# Spherical Harmonic coefficients of Degree 2

The spherical harmonic of degree 2 and order 0 - C(2,0) - is due to the flattening of the Earth. Its technical name is 'Earth’s dynamic oblateness'. C(2,0) (also known as 'J2', but they differ by a constant factor: J2 = -C(2,0)*sqrt(5)) is only a function of the difference between equatorial and polar radii of the equipotential surface of the Earth's gravity field that best fits mean sea level; more accurately, J2 is a function of the difference in principal moments of inertia (for more information on spherical harmonics, see the Wikipedia article & references therein).

A steady decrease in J2 has been observed by satellites since 1979, but also with historical eclipse data over the last 2500 years. This long-term decrease is mainly due to glacial isostatic adjustment. The steady decrease is modulated by ocean and ice mass redistribution [Cox and Chao, 2002; Dickey et al, 2002]. More recently, the accelerated ice mass loss from the Greenland and Antarctic ice sheets appears to increasingly offset the long-term GIA signal [Cheng et al., 2013].

We make two types of time series available for **download**:

- Variations of the C(2,0) spherical harmonic from 1976 to present [Cheng et al., 2013];
- Variations of the complete degree-2 spherical harmonic coefficients for the time period 2002 to present [Cheng et al., 2011]. This time series uses the same geophysical background fields as the latest GRACE RL06, in particular the AOD (atmosphere-ocean dealiasing) models.

Please be sure to check the **README** and individual file headers for more details about the SLR data and processing.

The estimates of C(2,0), C(2,1), S(2,1), C(2,2) and S(2,2) are obtained from the analysis of Satellite Laser Ranging (SLR) observations of five geodetic satellites: LAGEOS-1 and 2, Starlette, Stella, and Ajisai. The background gravity model used in the SLR analysis is consistent with the current GRACE **Release-06** processing, including use of the same Atmosphere-Ocean De-aliasing (AOD) model. However, the monthly mean of the AOD model (included in the file) has been restored, so that the monthly coefficients represent the full signal. These data are housed at the Center for Space Research at The University of Texas Austin.

#### Acknowledgement and Citation

When using these data, please acknowledge receiving the data from "http://grace.jpl.nasa.gov", and **cite** (depending on the product):

Cheng, M., J. C. Ries, and B. D. Tapley (2011), Variations of the Earth's figure axis from satellite laser ranging and GRACE, J. Geophys. Res., 116, B01409, doi:10.1029/2010JB000850.

Cheng, M., B. D. Tapley, and J. C. Ries (2013), Deceleration in the Earth's oblateness, J. Geophys. Res. Solid Earth, 118, 740-747, doi:10.1002/jgrb.50058.

#### Further References

C.M.Cox, B.F.Chao (2002), Detection of a Large-Scale Mass Redistribution in the Terrestrial System Since 1998, Science 297, p831.

J.O. Dickey et al (2002), Recent Earth Oblateness Variations: Unraveling Climate and Postglacial Rebound Effects, Science, p1975.