Chandrasekhar Limit

What is Chandrasekhar Limit?

There is always a line of separation when it comes to a bang and a whimper. In 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 in 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 preciseness of computation in papers. They named it after an Indian astrophysicist Subrahmanyan Chandrasekhar.


Fun Facts - In the beginning, the scientist community ignored this limit as it would mean legitimizing 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.


History of Chandrasekhar Limit

A decade before Chandrasekhar started his journey to England, i.e., by 1920, the astronomers had realized that Sirius B, a white dwarf companion to the bright star Sirius, had a million times more density that of 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 characterized, 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 alternate, 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 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.

FAQ (Frequently Asked Questions)

1. What are the Application of Chandrasekhar Limit?

Some of the application of Chandrasekhar limit is outlined here as follows:

  • Since the life of a star is characterized by thermonuclear fission, it plays an important role in studying stars.

  • The Chandrasekhar limit is said to be a threshold that makes life possible. In case of heavier elements than that of hydrogen and helium, lik nitrogen, oxygen, carbon; they would have trapped in stars if the supernovae explosion had not occured.

  • Dangerous circumstance occurs when iron amasses in the core, as obtaining energy through fusion is not possible in the case of iron ions. The mass will lower than the Chandrasekhar limit sooner or later if the star is less than eight solar masses.

  • Stars will get converted into a black hole which has more masses, as the pressure due to the electron degeneration will keep them from collapsing until the density is exceedingly high. When the protons capture electrons, neutrons are released, which take away the energy that was created due to the decreasing of potential energy, which would be around 1046 joules.

2. What is Chandrasekhar Unit?

In order to explain the maximum mass of a white dwarf star, Chandrasekhar unit is used. This is equivalent to 1.44 solar masses. A black hole or neutron star came into existence when the limit surpassed.

3. State the Chandrasekhar Limit for a Neutron Star?

3 Msun is the maximum Chandrasekhar limit for a neutron star.

4. State the Value of the Chandrasekhar Limit Value?

Chandrasekhar limit value- For a stable white dwarf star; it is the maximum mass. 1.4 M ☉ (2.765×1030 kg) is the currently accepted value for the Chandrasekhar limit.