Perturbation of hyperbolic orbit by oblate planet and escape hyperbola from eart

Sauer, C. G., Jr.

English

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Technical Report No. 32-731 
The Perturbations of a Hyperbolic 
Orbit by an Oblate Planet 
Carl G. Sauer, Jr. 
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PASADENA. CALIFORNIA 
January 15, 1963 
https://ntrs.nasa.gov/search.jsp?R=19630005405 2020-03-11T16:53:20+00:00Z
NATIONAL AERONAUTICS AND SPACE ADMINISTRATION 
CONTRACT No. NAS 7-100 
Tecbnical Report No. 32-731 
The Perturbations of a Hyperbolic 
Orbit by an Oblate Planet 
Carl G. Sauer, Jr. 
C. Gates, Chief 
Systems Analysis Section 
J E T  P R O P U L S I O N  L A B O R A T O R Y  
C A L I F O R N I A  I N S T I T U T E  O F  TECHNOLOGY 
PASADENA, CALIFORNIA 
January 15, 1963 
Copyright @ 1963 
Jet Propulsion laboratory 
California Institute of Technology 
The material in this Report appeared in the ARS Journal, Vol. 32, No. 5, pp. 714-717. 
JPL TECHNICAL REPORT N O  . 32-131 
CONTENTS 
1 . Introduction . . . . . . . . . . . . . . . . . . . . .  1 
II . The Perturbative Potential . . . . . . . . . . . . . . . .  2 
111 . Variation of the Orbital Elements . . . . . . . . . . . . .  2 
IV . Example of an Earth-Escape Mission . . . . . . . . . . . .  4 
References . . . . . . . . . . . . . . . . . . . . . . .  8 
1 . Example of an Earth-escape mission . . . . . . . . . . . . . .  5 
2 . Asymptotic value of the orbital elements . . . . . . . . . . . .  5 
6 
6 
7 
3 . Asymptotic perturbations of the hyperbolic excess velocity vector . . . .  
4 . Calculated and observed perturbations at  240 min . . . . . . . . .  
5 . Position and velocity a t  T = 240 min . . . . . . . . . . . . . .  
FIGURES 
1 . Perturbation of the semi-transverse axis . . . . . . . . . . . .  5 
2 . Perturbation of the eccentricity . . . . . . . . . . . . . . . .  5 
3 . Perturbation of the time of peri-focal passage . . . . . . . . . .  6 
4 . Perturbation of the inclination . . . . . . . . . . . . . . . .  6 
5 . Perturbation of the argument of perigee . . . . . . . . . . . . .  6 
6 . Perturbation of the longitudeof the ascending node . . . . . . . .  7 
7 . True anomaly vs time (e = 1.25) . . . . . . . . . . . . . . .  7 
JPL TECHNICAL REPORT NO. 32-131 
The perturbations of the hyperbolic orbital elements of a vehicle 
in the gravitational field of an oblate planet are derived as functions 
of the initial osculating elements. Assumptions are made that atmos- 
pheric drag is absent and that the gravitational potential of the planet 
may be represented adequately by the principal term and the second 
harmonic. An example of an Earth-escape mission is presented in 
which a comparison is made between calculated orbital perturbations 
and results from a numerical integration of the equations of motion. 
1. INTRODUCTION 
Although the variations of the orbital elements of Earth 
satellites have been derived numerous times, nevertheless 
a treatment of the variation of the orbital elements of a 
vehicle for an escape mission is lacking except for the 
case of perturbed hyperbolic motion in the equatorial 
perturbations of an interplanetary probe by an oblate 
planet at  encounter. 
'Since 1960, when this report was originally prepared, there have 
been two additional papers published concerning the perturbing 
jectory (Ref. 2 and 3). Of particular interest is Ref. 3 by G. Hori 
motion. The present paper uses the method of Poisson-LaGrange, 
as does Ref. 2. 
plane (see Ref. 1 ) There are several instances in which effects of an oblate Planet upon a vehicle on a hYFrbolic tra- 
the Of this paper are Of interest, the first being in which the Von-&ipel was used to derive the pefiw&d 
the perturbations Of the escape from Earth Of 
a lunar or interplanetary probe, and the second being the 
1 
JPL TECHNICAL REPORT NO. 32-131 
I I .  PERTURBATIVE POTENTIAL 
If an axially symmetric density distribution of the planet 
is assumed, the gravitational potential function U may be 
expanded in terms of surface zonal harmonics of the form 
(Ref. 4 and 5 ) .  
