How can we tell what the earth core is made of?
This a great question, the answer is we basically use methods that are similar to medical imaging. Just like we can diagnose tissue make up and pathology using Computer Tomography (CT) and X-Ray we can make out the earth inner making using seismic waves. Notice the modality, in medical imaging we are using electromagnetic waves in geo-imaging we use seismic waves.
In CT the main physics is the attenuation of beam as it traverse your body in seismic waves you look at chance of the velocity of propagation of the seismic waves (a little like ultrasound, which also measure change in velocity across the tissue for imaging)
Today, by using seismological and magnetic field data as well as other theoretical calculations, it's possible to get a sense of the actual size and composition of our planet's inner core. Obviously there is no way to get a sample of the Earth's core, so we use methods very similar to medical imaging to study, map and determine the inner making of our Earth's core.
Just like an medical modality say X-ray used in CT scanning, seismic waves bounce around, changing direction and speed based on the material they pass through. Seismologist observe changes in speed of the waves how they moves from one tracking station to another, they can get a pretty good sense of what the ground that wave is traveling through looks like. In fact that is how geophysics explore the innards of our Earth when searching for hydrocarbons.
Using seismology, scientist discovered (or confirm theoretical postulates) that Earth is broken into three layers all with slightly different structures. The core's heat is mostly due to the slow decay of radioactive elements left over from when the planet first formed. The molten iron outer core lies about 3000 kilometers below our feet, while the solid iron inner core is another 2000 kilometers further down. A few other elements, including oxygen and silicon, are thrown in for good measure. But for the most part, iron rules the Earth's underbelly.