Why is the Earths core warm

Jim, There's no Law of Conservation of Power like there is for momentum, mass-energy and so on. The more insulated a body, by absorbing-emitting layers, the less power it radiates from the overall surface, the cooler the insulated body's radiant temperature.

This is how fridges work and so on. The relevant radiant temperature in the case of a fridge is room temperature, not freezer temperature.

In the case of the Earth, the relevant radiant temperatures would be mostly freezing cloud top temperatures, with warm to cold patches of ocean and land surface shining through absorbant (grey) atmosphere. A grey body comprising grey patches shining through grey atmosphere.

A Hot Water Service (HWS) would have been a better analog.

Just as with a HWS, surface insulation greatly reduces heat flow and produces a strong temperature gradient across itself, so a similarly greatly reduced heat flow and temperature gradient are produced across the crusts and atmospheres of rocky planets such as the Earth.

The same analogy can be extended to include the mantle, thus explaining, in a simple way, the topic of this thread: "Why is the Earth's core warm?"

And again I agree with your ideas about how insulation works but that does not alter the fact the Earth radiates as a blackbody at a temperature of ~295 kelvin into the atmosphere and from there into space. The model says the atmosphere radiates into space at a blackbody temperature ~255 kelvin. The difference is a few hundred watts per square meter. The problem I have with this is if the surface of Earth radiates into the atmosphere more than 450 watts per square meter and the atmosphere radiates ~250w/m^2 into space what happens to the extra 200w/m^2 difference. If the atmosphere absorbs 450w then does it not have to radiate that much to maintain some sort of balance? Either the Earth does not radiate as a blackbody at 295 kelvin or the atmosphere is somehow using the energy that is being pumped into it. Or do you have another point of view on this detail?

And again I agree with your ideas about how insulation works but that does not alter the fact the Earth radiates as a blackbody at a temperature of ~295 kelvin into the atmosphere and from there into space. The model says the atmosphere radiates into space at a blackbody temperature ~255 kelvin. The difference is a few hundred watts per square meter. The problem I have with this is if the surface of Earth radiates into the atmosphere more than 450 watts per square meter and the atmosphere radiates ~250w/m^2 into space what happens to the extra 200w/m^2 difference. If the atmosphere absorbs 450w then does it not have to radiate that much to maintain some sort of balance? Either the Earth does not radiate as a blackbody at 295 kelvin or the atmosphere is somehow using the energy that is being pumped into it. Or do you have another point of view on this detail?

Jim, "the extra 200w/m^2 difference" must be reflected back, down into the Earth, just as HWS insulation reflects heat back into the water in the HWS.