EFFECTS OF SOLAR RADIATION PRESSURE ON SATELLITE ATTITUDE CONTROL

R. J. McElvain^{1 }

Space Technology Laboratories Inc., Los Angeles, Calif.

ABSTRACT

The effects of torques due to solar radiation pressure act- ing on satellite vehicles are discussed. The functional forms of the torques are developed for inclusion in the general equa- tions of motion of the vehicle. These combined equations are highly nonlinear; however, with certain restrictions, approxi- mate linear solutions are derived for the motion of the vehicle due to the action of the solar radiation torques. The torque equations are highly dependent on the vehicle configuration, the reflective properties of the surfaces exposed to the sun1s rays, and the vehicle orientation with respect to a sun refer- ence. The expressions are developed for spin - stabilized ve- hicles and for vehicles with solar arrays and an active atti- tude control system that orients the vehicle in an Earth - sun reference. Similar expressions for other configurations can be developed through use of the methods discussed. The results of this analysis indicate that solar radiation pressure may cause torques that are either periodic or constant with respect to inertial space, depending on the orbital inclination to the ecliptic plane, the orientation requirements, and the vehicle configuration. The approximate solutions developed here can be used to determine the vehicle angular momentum storage and/or removal requirements resulting from solar radiation torques.

INTRODUCTION

Many applications of satellites requiring attitude control have been discussed in scientific literature. In general, dis- turbance torques will act to produce deviations of the vehicle from the desired attitude. A number of the various sources of these attitude perturbation torques are summarized in Ref. 1 .

Presented at ARS Guidance, Control, and Navigation Confer- ence, Stanford, Calif., Aug. 7-9> 1 9 6 l .

Member of the Technical Staff.

**5 4 5 **

This paper is concerned with a more detailed discussion of the effects of one particular disturbance torque source, solar radiation pressure.

The model chosen to represent the solar pressure on a vehi- cle surface is one in which the force on the surface due to emission is neglected, the angle of reflection is assumed to equal the angle of incidence, and the coefficient of reflec- tivity for the surface is assumed independent of the spectrum or angle of incidence of the incident radiation. Torques will then act about the vehicle center of mass if the resultant center of pressure of the total solar radiation pressure forces is not coincident with the vehicle center of mass. The torque acting on the satellite from solar radiation pressure is therefore a function of the vehicle orientation with re- spect to the solar radiation field and the relative location and characteristics of the exposed satellite surfaces.

SOLAR RADIATION TORQUE ON AN ARBITRARY VEHICLE

A typical satellite vehicle may be divided into several dis- tinct surfaces, each with a characteristic area and surface reflective properties. The force on an arbitrary surface is due to the reaction from absorption and reflection of the ra- diation on the particular surface. Both magnitude and direc- tion of this force are affected by the surface properties. The torque about the vehicle center of mass will be due to the forces on the respective surfaces acting through a center of pressure determined by surface characteristics and geometry.

The unit vector ν is defined as the vector directed from the vehicle center of mass to the sun. For a vehicle orbiting Earth, this vector may be assumed coincident with the vector directed from the center of Earth to the sun. The force due to the solar radiation incident upon a surface A. of a vehicle

may be expressed by 1

where V is the coefficient of reflectivity over- the surface, η is a unit vector directed along the outward normal to dA., and

[1]

ι

V is the^sol^r radiation pressure constant for normal inci- dence (|v · n| = l) and complete absorptivity (ν = o). The total solar radiation torque about the vehicle center of mass is then

**? = Σ** Ί.^{ χ F.} [ 2 ]

Α. 1 1

ι

where is the vector directed from ttog vehicle center of .mass to1the center of pressure of the i surface.

COORDINATE SYSTEMS AND NOTATION

Definition of several reference coordinate sets is conven- ient in order to express the orientation of the vehicle rela- tive to inertial space and to derive the solar radiation torques using Eq. 2 . These coordinate sets are shown in Fig.

1 , ana are defined as follows :

1 ) Ecliptic reference (x.y.z.) - An inertial coordinate set
with origin at the center or** e a r t h ;** the x.z. plane the eclip-
tic plane; and the z. axis along the line^OÎ^{1} equinoxes, direc-
ted toward the autumnal equinox.

