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?
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.