Chapter 8 Part 1 Attitude Dynamics: Disturbance Torques AEO-43
Types of Disturbance Torques Solar Pressure Dominant torque for geosynchronous satellites Gravity Gradient Can be disturbance or control torque Atmospheric drag Magnetic Field Used for control
Solar Pressure Torque (1/7) The torque resulting from solar pressure is the major long-term disturbance torque for geosynchronous spacecraft. The dominant torque on LEO spacecraft is the gravity gradient or aerodynamic. The solar radiation force results from the impingement of photons on the spacecraft. A fraction, ρ s, are specularly reflected, a fraction, ρ d, are diffusely reflected and a fraction, ρ a, are absorbed by the surface n s A ψ ψ Specularly reflected photons Incoming photons The force created by the absorbed photons results from the transfer of momentum of the photons to the spacecraft and is given by F =ρ PA( n s) s =ρ P( Acos ψ) s (absorbed); P is the solar flux a a a Note that the force is in the direction along the sun line. 3
Solar Pressure Torque (/7) The fraction of the photons who are specularly reflected transfer twice the momentum and the direction is normal to the surface. There is no momentum transfer tangent to the surface F =ρ PA( n s ) s +ρ PA( n s ) s n sin = n sout = cos ψ F = ρ PA ψn s s s in in s out out s cos (Specularly reflected) For that portion that is diffusely reflected the photon's momentum may be considered stopped at the surface, and re-radiated uniformly into the hemisphere. Thus the force has a component due to the transfer of momentum plus a component due to the re-radiation. Since it is re-radiated uniformly, the re-radiation component will be normal to the surface. The force is Fd dpa( n s) =ρ s + n (Diffusely reflected) 3 4
Solar Pressure Torque (3/7) The total solar radiation force is F = Fa + Fs + Fd = PA( n s) (1 ρ s) s + ρs( n s) + ρd n 3 where ρ a + ρ s + ρ d = 1 has been used. The solar pressure is usually assumed to be constant with value P = 4.644 10 N/m 6 The total solar radiation force may be written as F = PA( n s)( F s + F n) S1 S where: F = 1 ρ and F = ρ ( n s) + ρ 3 S1 s S s d 5
Solar Pressure Torque (4/7) δ is the declination of the sun with respect to the orbit (δ is positive from the vernal equinox to the autumnal equinox), and K 0 α is the orbit angle measured from local noon, the unit vector to the sun is s = I0cos δ K0sin δ= = O1cos δcos α+ Ocos δsin α O3sin δ ( I0, J0, K0) Sun/Orbit F ( O1, O, O3) Orbital F I0 = O1, J0 = O (for α= 0) O3 = K0 (always) I 0 s and K 0 belong to a plane I 0 is the intersection of this plane with orbit plane. I0 = O1cos α Osin α J0 = O1sin α+ Ocos α K0 = O3 6 δ Orbit Plane s
Solar Pressure Torque (5/7) For momentum bias and three axis stabilized spacecraft the major portion of the solar pressure torque is usually due to the solar panels that track the sun. Usually the solar panels only have a single gimbal, which is about the 3-axis or pitch axis. Thus, they are always within 3 degrees of the Sun and the loss of power from not pointing directly at the sun is offset by not having the added complexity and weight of a double gimbal system to directly track the Sun. s = O1cos δcos α+ Ocos δsin α O3sin δ n = I0 = O1cos α+ Osin α therefore n s = cos δcos α+ cos δsin α= cos δ F = PA( K K K I ) 0 1 0 K1 = ( FS1cos δ+ FS)cosδ FS1 = 1 ρs K = FS1cos δsin δ FS = ρscos δ+ ρd / 3 7
Solar Pressure Torque (6/7) In the orbital frame the solar pressure force is F = PA K αo K α O + K O ( ) 1cos 1 1sin 3 Assuming no attitude errors the vector from the center of mass to the center of pressure of A is r = x O + y O + z O cp cp 1 cp cp 3 Then the solar pressure torque in the OF is ycpk zcpk1sin α T = rcp F = PA zcpk1cos α+ xcpk xcpk1sin α+ ycpk1cos α (Orbit F) 8
(, I JK, ) Solar Pressure Torque (7/7) In the 0 0 0 system, which rotates at approximately at 1 deg/day, the solar radiation torque is: I0 = O1cos α Osin α using: J0 = O1sin α+ Ocos α K0 = O3 ycpkcos α xcpksin α T = PA K( xcp cos α ycp sin α) zcpk1 (Sun/Orbit F) xcpk1sin α+ ycpk1cos α Note that the pitch torque is periodic, hence its effect over one orbit is zero. However, the roll and yaw torques have secular components which mean that over one orbit there will be a net change in the spacecraft angular momentum. To determine the critical or design conditions one must evaluate the torques at equinox and solstice to determine the maximum conditions. 9