2 ) Orbital reference (x y ζ ) - An inertial coordinate set with origin at the center of Earth; the χ ζ plane the orbit plane; and the ζ axis along the line of noâes, directed to- ward the ascending node. The angles ξ and β are, respec- tively, the inclination of the orbit plane to the ecliptic, and the angle in the x.z. plane between the z. and the ζ axes.

**1 1** ι ο

3) Sun reference (x1 y1 z1) - A slowly rotating coordinate
set with origin at the center of Earth; the** Χ**^ζ^ plane^the or-
bit plane; and the y!z! plane contains the sun vector v. The
* angles S'^and η are respectively the angle between the y*l

axis and v, and the angle between the ζ and z1 axes. 0

ο ο

* h) Vehicle reference (x y ζ ) - A rotating coordinate set *
with origin at the vehiele

^{r}**cenïer of mass (cm.); the χ ζ**plane the orbit plane; and the ζ axis directed along

**vector from the center of Earth Co the vehicle (cm.), r . The **
angle <x(t) is the angle between r and the ζ axis.

^The solar radiâtjon energy constant outside Earth^{1}s atmos-
phere is 1.9^^{ g m}" ^^{a} (Ref. 2 ) , corresponding to a pressure

**ο** cm- min
of 9 ^ x 1 0 lb/ft^{2 }

545

The -unit vector v, lies in the ecliptic plane; its position relative to the vernal equinox (-z. axis) is given by the angle S. When resolved to the vehicle reference set, ν is given by

-sin (a -** 77)** sin
ν xr

yr zr

cos S

cos (a -** 77)** sin S^{! }

where S^{l}** and 7j are defined from the geometry by **
cos S^{1} = sin ξ sin (S - β)

-cos £ sin (S - β)

tan** 77 = ? ** **—** **— **

' -cos (S - ßj

**C3] **

**C5] **

(The minus sign denotes the choice of the proper quadrant for

The body fixed coordinate set XYZ is defined with origin at the vehicle center of mass. The XYZ coordinate set may be chosen to be the principal axes of the vehicle if convenient.

For an attitude controlled satellite, the desired orientation
of the vehicle relative to a convenient reference is known as
a function of time. Since many applications of Earth satel-
lites require alignment of a body fixed reference with the ra-
dius vector r , a convenient choice for a reference is the
x y ζ coordinate set. The orientation of the body set rela-
tive £o the x y ζ set may be uniquely defined by three an-
gles, φ, Θ, and φ. These angles are represented by ordered
rotations about the ζ , the y^{1}, and the χ axes, where the

r r r primes refer to the new orientation of the coordinate axes after rotations. The matrix [A] which resolves vectors in the x y ζ set to the body set is given by

ai i ^{a}! 2 ai 3
[A] =

a2 1 ^{a}2 2 ^{a}2 3 ^{[6] }

a3 i ^{a}3 2 ^{a}3 3
where

= cos θ cos* φ *
a^2 = cos θ sin* φ *
a... = -sin θ

13

546

a = -sin* Φ* cos* φ* + sin* φ* sin θ cos* φ* [γ]

= cos φ cos** 0** + sin φ sin θ sin** φ **

= sin φ cos θ

a.

a. 3 2

**= sin**** φ**** sin*** φ*** + sin θ cos 0 cos*** φ *

**= -cos**_{ φ}** sin*** φ*^{ + cos}* φ*^{ sin θ sin}** φ **

**= cos θ COS ^ **

The angles 9^, Θ, and 0 can be written as functions of time or
orbital position α to define the prescribed vehicle orienta-
**tion relative to x^y z^. Then, given j^he desired orientation **
angles φ, θ, and φ, £he components of ν can be expressed in
body coordinates, and Eq. 2 will give the solar radiation
torques for a particular vehicle configuration.

Two particular cases of Earth satellites will be considered here: l) A nonspinning Earth-sun oriented satellite that is required to align one axis with the local vertical and point a body fixed solar cell array at the sun, and 2) a spin- stabilized vehicle requiring the spin axis to be parallel to the normal to the orbit. Other cases may be investigated us- ing the methods developed here; however, the two examples treated are typical of many Earth satellites (communication, surveillance, weather, navigation), and are therefore felt to be significant.

