University of Colorado at Boulder
Colorado Center for Astrodynamics Research
George H. Born
5/7/2001
General Expressions for the Gravitational Potential due to An Arbitrary Mass
George H. Born
May 7, 2001
Z
P1 ( X 1 ,Y1, Z1 )
1
o
d
r12
r1
M
dM

r2
2
P2 ( X 2 ,Y2 , Z 2 )
Y
1
2
X
Figure 1. Potential of an arbitrary shaped body
The geometry of the problem is shown in figure 1. It is desired to derive an expression for the
gravitational potential, which will exist at P2 due to a mass M of arbitrary shape and density
distribution. The orthogonal coordinate system X , Y , Z is located at an arbitrary point in
M and is inertial, i.e. it is undergoing neither acceleration nor rotation. The point P2 is defined by
X 2 , Y2 , Z 2 or r2 , 2 , 2 . Consider an element of mass dM , which is located at the point
P1 . The point P2 is an exterior point to M and it is understood that for any point of M , r2  r1
i.e., the potential function derived here is only valid outside the arbitrary mass, M. The mass
density of M , an arbitrary function of position, may be designated by
Gravitational Potential
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University of Colorado at Boulder
Colorado Center for Astrodynamics Research
George H. Born
5/7/2001
   r1 , 1 , 1 
(1)
also,
dM  dv  r1 cos 1 dr1 d1 d1
2
(2)
The gravitational potential at P2 due to mass dM is,
d 
P2  unit mass
GdM
r12
(3)
where G is the universal gravitational constant.
Hence,
  G
 dv
r12
V
 G 
r12 cos1
r12
dr1 d1 d1 .
(4)
By the law of cosines of plane triangles
1
1

r12
r12  r2 2  2r1r2 cos 


12
2
 r1 
 r1  
1


1  2  cos     
r2 
 r2 
 r2  

2

 r1 
 r1  
The quantity 1  2  cos     

 r2 
 r2  

1 2
.
(5)
1 2
is the generating function for an infinite series of
zonal solid harmonics (Hobson, page 105, 1965); hence Eq. (5) may be written as
1
1

r12 r2
Gravitational Potential

l
r 
 r1  Pl cos 
l  0 2 
(6)
2
University of Colorado at Boulder
Colorado Center for Astrodynamics Research
George H. Born
5/7/2001
where Pl cos   is the lth degree Legendre polynomial of the 1st kind. By the law of cosines
for spherical triangles,
cos   sin 1 sin 2  cos1 cos2 cos2  1  .
(7)
By the use of the addition theorem for Legendre polynomials (Whittaker and Watson, page 395,
1965) given by
 l  m !
Pl m sin 1 Pl m sin  2 cos m2  1  ,
m 1 l  m  !
l
Pl cos    Pl sin 1 Pl sin  2   2 
(8)
Eq. (6) may be expressed in terms of the polar angles  and  as
1
1

r12 r2

r 
  r1 
l  0 2 
1
 
r2 


l
l  l  m !


Pl m sin 1 Pl m sin  2 cos m2  1 
 Pl sin 1 Pl sin  2   2 


m 1 l  m  !
l

 r1 
   Pl sin 1 Pl sin 2   2 
l  0 r2 
l 0
 l  m  !  r1  l
  Pl m sin 1 Pl m sin  2 

m 1 l  m  !  r2 
l

(cos m 2 cos m1  sin m 2 sin m )  .

(9)
Here Pl m sin 1  is the associated Legendre polynomial of the 1st kind, and of the lth degree
and mth order, and where by definition of these polynomials, Pl m sin    0 for m  l .
Substituting this expansion for the distance between P1 and P2 into Eq. (4) the potential
becomes
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University of Colorado at Boulder
Colorado Center for Astrodynamics Research
GM 


r2 


l 0
George H. Born
5/7/2001
l
R
 1 
l 2
  Pl sin  2 
 r1 cos 1 Pl sin 1 dr1 d1 d1

