9873

Lane-Emden Equation in Stellar Structure

The Lane–Emden equation is , where dimensionless variables and relate to the density and radius (see Details). The equation is used in the study of polytropic gaseous spheres and the modeling of stars. Solutions of this second-order differential equation use polytropes to relate pressure and density as a function of , the radial coordinate measured from the center of the polytropic sphere (the star); and have the polytropic relation , where is the polytropic constant and is the polytropic index that you can choose with the control.
Here the equation is solved for the polytropic index value for the dimensionless function . Click the checkbox to see all solutions from 0 to the currently selected value. The first root of each solution, derivatives at that root value, and the critical/mean density ratio are also displayed.

SNAPSHOTS

  • [Snapshot]
  • [Snapshot]
  • [Snapshot]

DETAILS

Typically we are interested in finite solutions at . If we take the values and to be the densities, then .
The relationship between the radial coordinate and its dimensionless counterpart is
,
where is the gravitational constant. The relationship between the density and its dimensionless counterpart is .
Studies and derivations of these relations and the Lane–Emden equation can be found in [1] and [2].
References
[1] S. Chandrasekhar, An Introduction to the Study of Stellar Structure, Chicago: University of Chicago Press, 1939.
[2] R. Kippenhahn and A. Weigert, Stellar Structure and Evolution, Berlin: Springer-Verlag, 1994.

    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.









 
RELATED RESOURCES
Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Step-by-step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in Step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2014 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+