Astro- modelling a white dwarf via polytrope

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SUMMARY

The discussion focuses on solving the Lane-Emden equation numerically to model a white dwarf using polytropic indices. It establishes that the equation generates finite stars for low polytropic indices (0 to 4), with the maximum scaled radius being a constant multiple of the scale length. The formula provided for the maximum actual radius is S_{n} = sqrt{(n+1) k_{B}/(4 π G m) (T_{c}/ρ_{c})}. Central density and temperature are identified as free parameters in this modeling process.

PREREQUISITES
  • Understanding of the Lane-Emden equation
  • Knowledge of polytropic indices in astrophysics
  • Familiarity with thermodynamic principles (central density and temperature)
  • Basic grasp of numerical methods for solving differential equations
NEXT STEPS
  • Study the derivation and applications of the Lane-Emden equation
  • Explore numerical methods for solving differential equations in astrophysics
  • Research the implications of polytropic indices on stellar structure
  • Investigate the relationship between central density, temperature, and stellar evolution
USEFUL FOR

Astronomers, astrophysicists, and students studying stellar modeling, particularly those interested in white dwarf characteristics and polytropic processes.

laam
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have solved the lane - emden eqn numerically for scaled radius, and the derivative of scaled pressure with respect of scaled radius, are given the eqn of state, n, and mass, how would i go about working out the central density and the radius?
thanks
 
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Huh? Sorry, but can you restate that a little more clearly? Maybe include the equations you worked with too?
 
laam said:
have solved the lane - emden eqn numerically for scaled radius, and the derivative of scaled pressure with respect of scaled radius, are given the eqn of state, n, and mass, how would i go about working out the central density and the radius?
thanks

The Lane-Emden Equation only generates finite "stars" for low Polytropic Indices (0 to 4). For those indices, the maximum Scaled Radius is some constant (eg., Pi or sqrt(6)). Thus, the maximum "actual radius" is just that constant multiple of the Scale Length:
[tex]S_{n} = \sqrt{ \frac{(n+1) k_{B}}{4 \pi G m} \frac{T_{c}}{\rho_{c}} }[/tex]​
The central density & temperature are Free Parameters*.
* For derivation, see: https://www.physicsforums.com/showthread.php?t=278009 [post #8]​
 

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