Tolman-Oppenheimer-Volkoff equation

[Orion1]

I attempted to derive the TOV equation in modern physics notation, however my equation solution does not seem to match the equation solution derived by Tolman, Oppenheimer and Volkoff. (ref.1)

Also, the equation solution listed in (ref. 1) does not match the equation solution listed on Wikipedia, which listed (ref. 1) as the source of the equation. The TOV equation listed in (ref. 1) does not contain the 'mass function' listed on the Wikipedia page (ref. 2).

The (ref. 1) paper describes how the TOV equation was derived:

\tag{3} 8 \pi P(r) = e^{- \lambda} \left( \frac{1}{r} \frac{d \nu}{dr} + \frac{1}{r^2} \right) - \frac{1}{r^2}
\tag{8} e^{-\lambda} = r(r - 2u)
\tag{5} \frac{d \nu}{dr} = \frac{2}{P(r) + \rho(r) c^2} \left( \frac{dP}{dr} \right)

In Eq. (3) replace e^{- \lambda} by its value from (8) and \nu ' by its value from (5). It becomes:

Solve for: \frac{dP}{dr}
\tag{10} \frac{dP}{dr} = - (P(r) + \rho(r) c^2) [4 \pi r^3 P(r) + u] [r(r - 2u)]^{-1}

And here is my first attempt to derive the TOV equation:
8 \pi P(r) = e^{- \lambda} \left(\frac{1}{r} \frac{d \nu}{dr} + \frac{1}{r^2} \right) - \frac{1}{r^2}

Identity:
e^{-\lambda} = r(r - 2u) = 1 - \frac{2u}{r} = r(r - r_s) = 1 - \frac{r_s}{r}
\boxed{u = \frac{r_s}{2}}
\boxed{e^{-\lambda} = r(r - r_s)}
u(r) = \frac{1}{2} r(1 - e^{-\lambda})
e^{-\lambda} = r(r - 2u)
\frac{d \nu}{dr} = \frac{2}{P(r) + \rho(r) c^2} \left( \frac{dP}{dr} \right)
---
This is my first attempt to derive this equation.
Integration by substitution:
8 \pi P(r) = - ( r (r - 2u) ) ( ( \frac{2}{P(r) + \rho (r) c^2} ) ( \frac{dP}{dr} ) \frac{1}{r} + \frac{1}{r^2} ) - \frac{1}{r^2}

My equation solution:
\boxed{\frac{dP}{dr} = -(P(r) + \rho(r) c^2) \frac{r}{2} [(8 \pi P(r) + \frac{1}{r^2})[r(r - 2u)]^{-1} - \frac{1}{r^2}]}

TOV equation solution: ref. 1
\frac{dP}{dr} = - (P(r) + \rho(r) c^2) [4 \pi r^3 P(r) + u] [r(r - 2u)]^{-1}

TOV equation solution: ref. 2
\frac{dP}{dr} = - \frac{G}{r^2} [\rho(r) + \frac{P(r)}{c^2}][m(r) + 4 \pi r^3 \frac{P(r)}{c^2}][r(r - r_s)]^{-1}

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Reference:
http://home.comcast.net/~lambo1826/download/PHRVAO_55_4_374_1.pdf"
http://en.wikipedia.org/wiki/Tolman-Oppenheimer-Volkoff_equation"

 

[Orion1]

If the the stated equation solutions are equivalent, then the mass term in ref. 2 must have originated form the first 'u' term:

(P(r) + \rho(r) c^2) [4 \pi r^3 P(r) + u] = \frac{G}{r^2} [\rho(r) + \frac{P(r)}{c^2}][m(r) + 4 \pi r^3 \frac{P(r)}{c^2}]

Factoring out c^2 from the LHS results in:
c^2 (\frac{P(r)}{c^2} + \rho(r)) [4 \pi r^3 \frac{P(r)}{c^2} + \frac{u}{c^2}] = \frac{G}{r^2} [\rho(r) + \frac{P(r)}{c^2}][m(r) + 4 \pi r^3 \frac{P(r)}{c^2}]

Eliminate terms:
c^2 [4 \pi r^3 \frac{P(r)}{c^2} + \frac{u}{c^2}] = \frac{G}{r^2} [m(r) + 4 \pi r^3 \frac{P(r)}{c^2}]

Solve for u:
4 \pi r^3 \frac{P(r)}{c^2} + \frac{u}{c^2} = \frac{G}{c^2 r^2} [m(r) + 4 \pi r^3 \frac{P(r)}{c^2}]

\frac{u}{c^2} = \frac{G}{c^2 r^2} [m(r) + 4 \pi r^3 \frac{P(r)}{c^2}] - 4 \pi r^3 \frac{P(r)}{c^2}

u = c^2 [(\frac{G m(r)}{c^2 r^2} + 4 \pi G r^3 \frac{P(r)}{r^2 c^4}) - 4 \pi r^3 \frac{P(r)}{c^2}]

u = \frac{G m(r)}{r^2} + 4 \pi G r^3 \frac{P(r)}{r^2 c^2} - 4 \pi r^3 P(r)

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[Orion1]

\boxed{u = \frac{G m(r)}{r^2} + 4 \pi r^3 P(r) \left(1 - \frac{G}{r^2 c^2} \right)}
Is this equation solution correct?
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[Astronuc]

Still working through it, but equation 8 should be

\tag{8} e^{-\lambda} = r^{-1}(r - 2u)

so check the equations after that.

 

[Orion1]

I have not located a Mathematica command that can factor identities:
1 - \frac{2u}{r} = r^{-1} (r - 2u)

\tag{8} \boxed{e^{-\lambda} = r^{-1}(r - 2u)}
Affirmative that is correct.

e^{-\lambda} is a variable in the Schwarzschild metric and I calculate it has dimensionless SI units.

Identity:
\boxed{e^{-\lambda} = r^{-1} (r - 2u) = 1 - \frac{2u}{r} = r^{-1}(r - r_s) = 1 - \frac{r_s}{r}}

Integration by substitution:
8 \pi P(r) = - [ r^{-1} (r - 2u) ] \left[ \left( \frac{2}{P(r) + \rho (r) c^2} \right) \left( \frac{dP}{dr} \right) \frac{1}{r} + \frac{1}{r^2} \right] - \frac{1}{r^2}

My equation solution:
\boxed{\frac{dP}{dr} = -(P(r) + \rho(r) c^2) \frac{r}{2} \left[ \left(8 \pi P(r) + \frac{1}{r^2} \right) r [r - 2u]^{-1} - \frac{1}{r^2} \right]}

Mathematic 6 solution: (at this point)
\boxed{\frac{dP}{dr} = -(P(r) + \rho(r) c^2)(r + 4 \pi r^3 P(r) - u)[r(r - 2u)]^{-1}}

TOV equation solution: ref. 1
\frac{dP}{dr} = - (P(r) + \rho(r) c^2) (4 \pi r^3 P(r) + u) [r(r - 2u)]^{-1}

TOV equation solution: ref. 2
\frac{dP}{dr} = - \frac{G}{r^2} \left(\rho(r) + \frac{P(r)}{c^2} \right) \left(m(r) + 4 \pi r^3 \frac{P(r)}{c^2} \right) [r(r - r_s)]^{-1}

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Reference:
http://home.comcast.net/~lambo1826/download/PHRVAO_55_4_374_1.pdf"
http://en.wikipedia.org/wiki/Tolman-Oppenheimer-Volkoff_equation"
http://en.wikipedia.org/wiki/SI"
http://en.wikipedia.org/wiki/Schwarzschild_metric"

 

[Orion1]

Identity?:
\frac{r}{2} \left[ \left(8 \pi P(r) + \frac{1}{r^2} \right) r [r - 2u]^{-1} - \frac{1}{r^2} \right]} = (r + 4 \pi r^3 P(r) - u)[r(r - 2u)]^{-1}}

Mathematica 6 confirms identity as True.

