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The Limit Keeper

Chandrasekhar Limit Physics

\[M_{Ch} \approx \frac{5.8\,M_\odot}{\mu_e^{2}}\]
$21.99 USD
$21.99 USD
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Meet The Limit Keeper. In a white dwarf, electrons refuse to share states and push back. Above about 1.4 Suns, even that pressure loses to gravity. On the mug: M_Ch ≈ 5.8 M☉/μ_e² ≈ 1.4 M☉. Pixel-art full-wrap black ceramic mug in 11 or 15 oz; equation, symbols and sources in the physics panel below.

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Details

Size & material11oz · 15oz

Black glossy ceramic mug with a C-handle. Lead- and BPA-free.

  • 11oz: 0.33 l
  • 15oz: 0.44 l
CareDishwasher & microwave safe

Dishwasher safe (top rack) or hand wash. Microwave safe.

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Made to order: about 10 days of production and handling before it ships.

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The physics

Electron degeneracy pressure can hold up a white dwarf only up to M_Ch ≈ 5.8 M☉/μ_e². For a carbon–oxygen star (μ_e = 2) that is about 1.4 solar masses.

The Limit Keeper guards the heaviest a white dwarf can be.

A white dwarf is the hot core left when a Sun-like star dies. No fusion holds it up. Instead its electrons, packed so tightly that the Pauli exclusion principle forbids them to share states, push back with degeneracy pressure.

In 1931 Subrahmanyan Chandrasekhar showed that this support has a ceiling. As the mass grows the electrons are squeezed until they move near the speed of light, and their pressure can no longer keep up with gravity. In the ideal theory the radius shrinks to zero at the limiting mass M_Ch ≈ 5.8 M☉/μ_e², where μ_e is the number of nucleons per electron. He shared the 1983 Nobel Prize in Physics.

A worked number: a carbon–oxygen white dwarf has μ_e = 2, so M_Ch ≈ 5.8/4 ≈ 1.44 solar masses in the ideal theory. Realistic models give about 1.4 M☉.

“About 1.4 Suns,” says the Keeper. “One step further and even the electrons give way.”

Equations

\[M_{Ch} \approx \frac{5.8\,M_\odot}{\mu_e^{2}}\]
\[M_{Ch} = \frac{\omega_3^{0}\sqrt{3\pi}}{2}\,\left(\frac{\hbar c}{G}\right)^{3/2}\frac{1}{(\mu_e m_H)^{2}}\]
\[M_{Ch} \approx 1.4\,M_\odot\quad(\mu_e = 2)\]
SymbolMeaningUnit
\(M_{Ch}\) Chandrasekhar limiting mass \(\mathrm{kg}\)
\(M_\odot\) mass of the Sun \(\mathrm{kg}\)
\(\mu_e\) mean number of nucleons per electron (2 for carbon–oxygen) \(\mathrm{dimensionless}\)
\(\omega_3^{0}\) Lane–Emden constant for polytrope n = 3 (≈ 2.018) \(\mathrm{dimensionless}\)
\(\hbar\) reduced Planck constant \(\mathrm{J·s}\)
\(c\) speed of light in vacuum \(\mathrm{m/s}\)
\(G\) Newtonian constant of gravitation \(\mathrm{m³·kg⁻¹·s⁻²}\)
\(m_H\) mass of a hydrogen atom \(\mathrm{kg}\)

Sources

  1. Chandrasekhar (1931) The Maximum Mass of Ideal White Dwarfs, ApJ 74, 81
  2. NobelPrize.org — The Nobel Prize in Physics 1983 (Chandrasekhar, Fowler)
  3. NIST CODATA 2022 — Planck constant

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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