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uniform intervals along the celestial equator, in other words, the length of a solar day depends on the position of the earth relative to the sun. To overcome this difficulty a fictitious mean sun is introduced, which travels in uniform circular motion around the sun (this is similar in many ways to the mean anomaly defined in Sec. 2.5). The time determined in this way is the mean solar time. Tables are available in various almanacs which give the relationship between mean solar time and apparent solar time through the equation of time. The relevance of this to a satellite orbit is illustrated in Fig. 2.15. This shows the trace of a satellite orbit on the celestial sphere, (again keeping in mind that directions and not distances are shown). Point A corresponds to the ascending node. The hour angle of the sun from the ascending node of the satellite is as measured westward. The hour angle of the sun from the satellite (projected to S on the celestial sphere) is as b and thus the local mean (solar) time is tSAT 1 ( 15
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(2.56)
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To find b requires solving the spherical triangle defined by the points ASB. This is a right spherical triangle because the angle between the meridian plane through S and the equatorial plane is a right angle. The triangle also contains the inclination i (the angle between the orbital plane and the equatorial plane) and the latitude l (the angle measured at the center of the sphere going north along the meridian through S). The inclination i and the latitude l are the same angles already introduced in connection with orbits. The solution of the right spherical triangle
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The condition for sun synchronicity is that the local solar time should be constant.
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(see Wertz, 1984) yields for b arcsina tan l b tan i (2.57)
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The local mean (solar) time for the satellite is therefore tSAT 1 c 15
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arcsin a
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tan l bd tan i
(2.58)
Notice that as the inclination i approaches 90 angle b approaches zero. Accurate formulas are available for calculating the right ascension of the sun, but a good approximation to this is
d 360 365.24
(2.59)
where d is the time in days from the vernal equinox. This is so because in one year of approximately 365.24 days the earth completes a 360 orbit around the sun. For a sun-synchronous orbit the local mean time must remain constant. The advantage of a sun-synchronous orbit for weather satellites and environmental satellites is that the each time the satellite passes over a given latitude, the lighting conditions will be approximately the same. Eq. (2.58) shows that for a given latitude and fixed inclination, the only variables are as and . In effect, the angle ( as) must be constant for a constant local mean time. Let 0 represent the right ascension of the ascending node at the vernal equinox and the time rate of change of then tSAT 1 c 15 1 c 15
r d a r
d 360 365.24 360 b d 365.24
arcsina arcsina
tan l bd tan i
12 12 (2.60)
tan l bd tan i
For this to be constant the coefficient of d must be zero, or r 360 365.24 0.9856 degrees/day (2.61)
Use is made of the regression of the nodes to achieve sun synchronicity. As shown in Sec. 2.8.1 by Eqs. (2.12) and (2.14), the rate of regression of the nodes and the direction are determined by the orbital elements
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TABLE 2.5
Tiros-N Series Orbital Parameters 833-km orbit 870-km orbit 98.899 102.37 min 25.59 /day E 0.986 /day E 14.07
Inclination Nodal period Nodal regression Nodal precession Orbits per day
SOURCE:
98.739 101.58 min 25.40 /day E 0.986 /day E 14.18
Schwalb, 1982a and b.
a, e, and i. These can be selected to give the required regression of 0.9856 east per day. The orbital parameters for the Tiros-N satellites are listed in Table 2.5. These satellites follow near-circular, near-polar orbits.
2.11 Standard Time Local mean time is not suitable for civil time keeping purposes because it changes with longitude (and latitude), which would make it difficult to order day-to-day affairs. The approach taken internationally is to divide the world into 1-h time zones, the zonal meridians being 15 apart at the equator. The Greenwich meridian is used as zero reference and in the time zone that is 7.5 about the Greenwich meridian the civil time is the same as the GMT. Care must be taken, however, since in the spring the clocks are advanced by 1 h, leading to British summer time (BST), also known as daylight saving time. Thus BST is equal to GMT plus 1 h. In the first zone east of the GMT zone, the basic civil time is GMT + 1 h, and in the first zone west of the GMT zone, the basic civil time is GMT 1 h. One hour is added or subtracted for each additional zone east or west. Again, care must be taken to allow for summer time if it is in force (not all regions have the same summer time adjustment, and some regions may not use it at all). Also, in some instances the zonal meridians are adjusted where necessary to suit regional or country boundaries. Orbital elements are normally specified in relation to GMT (or as noted in Sec. 2.9.2, UTC), but results (such as times of equatorial crossings) usually need to be known in the standard time for the zone where observations are being made. Care must be taken therefore to allow for the zone change, and for daylight saving time if in force. Many useful time zone maps and other information can be obtained from the Internet through a general search for time zones.
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