The effects of solar radiation pressure on the motion of an artificial satellite are obtained, including the effects of the intermittent acceleration which results from the eclipsing of the satellite by the earth. Vectorial methods have been utilized to obtain the nonlinear equations describing the motion, and the method of Kryloff-Bogoliuboff has been applied in their solution

Bryant, Robert W.

English

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THE EFFECT OF SOLAR RADIATION
PRESSURE ON THE MOTION OF AN
ARTIFICIAL SATELLITE
by
Robert W. Bryant
Goddard Space Flight Center
Greenbelt, Maryland
NATIONAL AERONAUTICS AND SPACE ADMINISTRATION
WASHINGTON September 1961
https://ntrs.nasa.gov/search.jsp?R=19980227979 2020-06-15T22:57:25+00:00Z

lZ
THE EFFECT OF SOLAR RADIATION
PRESSURE ON THE MOTION OF AN
ARTIFICIAL SATELLITE
!
by
Robert W. Bryant
Goddard Space Flight Center
SUMMARY
The effects of solar radiation pressure on the
motion of an artificial satellite are obtained, includ-
ing the effects of the intermittent acceleration which
results from the eclipsing of the satellite by the
earth. Vectorial methods have been utilized to obtain
the nonlinear equations describing the motion, and
the method of Kryloff-Bogoliuboff has been applied in
their solution.

CONTENTS
D
!
Summary ...........................
INT RODUC TION ......................
THE KRYLOFF-BOGOLIUBOFF METHOD ....
THE CHANGE IN ENERGY PRODUCED BY
RADIATION PRESSURE ..............
THE COMPLETE SYSTEM OF EQUATIONS
FOR THE OSCULATING ELEMENTS ....
ACKNOWLEDGMENT ..................
References .........................
iii

