Читать книгу Foundations of Space Dynamics - Ashish Tewari - Страница 14

When can the effects of the atmosphere be considered negligible so that space flight can come into existence? The atmosphere of a planetary body – being bound by gravity – becomes less dense as the distance from the planetary surface (called altitude) increases, owing to the inverse‐square diminishing of the acceleration due to gravity from the planetary centre. For an atmosphere completely at rest, this relationship between the atmospheric density, , and the altitude, , can be derived from the following differential equation of aerostatic equilibrium (Tewari, 2006):

(1.1)

where refers to the atmospheric pressure, and the acceleration due to gravity prevailing at a given altitude. For a spherical body of radius , the gravity obeys the inverse‐square law discovered by Newton, given by

(1.2)

where is the acceleration due to gravity at the surface of the body (i.e., at ). When Eq. (1.2) is substituted into Eq. (1.1), and the thermodynamic properties of the atmospheric gases are taken into account, the differential equation, Eq. (1.1), can be integrated to yield an algebraic relationship between the atmospheric density, , and the altitude, , called an atmospheric model. For Earth's atmosphere, one such model is the U.S. Standard Atmosphere 1976 (Tewari, 2006), whose predicted density variation with the altitude in the range km is listed in Table 1.1. It is evident from Table 1.1 that the atmospheric density, , can be considered to be negligible for a flight for km around Earth. A similar (albeit smaller) value of is obtained on Mars at km. Hence, for both Earth and Mars, km can be taken to be the boundary above which the space begins.

The flight of a spacecraft around a large spherical body of radius is assumed to take place outside the atmosphere, (such as km for Earth and Mars), and is governed by the gravity of the body, with acceleration given by Eq. (1.2). Space‐flight trajectories are well defined orbits due to the simple nature of Eq. (1.2). However, since the atmospheric density in a very low orbit (e.g., km on Earth), albeit quite small, is not exactly zero, the flight of a spacecraft can be gradually affected, to cause significant deviations over a long period of time from the orbits predicted by Eq. (1.2). This is due to the fact that the atmospheric forces and moments are directly proportional to the flight dynamic pressure, , where is the flight speed. The high orbital speed, , required for space flight makes the dynamic pressure appreciable, even though the density, , is by itself negligible. The atmospheric drag (the force resisting the motion) causes a slow but steady decline in the flight speed, until the latter falls below the magnitude where an orbital motion can be sustained. Thus atmospheric drag can cause a low‐orbiting satellite to slightly decay in altitude after every orbit, and to ultimately enter the lower (dense) portions of the atmosphere, where the mechanical stress created by the ever increasing dynamic pressure, as well as the heat generated by atmospheric friction, lead to its destruction. Therefore, for predicting the life of a satellite in a low orbit, the atmospheric effects must be properly taken into account. Figure 1.1 shows an example of the decay in the orbit of a spacecraft initially placed into a circular orbit of km around Earth. In this simulation obtained by a Runge‐Kutta method (Appendix A), the spacecraft is assumed to be a sphere of 1 m diameter, with a constant free‐molecular drag coefficient of 2.0 (Tewari, 2006). As seen in the figure, the altitude decays quite rapidly as the number of orbits, , increases. The initial average rate of altitude loss seen in Fig. 1.1 – 1 km per 4 orbits – is likely to increase as the spacecraft descends lower, thereby encountering a higher density. When the spacecraft is placed in a circular orbit of km, its altitude decays very rapidly, and it re‐enters the atmosphere after only 3.5 orbits (Fig. 1.2). Hence, the life of the spacecraft is only about 3.5 revolutions in a circular orbit of altitude 180 km above Earth. As Figs. 1.1 and 1.2 indicate, a stable orbit around Earth for this spacecraft should have km at all times.

Table 1.1 Variation of density with altitude in Earth's atmosphere

Altitude, (km)	Density,
0	1.2252
1	1.1119
5	0.7366
10	0.4136
20	0.0891
30	0.0185
40	0.0041
50	0.0011
60
70
80
90
100
110
120
150
200
250

Apart from the atmospheric effects, there are other environmental perturbations to a spacecraft's flight around a central body, which is assumed to be spherical as required by Eq. (1.2). These are the gravity of the actual (non‐spherical) shape of the central body, as well as the gravity of other remote large bodies, and the solar radiation pressure. However, such effects are typically small enough to be considered small perturbations when compared to the spherical gravity field of the central body given by Eq. (1.2). Such effects can be regarded as small perturbations applied to the orbit governed by Eq. (1.2), and should be carefully modelled in order to predict the actual motion of the spacecraft.

Figure 1.1 Decay in the orbit due to atmospheric drag for a spacecraft initially placed in a circular orbit of km around Earth.

Figure 1.2 Decay in the orbit due to atmospheric drag for a spacecraft initially placed in a circular orbit of km around Earth.