EARTH-SUN ORIENTED SATELLITE

The vehicle described here is required to stabilize with re- spect to the sun and Earth. A typical configuration is shown in Fig. 2 . A body fixed antenna (Z axis) is to be aligned with the ζ axis, and the solar paddles (cells on one side only), which are free to rotate ±90° about the X axis, are to be normal to the sun's rays. The control system is assumed to include a means of storing angular momentum with a maximum storage capability, and a means of angular momentum removal.

Attitude error signals can be given by various sensing devices (horizon scanners, sun seekers, etc.)

A general approach is developed to determine the momentum storage and removal requirements due to disturbance torques acting on vehicles with this capability. This approach in- volves resolving the disturbance torques to an inertial ref- erence where the components can be integrated directly to ob- tain the total angular momentum of the vehicle as a function of time.

547

Newtons Second Law of Motion written for a rotating coordi- nate set is

Τ = Η + ω χ Η [8]

where Τ, Η, and ω are the disturbance torques, the total vehi- cle moment of momentum, and the angular velocity, respectively, expressed in vehicle body coordinates. The moment of momentum

(angular momentum) is given by

Η = I ω + Η [9] s

where H is^the angular momentum of the momentum storage de- vice, and I is the inertia tensor of the vehicle referred to the body coordinate set. Expressing the disturbance torques, the angular rate ω, and the inertia tensor Ϊ in the body coor- dinate set gives a set of three linear differential equations with time varying coefficients to solve for Η . In general, however, direct solution may be quite difficuît.

If the torques can be resolved to an inertial coordinate set, Eq. 8 reduces to

dH.

where the subscript i refers to the vectors resolved in an in- ertial coordinate system. The matrix [B] is taken to repre- sent a general transformation which resolves the vector from the body coordinate set to any inertial reference set. Then Eq. 10 may be written

dH. _

^ = [T.] = [B] [T] [11]

and, EL is given by

[H.] =

* J*

[ Β ] "1 [T] dr + [Ho.] [12]
ο

where Η . is a constant vector and represents the total angu- lar momentum of the vehicle at t = t .

ο

To obtain the angular momentum of the vehicle Η in body co- ordinates, the vector Hj_ must be resolved back to body coordinates

### r

^{ t }

### [H] = [B]^[H

^{O1}

### ] + y [Β

^{- 1}

### (τ)] [τ (τ)] άτ I [ 1 3 ]

Since the orbital position α is a function of time, Eq. 13 can be written

**[H(a)] = [ B****( a) ] | [ H**^{O I}] + j [Β"^{1}** (α)] [Τ (a)] ^ [lU] **

where ω = , and is constant for a circular orbit, ο dt^{ 7 }
In the absence of disturbance torques,the angular momentum
of the momentum storage device is given by

—» —»

**H =** S** if .** - Ï** ω** [ 1 5 ]

**SO 01**** L**** ^****J **

The components of Η along the body axes can be only constants or periodic functions of time.

The orientation requirements for the vehicle shown in Fig. 2 are described by

a) Antenna alignment: ^ = 0 = 0

b) Solar array alignment: (v ·** ê ) = 0 [ l 6 ] **
**X **

**<$ · ê**^{z}^{p )}** = ο [ I T ] **

### (0 · ô

^{w}

### ) =

1 [ 1 8 ] The paddle rotation angle φ relative to the vehicle body axesis shown in Fig. 2 along wi?h the x y ζ coordinate set. Sub- stitution of Eqs. 3 , 6 , and 7 into E§s? Ϊ 6 , 1 7 , and 18 gives the desired yaw and paddle motion

*ψ** ^{±}* = tan"

^{1}[sin (α-77)t a n s'l 0 < 0 < 2π [ 1 9 ]

^ = sin"^{1} [cos (α-77)s in S ' ] - - < ^ < ~ [20]

The solar radiation torques acting on this vehicle together with the components of ν along the body axes are developed in Appendix A. The case of ideal orientation will be chosen to demonstrate the approach of Eq. 8 through l 4 . Therefore, the

**549 **

solar radiation torques are given "by Eqs. A-11 and A -1 3 .
Since the orbital rate (da/dt) = ω (t) is usually much larg-
er than the rate of Earth's rotation about the sun (dS/dt) =
ou^, the angles S, S^{1}** and 77 may be assumed constant over sev-**
eral orbits. Hence the sun reference coordinate set can be
considered an 'inertial" coordinate set over several orbits.

