Emden–Chandrasekhar Equation
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astrophysics Astrophysics is a science that employs the methods and principles of physics and chemistry in the study of astronomical objects and phenomena. As one of the founders of the discipline, James Keeler, said, astrophysics "seeks to ascertain the ...
, the Emden–Chandrasekhar equation is a
dimensionless Dimensionless quantities, or quantities of dimension one, are quantities implicitly defined in a manner that prevents their aggregation into units of measurement. ISBN 978-92-822-2272-0. Typically expressed as ratios that align with another sy ...
form of the
Poisson equation Poisson's equation is an elliptic partial differential equation of broad utility in theoretical physics. For example, the solution to Poisson's equation is the potential field caused by a given electric charge or mass density distribution; with th ...
for the density distribution of a spherically symmetric
isothermal An isothermal process is a type of thermodynamic process in which the temperature ''T'' of a system remains constant: Δ''T'' = 0. This typically occurs when a system is in contact with an outside thermal reservoir, and a change in the sys ...
gas sphere subjected to its own gravitational force, named after Robert Emden and
Subrahmanyan Chandrasekhar Subrahmanyan Chandrasekhar (; 19 October 1910 – 21 August 1995) was an Indian Americans, Indian-American theoretical physicist who made significant contributions to the scientific knowledge about the structure of stars, stellar evolution and ...
. The equation was first introduced by Robert Emden in 1907. The equation readswhere \xi is the dimensionless radius and \psi is the related to the density of the gas sphere as \rho=\rho_c e^, where \rho_c is the density of the gas at the centre. The equation has no known explicit solution. If a
polytropic A polytropic process is a thermodynamic process that obeys the relation: p V^ = C where ''p'' is the pressure, ''V'' is volume, ''n'' is the polytropic index, and ''C'' is a constant. The polytropic process equation describes expansion and com ...
fluid is used instead of an isothermal fluid, one obtains the
Lane–Emden equation In astrophysics, the Lane–Emden equation is a dimensionless form of Poisson's equation for the gravitational potential of a Newtonian self-gravitating, spherically symmetric, polytropic fluid. It is named after astrophysicists Jonathan Homer L ...
. The isothermal assumption is usually modeled to describe the core of a star. The equation is solved with the initial conditions, :\psi =0, \quad \frac =0 \quad \text \quad \xi=0. The equation appears in other branches of physics as well, for example the same equation appears in the Frank-Kamenetskii explosion theory for a spherical vessel. The relativistic version of this spherically symmetric isothermal model was studied by Subrahmanyan Chandrasekhar in 1972.


Derivation

For an
isothermal An isothermal process is a type of thermodynamic process in which the temperature ''T'' of a system remains constant: Δ''T'' = 0. This typically occurs when a system is in contact with an outside thermal reservoir, and a change in the sys ...
gaseous
star A star is a luminous spheroid of plasma (physics), plasma held together by Self-gravitation, self-gravity. The List of nearest stars and brown dwarfs, nearest star to Earth is the Sun. Many other stars are visible to the naked eye at night sk ...
, the pressure p is due to the kinetic pressure and
radiation pressure Radiation pressure (also known as light pressure) is mechanical pressure exerted upon a surface due to the exchange of momentum between the object and the electromagnetic field. This includes the momentum of light or electromagnetic radiation of ...
:p = \rho\frac T + \frac T^4 where *\rho is the density *k_B is the
Boltzmann constant The Boltzmann constant ( or ) is the proportionality factor that relates the average relative thermal energy of particles in a ideal gas, gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin (K) and the ...
*W is the mean
molecular weight A molecule is a group of two or more atoms that are held together by Force, attractive forces known as chemical bonds; depending on context, the term may or may not include ions that satisfy this criterion. In quantum physics, organic chemi ...
*H is the mass of the proton *T is the temperature of the star *\sigma is the Stefan–Boltzmann constant *c is the speed of light The equation for equilibrium of the star requires a balance between the pressure force and gravitational force :\frac \frac \left(\frac\frac\right)= - 4\pi G \rho where r is the radius measured from the center and G is the
gravitational constant The gravitational constant is an empirical physical constant involved in the calculation of gravitational effects in Sir Isaac Newton's law of universal gravitation and in Albert Einstein's general relativity, theory of general relativity. It ...
. The equation is re-written as : \frac\frac \frac \left(r^2\frac \right) = - 4\pi G \rho Introducing the transformation :\psi = \ln \frac, \quad \xi = r \left(\frac\right)^ where \rho_c is the central density of the star, leads to :\frac \frac\left(\xi^2 \frac\right)= e^ The boundary conditions are :\psi =0, \quad \frac =0 \quad \text \quad \xi=0 For \xi\ll 1, the solution goes like :\psi = \frac - \frac + \frac + \cdots