l 
 MR 
l  0  r2 

R
 2   l  m !
l 2
  Pl m sin  2 
cos m 2  r1 cos 1


l
 MR   l  m  !
m 1  r2 
l

l

cos m1Pl m sin 1 dr1 d1 d1  sin m2  r1 l  2 cos1 sin m1Pl m sin 1 dr1 d1 d1 


R
GM  
  Al 

r2  l  0  r2

l


 Pl sin  2   


l 0
l

R





P
sin

C
cos
m


S
sin
m

 r  lm 2 lm
2
lm
2  (10)

m 1 2 

l
where


 1 
Al  
 r1 , 1 , 1 r1 l  2 cos1Pl sin 1 dr1 d1 d1
l  
 MR 
(11)
 2   l  m !
C lm  
 r1 , 1 , 1  r1 l  2 cos m1 cos 1 Pl m sin 1 dr1 d1 d1

l 
 MR   l  m !
(12)
 2   l  m !
S lm  
 r1 , 1 , 1  r1 l 2 sin m1 cos 1 Pl m sin 1 dr1 d1 d1 .

l 
 MR   l  m !
(13)
Note that in Eq. (10) the total mass M has been introduced and the summations, along with
quantities not participating in the integration, have been taken outside the integral. The quantity
R has been introduced; it is a dimensional parameter which is characteristic of the body of mass
M and which defines the ratio, r2 R , to be the distance of P2 from the origin as measured in
units of R . ( R is generally assumed to be the mean equatorial radius.)
The coefficients Al , C l m and S l m are functions of the size, shape and density distributions of
the body of mass M, are a set of constant characteristics of that body. If the shape and density
distributions are known, the integrations involved in these coefficients may be carried out
resulting in a set of theoretical values for these coefficients. When such information is lacking,
however, a theoretical determination of the coefficients of the potential function is impossible.
Gravitational Potential
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University of Colorado at Boulder
Colorado Center for Astrodynamics Research
George H. Born
5/7/2001
In the case of the Earth for instance the values of these coefficients have been estimated by
astronomical measurements and more recently by methods of satellite geodesy.
For convenience the coefficients J l m and the phase angles  l m , may be defined by
C l m  J l m cos m l m
S l m  J l m sin m l m
(14)
Al   J l .
Equation (15) gives the alternate expression for the potential

l


R
     J l   Pl sin    
r  l 0  r 
l 0

l

R




J
P
sin

cos
m



 l m  r  l m
lm 

m 1

l
(15)
where
  GM
and the subscript 2 which is no longer necessary has been omitted.
Equation (15) may be simplified since
P0 sin    1
giving, from Eq. (11), for l = 0
A0   J 0 
1
M
 
r12 cos1dr1 d1 d1 
1
M
 dm  1 .
(16)
M
This gives the alternate expression for  ,


l

R
  1   J l   Pl sin    
r  l 1  r 
l 1

l

R



  r  Pl m sin  C l m cos m  S l m sin m  . (17)
m 1

l
Or in terms of J l m and  l m ,
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University of Colorado at Boulder
Colorado Center for Astrodynamics Research

l


R
  1   J l   Pl sin    
r  l 1  r 
l 1

George H. Born
5/7/2001
l

R




J
P
sin

cos
m





 lm  r  lm
l m  . (18)

m 1

l
A node may be defined as a point, at which variation of an independent variable in a function
produces no variations in the value of the function with respect to that variable. The Legendre
polynomials are periodic on the surface of a unit sphere and vanish along l latitude nodes on the
surface, dividing it into (l+1) zones, thereby gaining the name, zonal harmonics. The functions,
Pl m sin  cos m and Pl m sin  sin m , also periodic on the surface of unit sphere, vanish along
 l  m latitudinal nodes and along 2m longitude nodes, thus dividing the spherical surface into
 l  m  1 zones and 2m sectors. These two families of nodal lines intersect orthogonally,
causing the surface divisions to be rectangular domains, or tesserae, and thus giving rise to the
name, tesseral harmonics, for these functions. The J l are termed the zonal coefficients of the
gravitational potential function and the J l m are known as the tesseral coefficients when m  l ,
and as the sectorial coefficients when m  l .
If the origin of the coordinate system, O , coincides with the center of mass of M, the expression
for  can be simplified still more. For m  l  1 , we substitute P1 sin 1   sin 1 into Eq. (11),
and P11 sin 1   cos 1 into Eqs. (12) and (13) yielding
A1   J 1 
1
 r1 3 cos 1 sin 1 dr1 d1 d1