where 
J, = the coefficients of the zonal hormonics P ,  ( sin 4 ) 
k2 = GM = gravitational constant of attracting body 
r = radial distance from gravicenter in equatorial radii 
0 = gravicenter latitude 
In an analysis of the perturbations of the bound orbit 
of an Earth satellite, it is necessary to consideral several of 
the harmonic terms to describe accurately the motion of 
the vehicle over an extended period of time (Ref. 6 and 
7) .  However, only the second harmonic J 2  contributes 
significantly to the perturbations of the elements of a non- 
bound or hyperbolic orbit, since the time spent in the 
vicinity of the planet is short and the effects of the higher 
order harmonics are negligible. 
The perturbing potential R due to the second harmonic 
J 2  is given by 
111. VARIATION OF 
The solution of the differential equations representing 
the variations of the orbital elements with respect to time 
is derived by a method analogous to that employed by 
Kozai (Ref. 8 )  in investigations of elliptic motion. The 
orbital elements E ,  h, h,, T,  O, and sz are employed, since 
they both form a canonical set and are also well behaved 
for eccentricities near unity. The elements not previously 
defined are 
where 
( 3 )  
The distance from the gravicenter T and gravicenter lati- 
tude + expressed as functions of the osculating elements of 
the vehicle are 
(4)  
1 P ,  ( s in+)  = -(3sin, 4 - 1 2 
T = p / ( l  + ecosv) 
and 
sin4 = sinisin ( W  + V )  ( 5 )  
where 
p 
a = semitransverse axis 
h = angular momentum 
e = eccentricity 
i 
w = argument of perigee 
v = trueanomaly 
= h2/k2 = ( -u) (e' - 1 ) = semilatus rectum 
= inclination to the equatorial plane 
The disturbing function R, when expressed in terms of 
the osculating elements of the orbit, becomes 
R = ] , ( k 2 / 2 p S )  (1 + BCOS v ) 3  [l - -sin2 3 .  i + 2 
3 ;z sin2 i cos (2,  + 2 ~ ) ]  ( 6 )  
ORBITAL ELEMENTS 
E = - k2/2a = undisturbed energy 
h, = h cos i = component of angular momentum along 
polar axis 
T = time of perifocal passage 
sz = longitude of the ascending node 
2 
~ ~~ 
J P L  TECHNICAL REPORT NO. 32-131 
The rate of change of the orbital elements with time 
may be expressed in terms of the differential coefficients 
of R as 
d d d t  = -?R/ah (9 )  
Since the orbital elements of the vehicle change only 
slightly due to the effects of the oblateness, the pertur- 
bations of the first order may be derived by considering 
the elements appearing in R on the right side of Eq. ( 7 )  
through (12)  constant. If the elements are assumed con- 
stant, the true anomaly may be regarded as a known 
function of time, and the independent variable may be 
transformed from time to the true anomaly by 
n( t - T ) ,  where the mean motion n is a function of the 
element E through the harmonic law n = ( 2E):’’2k-2. The 
variable A.1 corresponds to the mean anomaly for hyper- 
bolic motion and is a function of both the true anomaly 
and the eccentricity, and hence the disturbing function 
involves the element h both explicitly and also implicitly 
through the true anomaly. 
Since an axially symmetric density distribution is as- 
sumed, the disturbing function R is independent of 0, and 
Eq. ( 12) becomes 
(14) dhz/dt = 0 
and 
h, = h,(O) = const (15) 
the component of angular momentum along the polar 
axis being unaffected by the oblateness. 
The perturbations of the remaining five orbital ele- 
ments are given in Eq. ( 16) through (20)  where M and 
appearing in Eq. (20)  are the mean anomaly and mean 
motion, respectively. These expressions for the perturba- 
tions are to be evaluated between the limits v0 and V. 
