White-hot tungsten moon emitting heat and light to make my world habitable

Question

Never mind how the tungsten sphere formed. That's part of my pet projects, and I will deal with the question of how a nearly pure sphere of tungsten the diameter of the moon formed in space later.

So, if you read my question right, there is basically a tungsten moon orbiting my world at a distance of about 384,000km away. According to my calculations, a moon-sized sphere made of tungsten will be around 5.778 times as massive as our moon. So there might be powerful tides on this planet, but let's not consider that now. The host planet in question has the same size and mass as our Earth.

The host planet is basically a rogue planet that formed in a nebula, without a star. So I was looking for a heat source, and a tungsten moon seemed ideal.

So here's the real question. I want my tungsten moon to act like a "Sun", which means that it will radiate enough heat and light to keep my planet habitable. Meaning that the planet can host liquid water.

The tungsten moon is hot, like extremely hot. The surface temperature of the tungsten moon is about 3390°C. The crust of this moon is solid, but dangerously close to the verge of melting. This means that the tungsten moon is acting like a giant induction heater in the sky of my host planet, providing heat and light to my world.

However, I am at my wits' end trying to figure out a heat source hot enough to heat up the surface of my tungsten moon to the desired temperatures.

What is an ideal heat source to keep my tungsten moon hot enough to radiate enough heat and light for life to exist on my world?

Criteria:

• Needs to last for 3-5 billion years, not much.

• Sufficiently energetic enough to heat up the surface of my tungsten moon to 3390°C.

• Anything is allowed as long as it doesn't have any major side effects. For e.g. a core of uranium and other radioisotopes decaying and producing tremendous amounts of heat is okay, as long as it doesn't produce too much radiation that can damage life-forms on my host planet.

Answer 1

Barring any nuclear options or active heat sources, the answer is a hard NO.

I'll use a simple model for radiative cooling, brought to you by Hyperphysics, to gauge the timescale for the Moon-sized rock to cool below useful temperatures. (Here's a link to a Desmos calculator I made while preparing this answer.) The model is built from the theorem of equipartition of energy, which relates the temperature of a system to the average kinetic energy of its constituents.

$$KE=N\frac{3}{2}k_{B}T.$$

It is derived by dividing the equation of energy by its derivative w.r.t time and then integrating over temperature, solving for cooling time (simplified for a sphere):

$$t_{cooling}=\frac{Nk_{B}}{8\sigma \pi r^2}\left(\frac{1}{T_{final}^{3}}-\frac{1}{T_{initial}^{3}}\right).$$

• $T_{initial}$ and $T_{final}$ are the starting & ending temperatures of the sphere in Kelvin;
• $r$ is the radius of the sphere in meters; in this case, the lunar radius $\text{1,740,000 m}$
• $k_B$ is the Boltzmann constant, about $1.38\cdot10^{-23}$ $\text{J⋅K}^{-1}$;
• $\sigma$ is the Stefan-Boltzmann constant, about $5.67\cdot10^{-8}$ $\text{W⋅m}^{−2}\text{⋅K}^{−4}$;

$N$ is the number of particles in the sphere, and may be calculated by $N=\frac{mN_{A}}{M}$, where

• $N_A$ is the Avogadro constant, $6.022\cdot10^{23}$$\text{mol}^{-1}$;
• $M$ is the molar mass of the particles, in this case the molar mass of tungsten, $0.184$ $\text{kg mol}^{-1}$;
• and $m$ is the mass of the sphere in kilograms, and can be calculated by multiplying the density of tungsten by the volume of the Moon: $m=4.25\cdot10^{23}$ $\text{kg}$.

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The temperatures we're interested in are around 3,400 °C. Tungsten starts to vaporize at around 5,600 °C, so let's use that as our $T_{initial}$ and 3,000 °C as our $T_{final}$:

$$\text{105,000,000 seconds, or 3.3 years}$$

The model is probably accurate to within a few orders of magnitude, and so far it suggests the sphere will cool from its boiling point down to below the desired temperature in a millennium or less. (I don't know where you got the idea that 3-5 billion years is "not much".)

Let's say the tungsten moon forms a heavy tungsten atmosphere whose immense pressure raises the boiling point of tungsten, perhaps something like 16,000 °C at something like $10^9$ $\text{Pa}$:

$$\text{126,000,000 seconds, or 4.0 years}$$

Still within a thousand years.

We can't go much higher than this as the energy we've given this sphere already surpasses 1/10th the gravitational binding energy of the tungsten moon:

$$E=N\frac{3}{2}k_{B}T_{initial} = 4.70\cdot10^{29}\text{Joules}$$

$$E_{binding}=\frac{3Gm^{2}}{5r} = 4.15\cdot10^{30}\text{Joules}$$

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There's just no way to get what you're looking for without alternative energy sources. The rate of cooling follows an exponential decay. The Stefan-Boltzmann law states that the radiant power of an object is proportional to the fourth power of temperature. The tungsten moon will lose the vast majority of its internal heat initially, when it is hottest, and will cool more gradually onwards. Not even stellar remnants like neutron stars can do what you're asking.

Answer 2

Same reasons the earth's core is hot: 1) it got very hot a few billion years ago when it formed, and that heat has still not fully dissipated, 2) decay of naturally occurring radioactive isotopes (which I think is still under "Should not be something like nuclear fusion or antimatter").

PRIMORDIAL HEAT: Earth has 5-15TW of energy still flowing out of the core constantly caused by the intensity of the planetary accretion.

I have less confidence in this paragraph than the rest, others with more knowledge can chip in: I think that something made of tungsten coming together would have more primordial heat, because the overall density of the moon is very high and so the core was compressed more intensely with more heat when it formed.

RADIOGENIC HEAT: Earth's core has uranium-238 (238U), uranium-235 (235U), thorium-232 (232Th), and potassium-40 (40K), and these break down and create heat (that Icelanders then use to have hot showers).

You can, as a worldbuilder, put any amount of radioactive materials in the core of your "nearly pure sphere of tungsten", unless it's very important that it be so pure. Earth is 135 parts per million potassium (across all isotopes), 51.2 parts per billion Thorium, and 14.3 parts per billion uranium. (Morgan, J. W., & Anders, E. (1980). Chemical composition of Earth, Venus, and Mercury. Proceedings of the National Academy of Sciences, 77(12), 6973–6977. doi:10.1073/pnas.77.12.6973). So you're not really compromising on purity here hardly at all.

The core could be radioactive, the surface could be tungsten. Don't worry about harmful rays coming off the moon; gamma rays don't penetrate lead, let along kilometres of tungsten.

Answer 3

I think antimatter is the way to go tbh. There's no obvious way a tungsten moon could appear so you're not losing any scientific plausibility to make it an anti tungsten moon instead. Every asteroid and spec of dust annihilates on contact providing energy that will last as long the mass of the moon does. Easily billions of years if it's big enough to start with.

Otherwise uranium moon, but that would be massively over critical mass and blast itself and your planet into atoms. The bigger the mass the more explosive force so there's no way to overcome that, unlike with antimatter where you can have as much inert mass as you want as long as the regular matter colliding with it doesn't get too much at once.