My solution for u:
\boxed{u = \frac{G m(r)}{r^2} + 4 \pi r^3 P(r) \left(1 - \frac{G}{r^2 c^2} \right)}

Mathematica 6 solution for u: (ref. 1 = ref. 2)
u = \frac{4 c^4 \pi P(r) r^5+c^4 r^3-4 G \pi P(r) r^3-c^2 G m(r)}{c^4 r^2}

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[Orion1]

My solution for u:
u = \frac{G m(r)}{c^2 r^2} + \frac{4 \pi G r P(r)}{c^4} - 4 \pi r^3 P(r)
\boxed{u = \frac{G m(r)}{c^2 r^2} + 4 \pi r^3 P(r) \left( 1 - \frac{G}{r^2 c^4} \right)}

Mathematica 6 solution for u: (ref. 1 = ref. 2)
\boxed{u = \frac{4 c^4 \pi P(r) r^5+c^4 r^3-4 G \pi P(r) r^3-c^2 G m(r)}{c^4 r^2}}
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[Orion1]

Mathematic 6 solution: (at this point)
\boxed{\frac{dP}{dr} = -(P(r) + \rho(r) c^2)(r + 4 \pi r^3 P(r) - u)[r(r - 2u)]^{-1}}

u = \frac{r_s}{2} = \frac{G m(r)}{c^2}
\boxed{u = \frac{G m(r)}{c^2}}

Mathematic 6 solution: (at this point)
\boxed{\frac{dP}{dr} = -(P(r) + \rho(r) c^2) \left(r + 4 \pi r^3 P(r) - \frac{G m(r)}{c^2} \right) \left[ r \left( r - \frac{2G m(r)}{c^2} \right) \right]^{-1}}
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[Orion1]

Integration by substitution:
8 \pi P(r) = [ r^{-1} (r - 2u) ] \left[ \left( \frac{2}{P(r) + \rho (r) c^2} \right) \left( \frac{dP}{dr} \right) \frac{1}{r} + \frac{1}{r^2} \right] - \frac{1}{r^2}

My equation solution:
\boxed{\frac{dP}{dr} = -(P(r) + \rho(r) c^2) \frac{r}{2} \left[ \left(8 \pi P(r) + \frac{1}{r^2} \right) r [r - 2u]^{-1} - \frac{1}{r^2} \right]}

My identity:
\boxed{\frac{r}{2} \left[ \left(8 \pi P(r) + \frac{1}{r^2} \right) r [r - 2u]^{-1} - \frac{1}{r^2} \right] = (4 \pi r^3 P(r) + u)[r(r - 2u)]^{-1}}}

Mathematic 6 solution:
\boxed{\frac{dP}{dr} = -(P(r) + \rho(r) c^2)(4 \pi r^3 P(r) + u)[r(r - 2u)]^{-1}}

u = \frac{r_s}{2} = \frac{G m(r)}{c^2}
\boxed{u = \frac{G m(r)}{c^2}}

\boxed{\frac{dP}{dr} = -(P(r) + \rho(r) c^2) \left(4 \pi r^3 P(r) + \frac{G m(r)}{c^2} \right) \left[ r \left( r - \frac{2G m(r)}{c^2} \right) \right]^{-1}}
TOV equation solution: ref. 1
\frac{dP}{dr} = - (P(r) + \rho(r) c^2) (4 \pi r^3 P(r) + u) [r(r - 2u)]^{-1}

Unresolved issues at this point:
Mathematica 6 functionally confirms 'my identity' as True, however symbolic proof unresolved.
TOV equation solution: ref. 2 listed in Wikipedia is incorrect.
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[Orion1]

Symbolic proof of 'my identity' is now available:
https://www.physicsforums.com/showthread.php?p=1692563#post1692563"
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[Orion1]

TOV equation solution: ref. 2 listed in Wikipedia is incorrect.

\frac{dP}{dr} = - \frac{G}{r^2} \left( \rho(r) + \frac{P(r)}{c^2} \right) \left(m(r) + 4 \pi r^3 \frac{P(r)}{c^2} \right) \left( \frac{1}{r(r - r_s)} \right)

\frac{dF}{dL^2 \cdot dL} = \frac{dF \cdot dL^2}{dm^2 \cdot dL^2} \left( \frac{dm}{dL^3} + \frac{dF \cdot dt^2}{dL^2 \cdot dL^2} \right) \left(dm + \frac{dL^3 \cdot dF \cdot dt^2}{dL^2 \cdot dL^2} \right) \frac{1}{dL^2}

\frac{dF}{dL^3} = \frac{dF}{dm^2} \left( \frac{dm}{dL^3} + \frac{dF \cdot dt^2}{dL^4} \right) \left(dm + \frac{dF \cdot dt^2}{dL} \right) \frac{1}{dL^2}

\frac{dF}{dL^3} = \frac{dF \cdot dm}{dm^2 \cdot dL^2} \left( \frac{dm}{dL^3} \right)

\boxed{\frac{dF}{dL^3} \neq \frac{dF}{dL^5}}
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Reference:
http://en.wikipedia.org/wiki/Tolman-Oppenheimer-Volkoff_equation"

 

[Anony-mouse]

interesting...

 

[Orion1]

Mathematic 6 solution:
\frac{dP}{dr} = -(P(r) + \rho(r) c^2) \left(4 \pi r^3 P(r) + \frac{G m(r)}{c^2} \right) \left[ r \left( r - \frac{2G m(r)}{c^2} \right) \right]^{-1}

\frac{dF}{dL^2 \cdot dL} = \left( \frac{dF}{dL^2} + \frac{dm \cdot dL^2}{dL^3 \cdot dt^2} \right) \left( dL^3 \cdot \frac{dF}{dL^2} + \frac{dF \cdot dL^2 \cdot dm \cdot dt^2}{dm^2 \cdot dL^2} \right) \frac{1}{dL^2}

\frac{dF}{dL^3} = \left( \frac{dF}{dL^2} \right) \left(dF \cdot dL + \frac{dF \cdot dt^2}{dm} \right) \frac{1}{dL^2}


\frac{dF}{dL^3} = \frac{dF}{dL^4} \left( dF \cdot dL \right)

Unknown extra force derivative:
\boxed{\frac{dF}{dL^3} \neq \frac{dF^2}{dL^3}}

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[Astronuc]

Well there is a problem right here

\frac{dP}{dr} = - (P(r) + \rho(r) c^2) (4 \pi r^3 P(r) + u) [r(r - 2u)]^{-1}


specifically

(4 \pi r^3 P(r) + u)

because u is dimensionless per equation (8) of Oppenheimer-Volkoff paper.

and r³P(r) has units of energy! Pressure = F/L² = FL/L³ = energy density, where F = force = ML/T², and Energy = Force*L.


In the Oppenheimer-Volkoff paper, I believe there is an error in equation (1), actually an omission.

ds² = . . . . + e^{\nu}dt^2.

I believe it should be

ds² = . . . . + e^{\nu}\,c^2\,dt^2 so that it is dimensionally correct. That is more or less the form in the Wikipedia article on TOV.


Also, in the Oppenheimer-Volkoff paper.


Also, one must be careful between the Wikipedia article and the original OV paper.

In the OV paper, in the text following OV equation (2), it states that \rho(r) is the macroscopic energy density, and not the mass density, although energy density is related to mass density * c². So this [(P(r) + \rho(r) c^2)] could be problematic.

Also is [r(r - 2u)]^{-1} correct? Remember \tag{8} e^{-\lambda} = r^{-1}(r - 2u)


One has to be careful of units, consistency of terms and errors or omissions in the literature!