The matrix [B] which transforms vectors from the sun reference set to the body set for ideal orientation is

**[B] **

c o s (a-77) cos 0^ s i n φ^ - s i n (a-77) c o s φ^

- s i n 0^{i} c o s (a-77) **C** **S*** Φι***O** **N****S**** (****I** **A****"7?)** **S** **I****N **

s i n (a**-77) ^** **S**** (****C****Α****"??) ****O**

[ 2 1 ]

The assumption of ideal orientation implies that all changes of angular momentum must be stored in the momentum storage de- vice in order to maintain ideal orientation. This angular mo- mentum change in inertial space due to the solar torques is given by substitution of Eqs. 2 1 , A -1 1 , and A- 1 3 in the expression

α

[Β] - 1 ^{[} ^{T ]} da

**CA) **
**Ο **

[22]

where Η = Η + Β Η . gives the momentum of the momentum
storage^{S}device in boay coordinates.

For a circular orbit this results in terms periodic with frequencies ω and 2ou , and terms which increase monotonically with time. l£e maximum amplitude of the periodic terms gives the minimum momentum storage capability requirements. Integra- tion of Eq. 22 over one orbit gives the average build-up of angular momentum per orbit as

ΔΗ/orbit = ΔΗ , e , = Λ

' xo xo
4L^{Y}cos S^{1 }

ω

**f **

^{pn }

^{K(|sin S'}

^{ I}

^{ ) - E}( [ s i n S11 ) sin S '

+ 7 ** ^{I}** sin S
en 2 s i n S

**1 " . **

**3 ** 1 +

21 s i n S^{ 1 }

Eq. 23 continued on the next page.

550

χ (1 + s i n^{2}s ' U n **(lH-|sin S'l ) **
(1-1** sin** S!I**J **

[ 2 3 ]
jt** sin** S** cos** S

**CO **ο

(L**Y +**** R****o****) 7 ct **

h Ί Λ
2^{ 7}e t J f^{ Θ}:

where *J *

The sun reference set cannot he assumed an inertial refer- ence set for times longer than several orbits. Summation of Eq. 23 in an inertial reference using the angle 77 over one year shows that several components of the equation are peri- odic with frequency of 2ω^. However, if the momentum storage device is required also to store this low frequency periodic momentum, the minimum storage capability requirements are in- creased by approximately the factor 0^/20^. This may or may not be desirable based upon power and weight considerations, depending on the desired satellite lifetime. If it is not de- sirable, the momentum removal requirements per year will be the sum of the magnitudes of the ΔΗ over a year.

For this case, several observations can then be made about
the momentum removal requirements by investigation of Eq. 2 3 :
1 ) When the sun lies in the orbit plane (S^{1} = rt/2), no an-
gular momentum removal is necessary.

2) When the sun is always normal to the orbit plane

(S1 = 0 ) , no angular momentum removal is necessary. However, this can occur only for an orbit plane normal to the ecliptic plane and regressing at the rate of Earth's rotation about the sun ω .

k

**5 5 I **

A = 100 ft Ρ

0.95
**2 2 = 2k** hr

552

3) The moment-urn added per orbit is dependent only upon the c m . - c p . vector component along the Y axis.

A typical set of vehicle parameters is chosen to demonstrate the requirements for the removal of angular momentum for an equitorial orbit (ξ = 2 3 . ^ 5 ° )· The parameters are

L = 0.2 ft A = 12 f t2

Y e A = 20 f t2 V = θ Λ θ

c ρ R = 1 . 7 5 ft h = 5 . 0 ft

The magnitude of ΛΗ is plotted in Fig. 3 for one quarter of a year. The total momentum removal requirement per year is then four times the area under the curve of Fig. 3 , or 2 5 . 2 ft-lb-sec per year.

The example discussed here considered one particular scheme of Earth-sun orientation, namely, paddle motion constrained to ± (rt/2) and yaw motion unconstrained. The extension of these results to other vehicle configurations and orientation requirements changes only the form of T, which may be derived as in Appendix A. The extension to elliptic orbits requires the inclusion of the orbital rate as a function of time in Eq. 2 2 .

