Chandrasekhar Limit – Explanation, Limits, Units

Aug 13, 2022 | Turito Team

Chandrasekhar Limit

We all know that everything eventually dies, whether it’s a person, a plant, an object, or a star. With the aid of hydrogen, stars can maintain their distance. When their hydrogen runs out, even they fall apart. However, they don’t all decompose in the same way. Black holes and supernovae are the results of massive stars collapsing. 

Chandrasekhar was able to help make this discovery possible. Astronomers revere him for his findings and studies on the collision of stars and the subsequent collapse into white dwarfs.

This article will provide you with all information regarding the discovery by Chandrasekhar called the Chandrasekhar limit.

What is Chandrasekhar Limit?

Chandrasekhar Limit can be defined as the greatest amount of mass a white dwarf star can have. The maximum mass of a white dwarf star can only be achieved if it is steady.

As a result, many scientists were skeptical of this notion at first, fearing it would legitimize the existence of black holes. At the time, the Chandrasekhar Limit was set at 1.39 million megahertz.

E.C. Stoner and Wilhelm Anderson were the first to make the discoveries Chandrasekhar elaborated on.

History of Chandrasekhar Limit

Even before Chandrasekhar launched his journey to England in 1920, scientists had already established that Sirius B, a white dwarf companion to the brilliant star Sirius, has a million times more density than the sun. Only if the atoms that make up the star were so closely packed could an item acquire this density. Stars are made up of positively charged ions surrounded by an ocean of electrons due to the gravitational limitations put on the atoms.

Before the development of quantum mechanics, the field of physics lacked any understanding of the force required to keep even the most massive star in place in the face of such a gravitational pull. Quantum mechanics, on the other hand, has suggested an original method that a star could use to defy gravity. According to the principles of quantum mechanics, two electrons can’t exist simultaneously in the same state.

The scientific world initially overlooked this limit since it would legitimize the presence of a black hole. There were doubts about this since white dwarf stars, which resist gravitational collapse due to electron degeneration, were seen as implausible at the time.

At this point, the gravitational field’s self-attraction can’t be balanced by the mass of the pressure generated by electron degeneration.

To prove his hypothesis, Subrahmanyan Chandrasekhar, an Indian astronomer, employed Albert Einstein’s special theory of relativity and the laws of quantum physics.

Explanation

Astronomers have discovered that stars are formed when hydrogen and other lighter atoms are fused in the presence of thermonuclear radiation. Although thermonuclear fusion can produce iron, it cannot produce an element heavier than this. It is only through a supernova explosion that the metals that are essential to life can be generated.

In addition to carbon, nitrogen, and oxygen, which are all necessary for life, these elements are locked up in stars until a supernova explosion occurs and releases them. When a star is first created, the heavier elements hydrogen and helium, which account for most of the star’s mass, form its central core. This core is analogous to the iron core of the earth.

If, as Eddington postulated, stars are doomed to become white dwarfs, then the elements will be confined to the star’s interior, where they will be delivered to the rest of the cosmos in minute quantities via solar winds. A tremendous amount of rock is needed to support life as we know it, and there is no easy way to get it unless the stars can give it in bulk. However, supernovae can provide that.

Not only is the Chandrasekhar limit the maximum mass for an ideal white dwarf, but it is also the beginning point. After a star has passed a certain point, it can no longer safeguard the precious load of heavy elements it carries. They have the potential to be released into the universe in the case of a supernova. The fact that this allows for the possibility of existence also indicates that this life will eventually end.

Chandrasekhar Limit Applications

Cited below are the applications of the Chandrasekhar limit:

  1. The stars’ collision is prevented by the fusion of lighter nuclei elements with heavier nuclei elements. The core of a star heats up and condenses as its nuclei are exhausted.
  2. If the mass of a star went down to 8 or less, it would eventually fall below the lower limit of the Chandrasekhar Limit.
  3. Stars with a greater mass than others will transform into black holes due to the electrons’ degeneration, enabling the stars to resist collisions until the density increases.

