Question

A \(2 \mathrm{M}_{\odot}\) core of a star contracts after using its nuclear fuels. Explain why we can be sure that the star will not become a white dwarf.

Short answer

The star's core exceeds the Chandrasekhar limit, thus it won't become a white dwarf.

Step-by-step solution

Understanding Stellar Mass and Evolution

A star's fate after exhausting its nuclear fuel largely depends on its mass. The Chandrasekhar limit states that if a star's mass is less than approximately 1.4 times the mass of the Sun ( frac{M_{ ext{Ch}}}{M_{ ext{sun}}} imes M_{ ext{sun}} ext{ or } 1.4 M_{ ext{sun}}), it can become a white dwarf supported by electron degeneracy pressure.

Analyzing the Given Core Mass

Given that the stellar core in question is 2 solar masses ( frac{2M_{ ext{core}}}{M_{ ext{sun}}} = 2 M_{ ext{sun}}), we determine that it exceeds the Chandrasekhar limit (1.4 M_{ ext{sun}}). Therefore, the core cannot be supported by electron degeneracy pressure alone.

Considering Alternative Stellar Outcomes

When a stellar core exceeds the Chandrasekhar limit, it will continue to collapse under its gravity, possibly forming a neutron star or a black hole, instead of a white dwarf. These are the alternative outcomes for stars with core masses above this critical threshold.

Key concepts

Chandrasekhar Limit

The Chandrasekhar limit is a fundamental concept in astrophysics, named after the Indian-American scientist Subrahmanyan Chandrasekhar. This limit defines the maximum mass that a white dwarf star can have while being stabilized by electron degeneracy pressure. The value of this limit is approximately 1.4 times the mass of the Sun, known as the solar mass or \(1.4 M_{\odot}\).

If a star's mass exceeds this limit after it has burned out its nuclear fuel, it will eventually collapse beyond the white dwarf stage. If we consider a star with a core mass greater than \(1.4 M_{\odot}\), such as \(2 M_{\odot}\), it cannot remain as a stable white dwarf. The inevitable gravitational collapse will proceed further, leading to the formation of either a neutron star or a black hole.

The Chandrasekhar limit is crucial because it helps astronomers predict the ultimate fate of stars after their life cycle ends.

Electron Degeneracy Pressure

Electron degeneracy pressure plays a vital role in supporting white dwarf stars. But what exactly is it? In short, it's a quantum mechanical pressure. When a star depletes its nuclear fuel, it starts to collapse under gravity. Here, the electrons are forced into increasingly smaller spaces, resulting in a state where their energy levels start to become quantized, as described by the Pauli exclusion principle.

The principle states that no two electrons can occupy the same quantum state simultaneously, meaning they create a pressure even without thermal motion. This electron degeneracy pressure balances the gravitational forces in stars with masses at or below the Chandrasekhar limit, allowing them to become white dwarfs.

However, if a star's mass exceeds the Chandrasekhar limit, gravitational forces overpower this pressure. As a result, the star will undergo further contraction, leading to forms where other processes and pressures, like neutron degeneracy pressure, come into play.

Neutron Star

Neutron stars are fascinating celestial objects and form one of the possible endpoints of stellar evolution. When a star with a core mass greater than the Chandrasekhar limit collapses, electron degeneracy pressure cannot halt the collapse. As these stars gravitationally implode, they may eventually become neutron stars.

During the collapse, protons and electrons combine to form neutrons. The result is a dense core composed primarily of neutrons, hence the name "neutron star." These stars have incredibly high densities—so much so that a sugar-cube-sized amount of neutron star material would weigh as much as a mountain!

Neutron stars exhibit unusual characteristics, such as intense magnetic fields and rapid rotation, sometimes resulting in observable phenomena like pulsars. Their study helps scientists learn about nuclear matter under extreme conditions, expanding our understanding of fundamental physics.