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]
1
η
2
d
d
η
(
η
2
d
φ
d
η
)
+
(
φ
2
−
C
)
3
/
2
=
0
{\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
φ
(
0
)
=
1
,
φ
′
(
0
)
=
0
{\displaystyle \varphi (0)=1,\quad \varphi '(0)=0}
where
φ
{\displaystyle \varphi }
measures the density of white dwarf,
η
{\displaystyle \eta }
is the non-dimensional radial distance from the center and
C
{\displaystyle C}
is a constant which is related to the density of the white dwarf at the center. The boundary
η
=
η
∞
{\displaystyle \eta =\eta _{\infty }}
of the equation is defined by the condition
φ
(
η
∞
)
=
C
.
{\displaystyle \varphi (\eta _{\infty })={\sqrt {C}}.}
such that the range of
φ
{\displaystyle \varphi }
becomes
C
≤
φ
≤
1
{\displaystyle {\sqrt {C}}\leq \varphi \leq 1}
. This condition is equivalent to saying that the density vanishes at
η
=
η
∞
{\displaystyle \eta =\eta _{\infty }}
.
Derivation
edit
From the quantum statistics of a completely degenerate electron gas (all the lowest quantum states are occupied), the pressure and the density of a white dwarf are calculated in terms of the maximum electron momentum
p
0
{\displaystyle p_{0}}
standardized as
x
=
p
0
/
m
c
{\displaystyle x=p_{0}/mc}
, with pressure
P
=
A
f
(
x
)
{\displaystyle P=Af(x)}
and density
ρ
=
B
x
3
{\displaystyle \rho =Bx^{3}}
, where
A
=
π
m
e
4
c
5
3
h
3
=
6.02
×
10
21
Pa
,
B
=
8
π
3
m
p
μ
e
(
m
e
c
h
)
3
=
9.82
×
10
8
μ
e
kg/m
3
,
f
(
x
)
=
x
(
2
x
2
−
3
)
(
x
2
+
1
)
1
/
2
+
3
sinh
−
1
x
,
{\displaystyle {\begin{aligned}&A={\frac {\pi m_{e}^{4}c^{5}}{3h^{3}}}=6.02\times 10^{21}{\text{ Pa}},\\&B={\frac {8\pi }{3}}m_{p}\mu _{e}\left({\frac {m_{e}c}{h}}\right)^{3}=9.82\times 10^{8}\mu _{e}{\text{ kg/m}}^{3},\\&f(x)=x(2x^{2}-3)(x^{2}+1)^{1/2}+3\sinh ^{-1}x,\end{aligned}}}
μ
e
{\displaystyle \mu _{e}}
is the mean molecular weight of the gas, and
h
{\displaystyle h}
is the height of a small cube of gas with only two possible states.
When this is substituted into the hydrostatic equilibrium equation
1
r
2
d
d
r
(
r
2
ρ
d
P
d
r
)
=
−
4
π
G
ρ
{\displaystyle {\frac {1}{r^{2}}}{\frac {d}{dr}}\left({\frac {r^{2}}{\rho }}{\frac {dP}{dr}}\right)=-4\pi G\rho }
where
G
{\displaystyle G}
is the gravitational constant and
r
{\displaystyle r}
is the radial distance, we get
1
r
2
d
d
r
(
r
2
d
x
2
+
1
d
r
)
=
−
π
G
B
2
2
A
x
3
{\displaystyle {\frac {1}{r^{2}}}{\frac {d}{dr}}\left(r^{2}{\frac {d{\sqrt {x^{2}+1}}}{dr}}\right)=-{\frac {\pi GB^{2}}{2A}}x^{3}}
and letting
y
2
=
x
2
+
1
{\displaystyle y^{2}=x^{2}+1}
, we have
1
r
2
d
d
r
(
r
2
d
y
d
r
)
=
−
π
G
B
2
2
A
(
y
2
−
1
)
3
/
2
{\displaystyle {\frac {1}{r^{2}}}{\frac {d}{dr}}\left(r^{2}{\frac {dy}{dr}}\right)=-{\frac {\pi GB^{2}}{2A}}(y^{2}-1)^{3/2}}
If we denote the density at the origin as
ρ
o
=
B
x
o
3
=
B
(
y
o
2
−
1
)
3
/
2
{\displaystyle \rho _{o}=Bx_{o}^{3}=B(y_{o}^{2}-1)^{3/2}}
, then a non-dimensional scale
r
=
(
2
A
π
G
B
2
)
1
/
2
η
y
o
,
y
=
y
o
φ
{\displaystyle r=\left({\frac {2A}{\pi GB^{2}}}\right)^{1/2}{\frac {\eta }{y_{o}}},\quad y=y_{o}\varphi }
gives
1
η
2
d
d
η
(
η
2
d
φ
d
η
)
+
(
φ
2
−
C
)
3
/
2
=
0
{\displaystyle {\frac {1}{\eta ^{2}}}{\frac {d}{d\eta }}\left(\eta ^{2}{\frac {d\varphi }{d\eta }}\right)+(\varphi ^{2}-C)^{3/2}=0}
where
C
=
1
/
y
o
2
{\displaystyle C=1/y_{o}^{2}}
. In other words, once the above equation is solved the density is given by
ρ
=
B
y
o
3
(
φ
2
−
1
y
o
2
)
3
/
2
.
