Page 89 - Satellite Communications, Fourth Edition

Orbits and Launching Methods 69

(see Wertz, 1984) yields for b

arcsina tan l b (2.57)
tan i

The local mean (solar) time for the satellite is therefore
1 tan l
c arcsin a bd 12 (2.58)
t SAT s
15 tan i
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 (2.59)
s
365.24
where Δd 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 con-
stant. 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 a and Ω. In effect, the angle (Ω a ) must be con-
s
s
stant for a constant local mean time. Let Ω represent the right ascen-
0
sion of the ascending node at the vernal equinox and Ω the time rate
of change of Ω then
1 d tan l
r d 360 arcsina bd 12
t SAT c 0
15 365.24 tan i
1 360 tan l
c a r b d arcsina bd 12 (2.60)
0
15 365.24 tan i
For this to be constant the coefficient of Δd must be zero, or
360
r
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