THE EFFECT OF SOLAR RADIATION PRESSURE
ON THE MOTION OF AN ARTIFICIAL SATELLITE
by
cD
o
i
Robert W. Bryant
Goddard Space Flight Center
INTRODUCTION
In the motion of artificial satellites with large area-to-mass ratios, such
at the Echo I balloon satellite, the magnitude of the solar radiation pressure
effect is substantial and must be taken into account in the analysis of the track-
ing data. It is of fundamental importance in any study attempting to determine
atmospheric density from satellite drag data to distinguish the energy loss (or
gain) produced by radiation pressure from the loss due to drag. Furthermore,
the effect is important since the orbital lifetime of a satellite will be substan-
tially affected by the changes in energy resulting from radiation pressure. Sev-
eral authors have considered the effects of solar radiation pressure on the motion
of an artificial satellite (References i, Z, and 3); it is the purpose of this paper
to present these effects, including that of the earth's shadow, in the form of non-
linear equations derived vectorially and suitable for use with electronic computers.
THE KRYLOFF-BOGOLIUBOFF METHOD
H. Lass and J. Lorell (Reference 4) have applied the method of Kryloff-
Bogoliuboff (Reference 5) in the following low-thrust cases: constant radial,
constant transverse, and intermittent.
The method may be described simply: A nonlinear equation
Zwhere
dx
-- = #f(x, sin t)dt
is a small number, becomes, upon integration,
x(t+27r) - x(t)
27T
t+2_r
I f(x,
- 2_
t
sin r) dr. (1)
Since _z is small, x may be considered constant during the integration from
r = t to r = t+ 277. Since sin t is periodic, Equation 1 becomes
2_
/z _0 f(x, sin T) dr = _zF(x) . (2)x(t + 27r) - x(t) _27r
Replacing the left side of Equation 2 with dx/dt, since the slope of the secant
line is approximately that of the tangent, gives
2_T
d----t'dx_ _/z i f(x, sin _-) dr = _zF(x)
This result may be obtained simply by averaging the original equation over
one cycle of the motion with respect to time.
THE CHANGE IN ENERGY PRODUCED BY
RADIATION PRESSURE
The effect of radiation pressure, if the earth's shadow is neglected,
will provide only short period terms in the development for the semi-major
axis of the orbit (Reference 6). However, the inclusion of the shadow effect
results in quite a different phenomenon.
From the energy integral and Brown's _/dt operator (Reference 6), we
obtain
da 2a 2 v • F
d----t - _ (3)
The velocity vector v may be represented in terms of a unit vector P
directed from the center of the earth to perigee and a vector Q at right
angles to P such that P, Q, R, with R the unit vector in the direction of
the angular momentum vector, form a right handed system. The vector F
is the disturbing acceleration, a is the semi-major axis, and /_ = GM is the
gravitational constant multiplied by the mass of the earth.
We have
Q_/_a(l-e 2) cos E - P _/_--aasin E
0
I
t3
and
where
e =
E =
F =
u 0 =
r =
and we have
F - Fu 0 ,
the eccentricity,
the eccentric anomaly,
the magnitude of the disturbing acceleration,
the unit vector in the direction of the Sun, and
a(1- e cos E);
p
-- COS i
+ cos i
sin w sin fl + cos _0 cos _ Jsin w cos f) + cos a_ sin
+ sin i sin oJ
Q
- COS i COS OJ sin _ - sin oJ cos _J+ cos i cos oJ cos f) sin _o sin
+ sin i cos oJ
R _
+ sin i sin _1
- sin i cos
+ cos i
4Hence we obtain for Equation 3:
da _ -2a2F (Q.uOav__e 2 cos E- P-u° a sin E ).
dt r _V_-_-
(4)
Considering the slowly varying quantities to be constant during one revolu-
tion, and using
d r d
dE - an dt '
we obtain
Aa --- • cos E dE - P-u 0 sin E dE
n _x/_--a -1 _1
( /2a3F Q • u O v_ - e 2 sin E + P • u 0 cos E
/z E1
(5)
!
I-
¢
c
¢,
The quantities E 1 and E 2 are the eccentric anomaly at the exit and entrance
of the shadow, respectively, and are obtained from the equation
r..° = __/r 2_p2 (6)
or, in terms of E,
P. u 0 (cos E- e) + Q. u°v/1-e 2 sin E : -V (1-e cos E) 2 - (_Pa) 2 . (7,
where p is the radius of the earth. The solution of Equation 7 is readily
obtained by the iterative technique and the entrance and exit roots are easily
identifiable. We may write the expression for Aa as follows:
E2-E1 2( (El+E2 ¢ )Aa - 4a3F/x sin _ V(Q.u O) 1-e 2) + (P • uO) 2 sin 2 + , (8)
where
Q.u 0 v_-e 2
= arctan
_p. u 0
5An examination of this expression shows that: (1) if there is no shadowing,
sin (E2-E1)/2 = 0, and therefore Aa = 0; and (2) if e = 0, then ¢ becomes
- (E 1+E2)/2 from a consideration of Equation 7, and again Aa = 0.
THE COMPLETE SYSTEM OF EQUATIONS
FOR THE OSCULATING ELEMENTS
The equations for the elements may be obtained (Reference 7) as
follow s :
da 2a 2 v" F
d-t = _ '
dedt - V_ (l-e2) (T cos E + N) ,"
di rW
- cos (o_+ f) ;dt
_a(1 - e 2 )
d_
sin i _ :
rW
¢_za( 1 - e 2)
sin (oJ+ f) ;
dw dlq
d-Y- + cos i-_
where
T:F'Rx r O
N:F'Q,
W:F'R,
f = the true anomaly,
L = F" P, and
r 0 = the unit vector in the direction of r.
The complete procedure will be carried out for only one equation; the others
may be obtained in exactly the same way:
whe re
ded---t = (1-e 2) (Tcos E + N) , (9)
-Fu°'R× r
T = -Fu 0 .Rx r 0 =
(cos E - e) - P" u 0 V_-e 2 sin El,
N = _FuO .Q = _FQ.uO a (l-e cos E) ,
r
r = Pa (cos E- e) + Qav_-e 2 sin E .
I
O
O_
¢#0
Substituting the above values into Equation 9, we have
dtde _ V_ (l_e2) F . uO ar - 2e cos E + _ cos 2E
J-P" u° r sin E cos E ; (lo)
from which
e(t +r) - e(t)
9- T r 1 )2e cos E + _ cos 2E
]-P" uO r - sin E cos E dr,
where _ is the period of the satellite.
Since F is small we may consider slowly varying quantities constant
during the integration from t to t + r. By changing the independent vari-
able for the integration and replacing the secant by the tangent on the left
side of the above equation we obtain
7!
de 1_ _ _'_dt _ (i - e2) F •uO -
E 1
, )2e cos E + _- cos 2E dE
rE2 ]- P. uO V_-e 2 sin E cos E dE
E I
The limits are E 1 to E 2 since F is zero outside this range. Therefore
Equation 1 1 becomes
V ._
de
dt - 2_ _(i- ) F • , )E - 2e sin E +_-sin 2E
P.u ° _ _ E2
4 cos 2El EI
The complete set of equations for the osculating elements is:
da _ a2F •u0 - e2 sin E + P • uO cos ;
dt _V_- _ E1
(11)
de _ _/a(1 - e 2) F
dt 27r _- [_ (_ , )•uO - 2e sin E +_-sin 2E
p.u o ___2 "]_
+ 4 cos 2El ;
E 1
di aFR. uO {E_ _ e Eldr" = - 2zr_/_ za(1-e2) E + (l+e 2) sin E - -_-sin 2 cos w
+ os E - _-cos 2E sin ;
sin i _ = - 27r_ za(l-e2) E + (l+e 2) sin E - _-sin 2E sin co
- cos E --_-cos 2E cos ;
E I
d--t- + cos i dt -
2w _-- e
• u0 e cos E --_cos 2E
)1- P • u0 _ - e sin E - -_- sin 2E
E 1
The integration of this system of equations can now be carried out with a
large scale computer.
ACKNOWLEDGMENT
The author wishes to express his gratitude to Mrs. Ann Bailie for her
work in verifying the developments.
l•
2-°
.
.
•
o
REFERENCES
Parkinson, R. W., Jones, H. M., and Shapiro, I. I., "Effects of Solar
Radiation Pressure on Earth Satellite Orbits," Science 131(3404):920-
971, March 2-5, 1960
Musen, P., Bryant, R., and Bailie, A., "Perturbations in Perigee Height
of Vanguard I," Science 131(3404):935-936, March Z5, 1960
Musen, P., "The Influence of Solar Radiation Pressure on the Motion of
an Artificial Satellite," J. Geophys. Res. 65(5):1391-1396, May 1960
Lass, H., and Lorell, J., "Low Acceleration Takeoff from a Satellite
Orbit," Amer. Rocket Soc. J. 31:Z4-28, January 1961
Kryloff, N., and Bogoliuboff, N., "Introduction to Non-Linear Mechanics,"
(translation) Princeton: Princeton Univ. Press, 1947, Chapters I0-12
Musen, P., "Special Perturbations of the Vectorial Elements," Astron.
J. 59(7):262-267, August 1954
7. Stromgren, B., Pub. Med. Kobnhavns Obs. No. 65, 5, 1929
N*S^-Lm_.,.m, D-1063


The Effect of Solar Radiation Pressure on the Motion of an Artificial Satellite

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