Foundations of Space Dynamics. Ashish Tewari

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year is not the true period of revolution of Earth in its orbit around the sun. The true period of revolution is the sidereal year, which is measured by timing the passage of Earth against the background of distant stars, and equals 365.25636 mean solar days. Thus a tropical year is shorter than the actual year by 20 hr., 40 min., and 42.24 s.

       1.2.2 Celestial Frame

For a motion taking place inside the solar system, any two stars (except the sun) appear to be fixed for the duration of the motion. Hence, a reference frame constructed out of three mutually perpendicular axes, each of which are pointing towards different distant stars, would appear to be fixed in space, and can serve as a sidereal reference frame. A reference frame fixed relative to distant stars is termed a celestial reference frame. For example, the rate of rotation of Earth about its own axis can be measured by an observer standing astride the North Pole by timing the rate at which a straight line joining Earth to a distant star, called a celestial meridian (see Fig. 1.3), appears to rotate. This rate gives the true rotational time period of Earth, called the stellar day, which is measured by IERS to be 23 hr., 56 min., 4.0989 s. Hence, the sidereal day is shorter than the stellar day by about s.

       1.2.3 Synodic Frame

      When two objects orbit one another at nearly constant rates on a fixed plane, a reference frame can be defined by two of its axes on the plane of rotation and rotating at the constant rate, and the third axis normal to the plane. Such a rotating reference frame is called a synodic frame. An example of a synodic frame is the ecliptic frame, which is a reference frame constructed out of the ecliptic plane, such as the frame in Fig. 1.3. The motion of an object measured relative to a synodic frame must be corrected by a vector subtraction of the motion of the frame itself, as exemplified by the calculation of the sidereal day from the observed rotation in the ecliptic frame. The ecliptic frame has been used as a reference since the earliest days of astronomical observations. The division of the circle into arose out of the apparent motion of the sun per day, which subtends an arc of one diameter every 12 hours when seen from Earth. Since the moon's apparent diameter from Earth is roughly the same as that of the sun, the eclipses of the sun and the moon are observed in the ecliptic (thus the name). However, since the moon's orbital plane around Earth is tilted relative to the ecliptic, the eclipses happen only along the intersection (i.e., the line of nodes) of the two planes.

      The Earth‐moon line provides another synodic reference frame for space flight. The Earth and the moon describe coplanar circles about the common centre of mass (called the barycentre) every 27.32 mean solar days relative to the vernal equinox (called a sidereal month). This rotational period appears in the synodic frame to be 29.53 mean solar days (a synodic month) from one new moon to the next, which is obtained from the sidereal month by subtracting the rate of revolution of Earth‐moon system around the sun.

       1.2.4 Julian Date

Instead of the calendar year of 365 mean solar days, the tropical year of 365.242 mean solar days, and the sidereal year of 365.25636 mean solar days, it is much more convenient to use a Julian year of 365.25 mean solar days, which avoids the addition of leap years in carrying out astronomical calculations. A Julian day number () is defined to be the continuous count of the number of mean solar days elapsed since 12:00 noon universal time (UT) on January 1, 4713 BC. Universal time refers to the time taken as 12:00 noon when the sun is directly over the Greenwich meridian (which is defined to be zero longitude). The Julian day number 0 is assigned to the day starting at that time on the Julian proleptic calendar. The Julian date of a general time instant is expressed as the JDN plus the fraction of the 24‐hour day elapsed since the preceding noon UT. Julian dates are thus expressed as a Julian day number plus a decimal fraction. For example, the Julian date for 10:00 a.m. UT on April 21, 2020, is given by J2458960.91667, and the JDN is 2458960. Epochs are listed in ephemeris charts and nautical almanacs according to their Julian dates. Hence a Julian date serves as a common time measure for astronautical calculations involving two events separated in time.

      Computation of the Julian date (JD) from a Gregorian calendar date is complicated due to the three calendar cycles used to produce the Julian calendar, namely the solar, the lunar, and the indiction cycles of 28, 19, and 15 year periods, respectively (Seidelmann, 1992). A product of these gives the Julian period of 7980 years. The Julian period begins from 4713 BC, which is chosen to be the first year of solar, lunar, and indiction cycles beginning together. The next epoch when the three cycles begin together will happen at noon UT on January 1, 3268. The following conversion formula for the JDN, truncated to the last integer, uses the numbering of the months from January to December as ; the Gregorian calendar years are numbered such that the year 1 BC is the year zero, , (i.e., 2 BC is , 4713 BC is , etc.); and the day number, , is the last completed day of the month up to noon UT:

      (1.4)

      This formula calculates the JDN for 09:25 a.m. UT on June 25, 1975, by taking , , , and yields the last truncated integer value as . Then the time elapsed from noon UT on June 24 to 09:25 a.m. UT on June 25 is added as a fraction to give the following Julian date:

      An epoch in the Julian date is designated with the prefix , and the suffix being the closest Gregorian calendar date. For example, refers to 12:00 noon UT on January 1, 2000, and has the Julian date of 2451545. Similarly, the epoch , which occurs exactly 100 Julian years before 12:00 noon UT on January 1, 2000, must refer to 12 noon UT on January 0, 1900; hence its date in the Gregorian calendar is December 31, 1899, and its Julian date is 2415020. The difference in the epochs and is therefore mean solar days (which is exactly 100 Julian years).

      Since

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