SPIN STABILIZED SATELLITE

Spin stabilization is an effective means of attitude control provided that an orientation fixed in inertial space is de- sired. The effects of disturbance torques will-introduce a wobble motion of the spin stabilized vehicle. The vehicle cho-

sen as an example here is a cylindrical vehicle spinning about
its polar axis (see Fig. k-), and required to maintain the spin
axis normal to the orbit plane (z parallel with y^{1}) . The
allowable deviation from the nominal orientation and the mag-
nitude of the wobble will determine what type of control is
required. For example, if the wobble motion is bounded within
the allowable deviation cone, no control system will be nec-
essary; if the total angular momentum vector (H) motion is
bounded and within the allowable cone, but the wobble is not^

a simple passive damper which aligns the spin axis with the H vector will suffice; if neither condition will satisfy, an ac- tive control system will be required to fulfill the orienta- tion requirements by controlling the H vector.

The wobble motion due to the action of the solar radiation
torques is developed. The terms in the disturbance torques
dependent on the small angular deviations of the spin axis
from the nominal and the deviations from the nominal spin
speed relative to the^{ XT}*J** ^{TZT se}*"t> ω >ar e neglected.

The expression for the solar radiation torques on the ve- hicle is obtained from Eq. A-11 of Appendix A by neglecting

^he solar paddle terms (A = θ). However, the expressions for
ν will differ, for the veëicle Ζ axis is to be aligned with
the y^{!} axis rather than the ζ axis as in the first example.

This change is easily made by a simple rotation of -(rt/2) about the χ axis before resolving ν to the body coordinate set.

This results in X Y v„

Ζ _

-sin S^{!} sin (uü^t-rj)

-sin S^{1} cos (ω** t-77) ** [24]

*where u> t = (ω +ω )t. Eq. 2k is substituted in Eq. A-11 to *
r ο s

give the solar radiation torque for this case.

Equation 12 will be used to determine H.(t); however the matrix [B] for this case will be given by

cos (ω** t-77)**** 0**** "****s****i n t****"**^{r}**î ) **

[Β]** a. ** **-sin (ω t-77)** 0 "co s** (^t-77) ** [ 2 5 ]

Substitution of [Β]" in Eq. 12 along with the torque ex- pressions gives the expression for H.(t). Eq. B-8 of Appendix Β gives the resulting expression for S (t).

For a small orbital inclination (ξ-θ) Eq. B-8 reduces to

**—-**

H (t) xo H (t) H (t)

zov

L _ V ( l**+ V** )2Rh

**ZJ**** C **

Z r

sin ou. t k

L _ V ( l**+ V** )2Rh

Δ c (cos ou. t-l) ou,.

[26]

**555 **

For a vehicle with a passive damper, Eq.. 26 describes the
motion of the spin axis of the vehicle in inertial space. This
motion of the vehicle spin axis (z) will trace a cone passing
**through the y^ axis with frequency o^. and half cone angle of **

**L V(l+V ) 2 R h **
Z) c
_ I ω

k Z r

However, with no passive damping in the vehicle, the spin axis will not be coincident with the total angular momentum vector H. The equation of motion of the vehicle Eq. 1 3 , assuming that deviations of the spin axis from the nominal position are small, gives

Ιχ(^-ωΓθ) H (t) cos uu t-H (t) sin ω t-GI^uu

xo v r zoK r Ζ r

-H (t) sin a) t-H (t) cos ω t+czil^ju

xo r zo r r Ζ r [ 2 7 ]

Ζ r Ζ r

where φ and θ are deviation angles as seen by a body fixed observer. Solving Eq. 27 for ^(t) and 9(t) and resolving the vector into the orbital reference set gives the motion of the spin axis in inertial space as

Λ e„

Λ e xo

Λ e
**zo **
Λ e

Ι,ν(ΐ+ ν ) 2 R h

Z) c

sin ω.

k r

sin(lz / l)o)r t ω I / 2 _

r z/l

(cos o) t-l) - cos(l /l)u) t Κ. Z) r ω eu. r k 2 _

r Z/I [28]

Hence, compared with Eq. 26,_^Eq. 28 shows the free wobble mo- tion to be a cone about the H vector with frequency (i^/ljcu^

and half cone angle

*JL„ V(l+V ) 2 R h *
Zi c

I 2^{-}2
Z r

554

For this particular case, the effect of solar radiation torques on the spin stabilized vehicle results in a bounded wobble motion of known magnitude. As the spin frequency is increased, the deviation from the nominal orientation ap- proaches zero as a limit, hence the required spin frequency for a given maximum deviation can be calculated. However, as seen in Eq. B-8 of Appendix B, the H vector motion will become unbounded for ξ ^ 0, and the rate of buildup of the deviation from the y axis will increase as the inclination

ο

increases. However, this rate can be minimized by minimiz- ing the parameter (ν -V ), or by increasing the spin fre- quency 0) . From Eq. 28 it is seen that increasing the spin frequency will also decrease the bounded wobble motion.

