The evolution of stars, as discussed in Section 44–2, can lead to a white dwarf, a neutron star, or even a black hole, depending on the mass. (a) Referring to Sections 44–2 and 44–4, give the radius of (i) a white dwarf of 1 solar mass, (ii) a neutron star of 1.5 solar masses, and (iii) a black hole of 3 solar masses. (b) Express these three radii as ratios (r_i : r_ij : r_iii).

Verified step by step guidance

1. Step 1: Understand the context of the problem. The evolution of stars depends on their mass, and the final state (white dwarf, neutron star, or black hole) determines their radius. The radius of these objects can be estimated using specific physical principles and formulas.
2. Step 2: For a white dwarf of 1 solar mass, use the approximate radius formula for white dwarfs, which is on the order of Earth-like sizes. The radius can be estimated as approximately 10,000 km (or 10⁴ km).
3. Step 3: For a neutron star of 1.5 solar masses, use the typical radius range for neutron stars, which is about 10 km. This is derived from the balance of gravitational collapse and neutron degeneracy pressure.
4. Step 4: For a black hole of 3 solar masses, calculate the Schwarzschild radius using the formula: $r=\frac{2GM}{c^2}$, where G is the gravitational constant, M is the mass of the black hole, and c is the speed of light. Substitute M = 3 solar masses to find the radius.
5. Step 5: To express the radii as ratios (rᵢ : rᵢᵢ : rᵢᵢᵢ), divide each radius by the smallest radius (likely the neutron star's radius) to normalize the values. This will give the ratio of the radii for the white dwarf, neutron star, and black hole.

Key Concepts

Stellar Evolution

Stellar evolution refers to the life cycle of stars, which includes various stages such as main sequence, red giant, and the final remnants like white dwarfs, neutron stars, or black holes. The mass of a star plays a crucial role in determining its evolutionary path and ultimate fate, influencing the processes of nuclear fusion and gravitational collapse.

White Dwarfs and Neutron Stars

A white dwarf is the remnant of a star that has exhausted its nuclear fuel and shed its outer layers, typically having a mass similar to that of the Sun but a much smaller radius. In contrast, a neutron star is formed from the remnants of a supernova explosion, possessing a mass greater than the Sun but compressed into a radius of about 10 kilometers, resulting in extremely high densities due to the collapse of atomic structures.

Black Holes

A black hole is a region in space where the gravitational pull is so strong that nothing, not even light, can escape from it. Formed from the remnants of massive stars after a supernova, black holes are characterized by their event horizon, which defines the boundary beyond which escape is impossible. The mass of a black hole influences its size, with larger masses leading to larger event horizons.