where R, is the vector from the spacecraft center of mass to the center of pressure of the ith element. The center of pressure is at the intersection of the line of action of the single force which replaces the resultant radiation force and the plane passing through the center of mass of the spacecraft perpendicular to the line of action. The location of the center of pressure, r^, relative to the centroid of the geometrical sphere is given by

Solar radiation torques are reduced by the shadows cast by one part of the spacecraft on another. Shadowing reduces the total force and also shifts the center of pressure. The extent of shadowing is a function of the geometrical design of the spacecraft and the incident Sun angle. Examples of shadow modeling for DCSC II and AE-3 are given by Suttles and Beverly [1975] and Gottlieb el al., [1974]. Although the shadow modeling for AE-3 was used to evaluate aerodynamic torque, the same method can be applied to solar radiation torque.

The interaction of the upper atmosphere with a satellite's surface produces a torque about the center of mass. For spacecraft below approximately 400 km, the aerodynamic torque is the dominant environmental disturbance torque.

The force due to the impact of atmospheric molecules on the spacecraft surface can be modeled as an elastic impact without reflection [Beletskii, 1966]. The incident particle's energy is generally completely absorbed. The particle escapes after reaching thermal equilibrium with the surface with a thermal velocity equal to that of the surface molecules. Because this velocity is substantially less than that of the incident molecules, the impact can be modeled as if the incident particles lose their entire energy on collision. The force, df^, on a surface element dA, with outward normal N, is given by

where V is the unit vector in the direction of the translational velocity, V, of the surface element relative to the incident stream and p is the atmospheric density (Section 4.4). The parameter CD is the drag coefficient defined in Section 3.4 and js a function of the surface structure and the local angle of attack, arc cos (N-V) [Schaaf and Chambre, 1961]. For practical applications, CD may be set to 2.0 if no measured value is available.

The aerodynamic torque N^^, acting on the spacecraft due to the force dfAm, is

where rs is the vector from the spacecraft's center of mass to the surface element dA. The integral is over the spacecraft surface for which N-V>0. Note that the translational velocity of element dA for a spacecraft spinning with angular velocity <o is

where V0 is the velocity of the center of mass relative to the atmosphere. (Note that a is relative to the rotation of the atmosphere which approximately equals the Earth's rotational rate.) Because the linear surface velocity due to the spacecraft spin is generally small compared to Vq, second-order terms in <o can be neglected in substituting Eqs. (17-54) and (17-56) into Eq. (17-55). Thus, the total aerodynamic torque is

N^-iCopVgf (N • V0)(V0 X ra) dA + $CDpV0 f {N(«Xrr)(V0Xr,)

The first term in Eq. (17-57) is the torque due to the displacement of the spacecraft's center of pressure from the center of mass. The second term is the dissipation torque due to the spacecraft spin. For a spaoecraft in Earth orbit with w<£ Vq, the second term, is approximately four orders of magnitude smaller than the first and may be neglected.

The first term in Eq. (17-57) is evaluated in the same manner as the solar pressure torque. The surface area of the satellite is decomposed into simple geometric shapes and the total aerodynamic force is calculated by integrating Eq. (17-54) over the individual shapes. Table 17-3 lists the aerodynamic force for some simple geometric figures. The total torque about the center of mass of the spacecraft is the vector sum of the individual torques calculated by the cross product of the vector distance from the spacecraft's center of mass to the center of pressure of the geometric shapes and the force acting on the component.

Shadowing of one part of the spacecraft by another must also be considered in the torque evaluation. Because the aerodynamic torque increases as the spacecraft's altitude decreases, shadowing can be very important at low altitudes. The extent of shadowing is a function of the spacecraft's design and orientation relative to the velocity vector. Examples of shadowing models are given by Gottlieb, et al„ [1974] and Tidwell, [1970].

17.2 ENVIRONMENTAL TORQUES 575

Table 17-3. Aerodynamic Force for Some Simple Geometric Figures

17.2 ENVIRONMENTAL TORQUES 575

Table 17-3. Aerodynamic Force for Some Simple Geometric Figures

geometric figures |
force |

&herc OP radius r plane with surface area a ano normal unit vector ft right circular cylinder op length l and diameter 0. unit vector t is along cylinder axis |
"5 v^ol-ZT^Sfa |

17.2.4 Magnetic Disturbance Torque

Magnetic disturbance torques result from the interaction between the spacecraft's residual magnetic field and the geomagnetic field. The primary sources of magnetic disturbance torques are (1) spacecraft magnetic moments, (2) eddy currents, and (3) hysteresis. Of these, the spacecraft's magnetic moment is usually the dominant source of disturbance torques. The spacecraft is usually designed of material selected to make disturbances from the other sources negligible. Bastow [1965] and Droll and Iuler [1967] provide a survey of the problems associated with minimizing the magnetic disturbances in spacecraft design and development.

The instantaneous magnetic disturbance torque, N^ (in N-m), due to the spacecraft effective magnetic moment m (in A-m2) is given by

where B is the geocentric magnetic flux density (in Wb/m2) described in Section 5.1 and m is the sum of the individual magnetic moments caused by permanent and induced magnetism and the spacecraft-generated current loops. (See Appendix K for a discussion of magnetic units.)

The torques caused by the induced eddy currents and the irreversible magnetization of permeable material, or hysteresis, are due to the spinning motion of the spacecraft. Visti [1957] has shown that the eddy currents produce a torque which precesses the spin axis and also causes an exponential decay of the spin rate. This torque is given by

where « is the spacecraft's angular velocity vector and ke is a constant coefficient which depends on the spacecraft geometry and conductivity. Eddy currents are appreciable only in structural material that has a permeability nearly equal to that of free space. Table 17-4 lists values of ke for simple geometric figures. Tidwell [1970] has outlined an alternative procedure for calculating the torque due to eddy current interaction which involves the evaluation of three different constant coefficients.

In a permeable material rotating in a magnetic field, H, energy is dissipated in the form of heat due to the frictional motion of the magnetic domains. The energy loss over one rotation period is given by

where V is the volume of the permeable material and dB is the induced magnetic induction flux in the material. The integral is over the complete path of the

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