For the mathematical relationship between J2 (km^5/s^2) and dimensionless J2 - which one is derived from the other? - Space Exploration Stack Exchange most recent 30 from space.stackexchange.com 2019-11-18T23:35:16Z https://space.stackexchange.com/feeds/question/22976 https://creativecommons.org/licenses/by-sa/4.0/rdf https://space.stackexchange.com/q/22976 6 For the mathematical relationship between J2 (km^5/s^2) and dimensionless J2 - which one is derived from the other? uhoh https://space.stackexchange.com/users/12102 2017-09-10T06:14:51Z 2019-10-08T08:22:15Z <p>The term $J_2$ seen in discussions of nodal precession of orbits (e.g. <a href="https://en.wikipedia.org/wiki/Sun-synchronous_orbit" rel="noreferrer">sun-synchronous</a>) appears to come in two flavors. For example, in the Wikipedia article <a href="https://en.wikipedia.org/wiki/Geopotential_model" rel="noreferrer">Geopotential model</a> 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 $\tilde{J_2}=1.0826 \times 10^{-3}$.</p> <p>The article says that when many coefficients in the spherical harmonic expansion are used, they are normalized to the representative radius of the body <em>and</em> its standard gravitational parameter, in this case the Earth. It gives the expression:</p> <p>$$\tilde{J_n}=-\frac{J_n}{\mu R^n}$$</p> <p>where I believe $\mu$ is the standard gravitational parameter also written as the product $GM$, and $R$ is the normalizing radius.</p> <p>I also see Earth's $J_2$ written in normalized form, as in this article about the <a href="https://ipnpr.jpl.nasa.gov/progress_report/42-196/196C.pdf" rel="noreferrer">JPL Ephemerides DE430 &amp; DE431</a>, Tables 9 and 10 for the Sun and Earth parameters are shown below.</p> <p>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.</p> <p>Which value stands on its own, and which one is the <em>derived</em> value? This question becomes more important when considering less familiar bodies than the Earth or the Sun (or Jupiter or Saturn), especially <em>irregular rocky bodies such as some asteroids</em> where the definition of a radius may be difficult. </p> <p>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. </p> <p><em>extra credit:</em> 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 <a href="https://en.wikipedia.org/wiki/Geopotential_model" rel="noreferrer">Wikipedia article</a>)?</p> <hr> <p><strong>below x2:</strong> From <a href="https://en.wikipedia.org/wiki/Geopotential_model" rel="noreferrer">here</a>.</p> <blockquote> <p><a href="https://i.stack.imgur.com/6Bkdh.png" rel="noreferrer"><img src="https://i.stack.imgur.com/6Bkdh.png" alt="enter image description here"></a></p> <p><a href="https://i.stack.imgur.com/tcCMA.png" rel="noreferrer"><img src="https://i.stack.imgur.com/tcCMA.png" alt="enter image description here"></a></p> </blockquote> <p><strong>below:</strong> From <a href="https://ipnpr.jpl.nasa.gov/progress_report/42-196/196C.pdf" rel="noreferrer">here</a>.</p> <blockquote> <p><a href="https://i.stack.imgur.com/Zy8m3.png" rel="noreferrer"><img src="https://i.stack.imgur.com/Zy8m3.png" alt="enter image description here"></a></p> </blockquote>