Zonal and tesseral harmonic coefficients for the geopotential function are usually tabulated in normalized form to provide immediate information as to the relative significance of the coefficients in the gravity model. The normalized form of the geopotential coefficients cannot be used for computational purposes unless the gravity model has been modified to receive them. This modification is usually not done because the absolute or unnormalized form of the coefficients can be obtained from the simple mathematical relationship that relates the two forms. This computation can be quite tedious for hand calculation, especially for the higher order terms, and can be costly in terms of storage and execution time for machine computation. In this report, zonal and tesseral harmonic coefficients for the geopotential function are tabulated in absolute or unnormalized form. The report is designed to be used as a ready reference for both hand and machine calculation to save the user time and effort

Kirkpatrick, J. C.

English

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w(r
I	 k/^	 1r	 f
I	 1
JSC-09429
NASA TECHNICAI. MEMORANDUM	 NASA TM X- 58181
April 1976
ZONAL AND TESSERAL HARMONIC COEFFICIENTS FOR THE
GEOPOTENTIAL FUNCTION, FROM ZERO TO 18TH ORDER
(NASA-TC-X-`E1E1)	 ZCtAL ANC TESSERAL
HAFMCNIC CCEFFICli NIS FCR 1HE GECFCIFFTIAL
FUWCTICK, FFCM ZEMO TC 16TH CFCES (NASA)
1' p HC 13.`_C	 rECI 12A G3/v4
N -16- 27931
Unclas
42361
y
r
NATIONAL AERONAUTICS AND SPACE ADMINISTRATION
LYNDON A. JOHNSON SPACE CENTER
HOUSTON, TEXAS 77058
JUL 197P
RECEIVED
N"'3A STI FACIDIr 4
INPUT BRANCN `^^,^
I Report No	 2	 Government Accession No 3.	 Rec-p,ent's C.ataloq No
NASA TM X-5818'
4 Tale and Subtitle 5	 Report Date
April 1976
ZONAL AND TESSERAL IIARPION"O COEFFICIENTS FOR THE
GEOPOTENTIAL FUNCTION, FROM ZERO TO 18TH ORDER
6	 Performing Organitation Caw*
JSC-09429
7 Author(sl 8	 Performing Orgarotation Report No
James C. Kirkpatrick
10	 Work Unit No
986 16-00 00 729 Performing Organttahon Name and Address
Lyndon B. Johnson Space Center
11. Contract or Grant No
Houston, Texas
	
