## P2 4wvm V2

= 2.75118 x 10~8[6378.14+(AJ> + /i/,)/2]3 (3-15a)

or, equivalently,

where hP, hA, and a are the perigee and apogee heights, and semimajor axis, respectively, in kilometres and P is in minutes. The velocity, V, at any point depends on both the instantaneous altitude and the semimajor axis:

where A is the instantaneous altitude, a is the semimajor axis in kilometres, and V is the velocity in kilometres per second. We may also use the vis viva equation to determine the energy per unit mass which must be supplied to reach a given altitude, starting at rest on the Earth's equator. For a circular orbit at altitude h, Eq. (3-5) may be reformulated to yield:

where VB is the rotational velocity at the Earth's surface, h is in kilometres, and AE is the energy increase required in Joules per kilogram. Appendix M tabulates the period, velocity, energy required, and size of the Earth's disk as a function of altitude for Earth-orbiting satellites.

At an altitude of 35,786 km, the period of a satellite equals the sidereal rotation period of the Earth. (The sidereal period, or period relative to the stars, is 4 minutes less than the mean period of rotation with respect to the Sun (24 hours) because in 1 day the Sun has moved about 1 deg farther along the ecliptic and the Earth must rotate slightly more than 360 deg to follow the Sun.) Spacecraft at this mean altitude are called synchronous satellites because a 0-deg inclination satellite at this altitude will remain over the same location on the Earth's equator. A synchronous satellite in a circular orbit at nonzero inclination travels in a figure "8" relative to the surface of the Earth. -

Thus far, we have been concerned with two-body Keplerian orbits^ However, there is a simple class of three-body orbits, known as Lagrange point orbits,* which is of particular interest to spaceflight. As shown in Fig. 3-13, the Lagrange points, or libration points, for two celestial bodies in mutual revolution, such as the Earth and the Moon, are the five points such that an object placed at one of them will remain there. The three Lagrange points on the Earth-Moon line are positions of unstable equilibrium; i.e., any small change causes the object to drift away. However, L4 and L}, which form equilateral triangles with the Earth and the Moon in the plane of the Moon's orbit, are positions of stable equilibrium. A satellite placed near the Lagrange point (with an appropriate velocity) remains in essentially the same position relative to the Earth and the Moon. Because of the stability of Lagrange point orbits about L4 and Ls, they have been proposed as one possible location for permanent colonies in space [O'Neill, 1975]. As a natural example of this phenomenon, the Trojan asteroids are a group of 14 known asteroids which have collected at the stable Lagrange points of the Jupiter-Sun system.

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