WOLFRAM NOTEBOOK

Pure NS: Fermi Gas Model

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

Relativistic Structure Equations

[[TOV Equation]]

p[r]
r
=-
GM[r]ϵ[r]
2
c
2
r
1+
p[r]
ϵ[r]
1+
4π
3
r
p[r]
M[r]
2
c
-1
1-
2GM[r]
r
2
c

[[Name Don’t know]]

M[r]
r
=4π
2
r
ϵ[r]
2
c

[[EoS]]

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

Nonrelativistic Case

Pressure

Approximation

[
p
non
≈
5
k
F
15
2
π
3
ℏ
m
n
=
4
m
n
5
c
15
2
π
3
ℏ
5
k
F
m
n
c
]

In[]:=
Module{p},p=
4
m
n
5
c
3
2
π
3
ℏ
k
F
m
n
c
∫
0
4
u
u
In[]:=
5
k
F
15
2
π
3
ℏ
m
n
Out[]=
5
k
F
15
2
π
3
ℏ
m
n

Limits

such approximation can only be made when
k
F
m
n
c
<<1
, since
In[]:=
Plot
v
∫
0
4
u
u
v
∫
0
-1/2
(
2
u
+1)
4
u
u
,{v,0,2},AxesLabel
k
F
m
n
c
,
p
approx
p
real
Power
:Infinite expression
1
0.
encountered.
Out[]=

Energy Density

Approximation

ϵ
non
≈
2
c
3
k
F
m
n
3
2
π
3
ℏ
=
5
c
4
m
n
3
2
π
3
ℏ
3
k
F
m
n
c
]

In[]:=
Module{ϵ},ϵ=
4
m
n
5
c
2
π
3
ℏ
k
F
m
n
c
∫
0
2
u
u
Out[]=
2
c
3
k
F
m
n
3
2
π
3
ℏ

Limits

such approximation can only be made when
k
F
m
n
c
<<1
​and the limit imposed by approximation on ϵ is more restrict than by pressure.
In[]:=
Plot
v
∫
0
2
u
u
v
∫
0
1/2
(
2
u
+1)
2
u
u
,{v,0,2},AxesLabel
k
F
m
n
c
,
ϵ
approx
ϵ
real
Out[]=

NR Polytrope EoS

[[
p
non
=
2
ℏ
15
2
π
m
n
5/3
3
2
π
m
n
2
c
5/3
ϵ
non
=
K
non
5/3
ϵ
non
]]

In[]:=
Module{K,m=1.67*
-27
10
,ℏ=1.06*
-34
10
,c=3*
8
10
},K=
2
ℏ
15
2
π
m
5/3
3
2
π
m
2
c
Out[]=
3.03115×
-25
10

Dimensionless equations

Introduce
ϵ
0
and
R
0
by​
p=
ϵ
0
p
​​
ϵ=
ϵ
0
ϵ
​
p
=
K
γ
ϵ
where
K
=K
γ-1
ϵ
0
so
ϵ
=
1/γ
p
K
=
1/γ
p
γ-1
Kϵ
0
​
R
0
=
G
M
0
2
c
​
M
[r]=
M[r]
M
0
then the polytrope relation can be rewritten into:(1)
p
=
K
γ
ϵ
where
K
=K
γ-1
ϵ
0
​and TOV equation can be written into (simply substitute variables(except for r) with the product of its dimensionless form and dimension factor)​
p
[r]
r
​​=-
G
1/γ
M
0
p
[r]
γ-1
Kϵ
0
M
[r]
2
c
2
r
1+
p
[r]
1/γ
p
γ-1
Kϵ
0
1+
4π
ϵ
0
3
r
p
[r]
M
0
M
[r]
2
c
-1
1-
2G
M
0
M
[r]
r
2
c
​​=-
G
1/γ
M
0
(
p
[r])
M
[r]
2
c
1/γ
2
r
γ-1
Kϵ
0
1+
p
[r]
1/γ
p
γ-1
Kϵ
0
1+
4π
ϵ
0
3
r
p
[r]
M
0
M
[r]
2
c
-1
1-
2G
M
0
M
[r]
r
2
c
​Define Dimension Factor:
α=
R
0
1/γ
K
γ-1
ϵ
0
=
G
M
0
1/γ
2
c
K
γ-1
ϵ
0

[[*typing bars is a waste of time, just remember where dimensional factor α or β exists, all quantities should be considered as its dimensionless form except for r in km]]

it’s very weird that when representing α in the first form, there would be some bugs

Numerical Solution (NR)

[If we use Newtonian Structure Equations]

Relativistic Case

Pressure

Approximation

Limits

Energy Density

Approximation

Limits

NR Polytrope EoS

Dimensionless equations

[[*typing bars is a waste of time, just remember where dimensional factor α or β exists, all quantities should be considered as its dimensionless form except for r in km]]

it’s very weird that when representing α in the first form, there would be some bugs

Numerical Solution (NR)

[First Attempt]

Arbitrary Relativity

Goal

very close to those given by Reddy,w= which are 2.4216 and 2.8663, we’ll use Reddy’s values in later discussion

Rewrite TOV

[[Compare with Newtonian Equation]]

​
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