Question

Choose the best answer to each of the following. Explain your reasoning with one or more complete sentences. Which of these objects has the largest radius? (a) a \(1.2 M_{\text {Sun }}\) white dwarf (b) a \(1.5 M_{\text {Sun }}\) neutron star (c) a \(3.0 M_{\text {Sun }}\) black hole

Short answer

The 1.2 solar mass white dwarf has the largest radius.

Step-by-step solution

Understanding Size Comparison of Dense Objects

To determine which object has the largest radius, we have to understand how white dwarfs, neutron stars, and black holes differ in size based on their mass and formation. White dwarfs, neutron stars, and black holes are all remnants of massive stars, but they have very different densities and therefore different radii.

White Dwarf Radius

A white dwarf forms when a star with a mass similar to that of the Sun exhausts its nuclear fuel. It is very dense, but still has a relatively large radius compared to neutron stars and black holes. The radius of a white dwarf can be comparable to that of Earth, even for a mass as dense as 1.2 times the Sun's mass.

Neutron Star Radius

A neutron star forms when a massive star (typically between 1.4 and 3 solar masses) collapses. It is much denser than a white dwarf and thus has a smaller radius. A 1.5 solar mass neutron star can have a radius of about 10 kilometers, which is much smaller than a white dwarf.

Black Hole Radius

A black hole forms when a very massive star collapses. The term 'radius' for a black hole typically refers to the Schwarzschild radius, which is the radius of the event horizon. A 3.0 solar mass black hole has a very small Schwarzschild radius, smaller even than the neutron star's radius.

Comparing the Radii

Comparing the objects: (a) The white dwarf, (b) the neutron star, and (c) the black hole, the white dwarf will have the largest radius among the given options. This is because white dwarfs are less dense than neutron stars or black holes, allowing them to maintain a larger radius.

Key concepts

White Dwarf

White dwarfs are stellar remnants of stars that were once like our Sun. When such stars exhaust their nuclear fuel, they shed their outer layers and leave behind a core that becomes a white dwarf. These remnants are incredibly dense.
Despite their density, white dwarfs maintain a relatively large radius when compared to neutron stars and black holes. White dwarfs typically have a radius comparable to Earth's. For example, a white dwarf with a mass of 1.2 times the mass of the Sun, although incredibly dense, has a larger radius than the other stellar remnants.
White dwarfs do not undergo further fusion. Instead, they shine faintly as they slowly cool and fade over time, potentially becoming very cold and dim objects known as black dwarfs, a stage that the universe has not yet witnessed.

Neutron Star

Neutron stars are another type of stellar remnant, formed from the collapse of massive stars unable to become black holes. These stars typically have original masses between 8 and 40 times that of the Sun, but neutron stars themselves have masses between 1.4 and 3 solar masses. Despite their massive origin, neutron stars are incredibly compact.
The collapse causes protons and electrons to combine into neutrons, giving the star its name. A neutron star's radius is about 10 kilometers, which is extremely small compared to the Earth-sized radius of a white dwarf.

• Neutron stars are among the densest objects in the universe, second only to black holes.
• The intense gravitational force within a neutron star prevents further collapse and fusion reactions.

These stars can spin rapidly, emitting beams of radiation that, when observed from Earth, result in pulsating signals called pulsars.

Black Hole

Black holes are formed from the remnants of the most massive stars. Unlike white dwarfs and neutron stars, a black hole is not constrained to a solid surface; instead, it is defined by its event horizon. This is the boundary beyond which nothing, not even light, can escape its gravitational pull.
When a very massive star collapses, its core's gravity is overwhelming, leaving nothing but a singularity encased by an event horizon.

• The typical radius of this event horizon is known as the Schwarzschild radius.

For a black hole of 3.0 solar masses, the Schwarzschild radius is only a few kilometers—smaller than a neutron star's radius. This small size, despite its vast mass, makes black holes some of the most mysterious and intriguing objects in the universe.
Black holes can only be detected through their gravitational effects on nearby matter or by the radiation emitted from matter accelerating into them.