 

[Orion1]

The [r(r - 2u)]^{-1} term originates algebraically from 'my identity' equation which is solved symbolically in reference. 1 link below and by Mathematica 6:

\boxed{\frac{r}{2} \left[ \left(8 \pi P(r) + \frac{1}{r^2} \right) r [r - 2u]^{-1} - \frac{1}{r^2} \right] = (4 \pi r^3 P(r) + u)[r(r - 2u)]^{-1}}}

\tag{8} e^{-\lambda} = r^{-1}(r - 2u)

'My identity':
\frac{r}{2} \left[ \left(8 \pi P(r) + \frac{1}{r^2} \right) e^{\lambda} - \frac{1}{r^2} \right] = \frac{4 \pi r^3 P(r) + u}{r(r - 2u)}

Mathematica 6 solution for e^{-\lambda} based upon 'my identity':
e^{-\lambda} = \frac{r - 2u}{r}
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Reference:
https://www.physicsforums.com/showthread.php?p=1692563#post1692563"

 

[Orion1]

The overall SI terminology of of the TOV equation should be:
\frac{dF}{dL^3} = \left( \frac{dF}{dL^2} \right)(dL) \left( \frac{1}{dL^2} \right)

\boxed{u = dL}

Making the SI correction, the equation solution becomes:
\frac{dP}{dr} = -(P(r) + \rho(r) c^2) \left( \frac{4 \pi r^3 P(r)}{dF} + \frac{G m(r)}{c^2} \right) \left[ r \left( r - \frac{2G m(r)}{c^2} \right) \right]^{-1}

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[Orion1]

The overall SI terminology of of the Wikipedia TOV equation should be:
\frac{dF}{dL^3} = \left( \frac{dF \cdot dL^2}{dm^2} \right) \left( \frac{dm}{dL^3} \right)(dm) \left( \frac{1}{dL^2} \right)

The only known dimensionally functional solution for the Wikipedia TOV equation:
\boxed{\frac{dP}{dr} = - G \left( \rho(r) + \frac{P(r)}{c^2} \right) \left(m(r) + 4 \pi r^3 \frac{P(r)}{c^2} \right) \left( \frac{1}{r(r - r_s)} \right)}

TOV = TOV Wikipedia dimensional identity:
(P(r) + \rho(r) c^2) \left( \frac{4 \pi r^3 P(r)}{dF} + \frac{G m(r)}{c^2} \right) = G \left( \rho(r) + \frac{P(r)}{c^2} \right) \left(m(r) + 4 \pi r^3 \frac{P(r)}{c^2} \right)

Mathematica 6 solution for dF:
\boxed{dF = \frac{c^4}{G}}

The TOV equation should be:
\boxed{\frac{dP}{dr} = -(P(r) + \rho(r) c^2) \left( \frac{4 \pi G r^3 P(r)}{c^4} + \frac{G m(r)}{c^2} \right) \left[ r \left( r - \frac{2G m(r)}{c^2} \right) \right]^{-1}}

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[Orion1]

The TOV equation solution for a Neutron Star:
\boxed{u = \frac{r_s}{2}}

\boxed{\frac{dP}{dr} = -(P(r) + \rho(r) c^2) \left( \frac{4 \pi G r^3 P(r)}{c^4} + \frac{r_s}{2} \right) \left[ r \left( r - r_s \right) \right]^{-1}} \; \; \; (r > r_p) \; \; \; r \neq r_p

Integration by substitution:
\boxed{\frac{dP}{dr} = -(P(r) + \rho(r) c^2) \left( \frac{4 \pi G r^3 P(r)}{c^4} + \frac{G m(r)}{c^2} \right) \left[ r \left( r - \frac{2G m(r)}{c^2} \right) \right]^{-1}} \; \; \; (r > r_s) \; \; \; r \neq r_s

The TOV equation solution for a Black Hole:
\boxed{r_s = r_p}
\boxed{u = \frac{r_p}{2}}

\boxed{\frac{dP}{dr} = -(P(r) + \rho(r) c^2) \left( \frac{4 \pi G r^3 P(r)}{c^4} + \frac{r_p}{2} \right) \left[ r \left( r - r_p \right) \right]^{-1}} \; \; \; (r > r_p) \; \; \; r \neq r_p

Integration by substitution:
\boxed{\frac{dP}{dr} = -(P(r) + \rho(r) c^2) \left( \frac{4 \pi G r^3 P(r)}{c^4} + \frac{1}{2} \sqrt{\frac{\hbar G}{c^3}} \right) \left[ r \left( r - \sqrt{\frac{\hbar G}{c^3}} \right) \right]^{-1}} \; \; \; (r > r_p) \; \; \; r \neq r_p
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[Astronuc]

I haven't read through the details, but it looks good.

Reflecting on my previous post, the OV paper mentions just before equation (18) that Eqs. (3), (4) and (5) from which (16) and (17) are derived are stated in "relativistic units" which apparently Tolman used. In relativistic units, c = 1, so obviously c² = 1, and G = 1. So those factors do not show in the equations in the OV paper.

I now suspect that the c² is in equation 1, but has value 1, so it's not explicitly written. I never like systems that use c = 1, because while they might look nicer, it's easy to make a mistake in derivations.

The Wikipedia article apparently uses SI, so c² is explicitly used with the mass density, but then \rho is mass density, not energy density.

 

[Orion1]

\boxed{\frac{dP}{dr} = -(P(r) + \rho(r) c^2) \left( \frac{4 \pi G r^3 P(r)}{c^4} + \frac{1}{2} \sqrt{\frac{\hbar G}{c^3}} \right) \left[ r \left( r - \sqrt{\frac{\hbar G}{c^3}} \right) \right]^{-1}} \; \; \; (r > r_p) \; \; \; r \neq r_p

Planck Sphere surface pressure:
P_p = \frac{c^7}{4 \pi \hbar G^2}

Planck Sphere density:
\rho_p = \frac{3c^5}{4 \pi \hbar G^2}

Integration by substitution:
\frac{dP}{dr} = - \left[ \frac{c^7}{4 \pi \hbar G^2} + \left( \frac{3c^5}{4 \pi \hbar G^2} \right) c^2 \right] \left( \frac{4 \pi G r^3}{c^4} \left( \frac{c^7}{4 \pi \hbar G^2} \right) + \frac{1}{2} \sqrt{\frac{\hbar G}{c^3}} \right) \left[ r \left( r - \sqrt{\frac{\hbar G}{c^3}} \right) \right]^{-1} \; \; \; (r > r_p) \; \; \; r \neq r_p

The TOV equation solution for a Planck singularity:
\frac{dP}{dr} = - \left[ \frac{c^7}{4 \pi \hbar G^2} + \frac{3c^7}{4 \pi \hbar G^2} \right] \left( \frac{c^3 r^3}{\hbar G} + \frac{1}{2} \sqrt{\frac{\hbar G}{c^3}} \right) \left[ r \left( r - \sqrt{\frac{\hbar G}{c^3}} \right) \right]^{-1} \; \; \; (r > r_p) \; \; \; r \neq r_p

\boxed{\frac{dP}{dr} = - \frac{c^7}{\pi \hbar G^2} \left( \frac{c^3 r^3}{\hbar G} + \frac{1}{2} \sqrt{\frac{\hbar G}{c^3}} \right) \left[ r \left( r - \sqrt{\frac{\hbar G}{c^3}} \right) \right]^{-1}} \; \; \; (r > r_p) \; \; \; r \neq r_p

These equations predict 2 explosion types:
When a Neutron Star collapses into a Black Hole.
When a Black Hole collapses into a Planck singularity.
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In post #18 eq. 2, the limiting values should be: (30 min. PF edit limit)
r > r_s \; \; \; r \neq r_s
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Reference:
http://en.wikipedia.org/wiki/Planck_pressure"
http://en.wikipedia.org/wiki/Planck_density"

 

[Orion1]

According to Wikipedia (ref. 1, para. 2 listed below),

the pressure exerted by degenerate matter depends only weakly on its temperature. In particular, the pressure remains nonzero even at absolute zero temperature.

Adiabatic indexes:
Low pressure fully degenerate Fermi gas:
\gamma = \frac{5}{3}

High density quantum state relativistic degenerate Fermi gas:
\gamma = \frac{4}{3}

Polytropic degenerate Fermi gas pressure equation:
P(r) = K \rho(r)^{\gamma}

K - particle gas properties constant

K = \frac{P(r)}{\rho(r)^{\gamma}} = \left( \frac{dF}{dL^2} \right) \cdot \left( \frac{dL^3}{dm} \right) = \frac{dF \cdot dL}{dm}
\boxed{K = \frac{dF \cdot dL}{dm}}

TOV equation solution:
\boxed{\frac{dP}{dr} = -(P(r) + \rho(r) c^2) \left( \frac{4 \pi G r^3 P(r)}{c^4} + \frac{G m(r)}{c^2} \right) \left[ r \left( r - \frac{2G m(r)}{c^2} \right) \right]^{-1}}

Polytropic degenerate Fermi gas pressure equation:
P(r) = K \rho(r)^{\gamma}

Integration by substitution:
\frac{dP}{dr} = -(K \rho(r)^{\gamma} + \rho(r) c^2) \left( \frac{4 \pi G r^3 (K \rho(r)^{\gamma})}{c^4} + \frac{G m(r)}{c^2} \right) \left[ r \left( r - \frac{2G m(r)}{c^2} \right) \right]^{-1}}

