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Does the atmosphere depth (or how high the air molecules from the ground) of Earth or Mars differ gradually or can there be plumes of atmosphere that reaches into space? If I were able to travel a perfect circle around the equator would the atmosphere differ in elevation?

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    $\begingroup$ do you mean variance of atmospheric density with altitude? It's not clear what you are asking. $\endgroup$ – farrenthorpe Nov 20 '16 at 3:48
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    $\begingroup$ I think they are asking whether the height, and density gradient, of a planet's atmosphere is constant (ish) all over or whether the atmosphere may reach much higher in some places than others. I think the earth / Mars thing is a red herring. But I'm not sufficiently confident that this is the intent to edit the question. @muze, could you clarify please? $\endgroup$ – Semidiurnal Simon Nov 22 '16 at 8:26
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    $\begingroup$ @SimonW if i were able to travel a perfect circle around the equator would the atmosphere differ in elevation? $\endgroup$ – Muze Nov 22 '16 at 21:13
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    $\begingroup$ @Muze: You might this question & answer, posted on SE Space Exploration relevant to your question: Is there a calculation for determining the thickness of a planet's atmosphere? $\endgroup$ – Fred Nov 23 '16 at 1:45
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The atmosphere, as a whole, is approximately in hydrostatic equilibrium. This means that the gravity of the earth holds the atmosphere to the earth, preventing its escape, though few molecules may escape every so often.

Mathematically, this can be described by $$\frac{dP}{dr}=-\rho g$$ where P is the pressure, $\rho$ is the density, and $g$ is gravity. Using the Ideal Gas Law $$P=\rho R T$$, where $T$ is temperature and $R$ is the gas constant for air. Assuming that the temperature in the height of a column of the atmosphere is averaged ($\bar{T}$) an equation for the average height of the atmosphere can be found $$P(r,\phi,\lambda)=P_0(\phi,\lambda)exp(-\frac{(r-r_0)g}{R\bar{T}(\phi,\lambda)})=P_0(\phi,\lambda)exp(-\frac{gz}{R\bar{T}(\phi,\lambda)})$$ where $r_0$ is the radius of the earth,$P_0$ is the surface pressure, $r=z+r_0$, where $z$ is the height above the earth's surface, $\phi$ is the latitude, and $\lambda$ is the longitude.

To Summarize: As the average temperature of the atmosphere increases, the height of the atmosphere will generally increase. This means that the height of the atmosphere will generally be the lowest near the poles, but highest near the equator. There are certainly exceptions to this rule, but this generally works. If you were to go around the equator, it will likely not be a "perfect circle" since the average temperature would have to be exactly the same.

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The atmosphere has various layers. Going upwards they are - troposphere, stratosphere, mesosphere, thermosphere, exosphere. (There are "pause" layers between, like tropopause, stratopause etc.) The density gets less and less and the exact height to the top of the exosphere is difficult to determine because the gas is so thin. However, we normally talk about the height of the troposphere which varies from about 7 to 10 km at the poles to about 18 km above the Equator. The reason it is higher at the Equator has to do partly with centrifugal force being greater there. For example, if a jelly-like sphere is rotated it starts to bulge outwards in the central, faster moving region. Even the solid Earth itself bulges at the Equator due to centrifugal force. See this blog post for details.

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