# For the mathematical relationship between J2 (km^5/s^2) and dimensionless J2 - which one is derived from the other?

The term $J_2$ seen in discussions of nodal precession of orbits (e.g. sun-synchronous) appears to come in two flavors. For example, in the Wikipedia article Geopotential model it has a value of $1.7555 \times 10^{10} km^5s^{-2}$; in MKS units that's $1.7555 \times 10^{25} m^{5}s^{-2}$. It is also listed in the alternate, unitless form as $\tilde{J_2}=1.0826 \times 10^{-3}$.

The article says that when many coefficients in the spherical harmonic expansion are used, they are normalized to the representative radius of the body and its standard gravitational parameter, in this case the Earth. It gives the expression:

$$\tilde{J_n}=-\frac{J_n}{\mu R^n}$$

where I believe $\mu$ is the standard gravitational parameter also written as the product $GM$, and $R$ is the normalizing radius.

I also see Earth's $J_2$ written in normalized form, as in this article about the JPL Ephemerides DE430 & DE431, Tables 9 and 10 for the Sun and Earth parameters are shown below.

Since conversion between these two representation requires two separate numerical values (the characteristic radius defined by convention and the standard gravitational parameter) one of them can be used without $R$ and $GM$, but the other requires the value for both to be stated explicitly or implied.

Which value stands on its own, and which one is the derived value? This question becomes more important when considering less familiar bodies than the Earth or the Sun (or Jupiter or Saturn), especially irregular rocky bodies such as some asteroids where the definition of a radius may be difficult.

The answer may depend on the method of measurement. I am not sure if $GM$ and $J_2$ (the one with units) can be measured separately, or if it is a radiometric determination.

extra credit: To make sure I understand, is $\tilde{J}_{2 \ ☉} \approx -1.36 \times 10^{31} m^5s^{-2}$? Or is the minus sign in the equation in Wikipedia erroneous (shown above, and below, and just before Equation 10 in the Wikipedia article)?

below x2: From here.

below: From here.

• I am not positive of this so I am writing a comment and not an answer. But in that very same wikipedia article, this phrase is seen: "For large values of n the coefficients above (that are divided by r(n + 1) in (9)) take very large values when for example kilometers and seconds are used as units. In the literature it is common to introduce some arbitrary "reference radius" R close to Earth's radius and to work with the dimensionless coefficients". – Ross Presser Sep 30 '17 at 7:15
• @RossPresser thanks for your comment! I think the real issue might turn out to be how the multipole moments are experimentally determined. These are (usually) measured quantities; are they measured as a ratio to the monopole moment, ($GM$) or are they measured in absolute terms. The latter would suggest the "with-units" option, the former the "unitless" option. – uhoh Sep 30 '17 at 12:07