Transformation properties of coordinates in inverse power spherical harmonic expansion

Madden, S. J., Jr.

English

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I 
measurement systems laboratory 
rnassachusetts insti tute of technology, cambridge rnassachusetts 02139 
-- 
E 
RE - 68 
THE TRANSFORMATION OF EXTERN 
HARMONIC SERIES UNDER A TRANSLATION 
OF ORIGIN 
by 
Stephen J. Madden. Jr. 
https://ntrs.nasa.gov/search.jsp?R=19710027473 2020-03-11T21:40:56+00:00Z
THE TRANSFORMATION OF EXTERNAL 
HARMONIC SERIES UNDER A TRANSLATION 
OF ORIGIN 
Stephen J. Madden, Jr. 
June 1971 
Approved : 
Director 
Measurement Systems Lab 
ACKNOWLEDGMENT 
Acknowledgment is made to Mr. William N. Lee who con- 
tributed, through discussions, the impetus for this investiqa- 
tion. 
This report was prepared under DSR 70705 ,  spcmsored by 
the National Aeronautics and Space Administration through 
research grant NGR 22-009-262.  
ABSTRACT 
The behavior of inverse-power spherical harmonic 
expansions is presented when the origin of the coordinate 
system undergoes a translation. A seeming paradox, in-- 
volving large translations is discussed. 
THE TRANSFORMATION OF EXTERNAL 
HARMONIC SERIES UNDER A TRANSLATION 
OF ORIGIN* 
Stephen J. Madden, Jr. 
1. I n t r o d u c t i o n  
The o b j e c t  of  t h i s  r e p o r t  i s  t o  p r e s e n t  an elementary 
d e r i v a t i o n  of t h e  behavior  of e x t e r n a l  s p h e r i c a l  harmonics 
of t h e  form 
when t h e  o r i g i n  of t h e  coo rd ina t e  system d e f i n i n g  t h e  spheri- 
c a l  coo rd ina t e s  r , 8 , i $  undergoes a g e n e r a l  t r a n s l a t i o n ,  The 
corresponding problem f o r  s p h e r i c a l  harmonics of p o s i t i v e  
degree  has been so lved  by Aa~doom [ I ] ,  a s  has  t h e  r o t a t i o n a l  
problem, where t h e  s p h e r i c a l  coo rd ina t e  system undergoes 
a r o t a t i o n .  The r o t a t i o n a l  problem had been prev ious ly  solved 
by many a u t h o r s  and a most complete p r e s e n t a t i o n  i s  found 
i n  Courant and H i l b e r t  [ 2 ] ,  
The s p h e r i c a l  harmonics of t h e  form (1) a r e  of par- 
t i c u l a r  i n t e r e s t  i n  t h e  f i e l d s  of  s a t e l l i t e  t h e ~ r y  and 
geodesy which a r e  concerned wi th  t h e  g r a v i t a t i o n a l  potenti?&S 
e x t e r n a l  t o  t h e  body which g e n e r a t e s  it, A common potential 
* This  r e p o r t  ha s  been presen ted  a t  t h e  American Geoph, v s i c ~ l  
Meeting, Washington, D . C . ,  A p r i l ,  1971. 
representation in these fields is 
where y and a are the gravitational constant for the generating 
body and a characteristic radius for the body, respectively, 
The quantities r,0,$ are the spherical coordinates of the 
field point in a reference system. The angle 0 is the co- 
latitude and $ is the east longitude. The constants Cnm 
and Snm characterize the gravitational field of the body and 
its deviation from a point source field. 
The result of this report allows the series ( 2 ) ,  
usually expressed in a coordinate system located at the 
center of mass of the generating body, to be recast into 
a form which is valid in a coordinate system whose origin 
may be more conveniently located. An example of this 
procedure is given by Lee [6] where the translations des- 
cribed are used to compute the gravitational force between 
two bodies where neither body can be considered as a point 
mass. 
It is conceivable that the lunar potential can be ex- 
pressed in a coordinate system located at the center of 
mass of the earth and thus facilitate, in s2me cases, the 
earth satellite problem. 
