Home

Site Map

Suchen

Physik,Astronomie,Java,Applet,Unterricht,Schule,Sonne,Erde,Mond
GeoAstro
Applets

Astronomie
Physics,astronomy,
Chaos
Spiel
Physik,Astronomie,Java,Applet,Unterricht,Schule,Software
Java
Physik,Astronomie,Java-Applet,Informatik,Mathematik,Java,Applet,Mac
                    OS,Programm
Diverses

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

formula

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

Lane Emden differential
                      equation

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.

star
                          sun

The solution can only be obtained numerically by my Lane-Emden applet:

solution emden ode   polytropic index n=3

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

temperature
temperature

(2) Density:
rho/rho0 ∼ u3

density
density

(3) Pressure:
P/P0 ∼ u4

pressure
pressure

(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).
mass
mass
interior star sun

Relations of the standard model for the core values:

formula


The Sun:

In case of complete ionisation:

molar mass

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):

lane-emden result 3

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

result 2

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

polytropic index 4


core density


core pressure


core temperature

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

temperarure density core
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

eddington
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
R = radius = z

we have the relation (a):

gravity                     (1)

which is Newton's law of gravitation
 Newton law of gravitation
with gravitational constant G=1 in units of the Lane-Emden equation.

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

From (a) and (b):

density mass gravitaional constant

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

email