An investigation was performed to determine the feasibility of using a two plate model to approximate the Tracking and Data Relay Satellite (TDRS) in orbit propagation, taking into account the effects of solar radiation pressure. The two plate model comprises one plate which always points to the Earth, and the other which is hinged to an axis normal to the orbital plane and is always rotated so that its normal makes a minimum angle with the direction of the sun. The results indicate that it is sufficient to take three parameters, the areas of the two plates and the reflectivity of the Earth pointing plate, to achieve an accuracy of one meter during a 24 hour orbit propagation

Chan, F. K.

English

NASA Technical Reports Server

FEASIBILITY STUDY OF USING
A TWO-PLATE MODEL TO APPROXIMATE
THE TDRSS SOLAR PRESSURE EFFECTS
F. K. Chan
Systems and Applied Sciences Corporation
6811 Kenilworth Avenue Suite 500
Riverdale, Maryland 20840
ABSTRACT
An investigation was performed to determine the feasibility
of using a two plate model to approximate the Tracking and Data
Relay Satellite (TDRS) in orbit propagation, taking into account
the effects of solar radiation pressure. The two-plate model
comprises one plate which always points to the earth, and the
other which is hinged to an axis normal to the orbital pland and
is always rotated so that its normal makes a minimum angle with
the direction of the sun.
The results indicate that it is sufficient to take three par-
ameters (i.e., the areas of the two plates and the reflectivity
of the earth-pointing plate) to achieve an accuracy of one meter
during a 24-hout orbit propagation.
This work was supported by NASA Contract No. NAS5-25139.
The authors wish to acknowledge the programming assistance
provided by Michael Toporek.
I-i
https://ntrs.nasa.gov/search.jsp?R=19810002563 2020-03-21T08:35:05+00:00Z
SECTION 1 - INTRODUCTIOI_
Most of the work involving solar radiation pressure on
orbiting satellites has so far been limited to those which are
spherical and which have circular nominal orbits. The compara-
tively few studies which are less restrictive are still based
on very simplified models. For example, the work of Eliasberg (I)
deals with an elliptically orbiting spherical satellite and is
concerned with the first order perturbation effects expressed
in terms of Keplerian elements. The work of Fang (2) deals
with a circularly orbiting spherical satellite with a perfectly
reflecting earth-pointing disk. It is concerned with the first
order effects expressed in terms of along-track, cross-track
and radial components. Moreover, it also deals with the physical
insights into the modeling errors connected with tracking and orbit
determination of the Tracking and Data Relay Satellites (TDRS).
On the other hand, the work of Georgevic (3) deals only with the
computation of solar radiation force on a cylinder and on a
parabolic reflector, but does not deal with an orbiting satellite.
(Even then, the computation for the parabolic reflector is further
simplified by assuming that the ratio of the force on the illumin-
ated area of the reflector and the force on the whole area of the
reflector is the same as the ratio of the corresponding projected
areas. It is obvious that this assumption, introduced to eliminate
the cumbersome self-shadowing effects, is not really correct.)
The present work is concerned with the solar radiation
effects on the TDRS illustrated in Figure i.i, and modeled
as comprised of the 69 components listed in Table i_i. (In
the course of the present study, a novel method, simple in
comparison to other existing methods, for computing self-shadowing
was formulated but this consideration was not included in computing
the net solar radiation force on the satellite). The orbit of
the TDRS is taken to be representative of a realistic one in that
it is not nominally perfectly circular. The study also considers
the question of how accurately the 69-component TDRS can be
1-2
approximated by a two-plate model, i.e., one plate is hinged tc
