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Chandrasekhar Limit Tee: 1.4 M☉

\[M_{\mathrm{Ch}} = \frac{\omega_3^{0}\sqrt{3\pi}}{2}\left(\frac{\hbar c}{G}\right)^{3/2}\frac{1}{(\mu_e m_H)^{2}} \approx \frac{5.8\,M_\odot}{\mu_e^{2}}\]
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Meet The Limit Keeper, guardian of the heaviest white dwarf. A white dwarf is held up not by heat but by electrons that refuse to share quantum states. In 1931 Subrahmanyan Chandrasekhar showed that when those electrons move close to the speed of light, their pressure can no longer beat gravity above a certain mass: about 1.4 times the mass of the Sun for a carbon-oxygen star. The work was part of his 1983 Nobel Prize. The Limit Keeper on a pixel-art unisex tee; equations, symbols and sources in the physics panel below.

"One point four, and not a grain more."

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Details

Size & fitS–5XL

Unisex heavy cotton (Gildan 5000), classic fit.

SizeWidthLengthSleeve
S182815.1
M202916.5
L223018
XL243119.5
2XL263221
3XL283322.4
4XL303423.7
5XL323525

Ölçek için: M beden 20 in (50.80 cm) genişliğindedir, yaklaşık 3.143 × 1034 Planck uzunluğu (lP = 1.616255 × 10−35 m). Mezura daha hızlıdır.

Measurements in inches, ±1.5 in tolerance.

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The physics

In a white dwarf, electrons refuse to share quantum states and push back, holding the star up. But when the star is heavy enough the electrons move near light speed and that pressure can no longer win. Above about 1.4 times the Sun's mass, gravity always does.

The Limit Keeper knows exactly how much a dead star can carry, and never lets it carry more.

When a star like the Sun runs out of fuel, its core shrinks into a white dwarf: about a solar mass packed into something the size of Earth. No fusion holds it up. Instead the Pauli exclusion principle does: electrons cannot share a quantum state, so squeezing them forces them into ever higher momenta, and that creates degeneracy pressure.

In 1930, on a ship to England, the 19-year-old Subrahmanyan Chandrasekhar realised that in a heavy enough white dwarf those electrons move close to the speed of light. Relativistic electrons push back more weakly as the star is squeezed, and above a certain mass gravity always wins. His 1931 paper gave the formula on this page: M_Ch = (ω₃⁰√(3π)/2)(ħc/G)^{3/2}/(μ_e m_H)², where ħ is the reduced Planck constant, c the speed of light, G Newton's constant, m_H the hydrogen-atom mass, μ_e the number of nucleons per electron and ω₃⁰ ≈ 2.018 a number from the equation for the star's structure.

Worked number: with CODATA constants the formula gives M_Ch ≈ 5.7 M☉/μ_e², often rounded to 5.8 M☉/μ_e². Carbon and oxygen each have two nucleons per electron, so μ_e = 2 and M_Ch ≈ 5.7/4 ≈ 1.43–1.44 solar masses for the ideal model. Effects the ideal model leaves out, such as electron capture and Coulomb corrections, put real white dwarfs at about 1.4 M☉. In the ideal theory the star's radius shrinks to zero as its mass approaches the limit: the curve on the back of this shirt.

Above the limit, no cold white dwarf is stable. A white dwarf pushed over it collapses or explodes; the collapse of stellar cores past this point leads to neutron stars and black holes. Chandrasekhar shared the 1983 Nobel Prize in Physics for his studies of the structure and evolution of stars.

The Limit Keeper: "Electrons will hold up a star for you. Just not one gram past one point four."

Equations

\[M_{\mathrm{Ch}} = \frac{\omega_3^{0}\sqrt{3\pi}}{2}\left(\frac{\hbar c}{G}\right)^{3/2}\frac{1}{(\mu_e m_H)^{2}} \approx \frac{5.8\,M_\odot}{\mu_e^{2}}\]
\[\mu_e = 2 \;\Rightarrow\; M_{\mathrm{Ch}} \approx 1.44\,M_\odot\ \text{(ideal)},\ \approx 1.4\,M_\odot\ \text{(real white dwarfs)}\]
SymbolMeaningUnit
\(M_{\mathrm{Ch}}\) Chandrasekhar limit, maximum mass of a cold white dwarf \(\mathrm{kg}\)
\(\omega_3^{0}\) constant from the n = 3 Lane–Emden solution, ≈ 2.018 \(\mathrm{1}\)
\(\hbar\) reduced Planck constant \(\mathrm{J·s}\)
\(c\) speed of light \(\mathrm{m/s}\)
\(G\) Newtonian gravitational constant \(\mathrm{m³ kg⁻¹ s⁻²}\)
\(\mu_e\) mean molecular weight per electron (nucleons per electron; 2 for carbon and oxygen) \(\mathrm{1}\)
\(m_H\) mass of a hydrogen atom \(\mathrm{kg}\)
\(M_\odot\) solar mass, ≈ 1.989×10³⁰ kg \(\mathrm{kg}\)

Sources

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

Source Chandrasekhar, S. (1931). The Maximum Mass of Ideal White Dwarfs. The Astrophysical Journal 74, 81–82.doi.org/10.1086/143324

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