← Cosmology

# Stellar Evolution and Black Holes

Friday, April 22, 2022

## Stellar Collapse

As a star uses its fuel, it will begin to collapse due to the radiation pressure failing to oppose the strong gravitational pressure. Eventually, a white dwarf is reached which is stable due to the Pauli exclusion principle of electrons preventing further collapse.

### Neutron stars

If the star has a mass greater than 1.4 times that of the sun (called the Chandrasekhar limit), it will further collapse combining electrons and protons into neutrons:

$\text{e}^-+\text{p}\rightarrow\text{n}+\nu_\text{e}$

The Pauli exclusion principle applied to neutrons prevents even further collapse. A neutron star of mass 1.5 times that of the sun would have a radius of only $11~\text{km}$ and a density of $5\times 10^{17}~\text{kg}/\text{m}^3$.

In 1967, astronomers discovered a series of regular pulsations of $1.34~\text{s}$. Originally called LGM-1 (LGM standing for "little green men"), this was discovered to be caused by a pulsar.

Since rotating stars collapse into smaller neutron stars, neutron stars spin extremely fast (since angular momentum is conserved). The magnetic field produced by such an object is incredibly intense, trapping emitted charged particles and accelerating them to high speeds near the magnetic poles, letting off beams of radiation. This rotating star's beams sweep across the universe like a lighthouse, occasionally hitting Earth to be detected here.

Neutron stars are believed to be produced by supernova explosions of stars. First observed in 1054, the Crab Nebula was a supernova explosion visible during the daytime over many days. At the center of the Crab Nebula is a pulsar within a period of $0.033~\text{s}$.

## Black Holes

Unlike stars of a few stellar masses, more massive stars' gravitational attraction cannot overcome the Pauli principle, leading to all their material being collapsed to a single point in space.

Using general relativity, a spherically symmetric mass $M$ will curve spacetime by

$\left(ds\right)^2=c^2\left[1-\frac{2GM}{c^2r}\right]\left(dt\right)^2-\frac{\left(dr\right)^2}{\left[1-\frac{2GM}{c^2r}\right]}-r^2\left(d\theta\right)^2-r^2\sin^2{\theta}\left(d\phi\right)^2$

At a limit where the denominator of the $dr$ term becomes zero,

$r_S=\frac{2GM}{c^2}$

This length is called the Schwarzschild radius. For an observer crossing this radius towards an object $M$ would notice no difference crossing this radius. However, for an external observer, general relativity predicts a clock on the object would appear to run slower and slower until it freezes. The object completely stops when it reaches $r_S$, frozen forever at that location. As it falls deeper into the black hole, its light becomes red-shifted until eventually disappearing. The radius is called the event horizon since no external observer can see into it.

Once inside the event horizon, the falling object could not escape. The speed required to escape exceeds the speed of light, so nothing (not even light) can escape.

An object whose mass $M$ lies within its corresponding $r_S$ is called a black hole. It is predicted that black holes are the end product after stars reach the end of their lives. Black holes are present in binary systems with stars and at the center of galaxies like the Milky Way.

In 1974, Stephen Hawking showed black holes can emit particles. In quantum mechanics, particle-antiparticle pairs can spontaneously appear, so long as they borrow an energy $2mc^2$ and "repay" it (disappear) within $\Delta t\sim\hbar/2mc^2$. If a particle-antiparticle pair appears near the event horizon, the black hole can "repay" the energy, allowing the particles to exist. Usually the particles fall into the black hole (balancing the energy), but one particle might have enough energy to escape. It appears that the black hole is emitting particles, and the black hole loses some mass.