The method to be described is a direct generalization of 
one due to Hobson [ 4 1 ,  who has considered the special case of 
translations along the z-axis. 
2 ,  The ~armonic Representation 
~t is possible to put the potential expression (2) 
ilkto a fc,rm in which terms such as (1) appear, To do "chi-, 
we introduce complex coefficients 
and (2) becomes 
where 
This form of the series eliminates separate consideration 
of the sin m$ and cos m$ cases. 
A simple examination of the functions V especially 
nmf 
when one considers that 
with Pn(x) the Legendre polynomial, shows that any direct 
approach to the translation problem with a substitution of 
translated quantities for r, 0, and $ leads to an extremely 
complex expression. It is therefore desirable to find 
an alternate representation for V (r,0 ,$) which allows 
nm 
translations to be applied. Such a representation, due to 
Maxwell, can be found in Hobson [4] or Cunningham 133 
I f  t h e  n o t a t i o n  of Cunningham i s  modif ied t o  agree  w ~ t h  
( 6 1 ,  
n-m 
- (-1) a a m a  n  -m 
 Vnm (n-m) := +i -1 (-) (- 1 (7) ay a z  Irl 
w i t h  t h e  x , y ,  z d e r i v a t i v e s  taken i n  t h e  r e f e r e n c e  coo rd ina t e  
system, where r i s  t h e  p o s i t i o n  v e c t o r  of t h e  f i e l d  point 
- 
i n  t h i s  system. I t  w i l l  be shown i n  t h e  fo l lowing  para- 
graphs t h a t  t h i s  r e p r e s e n t a t i o n  f o r  t h e  s p h e r i c a l  har--  
rnonics i s  u s e f u l  f o r  t h e  t r a n s l a t i o n  problem and l e a d s  t o  
t r a c t a b l e  exp res s ions .  
3. Desc r ip t ion  of  t h e  T r a n s l a t i o n  
I n  a d d i t i o n  t o  t h e  s p h e r i c a l  harmonic r e p r e s e n t a t i o n ,  
we must s p e c i f y  t h e  t r a n s l a t i o n  of t h e  coo rd ina t e  system. 
Thi s  w i l l  be  used t o  t rans form t h e  s c a l a r  Irl and t h e  Zeri- 
- 
v a t i v e s  i n  ( 7 ) .  The t r a n s l a t i o n  i s  d e s c r i b e d  i n  F igure  1, 
Figure  1. ~ e s c r i p t i o n  of t h e  T r a n s l a t i o n  
The v e c t o r  6 g i v e s  t h e  p o s i t i o n  of t h e  o r i g i n  of t h e  old 
- 
system wi th  r e s p e c t  t o  t h e  new. A t y p i c a l  f i e l d  p o i n t ,  P, 
i s  desc r ibed  by t h e  v e c t o r s  r and R i n  t h e  o l d  and new sys- 
- - 
terns r e s p e c t i v e l y .  I n  what fo l lows  w e  w i l l  a s s o c i a t e  t h e  
s p h e r i c a l  coo rd ina t e s  i n  t h e  new system, R ,  0 8 ,  4 with  
R and 6 , 0 , +  with  6 where 8 and 8 '  a r e  c o l a t i t u d e s .  
- - 
A t  t h i s  p o i n t ,  a l l  t h e  necessary  n o t a t i o n  has  been 
in t roduced  wi th  t h e  except ion  of t h e  p o s i t i v e  degree  
s p h e r i c a l  harmonic corresponding t o  (5 )  
This  w i l l  be of u se  l a t e r .  
