[Physics Class Notes] on Chandrasekhar Limit Pdf for Exam

There is always a line of separation when it comes to a bang and a whimper. In the case of stars, these lines are known as Chandrasekhar Limit. In other words, this is the difference between dying supernaturally and going out in a slow fading on the verge of extinction. Here, in the universe, this line gives rise to a different cosmos formation where stars sow the seeds of life.

Chandrasekhar Limit Definition

A white dwarf star with the utmost mass limit that remains stable is known as the Chandrasekhar limit. EC Stone and Willhelm mentioned the discoveries on how to improve the preciseness of computation in papers. They named it after an Indian astrophysicist Subrahmanyan Chandrasekhar.

History of Chandrasekhar Limit

A decade before Chandrasekhar started his journey to England, i.e., by 1920, the astronomers had realised that Sirius B, a white dwarf companion to the bright star Sirius, had a million times more density than the Sun. This density could only be acquired by an object if the atoms forming the star were so firmly compressed that they were no longer separate entities. The gravitational pressures would compress the atoms so much that the star would consist of positively charged ions surrounded by a sea of electrons.

Before discovering quantum mechanics, physics didn’t understand the force capable of supporting any star against such gravitational force. But a new way was suggested by quantum mechanics, for a star to hold against gravity. As per the quantum mechanics rule, no two electrons can be in the same state. 

Explanation

With the help of thermonuclear fusion, a star is characterised, hydrogen merges to helium, helium merges to carbon, and so on, forming more massive and heavier elements. Still, thermonuclear fusion cannot create an element heavier than iron. Copper, gold, silver, and trace elements are created only by a supernova explosion, which is important for the process of life.

Oxygen, carbon, and nitrogen, which are lighter elements are also essential to life, but these elements will remain locked forever up in stars until a supernova explosion occurs. Similar to the iron-on earth that is locked up in the core, being heavier hydrogen and helium, which comprise most of the initial mass of the stars, they deposit to form the central core of the star.

If stars are destined to become white dwarfs, as Eddington believed, the elements will remain confined to the glamorous interior at best to be provided in minute quantities to the universe as a whole via solar winds. Rocky planet is required to form life as we know, and there is no simple method in which a large quantity of rock can be made available in the universe unless the stars can deliver the material in wholesale quantities, but supernovae can provide that.

Therefore, the Chandrasekhar limit is not just the upper limit for the maximum mass for an ideal white dwarf, but also the threshold. A star can no longer hoard its precious cargo of heavy elements once it crosses the threshold. As an alternative, it delivers them to the universe at large in a supernova. This allows the possibility of the existence of life but marks its death.

Chandrasekhar Limit Derivation

The value for the calculation of the limit depends on the nuclear composition of the mass. For an ideal Fermi gas, Chandrasekhar limit has provided the following expression which is based on the equation of the state: Chandrasekhar limit equation given as: 

[M_{limit} = frac{omega_{3}^{0} sqrt{3 pi}}{2} (frac{hbar c}{G})^{frac{3}{2}} frac{1}{( mu_{0} m_{H})^{2}}]

Where:

• ħ is reduced Planck constant

• c is the speed of light 

• G is gravitational constant

• μe is the average molecular weight per electron. This solely depends on the chemical composition of the star.

• mH is the hydrogen atom mass. 

• ω0

• 3 ≈ 2.018236 is a constant link with a solution to the Lane–Emden equation. 

As √ħc/G is Planck mass, the threshold is of the order of :

[frac{M_{Pl}^{3}}{m_{H}^{2}}]

This simple model requires adjustment for a variety of factors, including electrostatic interactions between electrons and nuclei and effects caused at nonzero temperature, for a more accurate value than a given range. Lieb and Yau give the thorough derivative of the limit from the relative multi-particle Schrödinger equation.

Fun Facts

In the beginning, the scientist community ignored this limit as it would mean legitimising the existence of a black hole. This was considered unrealistic at that time because the white dwarf stars oppose the gravitational collapse from the pressure of electron degeneration.

The Chandrasekhar limit is when the mass of the pressure from the degeneration of electrons is unable to balance the gravitational field’s self-attraction of 1.39 M☉limit.

The Chandrasekhar limit was found in 1930 by Subrahmanyan Chandrasekhar, an Indian astrophysicist and he used Albert Einstein’s special theory of relativity along with the principles of quantum physics to further prove his theory.