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An introduction to some ordinary differential equations governing stellar structures

di Giovanni, Yani LU (2019) In Bachelor’s Theses in Mathematical Sciences MATK01 20181
Mathematics (Faculty of Sciences)
Abstract
The Lane-Emden equation is a non-linear differential equation governing the equilibrium of polytropic stationary self-gravitating, spherically symmetric star models;

$${\frac {1}{\xi ^{2}}}{\frac {d}{d\xi }}\left({\xi ^{2}{\frac {d\theta }{d\xi }}}\right)+\theta ^{n}=0.$$

In the isothermal cases we have the Chandrasekhar equation:

$${\frac {1}{\xi ^{2}}}{\frac {d}{d\xi }}\left({\xi ^{2}{\frac {d\psi}{d\xi }}}\right)-e^{-\psi}=0$$

After having derived these models, we will go through all cases for which analytic solutions are achievable. Moreover, we will discuss the existence and uniqueness of positive solutions under specific boundary conditions by transforming the equations to autonomous ones. The analysis depends upon the... (More)
The Lane-Emden equation is a non-linear differential equation governing the equilibrium of polytropic stationary self-gravitating, spherically symmetric star models;

$${\frac {1}{\xi ^{2}}}{\frac {d}{d\xi }}\left({\xi ^{2}{\frac {d\theta }{d\xi }}}\right)+\theta ^{n}=0.$$

In the isothermal cases we have the Chandrasekhar equation:

$${\frac {1}{\xi ^{2}}}{\frac {d}{d\xi }}\left({\xi ^{2}{\frac {d\psi}{d\xi }}}\right)-e^{-\psi}=0$$

After having derived these models, we will go through all cases for which analytic solutions are achievable. Moreover, we will discuss the existence and uniqueness of positive solutions under specific boundary conditions by transforming the equations to autonomous ones. The analysis depends upon the value of the polytropic index $n.$ We also compute some solutions numerically. (Less)
Popular Abstract
The Lane-Emden equation is a nonlinear second order ordinary differential equation which models many phenomena in mathematical physics and astrophysics. This equation describes the equilibrium density distribution in a self-gravitating sphere of polytropic or isothermal gas and has crucial relevance in the field of radiative cooling and modeling of clusters of galaxies. Moreover, this equation has been considered quite versatile when examining aspects such as the analysis of isothermal cores, convective stellar interiors, and fully degenerate stellar configurations. In addition, recent observation lead to the conclusion that the density profiles of dark matter halos too are often modeled by the isothermal Lane-Emden equation with suitable... (More)
The Lane-Emden equation is a nonlinear second order ordinary differential equation which models many phenomena in mathematical physics and astrophysics. This equation describes the equilibrium density distribution in a self-gravitating sphere of polytropic or isothermal gas and has crucial relevance in the field of radiative cooling and modeling of clusters of galaxies. Moreover, this equation has been considered quite versatile when examining aspects such as the analysis of isothermal cores, convective stellar interiors, and fully degenerate stellar configurations. In addition, recent observation lead to the conclusion that the density profiles of dark matter halos too are often modeled by the isothermal Lane-Emden equation with suitable boundary conditions at the origin.

The Lane-Emden equation was first introduced in 1869 by the American astrophysicist Jonathan Homer Lane (1819-1880), who was interested in computing the temperature and the density of the solar surface. It is to be noted the contribution to this equation by the Swiss mathematician Robert Emden (1862-1940), who explained the expansion and compression of gas spheres through a mathematical model. Since Stefan's law was published a decade later, Lane used some other experimental results concerning the rate of emission of radiant energy by a heated surface, and the value that he obtained for the solar temperature was 30,000 degrees Kelvin, roughly five times larger than the actual number. Despite getting wrong results concerning the surface, Lane's temperature and density results for the stellar interior, turned out to be very reasonable, and his equation is still used nowadays for computing the inner structure of polytropic stars. The Emden-Chandrasekhar equation is the isothermal case of the Lane-Emden equation and was first introduced by Robert Emden in 1907; in this case, the polytropic index has an infinite value. Later on, Subrahmanyan Chandrasekhar (1910-1995) made important contributions by analyzing the system in the phase plane.

In this dissertation, we will go through explicit, numerical and qualitative aspects of the polytropic and the isothermal equation. (Less)
Please use this url to cite or link to this publication:
author
di Giovanni, Yani LU
supervisor
organization
course
MATK01 20181
year
type
M2 - Bachelor Degree
subject
publication/series
Bachelor’s Theses in Mathematical Sciences
report number
LUNFMA-4082-2018
ISSN
1654-6229
other publication id
2018:K29
language
English
id
8968382
date added to LUP
2019-07-15 11:00:28
date last changed
2019-08-14 14:22:54
@misc{8968382,
  abstract     = {The Lane-Emden equation is a non-linear differential equation governing the equilibrium of polytropic stationary self-gravitating, spherically symmetric star models;

$${\frac {1}{\xi ^{2}}}{\frac {d}{d\xi }}\left({\xi ^{2}{\frac {d\theta }{d\xi }}}\right)+\theta ^{n}=0.$$

In the isothermal cases we have the Chandrasekhar equation:

$${\frac {1}{\xi ^{2}}}{\frac {d}{d\xi }}\left({\xi ^{2}{\frac {d\psi}{d\xi }}}\right)-e^{-\psi}=0$$

After having derived these models, we will go through all cases for which analytic solutions are achievable. Moreover, we will discuss the existence and uniqueness of positive solutions under specific boundary conditions by transforming the equations to autonomous ones. The analysis depends upon the value of the polytropic index $n.$ We also compute some solutions numerically.},
  author       = {di Giovanni, Yani},
  issn         = {1654-6229},
  language     = {eng},
  note         = {Student Paper},
  series       = {Bachelor’s Theses in Mathematical Sciences},
  title        = {An introduction to some ordinary differential equations governing stellar structures},
  year         = {2019},
}