Polytropic degenerate Fermi gas pressure TOV equation:
\boxed{\frac{dP}{dr} = -(K \rho(r)^{\gamma} + \rho(r) c^2) \left( \frac{4 \pi G K r^3 \rho(r)^{\gamma}}{c^4} + \frac{G m(r)}{c^2} \right) \left[ r \left( r - \frac{2G m(r)}{c^2} \right) \right]^{-1}}}

Is this equation solution correct?
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The Adiabatic indexes for a degenerate Fermi gas are listed in Wikipedia (ref. 1), however not yet listed on Wikipedia (ref. 2 ) below.
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Reference:
http://en.wikipedia.org/wiki/Degenerate_matter"
http://en.wikipedia.org/wiki/Heat_capacity_ratio"

 

[Orion1]

According to PHYS 390 Lecture 19, the degenerate Fermi gas pressure of spin 1/2 particles is:
P(r) = \frac{\pi^3 \hbar^2}{15 m_n} \left( \frac{3N(r)}{\pi V(r)} \right)^{\frac{5}{3}}

Number density:
n(r) = \frac{N(r)}{V(r)} = \frac{\rho(r)}{m_n}
m_n - neutron mass

Integration by substitution:
P(r) = \frac{\pi^3 \hbar^2}{15 m_n} \left[ \frac{3}{\pi} \left( \frac{\rho(r)}{m_n} \right)\right]^{\gamma} = \frac{\pi^3 \hbar^2}{15 m_n} \left( \frac{3}{\pi m_n} \right)^{\gamma} \rho(r)^{\gamma} = K \rho(r)^{\gamma}

\boxed{P(r) = \frac{\pi^3 \hbar^2}{15 m_n} \left( \frac{3}{\pi m_n} \right)^{\gamma} \rho(r)^{\gamma}}

\boxed{K = \frac{\pi^3 \hbar^2}{15 m_n} \left( \frac{3}{\pi m_n} \right)^{\gamma}}

However, I failed to resolve the SI derivation for the pressure equation:
\frac{dF}{dL^2} = \left( \frac{dE^2 dt^2}{dm} \right) \left( \frac{1}{dm} \right) \left( \frac{dm}{dL^3} \right) = \left( \frac{dE^2 dt^2}{dm \cdot dL^3} \right)
[/Color]
Reference:
http://www.sfu.ca/~boal/385lecs/385lec19.pdf"

 

[Orion1]

SI derivation for the PHYS 390 Lecture 19 pressure equation:
\frac{dF}{dL^2} = \left( \frac{dE^2 dt^2}{dm \cdot dL^5} \right) = \left( \frac{dF^2 dL^2 dt^2}{dm \cdot dL^5} \right)

\frac{dF}{dL^2} = \frac{dF^2 dt^2}{dm \cdot dL^3}

Newton's second law:
\boxed{dF = \frac{dm \cdot dL}{dt^2}}

According to this solution, the PHYS 390 Lecture 19 pressure equation is correct:

P(r) = \frac{\pi^3 \hbar^2}{15 m_n} \left( \frac{3N(r)}{\pi V(r)} \right)^{\frac{5}{3}} \; \; \; \gamma = \frac{5}{3}
[/Color]
Reference:
http://en.wikipedia.org/wiki/Force"
http://www.sfu.ca/~boal/385lecs/385lec18.pdf"
http://www.sfu.ca/~boal/385lecs/385lec19.pdf"

 

[Orion1]

According to my SI derivations, the PHYS 390 Lecture 19 pressure equation is only valid for an Adiabatic index of \gamma = \frac{5}{3}
\frac{dF}{dL^2} \neq \left( \frac{dE^2 dt^2}{dm} \right) \left( \frac{1}{dm} \right) \left( \frac{dm}{dL^3} \right) = \left( \frac{dE^2 dt^2}{dm \cdot dL^3} \right) \; \; \; \gamma = 1

\frac{dF}{dL^2} = \left( \frac{dE^2 dt^2}{dm \cdot dL^{3 \gamma}} \right) = \left( \frac{dF^2 dL^2 dt^2}{dm \cdot dL^5} \right)\; \; \; \gamma = \frac{5}{3}

\frac{dF}{dL^2} \neq \left( \frac{dE^2 dt^2}{dm \cdot dL^{3 \gamma}} \right) = \left( \frac{dF^2 dL^2 dt^2}{dm \cdot dL^4} \right)\; \; \; \gamma = \frac{4}{3}
[/Color]

 

[Orion1]

According to Wikipedia ref. 1 listed below, the equation for Fermi energy is:

Fermi energy equation:
E_f = \frac{\hbar^2}{2m_n} \left( \frac{3 \pi^2 N}{V} \right)^{2/3}

Fermi pressure:
P_f = - \frac{dE_f}{dV_f}

Integration by substitution:
P_f = - \frac{\hbar^2}{2m_n} \left( \frac{3 \pi^2 N_f}{V_f} \right)^{2/3} \frac{1}{V_f} = - \frac{\hbar^2}{2m_n} \left( 3 \pi^2 N \right)^{2/3} \frac{1}{V_f^{\frac{5}{3}}} = K \rho_f^{\frac{2}{3}}

\boxed{P_f = - \frac{\hbar^2}{2m_n} \left( 3 \pi^2 N_f \right)^{2/3} \frac{1}{V_f^{\frac{5}{3}}}}
[/Color]
Reference:
http://en.wikipedia.org/wiki/Fermi_energy"

 

[Helios]

In the polytropic equation of state. the constant K is called " the entropy constant " in some literature. You have called it " the particle gas properties constant".

 

[Orion1]

According to Wikipedia ref. 1, para. 3 listed below:

...where K depends on the properties of the particles making up the gas.

where K' again depends on the properties of the particles making up the gas.

Greetings, Helios
Thanks for your important collaboration and correction.

That constant was copied from the Wikipedia ref.1, para. 3 paragraph which does not implicitly declare a key definition which states the SI name for that constant, however it does describe it twice in the same paragraph!

The Wikipedia ref. 1, still requires a lot of work with better, more implicitly written definition keys and definitional equations with more extensive referencing.

According to Wikipedia ref. 2, section 'Entropy in Astrophysics':

In astrophysics, what is referred to as "entropy" is actually the adiabatic constant derived as follows.

PV^{\gamma} = \text{constant} = K

This equation is known as an expression for the adiabatic constant, K, also called the adiabat.

K - adiabatic constant, K, also called the adiabat.

P = \frac{\rho k_{B}T}{\mu m_{H}}

\boxed{K = \frac{k_{B}T}{\mu m_{H} \rho^{2/3}}}
Is the adiabatic constant also called the 'entropy constant' in other literature?

Substituting this into the above equation along with V = [grams] / ρ and γ = 5 / 3 for an ideal monoatomic gas

PV^{\gamma} = K

\gamma = \frac{5}{3} - ideal monoatomic gas

P = \frac{\rho k_{B}T}{\mu m_{H}}

V = \frac{dm}{\rho}

Integration by substitution:
PV^{\gamma} = K

K = \left( \frac{\rho k_{B}T}{\mu m_{H}} \right) \left( \frac{dm}{\rho} \right)^{\gamma} = \left( \frac{k_{B} dm^{\gamma}}{\mu m_{H}} \right) T \rho^{1 - \gamma}

dm = m_H

Identities:
\frac{\rho}{\rho^{\gamma}} = \rho^{1 - \gamma}
\frac{m_H^{\gamma}}{m_H} = m_H^{\gamma - 1}

The adiabatic constant:
\boxed{K = \left( \frac{k_{B} dm_H^{\gamma - 1} }{\mu} \right) T \rho^{1 - \gamma}}
[/Color]
Reference:
http://en.wikipedia.org/wiki/Degenerate_matter"
http://en.wikipedia.org/wiki/Entropy"

 

[Orion1]

The adiabatic constant:
\boxed{K = \left( \frac{k_{B} dm_H^{\gamma - 1} }{\mu} \right) T \rho^{1 - \gamma}}

Standard SI derivation:
K = \frac{dE \cdot dm^{\gamma - 1}}{dT} \cdot \frac{amn}{amu} \cdot dT \left( \frac{dm}{dL^3} \right)^{1 - \gamma} = \frac{dE}{dL^{3(1 - \gamma)}} \cdot \frac{amn}{amu}

\boxed{K = \frac{dE}{dL^{3(1 - \gamma)}} \cdot \frac{amn}{amu}}
[/Color]

 

[Helios]

Yes, an adiabat is a curve of constant entropy on the P-V diagram. The adiabats are isentropic and the " entropy constant " is the value for one adiabat.