Chandrasekhar Unit

The Chandrasekhar Limit estimates the maximum mass a white dwarf star can have. It’s the weight of 1.44 solar masses. When this limit is exceeded, a star will become a black hole or a neutron star.

Derivation

The mass’s nuclear composition dictates the value used in the limit calculation. According to Chandrasekhar’s limit, the equation of state for an ideal Fermi gas can be summarized as follows:

The equation: M l i m i t = ω 3 0 3 π 2 (ℏ c G) 3 2 1 (μ 0 m H) 2

The Chandrashekhar limit, or the maximum mass of the biggest white dwarfs, must now be calculated. In addition, the “self-pressure” on the white dwarf to gravity will be calculated for the Chandrashekhar limit derivation. The self-pressure should be compared to the pressure of degeneracy.

In addition, the law of gravitation established by Newton is

F=GMmR2F=GMmR2

There is a perfect equilibrium between degeneracy pressure and gravity at the Chandrashekhar limit.

Consider a white dwarf with a mass of MM and a radius of RR, which is a spherical white dwarf. Is the gravitational force exerted on itself? Since the dwarf’s composition and other specifics are unknown, an exact answer cannot be given. However, dimensional analysis can be used to measure the magnitude.

Both masses must be adjusted to MM to comply with Newton’s law. This is because the white dwarf is feeling and exerting its gravitational attraction. Furthermore, the radius RR is used as the only unit of length; thus, it is the only unit of measurement.

Fgrav∼GM2R2.Fgrav∼GM2R2.

The gravitational self-pressure is this force that is divided by the sphere’s surface area because the pressure is a force that takes place by area, or

Pgrav∼FgravA∼GM2R4.

Protons make up the bulk of a white dwarf’s mass, with M=NmpM=Nmp, and V=R3V=R3. This means that the degeneracy pressure can be written as

PF∼ℏc(NV)4/3∼ℏc⋅(M/mp)4/3R4.

Setting this to the gravitational self-pressure, Pgrav=PFPgrav=PF, and then figuring out the mass is particularly important.

FgravA∼GM2R4⟹M2−4/3=M2/3∼ℏc⋅(M/mp)4/3R4∼ℏcG1m4/3p.

FgravA∼GM2R4∼ℏc⋅(M/mp)4/3R4⟹M2−4/3=M2/3∼ℏcG1mp4/3.

Chandrasekhar’s famous bound can be obtained by raising both sides to the power of 3/2.

MC∼(cℏG)3/21m2p.

Conclusion

Now, what is Chandrasekhar’s limit? It is the greatest amount of mass a white dwarf star can have. The maximum mass of a white dwarf star can only be achieved if it is steady. It is further used to estimate a white dwarf star’s maximum mass. It’s the weight of 1.44 solar masses. When this limit is exceeded, a star will become a black hole or a neutron star.

The topic is very important for studying quantum mechanics and helps you further your knowledge of physics.

FAQs

1. What do you understand by the term Chandrashekhar limit?

A. The Chandrashekhar limit is the greatest mass that may be contained within a dwarf star. In addition to that, the nature of this dwarf star is consistent and white. The value of the Chandrashekhar limit considered acceptable in the present day is approximately 1.4 M☉, which translates to 2.765×1030 kg.

2. What would happen if a neutron star collided with a black hole?

A. If a neutron star collided with a black hole, the resulting gravitational waves would cause a ridge to form in the fabric of space and time. Gold and platinum, both nuclear elements, will produce radiation when exposed to electromagnetic waves. The term “electromagnetic waves” actually refers to a category that also includes “light waves” and “gravitational waves.”

3. What is one of the effects of the Chandrashekhar limit?

A. The heat produced as a result of the fusion of lighter nuclei into nuclei of heavier elements helps to prevent the collapse of the star’s core that is undergoing this process. Over time, condensation occurs in the core.


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