Math 2.0

Astronomy and astrophysics Definition

Polytrope

Builds on Hydrostatic equilibriumLane–Emden equationEmden–Chandrasekhar equation

Definition

Let n≥0n \ge 0 be a real number and let K>0K > 0 be a constant. Consider a spherically symmetric, self-gravitating ball of fluid in hydrostatic equilibrium, with pressure PP and density ρ\rho. It is a polytrope of index nn if P=Kρ1+1/nP = K \rho^{1 + 1/n} at every point. The number nn is the polytropic index and γ=1+1/n\gamma = 1 + 1/n is the polytropic exponent.

Remarks

Write ρ=ρcθn\rho = \rho_c \theta^n with ρc\rho_c the central density, and r=αξr = \alpha \xi with α2=(n+1)Kρc1/n−1/(4πG)\alpha^2 = (n + 1) K \rho_c^{1/n - 1} / (4 \pi G), where GG is the gravitational constant. Then hydrostatic equilibrium becomes the Lane–Emden equation ξ−2(ξ2θ′)′=−θn\xi^{-2} (\xi^2 \theta')' = -\theta^n with θ(0)=1\theta(0) = 1 and θ′(0)=0\theta'(0) = 0, and the surface is the first zero ξ1\xi_1 of θ\theta. Example: for n=1n = 1 the solution is θ=sin⁡ξ/ξ\theta = \sin \xi / \xi, so ξ1=π\xi_1 = \pi and the radius R=παR = \pi \alpha does not depend on the central density. Example: n=0n = 0 gives a ball of uniform density, since ρ=ρcθ0\rho = \rho_c \theta^0 is constant. Non-example: an isothermal ideal gas, with PP proportional to ρ\rho, is the limit n→∞n \to \infty and is not a polytrope of finite index; it leads instead to the Emden–Chandrasekhar equation. A polytrope has finite radius when 0≤n<50 \le n < 5; for n=5n = 5 the radius is infinite but the mass is finite.

Reading path

To understand this page from the ground up, read these 87 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. Velocity concept
  17. Fréchet derivative definition
  18. Ordinary differential equation definition
  19. Cross product 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. Polytrope definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Polytrope
About this page
TagEWFSKZAM6Q
URIm2://EWFSKZAM6Q
Kinddefinition
WritingW0 stub
MSC85A15
DefinesPolytrope
UsesHydrostatic equilibrium, Lane–Emden equation, Emden–Chandrasekhar equation
Changed3 November 2026

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