Chandrasekhar limit

The Chandrasekhar limit (/ˌtʃændrəˈʃeɪkər/)^[1] is the maximum mass of a stable white dwarf star. The currently accepted value of the Chandrasekhar limit is about 1.4 M_☉ (2.765×10³⁰ kg).^[2][3][4] The limit was named after Subrahmanyan Chandrasekhar.

White dwarfs resist gravitational collapse primarily through electron degeneracy pressure, compared to main sequence stars, which resist collapse through thermal pressure. The Chandrasekhar limit is the mass above which electron degeneracy pressure in the star's core is insufficient to balance the star's own gravitational self-attraction.^[5]

Physics

Normal stars fuse gravitationally compressed hydrogen into helium generating vast amounts of heat. As the hydrogen is consumed, the stars' core compresses further allowing the helium and heavier nuclei to fuse, ultimately resulting in stable iron nuclei. The next step depends upon the mass of the star. Stars below the Chandrasekhar limit become stable white dwarf stars, remaining that way throughout the rest of the history of the universe absent external forces. Stars above the limit can become neutron stars or black holes.^[6]: 74

The Chandrasekhar limit is a consequence of competition between gravity and electron degeneracy pressure. Electron degeneracy pressure is a

As the mass of a model white dwarf increases, the typical energies to which degeneracy pressure forces the electrons are no longer negligible relative to their rest masses. The velocities of the electrons approach the speed of light, and special relativity must be taken into account. In the strongly relativistic limit, the equation of state takes the form P = K₂ρ^4/3. This yields a polytrope of index 3, which has a total mass, M_limit, depending only on K₂.^[8]

For a fully relativistic treatment, the equation of state used interpolates between the equations P = K₁ρ^5/3 for small ρ and P = K₂ρ^4/3 for large ρ. When this is done, the model radius still decreases with mass, but becomes zero at M_limit. This is the Chandrasekhar limit.^[9] The curves of radius against mass for the non-relativistic and relativistic models are shown in the graph. They are colored blue and green, respectively. μ_e has been set equal to 2. Radius is measured in standard solar radii^[10] or kilometers, and mass in standard solar masses.

Calculated values for the limit vary depending on the nuclear composition of the mass.^[11] Chandrasekhar^{[12]: eq. (36) [9]: eq. (58) [13]: eq. (43)} gives the following expression, based on the equation of state for an ideal Fermi gas:

$${\displaystyle M_{\text{limit}}={\frac {\omega _{3}^{0}{\sqrt {3\pi }}}{2}}\left({\frac {\hbar c}{G}}\right)^{\frac {3}{2}}{\frac {1}{(\mu _{\text{e}}m_{\text{H}})^{2}}}}$$

• ħ is the
  reduced Planck constant
• c is the speed of light
• G is the gravitational constant
• μ_e is the average
  molecular weight
  per electron, which depends upon the chemical composition of the star
• m_H is the mass of the hydrogen atom
• ω⁰
  ₃ ≈ 2.018236 is a constant connected with the solution to the Lane–Emden equation

As √ħc/G is the

$${\displaystyle {\frac {M_{\text{Pl}}^{3}}{m_{\text{H}}^{2}}}}$$

A more accurate value of the limit than that given by this simple model requires adjusting for various factors, including electrostatic interactions between the electrons and nuclei and effects caused by nonzero temperature.^[11] Lieb and Yau^[14] have given a rigorous derivation of the limit from a relativistic many-particle Schrödinger equation.

History

In 1926, the British

Chandrasekhar's work on the limit aroused controversy, owing to the opposition of the British

The star has to go on radiating and radiating and contracting and contracting until, I suppose, it gets down to a few km radius, when gravity becomes strong enough to hold in the radiation, and the star can at last find peace. ... I think there should be a law of Nature to prevent a star from behaving in this absurd way![28]

Eddington's proposed solution to the perceived problem was to modify relativistic mechanics so as to make the law P = K₁ρ^5/3 universally applicable, even for large ρ.

Chandra's discovery might well have transformed and accelerated developments in both physics and astrophysics in the 1930s. Instead, Eddington's heavy-handed intervention lent weighty support to the conservative community astrophysicists, who steadfastly refused even to consider the idea that stars might collapse to nothing. As a result, Chandra's work was almost forgotten.[30]^: 150

However, Chandrasekhar chose to move on, leaving the study of stellar structure to focus on stellar dynamics.^[26]: 51 In 1983 in recognition for his work, Chandrasekhar shared a Nobel prize "for his theoretical studies of the physical processes of importance to the structure and evolution of the stars" with William Alfred Fowler.^[37]

Applications

The core of a star is kept from collapsing by the heat generated by the fusion of nuclei of lighter elements into heavier ones. At various stages of stellar evolution, the nuclei required for this process are exhausted, and the core collapses, causing it to become denser and hotter. A critical situation arises when iron accumulates in the core, since iron nuclei are incapable of generating further energy through fusion. If the core becomes sufficiently dense, electron degeneracy pressure will play a significant part in stabilizing it against gravitational collapse.^[38]

If a main-sequence star is not too massive (less than approximately 8

A strong indication of the reliability of Chandrasekhar's formula is that the absolute magnitudes of supernovae of Type Ia are all approximately the same; at maximum luminosity, M_V is approximately −19.3, with a standard deviation of no more than 0.3.^{[45]: eq. (1)} A 1-sigma interval therefore represents a factor of less than 2 in luminosity. This seems to indicate that all type Ia supernovae convert approximately the same amount of mass to energy.

Super-Chandrasekhar mass supernovas

In April 2003, the

Since the observation of the Champagne Supernova in 2003, several more type Ia supernovae have been observed that are very bright, and thought to have originated from white dwarfs whose masses exceeded the Chandrasekhar limit. These include SN 2006gz, SN 2007if, and SN 2009dc.^[50] The super-Chandrasekhar mass white dwarfs that gave rise to these supernovae are believed to have had masses up to 2.4–2.8 solar masses.^[50] One way to potentially explain the problem of the Champagne Supernova was considering it the result of an aspherical explosion of a white dwarf. However, spectropolarimetric observations of SN 2009dc showed it had a polarization smaller than 0.3, making the large asphericity theory unlikely.^[50]

Tolman–Oppenheimer–Volkoff limit

Stars sufficiently massive to pass the Chandrasekhar limit provided by electron degeneracy pressure do not become white dwarf stars. Instead they explode as

• Bekenstein bound
• Chandrasekhar's white dwarf equation
• Schönberg–Chandrasekhar limit

References