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Applet

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 assuming a polytropic model

(constant K, polytropic index n=3) is given by:

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 by my Lane-Emden 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 for the core values:

The Sun:

In case of complete ionisation:

Using the mass fractions for the solar composition (Hydrogen x=0.734, Helium y=0.25, heavier elements: 1-x-y = 0.016) the mean molar mass, relative to hydrogen, is:

μ =  0.60114
or, in absolute units:
μ =  0.60114 · 1.0079·10-3 kg/mol = 6.059·10-4 kg/mol

The results for the Sun
(polytropic index n=3):

The results for the Sun (polytropic index n=2):

The results for the Sun (polytropic index n=4):

In the book of Vogt we find the formulae for the standard model:

which agree very well with the results of my applet.

 Temperature 106 K Density 103 kg/m3 Pressure 1015 N/m2 Lane-Emden model (applet) n=3 12 76 12 Lane-Emden model (applet) n=3.25 14 124 24 Web 15 160 14 150 23-35 15.6 34 15.7 15.7 162 25

Table from the book of Eddington

"The successive colums give the following physical quantities, expressed in each
case in terms of a unit which will depend on the star considered:
1. Distance from the centre.
2. Gravitational potential. Temperatute (for a perfect gas of constant molecalar weight).
3. Density.
4. Pressure.
5. Acceleration of gravity.
6. Reciprocal of mean density to the point considered.
7. Mass interior to the point considered."

Using the symbols g, M, and R for the values at the surface (columns 5, 7, 1):
g = acceleration of gravity = - du/dz
M = mass = -z2 du/dz

we have the relation (a):

(1)

which is Newton's law of gravitation

with gravitational constant G=1 in units of the Lane-Emden equation.

The mean density (column 6) is (b):

From (a) and (b):

 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 Lecture 23: The Lane-Emden Equation Polytropes - Derivation and Sulution of the Lane-Emden Equation Lecture 7: Polytropes Lane-Emden Equation in Stellar Structute (Wolfram Demonstrations Project) Lecture 5: Polytropic Models Robert Emden: Gaskugeln : Anwendungen der mechanischen Wärmetheorie auf kosmologische und meteorologische Probleme. Leipzig, Berlin: Teubner, 1907. Books Search Amazon A. S. Eddington: The Internal Constitution of the Stars, Cambridge University Press, 1926. H. Vogt: Aufbau und Entwicklung der Sterne, Akadem. Verlagsgesellschaft, Leipzig 1957. Robert Emden: Gaskugeln: Anwendungen der mechanischen Wärmetheorie auf kosmologische und meteorologische Probleme. Leipzig, Berlin: Teubner, 1907. Amazon: Books on Demand, ISBN 978-5875749025 Dermott J. Mullan: Physics of the sun: A First Course; CRC Press, Boca Raton - London - New York, 2009; ISBN 978-1420083071 https://www.crcpress.com/Physics-of-the-Sun-A-First-Course/Mullan/9781420083071

Last update: 2023, Oct 07