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

Understand the Objects

We have three objects: a white dwarf, a neutron star, and a black hole, each with different masses given in solar masses.

Recall Properties of White Dwarfs

A white dwarf has a similar radius to Earth, typically around 7,000 km, despite their high density and smaller masses relative to other stellar remnants.

Recall Properties of Neutron Stars

Neutron stars have a radius of about 10-15 km, which makes them much smaller than white dwarfs, although they are more massive and denser.

Recall Properties of Black Holes

A black hole's size is determined by its event horizon, which increases with mass. For the given mass of 3.0 solar masses, the radius would be roughly 9 km, smaller than both a white dwarf and a neutron star.

Compare Radii

Comparing the rough radii determined, white dwarfs (around 7,000 km) are much larger than neutron stars (10-15 km) and black holes (9 km).

Conclusion

Based on the comparison, the 1.2 solar mass white dwarf has the largest radius among the given objects.

Key concepts

White Dwarf Properties

White dwarfs are the fascinating remnants of stars like our Sun that have exhausted their nuclear fuel. Despite being devoid of any nuclear reactions, these stellar objects are characterized by their extreme density. Imagine compressing the mass of our Sun into a sphere with a size similar to Earth, typically about 7,000 km in radius.
Because of their high density, white dwarfs can't continue to contract indefinitely. They are supported against collapse by what's known as electron degeneracy pressure, a quantum mechanical principle that arises due to the rules governing subatomic particles called electrons. Essentially, the electrons are packed so tightly that they resist further compression, preventing the white dwarf from collapsing under its own gravity.
Astronomers can learn a lot from studying white dwarfs. Their composition, temperature, and luminosity offer insights into the life cycles of stars, cosmic distances, and the ultimate fate of stellar systems.

Neutron Star Characteristics

Neutron stars are born from the cataclysmic death of massive stars. When such a star exhausts its fuel, it undergoes a supernova explosion, leaving behind a core that collapses to form a neutron star. These remnants are incredibly dense, packing more mass than the Sun into a sphere with a radius of just about 10-15 km.
One of the most remarkable features of neutron stars is their intense magnetic fields which can be billions of times stronger than Earth's magnetic field. Moreover, they often rotate at phenomenal speeds, sometimes spinning hundreds of times per second due to the conservation of angular momentum during their collapse.
Neutron stars are composed almost entirely of neutrons, as protons and electrons have combined under the immense gravitational pressure. Observing neutron stars helps astronomers understand the extreme conditions in the universe, like high-density matter and strong magnetic fields.

Black Hole Event Horizon

The event horizon of a black hole is one of the most intriguing concepts in astrophysics. It represents the boundary beyond which nothing, not even light, can escape the gravitational pull of the black hole. This makes black holes invisible and detectable only through their gravitational effects on nearby matter.
The radius of the event horizon, also known as the Schwarzschild radius, is proportional to the mass of the black hole. For a black hole with a mass of 3 solar masses, the event horizon would have a radius of around 9 km, making it extremely compact.
Despite their mysterious nature, black holes play a critical role in our understanding of gravity and general relativity. They challenge our comprehension of spacetime and serve as natural laboratories for testing the limits of physical laws.

Stellar Remnants Comparison

Comparing stellar remnants like white dwarfs, neutron stars, and black holes showcases the diverse ends that stars can meet. This comparison gives insight into cosmic evolution and stellar dynamics.

• **White Dwarfs**: With radii similar to Earth's but with much greater density, these remnants provide clues about electron degeneracy pressure.
• **Neutron Stars**: Featuring smaller radii of 10-15 km, yet more massive and denser than white dwarfs, they allow scientists to explore matter under extreme pressure.
• **Black Holes**: Defined by their event horizons, they are even more compact, with a 3 solar mass black hole having a horizon at just 9 km radius.

Each type of remnant offers different insights into stellar physics and gravitational dynamics. White dwarfs and neutron stars can be observed with telescopes, providing direct data about their properties. In contrast, black holes require us to infer their presence through their impact on surrounding matter and light. Understanding these remnants enhances our comprehension of the cosmos's lifecycle and the fundamental forces at play.