The extension to other configurations and orientation re- quirements follows directly from this example and the appendices.

APPENDIX A: SOLAR RADIATION TORQUE FOR AN EARTH-SUN ORIENTED SATELLITE

Eq. 2 of the text may be used to derive the expression for the solar radiation torque on the vehicle shown in Fig. 2 . The following assumptions are made

1) The time spent by the vehicle in the shadow of Earth is neglected in this development.

2) The paddle control is perfect such that the solar radia- tion vector is in the plane normal to the face of the paddle containing the paddle control axis. (Yaw attitude control may be imperfect.)

**3 ) The solar paddles are thin such that the edge effects **
may be neglected, and the paddle c.p. lies at the paddle cen-
troid.

**h) The psCddles cast no shadows on the body and vice versa. **

5) The vehicle c m . does not lie at the centroid; the pad- dles are symmetrical to the vehicle centroid.

The author defines the vector L fixed in the vehicle as the vector from the vehicle c m . to the centroid. Then from As- sumption ( 2 ) , the terms · ô) η and η χ ν χ η become for the paddles

/ Λ Λ \ Λ Λ , **Λ Γ Λ Ι****Ί **

(ν · η) η = v^{y}e^{Y} + ν ^ [Α-1]

555

η χ ν χ η = vx X ** ê** [A-2]

Since the c.p. of the two paddle set lies at the centroid, Eq. 2 gives for the paddle contribution

Τ =** 2 L** x V [(v e + ν_ ê_)(l +V ) A

p Y Y Ζ Z^{ s} Ρ Ρ

Ρ Ρ

The vehicle body may be broken into two surfaces - the end faces and the cylindrical portion.

The c.p. locations relative to the vehicle center of mass for each surface can be calculated by

c T p ^ χ Γ F d A , = Γ (S x F) dA. [Α-Λ]

Α.

ι ι

where S is a vector from the vehicle centroid to a point on
*the ΛΡ** ^{1}* surface. For the cylindrical body considered here,
both ends are taken to be identical with V

^{e}and V

^{c}con-

stant, where the subscripts e and c refer to the end and the curved surfaces respectively. Then, applying Eq. A-h the c.p.

locations are

cTp-^{e} = I (sgn v^{z}) ê ** ^{z}** + L [A-5]

**^ • c =**

* ^rr=*

^{ [ ν}

^{χ}

^{ê}

^{x}

^{+}

^{ V}

^{Y}

^{ ê}

^{i}

^{] + 1}

^{>}

^{ [ A}

^{-}

^{6 ] }

**i T V i - v**z2

where R is the radius of the cylindrical portion and h is the height. The component of Φ along the Ζ axis determines which end is illuminated, hence the coefficient (sgn v^) for c.p.^.

**The terms (Φ · η) η and α χ Φ x â for these surfaces become **
(v • ^{Λ}** ^{Ν}** Λ

η; η

e =^{ ν}ζ

**κ **

**κ**

^{[Α-7] }

(ν ^{• }^{Λ}** ^{Ν}** Λ
η; η

c = νχ

**è**

**è**

^{ + }χ

ν e Λ

Υ Υ [Α-8]

(η χ Λ
ν χ* η) *

e = νχ Λ

ν^{γ}β^{γ } [Α-9]

(η χ Λ
ν χ η; ^{Λ}^

c ^{= Ύ}**ζ **

**Λ **

ez [Α-10]

556

Substitution of Eqs. A-5 through A- 1 0 into Eq. 2 , and com- bining with A-3 gives the solar radiation torque acting on the vehicle

^ = Lz * R2( 1 - V**e****) | ν** ^{ζ}| νγ + § * R2( l - Ve) vzvY

~2

T.

