Explicit and recursive formulas for acceleration and gravity gradient derived from spherical harmonic

Heffron, W. G.

English

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6^S
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Bellcmim
date August 30, 1971
to Distribution
t ►om W. G. 4effron
.abject Gravity and Gravity Gradient frot
Spherical Harmonics - Case 310
S'j55 L' Entant Pi um North, S W
Washington, D C. 20024
B711 08040
ABSTRAC'
Beginning with the spheric,;-*. :larnton,_ expansie:. for
the giavi.tational potential of a planet., (-xp.'f icit and recursive
formulas are presented for the acceleration and gradient of
gravity. Figures are given showing the effects on the potential
and acceleration of several individual coefficients in the expan-
sion, and also of the potential. and acceleration resulting from
the sets of coefficients in cLrrent use for the earth and moon.
I III
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date	 August 1'k0 , 19 71
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from	 IN. G. fle f f ron
	
%ub f ect	 Gravity and Gravity Gradient from
Spherical Harmonics - 'Case :310
MEMOI2AN" ' N r'O R FIDE
INTRODUCTION
955 L Entdnt P l aza North, S W
Wa!hington, D. C. 20020
A71 OP '.10
e
mho memorandum documents explicit and recursi^re
formulas for gravity and gravity gradient using spherical
harmonics. It is a companion paper to [1] which is a com-
pendium on the properties of the Legendre polynomials used
4	 in the spherical harmonics.
THE POTENTIAL
The spherical harmonic expansion for the gravita-
tional potential of a planet is commonly given as
C	 v
U = rZ L R1 Pem (sin4l) (C^ m cosma +SQm sinma )
e=0 m=0
wh are U is the potential, u equcls GM where G is the universal
gravitational constant, M is the planet's mass, r is the radius
from the origin of the plariet-•f ixed axes to a field point; ^ and
X are the field point's latitude and longitude, R is a "refE!rence"
radius for the planet, P am is the unnormaiized associated Legendre
polynomial of degree f- and c,rder m, and C Qm and Stm are the con-
stants which describe the planet.
N
M,
6
- 2 -
	 0
C 00 - 1 describes the central body (spherical)
potential, as the planet would appear from a far distant p^int.
All S fols are taken as zero. C10, C11, and S 11 are associated
with the center of mass location relative to the origin of the
axes:
__ z C. M.
C 10	 R
C 11	 2
__	 xC.M.
 R
__	 yC.M.
S 11.	 2	 R
Usually these are assumed to be zero.
If the planet's inertia tensor is
A -D -E
-D B -F
-L -fCL
where, typically
A =
	 (y2 + z1)dM
D = +	 xy dM
.,.
•then
_ 1	 A+B
C20 - R ,^ 	
2 - C'
1
C 21 - R2M 
E
_ 1
S21	 R 2 M
F
C 22 = 1 ^- (B-A)
4MR
__	 1
S22	 2 
D
2MR 
If the planet is symmetric about its rotational (z)
axis, then Cem = 0 for m>0 and all S Q,n = 0.
GRAVITY
The acceleration of gravity g is the gradient of the
potential. Two decompositions of the acceleration vector are of
interest. One projects it on planet axes xyz and 'Che other on
radial, East (and horizontal), and North (and oo-izonta lk) axes at
the field point r,m,a. Explicit forms are given in Table 1.
Figures 1 - 7 show plots simplified forms of [land
of g r , gE , and gN for the coefficients t and m from 2,0 to 3,3.
Specifically, what is plotted is
U (Z,m) = P i m (sink) cos ma
g r (f-, m) _ - (t+1) Pt  (sin0 cos ma
U E
Q=0 m=0
(Cf-Mutm + S imVQ m )
6
,is
4-	 A
g N ►n) _ - ^ m CV .-3 	 P^m (sinf) - p^ ►n+l(sin$)^ cos ma
gE(t,m) = - m Pe m sin ma/cosm
The results for g r , gE , and gN are thus in a is 	 scaling,
and apply for Cam . For S tm add 90° Fast to the longitude. A
satellite's orbit could be affected by g E and g N both in-plane
and normal to the orbit plane, both horizontally, and in varying
amounts . if. the orbit is -ccpntri_c.
Table 2 lists curr.ent.Ly used coefficients for the
earth (4] and moon [5] and Figures R and 9 give Mots of T7, g ,
r
gF , and gN for these fields.
Recursive formulas for the calculation of g , g , andX,,	 y
9Z have been developed by DeWitt [2], superior for such calcula-
tions to those developed by James [3]. DeWitt writes
m _ URQ
	