77058
13	 Type of Report and Per,ud Covered
Technical Memorandum12 Sponsoring Agency Name and Address
National Aeronautics and Space Administration
Washington, D.C. 	 10546
14	 SpO4)6or,n9 Agttncy code
15 Supplementary Notes
16 As.adarct
Zonal and tesseral harmonic coefficients for the geopotential function are usually tabulated in
normalized form to provide immediate information as to the relative significance of the coefficients
in the gravity model.	 The normalized form of the geopotential coefficients cannot be used for com-
putational purposes unless the gravity model has been modified to receive then,. 	 This modifica-
tion is usually not done because the absolute or unnorm:°,lined form of the coefficients can be
obtained from the simple mathematical relationship that relates the two forms. 	 This computation
can b.2 quite tedious for hand calculation, especially for the higher order terms, and can be
costly in terms of storage and execution time for machine computation. 	 In this report, zonal and
tesseral harmonic coefficients for the geopotential function are tabulated in absolute or unnormal-
ized form.
	 The report is designed to be used as a ready reference for both hand and machine
calculation to save the user time and effort.
11. Key Words (Suggested by Author(s)) ('tl	 D,str,but,on Statement
Zonal harmonics	 Geopotential I	 STAR Subject Category:
Tesseral harmonics	 Harmonic functions 64 (Numerical Analysis)
Coefficients
Legendre functions
19, Security	 Ciassif. !of this report! 20	 Security Classi f	(of this page! 21	 No	 of Page. 22	 P rr
Unclassified Unclassified 11 $3.50
'For sale by the National Technical Information Service, Springfield, Virginia 22161
JSC torn 1424 IPev Nov 751
	 NASA - JSC
NASA TM X-58181
ZONAL AND TESSERAL HARMONIC COEFFICIENTS .'OR THE
GEOPOTENTIAL FUNCTION. FROM ZERO TO 18TH ORDER
James C. Kirkpatrick
Lyndon B. Johnson Space Center
Houston, Texas 71058
ZONAL AND TESSERAL HARMONIC COEFFICIENTS FOR THE
GEOPOTENTIAL FUNCTION, FROM ZERO TO 18TH ORDER
By James C. Kirkpatrick
Lyndon B. Johnson Space Center
SUMMARY
The zonal and tesseral harmonic ; oefficients for the geopotential function are
given in unnormalized form, from zero to 18th order.
INTRODUCTION
The coefficients of the geopoten .ial function, both zonal and !esseral , are
usually published in normalized form so that all coefficients have about the same
order of magnitude. Although this practice is useful, it obscures the actual mag-
nitude of the coefficients. This type of publication results in an unwieldy reference
for computation purposes, and it requires the evaluation of lengthy factorial
expressions to arrive at useful data. This report has been prepared to provide
the user of the coefficients of the geopotential function with a ready reference for
the zonal and tesseral harmonic coefficients. The data for this work were obtained
from reference 1.
DISCUSSION
The geopotential function V is given as
00	 n
V =gr I
n=0
n
EPnm (sin (p) ICnm cos (mX) + Snmsin (mX)l
m=0 	i
where gF is the gravitati;rial parameter, having units of length "/time 2 ; r is the
magnitude of the pos<t'.on vector to the point in question, having units of length;
RE is the equatorial radius of the attracting hvdy (in this case, the Earth), having
units of length; Cnm and Snrr rare the harmonic coefficients; Pnm (sin (p) is the
Legen-Ire polynomial function of degree n and order m given by the expression,
rn	 1	 (-1 ) tOn - 2t)!	 n-m 2tP	 (cp)nm (sin cp) =cos	 n ^ ?(n - t) , (n m	 t) , sin	 (^)
2 t=0
where Q is the greatest integer eq-1:Z.i to (n - m)/2 if n - m i3 even and the
greatest integer equal to (n - m - 1)/2 if n - m is odd, y is the latitude of the
point in question, and X is the east longitude of the point in question.
The ;.armonic coefficients are further classified as follows. When m = 0,
the harmonic coefficients are called zonal; when n # m, the harmonic coefficients
are called tesseral. For the particular case in which n = m, the harmonic
coefficients are called sectorial. It may be seen from equation (1) that there are no
zol.al Sn0 coefficients. Further, C n0	 Jn0 "here Jn0 refers to the usually
published zonal harmonic coefficients. The term C 00 = +1 as the geopotential
function is associateL, .,fith an inverse square law of attraction. Tables I and 11 con-
tain the unnormalized harmonic coefficients C nm and S nm ,
 
respectively. The
normalized barred values are converted to unnormalized values as follows.
1
(2-S )(2n+1)(n m)!^
C nm' S	
m0
nm	 (n + m)!	 Snm' Snm	 (3)
where Sm0 = 1 for m = 0 and Sm0 - 0 for m # 0.
CONCLUDING REMARK
Thy information presented in this report is in a form designed for immediate
use for either hand or machine computation of the geopotential function end its
gradient.
Lyndon B. Johnson Space Center
National Aeronautics and Space Administration
Houston, Texas, April 21, 1976
986-16--00-00-72
REFERENCE
1. Gaposchkin , E. M.: 1973 Smithsonian Standard Earth (1I I) . Special Rep.
No. 353, Smithsonian Astrophysical Observatory (Cambridge, Mass.),
Nov. 28, 1973,  pp. 282-293.
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Zonal and tesseral harmonic coefficients for the geopotential function, from zero to 18th order

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