How does the radius of a white dwarf depend on its mass? How is this different from other types of stars?

In the depths of a white dwarf, a remarkable quantum mechanical phenomenon known as electron degeneracy pressure comes into play, preventing the star from collapsing under its own gravity. Imagine the electrons within the star as a sea of particles that are forced to occupy very close quarters. According to quantum mechanics, specifically the Pauli Exclusion Principle, two electrons cannot inhabit the same quantum state simultaneously.

This means that as a white dwarf's mass increases and gravity attempts to squeeze the star tighter, not all the electrons can be compressed into a smaller space – there's a limit to how close they can get. This resistance, stemming from their quantum mechanical properties, is what creates the electron degeneracy pressure. It is this pressure that influences the star's mass-radius relationship, contributing to the intriguing fact that more massive white dwarfs have smaller radii.

The Pauli Exclusion Principle is a rule arising from the realm of quantum physics, bearing the name of physicist Wolfgang Pauli. It states that no two fermions (particles like electrons, protons, and neutrons, which have a half-integer spin) can occupy the same quantum state within a quantum system simultaneously.

In the context of white dwarfs, this principle explains why electrons resist being squeezed into the same energy levels. When the white dwarf is compressed by gravity, each electron must find a unique quantum state, but as these states become filled, further compression requires electrons to occupy higher energy levels. Eventually, a point is reached where the energy required to compress the electrons further is greater than the force of gravity – this is the essence of electron degeneracy pressure.

Stellar evolution is the life cycle of a star, from its birth in a cloud of dust and gas to the final stages of its existence. This journey is marked by a balance of forces, nuclear reactions, and changes in composition and structure.

In the case of white dwarfs, their place in the stellar life cycle is at the end stage, after a star has burned through its nuclear fuel and shed its outer layers. Following the red giant phase, the core that's left behind becomes a white dwarf. White dwarfs are no longer undergoing nuclear fusion; instead, their glow is from the residual heat of past nuclear reactions. Unlike their progenitor stars, white dwarfs do not follow the typical mass-radius relationship, showcasing unique behavior due to electron degeneracy pressure.

Hydrostatic equilibrium is a state in which a star's inward gravitational pull is precisely balanced by the outward force of pressure within the star. This delicate balance determines the star's structure and whether it will expand, contract, or remain stable in size.

Most stars achieve hydrostatic equilibrium through thermal pressure, caused by the heat from nuclear fusion reactions in their cores. In contrast, white dwarfs are no longer powered by fusion but by electron degeneracy pressure. As such, even though the general principle of hydrostatic equilibrium still applies to white dwarfs, the nature of the pressure involved is fundamentally different, leading to the observed inverse relationship between mass and radius in these compact remnants of stellar evolution.