4 .  The Transformat ion Process  
I n  o r d e r  t o  use  on ly  q u a n t i t i e s  which a r e  de f ined  w i t h  
r e s p e c t  t o  t h e  new system, w e  f i r s t  examine t h e  q u a n t i t y  
I f  $ i s  t h e  ang le  between R and 6 ,  t hen  accord ing  t o  t h e  
- - 
g e n e r a t i n g  f u n c t i o n  f o r  Legendre polynomials,  
If we use the addition theorem for spherical harmonics, 
Jackson [5], and some elementary manipulations, then 
where P:(B) = ~:(cos 8) , for the sake of brevity. This depends 
also on the definition of the associated Legendre polynomials 
for negative order, MacRobert [ 7 ] ,  
-m (n-m) ' m 
Pn !x) = .e Pn (x) r m > 0. 
If the definitions (5) and (8) are used, we find the result, 
1 "3 n 
- -  
(n-m) ' 
- 1  1 . e  
R-6 1 n=O m=-n I _  - 
The asterisk superscript denotes the complex conjugate, 
It is at this point that the convenience of the represen- 
tation (7) becomes more apparent. With the result given in 
(9) , the term l/lr - 1 in (7) has been dealt with. All that re- 
mains is the transformation of derivatives, From the def;ni- 
ti3n of the transformation, 
and since the coordinate axes remain parallel, it is a simple 
process to transform derivatives from those with respect 
to the old coordinates, x,y,z to those with respect to the 
new quantities, components of R or 6. 
- - 
Consider first the case where R > 6 .  Symbolically, 
from (lo), 
and if we look at (7) the derivatives of interest are 
a a a a m a n-m m a k-m (= +i -1 (=) = (= +i -) a Y 
3~ I 
where the derivatives with respect to capital letters refer 
to derivatives with respect to components of R. We can B:hus 
- 
use this result, and the first part of equation ( 9 ) ,  to find 
(-ilk-a n (n-m) I a a I, a k- P,
Vka.(~) = 1 TGGJ-! Him 6 -  - ax ay - az vnmS~:* 
n=O m=-n 
The derivatives acting on Vnm(R). - can be simplified with the 
additional use of (7) , and finally 
This result holds for R > L .  
S i m i l a r l y ,  i f  R < 6, w e  f i n d  from ( 1 0 )  t h a t  
I f  an analogous procedure i s  fol lowed,  we f i n d  
This  ho lds  i f  R < 6 .  
5. Discuss ion  
I f  t h e  exp res s ions  (11) and ( 1 2 )  are examined, t hen  we 
s e e  t h a t  t h e  c h a r a c t e r  of  an i n v e r s e  harmonic s e r i e s  such as 
( 2 )  may change i f  t h e  t r a n s l a t i o n  d i s t a n c e  i s  l a r g e  enough, 
The s e r i e s  may change from one i n  i n v e r s e  powers t o  one 
i n  p o s i t i v e  powers. However, a s  was po in t ed  o u t  by Lee 
( p r i v a t e  communication), t h i s  i s  a seeming paradox. If w e  
t a k e  a s i n g l e  l a r g e  ( R  < 6) t r a n s l a t i o n ,  t hen  t h e  s e r i e s  
changes from one i n  nega t ive  powers t o  one i n  p o s i t i v e  powers, 
But i f  we c o n s i d e r  a f i n i t e  sequence of  sma l l  t r a n s l a t i o n s ,  
each of which keeps R > 6 ,  w e  may o b t a i n  an expansion with 
t h e  same o r i g i n  b u t  w i t h  an i n v e r s e  power expansion.  The 
exp lana t ion  f o r  t h i s  can be  found i n  F igu re  2 .  There a r e  
i n  c e r t a i n  c i rcumstances ,  two series which a r e  meaningful ,  
and a choice  must be  made between them depending on t h e  
c i rcumstances  under cons ide ra t ion .  I n  p a r t  A of F igure  2 
w e  see t h e  u s u a l  s i t u a t i o n  where a s p h e r i c a l  harmonic ex- 
pansion converges o u t s i d e  t h e  s m a l l e s t  sphe re ,  cen t e red  at 
