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Wo jetzt nun, wie unsre Weisen sagen, Seelenlos ein Feuerball sich dreht, Lenkte damals seinen goldnen Wagen Helios in stiller Majestät. aus Friedrich Schillers Gedicht "Die Götter Griechenlands" The Standard Stellar Model Emden's differential equation arising in the study of stellar interiors is given by (polytropic index n=3): u is a (dimensionless) temperature, the (dimensionless) variable z is related to the distance r from the center. The boundary conditions are:     at the center (z=0, r=0): u=1, du/dz=0,     at the surface (r=R): u(z0)=0. The solution can only be obtained numerically (my applet): Tools: ODE Toolkit (online), Berkeley Madonna (Mac, Windows trial). The first zero of u(z) is found to be z0=6.897. The relative distance is r/R=z/z0. My results for the variables temperature, density, pressure, mass: (1) Temperature: T ∼ u
(2) Density: rho/rho0 ∼ u3
(3) Pressure: P/P0 ∼ u4
(4) Mass: m(r)/M ∼ -u2*du/dz m(r)/M is the fraction of the total mass within the radius r. About half of the mass is included by the sphere of radius r=0.3*R (2,7% of the total volume).
Relations of the standard model: The Sun: Using the mass fractions for the solar composition (Hydrogen x=0.734, Helium y=0.25) the mean molecular mass is (in case of complete ionisation): μ=  0.60114 The results for the Sun are (polytrope index n=3): polytrope index n=2: polytrope index n=4: Temperature 106 K Density 103 kg/m3 Pressure 1015 N/m2 Standard model 17 76 12 web 15 160 14 150 25 15.6 34 15.7

Web Links

Polytropic Process Lane-Emden equation (wikipedia) Emden's Equation (R. Baretti) Lane-Emden Differential Equation (Wolfram MathWorld) Program to solve the Lane-Emden equation numerically Lane, Jonathan Homer (wikipedia) Emden, Robert (wikipedia) Stellar Structure and the Lane-Emden Function The Solar Composition The Astrophysics Spectator

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Last update: 2010, Apr 05