• S. Chandrasekhar White Dwarf pixel-art physics sticker, 3" × 3" / White

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Subrahmanyan Chandrasekhar Sticker

No Seconds. | Physics

\[\begin{aligned} M_{\text{Ch}} &= \frac{\omega_3^{0}\sqrt{3\pi}}{2}\left(\frac{\hbar c}{G}\right)^{3/2} \\ &\quad\times\frac{1}{(\mu_e m_H)^{2}} \\ &\approx \frac{5.73}{\mu_e^{2}}\,M_\odot \end{aligned}\]
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Dead stars have a weight limit. In 1931 Chandrasekhar derived a maximum mass for white dwarfs; for carbon–oxygen stars it is about 1.4 solar masses: the Chandrasekhar limit. A pixel-art portrait on a 3" × 3" sticker; the physics panel below has the equation, every symbol and the sources.

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3″ × 3″ kiss-cut sticker.

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The physics

A white dwarf is held up by electron degeneracy pressure. Chandrasekhar showed that once the electrons become relativistic, this pressure can only support a limited mass, the Chandrasekhar limit. Above that, collapse cannot be stopped.

The sticker says No Seconds. A white dwarf can't go back for more.

A white dwarf is the leftover core of a star like the Sun, held up not by heat but by electrons that refuse to share quantum states. In 1930, aged 19 and on the ship from India to England, Subrahmanyan Chandrasekhar asked what happens when those electrons are squeezed until they move close to the speed of light. His answer, published in 1931, was a ceiling: above a certain mass, no white dwarf can hold itself up.

In the idealised model the ceiling depends on how many nucleons there are per electron. For carbon and oxygen, two per electron, it comes to about 1.44 solar masses, the figure usually quoted and rounded to 1.4 on the scale here. His 1931 paper assumed 2.5 per electron and got 0.91. Real white dwarfs top out a little lower once effects the simple model leaves out are included.

He was not the first to find a limit. Wilhelm Anderson in 1929 and Edmund Stoner in 1930 had already combined relativity with electron degeneracy and found maximum masses for stars of uniform density. Chandrasekhar's version, for a star in proper equilibrium, became the standard one. In January 1935 Arthur Eddington publicly rejected the idea at the Royal Astronomical Society, saying there should be a law of Nature to stop a star behaving so absurdly.

A white dwarf gaining mass from a companion usually ignites just short of the limit and can explode as a Type Ia supernova. Whether most of those explosions come that way or from two white dwarfs merging is still an open question. Chandrasekhar shared the 1983 Nobel Prize in Physics with William Fowler.

No seconds.

Equations

\[\begin{aligned} M_{\text{Ch}} &= \frac{\omega_3^{0}\sqrt{3\pi}}{2}\left(\frac{\hbar c}{G}\right)^{3/2} \\ &\quad\times\frac{1}{(\mu_e m_H)^{2}} \\ &\approx \frac{5.73}{\mu_e^{2}}\,M_\odot \end{aligned}\]
\[\begin{gathered} \mu_e = 2\ (\text{carbon/oxygen}) \\ \Rightarrow\; M_{\text{Ch}} \approx 1.43\,M_\odot \\ (\text{usually quoted as } 1.44) \end{gathered}\]
\[\begin{gathered} \mu_e = 2.5\ (\text{assumed in 1931}) \\ \Rightarrow\; M_{\text{Ch}} \approx 0.91\,M_\odot \end{gathered}\]
SymbolMeaningUnit
\(M_{\text{Ch}}\) Chandrasekhar mass limit (idealised: cold, non-rotating, ideal electron gas) \(\mathrm{kg}\)
\(\omega_3^{0}\) constant from the Lane–Emden n = 3 solution (about 2.018) \(\mathrm{1}\)
\(\hbar\) reduced Planck constant \(\mathrm{J·s}\)
\(c\) speed of light \(\mathrm{m/s}\)
\(G\) gravitational constant \(\mathrm{m³·kg⁻¹·s⁻²}\)
\(\mu_e\) mean molecular weight per electron \(\mathrm{1}\)
\(m_H\) mass of a hydrogen atom \(\mathrm{kg}\)
\(M_\odot\) solar mass \(\mathrm{kg}\)

Sources

  1. Chandrasekhar (1931) The Maximum Mass of Ideal White Dwarfs, ApJ 74, 81
  2. Chandrasekhar, Nobel Lecture (1983) On Stars, Their Evolution and Their Stability (eq. 43)
  3. NobelPrize.org — The Nobel Prize in Physics 1983 (summary)
  4. Blackman — Anderson and Stoner published white dwarf mass limits before Chandrasekhar, arXiv:1103.1342
  5. Anderson (1929) Z. Phys. 56, 851
  6. Stoner (1930) The equilibrium of dense stars, Phil. Mag. 9, 944
  7. MacTutor — Chandrasekhar and Eddington
  8. Maoz, Mannucci & Nelemans (2014) Observational Clues to the Progenitors of Type Ia Supernovae, ARA&A 52, 107
  9. Wikipedia — Chandrasekhar limit

Source Chandrasekhar, S. (1931). The Maximum Mass of Ideal White Dwarfs. The Astrophysical Journal 74, 81.doi.org/10.1086/143324

Field notes

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