Pure NS: Fermi Gas Model

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=

ℏ

∫

u

In[]:=

ℏ

Out[]=

ℏ

Limits

such approximation can only be made when

<<1

, since

In[]:=

Plot



∫

u

∫



-1/2

(

+1)

u

,{v,0,2},AxesLabel

approx

real



Power

:Infinite expression

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=

ℏ

∫

u

Out[]=

ℏ

Limits

such approximation can only be made when

<<1

and the limit imposed by approximation on ϵ is more restrict than by pressure.

In[]:=

Plot



∫

u

∫



1/2

(

+1)

u

,{v,0,2},AxesLabel

approx

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

,ℏ=1.06*

-34

,c=3*

},K=

ℏ

5/3



Out[]=

3.03115×

-25

Dimensionless equations

Introduce

and

ϵ=

where

γ-1

1/γ

γ-1

Kϵ

[r]=

M[r]

then the polytrope relation can be rewritten into:(1)

where

γ-1

and TOV equation can be written into (simply substitute variables(except for r) with the product of its dimensionless form and dimension factor)



[r]

r

1/γ

[r]

γ-1

Kϵ

[r]

1/γ

γ-1

Kϵ

4π

[r]

-1

[r]

1/γ

(

[r])

[r]

1/γ



γ-1

Kϵ



[r]

1/γ

γ-1

Kϵ

4π

[r]

-1

[r]

Define Dimension Factor:

α=

1/γ

K

γ-1



1/γ

K

γ-1



[[*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

Pure NS: Fermi Gas Model

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

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
]

Limits

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
]

Limits

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
]]

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]]

Numerical Solution (NR)

[If we use Newtonian Structure Equations]

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]]

Numerical Solution (NR)

[First Attempt]

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]]

Pure NS: Fermi Gas Model

Relativistic Structure Equations

[[TOV Equation]]

p[r]r=-GM[r]ϵ[r]2c2r1+p[r]ϵ[r]1+4π3rp[r]M[r]2c-11-2GM[r]r2c

[[Name Don’t know]]

M[r]r=4π2rϵ[r]2c

[[EoS]]

ϵn=4mn5c2π3ℏkFmnc∫01/2(2u+1)2uu​​p=4mn5c32π3ℏkFmnc∫0-1/2(2u+1)4uu​​

Pressure

Approximation

[pnon≈5kF152π3ℏmn=4mn5c152π3ℏ5kFmnc]

Limits

Energy Density

Approximation

ϵnon≈2c3kFmn32π3ℏ=5c4mn32π3ℏ3kFmnc]

Limits

NR Polytrope EoS

[[ pnon=2ℏ152πmn5/332πmn2c5/3ϵnon=Knon5/3ϵnon ]]

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]]

Numerical Solution (NR)

[If we use Newtonian Structure Equations]

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]]

Numerical Solution (NR)

[First Attempt]

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]]

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

M[r]
r
=4π
2
r
ϵ[r]
2
c

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

[
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
]


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
]

[[
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
]]