earth, it has been agreed to adopt the Greenwich local time as the reference time. \u00a0The universal time (UT) is then defined as follows:<\/p>\n

           <\/p>\n

          UT = HAMS at Greenwich + 12 h.\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 (5.3)<\/p>\n

           <\/p>\n

          The relation then between the local mean time at any place and the universal time is<\/p>\n

           <\/p>\n

          UT = LMT \u2213 longitude of the place in hours,\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 (5.4)<\/p>\n

           <\/p>\n

          the minus sign being for places east of Greenwich and the plus sign for places west of Greenwich. For example, at a place at longitude 15\u00b0 East, the local mean time is ahead of the universal time by one hour. By the same token, the local mean time at a place at 15\u00b0 West is one hour behind the universal time. In order to avoid continuous setting of clocks as people travel from one place to another, Standard Times and Zone Times, spanning whole countries or their major parts, have been adopted. The Indian standard Time (IST) is the local mean time of a place (near Allahabad) whose longitude is 82.5\u00b0 East.<\/p>\n

           <\/p>\n

          7.\u00a0 Summary<\/span><\/strong><\/p>\n<\/div>\n

            \n
          • Interval between two successive passages of the Sun over the observer\u2019s meridian defines a solar day.<\/li>\n
          • Length of the solar day is not constant.<\/li>\n
          • The two reasons are (i) variation in the speed of the earth in its elliptic path round the Sun, and (ii) obliquity of the plane of the ecliptic to the celestial equator.<\/li>\n
          • To overcome these problems, a mean Sun is imagined, travelling along the equator at a uniform speed.<\/li>\n
          • Mean solar time is defined by the mean Sun. Our clocks and watches keep the mean solar time.<\/li>\n
          • The difference between the mean solar time and the apparent solar time is given by the Equation of Time.<\/li>\n
          • Sidereal time is defined as the interval between two successive passages of the vernal equinox over the observer\u2019s meridian.<\/li>\n
          • Vernal equinox is not a fixed point in the sky; it drifts irregularly due to precession and nutation of the earth.<\/li>\n
          • Precession of the earth is caused by the inclination of its axis of rotation to the plane of the ecliptic.<\/li>\n
          • Precession has three major effects: (i) it makes the sidereal year shorter than the tropical year, (ii) it changes the equatorial coordinates of celestial objects, and (iii) it causes a slow apparent drift in the zodiacal constellations.<\/li>\n
          • Sidereal day is shorter than the solar day by four minutes.<\/li>\n
          • Every day a star rises four minutes earlier than the previous day. Similarly, a star sets four minutes earlier than the previous day.<\/li>\n
          • Early rising and setting of stars than the previous day causes the sky to change slowly every night. The effect is apparent over a time of a few weeks.<\/li>\n
          • Local Mean Time (LMT) is defined as hour angle of the mean Sun + 12 hours.<\/li>\n
          • LMT of various places is coordinated by the LMT of Greenwich, which is called the Universal Time.<\/li>\n
          • Standard and Zone times are defined with respect to the Universal Time.<\/li>\n<\/ul>\n

             <\/p>\n

            The material covered in this module and related material can be found in the following websites:<\/p>\n

              \n
            • https:\/\/en.wikipedia.org\/wiki\/Sidereal_time<\/li>\n
            • http:\/\/www.astro.cornell.edu\/academics\/courses\/astro201\/sidereal.htm<\/li>\n
            • http:\/\/astro.wsu.edu\/worthey\/astro\/html\/lec-precession.html<\/li>\n
            • http:\/\/davidpratt.info\/pole1.htm<\/li>\n
            • http:\/\/www.heavens-above.com\/SkyChart.aspx?lat=0&lng=0&loc=Unspecified&alt=0&tz=UCT<\/li>\n
            • http:\/\/www.thetropicalevents.com\/<\/li>\n
            • https:\/\/en.wikipedia.org\/wiki\/Axial_precession<\/li>\n
            • http:\/\/www.universetoday.com\/113328\/having-fun-with-the-equation-of-time\/<\/li>\n
            • https:\/\/en.wikipedia.org\/wiki\/Equation_of_time<\/li>\n
            • http:\/\/www.intmath.com\/blog\/mathematics\/the-equation-of-time-5039<\/li>\n<\/ul>\n

               <\/p>\n

              Most books on classical mechanics discuss phenomena of precession in the context of tops. It is directly applicable to the earth.<\/p>\n","protected":false},"author":4,"menu_order":4,"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-66","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\/66","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\/66\/revisions"}],"predecessor-version":[{"id":83,"href":"https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-json\/pressbooks\/v2\/chapters\/66\/revisions\/83"}],"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\/66\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-json\/wp\/v2\/media?parent=66"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-json\/pressbooks\/v2\/chapter-type?post=66"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-json\/wp\/v2\/contributor?post=66"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-json\/wp\/v2\/license?post=66"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}