U,e 	 +1
r
V m = uR^
	t 	 ti+l
r
PZm (sinm) cos ma
PZ m (sinf) sin ma
so that
and develops the recursive formulas
*Note that with u, R, r all equal 1 and a equal zero, then
U m = P m . The recursive formulas can thus be used to calculate
P t m , if desired.
- 5 -
6
U
^+1	 ( r i'' m+ 1 1 cos m (2Q+1) J^ in - (Q +m) U Q m I }
V
Q+1	 (r)	 {cosh (2Q+1)V^ m - (4' + ►n)VZ-1
and
UQ+1
	
i r`) (2t+1 ) (.U^^Cosa - V^ Q sina )
V Q
+1	 (r) (2Q+1) (VQ ^cosa - Vsina)
One starts with U 0 0 = ^,/r, V 0 0 = 0 and applies the last two
formulas to the maximum t desired, and then uses the other two
to obtain the remaining needed terms.
The gradient of the potential, in xyz axes, is the
vector
9x
VU =
	 9y	 =
Z=0 m=0
9z
(Ctm VUtM + StM VVzM)
where
1
- 6 -
0
A,
	
(1-m4-1) (Q-m+2) UM+1	 UM -
VU	 2R= R	 - ( Q-m+l) (f--m+2) VZ+1	 VQ+1
-2 (i -m+1) Ue f l
vVem	 2R	 (e-ri+l) (t-m+2) UM-1 + UM+1
-2 (f.-m+l) VF-+1
Thus if Q, m are the maximum indices for Cfm and SQm , then
e+l, m+l are the necessary maximum indices for U and V.
GRAVITY GRADIENT
The gravity gradient is a 3 x 3 matrix here called G.
Explicit formulas for the elements of G are given in
Table 3. Recursive formulas for G were also developed by DeWitt
ane are
00	
e
G = vv u = X)	 E (C 
f-m 
ovUi m + S em OOVt m )
Q=0 m=0
where
VIP-
i7 -
	 0
(t-m+l) (e-m+2) VUM+1 - VU M+1
VVU^ m
 =	 - (C-- m +1 } (Q - m+2) VVM -1 - VV M+1
- 2 (t-m+l) VUe
+1
r
VVV
t m =
(t-m+1)
	
(e-m+2)
	 VVm+1	 - VV1+1
(l.-m+l)	 (t-m+21,	 VUM+1	 - VU M
+1
-2 (t-m+l)	 VV^*i
In this VV and VU become row vectors instead of column
vectors so that, for example, -2(f-m+1) VU t	is the last row of
VVTj t	Note that since 
vUZ+1 involves Ut+2, more coefficients
must be computed if the gradient is also required. 	 ,
2014-WGEi-j f	 W. G. Hfo
4i
0
TABLE I
EXPLICIT FORMULAS FOR GRAVITY
!D
g x `	 (r)
	
2 ( (C^m cos (m+1) a + S t m Sin ( m +l) a) pm+i (sinm) -
r t=0 m=0
-- (I-m+1) (t-m+2) (Ck m cos (m-1) a + SQm sin (m-1) a) P^ + i (sinf) )
co	 i
9	 2 E E (r) 2 ( (C^m sin (m+l) a - S im cos ( m +1) A) Pm +1 (sin m) +Y	
r .^=0 m=0
+ (Z-m+1) (Q-m+2) (C em sin (m-1) a - S im cos (m-1) a) p in+i (sinm) )
_ ._ u	 ^  ^ 	 R I ^	 m
g z 	 rl	 L (r) (t-m+l) (C'm cosma + S im sin ►na) p1+l (sink)
r 1=0 m=0
g r `	 2 E E ( r) (Z+1) ( CQm cosma + S im sin-na) P m (sinm )r 1
=0 m=0
g E	 `1 Z	 (r) cosm (Cp m sin:-a - SQm cosma) p m (sinm )
r t=0 m=0
O
r^
9N	 r	 L (r) (Cf-m cosma + S Qm sinma) ( m cosp P^ (sinm) -
r t=0 m=0
•
pe+1 (sino) )
I
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TABLE III
GRAVITY GRADIENT FORMULAS
e 1
GXx = + -^ ^.
	
