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White dwarfs are fascinating stellar remnants left behind when stars, like our sun, exhaust their nuclear fuel. At the end of its life, a star like the sun will shed its outer layers, leaving a hot core behind. This core becomes a white dwarf. What makes white dwarfs particularly interesting is their dense nature. Despite having a mass similar to that of the sun, they are drastically smaller in size.
This is because the mass of the sun compacts into a volume as small as about 15,000 kilometers in diameter. To provide some perspective, this is nearly comparable to Earth's diameter! Due to this extreme reduction in size while maintaining mass, white dwarfs have exceptionally high densities. Calculating the density involves dividing the sun's unchanged mass by its new volume as a white dwarf. The result is an astonishingly high density, which poses interesting questions about the physics and internal structure of these remnants.

Neutron stars are even denser than white dwarfs. They form from the remnants of massive stars after a supernova explosion. Despite having a mass comparable to the sun, a neutron star's diameter shrinks to just about 20 kilometers! Imagine all that mass packed into such a small space. This incredible density arises due to the immense gravitational forces at play during the star's collapse after the supernova.
Within a neutron star, gravity is so strong that it compresses atoms tightly. Electrons and protons combine to form neutrons, leaving a dense core made almost entirely of neutrons, hence the name. The typical density of a neutron star can be calculated using the same approach as for white dwarfs: divide the sun's mass by the neutron star's remarkably small volume. This calculation gives rise to densities that are the highest known in the universe, illustrating the extreme conditions these stars embody.

Density calculations are crucial for understanding the properties of stellar remnants like white dwarfs and neutron stars. The basic principle is to divide the mass of the object by its volume using the formula:
\[ \text{Density} = \frac{\text{Mass}}{\text{Volume}} \]
For spheres, which include stars and planets, the volume is determined via the formula:
\[ V = \frac{4}{3} \pi r^3 \]
where \( r \) is the radius. Each step in the density calculation process involves substituting the known mass and the calculated volume into the density formula. For example, to find the density of a white dwarf, one uses the sun's mass and its volume when it becomes a white dwarf.
The same procedure applies to neutron stars, where extremely small radii are used, leading to remarkably high densities. Understanding these calculations helps unlock the mysteries of stars and provides insights into their life cycles and remnants, turning abstract concepts into comprehensible numbers.