Page 91 - Satellite Communications, Fourth Edition

Orbits and Launching Methods 71

2.12 Problems

2.1. State Kepler’s three laws of planetary motion. Illustrate in each case their
relevance to artificial satellites orbiting the earth.

2.2. Using the results of App. B, show that for any point P, the sum of the focal
distances to S and S′ is equal to 2a.

2
2.3. Show that for the ellipse the differential element of area dA r d /2,
where d is the differential of the true anomaly. Using Kepler’s second law,
show that the ratio of the speeds at apoapsis and periapsis (or apogee and
perigee for an earth-orbiting satellite) is equal to

(1 e)/(1 e)

2.4. A satellite orbit has an eccentricity of 0.2 and a semimajor axis of 10,000
km. Find the values of (a) the latus rectum; (b) the minor axis; (c) the distance
between foci.

2.5. For the satellite in Prob. 2.4, find the length of the position vector when
the true anomaly is 130°.

2.6. The orbit for an earth-orbiting satellite has an eccentricity of 0.15 and a
semimajor axis of 9000 km. Determine (a) its periodic time; (b) the apogee
height; (c) the perigee height. Assume a mean value of 6371 km for the earth’s
radius.

2.7. For the satellite in Prob. 2.6, at a given observation time during a south
to north transit, the height above ground is measured as 2000 km. Find the
corresponding true anomaly.

2.8. The semimajor axis for the orbit of an earth-orbiting satellite is found to
be 9500 km. Determine the mean anomaly 10 min after passage of perigee.
2.9. The exact conversion factor between feet and meters is 1 ft 0.3048 m.
A satellite travels in an unperturbed circular orbit of semimajor axis a
27,000 km. Determine its tangential speed in (a) km/s, (b) ft/s, and (c) mi/h.

2.10. Explain what is meant by apogee height and perigee height. The Cosmos
1675 satellite has an apogee height of 39,342 km and a perigee height of 613
km. Determine the semimajor axis and the eccentricity of its orbit. Assume a
mean earth radius of 6371 km.

2.11. The Aussat 1 satellite in geostationary orbit has an apogee height of
35,795 km and a perigee height of 35,779 km. Assuming a value of 6378 km for
the earth’s equatorial radius, determine the semimajor axis and the eccentricity
of the satellite’s orbit.