How are the other planets like - and unlike - our Earth? To answer that very fundamental question, at the dawn of the Space Age my friend and fellow USGS scientist Gene Shoemaker founded the Branch of Astrogeology in Flagstaff, Arizona. He and his many fellow scientists there have figured a LOT out about the other planets by using data and imagery provided by NASA.
I am told that the core of the earth is as big as the moon and as hot as the surface of the sun, and that the mantle is pretty darn hot too...
Why doesn't all of this heat transfer, move through, conduct up through the crust so that the surface of the earth would at least be very warm to the touch?
The Earth's core is not quite as large as the Moon - it's about 70% of the Moon's radius using evidence accumulated from the seismic tomography studies over the past half century or so. Think: earthquakes send sound waves downwards, and sophisticated calculations convert the refracted waves and their arrival times into an image of the Mantle and the Outer Core and the Inner Core. No one has ever measured the Core's temperature directly, of course, but laboratory high-pressure experiments, along with theoretical calculations, suggest that the temperature may be in the 5,400C/9,800F range - pretty close to the surface temperature of the Sun.
Actually the Core's heat DOES transfer outwards, and in some pretty spectacular ways: parts of the Earth's crust (Kamchatka, for instance) are moving as much as 8 cm/3 inches per year under the convective force of that heat trying to escape. Think: this crustal movement is analogous to the skin moving on the surface of a pot of cooking Cream of Wheat. This is the reason we are seeing those monstrous earthquakes off the coast of Chile, Japan, and in Haiti. The fact that heat escapes from the core is also manifested in the hundreds of volcanoes we see, for instance, all around the Pacific Ring of Fire. The continental crust rides up and over the down-going (denser) oceanic crust, which melts as it goes deeper and gets hotter, and the lighter water-and-gas-saturated components work their way upward through that continental crust to give us things like Mount St Helens.
Heat flow is actually a venerable (old and respected) field of geoscience. There are specialists who study heat flow all their professional lives - you have to put sensors deep in wells and block the fluids from convecting in order to get accurate numbers. There are places like Battle Mountain, Nevada, where the heat flow - the amount of heat escaping through a square meter of the surface - is several times higher than it is, on average, elsewhere on the Earth's crust. Another manifestation of that heat flow is the fact that no matter where you are on the Earth's surface, you can go down in a mine a few tens of meters/yards, and the temperature will almost always be about 55 degrees F (12 degrees C). It could be 122F/50C on the surface, and it will still be that cool at depth. It could be 'way below freezing on the surface in the Arctic, and it will STILL be 55F/12C at a drill-able depth. As you go much deeper, perhaps 4,000m/12,000 feet deep in some of the South African gold mines, the temperature gets hotter and hotter the deeper you go. A friend told me that in one South African gold mine, the temperature at the rock face at those depths can be 140F/60C. The deeper you go, the hotter it gets.
Why isn't the crust hot to the touch? For the same reason that a cinder-block wall is good insulation against the heat of the day-time sun. Rock is just not thermally conductive like a metal is - it's usually a pretty good insulator, in fact. For this reason, when the heat can't easily get out by conduction, it gets out by convection, but at a much slower rate. Think of that pot of Cream of Wheat again - that's convective heat transfer going on. On the global scale of the Earth's crust, this is the same thing as continental drift... which gives us huge subduction earthquakes and volcanoes.