Parameters of Common Relevance of
Astronomy, Geodesy, and Geodynamics
By E. Groten (President of IAG Subcommission 3)
At present, systems of fundamental constants are in a state of transition. Even though the uncertainties of many constants have substantially decreased, the numerical values themselves did not substantially change. On the other hand, relativistic reductions and corrections underwent a variety of substantial revisions that, however, did not yet find final agreement within the scientific working groups of international committees in charge of evaluating relevant quantities and theories. Consequently, substantial changes and revisions still have to be expected in IAU, IERS, IUGG etc. within the next few years.
Therefore SC 3, after lengthy discussions and considerations, decided not to propose, at this time, any change of existing geodetic reference systems such as WGS 84 (in its recent form updated by NIMA, 1997) and GRS 80. This would only make sense in view of relatively small numerical changes which would not justify, at this moment, complete changes of systems and would rather produce more confusion within user communities – as soon as working groups within IAU, IERS etc. have made up their minds concerning the background of new systems and will be prepared to discuss new numerical values. This should be around the year 2001.
The present situation is also reflected by the fact that in view of substantial progress in evaluating temporal changes of fundamental „constants" and related accuracies, we should better speak about „fundamental parameters" instead of „fundamental constants"; however, the majority of members of SC 3 preferred to preserve the traditional name of SC 3.
In view of this situation and of the fact that IERS in its „conventions" which are edited at regular intervals SC 3 cannot and should not act independently in proposing changes of fundamental parameters,  there will consequently be relatively small changes in the following part on „current best estimates" and only minimal changes in the part on „official numerical values" within this report. It is, moreover, proposed to strengthen the interrelations between IERS and SC 3.
Interrelations between IERS, IAU, IAG etc. make it, however, more difficult to implement necessary changes in fundamental systems. This was particularly realized in discussing adoption of new fundamental constants. This fact may be explained by the discussion of small changes inherent in the adoption of particular tidal corrections which became relevant in view of higher accuracies of ± 10^{8} or ± 10^{9}. It turns out to be almost impossible to explain to other scientific bodies the modern relevance of the dependence of the numerical value of the semimajor axis „a" of the Earth on specific tidal corrections. Other temporal variations imply similar difficulties.
From the view point of SC 3, i.e. in deriving fundamental parameters, it is, to some extent, confusing that a variety of global or/and regional systems exist; it would be best to use only one global terrestrial and one celestial system such as ITRF, referred to a specific epoch, and an associated celestial system, unless precise transition and transformation formulae are available such as those between ETRF, ITRF, EUREF, and perhaps WGS 84 (in updated form), IGS, GRS 80 etc. where IERSsystems, in general, could serve to maintain transformation accuracy and precision.
However, the consequent replacement of „a" by a quantity such as the geopotential at the geoid W_{0} (which is independent of tides) in a geodetic reference system (or a similar system) was not well understood and not supported by other working groups so that we finally gave up the idea of a reformation of systems of fundamental constants in this way even though quantities such as W_{0} are now very precisely determined by satellite altimetry etc. Whether seasonal variations (Bursa et al. 1998a) of W_{0} are significant or not is still an open question, when expressed in R_{0} = GM/W_{0} they amount to a few centimeters in global radius.
I Current (1999) best estimates of the parameters of common relevance to astronomy, geodesy, and geodynamics
SI units are used throughout (except for the TDBvalue (value below (4))
(SIvalue can be associated with TCB or TCG)
velocity of light in vacuum 
c = 299 792 458 m s^{1}. (1)
Newtonian gravitational constant 
G = (6 672.59 ± 0.30) ´ 10^{14} m^{3} s^{2} kg^{1}. (2)
Geocentric gravitational constant (including the mass of the Earth’s atmosphere); reconfirmed by J. Ries (1998, priv. comm.) 
GM = (398 600 441.8 ± 0.8) ´ 10^{6} m^{3} s^{2}. (3)
For the new EGM 96 global gravity model
GM = 398 600 441.5 ´ 10^{6} m^{3} s^{2 }was adopted.
In TT units (Terrestrial Time) the value is
GM = (398 600 441.5 ± 0.8) ´ 10^{6} m^{3} s^{2}. (4)
Note that if expressed in old TDB units (solar system Barycentric Dynamical Time), the value is
GM = 398 600 435.6 ´ 10^{6} m^{3} s^{2}.
