A burning Moon does not have enough fuel to sustain life on Earth for more than a few years, and it would have to be so hot it would instantly blow itself apart. Here's the rough calculations.
This is a problem of energy density, and an important historical one.
Before nuclear fusion was discovered, the new science of geology was in conflict with the known physics and chemistry. Geology said the Earth had to be very old, hundreds of millions or billions of years old! But there was no known mechanism to have something like the Sun burning for that long. Even radioactive decay and nuclear fission were insufficient. Only nuclear fusion could provide energy for the 5 billion required years. If you want to read more about that, I'd suggest Bill Bryson's pop-sci book A Short History Of Nearly Everything.
The most energy dense, naturally occurring, chemical reaction is the oxidation of hydrocarbons: ie. burning methane, oil, fat, kerosene, etc... that's why we use them in cars, it's very energy dense and very safe. Burning hydrocarbons have an energy density of roughly 5e7 J/kg.
In contrast, uranium and thorium in a nuclear reaction have an energy density of 8e13 J/kg. The energy density of fusion is even higher, 6e14 J/kg.
So you can see, a burning Moon is roughly 10 million times less energy dense than a star. This will effect how long it can burn, and how much energy it can give off.
Next problem is one of surface area. While we could, theoretically, ensure our burning moon is fully oxidized and can burn all the way to the core, only the energy on the surface will radiate into space. The moon's surface area, and thus its radius, limits how much energy it can radiate.
And finally, its distance from the Earth is important. Since the burning moon radiates in all directions, only a tiny fraction of its energy will reach the Earth. The closer the Moon is to the Earth, the larger percentage of its energy will reach the Earth.
How Long Can A Burning Moon Heat The Earth?
Let's try this out for our Moon. Let's assume the Moon magically became a burning ball of hydrocarbons and oxygen. How much energy would reach the Earth, and how long would it last?
First, some important attributes of our Moon. I'll use approximate numbers to make the calculations simpler.
- Radius: 1700 km
- Surface area: 3.8e7 km^2
- Mass: 7e22 kg
- Distance: 3.8e5 km
We can preserve the Moon's mass, or the Moon's radius. I'll do mass, it's easier and since the Moon is about 5 times more dense than gasoline it provides more fuel giving this a better chance of working. 7e22 kg at 5e7 J/kg is 3.5e29 J of energy available. This assumes the entire mass of the Moon burns.
How much energy is that? Using the handy Orders of Magnitude (energy) list, we find that the Earth receives about 5e24 J of energy from the Sun per year. This is nearly 100,000 times that, so it's potentially enough energy to heat the Earth for 100,000 years. But not so fast, that's the total energy output by the Moon, but how much reaches the Earth?
To know that we need to know what percentage of the Moon's sky the Earth covers. We can figure that out by imagining a sphere around the Moon with a radius that's the distance to the Earth, that's all the Moon's energy radiating outward into space. The surface of that sphere is 4πr^2. r is the distance from the Earth to the Moon, 3.8e5km, giving us a surface area of 1.8e12 km^2.
The Earth can be thought of as a disk on the surface of that sphere. Its surface area is πr^2. r is 6.4e3 km giving us a surface of 1.3e8 km^2.
1.3e8 km^2 / 1.8e12 km^2 is 7e-5. The Earth receives only 1/14,000th of the burning Moon's radiated energy. This means of the 3.5e29 J available, only 2.5e25 J will reach the Earth ever.
The Earth needs 5e24 J per year to sustain its current environment. Only 2.5e25 J will reach the Earth. The burning Moon can only heat the Earth for 5 years and that's an upper bound.
Now that I've done the calculations once, you can change the parameters and do them again. Moving the Moon closer or making it larger will help.
How Hot Does The Surface Of The Moon Have To Be?
The next problem is just how hot the surface of the Moon would have to be. How much power is each square meter of the Moon radiating? Is it feasible? Since the Moon is so small, it might need to be absurdly hot.
Let's, again, assume everything is the same as now, and the Earth is receiving its 5e24 J per year. Power is normally measured in Watts which is J/s. 1 year has 3.15e7 seconds, so the Earth receives about 1.6e17 Watts from the Sun.
We calculated above that's just 1/14,000th of what the Moon is putting out, so the total power output of the burning Moon is about 2e21 Watts. That's about 1/1000th of what the Sun produces, or roughly the same as a very, very small red dwarf star.
The Moon has a surface area of 3.8e7 km^2 giving us a power per unit area of 5e13 W/km^2. Is that a lot? The Sun has a surface area of 6e12 km^2 and puts out 3.8e26 Watts giving a power per unit area of 6e13 W/km^2. Nearly the same!
Somehow the burning Moon has to put out the same energy per per unit area as the Sun using fuel that's 10 million times less energy dense. That's a problem, no fire is going to burn that intensely.
Worse, that level of energy output will produce a great force wanting to blow the Moon apart. Stars cope with this by being very, very massive; gravity balances this force wanting to blow it apart. Our burning Moon has nothing like the gravity of the Sun and will be instantly blown apart, the Earth will be showered with extremely hot fragments of burning hydrocarbons.
This part is particularly important because it means no amount of high-energy unobtanium will work. A burning Moon has to radiate too much energy for its gravity to hold itself together; it will blow itself apart.
Again, now that I've done the calculations, you can play with the parameters to try and make it work.