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Emden–Chandrasekhar equation

Builds on Ordinary differential equationHydrostatic equilibrium

Definition

The Emden–Chandrasekhar equation is the ordinary differential equation 1ξ2ddξ(ξ2dψdξ)=e−ψ\frac{1}{\xi^2} \frac{d}{d\xi} \left( \xi^2 \frac{d\psi}{d\xi} \right) = e^{-\psi} for a real function ψ\psi of ξ>0\xi > 0, with the initial conditions ψ(0)=0\psi(0) = 0 and ψ′(0)=0\psi'(0) = 0.

Remarks

It comes from a spherically symmetric, self-gravitating gas in hydrostatic equilibrium with isothermal equation of state P=cs2ρP = c_s^2 \rho, where csc_s is the constant isothermal sound speed. With central density ρc\rho_c and gravitational constant GG, one sets ρ=ρce−ψ\rho = \rho_c e^{-\psi} and r=aξr = a \xi with a2=cs2/(4πGρc)a^2 = c_s^2 / (4 \pi G \rho_c). Example: the function ψ(ξ)=ln⁡(ξ2/2)\psi(\xi) = \ln(\xi^2 / 2) satisfies the differential equation for ξ>0\xi > 0. Indeed, ξ2ψ′=2ξ\xi^2 \psi' = 2 \xi, so the left side is 2/ξ22/\xi^2, and e−ψ=2/ξ2e^{-\psi} = 2/\xi^2. It is the singular isothermal sphere, and it fails the initial condition ψ(0)=0\psi(0) = 0, since ψ→−∞\psi \to -\infty as ξ→0\xi \to 0. The regular solution with ψ(0)=ψ′(0)=0\psi(0) = \psi'(0) = 0 has no closed form; it starts as ψ=ξ2/6−ξ4/120+⋯\psi = \xi^2/6 - \xi^4/120 + \cdots. An equivalent form uses u=−ψu = -\psi, which satisfies ξ−2(ξ2u′)′+eu=0\xi^{-2} (\xi^2 u')' + e^{u} = 0.

Reading path

To understand this page from the ground up, read these 86 pages in order. The foundations come first.

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

Further reading

  1. Web https://en.wikipedia.org/wiki/Emden%E2%80%93Chandrasekhar_equation
About this page
Tag1GN7MP5BRQ
URIm2://1GN7MP5BRQ
Kinddefinition
WritingW0 stub
MSC85A15, 34A34
DefinesEmden–Chandrasekhar equation
UsesOrdinary differential equation, Hydrostatic equilibrium
Changed3 November 2026

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