WOLFRAM NOTEBOOK

​

WDs: Lead in

CopyRight@Zheng SHEN
E-mail: kantopenhauer@whu.edu.cn
In[]:=
Clear["Global`*"]
Basic Formulae

Newtonian Structure Equations

p
x
[r]
r
=-
Gϵ
x
[r]
M
x
[r]
2
c
2
r
​​
M
x
[r]
r
=4π
ϵ
x
[r]
2
c

EoS under Fermi Gas Model of Degenerated Electrons

Energy Density

n=
3
k
F
3
2
π
3
ℏ
​​
ϵ
electron
=
4
m
e
5
c
2
π
3
ℏ
k
F
m
e
c
∫
0
1/2
(
2
u
+1)
2
u
u
​​​
ϵ
total
=
ϵ
electron
+
n
m
N
2
c
A
Z
=
ϵ
electron
+
m
N
A
Z
3
m
e
5
c
3
2
π
3
ℏ
3
k
F
3
m
e
3
c

[Energy density is dominated by baryon rest mass term]

it can be shown, for WDs composed of degenerated electrons and baryons, the total energy density is always dominated my the rest mass term
n
m
N
2
c
A
Z
​(1)the factor in front of the integral in
ϵ
electron
is about
1.4*10^23J/
3
m
, while the integral itself is obviously dimensionless
In[]:=
Module{m=9.1093837015*
-31
10
,c=3*
8
10
,ℏ=1.054517871*
-34
10
},
4
m
5
c
2
π
3
ℏ
Out[]=
1.44577×
23
10
(2) the integral take values as plotted if upper limit of integral set from 0 to 2 (when taking upper limit 2, the Fermi momentum is sufficiently relativistic
In[]:=
Plot
v
∫
0
1/2
(
2
u
+1)
2
u
u,{v,0,2}
Out[]=
(3)while the scale of
m
N
3
m
e
3
c
3
2
π
3
ℏ
(the factor with dimension of energy density) is about
8.8×
25
10
J/
3
m
, is over 100 times larger than degenerated electrons’ contribution
In[]:=
Module{m=9.1093837015*
-31
10
,c=3*
8
10
,ℏ=1.054517871*
-34
10
,M=1.67*
-27
10
},M
3
m
5
c
3
2
π
3
ℏ
Out[]=
8.83501×
25
10

[#Domination? A More Precise Evaluation]

The portion of degenerated-electron energy density in total energy density, i.e.
​
ϵ
electron
ϵ
electron
+
ϵ
baryon
when
k
F
m
e
c
takes value from 0 to 100
In[]:=
Module{m=9.1093837015*
-31
10
,c=3*
8
10
,ℏ=1.054517871*
-34
10
,M=1.67*
-27
10
,A=56,Z=26},Plot​​
4
m
5
c
2
π
3
ℏ
v
∫
0
1/2
(
2
u
+1)
2
u
u
MA
3
m
5
c
3Z
2
π
3
ℏ
3
v
+
4
m
5
c
2
π
3
ℏ
v
∫
0
1/2
(
2
u
+1)
2
u
u
​​,{v,0,10000}
Out[]=
In[]:=
Module{m=9.1093837015*
-31
10
,c=3*
8
10
,ℏ=1.054517871*
-34
10
,M=1.67*
-27
10
,A=56,Z=26},Plot​​
4
m
5
c
2
π
3
ℏ
v
∫
0
1/2
(
2
u
+1)
2
u
u
MA
3
m
5
c
3Z
2
π
3
ℏ
3
v
+
4
m
5
c
2
π
3
ℏ
v
∫
0
1/2
(
2
u
+1)
2
u
u
​​,{v,0,1000}
Out[]=
Thus,
ϵ
electron
is relevantly small, compared with the contribution of rest mass of baryons, even when Fermi momentum takes extremely relativistic values.But we shall still be aware that when
k
F
m
e
c
takes values over
3
10
this approximation will be no longer practical.

Pressure

p=
4
m
e
5
c
3
2
π
3
ℏ
k
F
m
e
c
∫
0
-1/2
(
2
u
+1)
4
u
u

[ParametricPlot: a Panoramic on p~ϵ Relation]

Integral p and ϵ numerically with Fermi momentum
k
F
as parameter, we can estimate the relation between p and ϵ under the limit
0<
k
F
m
e
c
<2
; by observation the curve has a shape like
p~
4/3
ϵ
In[]:=
Module{m=9.1093837015*
-31
10
,c=3*
8
10
,ℏ=1.054517871*
-34
10
,M=1.67*
-27
10
,Z=26,A=56},ParametricPlotM
A
Z
3
m
5
c
3
2
π
3
ℏ
3
v
,
4
m
5
c
3
2
π
3
ℏ
v
∫
0
-1/2
(
2
u
+1)
4
u
u,{v,0,2}
Out[]=
Polytrope EoS

Extreme Relativistic (
k
F
>>
m
e
)

Pressure:[
p
rel
≈
c
4
k
F
12
2
π
3
ℏ
=
4
m
e
5
c
12
2
π
3
ℏ
4
k
F
m
e
c
]

Dimensionless Equations

[Failed Attempt]

Dimensionless Equations

Numerical Integration

the result of R is unexpectedly small, that is because the choice of initial condition of pressure p is not realistic

[[another attempt]]

[[Another Attempt: Try Something New]]

For WDs, once EoS has been determined, pressure~radius and mass~radius relation depend only on the choice of initial condition (central pressure). Thus in this subsubsection I use Plot3D function to show the pressure~(radius, central pressure) and mass~(radius, central pressure) dependence graphically.
*The results are beyond my expectation ( I didn’t expect MMA can do this) but still primitive. Since the results are of no importance in our research into NS, the adjustment of arguments is postponed until we deal with NS models.
​
**what inspired me to use Plot3D is that peaks of some diagram might have some relationship with the behavior of system under disturbance, and this might be represented more explicitly on 3D diagrams than 2Ds.

Dimensionless Equations

Dimensionless Equations

Numerical Integration

[realistic initial condition]

Rel vs NR Approximations: Conclusions

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