# If the moon still had a magnetic field how long would have been possible to keep an atmosphere?

Let's assume the Moon had a considerable magnetic field to prevent the atmosphere dispersion caused by solar wind. Given the low gravity of the Moon, how many years would pass before all the gases (of an Earth like atmosphere) would escape its body? Would its gravity be sufficient to trap heavier gases like CO2?

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I'll give a slightly different answer: The escape velocity at the surface of our Moon is about 2.38 km/s. Derived from this paper, as a rule of thumb, an atmosphere can survive 4.5 billion years, if it's average molecule velocity is below 1/6 of the escape velocity of the planet/moon. Applied to our Moon, it's 2380 m/s / 6 = 396.67 m/s.

Carbon dioxide has a molecule mass of 44u. The according temperature for an average molecule velocity for carbon dioxide is hence $$\mbox{temperature}=(v_{\mbox{gas}}/157)^2 \cdot \mbox{molecule mass}\mbox=(396.67/157)^2\cdot 44\mbox{ K}=280.87\mbox{ K}.$$ Moon's mean surface temperature at the equator is about 220 K. That's well below the allowed 280.87 K. A carbon dioxide atmosphere could have survived 4.5 billion years according to these oversimplified assumptions.

When looking closer to the atmosphere, it turns out, that the higher layers of the thermosphere and the exosphere can reach more than 1000 K. Even carbon dioxide can escape from Moon at this temperature over time: The average molecule velocity for carbon dioxide at 1000 K is $$157\cdot \sqrt{\frac{1000}{44}}\mbox{ m/s}= 748.5\mbox{ m/s}.$$

This results in a Jeans escape parameter of $$\lambda_0=\left(\frac{v_{\mbox{esc}}}{v_{\mbox{gas}}}\right)^2 =\left(\frac{2380}{748.5}\right)^2= 10.1,$$ resulting in an escape rate of about $10^{-4}$ relative to a free molecular flow.

From this it isn't yet straightforward to calculate the period of time, Moon would loose a carbon dioxide atmosphere, since it's a function of the height-dependent temperature curve of the atmosphere. But it's at least clear, that Moon wouldn't loose a carbon dioxide atmosphere suddenly, since this is likely just for $\lambda_0 < 3$.

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Your question could be interpreted in two ways: Could a strong magnetic field be enough to hold an atmosphere in place, or could a magnetic field provide enough shielding for an atmosphere to be kept.

Firstly, a magnetic field could never be enough to keep an atmosphere on a body.

Secondly, a magnetic field would stop an atmosphere from being heavily irradiated and blown away, but the Moon's gravity is just too low.

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sorry about the confusion, I edited the question. I didn't mean to imply that a magnetic field can "hold in place" an atmosphere – symbiotech Apr 27 '14 at 10:41