Math 2.0

Astronomy and astrophysics Definition

Virial mass

Builds on Friedmann equationsHubble's lawSpherical collapse modelVirial theorem

Definition

Let H(z)H(z) be the Hubble parameter at redshift zz, as in Hubble's law, and let ρc(z)=3H(z)2/(8πG)\rho_c(z) = 3 H(z)^2 / (8 \pi G) be the critical density of the Friedmann equations. Fix an overdensity Δ>0\Delta > 0. Let a roughly spherical object have mass density ρ(r)\rho(r) at distance rr from its centre. Its virial radius rΔr_\Delta is the radius at which the mean density inside equals Δρc(z)\Delta \rho_c(z): 34πrΔ3∫0rΔ4πr2ρ(r) dr=Δ ρc(z).\frac{3}{4 \pi r_\Delta^3} \int_0^{r_\Delta} 4 \pi r^2 \rho(r) \, dr = \Delta \, \rho_c(z). The virial mass is MΔ=4π3Δ ρc(z) rΔ3.M_\Delta = \frac{4 \pi}{3} \Delta \, \rho_c(z) \, r_\Delta^3.

Remarks

Common choices are Δ=200\Delta = 200, giving M200M_{200}, and Δ=18π2≈178\Delta = 18 \pi^2 \approx 178, the value the spherical collapse model predicts for a collapsed object in a universe of matter only. Some authors use the mean matter density instead of ρc\rho_c. The name comes from the virial theorem, which the matter inside rΔr_\Delta approximately satisfies.

Example: a singular isothermal sphere ρ(r)=σ2/(2πGr2)\rho(r) = \sigma^2 / (2 \pi G r^2) has mean density 3σ2/(2πGr2)3\sigma^2 / (2 \pi G r^2) inside rr. Setting this equal to Δρc\Delta \rho_c gives rΔ=σ3/(2πGΔρc)r_\Delta = \sigma \sqrt{3 / (2 \pi G \Delta \rho_c)} and MΔ=2σ2rΔ/GM_\Delta = 2 \sigma^2 r_\Delta / G.

Non-example: a density that is constant out to infinity has mean density equal to that constant at every radius. If the constant is below Δρc\Delta \rho_c, no radius satisfies the condition, and no virial mass is defined.

Reading path

To understand this page from the ground up, read these 91 pages in order. The foundations come first.

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

Further reading

  1. Web https://en.wikipedia.org/wiki/Virial_mass
About this page
Tag8K5AM6EMWH
URIm2://8K5AM6EMWH
Kinddefinition
WritingW0 stub
MSC85A15, 85A40
DefinesVirial mass
UsesFriedmann equations, Hubble's law, Spherical collapse model, Virial theorem
Changed3 November 2026

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