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Others have covered the fact it's because of air pressure but haven't fully answered why that is the way it is.
It's simple really.
The force of gravity is also at play. As you go higher up, gravity gets weaker as you get farther from the earth's centre.
And it is that gravitational force that increases the air's density, same reason why if you keep going down in the water, the water gets denser.
For the heat to move around you need to be in a sort of goldilocks zone of density.
It needs to be dense enough that the fluid molecules can move around and spread the convection energy around... but not so dense they can't move much either.
Furthermore there's actually a couple different layers of our atmosphere.
First at our level is the troposphere, where heat is absorbed into the ground itself and radiated back out, as well as the perpetual heat from the earth's core, and reflected off the ground too (visible light).
The troposphere is warm and gets colder as you get farther away from the earth's surface, naturally. That heat is absorbed by the air itself so, as you get farther away it gets colder as it has more air to travel through.
Up higher is the Stratosphere, where it's ice cold and the air thins out.
However we get a sudden uptick in temp as we go even higher into what is called the Stratopause, back to briefly warm temperatures between the Stratosphere and the Mesosohere. Why? How?
Simple, this is the little sweet spot Ozone molecules hang out, forming a protective convenient bubble around the earth. Ozone absorbs Ultraviolet light from the sun and turns out that stuff is HOT, so there's a band of a hot zone right above and below the Ozone layer. Think of it as a toasty little bubble around us.
Above is the mesosphere which cools off again and gets back to being really frosty quickly, for the same reason the Stratosphere did, distance.
Then we hit the mesosphere, which is effectively the point when the atmosphere is so thin it stops protecting and is the "outside" of our protective blanket.
You can imagine this like earth being wrapped in a blanket, and the mesosphere is everything outside the blanket. Without any protection you are subject to the unbridled radiation of the sun which means you go back to being really toasty, as you get a bit higher you are effectively in space now and will soon enough hit temps that just cook you alive in a minute or two. Really bad sunburn zone.
So to answer the question overall:
Hot air rises... but only when there is air to rise.
Top of the mountains just don't have enough air anymore for it to really rise much more. It still does but the hot air rising effect just gets weaker and weaker as the air gets thinner due to less gravity.
The strength of the gravity field at the ISS orbit (400km) is still 90% of what it is at sea level. The air thins out at high altitudes because there's less air above it pushing down, not because of weaker gravity.
To add to this, the force of gravity at the top of Mt Everest is about 99.7% as strong as sea level. So you're right that it's not about the strength of gravity itself at that particular point, but about the weight of all the air above that point.
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It's mostly true, but the basic premise is not. Gravity is not significantly lower in the upper atmosphere.
Calvin's Dad: "Because the hot air rose all the way to space and isn't on the mountain anymore."
That must be why it's still warm in aeroplanes that fly higher than Everest.
Also why cold air balloons never took off
Oh, they did take off.... They just went "ppthhlppppsssssssspplpssseeeeeeeethstsplpssseeeeeeeesssssssss" and fell to the ground again.
I saw a great one-liner, and two megalogs, but no Goldilocks-sized answer, so here's my attempt.
As air rises, the weight of air above it (all the way to space) is less, so it's less squashed, letting it expand.
It expands by pushing out on all the air around it, and every time an air molecule bumps a neighbouring bit of air away, but isn't bumped back so hard (so it expands), it loses a bit of energy - i.e. heat.
So as some air goes up, it expands and loses heat; or as it sinks, it squashes and gets more heat.
This is adiabatic expansion.
Appendix:
This might beg the question of why higher air isn't just heated by neighbouring expanding air, making up for its original loss. I think that can be answered by saying overall the top air is squashing the bottom air, so overall the top is cooler. Is that fully right? Right now I feel there's multiple ways to think about it and I can't write any clearly without long rambling!
Fun fact: the temperature of space is actually thousands of degrees, but you would still freeze to death without protection.
(The actual answer is that atmospheric pressure is just as important as temperature in determining how "cold" something is)
The temperature of space actually is close to absolute zero, so quite cold. As the heat balance of an object there is mainly dominated by radiation, the object looses heat (~T⁴) but almost has no heat input from the surrounding if not directed to a star in sufficient proximity, e.g. the sun. The surface exposed to sunlight however, can become really hot.
It really depends on what you mean by temperature. You're both right, but both wrong depending on context.
Individual atoms and particles tend to have a lot of energy, but also there's almost no heat transfer into larger bodies because of the low density of those particles, so you lose more heat to radiation than you take in (unless you are in direct sunlight.)
Yes. One place in space has different temperatures. I would assume even individual particles are not distributed by a Maxwell distribution, so the concept of temperature is hard to apply. The background radiation has one temperature. If you add the sun, however, you already have a problem as the sun radiation is not in thermal equilibrium. So depending on how you look at it, you get different temperatures. The particles have a high energy, so also a high temperature. But they are so rare, that radiation is the dominant mode of heat transfer and determines the temperature of a thermometer placed in space.