Question

Describe the mass, size, and density of a typical white dwarf. How does the size of a white dwarf depend on its mass?

Short answer

White dwarfs are very dense, with mass similar to the Sun and size like Earth. They shrink in size as their mass increases.

Step-by-step solution

Understanding White Dwarfs

A white dwarf is a stellar remnant formed when a low to intermediate-mass star has exhausted its nuclear fuel and shed its outer layers. These stars are very dense and do not have any nuclear fusion occurring in their core, so they are no longer emitting energy and are slowly cooling and fading away over time.

Determining the Mass

A typical white dwarf has a mass approximately similar to that of the Sun, which is about 0.5 to 1.4 solar masses ( M_ ext{solar} ). The Chandrasekhar limit, which is approximately 1.4 solar masses, is the maximum mass a stable white dwarf can have before it collapses into another form of compact object, such as a neutron star.

Evaluating the Size

Despite having a mass close to that of the Sun, a white dwarf is much smaller in size. Typically, the radius of a white dwarf is about the same as Earth's radius, approximately 10,000 kilometers. This small size is due to the effects of electron degeneracy pressure, which supports the star against further gravitational collapse.

Analyzing Density

Given the mass and the small size of a white dwarf, its density is incredibly high. Densities of white dwarfs range from 10,000 kg/m³ to 10,000,000 kg/m³ or more, making them one of the densest forms of matter kept from further collapse by electron degeneracy pressure.

Exploring the Mass-Size Relationship

The size of a white dwarf inversely depends on its mass. As the mass of a white dwarf increases, its size decreases. This inverse relationship is due to the fact that additional mass compresses the star further because of stronger gravitational forces, and degeneracy pressure counters the gravitational forces, making the white dwarf smaller as it gets more massive.

Key concepts

Stellar Remnants

When a star reaches the end of its life and exhausts its nuclear fuel, it sometimes transforms into what we call a stellar remnant. One of the most fascinating types of stellar remnants is the white dwarf. These remnants are the leftover cores of stars that were once sun-like. When the outer layers of the star are shed, the core, having no more nuclear reactions to support it, becomes a dense and compact object. White dwarfs are incredibly dense because they contain mass comparable to our Sun but packed into a volume similar to that of Earth. They are fascinating as they give us a glimpse into the future of our own Sun, which will eventually become a white dwarf tens of billions of years from now.

Electron Degeneracy Pressure

An essential concept to understand white dwarfs is electron degeneracy pressure. This is a quantum mechanical effect that occurs when electrons are packed together very closely in a small space. According to the Pauli exclusion principle, no two electrons can occupy the same quantum state simultaneously. Therefore, when a white dwarf's core becomes ultra-dense, electrons are forced into higher energy states, providing a type of pressure that counteracts gravitational collapse. This pressure is what supports a white dwarf against its self-gravity despite having no ongoing nuclear fusion. Unlike normal gas pressure, electron degeneracy pressure does not depend on temperature, which is why white dwarfs can continue to exist long after they have cooled.

Chandrasekhar Limit

The Chandrasekhar Limit is a fundamental concept when discussing white dwarfs. It refers to the maximum mass that a white dwarf can have, about 1.4 times the mass of the Sun. If a white dwarf exceeds this mass, electron degeneracy pressure can no longer hold it up against gravitational collapse, and it will become unstable. Once this limit is surpassed, the white dwarf will likely collapse into a more compact object, such as a neutron star or possibly initiate a type Ia supernova if it is accreting mass from a companion star. This theoretical limit is named after the astrophysicist Subrahmanyan Chandrasekhar, who, through his pioneering work, helped define the boundary conditions of these fascinating celestial objects.

Mass-Size Relationship in Stars

The relationship between mass and size in stars, specifically in white dwarfs, is quite intriguing and counterintuitive. As the mass of a white dwarf increases, its size actually decreases. This is due to the balance between gravity and electron degeneracy pressure. With more mass, there is a stronger gravitational pull drawing the star's material inward, but degeneracy pressure resists this pull. Thus, more massive white dwarfs are smaller in size compared to less massive ones, leading to a fascinating inverse relationship: larger mass results in a smaller volume. This paradoxical mass-size relationship helps us understand the incredible densities found in white dwarfs and deepens our appreciation for these remnants of stellar evolution.