IF
ft
) 4 { (Cem cos (m+2) x + S tm sin (m+2) a ) P
m
+
+2
2 (sin f) -
r i=0 ► n=0
- 2 (e-m+l) (1-m+2) ( Cem cosmx + Sim sinmA) P`+2 (sino) +
+ (t-m+l) (£-m+2) (e-m+3) (t-m+4) (C f m cos (m-2) ^ +
+ Sim sin (m-2) OP M  2 (sinf) }
e	
e
G	 = G	 = +u	 R	 1 { (C
	
sin (m+2) x -xY	 Yx	 -T	 4 em
e=0 rn=0
- S im cos (m+2) a) Pm+2 (sink) - (e-m+l) (.Q-m+2) (e-m+3) (e-m+4) times
(Cem sin (m-2) a - S e m cos (m - 2) a) P f-+ 2 (sinm) }
- u	
W
	
p
	
R e 1	 ,Gxz _ G zx	 1 E
	
(r I 2 { (e-m+1,^ tC.^m c^^s ^Yi^-)
P=0 m=0
i	 + Sem sin (i+).) a) PM+2 (sinm)
	 (e-n.+1) (e- ►n+2) (e-m+3) times
•
N
(C em cos (m-1) a + S^^ sin (m-1 *) a; P"l+2 (sin-) }
•TABLE III (continued)
	 A
O	 Q
GYy
r 1=0 m=0
QR) 
4' 4 - ( Ctm cos (m+2) a + Sim sin (m+2)a ? Pe+2 (sins) -
- 2 (t.-m+l) ( 1- m +2) ( Ctm cosma + Sim sinma) Pt+2 (sins) -
- (t-m+l) ( P -m+2) It-m+3) (Y-m+4) ( Cim cos (m-2) A +
+ Sim sin (m-20) Pm+2 (sinm) )
f	 -
Gyz	 Gzy	 3 E Z ( r I j ( (t-m+2) (C^m sin (m+l) a -
r
t.=0 m=0
- S	 cos (m+lM Pm+1 (sins) + (Z-m+l) (t.-m+2) (t-m+3) timesQm	 ,t+2
(C	 sin si  ( m -?.) a - Sim cos ( m - 1) x) Pm+2 (sinf) )
G	 (i-m+1) (t-m+2) ( Ctm cosma + SQm sinma) Pk.+2 (sins)zz	 u3 7 T, (r
r Q=0 M=0
Note: GXX + Gyy + GzZ = 0 (LaPlace`s equation)
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REFERENCES
1. Neffron, W. G. and Levie, S. L., "Unnormalized Associated
Legendre Functions: A Compendium of Graphs, Expansions,
Properties and Integrals," Bellcomm, Inc., Memorandum for
File B71 08018, Washington, D. C., August 16, 1971.
2. De Witt, P. N., "Derivation of Fxpressions Describing the
Gravitational Field of the Earth," U. S. Naval Weapons
Laboratory, Technical Memorandum K-35/62, November 1962
(with Addendum of 12 Feb. 1963).
3. James, R. W., "Transformation of Spherical Harmonics Under
Change of Reference Frame," Geophys. J.R. Astr. Soc., 17,
305-316, 1969.
4. Melbourne, W. G., et al, "Constants and Related in Formation
for Ostrodynamic Calculations, 1968" Jet Propulsion Labora-
tory 'Technical Report 32-1306, Pasadena, Calif., July 15, 1968.
5. Wollenhaupt, W. R., "Apollo Orbit Determination and Navigation"
AIAA Paper 70-27, Aerospace Sciences Meeting, Jan 19-21, 1970.
•
A


Gravity and gravity gradient from spherical harmonics, case 310

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