Natural ways to acquire gravity and heat for a colony on earth's moon
Is a colony on earth's moon must be deeply underground to ensure more natural gravity (by getting closer to its core) as well as more natural heat?
By natural heat I mean "without the need of artificial heating".
I have splitted this question into two distinct question in Physics CD:
I encourage answerers to copy-paste and edit where needed, their answers, into these two posts.
You seem to assume that the Moon's mass is mostly concentrated in its core. While I didn't find detailed data on the Moon's density distribution, this Wikipedia page tells that the density of the outer core is about $5\,\rm g/cm^3$ (unfortunately that value is not referenced), while the average density of the moon is about $3.3\cdot 10^3\,\rm kg/m^3$ (which is confirmed by NASA's Moon fact sheet). Given that $1\,\rm g/cm^3 = 10^3\,kg/m^3$ this means that the density of the moon does not vary too much depending on depth, thus an uniform density is a better approximation than all mass concentrated in the core.
Now with uniform density, gravitation is getting lower the deeper you go. This is easily understood from the fact that when you are at the surface, all the mass is pulling you down, while as soon as you go deeper, the mass above you is pulling you up.
Thus by going down on the moon, you don't get increasing, but decreasing gravity. Therefore gravity-wise the best you can do is stay on or near the surface.
Also on the Wikipedia page mentioned above, you find that the temperature in the core is about $1600\,K$ to $1700\,K$ (which is a bit more than $1/4$ of the Sun's surface temperature). Given that the surface temperature of the moon varies very much depending on whether you are on the Sun-facing side (NASA gives a range from $95\,\rm K$ to $390\,\rm K$), and the average surface temperature is definitely far below the freezing point of water, temperature-wise it may indeed make sense to go to a specific depth inside the moon (assuming you've got the necessary mining technology) in order to get a constant heat in the range suitable for human life.
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There seem to be two separate questions here:
Would a moon colony need to be deep underground to get more gravity?
No. This is basic physics.
Going deeper into the a gravitational body reduces gravity. Think of the limiting case where you are in a hollow chamber in the middle of the moon. Gravity would pull equally in all directions, cancelling out. You would be weightless there.
Would a moon colony need to be deep underground to get more natural heat?
Huh? What does "natural" heat mean? What would be unnatural heat?
The answer is probably "no". The main source of heat on the moon is sunlight. Going deeper would reduce fluctuations to get closer to the average, not not really much more heat. The moon is too small to retain much fossil heat. The small volume relative to the surface area also means that heat from radioisotope decay is much less significant than on Earth.
A lunar colony would likely use solar collectors as a significant source of power, including heat. The main problem with that is the long nights, and therefore need for a large amount of storage, or additional power generation, like nuclear reactors.
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