Those experiments provide a stiff challenge, but our estimates for the melting temperature of iron at these conditions range from about 4,500 to 7,500 kelvins (about 7,600 to 13,000 degrees F). As the outer core is fluid and presumably convecting (and with an additional correction for the presence of impurities in the outer core), we can extrapolate this range of temperatures to a temperature at the base of Earth's mantle (the top of the outer core) of roughly 3,500 to 5,500 kelvins (5,800 to 9,400 degrees F) at the base of the earth's mantle.
The bottom line here is simply that a large part of the interior of the planet (the outer core) is composed of somewhat impure molten iron alloy. The melting temperature of iron under deep-earth conditions is high, thus providing prima facie evidence that the deep earth is quite hot.
Gregory Lyzenga is an associate professor of physics at Harvey Mudd College. He provided some additional details on estimating the temperature of the earth's core:
How do we know the temperature? The answer is that we really don't--at least not with great certainty or precision. The center of the earth lies 6,400 kilometers (4,000 miles) beneath our feet, but the deepest that it has ever been possible to drill to make direct measurements of temperature (or other physical quantities) is just about 10 kilometers (six miles).
Ironically, the core of the earth is by far less accessible more inaccessible to direct probing than would be the surface of Pluto. Not only do we not have the technology to "go to the core," but it is not at all clear how it will ever be possible to do so.
As a result, scientists must infer the temperature in the earth's deep interior indirectly. Observing the speed at which of passage of seismic waves pass through the earth allows geophysicists to determine the density and stiffness of rocks at depths inaccessible to direct examination. If it is possible to match up those properties with the properties of known substances at elevated temperatures and pressures, it is possible (in principle) to infer what the environmental conditions must be deep in the earth.
The problem with this is that the conditions are so extreme at the earth's center that it is very difficult to perform any kind of laboratory experiment that faithfully simulates conditions in the earth's core. Nevertheless, geophysicists are constantly trying these experiments and improving on them, so that their results can be extrapolated to the earth's center, where the pressure is more than three million times atmospheric pressure.
The bottom line of these efforts is that there is a rather wide range of current estimates of the earth's core temperature. The "popular" estimates range from about 4,000 kelvins up to over 7,000 kelvins (about 7,000 to 12,000 degrees F).
If we knew the melting temperature of iron very precisely at high pressure, we could pin down the temperature of the Earth's core more precisely, because it is largely made up of molten iron. But until our experiments at high temperature and pressure become more precise, uncertainty in this fundamental property of our planet will persist.