What are the scale heights of the atmospheres of the Solar System worlds?

Question

What causes the relative scale heights of the atmospheres in the Solar System?

In various scientific contexts, a scale height, usually denoted by the capital letter H, is a distance over which a quantity decreases by a factor of e (the base of natural logarithms, approximately 2.718).

For planetary atmospheres, scale height is the increase in altitude for which the atmospheric pressure decreases by a factor of e. The scale height remains constant for a particular temperature.

_{Scale height - Wikipedia}

And I imagine that the scale heights of planetary atmospheres might be determined by the surface gravities of those worlds, as the varying strength of surface gravity compresses the atmosphere more or less.

Is that correct?

Answer 1

The pressure of an isothermal atmosphere varies as $$ P = P_0 \exp(- h/H)\ ,$$ where $h$ is the height in the atmosphere and the scale height $H$ is approximately $$ H \simeq \frac{k_B T}{\mu g}\ .$$ $T$ is the temperature of the atmosphere (in Kelvins), $\mu$ is the average mass of an atmospheric particle/molecule, $g$ is the surface gravity (in m/s$^2$) and $k_B$ is the Boltzmann constant ($1.38\times 10^{-23}$ J/K).

Thus the scale height is proportional to the atmospheric temperature and inversely proportional to the surface gravity (which itself depends on the mass of the planet/moon divided by its radius squared) and the mass of a typical molecule in the atmosphere.

e.g. for Earth assume $T \sim 300$ K, $\mu \simeq 30\times 1.67\times10^{-27}$ kg (for a mix of O$_2$ and $N_2$) and $g=9.8$ m/s$^2$. This gives $H \simeq 8.4$ km - roughly the height of Mt Everest.

Apply to other situations as needed. Note that the scale height varies with height in the atmosphere, since both the temperature and molecular composition can change with height.

Answer 2

Here is my attempt to check my idea that the surface gravity of a planet might determine its scale height.

Other factors which might possibly determine the scale height of a planet are the density of the atmosphereas a whole, the different gases it is composed of, and the temperatures in the atmosphere, etc..

So I take the surface gravities in meters per second per second (m/s2) from this list:

https://en.wikipedia.org/wiki/List_of_Solar_System_objects_by_size

And the scale heights from this list:

https://en.wikipedia.org/wiki/Scale_height#Planetary_examples

For comparison, Earth's surface gravity is 9.80 m/s2, and scale height is 8.5 kilometers (km).

Jupiter surface gravity 24.79 m/s2, 2.529 Earth's, and scale height is 27 km, 3.176 Earth's.

Neptune surface gravity 11.15 m/s2, 1.137 Earth's, & scale height 19.1-20.3 km, 2.247-2.388 Earth's.

Saturn surface gravity 10.44 m/s2, 1.065 Earth's, & scale height 59.5 km, 7 Earth's.

Earth surface gravity 9.80 m/s2, 1 Earth's, & scale height 8.5 km, 1 Earth's.

Uranus surface gravity 8.87 m/s2, 0.905 Earth's, & scale height 27.7 km, 3.258 Earth's.

Venus surface gravity 8.87 m/s2, 0.905 Earth's, & scale height 18.9 km, 2.223 Earth's.

Mars surface gravity 3.71 m/s2, 0.378 Earth's, & scale height 11.1 km, 1.132 Earth's.

Titan surface gravity 1.534 m/s2, 0.156 Earth's, & scale height 21 km, 2.470 Earth's.

Pluto surface gravity 0.620 m/s2, 0.063 Earth's, & scale height about 50 km, about 5.108 Earth's.

Triton surface gravity 0.782 m/s2, 0.078 Earth's, but its scale height is unknown to me.

I see that theplanets can't all have the same relationship between surface gravity and scale height. Uranus and Venus have the same surface gravity but the scale height of Uranus's atmosphere is much higher than that of Venus.

But if the worlds are divided into four sets of similar worlds, the concept that scale height increases with decreasing surface gravity seems mostly correct.

It is correct with the two gas giants Jupiter and Saturn, and with the two ice giants Uranus and Neptune, and hte two small mixed icey and rocky worlds in the outer solar system, Titan and Pluto.

In the case of the Earth like planets, Mercury doesn't have a significant atmosphere, and Venus has a much highter scale height than is proprtional to its surface gravity, only slightly less than Earth's.. The scale heights of Earth and Mars fit the pattern of increasing scale height with decreasing surface gravity.

And it is possible that the scale heights of Earth and Mars fit on the same curve as the scale heights of Titan and Pluto.

So possibly the scale height depends mostly on the surface gravity within worlds of a specific type, though not between worlds of different types, with Venus being a notable exceptionw within its type.