It seems obvious why mercury has no atmosphere, given its proximity to the sun–but yet Venus is also fairly close, and has an extremely dense atmosphere. Titan is a large moon with an atmosphere thicker than Earth's, and then Triton, a more distant large moon, has only a tenuous Nitrogen atmosphere. Why aren't more rocky bodies able to maintain an atmosphere? What factors affect whether a planet or moon is able to hold on to its atmosphere? What factors determine whether a planet or moon develops an atmosphere to begin with?
Planetary Science – Why Don’t More Rocky Planets/Moons Have Appreciable Atmospheres?
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Because orbits are general conic sections. Why this is true is another fascinating question in and of itself, but for now I'll just assume it. The point is that circular orbits are special examples of general orbits. It's perfectly possible to get a circular orbit, but the relationship between the bodies' velocities and separation needs to be exactly right. In practice it rarely is, unless we plan it that way (e.g, for satellites).
If you threw a planet around the sun really hard its path would be bent by the sun's gravity, but it would still eventually fly off at a tangent. Throwing it really hard would make it almost go straight, since it moves by the sun so quickly. As you reduce the speed, the sun gets to bend it more and more, and so the tangent is flies off on gets angled more and more towards moving backwards. So general hyperbolas are possible orbits. If you move it at the right speed, then it'll be just slow enough that other tangent points 'exactly backwards', and here the motion will be a parabola. Less than this and the planet will be captured. It doesn't have enough energy at this point to escape at all.
A key realization here is that the path should change continuously with the initial speed. Imagine the whole path traced out by a planet with a high velocity. An almost-straight hyperbola, say. Now as you continuously lower the velocity, the hyperbola bends more and more (continuously) until it bends "all the way around" and becomes a parabola. After this point, you'll have captured orbits. But they have to be steady changes from the parabola. All captured orbits magically being circles (of what size anyway, since they have to start looking like parabolas at some point?) wouldn't make any sense. Instead you get ellipses that get shorter and shorter as you get slower. Keep doing this, and those ellipses will come to a circle at some critical speed.
So circular orbits are possible, they're just not general. In fact, I'd say the real question is why the orbits are often so close to circular, since there are so many other options!
Some of your requirements are conflicting. The surface pressure of any atmosphere is the weight of the column of air (whatever mixture of gasses your atmosphere is made up of) of unit area extending upwards to the end of the atmosphere.
The surface pressure is therefore a function of the gravity and how much total gas is in the atmosphere. For example, Earth and Venus have about the same gravity, but the surface pressure of Venus is about 90x more because Venus has about 90x more stuff in its atmosphere.
You can generally assume that any planetary atmospheric layer will be thin enough that gravity is constant over that layer to a reasonable approximation. Therefore, for any chosen gravity, how fast the pressure decreases only has to do with the density of the gas. The denser it is, the faster you have a smaller weight of air column above you as you go up in altitude. To have a planet with 1 g surface gravity, for example, that has a more rapid falloff in pressure than earth, you need its atmosphere to be made of a more dense gas. The denser the gas, the thinner the atmospheric layer for the same gravity and surface pressure.
This is why the larger planet in your description just can't be. You say it has less gravity than Earth, a thinner atmosphere (presumably you mean lower surface pressure), but yet you want the pressure to fall off more rapidly than on Earth. This simply can't happen. Picture one inch square on the ground and the colum of air above it to the end of the atmosphere. With less gravity it will be less squished, so the column will be higher. This in turn makes the pressure fall off more slowly with altitude. If you also say it is a nitrogen-oxygen mixture, then it's density is pretty much defined over a narrow range (N2 has a molecular weight of 28 and O2 of 32).
Best Answer
There are a lot of factors that go into whether or not a planet has an atmosphere.
First, the mass and size of the planet. Really what it comes down to is the escape velocity. The higher the escape velocity (ve), the easier it is for a planet (or moon) to retain any atmosphere it gets as the gases that make up the atmosphere have to be moving faster to escape. Now ve is proportional to the square root of mass divided by radius and mass is proportional to density times radius cubed. Since the density of the rocky bodies in the Solar System are basically the same (and the icey bodies only differ by a factor of 2-3) density is roughly constant and that means that ve is basically just proportional to the radius of the object. So larger objects have a higher escape velocity and are more likely to retain an atmosphere.
Next is temperature which is just a measure of the average energy of the particles. So the gas molecules that make up the atmopherers of hot planets have a higher average energy than the molecules in a planet that is cooler. Since the average energy translates into the speed of the gas particles, gasses on warmer planets are more likely to be close to or over the escape velocity of the planet and as such can fly off into space. So cooler planets have a higher chance of holding on to their atmospheres since the particles are moving slower.
Next we have atmopheric composition. At a given temperature more massive particles are going to be moving slower. Thus if you have Hydrogen gas, with a molecular weight of 2, compared to oxygen gas, molecular weight 32, the Hydrogen will be moving on average 16 times faster and is much more likely to escape. CO2, with a molecular weight of 44, will be moving even slower still. Thus for a given planet with a fixed mass and temperature, the lighter gasses will escape first.
Finally we have availability of materials. If there are no gasses there to start with and no way to acquire them, it doesn't matter how massive or how cold the planet is, there is nothing there to hold on to.