T,

+ L^{Z}2 R h ( l + V^{c}) v^{Y}* J l -* v^{z }

**+ 2**

**V p**

**+ VP) VY s i**

^{( l}

^{1}*'*

** V -**

**2**

**V p**

^{( l + V}**p**

^{) V}

### Z

- νκ2( ΐ+ν6) | νζ| νζ - LY 2R h ( 1 - Vc) vz ^v z2

- 2R h ( l - V^{c}) v^{Y} v^{z }

L^{Z}« R ( 1 - V^{e}) | v^{z}| v^{x} - - * R ( l - V^{e}) v^{z} ν ^{χ }

2 LZ** s**(I-VP)VX

** y ^ ?** ^{ t}

^{A}

^{-}

^{l l ] }

L^{Z}2 R h ( l^{+}V^{c}) v^{x > /}**r i ^ - 2** + ^ Λ2Κ(1 +νβ) | νζ| νζ

**+ 2**

**V p (**

^{1 + V}**p**

^{) V}

**Z >P?**

^{ +}

### Τ

^{2}

^{R h}** k - V**

^{ V}

### X

^{ V}

### Z

+ I^2R h ( 1 - Vc) vz** / î ^ f **

- | =^{ L y}2 R h ( l+Vc) vx / ï ^ ^ LY Jr R2( l - Ve) vx| vz|

*- h « R*^{2}**a -**^{v e}**) v**^{Y}**| v**^{z}**i - 2 i**^p( i+v^{p )}**v**^{y}** y î** ^ 2

**- V**

**R h ( l + V**

**c**

^{} V}### Y

* 2 *
X

2

The expression for the components of ν along the body axes is obtained by resolving Eq. 3 to the body axes using Eqs. 6 and 7 for φ and θ small

557

**- cos 0 sin (α-77)**sin S! + s^n* Φ* ^{c}^{o s}* &*^{l }**- θ cos (α-77)**s i n S'

**sin 0 sin (α-77)**** s****^-*** n* S! + c os

**0 cos S**l

**+ ^ cos (α-77)****s** **i n s***** **

**θ [-cos 0 sin (α-77)****s** **i n s****'**** + ** **s** **in*** Φ*^{c} ^{o s s}'l
**+ cos (α-77)****s** **i n s****' **

**- <j>**** [sin 0 sin (α-77)****s** **i n s****'**** + ** **c** **os**** 0 cosS'] **

For ideal control as defined by Eqs. 8, 9 and 10,the compo- nents of § along the body axes become

[A-12]

### y

^{2 2 }

1** - cos (α-77)**s i n S*

**cos (α-77)** si-n s'

[A-13]

Either Eq. A- 1 2 or A- 1 3 may be substituted in Eq. A -11 to give the expression for solar radiation torques on the vehicle.

APPENDIX B: EFFECTS OF SOLAR TORQUES ON SPIN STABILIZED VEHICLE

The expression for the solar torques resolved along the sun reference axes is obtained using Eqs. 2k, 2 5 , and A-11 and is given by

Τ , xo

zo

**A ( s ' ) + λ****1** (S')** [-1^** sin** (üJ****r** t-77)
- cos (u)^{r}** t-77)] **

X^{1}( S^{l}) [-Ly sin (u^t-77) + 1 ^ cos (u^t-77)]

X^{2}( S^{l}) [L^{y} sin** (uJ**^{r}**t-7?)**** - 1 ^** cos (a^t-77)]

[B-l]

**55S **

where

A(S') ^{- L}^{z} V s i n S1 [2R h ( l + Vc) | s i n S'|

+ *R2 ( l - Ve) | c o s S'|]

+ *r 2 h V (V . ν ) c o s S^{1} s i n S'

2 N e c

[B-2]

**X****1****( s ' ) **

**X****2****( s ' ) **

[«R ( l + Ve) | c o s S'| + 2R h ( l - V ^ s i n S'|]V c o s S1

[2R h (l+V** )I** sin S

**+**** J Î**R^{2} (l-V^{e})|cos S'|] V sin S^{1 }
Substitution of Eq. B-l into 12 with t
results in

[B-3]

