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Critical Spacecraft Velocities

(orbital, escape, and solar system escape velocities), critical values of velocities for a spacecraft at the moment of entering an orbit (that is, upon shutdown of the launch vehicle’s engine) in a gravitational field. Each critical spacecraft velocity is calculated according to specific formulas and may be physically interpreted as the minimal initial velocity required for a spacecraft launched from the earth to become an artificial earth satellite (orbital velocity), to leave the earth’s sphere of gravitational action (escape velocity), or to overcome the sun’s attraction and leave the solar system (solar system escape velocity).

Two variants of the mathematical definition of the critical velocities are found in the literature. In the first variant the velocities may be calculated for any height above the earth’s surface or for any distance from the center of the earth. The orbital velocity v_I for a distance r from the center of the earth is given by the formula, where f is the gravitational constant and M is the earth’s mass. The value for fM is taken as 398,603 km³/sec². In celestial mechanics this velocity is also called circular velocity because in the two-body problem the motion of a body with mass m in a circle of radius r about another body having an incomparably greater mass M (for M≫) takes place with just this velocity. If at the moment of entry into orbit a spacecraft has the velocity v₀ = v₁ perpendicular to the direction of the center of the earth, then its orbit (in the absence of perturbations) will be circular. For v₀ < v₁, the orbit will be elliptical, and the point of entry into orbit will be at the apogee. If this point is at a height of about 160 km, then the artificial satellite will immediately descend into the lower-lying dense layers of the atmosphere and burn up. Thus, the orbital velocity for this height is the minimal possible velocity for a spacecraft to become an artificial earth satellite. At greater heights, the spacecraft may become an artificial satellite even for V₀ somewhat less than v_I calculated for this height. Thus, at a height of 300 km, it is sufficient for the spacecraft to have a velocity 45 m/sec less than v_I.

The escape velocity v_II at a distance r from the center of the earth is defined by the formula v_II = Escape velocity is also called parabolic velocity, since at an initial velocity v₀ = v_II in the two-body problem a body with mass m will move with respect to a body with mass M (for M≫m) in a parabolic orbit and recede from it indefinitely, thereby escaping in a certain sense from the gravitational effect of M. Velocities less than the parabolic velocity are called elliptical velocities and velocities greater than the parabolic velocity are called hyperbolic velocities, because for such initial velocities the motion in the two-body problem, with bodies of masses m and M (for M ≫m), will be in elliptical or hyperbolic orbits, respectively.

Table 1 gives values for the orbital and escape velocities for various heights h above sea level at the equator (h = r – 6,378 km).

Table 1. Orbital (v_I) and escape (v_II) velocities for various heights (h) above sea level
h (km)	v_I (km/sec)	v_II (km/sec)
0......................	7.90	11.18
100......................	7.84	11.09
200......................	7.78	11.01
300......................	7.73	10.93
500......................	7.62	10.77
1,000......................	7.35	10.40
5,000......................	5.92	8.37
10,000......................	4.94	6.98

The concept of critical velocities is also used in analyzing the motion of spacecraft in the gravitational fields of other planets or the planets’ natural satellites, as well as in the gravitational field of the sun. Thus critical velocities may be determined for Venus and other planets, the moon, and the sun. These velocities are calculated from the formulas given above, in which M is taken as the mass of the celestial body in question. The values of fM for some celestial bodies are given in Table 2.

Table 2. Values of the gravitational constant for celestial bodies
	fM (km³sec²)
Moon...............................	4.903 × 10³
Sun...............................	1.327 × 10¹¹
Mercury...............................	2.169 × 10⁴
Venus...............................	3.249 × 10⁵
Earth...............................	3.986 × 10⁵
Mars...............................	4.298 × 10⁴
Jupiter...............................	1.267 × 10⁸
Saturn...............................	3.792 × 10⁷
Uranus...............................	5.803 × 10⁶
Neptune...............................	7.026 × 10⁶
Pluto...............................	3.318 × 10⁵

The solar system escape velocity v_III is determined by the condition that the space vehicle, upon reaching the limits of the earth’s sphere of gravitational action (about 930,000 km from the earth), has the parabolic velocity relative to the sun (in the vicinity of the earth’s orbit, this velocity is 42.10 km/sec). At this moment, the velocity of the space vehicle with respect to the earth cannot be less than 12.33 km/sec. In order to reach this velocity, according to the laws of celestial mechanics, the velocity of the spacecraft on launching in the vicinity of the earth’s surface (at a height of 200 km) must be about 16.6 km/sec.

In the second variant of the mathematical definition, the orbital, escape, and solar system escape velocities are calculated using the same formulas but only for the surface of a uniform spherical model of the earth (with a radius of 6,371 km). In this sense, the orbital velocity is the circular velocity and the escape velocity is the parabolic velocity calculated for the earth’s surface. For these conditions, the critical velocities have unique values: the orbital velocity is 7.910 km/sec; the escape velocity, 11.186 km/sec; and the solar system escape velocity, 16.67 km/sec. For the hypothetical launching of a spacecraft from the surface of this model of the earth, which is taken to be absolutely smooth and without an atmosphere, the critical velocities correspond exactly to the physical interpretations given at the beginning of this article.

Analogously, critical spacecraft velocities may also be calculated for the surfaces of other celestial bodies. Thus, for the moon, the orbital velocity is 1.680 km/sec and the escape velocity is 2.375 km/sec. The escape velocities for Venus and Mars are 10.4 and 5.0 km/sec, respectively.

REFERENCES

Duboshin, G. N. Nebesnaia mekhanika. Osnovnye zadachi i melody. Moscow, 1963.
Levantovskii, V. I. Mekhanika kosmicheskogo poleta v elementarnom izlozhenii. Moscow, 1970.
Ruppe, H. O. Vvedenie v astronavtiku, vol. 1. Moscow, 1970. (Translated from English.)

IU. A. RIABOV

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