 

[Orion1]

According to Wikipedia ref. 1 listed below, the equation for Fermi energy is:

Fermi energy equation:
E_f = \frac{\hbar^2}{2m_n} \left( \frac{3 \pi^2 N}{V} \right)^{2/3}

Fermi pressure:
P_f = - \frac{dE_f}{dV_f}

Integration by differentiation substitution:
P_f = - \frac{\hbar^2}{2m_n} \left( 3 \pi^2 N_f \right)^{2/3} \left( \frac{V^{- \frac{2}{3}}}{dV} \right)

Differentiation identity:
\frac{V^{-n}}{dV} = -n V^{-n - 1} = \left(-\frac{2}{3} \right) V^{-\frac{2}{3} - \frac{3}{3}} = \left(- \frac{2}{3} \right) V^{-\frac{5}{3}}

P_f = - \left(- \frac{2}{3} \right) \frac{\hbar^2}{2m_n} \left( 3 \pi^2 N \right)^{2/3} \frac{1}{V_f^{\frac{5}{3}}} = \frac{2 \hbar^2}{6 m_n} \left( 3 \pi^2 N \right)^{2/3} \frac{1}{V_f^{\frac{5}{3}}}

\boxed{P_f = \frac{\hbar^2}{3 m_n} \left( 3 \pi^2 N \right)^{2/3} \frac{1}{V_f^{\frac{5}{3}}}}
[/Color]

Post #25 requires integration by differentiation substitution.
[/Color]
Reference:
http://en.wikipedia.org/wiki/Fermi_energy"
http://www.sfu.ca/~boal/385lecs/385lec19.pdf"

 

[Orion1]

My first attempt to integrate the Adiabatic constant and Adiabatic index with degenerate Fermi pressure.

Adiabatic constant:
K = P V^{\gamma}

Degenerate Fermi pressure:
P = - \frac{dE}{dV}

Degenerate Fermi energy:
E_f = \frac{\pi^3 \hbar^2}{10 m_n} \left( \frac{3 N}{\pi} \right)^{\frac{5}{3}} \frac{1}{V^{\frac{2}{3}}}

Adiabatic constant:
K = \left( - \frac{dE}{dV} \right) V^{\gamma} = - \frac{\pi^3 \hbar^2}{10 m_n} \left( \frac{3 N}{\pi} \right)^{\frac{5}{3}} \left( \frac{V^{-\frac{2}{3}}}{dV} \right) V^{\gamma}

Differentiation identity:
\frac{V^{-n}}{dV} = -n V^{-n - 1} = \left(-\frac{2}{3} \right) V^{-\frac{2}{3} - \frac{3}{3}} = \left(- \frac{2}{3} \right) V^{-\frac{5}{3}}

Integration by differentiation substitution:
K = - \left( - \frac{2}{3} \right) \frac{\pi^3 \hbar^2}{10 m_n} \left( \frac{3 N}{\pi} \right)^{\frac{5}{3}} \frac{V^{\gamma}}{V^{\frac{5}{3}}} = \left( \frac{2 \cdot 3 \cdot 3^{\frac{2}{3}}}{30} \right) \frac{\pi^{ \left( \frac{9}{3} - \frac{5}{3} \right)} \hbar^2}{m_n} \left( \frac{N}{V} \right)^{\frac{5}{3}} V^{\gamma} = \frac{3^{\frac{2}{3}} \pi^{\frac{4}{3}} \hbar^2}{5 m_n} \left( \frac{N}{V} \right)^{\frac{5}{3}} V^{\gamma}

Number density:
n(r) = \frac{N(r)}{V(r)} = \frac{\rho(r)}{m_n}

Adiabatic volume:
V^{\gamma} = \left( \frac{m_n}{\rho(r)} \right)^{\gamma}

K = \frac{3^{\frac{2}{3}} \pi^{\frac{4}{3}} \hbar^2}{5 m_n} \left( \frac{\rho(r)}{m_n} \right)^{\frac{5}{3}} \left( \frac{m_n}{\rho(r)} \right)^{\gamma} = \left( \frac{3^{\frac{2}{3}} \pi^{\frac{4}{3}} \hbar^2}{5} \right) m_n^{\gamma - \frac{5}{3} + \frac{3}{3}} \rho(r)^{\frac{5}{3} - \gamma} = \left( \frac{3^{\frac{2}{3}} \pi^{\frac{4}{3}} \hbar^2}{5} \right) m_n^{\gamma - \frac{8}{3}} \rho(r)^{\frac{5}{3} - \gamma}

Degenerate adiabatic constant with adiabatic index:
\boxed{K = \left( \frac{3^{\frac{2}{3}} \pi^{\frac{4}{3}} \hbar^2}{5} \right) m_n^{\gamma - \frac{8}{3}} \rho(r)^{\frac{5}{3} - \gamma}}

Is this equation solution correct?

Standard SI derivation:
K = \frac{dE^2 \cdot dt^2}{dm \cdot dL^5} = \frac{dF^2 \cdot dL^2 \cdot dt^2}{dm \cdot dL^5} = \frac{dF^2 \cdot dt^2}{dm \cdot dL^3} = \frac{dF}{dL^2}

\frac{dF}{dL^2} = \frac{dF^2 \cdot dt^2}{dm \cdot dL^3}

Newtons second law:
\boxed{dF = \frac{dm \cdot dL}{dt^2}}

Adiabatic constant SI units:
\boxed{K = \frac{dF}{dL^2}}

The SI adiabat is a unit of pressure:
1 \; \text{adiabat} = \frac{1 \; \text{Newton}}{1 \; \text{meter}^2}
[/Color]
Reference:
http://en.wikipedia.org/wiki/Force"
http://en.wikipedia.org/wiki/Entropy"
http://en.wikipedia.org/wiki/Fermi_energy"
http://en.wikipedia.org/wiki/Heat_capacity_ratio"
http://en.wikipedia.org/wiki/Degenerate_matter"
http://www.sfu.ca/~boal/385lecs/385lec18.pdf"
http://www.sfu.ca/~boal/385lecs/385lec19.pdf"

 

[Orion1]

TOV equation solution:
\frac{dP}{dr} = -(P(r) + \rho(r) c^2) \left( \frac{4 \pi G r^3 P(r)}{c^4} + \frac{G m(r)}{c^2} \right) \left[ r \left( r - \frac{2G m(r)}{c^2} \right) \right]^{-1}

Degenerate Fermi pressure:
P(r) = \frac{3^{2/3} \pi^{4} \hbar^2 \rho (r)^{5/3}}{5 m_n^{8/3}}

Integration by substitution:
\frac{dP}{dr} = - \left( \frac{3^{2/3} \pi^{4/3} \hbar^2 \rho (r)^{5/3}}{5 m_n^{8/3}} + \rho(r) c^2 \right) \left[ \frac{4 \pi G r^3}{c^4} \left( \frac{3^{2/3} \pi^{4/3} \hbar^2 \rho (r)^{5/3}}{5 m_n^{8/3}} \right) + \frac{G m(r)}{c^2} \right] \left[ r \left(r - \frac{2G m(r)}{c^2} \right) \right]^{-1}

Degenerate Fermi-TOV equation solution:
\boxed{ \frac{dP}{dr} = - \left( \frac{3^{2/3} \pi^{4/3} \hbar^2 \rho (r)^{5/3}}{5 m_n^{8/3}} + \rho(r) c^2 \right) \left[ \left( \frac{4 \cdot 3^{2/3} \pi^{7/3} \hbar^2 G r^3 \rho (r)^{5/3}}{5 c^4 m_n^{8/3}} \right) + \frac{G m(r)}{c^2} \right] \left[ r \left( r - \frac{2G m(r)}{c^2} \right) \right]^{-1} }

Tolman VII density solution:
\rho(r) = \rho_c \left[1 - \left( \frac{r}{R} \right)^2 \right] \; \; \; \rho(R) = 0