an axis normal to the orbital plane and is always rotated to
make a maximum angle with the sun, while the other plate is
always earth-pointing. This two-plate model has the capability
of handling up to four solve-for parameters, i.e., the area and
reflectivity of each of the two plates.
Section 2 is concerned with the analysis of a differential
correction procedure to obtain the values of these four solve-for
parameters. A reference orbit for the 69-component TDRS is
first generated. Its orbital position at regular intervals of
one hour is then used as epoch elements of the two-plate model
to obtain the values of the parameters which yield the best
approximating orbit over the next 24 hours.
Section 3 summarizes the numerical results obtained in
this feasibility study, and presents tabulated and graphical
results for rapid comparisons.
Section 4 discusses the quality of the results, and the
applicability of the two-plate model for use in orbit determination
purposes.
1-3
FIGURE i.i THE TDRS SATELLITE
1-4
Component
1
2
3
4
5
6
7
9
I0
II
12
13
14
15
16
17
18
19
20-69
TABLE 1.1
DETAILS OF 69-COMPONENT MODEL
Description Area (m2) Reflectivir 7
Solar Panel 1 14. 7553 0.0
Solar Panel 2 14.7553 0.0
SCL Antenna 3.13761 0.5
C-Band Antenna 2.67112 0.5
Solar Sail 0.90593 1.0
Antenna Feed
1 (top) 0.00462 1.0
Antenna Feed
2 (top) 0.00462 1.0
Antenna Feed
i (bottom) 0.29570 1.0
Antenna Feed
2 (bottom) 0.29570 1.0
Antenna Feed
i (side) 1.11771 1.0
Antenna Feed
2 (side) 1.11771 1.0
Main Body (top) 4.03665 0.5
Main Body
(bottom) 4.03665 0.5
Main Body
(side I) 1.1599 0.5
Main Body
(side 2) 1.1599 0.5
Main Body
(side 3) 1.1599 0.5
Hain Body
(side 4) 1.1599 0.5
Main Body
(side 5) 1.1599 0.5
Main Body
(side 6) 1.1599 0.5
Sections of Computed by
POLYN 1.0
stationary
antennae
Components of _:orma!
X Y z
Normal makes
minimum angle
with sun
II I, _0
Points to
White Sands
Points to
Los Angeles
0 0 i
0 0 1
0 0 1
0 0 -I
0 0 -I
0 0 1
0 0 -I
0 1 0
.866 .5 0
.$66 -.5 0
0 -I 0
-.866 -.5 0
-.866 .5 0
Computed by POLIq_
1-5
SECTION 2 - ANALYSIS
This section is concerned with the analysis of a
differential correction procedure to obtain the values of
the four solve-for parameters incorporated into the two-plate
model.
For convenience, let us introduce the following
notation:
f(t) = position vector of 69-component TDRS
at time t
_(t, _) = position vector of two-plate model at time t
= (_,_,_,,_4_ = 4 parameters for two-plate model
Q = loss function defined
25
as Z
£--!
2
The problem is then to obtain the values of _ such that Q
is a minimum. It is obvious that the minimum of Q is given by
the necessary condition _. -- 0 where j = i, 2, 3, 4.J
The loss function Q may also be written as
2£
(2-1)
The necessarycondition for minimum is
1,2,3,4
(2-2)
1-6
The function _(ti,_ ) may be expanded in a Taylor series about
an a priori value _, .
,_ + ....a_Xk-- _'Xk -+ (2.3)
where _k is defined as
Let
L_o(}_ = o4 k - cx_s k k=i,2,3,4
(2.4)
(2.5)
Substitution of equations (2.3) and (2.5) into (2.2) yields
to first order
6:I k=_
j=i,2,3,4 (2.6)
Interchanging the order of summation yields
Z
Let
j=I,2,3,4
j,k=l,2,3,4
(2.7)
(2.8)
I.-|
1-7
(2.9)
Equation (2.7) then becomes
4
k--I
j=i,2,3,4 (2.10)
It remains to solve for _Wk where k=i,2,3,4. This constitutes
the first iteration in the differential correction procedure to
solve for the value of _ such that Q is a minimum.
1-8
SECTION 3 - RESULTS
This section summarizes the numerical results in this
feasibility study, and presents tabulated and graphical results
for rapid comparisons.
Numerous computer runs were made for the TDRS with epoch
elements:
XO = 31,662,513.0 m
Yo = -27,523,890.0 m
Zo = 0.0m
= 2,012.15997 m/sec
o
Y = 2,314.7253 m/sec
o
Z = 376.58528 m/sec
o
One set of runs had epoch time set at Day 183.0, Year 1980 (i.e.,
July i, 1980 which is close to the summer solstice), and another
set had epoch time set at Day 275.0, Year 1980 (i.e., October i,
1980 which is close to the autumnal equinox). Each of these sets
of runs was made for the cases of N=I,2,3 and 4 parameters. The
parameters were aligned in the following sequence which is
