In The Event One Wants a Gas Titan in His Worldbuilding Story

Question

In recent years, we have discovered exoplanets that defy our traditional perspectives on how planets work. WASP-17b is twice as wide as Jupiter, yet half as massive, probably because of its orbital proximity to its sun.

Saturn itself is very confusing. It is second to Jupiter in diameter and mass. Despite that, Saturn is the lightest planet in the solar system, at only 0.687 grams per cubic centimeter. Its atmosphere is 75% hydrogen and 25% helium--could this have played a part in determining Saturn's record density?

So in the event that some worldbuilder creates a five-billion-year-old alternate Jupiter that is 11x greater in mass (much like HD 106906 b) yet lighter in density and still in orbit 483.8 million miles from the sun, would the overall structure be realistically sound? Would a super-Jupiter with an atmosphere 75% hydrogen and 25% helium help reduce its density to below regular-Jupiter?

Answer 1

Wikipedia says:

Thus, Jupiter's atmosphere is approximately 75% hydrogen and 24% helium by mass, with the remaining one percent of the mass consisting of other elements. The interior contains denser materials, such that the distribution is roughly 71% hydrogen, 24% helium, and 5% other elements by mass.

Note the use of the "by mass" comparison, meaning that even in these results hydrogen is only weighted 1 against helium's $\approx$4 (depending on isotope).

For a more diffuse gas giant you need something pushing the gas out such that hydrostatic equilibrium is reached a little further out.

Now, to do this we can look to a couple of different equations.

$$ \frac{dp}{dz} = -\rho g = -\frac{4G\rho^{2}\pi z}{3} $$ Where we balance the force from the gas pressure ($p$) with the gravitational force at a height $z$.

We can relate the pressure and density through the volume and the ideal gas equation, given $\rho = \frac{M}{V} = \frac{\nu\mu}{V} \therefore V = \frac{\nu \mu}{\rho}$ and $pV=\nu RT$ such that: $$ p \mu = \rho R T $$ Where $\mu$ is the atomic weight of our gas. Now we can put it all together as: $$ \frac{dp}{dz} = -\frac{4G\pi z}{3} (\frac{\mu p}{RT})^{2} $$

However, we now reach an impasse. We have no fixed temperature and no fixed composition of the planet. You do, however, have a freedom to choose.

• You could decide your core is still hot, through radioactive decays, collisions, perhaps some artificial source. Regardless, if there is a sufficient heat source you can pump your gas giant up pretty big.

• You can play about with your density a little more.

• You could have some orbiting moons tugging the atmosphere out (though this wouldn't be stable).

You have the freedom to decide which of these you would like. If you are looking to write a story I doubt your readers want any particularly in depth analysis on just how the planet works so going much further, unless you particularly desire to, would be crunching numbers for the sake of it.

TLDR: Higher temperatures can counter for higher densities. Establishing a hydrostatic equilibrium further out can be a matter of increasing the temperature (and so gas pressure).