Based on well known transformation formulas we may relate GM in SIunits to TT/TCG/TCB; see IERSConvention 1996 p. 85. The well known secular term was not originally included in the GM(E)analysis, therefore it was related to TT, neither to SI nor (TCG, TCB); as still satellite analysis occurs without the secular term, GM(E) in TT is still of geodetic interest; GM(E) = GM of the Earth.
Mean angular velocity of the Earth’s rotation 
w = 7 292 115 ´ 10^{11} rad s^{1}. (5)
Table 1. Mean angular velocity of the Earth’s rotation 19781994
Year w [10^{11} rad s^{1}] 
Year w [10^{11} rad s^{1}] 
DLOD [ms] 
Min: 1978 7 292 114.903 max: 1986 292 115.043 
7.292 114.964 
2.17 2.31 1.83 1.84  
Longterm variation in w 
(6)
This observed average value is based on two actual components:
(7)
This value is commensurate with a tidal deceleration in the mean motion of the Moon n
(8)
(9)
 Seconddegree zonal geopotential (Stokes) parameter (tidefree, conventional, not normalized, Love number k_{2} = 0.3 adopted)
J_{2} = (1082 626.7 ± 0.1) ´ 10^{9}. (10)
To be consistent with the I.A.G. General Assembly Resolution 16, 1983 (Hamburg), the indirect tidal effect on J_{2} should be included: then in the zerofrequency tide system
J_{2} = (1082 635.9 ± 0.1) ´ 10^{9}. (11)
Table 2. The Stokes seconddegree zonal parameter; marked with a bar: fully normalized; k_{2} = 0.3 adopted for the tidefree system
Geo Poten Tial model 
Zerofrequency tide system J_{2} [10^{6}] [10^{6}] 
Tidefree J_{2} [10^{6}] [10^{6}] 

JGM3 EGM 96 
484.16951 
1082.6359 
484.16537 484.16537 
1082.6267 
Longterm variation in J_{2} 
(12)
seconddegree sectorial geopotential (Stokes) parameters (conventional, not normalized, geopotential model JGM3) 
(13)
(14)
(15)
Table 3. The Stokes seconddegree sectorial parameters; marked with a bar: fully normalized
Geopotential model 
[10^{6}] 
[10^{6}] 
JGM3 EGM 96 
2.43926 2.43914 
1.40027 1.40017 
Only the last decimal is affected by the standard deviation.
For EGM 96 Marchenko and Abrikosov (1999) found more detailed values:
Table 3. Parameters of the linear model of the potential of 2nd degree
Harmonic coefficient 
Value of coefficient ´ 10^{6} 
Temporal variation ´ 10^{11}[yr^{1}] 
= =  
484.165371736 0.00018698764 0.00119528012 2.43914352398 1.40016683654 
1.16275534 0.32 1.62 0.494731439 0.203385232 
Coefficient H associated with the precession constant
(16)
The geoidal potential W_{0} and the geopotential scale factor R_{0} = GM/W_{0} recently derived by Bursa et al. (1998) read
W_{0} = (62 636 855.611 ± 0.5) m^{2}s^{2}, (17)
R_{0} = (6 363 672.58 ± 0.05) m.
W_{0} = (62636856.4 ± 0.5) m^{2}s^{2} J. Ries (priv. comm, 1998) found globally.
If W_{0} is preserved as a primary constant the discussion of the ellipsoidal parameters could become obsolete; as the Earth ellipsoid is basically an artefact. Modelling of the altimeter bias and various other error influences affect the validity of W_{0}determination. The variability of W_{0} and R_{0} was studied by Bursa (Bursa et al. 1998) recently; they detected interannual variations of W_{0} and R_{0} amounting to 2 cm.
The relativistic corrections to W_{0} were discussed by Kopejkin (1991); see his formulas (67) and (77) where tidal corrections were included. Whereas he proposes average time values, Grafarend insists in corrections related to specific epochs in order to illustrate the timedependence of such parameters as W_{0}, GM, J_{n}, which are usually, in view of present accuracies, still treated as constants in contemporary literature.
Based on recent GPS data, E. Grafarend and A. Ardalan (1997) found locally (in the Finnish Datum for Fennoscandia): W_{0} = (6 263 685.58 ± 0.36) kgal m.
The temporal variations were discussed by Wang and Kakkuri (1998), in general terms.
