<p>The pressure of an isothermal atmosphere varies as
<span class="math-container">$$ P = P_0 \exp(- h/H)\ ,$$</span>
where <span class="math-container">$h$</span> is the height in the atmosphere and
the scale height <span class="math-container">$H$</span> is approximately
<span class="math-container">$$ H \simeq \frac{k_B T}{\mu g}\ .$$</span>
<span class="math-container">$T$</span> is the temperature of the atmosphere (in Kelvins), <span class="math-container">$\mu$</span> is the average mass of an atmospheric particle/molecule, <span class="math-container">$g$</span> is the surface gravity (in m/s<span class="math-container">$^2$</span>) and <span class="math-container">$k_B$</span> is the Boltzmann constant (<span class="math-container">$1.38\times 10^{-23}$</span> J/K).</p>
<p>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.</p>
<p>e.g. for Earth assume <span class="math-container">$T \sim 300$</span> K, <span class="math-container">$\mu \simeq 30\times 1.67\times10^{-27}$</span> kg (for a mix of O<span class="math-container">$_2$</span> and <span class="math-container">$N_2$</span>) and <span class="math-container">$g=9.8$</span> m/s<span class="math-container">$^2$</span>. This gives <span class="math-container">$H \simeq 8.4$</span> km - roughly the height of Mt Everest.</p>
<p>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.</p>

CC BY-SA 4.0
2022-02-07T22:39:16.120Z
2022-02-07T22:46:31.707Z
2022-02-07T22:46:31.707Z
ProfRob
<p>Here is my attempt to check my idea that the surface gravity of a planet might determine its scale height.</p>
<p>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..</p>
<p>So I take the surface gravities in meters per second per second (m/s2) from this list:</p>
<p><a href="https://en.wikipedia.org/wiki/List_of_Solar_System_objects_by_size" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/List_of_Solar_System_objects_by_size</a></p>
<p>And the scale heights from this list:</p>
<p><a href="https://en.wikipedia.org/wiki/Scale_height#Planetary_examples" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Scale_height#Planetary_examples</a></p>
<p>For comparison, Earth's surface gravity is 9.80 m/s2, and scale height is 8.5 kilometers (km).</p>
<p>Jupiter surface gravity 24.79 m/s2, 2.529 Earth's, and scale height is 27 km, 3.176 Earth's.</p>
<p>Neptune surface gravity 11.15 m/s2, 1.137 Earth's, &amp; scale height 19.1-20.3 km, 2.247-2.388 Earth's.</p>
<p>Saturn surface gravity 10.44 m/s2, 1.065 Earth's, &amp; scale height 59.5 km, 7 Earth's.</p>
<p>Earth surface gravity 9.80 m/s2, 1 Earth's, &amp; scale height 8.5 km, 1 Earth's.</p>
<p>Uranus surface gravity 8.87 m/s2, 0.905 Earth's, &amp; scale height 27.7 km, 3.258 Earth's.</p>
<p>Venus surface gravity 8.87 m/s2, 0.905 Earth's, &amp; scale height 18.9 km, 2.223 Earth's.</p>
<p>Mars surface gravity 3.71 m/s2, 0.378 Earth's, &amp; scale height 11.1 km, 1.132 Earth's.</p>
<p>Titan surface gravity 1.534 m/s2, 0.156 Earth's, &amp; scale height 21 km, 2.470 Earth's.</p>
<p>Pluto surface gravity 0.620 m/s2, 0.063 Earth's, &amp; scale height about 50 km, about 5.108 Earth's.</p>
<p>Triton surface gravity 0.782 m/s2, 0.078  Earth's, but its scale height is unknown to me.</p>
<p>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.</p>
<p>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.</p>
<p>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.</p>
<p>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.</p>
<p>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.</p>
<p>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.</p>

CC BY-SA 4.0
2022-02-07T21:29:25.277Z
Note that in the wikipedia link you gave in the question the scale height formula is given by $kT/mg$, dependent on gravity but also temperature and mean mass.
Anders Sandberg
CC BY-SA 4.0
2022-02-08T14:44:35.143Z
"Here is my attempt to check my idea that the surface gravity of a planet might determine its scale height." How is it not evident that this is wrong from just the definition of the scale height??? I don't get the entire question.
AtmosphericPrisonEscape
CC BY-SA 4.0
2022-02-07T20:23:30.290Z
2022-02-07T20:23:30.290Z
M. A. Golding
<p>What causes the relative scale heights of the atmospheres in the Solar System?</p>
<blockquote>
<p>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).</p>
<p>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.</p>
<p><sub><a href="https://en.wikipedia.org/wiki/Scale_height" rel="nofollow noreferrer">Scale height - Wikipedia</a></sub></p>
</blockquote>
<p>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.</p>
<p>Is that correct?</p>

<planetary-atmosphere>
What are the scale heights of the atmospheres of the Solar System worlds?
M. A. Golding
# What are the scale heights of the atmospheres of the Solar System worlds?

(Asked by: username_0 on 2022-02-07)

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](https://en.wikipedia.org/wiki/Scale_height)

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?

**Tags:** planetary-atmosphere


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## 2 answers

### Answer by username_1 on 2022-02-07. Score: 5

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.


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### Answer by username_0 on 2022-02-07. Score: 1

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.

> **Comments:**
> **username_2**: Note that in the wikipedia link you gave in the question the scale height formula is given by $kT/mg$, dependent on gravity but also temperature and mean mass.  
> **username_3**: "Here is my attempt to check my idea that the surface gravity of a planet might determine its scale height." How is it not evident that this is wrong from just the definition of the scale height??? I don't get the entire question.  

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