Astronomy

Chandrasekhar Limit

Chandrasekhar limit refers to the maximum mass of a dwarf star that is of a stable white nature. Furthermore, the value that is currently accepted by scientists of the Chandrasekhar limit is about 1.4 M or in other words 2.765×1030 kg. Moreover, the Chandrasekhar limit is related to solar masses.

Introduction to Chandrasekhar Limit                               

Chandrasekhar limit refers to the utmost mass that a stable white dwarf star can have. Furthermore, an expert like Willhelm Anderson and E.C. Stoner have termed it after the famous Indian astrophysicist, Subrahmanyan Chandrasekhar, who made major independent discoveries on enhancing the computation.

Initially, there was an avoidance of Chandrasekhar limit by the scientist community for fear of legitimizing the existence of black holes. This is because the black holes were considered unrealistic at that time.

chandrasekhar limit

Chandrasekhar Limit

How do we Measure Chandrasekhar Limit?

Stars produce energy via the process of nuclear fusion, thereby resulting in the production of heavier elements from lighter ones. Furthermore, the generation of heat from these reactions prevents a star’s gravitational collapse. Gradually, the star would build up a central core involving various elements.

For main-sequence stars that are characterized with a mass below approximately 8 solar masses, the core’s mass will be below the Chandrasekhar limit. Consequently, they will eventually lose mass until only the core. Most importantly, this core would become a white dwarf.

For stars with higher mass, there will be a development of a degenerate core. Furthermore, the mass of this degenerate core will keep on growing until it exceeds the limit. At this point, an explosion will take place which converts the star into a supernova, leaving behind a black hole or neutron star.

Electron degeneracy pressure is a quantum-mechanical effect which arises due to the Pauli Exclusion Principle. Furthermore, it is not possible for two electrons to be in the same state. Consequently, not all electrons would be present in the minimum-energy level, rather they must occupy a band consisting of energy levels.

The number of electrons in a given volume increases due to the compression of the electron gas.  This results in raising the occupied band’s maximum energy level.

As such, an increase in the energy of the electrons will take place upon compression, so there must be exertion of pressure on the electron gas to compress it. Most noteworthy, this is the origin of electron degeneracy pressure’s origin.

The measurement of the Chandrasekhar limit involves the radius-mass relations for a model white dwarf. The blue curve is for a non-relativistic ideal Fermi gas while the green curve makes use of the general pressure law for an ideal Fermi gas. Moreover, the black line marks the ultra-relativistic limit.

In the non-relativistic case, an equation of state of the form P=K1ρ5/3 would be due to the electron degeneracy pressure. Furthermore, solving the hydrostatic equation would lead an individual to a model white dwarf which is an index 3/2 polytrope. Consequently, this particular index has a radius that is inversely proportional to its mass’s cube root, and its volume is inversely proportional to its mass.

As the model white dwarf mass shows an increase, the typical energies are no longer negligible with relation to their rest masses. Moreover, these typical energies are those to which degeneracy pressure forces the electrons.

The velocities of the electrons would certainly approach the speed of light. This would require special relativity to be taken into consideration.

Furthermore, in the strongly relativistic limit, the equation of state would take the form P=K2ρ4/3. This will yield a polytrope of index 3, which will depend only on K2 and have a total mass, Mlimit.

Interpolation between the equations P=K1ρ5/3 for small ρ and P=K2ρ4/3 for large ρ will take place due to the equation of state, for a fully relativistic treatment. When this happens, the model radius would decrease with mass, and at Mlimit it would become zero. Most noteworthy, this is what is meant by the Chandrasekhar limit.

Formula of Chandrasekhar Limit

One may set the central pressure’s estimate with r = 0, with ρ = Mwd/ 4 3 πR3 wd, with Z/A = 0.5 in order to obtain the approximate value for the maximum white dwarf mass. Furthermore, MCh ∼ 3 √ 2π 8  ~c G 3 2 Z A  1 mH 2 = 0.44 M.

There are three fundamental constants ~, c and G; which represent the combined effect of Newtonian gravitation, quantum mechanics, and relativity on the structure of a white dwarf. Moreover, a precise derivation with Z/A = 0.5 would give us the result in a value of MCh = 1.44, known as the Chandrasekhar limit.

Derivation of the Formula of Chandrasekhar Limit

Now one must calculate the mass of the largest white dwarfs, known as the Chandrasekhar limit. Furthermore, the goal is to calculate the “self-pressure” on the white dwarf to gravity for the purpose of Chandrasekhar limit derivation. Moreover, one must compare this self-pressure to the degeneracy pressure.

Furthermore, Newton’s law of gravitation is

F=GMmR2F=GMmR2

where the separation of  M, mM, and m takes place by RR, and G=6.7 × 10 − 11 m3/kg s2G=6.7×10−11 m3/kg s2 is known by experts as Newton’s constant.

At the Chandrasekhar limit, degeneracy pressure and gravity happen to be in perfect balance.

Consider a spherical white dwarf that has mass MM and it has a radius RR. Is gravitational force exertion on itself? Obviously,the accurate answer will be dependent on the dwarf’s composition and other details. However, it is possible to make an estimate of order-of-magnitude from dimensional analysis.

In Newton’s law, the setting of both the masses should be equal to MM. This is because, the white dwarf is both experiencing the pull as well as doing the pulling. Similarly, the only length scale is the radius RR, which means

Fgrav∼GM2R2.Fgrav∼GM2R2.

Since pressure is a force who’s division takes place by area, the gravitational self-pressure is going to be this force whose division takes place sby the sphere’s surface area of the sphere, or

Pgrav∼FgravA∼GM2R4.Pgrav∼FgravA∼GM2R4.

Finally, calculation of the Chandrasekhar limit would become possible. A white dwarf’s mass is mostly protons, with M=NmpM=Nmp, while V∼R3V∼R3. Thus, it is possible to write the degeneracy pressure as

PF∼ℏc(NV)4/3∼ℏc⋅(M/mp)4/3R4.

Most noteworthy, setting this equal to the gravitational self-pressure, Pgrav=PFPgrav=PF, and finally solving for the mass:

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.

Raising both sides to the power 3/2, one can obtain Chandrasekhar’s famous bound:

MC∼(cℏG)3/21m2p.

FAQs on Chandrasekhar Limit

Question 1: What is meant by the Chandrasekhar limit?

Answer 1: Chandrasekhar limit refers to the maximum mass of a dwarf star. Moreover, this dwarf star is of a stable white nature. Furthermore, the value that is currently acceptable of the Chandrasekhar limit is about 1.4 M or in other words 2.765×1030 kg.

Question 2: Explain one effect of the Chandrasekhar limit?

Answer 2: When the nuclei of lighter elements fuse into the nuclei of a heavier one, the collapsing of the star’s core is prevented due to the resultant heat.  Gradually, the condensation of the core will happen.

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