What determines the magnitude of the atmospheric scale height of a planet? What determines the magnitude of the atmospheric scale height of a planet?
https://en.wikipedia.org/wiki/Scale_height says that:
"Approximate atmospheric scale heights for selected Solar System bodies follow.
Venus: 15.9 km
Earth: 8.5 km
Mars: 11.1 km
Jupiter: 27 km
Saturn: 59.5 km
Titan: 21 km
Uranus: 27.7 km
Neptune: 19.1–20.3 km
Pluto: ~50 km

I would have thought that a massive planet like Jupiter, with it's very strong surface gravity would have a scale height that with a smaller magnitude. But it's scale height is just over three times that of earth's. And Saturn, another massive planet has an even bigger scale height, that is just over seven times that of earth's.
If it's possible, please put it in layman's terms or easy (elementary or middle school)math while remaining rigorous.
 A: In hydrostatic equilibrium,
$$\frac{dP}{dz} = - \rho g.$$
This means the scale height, which is the height over which the pressure changes substantially, is
$$H \sim \frac{P}{\rho g}.$$
According to the ideal gas law,
$$P = \frac{\rho R T}{\mu}$$
where $\mu$ is the molar mass. Thus,
$$H \sim \frac{RT}{\mu g}.$$
Jupiter has a somewhat higher surface gravity $g$ and somewhat lower temperature. But its $\mu$ is much lower, because its atmosphere is primarily $\text{H}_2$, while the Earth's atmosphere is primarily $\text{N}_2$, which has $14$ times the molar mass. That's why the scale height turns out to be larger.
A: The same Wikipedia article gives a formula for scale height (which is valid in approximation that temperature of the atmosphere does not change with height, which is not the best assumption but it gives reasonable predictions):
$$H = \frac{RT}{Mg}$$
Here $R$ is universal gas constant which is the same on each and every planet. $T$ is temperature measured in Kelvins, and it decreases for planets further away from the Sun. $g$ is free fall acceleration, which is larger on the big planets. So it would seem that large and far from the Sun planets should have smaller scale height, if we ignored $M$. However $M$ - molar mass of the gas - is the most important factor. Jupiter's and Saturn's atmospheres consist primarily of hydrogen with molar mass 2 g/mole, as opposed to Earth's average molar mass of the atmosphere ~ 29 g/mole. So this would explain why Jupiter and Saturn have actually larger scale heights, than Earth.
Because of similar atmosphere composition the difference between scale heights for Jupiter and Saturn could be mostly attributed to the difference of free fall accelerations: $2.5g_E$ on Jupiter against $1.1g_E$ on Saturn, where $g_E$ is Earth's free fall acceleration.
A: Although it is true that Jupiter is very massive, at 317.8 times the mass of Earth, it's density is much less, and as a result, it's surface gravity, with the surface defined as the cloud tops, is 24.79 m/s, or 2.528 g which although greater is not enormously greater than that at the surface of the Earth.
As for Saturn, for similar reasons,it has about the same surface gravity as Earth has: 10.44 m/s/s (or 1.065 g). Source:https://phys.org/news/2016-01-strong-gravity-planets.html
So the difference in surface gravity is not that important. What is important is the different composition of the atmosphere, which results in a much less dense atmosphere and hence a much greater scale height.
