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,^[1] in his study of the gravitational potential of completely degenerate white dwarf stars. The equation reads as^[2]

$${\displaystyle {\frac {1}{\eta ^{2}}}{\frac {d}{d\eta }}\left(\eta ^{2}{\frac {d\varphi }{d\eta }}\right)+(\varphi ^{2}-C)^{3/2}=0}$$

with initial conditions

$${\displaystyle \varphi (0)=1,\quad \varphi '(0)=0}$$

where ${\displaystyle \varphi }$ measures the density of white dwarf, ${\displaystyle \eta }$ is the non-dimensional radial distance from the center and ${\displaystyle C}$ is a constant which is related to the density of the white dwarf at the center. The boundary ${\displaystyle \eta =\eta _{\infty }}$ of the equation is defined by the condition

$${\displaystyle \varphi (\eta _{\infty })={\sqrt {C}}.}$$

such that the range of ${\displaystyle \varphi }$ becomes ${\displaystyle {\sqrt {C}}\leq \varphi \leq 1}$. This condition is equivalent to saying that the density vanishes at ${\displaystyle \eta =\eta _{\infty }}$.