# How can you determine the initial volume of a planet's atmosphere?

Since the surface pressure of a planet is determined by the mass of the column of gasses above it one would surmise that to determine the pressure you must know the volume and mass of the atmosphere.

Knowing the composition and volume of the gasses in the atmosphere would allow you to calculate their mass but how do we determine what that volume is?

Assuming we are only concerned with initial conditions when the primary atmosphere (created through accretion) was formed would the volume be arbitrary?

I hope this isn't too broad of a question. I'm trying to pin down a possible initial state that would be created when a new planet is formed and the conditions stabilize.

• Defining the volume of an atmosphere is somewhat arbitrary, since it doesn't suddenly stop, but rather decreases exponentially. You could define it by the area of the planet, times the atmosphere's scale height, which is where the pressure has decreased by a factor of e ~ 2.72. The scale height is a function of temperature, gravity, and molecular mass. – pela Mar 27 '15 at 10:52
• Also if you want a initial atmosphere, its link with the protosolar nebula is continuous, maybe separated by an accretion shock onto the planet. There you could define the volume, but in later eras it doesn't make sense to talk about the volume. @td-lambda: Why are you interested in that? If you're really about initial conditions then why not rather define surface pressure and temperature? – AtmosphericPrisonEscape Mar 28 '15 at 21:54
• The planet temperature should be mainly related to the distance to its star but pressure would be dependent on the height (volume) of the atmosphere and its composition. I would rather arrive at the pressure than to arbitrarily assign it. – Marcin Mar 29 '15 at 6:18
• @pela So based on that to calculate avg surface pressure could i use the following? V = H x A (A=1m^2 this gives column of gas) then could i get the pressure from the ideal gass equation p = nRT / V? How could i get the n (number of moles)? ref: Ideal Gas – Marcin Apr 5 '15 at 9:58
• That would only give you an order-of-magnitude estimate, I think. You need to integrate the total column, not only to H. But you need some more information. Even if you neglect solar winds and magnetic fields, a given planet could have any amount of gas molecules up to its maximums given by its graviational field, I suppose. But this is really outside my field. – pela Apr 5 '15 at 20:06

## 1 Answer

In the Q on Earth Sciences SE What is the mass of our planet's atmosphere, hydrosphere, and cryosphere combined? this formula was derived: $$M_{\mathrm{atm}_\bigoplus} \approx \frac{4\pi R^2_{\scriptscriptstyle \bigoplus} P_{\mathrm{sea\,level}}}{g_{_\bigoplus}}$$ which can be rearranged to derive pressure from the total mass of the atmosphere. $$P_{\mathrm{sea\,level}} \approx \frac{g_{_\bigoplus}M_{\mathrm{atm}_\bigoplus}}{4\pi R^2_{\scriptscriptstyle \bigoplus}}$$