Mean equatorial gravity in the zerofrequency tide system 
g_{e} = (978 032.78 ± 0.2) ´ 10^{5}m s^{2}. (18)
Equatorial radius of the Reference Ellipsoid (mean equatorial radius of the Earth) in the zerofrequency tide system (Bursa et al. 1998) 
a = (6 378 136.62 ± 0.10) m. (19)
The corresponding value in the mean tide system (the zerofrequency direct and indirect tidal distortion included) comes out as 
a = (6 378 136.72 ± 0.10) m (20)
and the tidefree value
a = (6 378 136.59 ± 0.10) m. (21)
The tide freevalue adopted for the new EGM96 gravity model reads a = 6 378 136.3 m.
Polar flattening computed in the zerofrequency tide system, (adopted GM, w , and J_{2} in the zerofrequency tide system) 
1/f = 298.25642 ± 0.00001 (22)
The corresponding value in the mean tide system comes out as
1/f = 298.25231 ± 0.00001 (23)
and the tidefree
1/f = 298.25765 ± 0.00001 (24)
Equatorial flattening (geopotential model JGM3). 
1/a _{1 }= 91 026 ± 10. (25)
Longitude of major axis of equatorial ellipse, geopotential model JGM3 
L _{a} = (14.9291° ± 0.0010°) W. (26)
In view of the small changes (see Table 3) of the second degree tesserals it is close to the value of EGM 96. We may raise the question whether we should keep the reference ellipsoid in terms of GRS 80 (or an alternative) fixed and focus on W_{0} as a parameter to be essentially better determined by satellite altimetry, where however the underlying concept (inverted barometer, altimeter bias etc.) has to be clarified.
Table 4. Equatorial flattening a _{1} and L _{a} of major axis of equatorial ellipse
Geopotential Model 
L _{a} [deg] 

JGM3 
91026 
14.9291 W 
Coefficient in potential of centrifugal force 
(27)
Computed by using values (3), (5) and a = 6 378 136.6
Principal moments of inertia (zerofrequency tide system), computed using values (11), (15), (3), (2) and (16) 
(28)
(29)
(a_{0} = 6 378 137 m);
C A = (2.6398 ± 0.0001) ´ 10^{35} kg m^{2}, (30)
C B = (2.6221 ± 0.0001) ´ 10^{35} kg m^{2},
B A = (1.765 ± 0.001) ´ 10^{33} kg m^{2};
(31)
(32)
A = (8.0101 ± 0.0002) ´ 10^{37} kg m^{2},
B = (8.0103 ± 0.0002) ´ 10^{37} kg m^{2}, (33)
C = (8.0365 ± 0.0002) ´ 10^{37} kg m^{2},
,
.
II Primary geodetic Parameters, discussion
It should be noted that parameters a, f, J_{2}, g_{e}, depend on the tidal system adopted. They have different values in tidefree, mean or zerofrequency tidal systems. However, W_{0} and/or R_{0} are independent of tidal system (Bursa 1995). The following relations can be used:
a (mean) = a (tidefree) + (34)
a (mean) = a (tidefree) +
a (zerofrequency) = a (tidefree) + (35)
a (zerofrequency) = a (tidefree) +
k_{s} = 0.9383 is the secular Love number, d J_{2} is the zerofrequency tidal distortion in J_{2}. First, the internal consistency of parameters a, W_{0}, (R_{0}) and g_{e} should be examined:
(i) If
a = 6 378 136.7 m
is adopted as primary, the derived values are
W_{0} = 62 636 856.88 m^{2} s^{2}, (36)
(R_{0} = 6 363 672.46 m), (37)
g_{e} = 978 032.714 ´ 10^{5} m s^{2}. (38)
(ii) If
W_{0} = (62 636 855.8 ± 0.5) m^{2} s^{2},
R_{0} = (6 363 672.6 ± 0.05) m,
is adopted as primary, the derived values are (mean system)
a = 6 378 136.62 m, (39)
g_{e} = 978 032.705 ´ 10^{5} m s^{2}. (40)
(iii) If (18)
g_{e} = (978 032.78 ± 0.2) ´ 10^{5} m s^{2},
is adopted as primary, the derived values are
a = 6 378 136.38 m, (41)
W_{0} = 62 636 858.8 m^{2} s^{2} (42)
(R_{0} = 6 363 672.26 m). (43)
There are no significant discrepancies, the differences are about the standard errors.