{\displaystyle \rho =By_{o}^{3}\left(\varphi ^{2}-{\frac {1}{y_{o}^{2}}}\right)^{3/2}.}
The mass interior to a specified point can then be calculated
M
(
η
)
=
−
4
π
B
2
(
2
A
π
G
)
3
/
2
η
2
d
φ
d
η
.
{\displaystyle M(\eta )=-{\frac {4\pi }{B^{2}}}\left({\frac {2A}{\pi G}}\right)^{3/2}\eta ^{2}{\frac {d\varphi }{d\eta }}.}
The radius-mass relation of the white dwarf is usually plotted in the plane
η
∞
{\displaystyle \eta _{\infty }}
-
M
(
η
∞
)
{\displaystyle M(\eta _{\infty })}
.
Solution near the origin
edit
In the neighborhood of the origin,
η
≪
1
{\displaystyle \eta \ll 1}
, Chandrasekhar provided an asymptotic expansion as
φ
=
1
−
q
3
6
η
2
+
q
4
40
η
4
−
q
5
(
5
q
2
+
14
)
7
!
η
6
+
q
6
(
339
q
2
+
280
)
3
×
9
!
η
8
−
q
7
(
1425
q
4
+
11346
q
2
+
4256
)
5
×
11
!
η
10
+
⋯
{\displaystyle {\begin{aligned}\varphi ={}&1-{\frac {q^{3}}{6}}\eta ^{2}+{\frac {q^{4}}{40}}\eta ^{4}-{\frac {q^{5}(5q^{2}+14)}{7!}}\eta ^{6}\\[6pt]&{}+{\frac {q^{6}(339q^{2}+280)}{3\times 9!}}\eta ^{8}-{\frac {q^{7}(1425q^{4}+11346q^{2}+4256)}{5\times 11!}}\eta ^{10}+\cdots \end{aligned}}}
where
q
2
=
C
−
1
{\displaystyle q^{2}=C-1}
. He also provided numerical solutions for the range
C
=
0.01
−
0.8
{\displaystyle C=0.01-0.8}
.
Equation for small central densities
edit
When the central density
ρ
o
=
B
x
o
3
=
B
(
y
o
2
−
1
)
3
/
2
{\displaystyle \rho _{o}=Bx_{o}^{3}=B(y_{o}^{2}-1)^{3/2}}
is small, the equation can be reduced to a Lane–Emden equation by introducing
ξ
=
2
η
,
θ
=
φ
2
−
C
=
φ
2
−
1
+
x
o
2
+
O
(
x
o
4
)
{\displaystyle \xi ={\sqrt {2}}\eta ,\qquad \theta =\varphi ^{2}-C=\varphi ^{2}-1+x_{o}^{2}+O(x_{o}^{4})}
to obtain at leading order, the following equation
1
ξ
2
d
d
ξ
(
ξ
2
d
θ
d
ξ
)
=
−
θ
3
/
2
{\displaystyle {\frac {1}{\xi ^{2}}}{\frac {d}{d\xi }}\left(\xi ^{2}{\frac {d\theta }{d\xi }}\right)=-\theta ^{3/2}}
subjected to the conditions
θ
(
0
)
=
x
o
2
{\displaystyle \theta (0)=x_{o}^{2}}
and
θ
′
(
0
)
=
0
{\displaystyle \theta '(0)=0}
. Note that although the equation reduces to the Lane–Emden equation with polytropic index
3
/
2
{\displaystyle 3/2}
, the initial condition is not that of the Lane–Emden equation.
Limiting mass for large central densities
edit
When the central density becomes large, i.e.,
y
o
→
∞
{\displaystyle y_{o}\rightarrow \infty }
or equivalently
C
→
0
{\displaystyle C\rightarrow 0}
, the governing equation reduces to
1
η
2
d
d
η
(
η
2
d
φ
d
η
)
=
−
φ
3
{\displaystyle {\frac {1}{\eta ^{2}}}{\frac {d}{d\eta }}\left(\eta ^{2}{\frac {d\varphi }{d\eta }}\right)=-\varphi ^{3}}
subjected to the conditions
φ
(
0
)
=
1
{\displaystyle \varphi (0)=1}
and
φ
′
(
0
)
=
0
{\displaystyle \varphi '(0)=0}
. This is exactly the Lane–Emden equation with polytropic index
3
{\displaystyle 3}
. Note that in this limit of large densities, the radius
r
=
(
2
A
π
G
B
2
)
1
/
2
η
y
o
{\displaystyle r=\left({\frac {2A}{\pi GB^{2}}}\right)^{1/2}{\frac {\eta }{y_{o}}}}
tends to zero. The mass of the white dwarf however tends to a finite limit
M
→
−
4
π
B
2
(
2
A
π
G
)
3
/
2
(
η
2
d
φ
d
η
)
η
=
η
∞
=
5.75
μ
e
−
2
M
⊙
.
{\displaystyle M\rightarrow -{\frac {4\pi }{B^{2}}}\left({\frac {2A}{\pi G}}\right)^{3/2}\left(\eta ^{2}{\frac {d\varphi }{d\eta }}\right)_{\eta =\eta _{\infty }}=5.75\mu _{e}^{-2}M_{\odot }.}
The Chandrasekhar limit follows from this limit.
See also
edit
References
edit