The method circumvents many of the existing difficulties in computational logic presently encountered in the direct analytical or numerical evaluation of the appropriate surface integral. It may be applied to complex spacecraft structures for computing the total force arising from either specular or diffuse reflection or even from non-Lambertian reflection and re-radiation

Chan, F. K.

English

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A GENERAL METHOD FOR COMPUTING THE TOTAL SOLAR
RADIATION FORCE ON COMPLEX SPACECRAFT STRUCTURES
F. K. Chan
Scientific Analysts and Consultants, Inc.
4114 Heathfield Road, Rockville, Md. 20853
ABSTRACT
A general approach has been developed for computing
the force due to solar radiation on an object of arbitrary shape.
This method circumvents many of the existing difficulties in
computational logic presently encountered in the direct analytical
or numerical evaluation of the appropriate surface integral. It may
be applied to complex spacecraft structures for computing the total
force arising from either specular or diffuse reflection or even
from non-Lambertian reflection and re-radiation.
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https://ntrs.nasa.gov/search.jsp?R=19820002193 2020-03-21T11:58:32+00:00Z
SECTION I - INTRODUCTION
The problem of computing the total force or total torque
on a spacecraft due to solar radiation is, in general, very difficult.
Mathematically, itrequires the evaluation of a surface integral
over only the illuminated region of the surface. Even if the illu-
minated region is known by some other means, the evaluation of
the surface integral can still be very difficult analytically
in the case of complexspacecraft structures. Moreover, if the
illuminated region is not known a priori, the difficulties are
compounded by having to determine self-shadowing. For non-convex
objects, it is not trivially governed by a condition such as
cos e_O where e is the angle between the sun vector and the out-
ward vector normal to the surface. In fact, the logic in the present
methods becomes extremely complicated and is also not fool-proof.
Additional difficulties are introduced by choosing a set of points
(vertices) on the surface to form a network in approximating it;
this inadvertently leads to book-keeping problems associated with
selecting appropriate sets of points for computing surface elements.
This paper presents a general method for performing the
computations without encountering the difficulties described above.
It does not attempt to evaluate the surface integral directly as
it presents itself as done in the usual methods, but considers
the same problem from a slightly different point of view which
leads to the same results.
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SECTION 2 - ANALYSIS
Consider an arbitrarily shaped object as illustrated in Figure 1.1.
f
(# t.o;_,t,_..d ..,,.t.,)
Figure 1.1 - Illustration of an Arbitrarily Shaped Object
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For convenience, let us use the following notation:
^
u = unit vector along a specified direction
^ A A A
v = any unit Vector orthogonal to u, i.e., u • v = 0
w = third unit vector forming orthogonal triad, i.e., w = u x v
0 = origin of coordinate system
P = any point On object's surface
r = vector from 0 to P
P'= projection of P onto (v,w)-plane
_'= projection of_onto (v,w)-plane
(x,y,z) = reference orthogonal system for describing object's
surface.
^
In the present analysis, it is advantageous to choose u to be opposite
in direction to the incident solar radiation. (Alternatively, it can
also be chosen to be in the same direction.)
A
The vectors u and r are known in the (x,y,z) system.
In general, if V is any vector, then it may be more explicitly
written in the (x,y,z)-space as V_x,y,zJ and has components Vx, _V' Vz.
That is, we implicitly mean
In view of the definition of the vector v, we may choose
v = O. Then, it may be shown that the other two components are given byg
From the definition of w, we obtain
Therefore, any vector V(x,y,z) can be transformed to V(u,v,w)
by the equation
Vc.,_,.._= T Vc_,_'_) 0"_)
where the transformation matrix T is given by
T = r= _ o (,,,€)
,.,&
Then, using equation (1.6), the vector r(x,y,z) is transformed
and we obtain
Consequently, the projection vectorS' is simply given by
The component r of the vector r is particularly important because,u
for a complex spacecraft structure, it can be used to yield the surface
element which is directly exposed to solar radiation. This can be seen
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as follows: For any given point on the (v,w)-plane (i.e., for any
given vector _' ), the point on the spacecraft which is not shadowed
is the one which has the maximum value of ru, independent of where
the origin of the (u,v,w) coordinate system is chosen. (It would be
the minimum value of r if the vector _ had been chosen to be in
u
the same direction as the incident solar radiation. ) To find the
illuminated surface of the spacecraft, we proceed by dividing
the (v,w)-plane into cells of area _v_w with cell centers (vi,wj).
At these cell centers, the illuminated surface element is the one
which has the maximum value of r . In this way, the logic of deter-
U
mining self-shadowing is extremely simple as compared to other methods
which encounter considerable difficulty conceptually and computationally.
Thus, given a vector _' = (0,vi,wj), the vector _ _ ([ru]max,Vi,Wj)
corresponding to the illuminated point is determined. It is then
transformed to the (x,y,z)-space by the equation
= "T"r (/,H)
n(x,y,z ) normal to the surfaceAt this point r(x,y,z), the unit vector A
is then obtained by
