Math 2.0

Astronomy and astrophysics Stellar structure Fact

Tolman–Oppenheimer–Volkoff limit

Builds on Einstein field equationsHydrostatic equilibriumChandrasekhar limitSchwarzschild metric

Statement

Consider static, spherically symmetric stars made of a perfect fluid with a given cold equation of state P=P(ρ)P = P(\rho), in general relativity. Let m(r)m(r) be the gravitational mass inside the radius rr, GG the gravitational constant and cc the speed of light. By the Einstein field equations, such a star satisfies the Tolman–Oppenheimer–Volkoff equation dPdr=−Gr2(ρ+Pc2)(m+4πr3Pc2)(1−2Gmc2r)−1,dmdr=4πr2ρ.\frac{dP}{dr} = -\frac{G}{r^2} \left( \rho + \frac{P}{c^2} \right) \left( m + \frac{4 \pi r^3 P}{c^2} \right) \left( 1 - \frac{2 G m}{c^2 r} \right)^{-1}, \qquad \frac{dm}{dr} = 4 \pi r^2 \rho. For the equations of state of neutron-star matter, the mass of these stars, as a function of the central density, has a largest value MTOVM_{\mathrm{TOV}}. This maximum is the Tolman–Oppenheimer–Volkoff limit: no static, cold, non-rotating star with that equation of state has a larger mass.

Remarks

When P≪ρc2P \ll \rho c^2 and Gm≪c2rG m \ll c^2 r, the TOV equation reduces to Newtonian hydrostatic equilibrium. The three relativistic factors all strengthen gravity, so pressure helps less than in Newtonian theory, and the mass curve reaches a maximum. Configurations with central density beyond the maximum are unstable to radial perturbations. Oppenheimer and Volkoff found a maximum of about 0.70.7 solar masses for an ideal gas of free neutrons in 1939. With realistic nuclear forces the value is not known exactly. It is believed to lie roughly between 22 and 33 solar masses; it is at least about 22, since neutron stars of that mass are observed. The limit plays for neutron stars the role that the Chandrasekhar limit plays for white dwarfs.

Going deeper

For a student

The TOV equation is the general-relativistic form of hydrostatic equilibrium. For a given equation of state, one integrates it outward from a chosen central density until the pressure vanishes, which gives the mass and radius. Varying the central density traces the mass–radius curve. The maximum of this curve is MTOVM_{\mathrm{TOV}}, and its value tests models of matter at nuclear densities. Outside the star the metric is the Schwarzschild metric.

Reading path

To understand this page from the ground up, read these 92 pages in order. The foundations come first.

  1. Chandrasekhar's white dwarf equation concept
  2. Derivative concept
  3. Differential equation concept
  4. Einstein field equations concept
  5. Euclidean space object
  6. Euclidean vector concept
  7. Euler equations concept
  8. Function concept
  9. Interval concept
  10. Lane–Emden equation concept
  11. Limit point of a set definition
  12. Local maximum definition
  13. Ordinary differential equation concept
  14. Partition of an interval definition
  15. Poisson's equation concept
  16. Polytrope concept
  17. Ricci curvature concept
  18. Sequence definition
  19. Velocity concept
  20. Fréchet derivative definition
  21. Ordinary differential equation definition
  22. Cross product definition
  23. Function of several variables concept
  24. Vector-valued function concept
  25. Lipschitz condition definition
  26. Classical mechanics concept
  27. Limit of a function definition
  28. Local minimum definition
  29. Mesh of a partition definition
  30. Riemann sum definition
  31. Schwarzschild metric example
  32. Real sequence definition
  33. Subsequence definition
  34. Momentum concept
  35. Limit of a function of several variables concept
  36. Partial derivative concept
  37. Solution of an ordinary differential equation definition
  38. Inertial frame of reference concept
  39. Continuity at a point definition
  40. Derivative definition
  41. Riemann integrable function definition
  42. Bounded sequence definition
  43. Limit of a sequence definition
  44. Monotone sequence definition
  45. Continuity in R^n definition
  46. Differentiable function of several variables definition
  47. Initial value problem definition
  48. Partial derivative definition
  49. Newton's laws of motion statement
  50. Continuous function definition
  51. Antiderivative definition
  52. Riemann integral definition
  53. Fermat's theorem on stationary points statement
  54. Convergent sequence definition
  55. Monotone convergence theorem for sequences statement
  56. Every sequence has a monotone subsequence statement
  57. Differentiable on an open set definition
  58. Gradient definition
  59. Higher partial derivatives definition
  60. Picard–Lindelöf hypotheses definition
  61. Intermediate value theorem statement
  62. Definite integral definition
  63. Bolzano–Weierstrass theorem statement
  64. Iterated derivative definition
  65. Picard operator definition
  66. Additivity of the integral over intervals statement
  67. Boundedness theorem statement
  68. Monotonicity of the integral statement
  69. C^k function characterised by continuous derivatives statement
  70. Integral equation form of an initial value problem statement
  71. An iterate of the Picard operator is a contraction statement
  72. Extreme value theorem statement
  73. Vector field definition
  74. Picard–Lindelöf theorem statement
  75. Mean value theorem for integrals statement
  76. Rolle's theorem statement
  77. Conservative vector field definition
  78. Mean value theorem statement
  79. First fundamental theorem of calculus statement
  80. Classical mechanics definition
  81. Potential energy definition
  82. Momentum definition
  83. Second fundamental theorem of calculus statement
  84. Kinetic energy definition
  85. Angular momentum definition
  86. Fundamental theorem of calculus statement
  87. Central force definition
  88. Virial theorem statement
  89. Newton's law of universal gravitation statement
  90. Hydrostatic equilibrium statement
  91. Chandrasekhar limit statement
  92. Tolman–Oppenheimer–Volkoff limit statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Tolman%E2%80%93Oppenheimer%E2%80%93Volkoff_limit
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Changed3 November 2026

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