Integration by substitution:
\frac{dP}{dr} = - \left[ \frac{3^{2/3} \pi^{4/3} \hbar^2}{5 m_n^{8/3}} \left( \rho_c \left[1 - \left( \frac{r}{R} \right)^2 \right] \right)^{5/3} + \left( \rho_c \left[1 - \left( \frac{r}{R} \right)^2 \right] \right) c^2 \right] \left[ \left( \frac{4 \cdot 3^{2/3} \pi^{7/3} \hbar^2 G r^3}{5 c^4 m_n^{8/3}} \left( \rho_c \left[1 - \left( \frac{r}{R} \right)^2 \right] \right)^{5/3} \right) + \frac{G m(r)}{c^2} \right] \left[ r \left( r - \frac{2G m(r)}{c^2} \right) \right]^{-1}

Degenerate Fermi-TOV equation solution VII:
\boxed{\frac{dP}{dr} = - \left[ \frac{3^{2/3} \pi^{4/3} \hbar^2}{5 m_n^{8/3}} \left( \rho_c \left[1 - \left( \frac{r}{R} \right)^2 \right] \right)^{5/3} + \rho_c c^2 \left[1 - \left( \frac{r}{R} \right)^2 \right] \right] \left[ \left( \frac{4 \cdot 3^{2/3} \pi^{7/3} \hbar^2 G r^3}{5 c^4 m_n^{8/3}} \left( \rho_c \left[1 - \left( \frac{r}{R} \right)^2 \right] \right)^{5/3} \right) + \frac{G m(r)}{c^2} \right] \left[ r \left( r - \frac{2G m(r)}{c^2} \right) \right]^{-1}}

[/Color]

 

[Orion1]

Tolman VII density equation solution:
\boxed{\rho(r) = \rho_c \left[1 - \left( \frac{r}{R} \right)^2 \right] \; \; \; \rho(R) = 0}

Mass integration:
m(r) = 4 \pi \int_r^R r^2 \rho(r) dr

m(r) = 4 \pi \rho_c \int_r^R r^2 \left[ 1 - \left( \frac{r}{R} \right)^2 \right] dr = 4 \pi \rho_c \left( \frac{r^5}{5R^2} + \frac{2R^3}{15} - \frac{r^3}{3} \right) = \frac{4 \pi \rho_c}{15} \left( \frac{3r^5}{R^2} + 2R^3 - 5r^3 \right)

Tolman VII mass equation solution:
\boxed{m(r) = \frac{4 \pi \rho_c}{15} \left( \frac{3r^5}{R^2} + 2R^3 - 5r^3 \right) \; \; \; m(R) = 0}

Integration by substitution:
Degenerate Fermi-TOV equation solution VII:
\frac{dP}{dr} = - \left[ \frac{3^{2/3} \pi^{4/3} \hbar^2}{5 m_n^{8/3}} \left( \rho_c \left[1 - \left( \frac{r}{R} \right)^2 \right] \right)^{5/3} + \rho_c c^2 \left[1 - \left( \frac{r}{R} \right)^2 \right] \right]...

...\left[ \left( \frac{4 \cdot 3^{2/3} \pi^{7/3} \hbar^2 G r^3}{5 c^4 m_n^{8/3}} \left( \rho_c \left[1 - \left( \frac{r}{R} \right)^2 \right] \right)^{5/3} \right) + \frac{G}{c^2} \left( \frac{4 \pi \rho_c}{15} \left( \frac{3r^5}{R^2} + 2R^3 - 5r^3 \right) \right) \right]...

... \left[ r \left( r - \frac{2G}{c^2} \left( \frac{4 \pi \rho_c}{15} \left( \frac{3r^5}{R^2} + 2R^3 - 5r^3 \right) \right) \right) \right]^{-1}


[/Color]

 

[Orion1]

Degenerate Fermi-TOV equation solution VII:

\frac{dP}{dr} = - \left[ \frac{3^{2/3} \pi^{4/3} \hbar^2}{5 m_n^{8/3}} \left( \rho_c \left[1 - \left( \frac{r}{R} \right)^2 \right] \right)^{5/3} + \rho_c c^2 \left[1 - \left( \frac{r}{R} \right)^2 \right] \right]...

...\left[ \left( \frac{4 \cdot 3^{2/3} \pi^{7/3} \hbar^2 G r^3}{5 c^4 m_n^{8/3}} \left( \rho_c \left[1 - \left( \frac{r}{R} \right)^2 \right] \right)^{5/3} \right) + \frac{4 \pi G \rho_c}{15 c^2} \left( \frac{3r^5}{R^2} + 2R^3 - 5r^3 \right) \right]...

... \left[ r \left( r - \frac{4 \pi G \rho_c}{15 c^2} \left( \frac{3r^5}{R^2} + 2R^3 - 5r^3 \right) \right) \right]^{-1}

key:
\rho_c - neutron star core density
R - neutron star radius

Reference:
http://en.wikipedia.org/wiki/Neutron_star"
[/Color]

 

[Orion1]

Neutron average density:
\rho_n = \frac{m_n}{V_n} = \frac{3 m_n}{4 \pi r_n^3}

Neutron radius:
r_n = r_0 A_n^{\frac{1}{3}}
r_0 = 1.25 \cdot 10^{-15} \; \text{m}

Neutron atomic mass unit mass:
A_n = 10^3 N_A m_n
N_A - Avogadro's number

Integration by substitution:
\rho_n = \frac{3 m_n}{4 \pi ( r_0 A_n^{\frac{1}{3}} )^3} = \frac{3 m_n}{4 \pi A_n r_0^3} = \frac{3 m_n}{4 \cdot 10^3 \pi N_A m_n r_0^3}

Neutron density:
\boxed{\rho_n = \frac{3 m_n}{4 \cdot 10^3 \pi N_A m_n r_0^3}}

\boxed{\rho_n = 2.0297 \cdot 10^{17} \; \frac{\text{kg}}{\text{m}^3}}

Neutron star density function parameters:
\rho(r) \; \propto \; \int_{\text{1E9}}^{\text{8E17}} \; \frac{\text{kg}}{\text{m}^3}

Neutron star average density parameters:
\rho_t \; \propto \; \int_{\text{8.4E16}}^{\text{1E18}} \; \frac{\text{kg}}{\text{m}^3}

Neutron average density:
\boxed{\rho_n = 2.0297 \cdot 10^{17} \; \frac{\text{kg}}{\text{m}^3}}
[/Color]
Reference:
http://en.wikipedia.org/wiki/Neutron"
http://en.wikipedia.org/wiki/Nuclear_size"
"[URL number - Wikipedia[/URL]
http://en.wikipedia.org/wiki/Neutron_star"

 

[Orion1]

Neutron average density:
\boxed{\rho_n = \frac{3}{4 \cdot 10^3 \pi N_A r_0^3}}

Neutron star core density:
\boxed{\rho_c = \rho_n}

Integration by substitution:
\frac{dP}{dr} = - \left[ \frac{3^{2/3} \pi^{4/3} \hbar^2}{5 m_n^{8/3}} \left( \left( \frac{3}{4 \cdot 10^3 \pi N_A r_0^3} \right) \left[1 - \left( \frac{r}{R} \right)^2 \right] \right)^{5/3} + \left( \frac{3}{4 \cdot 10^3 \pi N_A r_0^3} \right) c^2 \left[1 - \left( \frac{r}{R} \right)^2 \right] \right]...

...\left[ \left( \frac{4 \cdot 3^{2/3} \pi^{7/3} \hbar^2 G r^3}{5 c^4 m_n^{8/3}} \left( \left( \frac{3}{4 \cdot 10^3 \pi N_A r_0^3} \right) \left[1 - \left( \frac{r}{R} \right)^2 \right] \right)^{5/3} \right) + \frac{4 \pi G }{15 c^2} \left( \frac{3}{4 \cdot 10^3 \pi N_A r_0^3} \right) \left( \frac{3r^5}{R^2} + 2R^3 - 5r^3 \right) \right]...

... \left[ r \left( r - \frac{4 \pi G}{15 c^2} \left( \frac{3}{4 \cdot 10^3 \pi N_A r_0^3} \right)} \left( \frac{3r^5}{R^2} + 2R^3 - 5r^3 \right) \right) \right]^{-1}

Degenerate Fermi-TOV equation solution VII:
\frac{dP}{dr} = - \left[ \frac{9 \hbar^2 }{4 \cdot 10^6 m_n^{\frac{8}{3}}} \left( \frac{3}{2 \pi} \right)^{\frac{1}{3}} \left( \frac{1}{N_A r_0^3} \right)^{\frac{5}{3}} \left[1 - \left( \frac{r}{R} \right)^2 \right]^{5/3} + \frac{3 c^2}{4 \cdot 10^3 \pi N_A r_0^3} \left[1 - \left( \frac{r}{R} \right)^2 \right] \right]...