MR

1
r1 sin 1dm
MR M

1
Zdm
MR M

Z
R
(where Z is the distance between the c.m. and the coordinate
origin along the Z axis)
0
Gravitational Potential
(by definition of the c.m.)
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University of Colorado at Boulder
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C11 
George H. Born
5/7/2001
1
 r13 cos 1 cos 2 1dr1 d1 d1

MR

1
r1 cos 1 cos1dm
MR M

X
R
0
S11 
1
 r13 sin 1 cos 2 1dr1 d1 d1

MR

1
r1 sin 1 cos1dm
MR M

Y
R
0
In summary, if the origin coincides with the c.m., each of the integrals vanishes leaving
A1   J 1  C11  S11  J 11  0 .
Inclusion of these modifications allows the lower limit of the summation index for l to be raised
to 2 and eqs. (17) and (18) become


l
R
  1   J l   Pl sin   
r  l 2  r 



l


l 2

R
  1   J l   Pl sin    
r  l 2  r 
l 2

l

R




P
sin

C
cos
m


S
sin
m




lm
lm
lm

m 1 r 

l
l

R
 J l m  r  Pl m sin  cos m   l m  .
m 1

(19)
l
(20)
By consideration of various other types of symmetry such as symmetry about the xy plane etc.,
the expression for the potential may be reduced still further.
Gravitational Potential
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University of Colorado at Boulder
Colorado Center for Astrodynamics Research
George H. Born
5/7/2001
Another common means of writing Eq. (15) is


R l
l
 
r l 1
l 0 m0
Pl m sin  C l m cos m  S l m sin m 
(20a)
Consider for example the case of symmetry about the xy-plane. In this case the potential at a
given point must equal the potential at the reflection of this point, i.e.,

 r,  , 


  r, , 

Substituting into Eq. (18) the relations
Pl sin(  )    1 Pl sin  
l
and
(21)
Pl m sin(  )   1
l m
Pl m sin   ,
we have
l


R
 J l  r  Pl sin    
l 1
l 1

l

l
l
l
R
 J l m  r  Pl m sin  cos m   l m 
m 1

R
  J l   Pl sin(  )   
r
l 1
l 1
l
R
J l m   Pl m sin(  )  cos m   l m 

r
m 1
l

R
l
  J l    1 Pl sin    
r
l 1
l 1
l
R
 l -m 
J l m    1
Pl m sin   cos m   l m  . (22)

r
m 1
l
Since corresponding polynomials of equivalent polynomials must be equal, it must be concluded
that J l  0 for l odd and J l m  0 for l  m odd. Thus for symmetry about the xy-plane all
zonal harmonics must vanish for l odd and only those tesseral harmonics for which l  m is even
may exit.
Gravitational Potential
8
University of Colorado at Boulder
Colorado Center for Astrodynamics Research
George H. Born
5/7/2001
Now consider the case of axial symmetry with respect to the Z axis. For this situation, the
density is independent of  and the integration with respect to  becomes separable in Eqs.
(11), (12), and (13) and from Eqs. (12) and (13) we may separate the following integrals
2
 cos m1 d1  0
0
2
 sin m1 d1  0
.
0
Hence,
C lm  S lm  J lm  0 .
Thus for the case of axial symmetry only the zonal harmonics exist, i.e. for the case of the
surface of mass M being a surface of revolution with respect to the z-axis and when the
distribution of mass density is rotationally symmetric with respect to the z-axis, the
gravitational potential at an exterior point will be given by

l

R

  1   J l   Pl sin   .
r  l 2  r 




(23)
For the oblateness, for example

2

R 1
  1  J 2  
3 sin 2   1  .
r 
r 2



(24)
References:
1. Hobson E. W., The Theory of Spherical and Ellipsoidal Harmonics, Chelsea Publishing Co.,
New York, 1965.
2. Whittaker E. T., and G. N. Watson, A Course of Modern Analysis, 4th edition, Cambridge
Press, 1965.
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University of Colorado at Boulder
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Gravitational Potential
George H. Born
5/7/2001
10
University of Colorado at Boulder
Colorado Center for Astrodynamics Research
Gravitational Potential
George H. Born
5/7/2001
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