SE = R1’ 
JVO 
3e cos 
h sin2 i 
Sh = J, -4p? + ”) + 3 cos (2w + 2”) + e cos (20 + 3”) I:, (17) 
>:, cos i 6 0  = -I., -i 6 ( ”  + e sin “) - [3e sin (2, + v) + 3 sin (2” + 2”) + e sin ( 2 ~  + 3v)] - 4pX 1 
e 
6” + cos is0 = J, 
3e3 sin (2w - v )  + (12 - 21e’) sin (2w + v )  - 36e sin (2, + 2v) - 48e 
(28 + l le?) sin (2, + 3“) - 18e sin (2, + 4”) - 3e2 sin (20~ + 5 ~ )  
The perturbations of the semitransverse axis a, eccen- 
(13)  tricity e, and orbital inclination i are found from d/dt = h/r2 d/dv 
The disturbing function involves the element E and ‘T 
implicity through the true anomaly from the relation M = 
3 
JPL TECHNICAL REPORT NO. 32-131 
8i = cot i (6h/h) (23) 
The expressions for the variations of the elements E ,  
h, h,, and 0 are well behaved for eccentricities near unity; 
however, the expression for the variation of the time of 
perifocal passage breaks down when e = 1. Fortunately, 
in most cases of interest, the change in the time of peri- 
focal passage is of not great importance. 
Of additional importance in the analysis of interplan- 
etary escape trajectories is the velocity vector of the 
vehicle at a great distance from Earth. The magnitude of 
this hyperbolic excess velocity vector is given by the 
following function of the energy: 
The unit vector S in the direction of the hyperbolic excess 
velocity vector and also in the direction of the outgoing 
asymptote of the hyperbola has direction cosines of the 
form 
S, = cos C#3h sin a h  = sin 0 cos ( W  + vI l )  + 
cos 0 sin (W + v I I )  cos i (26) 
S, = sin 41, = sin ( W  + V h )  sin i (27) 
a h  and 4h being the right ascension and declination of the 
hyperbolic excess velocity vector and Vh being the angle 
between the hyperbolic excess velocity vector and peri- 
apsis, 
The angle Vh is related to the eccentricity of the hyper- 
bola by 
cos Vh = - l/e (28) 
The perturbation of the magnitude of the hyperbolic 
8 v h  = 8 E / V h  (29) 
excess velocity vector is given by 
whereas the perturbations of the right ascension and dec- 
lination are given by 
cos i sin ( w  + V I , )  sin i cos ( 0  + Vh) 
( 8 0  f 8Vh) cos $‘h si + cos 4 h  84h = 
The perturbation of the angle between S and periapsis 
is given by 
IV. EXAMPLE OF AN EARTH-ESCAPE MISSION 
As an example of the application of the deriviation of 
the perturbations, an Earth-escape mission with the initial 
parameters shown in Table 1 was used. The perturbations 
of the elements a, e, i, T, W, and R were calculated from 
Eq. (16) through (20) and are shown in Fig. 1 through 
6 as functions of the true anomaly. The time along the 
trajectory may be determined from Fig. 7 as a function of 
the true anomaly. The maximum true anomaly is approxi- 
mately 2.5 rad; in the following discussion the time of 
240 min corresponds to a true anomaly of about 2.3 rad. 
4 
JPL TECHNICAL REPORT NO. 32-131 
k', kmJ/sec' 
re, km 
1, 
1,  
I4 
Table 1. Example of an Earth-escape mission 
0.398602 X 10" 
1.08228 x lo" 
-0.00230 X lo" 
-0.00212 x le 
6378.150 
X, km 3826.8900 
r. km -4418.9120 
I Initial position and velocity 1 
J = 1 . 6 ~ ~ 1  x 0-3 
H = -0.00604 x 10-3 
selected points along the path, the osculating elements 
were calculated and the perturbations of the initial ele- 
ments determined. These values are shown as individual 
points in Fig. 1 through 6. In the numerical integration 
the second, third, and fourth harmonic terms were re- 
tained in the potential function. The values of these coeffi- 
cients are shown in Table 1 and correspond to the 
following values for the commonly used coefficients for 
the second, third, and fourth zonal harmonics: 
X, kmlsec 
i, km/sec 
2. km/rec 
9.4864475 The asymptotic values of the elements for t + co are 
shown in Table 2. In addition, the asymptotic values for 6.1616282 
3.5574179 
I 2. km I -22551.2600 I K = 0.00637 X ~ ~~ 
i 
the perturbations of the hyperbolic excess velocity vector 
I Initial osculating elamentr I are shown in Table 3. 
