# Total rotational angular momentum estimates for Jupiter?

I wrote a quickie answer where I estimate the total rotational angular momentum of Jupiter as about 7E+38 kg m^2/s. However it bothers me because one of the sources gets this number assuming uniform density, and the other two cite 6.9E+38 kg m^2/s but it's not clear to me where this comes from.

Are there better estimates that use reasonable models for Jupiter's density distribution?

update: If not answerable due to uncertainties in density distribution and/or gradients in rotation rate (i.e. non rigid-body rotation) which I sort-of remember reading about recently, then an explanation why a better value isn't available would be an acceptable answer.

• Not sure you can do the $n=3/2$ polytrope analytically. – ProfRob Jan 1 '20 at 8:41

A better estimate might be to use the moment of inertia of a $$n=3/2$$ (fully convective) polytrope, which will be a good approximation in an object like Jupiter, even when it approaches electron degeneracy in its interior. You can look this up and it is given by $$I = kMR^2$$, with $$k=0.205$$ (almost exactly a factor of two smaller than a uniform sphere because the mass is concentrated towards the centre).
If we use a mass $$M$$ of $$1.898 \times 10^{27}$$ kg, the equatorial radius of Jupiter ($$R=71500$$ km) and the rotation rate determined from its magnetosphere ($$P= 9.93$$ hours), then the angular momentum ($$2\pi I/P$$) is $$3.496 \times 10^{38}$$ kg m$$^2$$ s$$^{-1}$$.
If instead, we use the volumetric average radius of $$R=69900$$ km, this reduces to $$3.34\times 10^{38}$$ kg m$$^2$$ s$$^{-1}$$.
This simple approach neglects the more complicated equation of state of Jupiter, it's non-sphericity and the probable presence of a solid core. Consideration of these, together with constraints provided by the measured even harmonics of the gravitational field (referred to as $$J_2$$ and $$J_4$$, led Helled et al. (2011) to suggest that $$k=0.264$$, with an uncertainty below 1%. Combined with the average radius, this gives $$4.30\times 10^{38}$$ kg m$$^2$$ s$$^{-1}$$.
More recent work by Ni (2018) has used refinements to the gravitational harmonics from Juno measurements, along with a more sophisticated interior model to suggest $$k=0.274$$ with 0.5% uncertainty.