Gravity Gradient Torque (1/3) The gravitational torque is: where: ag = GM if a g = const then M = 0 + r + r 3 M = r agdm a g dm r CM r ( + r) = r In the principal-axis body frame: T T r = { x y z} and = { X Y Y} r = ( yz zy) b + ( zx xz) b + ( xy yx) b 1 3 10
Gravity Gradient Torque (/3) Since r << 3 3/ 3 r + r = ( + r + r ) = 1 + + using the binomial theorem 3 3 3( xx + yy + zz) + r 1 + GM { } T1 = Z ydm Y zdm + 3 3GM { XZ xydm YZ ( y z ) dm} + 5 3GM + + 5 { Z yzdm XY xzdm Y zydm} 3/ non zero 11
Gravity Gradient Torque (3/3) The products of inertia integrals are also zero because we have chosen the principal axis body frame. Therefore, it remains... 3GM YZ 3GM YZ T1 = ( y z ) dm= ( x y ) ( x z ) dm 5 5 + + = 3GM = ( 5 YZ I 3 I ) and the other two torque components are: 3GM 3GM T = XZ( I 5 1 I3) and T3 = XY( I 5 I1) Note that a fully symmetric body will NOT experience any gravitational torque. 1
Gravity Gradient Torque (4/4) The Euler s equations for a rigid body subject to gravity gradient Torque are 3GM I1 ω+ 1 ( I3 I) ωω 3 = YZ( I 5 3 I) 3GM I ω + ( I 1 I3) ωω = 1 3 XZ( I 5 1 I3) 3GM I ω + 3 3 ( I I1) ω ω = 1 XY( I 5 I1) Note that X, Y and Z are expressed in the BF, while they are more conveniently evaluated in the IF. These two different representations of the same vector are related through the attitude matrix T { } { } = X Y Z = C X Y Z = C B I I I I T 13
Aerodynamic drag (1/3) The torque is originated by the interaction between the satellite s surfaces and the upper atmosphere. Model: elastic impact without reflection (energy completely absorbed). Mechanism: the particle arrived with v 0, reaches thermal equilibrium (velocity v 1 ) with molecular surface and escapes with v 1 << v 0. The torque is about the center of mass. Aerodynamic drag is the dominant perturbation for h<400 Km. 1 1 df ˆ ˆ ˆ ˆ drag = CDρV ( n v) vda= CDρV cosαvda where: C = f(surface structure, local angle of attack α) D T ˆ ˆ drag = r dfdrag (only where n v= cos α> 0 is satisfied) 14
Aerodynamic drag (/3) The wind velocity, V = Vvˆ, is wrt the atmosphere, therefore: ω r V ˆ 0 = V0v0 = CM velocity vˆ= vˆ0 + V = V0 +ω r where V ω 0 = relative to the atmosphere 1 T ˆ ˆ ˆ drag = CDρV 0 ( n v0 )( v0 r) da+ 1 + C ˆ ˆ ˆ ˆ DρV 0 {[ n ( ω r)]( v0 r) + ( n v0)[( ω r) r]} da usually: V ω r 0 First term: displacement between CM and center of pressure. Second term: torque due to the S/C spin (wrt the atmosphere). 15
Aerodynamic drag (3/3) For an Earth orbiting satellite, since V ω r, the nd term is four order of magnitude less than the first one. 1 T C ρv ( nˆ vˆ )( vˆ r) da drag D 0 0 0 Aerodynamic drag is evaluated similarly to the solar pressure. Satellite surface is split in smaller geometrical shapes. Evaluate the torque for each shape. The overall torque is then the vectorial sum. Shadowing is very important, especially, at lower altitudes (because ρ increases). 0 FIGUE Sphere (radius r) Plane (area A and normal n) Cylinder (length l, diameter d, axis a) FOCE F = 0.5 πc ρ( V) drag D F 0.5 ( ˆ ˆ drag = CDρAV n v) F = 0.5C ρv DL 1 ( a vˆ ) drag D 16
Magnetic Torque (1/) Torque caused by the interaction between the geomagnetic field B and the satellite s residual magnetic fields m. T = m B mag a) S/C magnetic moments (dominant), b) eddy currents (parasite currents), and c) hysteresis. B is the geocentric magnetic flux density (Wb/m ) m is the sum of permanent and induced magnetism, and current loops. Torque caused by eddy currents and/or hysteresis are due to ω. Teddy = ke( ω B) B (k e depends on S/C geometry and conductivity) 17
Magnetic Torque (/) FIGUE Thin spherical shell (radius r, thickness d, conductivity s) Circular loop (radius r, cross-sectional area S on a plane containing spin axis) Thin-walled cylinder (length l, radius r, thickness d) πσ Coefficient k e 4 πr σd/3 3 πr σs/4 3 rld d l l d {[1 ( / ) tanh( / )]} In a permeable material rotating in a magnetic field, H, the energy is dissipated in the form of the heat due to the frictional motion of the magnetic domains. The energy loss over one rotation period is: where: EH = V H db E T H V is the volume of the permeable material, hyst = ω ω t db is the induced magnetic induction flux, Integral over the complete path of the hysteresis loop. Hysteresis effects are appreciable only in very elongated soft magnetic material (changes in the ambient field cause large changes in magnetic moment). t is the time over which the torque is evaluated. 18