Limitations of the model

Assuming isothermal sphere has some disadvantages. Though the density obtained as solution of this isothermal gas sphere decreases from the centre, it decreases too slowly to give a well-defined surface and finite mass for the sphere. It can be shown that, as \xi \gg 1, :\frac=e^=\frac \left +\frac \cos\left(\frac\ln \xi + \delta\right) + O(\xi^)\right/math> where A and \delta are constants which will be obtained with numerical solution. This behavior of density gives rise to increase in mass with increase in radius. Thus, the model is usually valid to describe the core of the star, where the temperature is approximately constant.Henrich, L. R., & Chandrasekhar, S. (1941). Stellar Models with Isothermal Cores. The Astrophysical Journal, 94, 525.


Singular solution

Introducing the transformation x=1/\xi transforms the equation to :x^4 \frac = e^ The equation has a
singular solution A singular solution ''ys''(''x'') of an ordinary differential equation is a solution that is singular or one for which the initial value problem (also called the Cauchy problem by some authors) fails to have a unique solution at some point on the so ...
given by :e^ = 2x^2, \quad \text \quad -\psi_s = 2 \ln x+ \ln 2 Therefore, a new variable can be introduced as -\psi = 2 \ln x + z, where the equation for z can be derived, :\frac-\frac+ e^z -2 =0, \quad \text \quad t=\ln x This equation can be reduced to first order by introducing :y=\frac = \xi \frac - 2 then we have :y\frac - y + e^z- 2 = 0


Reduction

There is another reduction due to
Edward Arthur Milne Edward Arthur Milne FRS (; 14 February 1896 – 21 September 1950) was a British astrophysicist and mathematician. Biography Milne was born in Hull, Yorkshire, England. He attended Hymers College and from there he won an open scholarshi ...
. Let us define :u = \frac, \quad v = \xi \frac then :\frac\frac = -\frac


Properties

*If \psi(\xi) is a solution to Emden–Chandrasekhar equation, then \psi(A\xi)-2\ln A is also a solution of the equation, where A is an arbitrary constant. *The solutions of the Emden–Chandrasekhar equation which are finite at the origin have necessarily d\psi/d\xi=0 at \xi=0


See also

*
Lane–Emden equation In astrophysics, the Lane–Emden equation is a dimensionless form of Poisson's equation for the gravitational potential of a Newtonian self-gravitating, spherically symmetric, polytropic fluid. It is named after astrophysicists Jonathan Homer L ...
*
Frank-Kamenetskii theory In combustion, Frank-Kamenetskii theory explains the Thermal runaway, thermal explosion of a homogeneous mixture of reactants, kept inside a closed vessel with constant temperature walls. It is named after a Russian scientist David A. Frank-Kamene ...
*
Chandrasekhar's white dwarf equation In astrophysics, Chandrasekhar's white dwarf equation is an initial value ordinary differential equation introduced by the Indian American astrophysicist Subrahmanyan Chandrasekhar, in his study of the gravitational potential of completely degener ...


References

{{DEFAULTSORT:Emden-Chandrasekhar equation Equations of physics Fluid dynamics Stellar dynamics Ordinary differential equations