...\left[ \frac{9 \pi^{\frac{2}{3}} \hbar^2 G r^3}{10^6 c^4 m_n^{\frac{8}{3}}} \left( \frac{3}{2} \right)^{\frac{1}{3}} \left( \frac{1}{N_A r_0^3} \right)^{\frac{5}{3}} \left[1 - \left( \frac{r}{R} \right)^2 \right]^{5/3} + \frac{G}{5 \cdot 10^3 c^2 N_A r_0^3} \left( \frac{3r^5}{R^2} + 2R^3 - 5r^3 \right) \right]...

... \left[ r \left( r - \frac{G}{5 \cdot 10^3 c^2 N_A r_0^3} \left( \frac{3r^5}{R^2} + 2R^3 - 5r^3 \right) \right) \right]^{-1}

[/Color]

 

[Orion1]

Note: physicsforums only allows for 30 minutes for editing posts.

Corrections:
The third factor on post #34 should read:
... \left[ r \left( r - \frac{8 \pi G \rho_c}{15 c^2} \left( \frac{3r^5}{R^2} + 2R^3 - 5r^3 \right) \right) \right]^{-1}

The boxed solution for Neutron density on post #35 should read:
\boxed{\rho_n = \frac{3}{4 \cdot 10^3 \pi N_A r_0^3}}

The first third factor on post #36 should read:
... \left[ r \left( r - \frac{8 \pi G}{15 c^2} \left( \frac{3}{4 \cdot 10^3 \pi N_A r_0^3} \right)} \left( \frac{3r^5}{R^2} + 2R^3 - 5r^3 \right) \right) \right]^{-1}

The second third factor on post #36 should read:
... \left[ r \left( r - \frac{2 G}{5 \cdot 10^3 c^2 N_A r_0^3} \left( \frac{3r^5}{R^2} + 2R^3 - 5r^3 \right) \right) \right]^{-1}
[/Color]

 

[Orion1]

The attached image is my first attempt to plot the degenerate Fermi-TOV equation solution VII function based upon this model.

These are the results of a 10 km radius static model test.

The function projects semi-asymptoticly until the core radius is reached and then projects a semi-parabolic curve.

Degenerate Fermi-TOV neutron star properties:
Core Radius:
m = 1 - slope
sgn \left( \frac{dP}{dr} \right) = -1 \; \; \; r < r_c - sign
\frac{dP}{dr} = 0 \; \; \; r = r_c
r_c = .3421 \; \text{km}

Shell 1:
m = 1 - slope
sgn \left( \frac{dP}{dr} \right) = 1 \; \; \; r_c < r < r_1 - sign
m = 0 \; \; \; r = r_1 - peak resonance
\frac{dP}{dr} = \text{3.638} \cdot \text{10}^{28} \; \frac{\text{N}}{\text{m}^3} \; \; \; r = r_1
r_1 = 4.8673 \; \text{km}

Shell 2:
m = -1 - slope
sgn \left( \frac{dP}{dr} \right) = 1 \; \; \; r_1 < r < R - sign
r_2 = 10 \; \text{km}

Shell 3: Degenerate Iron crust

Total core pressure:
P_c = \int_0^R \left( \frac{dP}{dr} \right) dr
[/Color]
Reference:
http://en.wikipedia.org/wiki/Slope"
http://en.wikipedia.org/wiki/Sign_function"

 

[Orion1]

Mass integration function equation:
m(r) = 4 \pi \int_r^R r^2 \rho(r) dr

Tolman density equation solution VII:
\rho(r) = \rho_c \left[ 1 - \left( \frac{r}{R} \right)^2 \right] \; \; \; \rho(R) = 0

Integration by substitution:
m(r) = 4 \pi \rho_c \int_r^R r^2 \left[ 1 - \left( \frac{r}{R} \right)^2 \right] dr = 4 \pi \rho_c \left( \frac{r^5}{5R^2} + \frac{2R^3}{15} - \frac{r^3}{3} \right) = \frac{4 \pi \rho_c}{15} \left( \frac{3r^5}{R^2} + 2R^3 - 5r^3 \right)

Tolman mass function equation solution VII:
\boxed{m(r) = \frac{4 \pi \rho_c}{15} \left( \frac{3r^5}{R^2} + 2R^3 - 5r^3 \right) \; \; \; m(R) = 0}

Total mass function integration:
M_0 = 4 \pi \int_0^R r^2 \rho(r) dr

Tolman density equation solution VII:
\rho(r) = \rho_c \left[ 1 - \left( \frac{r}{R} \right)^2 \right] \; \; \; \rho(R) = 0

Integration by substitution:
M_0 = 4 \pi \rho_c \int_0^R r^2 \left[ 1 - \left( \frac{r}{R} \right)^2 \right] = \frac{8 \pi \rho_c R^3}{15}

Total Tolman mass equation solution VII:
\boxed{M_0 = \frac{8 \pi \rho_c R^3}{15}}
[/Color]

Correction:
The declaration equations for slope on post# 38 should be:
sgn ( m ) = -1,0,1 instead of m = -1,0,1
[/Color]

 

[Orion1]

Total Tolman mass equation solution VII:
M_0 = \frac{8 \pi \rho_c R^3}{15}

Neutron average density:
\rho_n = \frac{3}{4 \cdot 10^3 \pi N_A r_0^3}}

Core density:
\rho_c = \rho_n

Integration by substitution:
M_0 = \frac{8 \pi}{15} \left( \frac{3}{4 \cdot 10^3 \pi N_A r_0^3}} \right) R^3 = \frac{1}{2.5 \cdot 10^3 N_A} \left( \frac{R}{r_0} \right)^3

\boxed{M_0 = \frac{1}{2.5 \cdot 10^3 N_A} \left( \frac{R}{r_0} \right)^3}

Static model radius:
R = 10 \; \text{km}

Static model mass:
\boxed{M_0 = 3.4007 \cdot 10^{29} \; \text{kg}}

M_0 = 0.1709 \cdot M_{\odot}
[/Color]
Reference:
http://en.wikipedia.org/wiki/Sun"

 

[Orion1]

The TOV equation can only be numerically integrated using Riemann sum rules.

Total core pressure:
P_c = \int_0^R \left( \frac{dP}{dr} \right) dr

Midpoint Rule numerical integration:
P_c = \int_0^R \left( \frac{dP}{dr} \right) dr = \int_0^R f(r) dr = \sum_{i = 1}^n f(\overline{r_i}) \Delta r \; \; \; \Delta r = \frac{R - 0}{n} \; \; \; \overline{r_i} = \frac{1}{2} (r_{i - 1} + r_i)

\boxed{P_c = \int_0^R f(r) dr = \sum_{i = 1}^n f(\overline{r_i}) \Delta r \; \; \; \Delta r = \frac{R}{n} \; \; \; \overline{r_i} = \frac{1}{2} (r_{i - 1} + r_i)}

Degenerate Fermi-TOV equation solution VII
Static model test results for R = 10 km.
Total core pressure:
\boxed{P_c = -1.8635 \cdot 10^{15} \; \frac{\text{N}}{\tex{m}^2}}
[/Color]
Reference:
http://en.wikipedia.org/wiki/Riemann_sum"

 

[Orion1]

Correction, the numerical solution for core pressure on post # 41 is incorrect as a result of incomplete integration.
The correct solution is:[/Color]

Degenerate Fermi-TOV equation solution VII
Static model test results for R = 10 km.
Total core pressure:
\boxed{P_c = 1.8407 \cdot 10^{36} \; \frac{\text{N}}{\tex{m}^2}}
[/Color]

 

[Orion1]

Calculating the maximum mass of a neutron star.

