Astronomy Principles and Practice Fourth Edition.pdf

426 Practical projects
N
φ
Earth
1
α max
Horizon
Equatorial plane
Satellite
R=6 . 62
Figure 24.17. The apparent position of a geostationary satellite affected by parallax with its maximum altitude,
α max , being dependent on the observer’s latitude, φ.
Figure 24.18. The altitudes of geostationary satellites at different latitudes showing the parallactic displacement
of the Clarke belt.
For a body in orbit, the gravitational force is matched by the centripetal force. Hence,
GM ⊕ m
R 2 s
= mω 2 R s (24.4)
where ω is the angular frequency (rad s −1 ) of the satellite’s motion.
By combining equations (24.3) and (24.4),
( )1
R s
g 3
= R = . (24.5)
R ⊕ ω 2 R ⊕
By using the determined value of ω and with g = 9·8ms −2 and R ⊕ = 6371 km, R ≈ 6·62.
All geostationary satellites lie directly above the Earth’s equator and are characterized by their
longitude, λ s , east or west of the prime (Greenwich) meridian. Their apparent positions, however,
suffer from parallax because of their proximity and they appear to be below the celestial equator. With
reference to figure 24.17, the altitude of the celestial equator is given by 90 ◦ − φ, whereφ is the
observer’s latitude, whereas the maximum altitude for a satellite on the meridian is depicted as α max .
Looking at the sky (figure 24.18), we may note that the satellites occupying the Clarke belt are close
to but slightly below the celestial equator. By plotting the observed altitudes of a number of satellites
on this belt, we can determine α max and, therefore, obtain a value for latitude of the observing station