About escape velocity and temperature

[kikinho]

Hi, I'm creating a fictional planet that have size a bit larger than Mars, gravity of 5m/s?? (between Mars and Earth gravity). The atmosphere gases are methane, carbon monoxyde and some sulfur. The air pressure is 13 atmospheres. The surface temperature is about 675K.
One more thing, the star in which the planet orbit is a Mdwarf star M0V.
What I want to know is...
Is this planet able to hold a dense atmosphere of the gases as I mentioned above, even having that gravity and being so hot?

How can I calculate this? If possible, can someone calculate this to me?

[ajtribick]

You haven't provided enough information to work out the escape velocity. Two planets can have the same surface gravity but different escape velocities.

Relationship between the two quantities is

v_escape = ???(2gr)

where g is the surface gravity and r is the radius of the planet.

You can work out the root mean square (r.m.s.) velocity of a gas particle with the formula (assuming it is an ideal gas!):

v_RMS = ???(3kT/m)

Where k is the Boltzmann constant, T is the temperature and m is the mass of the particle.

But you have to be careful! You can't just equate the two to work out how heavy the molecules have to be to stay in the atmosphere. For example, for protons (H atoms - bear in mind that molecules may get broken up in the upper atmosphere) at 300 K, the r.m.s. velocity is about 3 km/s, while the escape velocity of the Earth is 11 km/s. Yet the Earth has not retained its hydrogen!

What's going on?

One point to consider is that the relevant temperature is that of the exosphere, not the surface. The exosphere temperature depends on what kinds of atoms/molecules are around, how they absorb and radiate light, the spectrum of the star, etc and is not simple to calculate! For Earth, the exosphere is somewhere around 1300 K. However that would only put the gas velocity up to 6 km/s. Still not enough to lose the Earth's hydrogen!

The other problem is that there are always molecules going faster than the r.m.s. velocity. So what's important is the number of molecules in the tail of the distribution going faster than escape velocity.

The problem is further compounded by reactions going on in the atmosphere, sources/sinks at the planet's surface, etc.

So it isn't easy to solve!

[kikinho]

Ok, but what about my fictional planet? Can it have a dense atmosphere of methane having such gravity and the temperature I mentioned?
Venus, for example, have 90atm despite the lower gravity than Earth. But my planet have gravity between Mars and Earth and I want to know if its possible to keep 13atm air pressure of methane.
I'll talk more about this planet.
My fictional planet is a mars-sized carbon planet. This huge methane atmosphere is caused by the abundance of carbon in the stellar disk. The core of the planet is big relative to the size of the planet, giving it a high density that give this planet a gravity of 5m/s??. As with carbon, this planet is also rich in heavy metals and it will form 13 billion years in the future, after many generations of stars. As the Universe age, more carbon and metals are produced by stars. One day most planets will be made mostly of carbon and will be metal-rich.

[ajtribick]

Depends on the environment.

How old is the planet? What's the magnetic field like? How violent is the solar activity?

Scale height for methane at those temperatures is about 7 times that for nitrogen in Earth's atmosphere or the carbon dioxide at Venus. The planet's atmosphere would extend for quite a way above the surface, which is not good for retaining it.

If the r.m.s. velocity is at least an order of magnitude below escape velocity, you might be ok.

BTW I don't think that atmosphere is hot enough for gaseous sulphur, in any case it would probably be in the form of compounds, e.g. sulphur dioxide, hydrogen sulphide, etc.

[kikinho]

The planet age is 12 billion years (not in our current time, but in the far future). This planet orbit 2 very near stars, that are M0V with 50% the Sun diameter, but the stellar wind is still weaker than our Sun. The magnetic field is strong, since the core of this planet is big for the planet size and this planet is made of a mixture of carbides on the surface and mantle and part of the mantle and the core is rich in iron, nickel and other metals in smaller amounts.
Let's raise the temperature to 675K.
Clouds of sulfur can form in this planet as clouds of Water on Earth. Earth temperature colder than 373K and still have water vapour that condenses in clouds to form rain.
The same thing happen on this fictional planet, that is full of life that breathe methane instead of oxygen and is not based on DNA or RNA or proteins, but on fluorocarbons that use sulfur a lot in its molecules.

[kikinho]

Are the informations I posted enough to know if this planet can hold a dense methane atmosphere?