- (/,1.0
where _(x,y,z) = 0 denotes the equation of the surface in a region
"_ n is chosen such thatcontaining r. For convenience, the direction of ^
A A
>i0 (11 )
This choice of direction automatically makes _ the outward unit normal
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if the surface element belongs to a closed surface. Moreover,
.
it establishes a direction for n In the case of a surface for which
an outward unit normal is meaningless (such as a finite planar surface).
A
The vector n(x,y,z ) is then transformed to the (u,v,w)-space using
the equation
The cell (vi,wj) whose area is _v_w corresponds to
a surface element whose area is denoted by _A. It is evident that
we have
_A = = (/,_s)
Therefore, the force_F exerted on this surface element is given by
ziF = p AA (/,1_,)
°
where p xs the solar radiation pressure vector acting on the surface
element. Under very general conditions of surface reflection and
re-radiation, it can be shown that this pressure vector has the form
Lc, s. c,) o,,7)
where S is the solar radiation flux per unit area normal to the flux,
c is the velocity of light, and _is the angle between the sun vector
and the normal to the surface element, i.e.,
A
o_# -- n._ CI'/_')
The coefficients CI, C2 and C3 may change with time due to aging of
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of the surface material by some complex process.
For the case of specular reflection and diffuse (I_mbertian)
reflection, the Ck'S are given by (I)
where kI - the fraction of incident radiation reflected specularly
k2 = the fraction of incident radiation reflected diffusely
by a Lambertian surface.
It is to be noted that in equations (1.19) - (1.21), it is not
implicitly assumed that the surface is radiating the entire energy
incident on it, i.e., it is not necessary that we require the
condition kI + k2 - I in order to obtain these equations.
For the case of specular reflection and non-Lambertian
reflection and re-radiation (2), a little consideration will reveal
that the Ck'S are given by
c, = (/- I
= afr 0. 3)
where Y - the fraction of incident radiation reflected (specularly
and otherwise)
- the fraction of reflected radiation that is specular
2-8
Bf, _ = non-Lambertian coefficients for front and back
surfaces respectively
ef, eb = emission values for front and back surfaces
respectively.
In passing, it may be noted that we have the relations
4_= _(/-77 Ct,2_)
Moreover, it may be remarked that the form of equation (1.17) is
valid for the more general non-Lambertien reflection and re-radiation
which have a period of 7fin the azimuthal variable. In other words,
Lambertian reflection means that the intensity I of the reflection
is given by
Then, the case of non-Lambertian reflection and re-radiation expressed
by equation (1.22) would correspond to an intensity which is indepen-
dent of the azimuthal variable _ and is of the form
where implicitly we exclude the case of Lambert's law, i.e.,
/(_) # _ _ C,',._q)
The even more general case means that we can have reflection and
re-radiation for which the intensity is of the form
I = Io j:(o_.) 0,3o)
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T"
Finally, to compute the total force F due to solar radiation,
we obtain from equations (I.15) - (I.19) the following expressions
It is also trivial to compute the total torque M on the spacecraft
by using the equations
4/4 = ,n. x _ F CI'3s')
but this will not be done here.
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SECTION 3 - DISCUSSION
It is obvious that the method just discussed does not
encounter logic problems in determining self-shadowing. Moreover,
because the points _vi,wj_ are first chosen on the projection plane,
it circumvents the difficulties in book-keeping experienced in
the other method of choosing vertices on the surface of the object.
Furthermore, it does not require excessive core for storing the
vertex data such as coordinates, area of surface element, normal
vector, solar incidence angle, etc. This advantage becomes evident
by evaluating the expressions in equations (1.32) - (1.32) using
three accumulators (one for each force component), not having to
store the set of points {vi,wjl . Finally, if greater accuracy is
desired, it suffices only to choose smaller values _v*_w*, multiply
the previous result by the factor (_v*_w*)Av A w , and then perform
computations only for the additional points newly introduced into
the set Iv.,w I. This advantage cannot be realized in the other] j
method of choosing vertices on the surface of the object. In that
case, in going to a refined model with additional vertices, it is
necessary to perform the entire computations starting from the
beginning each time.
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SECTION4 - CONCLUSION
From the foregoing discussion, it may be concluded that
the present method has the following advantages:
I. It does not experience logic problems in determining
self-shadowing.
2. It does not encounter the book-keeping problems arising
in the case of choosing vertices on the surface of the
object.
3. It does not require excessive core for storing vertex
data.
4. It can utilize previously0btained results in going to
progressively more refined models.
REFERENCES
(I) Eliasberg, P. E., "Introduction to the Theory of Flight of
Artificial Earth Satellites," Moscow (1965). Translation,
Jerusalem (1967).
(2) Wright, J., "Solar Sail Force Model," Jet Propulsion Laboratory
Interoffice Memorandum 312/77.3-201 (28 March 1977).
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A general method for computing the total solar radiation force on complex spacecraft structures

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