t h e  c e n t e r  of mass, con ta in ing  a p l a n e t .  I n  p a r t  B ,  we 
s e e  t h e  c a s e  where t h e  o r i g i n  has  been s h i f t e d  through a 
d i s t a n c e  which i s  s m a l l e r  t han  t h e  r a d i u s  of t h e  sphere  of 
convergence.  I n  t h i s  c a s e  t h e r e  i s  s t i l l  only  one s e r i e s ,  

which i s  a p p l i c a b l e .  I f ,  however, t h e  o r i g i n  i s  s h i f t e d  
through a  d i s t a n c e  which i s  g r e a t e r  t han  t h e  r a d i u s  of the 
o r i g i n a l  sphere  of convergence, t h e r e  a r e  two a p p l i c a b l e  series, 
One s e r i e s  converges o u t s i d e  t h e  s m a l l e s t  sphere  cen te red  
a t  t h e  new o r i g i n  which con ta ins  t h e  p l a n e t  and t h e  o t h e r  
converges i n s i d e  t h e  s ~ h e r e  about t h e  new o r i g i n  which i s  
t angen t  t o  t h e  o r i g i n a l  sphere  of  convergence. Whether 
t h i s  second s e r i e s  converges down t o  t h e  s u r f a c e  of t h e  
p l a n e t  seems t o  be  an open q u e s t i o n  a t  t h e  moment i n  s p i t e  
of  a  d i s c u s s i o n  due t o  Moritz [ 8 ] .  
The a n a l y t i c a l  exp res s ions ,  (11) and ( 1 2 )  , can be  
s u b s t i t u t e d  i n t o  t h e  series ( 2 )  and t h e  summations i n t e r -  
changed s o  t h a t  a  new s p h e r i c a l  harmonic expansion,  r e f e r r e d  
t o  a  new o r i g i n ,  i s  ob ta ined .  This  i n t e r change  of sununations 
i s ,  however, v a l i d  only  i n  t hose  r eg ions  where convergence 
of t h e  o r i g i n a l  series i s  guaranteed.  I n  any p r a c t i c a l  
c a s e ,  t h e  numerical  behavior  of t h e  summations i nvo lv ing  
t h e  o r i g i n a l  c o e f f i c i e n t s  should be i n v e s t i g a t e d  t o  
i n s u r e  t h a t  measurement e r r o r s  a r e  n o t  magnified t o  t h e  
p o i n t  where t h e  r e s u l t s  a r e  i n s i g n i f i c a n t .  
Ref ereizces 
Aardoom, L., "Some Transformation Properties for the 
Coefficients in a Spherical Harmonics Expansion of the 
Earth's External Gravitational Potential", Tellus XXI 
(1969), 4, pgs. 572-584. 
Courant, R. and Hilbert, D., Methods of Mathematical 
Physics, V1, Interscience, New York, 1953. 
Cunningham, L.E., "On the Computation of the Spherical 
Harmonic Terms Needed During the Numerical Integration 
of the Orbital Motion of an Artificial Satellite", 
Celestial Mechanics, V2, No. 2, July, 1970, pgs, 
207-216. 
Hobson, E.W., The Theory of Spherical and Ellipsoidal -- 
Harmonics, Chelsea, New York, 1965. 
Jackson, J.D., Classical Electrodynamics, Wiley, New 
York, 1962. 
Lee, W.N., "The Transformation of a Series Expansion 
in Solid Spherical Harmonics Under Translation and 
Rotation of Coordinates", M.I.T. Measurement Systems 
Laboratory Report TE-47, June, 1971. (M.S. Thesis 
M.I.T. Aeronautics and Astronautics Department). 
MacRobert, J.M,, Spherical Harmonics, 3rd ed., 
Pergammon, New York, 1967. 
. . 
Moritz, H., "Uber die Konvergenz der Kugelfunctions- 
entwicklung fur"das Aussenraumpotential an der 
~rdoberflache", Osterreichischen Zeitschrift fur 
Vermessungswesen, V.49, No. 1, 
Robin, L., Fonctions sph6riques de Legendre et 
Fonctions Sphgroidales, 3 Vols,, Gauthier-Villars, 
Paris, 1957. 


The transformation of external harmonic series under a translation of origin

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