Neutron star total gravitational mass:
M_G = m(R) = \int_0^R 4 \pi r^2 \rho(r) dr

Schwarzschild metric spherical layer differential volume:
dV = \frac{4 \pi r^2 dr}{ \sqrt{\left[ 1 - \frac{r_s}{r} \right]}}
r_s - Schwarzschild radius

Baryon number density:
n(r) = \frac{N(r)}{V(r)} = \frac{\rho(r)}{m_n}

Neutron star total baryon number:
N_B = \int_0^R n(r) dV = \int_0^R \frac{4 \pi r^2 \rho(r) dr}{ m_n \sqrt{\left[ 1 - \frac{r_s}{r} \right]}}

\boxed{N_B = \int_0^R \frac{4 \pi r^2 \rho(r) dr}{ m_n \sqrt{\left[ 1 - \frac{r_s}{r} \right]}}}

Neutron star total baryonic mass:
M_B = m_n N_B = \int_0^R \frac{4 \pi r^2 \rho(r) dr}{ \sqrt{\left[ 1 - \frac{r_s}{r} \right]}}}

\boxed{M_B = \int_0^R \frac{4 \pi r^2 \rho(r) dr}{ \sqrt{\left[ 1 - \frac{r_s}{r} \right]}}}}

Neutron star total binding energy:
B = (M_B - M_G)c^2
[/Color]
Reference:
http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?1996A%26A...305..871B&data_type=PDF_HIGH&whole_paper=YES&type=PRINTER&filetype=.pdf"
http://en.wikipedia.org/wiki/Schwarzschild_radius"

 

[malawi_glenn]

Be careful on what radius you use for the neutron.

r = r_0*A^(1/3) is an emperical formula for middle mass nuclei, not a 'law'.

It should be better to use the same radius as for the proton, i.e 0.87fm (charge radius of proton, r.m.s)

See also this thread: https://www.physicsforums.com/showthread.php?t=241465

Also, it is quite naive that a neutron star is 100% neutrons.

 

[Orion1]

Proton charge radius...


Proton charge radius:
r_p = \sqrt{\left< r^2 \right>} = \sqrt{-6 \lim_{Q \rightarrow 0} \frac{d}{dQ^2} F(Q)}

ref. 1 Proton charge radius:
r_p = 0.805 ± 0.011 and 0.862 ± 0.012 femtometer (Stein 1995).

Wikipedia Proton charge radius:
r_p = 0.875(7) fm

I found reference for the Proton charge radius, unfortunately Wikipedia does not cite a reference, which radius is more accurate?
[/Color]
Reference:
Stein, B. P. "Physics Update." Physics Today 48, 9, Oct. 1995.
http://scienceworld.wolfram.com/physics/Proton.html"
http://en.wikipedia.org/wiki/Proton"
http://cnr2.kent.edu/~pichowsk/IntroNuc/hw3.pdf"

 

[malawi_glenn]

Particle data group: http://pdg.lbl.gov/2007/tables/bxxx.pdf

But what you really want is the nuclear matter distribution for nucleons, i.e you want to know the 'size' of the 'quark cloud'. In the first approximation, this should be the same as the charge radius distrubution. Since the neutron is overall neutral, the charge radius is not a good measurement (it becomes negative hehe..), so the first approximation of neutron radius is the same the proton radius.

 

[Orion1]

r_0 = 1.25 \cdot 10^{-15} \; \text{m} - empirical nuclear radius

Empirical neutron density:
\rho_n = \frac{3}{4 \cdot 10^3 \pi N_A r_0^3}} = 2.0297 \cdot 10^{17} \; \frac{\text{kg}}{\text{m}^3}

r_p = 0.8757 \cdot 10^{-15} \; \text{m} - Proton charge radius

Proton charge radius neutron density:
\rho_n = \frac{3 m_n}{4 \pi r_p^3} = 5.954 \cdot 10^{17} \; \frac{\text{kg}}{\text{m}^3}

\boxed{\rho_n = 5.954 \cdot 10^{17} \; \frac{\text{kg}}{\text{m}^3}}

Proton charge radius neutron density:
\rho_n = \frac{3 m_n}{4 \pi r_p^3}

Neutron star core density equivalent to Proton charge radius neutron density:
\rho_c = \rho_n

Degenerate Fermi-TOV equation:
\frac{dP}{dr} = - \left[ \frac{3^{2/3} \pi^{4/3} \hbar^2}{5 m_n^{8/3}} \left( \frac{3 m_n}{4 \pi r_p^3} \left[1 - \left( \frac{r}{R} \right)^2 \right] \right)^{5/3} + \left( \frac{3 m_n}{4 \pi r_p^3} \right) c^2 \left[1 - \left( \frac{r}{R} \right)^2 \right] \right]...

...\left[ \left( \frac{4 \cdot 3^{2/3} \pi^{7/3} \hbar^2 G r^3}{5 c^4 m_n^{8/3}} \left( \frac{3 m_n}{4 \pi r_p^3} \left[1 - \left( \frac{r}{R} \right)^2 \right] \right)^{5/3} \right) + \frac{4 \pi G}{15 c^2} \left( \frac{3 m_n}{4 \pi r_p^3} \right) \left( \frac{3r^5}{R^2} + 2R^3 - 5r^3 \right) \right]...

... \left[ r \left( r - \frac{8 \pi G}{15 c^2} \left( \frac{3 m_n}{4 \pi r_p^3} \right) \left( \frac{3r^5}{R^2} + 2R^3 - 5r^3 \right) \right) \right]^{-1}

Degenerate Fermi-TOV equation charge radius equation:
\frac{dP}{dr} = - \left[ \frac{3^{\frac{7}{3}} \hbar^2}{4^{\frac{5}{3}} \pi^{\frac{1}{3}} m_n r_p^5} \left[ 1 - \left( \frac{r}{R} \right)^2 \right]^{5/3} + \frac{3 c^2 m_n}{4 \pi r_p^3} \left[1 - \left( \frac{r}{R} \right)^2 \right] \right]...

...\left[ \left( \frac{3^{7/3} \pi^{4/3} \hbar^2 G r^3}{5 \cdot 4^{\frac{2}{3}} c^4 m_n r_p^5} \left[1 - \left( \frac{r}{R} \right)^2 \right]^{5/3} \right) + \frac{G m_n}{5 c^2 r_p^3} \left( \frac{3r^5}{R^2} + 2R^3 - 5r^3 \right) \right]...

... \left[ r \left( r - \frac{2 G m_n}{5 c^2 r_p^3} \left( \frac{3r^5}{R^2} + 2R^3 - 5r^3 \right) \right) \right]^{-1}

The first attached photo is the static model based upon the empirical neutron radius, the second attachment is the static model based upon the Proton charge radius.

Replacing the empirical neutron density with proton charge radius neutron density in a R = 10 km static model resulted in a peak change in pressure increase by a factor of 8.6. The core radius increased by 15 meters and the first shell radius decreased by 112 meters.

The total core pressure became negative, which I believe is required for a neutron star with a stable core.

Total core pressure:
\boxed{P_c = -1.713 \cdot 10^{36} \; \frac{\text{N}}{\tex{m}^2}}

key:
\rho_c - neutron star core density
R - neutron star radius
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Reference:
http://en.wikipedia.org/wiki/Neutron_star"

 

[malawi_glenn]

Orion1: the 'emperical nuclear radius' is only applicable to medium size nuclei... how many times do I have to tell you?

 

[K.J.Healey]

Maybe I'm misunderstanding this but why would the intra-nucleon distance be related to the "charge radius" of a proton? I guess I could see them being both on the order of ~1fm. But shouldn't it be a force balance between the neutrons with the strong interaction and the gravitic pressure that determines the separation distance?

Seems wrong to use anything from a single atom to describe the core of a neutron star.

 

[Orion1]

The nuclear density of nuclei is described by the empirical nuclear radius, which describes the radii of large nuclei, however it does not accurately describe the radii of light nuclei or nuclear particles, because it is not a natural law. The 'charge radius' of a proton is a natural law and used to describe the radius of a neutron in this model. (ref. 3)

Using the nuclear density properties of a single neutron to describe a neutron star nuclear core density is correct.

The neutron star itself is a balance between positive neutron degeneracy pressures generated from the Pauli exclusion principle, which states that no two identical fermions may occupy the same quantum state simultaneously and indicating repulsion between neutrons, and the negative internal and surface gravitational pressures, calculated from the TOV equation of state generated from Relativity.

Neutrons are the most 'rigid' objects known - their Young modulus (or more accurately, bulk modulus) is 20 orders of magnitude larger than that of diamond.
[/Color]
Reference:
http://en.wikipedia.org/wiki/Pauli_exclusion_principle"
http://en.wikipedia.org/wiki/Neutron_star"
https://www.physicsforums.com/showpost.php?p=1789174&postcount=45"