However, the inaccuracy in (iii) is much higher than in (i) and/or (ii). That is why solution (iii) is irrelevant at present.
If the rounded value
W_{0} = (62 636 856.0 ± 0.5) m^{2} s^{2} (44)
R_{0} = (6 363 672.6 ± 0.1) [m] (45)
is adopted as primary, then the derived length of the semimajor axis in the mean tide system comes out as
a = (6 378 136.7 ± 0.1) m, (for zerotide: 6 378 136.6) (46)
which is just the rounded value (20), and (in the zero frequency tide system)
g_{e} = (978 032.7 ± 0.1) ´ 10^{5} m s^{2}. (47)
However, SC 3 recommends that, at present, GRS 1980 should be retained as the standard.
III Consistent set of fundamental constants (1997)
Geocentric gravitational constant (including the mass of the Earth’s atmosphere) 
GM = (398 600 441.8 ± 0.8) ´ 10^{6} m^{3} s^{2},
[value (3)]
Mean angular velocity of the Earth’s rotation 
w = 7 292 115 ´ 10^{11} rad s^{1} [value (5)]
Seconddegree zonal geopotential (Stokes) parameter (in the zerofrequency tide system, Epoch 1994) 
J_{2} = (1 082 635.9 ± 0.1) ´ 10^{9 }[value (11)]
Geoidal potential 
W_{0} = (62 636 856.0 ± 0.5) m^{2} s^{2}, [value (44)]
Geopotential scale factor 
R_{0} = GM/W_{0} = (6 363 672.6 ± 0.05) m
[value (45)]
Mean equatorial radius (mean tide system) 
a = (6 378 136.7 ± 0.1) m [value (46)]
Mean polar flattening (mean tide system) 
1/f = 298.25231 ± 0.00001 [value (23)]
Mean equatorial gravity 
g_{e} = (978 032.78 ± 0.1) ´ 10^{5} m s^{2},
[value (18)].
Grafarend and Ardalan (1999) have evaluated a (consistent) normal field based on a unique set of current best values of four parameters (W°, w , J_{2} and GM) as a preliminary "followup" to the Geodetic Reference System GRS 80. It can lead to a levelellipsoidal normal gravity field with a spheroidal external field in the SomiglianaPizetti sense. By comparing the consequent values for the semimajor and semiminor axes of the related equipotential ellipsoid with the corresponding GRS80 axes (based on the same theory) the authors end up with axes which deviate by  40 and  45 cm, respectively from GRS 80 axes and within standard deviations from the current values such as in (21); but no gvalues are given until now.
IV Appendix
A1. Zerofrequency tidal distortion in J_{2}
(J_{2} = C_{20})
GM_{L} = 4 902.799 ´ 10^{9} m^{3}s^{2}
(selenocentric grav. Const.),
GM_{S} = 13 271 244.0 ´ 10^{13} m^{3}s^{2},
D Ĺ _{L} = 384 400 km
(mean geocentric distance to the Moon),
D Ĺ _{S} = 1 AU = 1.4959787 ´ 10^{11} m,
a_{0 } = 6 378 137 m
(scaling parameter associated with J_{2}),
e _{0 } = 23°26’21.4" (obliquity of the ecliptic),
e_{L} = 0.05490
(eccentricity of the orbit of the Moon),
i_{L} = 5°0.9’
(inclination of Moon’s orbit to the ecliptic),
e_{S} = 0.01671
(eccentricity of the heliocentric orbit of the EarthMoon barycenter),
n = a_{0}/R_{0} = 1.0022729;
k_{s} = 0.9383
(secularfluid Love number associated with the zerofrequency second zonal tidal term);
d J_{2} =  d C_{20} = (3.07531 ´ 10^{8}) k_{s} (conventional);
(fully normalized).
L = Lunar
S = Solar
A2. Definition
Because of tidal effects on various quantities, the tidefree, zerofrequency and mean values should be distinguished as follows:
A tidefree value is the quantity from which all tidal effects have been removed.  
A zerofrequency value includes the indirect tidal distortion, but not the direct distortion.  
A mean tide value included both direct and indirect permanent tidal distortions. 
Acknowledgement: This report is basically an updated version of M. Bursa’s SC 3 report presented in 1995 with some new material added.
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