How does the density of material differ from the inner core to the outer core?

Answer 1

When considering a relatively small object, like a sphere of molten iron the size of a Basketball, there is hardly a density gradient from center to edge. If the size of the object we are considering grows to the scale of our planet, or larger, gravity plays a strong role in the density variation from center to edge.

For instance, it is unknown for certain in what state of the core of Jupiter resides. Jupiter is mostly hydrogen and is the largest planet in our system. Even though it is a gas giant, it is speculated that the core is molten hydrogen, possibly solid at the very center. This is considered because of the tremendous weight present at the center of the huge ball of gas. The Sun takes it a step further. It, too, is mostly hydrogen, but it is such a tremendously large body that the core is under enough pressure to start and sustain nuclear fusion.

Two things play a role in determining the state of matter; temperature and pressure. Scale is what allows us to see density gradients within a volume of the same substance.