What are the gravitational binding energies of the planets in our solar system? In particular, interested in the giant planets: Jupiter, Saturn, Uranus, and Neptune. Ideally the information would be from a paper or other peer-reviewed source, but I'll take what I can get. What I'd like to avoid is the approximation
$$
U_\circ=\frac{3G}{5}\cdot\frac{m^2}{r}=\frac{16\pi^2G}{15}r^5\rho^2
$$
which comes from assuming that the planet is a sphere of uniform density. This isn't all that realistic for rocky planets, but it's far worse for ice and gas giants.
I know that there's been a lot of revision recently on the composition and layers of Jupiter and Saturn, and a lot is based on our ability to model metallic hydrogen about which we know very little. So recent sources would be better than old sources (though I'll take what I can get).
Attempt
Arturo don Juan and A. C. A. C. asked how much this effect matters, so here are some calculations. I'll use Jupiter as my example. Its mass is 1.8986e27 kg and its mean radius is 69911 km, giving it $U_\circ = 2.065 \cdot 10^{36}\text{ J}.$ Using the crude approximation
$$
\rho_x=\begin{cases}
25 & \text{ if }x \le .14\\
6.0533 – 6.3166x & \text{ if }.14<x<.8\\
\frac52(1-x) & \text{ if }x \ge .8
\end{cases}
$$
(where 0 is the core and 1 is the edge of the clouds and the density is in g/cm^3) based on the diagram here I compute
$$
U \approx 1.775 \cdot 10^{36}\text{ J}.
$$
Background
I recently answered a question about the Death Star's destructive ability in Star Wars. My answer relied heavily on the uniform density approximation for scaling, and I'm not sure how much that affected it. I hoped that starting from reasonable baselines would help, but I wasn't really able to do that because I couldn't find any information on the GBE for planets other than Earth. (I could find plenty of exercises asking students to apply the above approximation of course.)
I have to imagine there's something out there, maybe just giving the best-guess density $\rho_x$ at a given depth $x$, from which one can integrate
$$
U = \frac{16\pi^2G}{3}\int_0^r x^4\rho_x^2 dx
$$
to get the GBE.
Best Answer
The gravitational potential energy of a polytropic sphere of gas (i..e governed by a polytropic equation of state), with a polytropic index $n$ is given by $$ \Omega = -\frac{3}{5-n}\ \frac{GM^2}{R}$$ See for example here.
A constant density sphere would have to have to be incompressible. Since a polytrope has a pressure $p \propto \rho^{1 +1/n}$, this corresponds to $n=0$ and gives you your result for a uniform sphere.
Higher values of $n$ have more centrally condensed profiles and larger binding energies.
The appropriate value of the polytropic index for brown dwarfs and gas giants, where energy transport is convective or where the gas is (non-relativistically) degenerate is $n=3/2$. In which case, your leading coefficient increases from 3/5 to 6/7. i.e. No big deal for an approximate calculation.