At non-relativistic speed all of the classical orbital mechanics from near-Earth to the edge of our solar system and beyond are based on Newton's fundamental law of gravitational attraction. The assumption is that the gravity force, Fg, acts throughout the universe in the same way. Newton's law of universal gravitation is:
Fg = Universal gravitational force between two bodies mMG
where m, M = mass of the two bodies, G = universal gravitational constant = 6.67 x 10~nm3kg_1 s_1, and d = distance between the center of mass of the two bodies.
Gravity is probably one of the most mysterious forces in the universe. In fact, while our everyday experience of gravity is commonplace, our understanding is very limited. The law has been well tested on Earth and in the vicinity of the Earth. However, when astronomers attempt to use Newton's fundamental law of gravitational attraction to predict the motion of stars orbiting the center of the Galaxy, they get the wrong answers. The most distant man-made objects are Pioneer 10, launched in 1972 and Pioneer 11, launched in 1973. Pioneer 10 is now more than 8 billion miles from Earth. On January 23, 2003 the tracking stations picked up the last feeble transmission from the probe's radioactive isotope (plutonium) powered transmitter [Folger, 2003]. As Pioneer 10's feeble signal faded from detection, the spacecraft seemed to be defying Newton's law of gravity because it was slowing down as if the gravitational attraction from the Sun was growing stronger the farther away it traveled. Pioneer 11 also slowed down in a similar manner. The Ulysses spacecraft, which has been orbiting the Sun for 13 years, has also behaved in a manner characteristic of an unknown force slowing it down. This chapter will not attempt to explain the behavior (the so-called Pioneer anomaly), but there is scant but growing evidence that perhaps gravity does not act in the same way on a galactic scale. Our Galaxy makes one rotation in about the time from when dinosaurs began to inhabit the Earth to now. Perhaps on that time and distance scale gravity acts differently. Until more is known, we will continue with the traditional assumption of gravity acting the same throughout the universe, but also acknowledge that the farther we travel and the longer we are in space we may be departing slightly more from the expected.
The law of gravity rules the attraction between two masses. When we put them into motion, then the laws that govern the two-body problem (that is, a large central body and a moving smaller body) yield Kepler's three laws of motion. Although these laws can be formulated for N number of bodies, the only analytic (closed-form) solutions found are for N = 2. Numerical solutions are possible, but these involve the use of the largest computers and are used only when the two-body problem is suspect (such as a Mercury orbiter) or high navigational accuracy is required [Brown, 1988]; the Keplerian circular orbit between two bodies is as given below [Koelle, 1961].
where ^ = gravitational constant = MG, M = mass of the central body, r = radius from the spacecraft center of mass to the center of mass of the central body, R0 = planet radius, and h = altitude above surface.
The gravitational parameters and the orbital speeds for a 200-km orbit and escape are given in Table 5.2 for selected bodies.
From equation (5.1), the orbital velocity decreases and the orbital period increases as the spacecraft altitude is increased (see Figures 3.5 and 3.6). The km/s km/s s s
Table 5.2. Gravitational characteristics of nearby planets and Earth's Moon.
Venus Earth Moon Mars Jupiter
^ (km3/sec2) 324,858.8 398,600.4 4,902.8 42,828.3 126,711,995.4
two-body equations assume non-rotating masses. If the central body is rotating, then its rotation can add a velocity vector increment to the launcher vehicle depending on the latitude of the launch site and the launch azimuth. Figure 5.2 shows the required velocity increment from the Earth's surface to the orbital altitude (in nautical miles).
Both the non-rotating Earth and rotating Earth (launch site at the Equator) velocity increments required are shown in Figure 5.2. These are not velocity in orbit, but the velocity increment (energy increment) that determines the mass ratio to reach simultaneously the given orbital altitude and required orbital speed. The speed of the Earth's surface at the Equator is 1,521 ft/s. That reduces the launch speed increment (A V) to 24,052 ft/s if the launcher is launched due east (90° from true north) at the Equator. If the launcher is launched due west, then the launcher must cancel out the easterly motion, so the launch speed increment (AV) is 27,094 ft/s. For a true east launch, the launch velocity increment as a function of the launch site
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