Math 2.0

Astronomy and astrophysics Definition

Bonnor–Ebert mass

Builds on Hydrostatic equilibriumEmden–Chandrasekhar equationSpeed of sound

Definition

Let an isothermal gas with sound speed cs>0c_s > 0 form a ball in hydrostatic equilibrium under its own gravity. Let GG be the gravitational constant, and let the boundary be held at a fixed external pressure Pext>0P_{\text{ext}} > 0. Such equilibria are the solutions of the isothermal Emden–Chandrasekhar equation truncated at a dimensionless radius ξ0\xi_0. For fixed csc_s and PextP_{\text{ext}} their mass MM is a function of ξ0\xi_0. The Bonnor–Ebert mass MBEM_{\text{BE}} is the maximum of this function over ξ0>0\xi_0 > 0. It is attained at ξ0≈6.45\xi_0 \approx 6.45, and MBE≈1.18 cs4G3/2Pext1/2.M_{\text{BE}} \approx 1.18 \, \frac{c_s^4}{G^{3/2} P_{\text{ext}}^{1/2}}.

Remarks

Example: doubling the temperature at fixed PextP_{\text{ext}} doubles cs2c_s^2, so MBEM_{\text{BE}} grows by the factor 44. At the critical sphere the ratio of central to boundary density is about 1414; equilibria with a larger density contrast, ξ0>6.45\xi_0 > 6.45, are unstable to collapse. Non-example: a sphere of mass 2MBE2 M_{\text{BE}} at the same csc_s and PextP_{\text{ext}} has no hydrostatic equilibrium at all, so it has no Bonnor–Ebert structure and must collapse. The isothermal equation is 1ξ2ddξ(ξ2dψdξ)=e−ψ\frac{1}{\xi^2} \frac{d}{d\xi}\left(\xi^2 \frac{d\psi}{d\xi}\right) = e^{-\psi}, with density ρ=ρce−ψ\rho = \rho_c e^{-\psi}, central density ρc\rho_c and radius r=ξ cs/4πGρcr = \xi \, c_s / \sqrt{4\pi G \rho_c}. Letting ξ0→∞\xi_0 \to \infty approaches the singular isothermal sphere at large radii.

Reading path

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

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

Further reading

  1. Web https://en.wikipedia.org/wiki/Bonnor%E2%80%93Ebert_mass
About this page
TagD67W7FNRDW
URIm2://D67W7FNRDW
Kinddefinition
WritingW0 stub
MSC85A30, 85A15, 76E20
DefinesBonnor–Ebert mass
UsesHydrostatic equilibrium, Emden–Chandrasekhar equation, Speed of sound
Changed3 November 2026

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