Math 2.0

Astronomy and astrophysics Stellar structure Fact

Mass–luminosity relation

Builds on Stellar structureHydrostatic equilibriumVogt–Russell theorem

Statement

Let LL and MM be the luminosity and the mass of a main-sequence star, and L⊙L_\odot and M⊙M_\odot those of the Sun. Then approximately L/L⊙≈(M/M⊙)aL / L_\odot \approx (M / M_\odot)^a, where the exponent aa depends on the range of masses. For stars of roughly solar mass, aa is between 33 and 44, and a=3.5a = 3.5 is the value usually quoted.

Remarks

The relation is empirical, and it is explained by the equations of stellar structure through scaling arguments. For a star of radius RR in hydrostatic equilibrium made of ideal gas, the central temperature scales like T∝M/RT \propto M/R. If energy is carried by radiation through gas of constant opacity, the luminosity scales like L∝R4T4/(κM)∝M3L \propto R^4 T^4 / (\kappa M) \propto M^3, independent of RR. Varying opacity and the change from radiative to convective transport bend the power law, which is why the exponent changes along the main sequence. Example: a star of 1010 solar masses with a=3.5a = 3.5 has luminosity about 103.5≈300010^{3.5} \approx 3000 solar luminosities. The relation does not hold for giants or white dwarfs, whose luminosity is not fixed by their mass alone.

Going deeper

For a student

Since the fuel of a star is proportional to MM and the rate of burning is LL, the main-sequence lifetime scales like M/L∝M1−aM / L \propto M^{1 - a}, about M−2.5M^{-2.5}. The relation is a direct consequence of the Vogt–Russell theorem: on the main sequence, mass is the only free parameter. It is used to estimate the masses of single stars from their luminosity, after calibration with binary stars, whose masses can be measured directly.

Reading path

To understand this page from the ground up, read these 88 pages in order. The foundations come first.

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

Further reading

  1. Web https://en.wikipedia.org/wiki/Mass%E2%80%93luminosity_relation
About this page
TagAN3ZQVNG7Z
URIm2://AN3ZQVNG7Z
Kindstatement, fact
WritingW0 stub
MSC85A15
StatusThis copy of the site has no build data. Run m2 status AN3ZQVNG7Z.
AboutStellar structure
UsesStellar structure, Hydrostatic equilibrium, Vogt–Russell theorem
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Changed3 November 2026

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