a, km 
e 
T, sec 
i, rad 
w, rad 
R, rad 
- 25.51 2.6000 
1.25000000 
0.0 
0.52359881 
5.35589010 
0.0 
0 I 2 
TRUE ANOMALY, rodions 
Fig. 1. Perturbation of the semi-transverse axis 
3 
Table 2. Asymptotic value of the orbital elements 
-25,570.033 
1.2495 1 297 
T, sec 0.160659 
0.52367376 i, rad 
w. rad 5.35710409 
Q, rad - 0.00047714 
TRUE AN0MN-Y. rodions To check the results that were obtained, a numerical 
integration of the equations of motion was performed. At Fig. 2. Perturbation of the eccentricity 
5 
JPL TECHNICAL REPORT NO. 32-131 
6 v h ,  kmfsec 
6ah, rad 
8$h, rad 
An indication of the accuracy of the analysis calculation 
of the perturbations may be had by observing the calcu- 
lated and observed perturbations of the orbital elements 
at t = 240 min in Table 4. An observation of the variation 
0.004449 
1.52565 X 10” 
0.07498 X 1 0 3  
Observed Calculated 
value, value, 
numerical int. 1st iteration 
Perturbed 
element 
Table 4. Calculated and observed perturbations 
at 240 min 
Calculated 
value, 
2nd iteration 
6a, km 
6e 
67, sec 
ai, mrad 
6w, mrad 
60, mrad 
- 57.71 3 
- 0.0004887 
0.1 660 
0.07499 
1.2100 
- 0.4695 
- 57.387 
- 0.0004868 
0.1 635 
0.07558 
1.2076 
- 0.4683 
- 57.729 - 0.0004876 
0.1 699 
0.07501 
1.2071 
- 0.4678 
of the elements in Fig. 1 through 6 would indicate that the 
initial osculating elements are not necessarily the best to 
I I 
TRUE ANOMALY, radians 
Fig. 4. Perturbation of the inclination 
0 t 
oo c 
TRUE ANOMALY, radians 
Fig. 3. Perturbation of the time of peri-focal passage 
TRUE ANOMALY, radians 
Fig. 5. Perturbation of the argument of perigee 
6 
~~ ~ ~ 
JPL TECHNICAL REPORT NO. 32-131 
Table 5. Position and velocity at T = 240 min 
Observed I Coordinate value, 
X. km 
Y, km 
2. km 
X, kmfsec 
r, kmfsec 
Z, kmfsec 
numerical int. 
16,781,044 
72,067.631 
41,620.015 
0.098739 
4.327236 
2.498790 
I 1 
1 
Calculated 
value, 
2nd iteration 
Unperturbed 
16,781.333 
72,067.541 
41,619.951 
0.090754 
4.327235 
2.490790 
16.876.470 
72,092.005 
41.622.339 
0.10501 0 
4.329907 
2.499919 
TIME, min. 
I 
use in the expressions for the perturbations. If a second 
iteration is made in which the elements used in calcu- 
lating the perturbations are the asymptotic values from 
Table 2, then slightly better values are realized for the 
perturbations as shown in Table 4. A comparison between 
the results of the second iteration and the numerical 
integration in the position and velocity of the vehicle at 
t = 240 min is shown in Table 5. Also shown in Table 5 
are the unperturbed values of position and velocity at 
240 min. 
Fig. 6. Perturbation of the longitude of the 
ascending node 
7 
JPL TECHNICAL REPORT NO. 32-131 
REFERENCES 
1. Anthony, M. L., and G. E. Fosdick, "Escape in the equatorial plane of an oblate 
planet," ARS Journal, 30, 898-900, 1960. 
2. Anthony, M., G. Fosdick, and L. Perko, "Escape from an oblate planet," Advances 
in the Astronautical Sciences, vol. 7, p. 175. Plenum Press Inc., New York, 1961. 
3. Hori, Gen-lchiro, "The motion of hyperbolic artificial satellite around the oblate 
earth," Astronomical Journal, 66, 258-263, 1961. 
4. Bomford, B. G., Geodesy, Clarendon Press, Oxford, 1952, Chap. 7. 
5. Garfindel, B., "The orbit of a satellite of an oblate planet, Astronomical Journal, 
64, 353-367, 1959. 
6. Brouwer, D., "Solution of the problem of artificial satellite theory without drag," 
Astronomical Journal, 64, 378-397, 1959. 
7. O'Keefe, J. A., A. Eckels, and R. K. Squires, "The gravitational field of the earth," 
Astronomical Journal, 64, 245-253, 1959. 
8. Kozai, Y., "The motion of a close earth satellite," Astronomical Journal, 64, 367-377, 
1959. 
8 


The Perturbations of a Hyperbolic Orbit by an Oblate Planet

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