probably the order of decreasing importance:
_i = area of sun-pointing plate (M 2)
_2 = area of earth-pointing plate (M 2)
e3 = reflectivity of earth-point plate
e 4 = reflectivity of sun-pointing plate
The initial values of these parameters used in the first iteration
of the differential correction procedure were taken to be the
following:
N = 1 2 3 4
= 36.6 29.5 29.5 29.5
a0,1
= 0.0 18.81 18.81 18.81
s0,2
= 0.0 0.0 0.74 0.74
0,3
= 0.0 0.0 0.0 0.0
_0,4
The tolerance e for testing convergence of the iterations
1-9
(i.e.,l_l<£) was taken to be 0.01. The values for the stepsize
for computing the partial derivatives was taken to be 0.i. In the
runs described above, convergence was achieved after two or three
iterations. Figures 3.1 and 3.2 summarize the results of these
computer runs. The symbols appearing on the horizontal axis in
these figures have the following connotations:
0 = The sun is overhead with respect to the satellite
U = The sun is underfoot with respect to the satellite
A = The satellite is moving away from the sun
T = The satellite is moving toward the sun
1-10
e,,I
2 2
I
o
I
Q
I
IY o/._ VIA _(Z
..g
11
z
!
i-ii
II It
2
I
O
I
0
f
• tVC/J-V/A_"
i I
:
?
°\ :
0
OX
1-12
SECTION 4 - CONCLUSION
This section is concerned --_ - _-_ _ .... =_{_n _ eh_
results obtained in Section 3, and also an attempt to relate
them to results obtained in other investigations.
From Figures 3.1 and 3.2, it is seen that the following
observations may be made:
i. The maximum deviation of the two-plate model from
the 69-component TDRS is essentially cyclical with
a 12-hour period.
2. The magnitude of these deviations decreases as the
number N of parameters increases, as would be
expected.
3. There is a pronounced phase change of this period
curve in going from the case of N=I to N=2.
4. The curves for the cases of N=3 and N=4 are essentially
identical. Moreover, the amplitude of oscillation is
so small that they are almost constant.
A computer run was also made comparing the 69-component TDRS
with and without solar radiation pressure effects. The results
are plotted in Figure 4.1.
It is noted that in this case the curve is essentially sinusoidal,
unlike those in Figure 3.1 and 3.2. Moreover, the maximum deviation
occurs when the sun is overhead or underfoot in Figure 4.1, unlike
the cases of N=I and N=2 in Figures 3.1 and 3.2.
Finally, it is interesting to recall the results obtained by
Fang (2) who observed that:
i.
•
In orbit propagation, the least perturbation occurs
when the sun vector is parallel to the satellite
velocity vector in the beginning, and the worst
perturbation occurs when the sun is overhead or
underfoot in the beginning.
TDRS orbits determined from a one-day tracking arc
tend to be less sensitive to solar pressure errors
if the tracking arc begins when the sun is directly
overhead or underfoot. This is contrary to the
(previous) result for solar pressure perturbations
in the absence of tracking•
1-13
"<---0
l----_
¢r-- _
5
NO/J. V /A3 G
¢>
-Pi
1-14
From the numerical results of this feasibility study, it is
seen that the 69-component TDRS can be accurately replaced by
the two-plate model. It suffices to take only three parameters
to achieve an accuracy to within about one meter. Moreover, it
is sufficient to use only one approximating orbit throughout
the 24 hour period, instead of 24 approximating orbits regularly
spaced throughout the day.
REFERENCES
(i) Eliasberg, P.E., "Introduction to the Theory of Flight of
Artificial Earth Satellites," Moscow (1965). Translation,
(2) Fang, B.T., "Short-Term Solar Pressure Effect and GM
Uncertainty on TDRS Orbital Accuracy--A Study of the Inter-
action of Modeling Error with Tracking and Orbit Determina-
tion," EG&G/Washington Analytical Services Center, Inc.,
Report No. 002-78 (1978).
(3) Georgevic, R.M., "The Solar Pressure Force and Torque Model,"
J. Astron. Sc., Vol. XX, No. 5, pp. 257-274. (1973).
1-15


Feasibility study of using a two-plate model to approximate the TDRSS solar pressure effects

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