**[B-V| **

0 and** H . = I ω ê , **

οι Z r yo

**H , **

xo

**Η , **

**Η**** t **

ζο

**I ω **Ζ r

At 1

**+ — [ 1 ^ cos (ayt-rç) - Ιγ sin(ayt-7?) **
r

**- cos 77-L**Y** sin 7?] **

λ1

**— [ 1 ^** cos(œ^{r}t-r?)** +**** 1^ Β ΐ η****( ω****Γ****ΐ - η ) **

r

- cos** 77 + 1 ^** sin

** η] **

**η]**

— [-LY cos (ü)t-t|) - L sin (ω t-r?) r

+ Ly cos rj-I^ sin** 77] **

The terras with coefficients (λ./ωΓ)ar e periodic with the
spin frequency relative to inertial space, ω ; hence in η
**Earth orbits, At = Δ (2τΐη/ω ). Therefore since |λ | == |Δ|, **
Eq. B-5 may be approximated by

**H**** t ** *At *
xo

**H**** t **

yo Z r

**Η , ** 0

**_ **

t « 1 year [B-6]

The average angular momentum change per orbit is then just
(2αΐ/ω ) A(S^{f}) e 1 . Since the sun reference set is slowly
rotating relative to the orbital reference set, the average
angular momentum of the vehicle relative to inertial space

becomes

559

m » l

**I. = La) e + f — A(s')[cos ne - sin 7?ê ] dk **
ι Ζ r yo / ω v /L ' xo ' zoJ

**Λ****= ° ° CB-7] **

where k is the number of complete orbits the integration in- cludes, given by

k = t/(2*/a> ) [B-8]

*Substitution of Eqs. k, 5*^{Λ} B- 2 , and B-8 in Eq. B-7 gives the
average angular momentum in inertial space

k

L^{z} V(l+vJ 2R h

+ 2αι sin k

+ Vc^{} } ^{2 R h} / 2 2

**τ^-ξ [sin ξ sin cu^t ^1-sin ξ sin co^t **

+ sin""1"(sin ξ sin
L^{z} VrtR^{2}(l-V^{e}**)|sin ξ I **

+ g- sinco^{k}t|sina^t|

k H (t) = Τ ω

yo^ Ζ r 2

H (t) = **f**** h****n**** V**** ( V - V** )cos ξ sin ξ (2ω t - sin 2ai t)

zo ta,** e** c k k

k

L ^ V U**+ V )2R h . 2**^{ t }

Z c^{ i .}** Γ**** / , _ \ /_ s i n ξ **

**+ cos ξ [(cos (i^ t - D l l 2~"^ **

k

5** s i n ξ \ η**** r **

**-** **.** **.** **.** **)****] L**B- 9]

L V:tR2(l-V )|sin £|cos ξ

**+ gsr ^ "****c o s**** ^ l** **8** **1** **1 1**** < ν ' **
k

\ sin** a i** t

**I**** I I sin a^t I **

where η is even and chosen such that I t - — I < —

REFERENCES

1 Roberson, R.E., "Attitude control of a satellite vehicle - an outline of the problems", presented at the Vlllth IAF Con- gress, Barcelona, Spain, 1 9 5 7 .

2 S°lar Constant, Handbook of Chemistry and Physics, 3 4 t h

ed., ρ. 2 8 7 5 ( Chemical Rubber Publishing Co, Cleveland, Ohio).

3 Roberson, R.E., "Methods of attitude sensing," Proc. IAS Manned Space Stations Symposium, Los Angeles, April 2 0 - 2 2 ,

I 9 6 0 .

4 Richtmeyer, F.K. and Kennard, E.H., Introduction to Modern Physics (McGraw-Hill Book Co., New York, 1 9 4 7 ) , 4 t h ed.

5 Goldstein, H., Classical Mechanics(Addison-Wesley Pub- lishing Co., Reading, Mass., 1 9 5 7 J L

6 Katz, A.J., "Determination of thermal radiation incident upon the surfaces of an earth satellite in an elliptic orbit, presented at IAS National Summer Meeting, Los Angeles, June 2 8 - July 1 , I 9 6 0 .

7 McElvain, R.J., "Solar radiation torques acting on an earth-sun oriented satellite," Tech. Rep. T R- 6 0 - 0 0 0 0 - 0 4 l 6 4 , Space Technology Laboratories, Inc., El Segundo, Calif., June i 9 6 0 .

8 McElvain, R.J., "Estimation of gas consumption due to solar radiation torques acting on an earth-sun oriented ve- hicle," Internal Document 7 ^ 3 1 . 1 - ^ 1 , Space Technology Labo- ratories Inc., El Segundo, Calif., June 2 9 , i 9 6 0 .

56I

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