The mean momentum flux, P, acting on a surface normal to the Sun's radiation, is given by

where Fe is the solar constant (see Section 5.3) and c is the speed of light. The solar constant is wavelength dependent and undergoes a small periodic variation for an Earth-orbiting spacecraft because of the eccentricity of the Earth's orbit about the Sun. If the momentum flux incident on the spacecraft's surface is known, Edwards and Bevans [1965] have shown that the reflected flux can be described analytically by the reflection distribution function and the directional emissivity. However, these properties of the irradiated surface are generally not known in sufficient detail to evaluate the required functions.

For most applications, the forces may be modeled adequately by assuming that incident radiation is either absorbed, reflected specularly, reflected diffusely, or some combination of these as shown in Fig. 17-2. Let P be the momentum flux incident on an elemental area dA with unit outward normal N. (Each area consists of two surfaces with oppositely directed outward normal vectors.) The differential

Fig. 17-2. Absorption and Reflection of Incident Radiation

Fig. 17-2. Absorption and Reflection of Incident Radiation radiation force (momentum transferred per unit time) due to that portion of the radiation that is completely absorbed is d" PCacosBSdA (0 < 9 < 90°) (17-46)

where S is the unit vector from the spacecraft to the Sun, 9 is the angle between S and N, and Ca is the absorption coefficient. If cos9 is negative, the surface is not illuminated and will not experience any solar force. The differential radiation force due to that portion of the radiation which is specularly reflected is

Mspmûar~ ~2PC, cos'flN dA (0<9<90°) (17-47)

where the reflected radiation is in the direction (—S+2Ncos0). The coefficient of specular reflection, Cs, is the fraction of the incident radiation that is specularly reflected. For a diffuse surface, the reflected radiation is distributed over all directions with a distribution proportional to cos<j>, where <t> is the angle between the reflected radiation and N. The differential radiation force for diffusely reflected radiation is determined by integrating the contribution of the reflected radiation over all angles to obtain df<4j^=PQ(-fcos0N-cos0S)dv4 (0<9<90°) (17-48)

where the coefficient of diffuse reflection, Cd, is the fraction of the incident radiation that is diffusely reflected. Assuming that absorption, specular reflection, and diffuse reflection all play a part (without any transmission), then the total differential radiational force is df,OJa/=-/»J[(l-Ci)S + 2(Cicos0 + iQ)N]cos0d/l (17-49)

where Ca+ Cs + Cd= 1. For surfaces that are not completely opaque, the incident momentum flux, P, can be modified to account for the radiation that does not impinge or interact with the surface. The differentia] radiation force can be written to include secondary reflections, but this is normally not a significant factor in the total radiation force [McElvain, et al., 1966].

The solar radiation torque, acting on a spacecraft is given by the general expression

where R is the vector from the spacecraft's center of mass to the elemental area dA. df,0M/ is given by Eq. (17-49), and the integral is over the spacecraft's irradiated surface. Because of the difficulty in evaluating the radiation torque directly from Eq. (17-50) for 'arbitrary surfaces, the spacecraft configuration is frequently approximated by a collection of simple geometrical elements (e.g., plane, cylinder, sphere). The solar radiation force, F,, on each element is determined by evaluating the integral of Eq. (17-49) over the exposed surface area, that is,

Table (17-2) lists the solar radiation force F,- for some simple geometrical shapes. The torque on the spacecraft is the vector sum of the torques on the individual

Table 17-2. Solar Radiation Force for Some Simple Geometrie Figures

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