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Oppenheimer-Volkov degeneracy

  1. Jul 19, 2004 #1


    Tolman, Oppenheimer, Volkov (1939)

    Degenerate Neutron Star Equasion of State:
    [tex]\inline{ \frac{dP}{dr} = - \frac{Gm \rho}{r^2} \left( 1+ \frac{P}{ \rho c^2} \right) \left( 1+ \frac{4 \pi r^3 P}{mc^2} \right) \left( 1- \frac{2Gm}{c^2r} \right)^{-1}}[/tex]

    Conditions:
    [tex]\frac{dm}{dr} = 4 \pi r^2 \rho[/tex]
    [tex]\rho = \frac{ \epsilon}{c^2}[/tex] - ideal fluid
    [tex]m = m(r) = \frac{ E(r' < r)}{c^2}[/tex]
    [tex]P = P(\rho)[/tex] - strong degenerate matter
    [tex]\frac{dP}{dr} < 0[/tex]

    Integration:
    [tex]\int_{r = 0(P = P_c)}^{r = R(P = 0) \rightarrow R}[/tex]
    [tex]M = m(R) = \frac{E}{c^2}[/tex]
    [tex]\rho = \rho(P) \rightarrow \rho(r)[/tex]
    [tex]\rho(0) = \rho_c[/tex]

    Neutron Stars Configurations:
    [tex]M = M( \rho_c ), R = R( \rho_c ),...[/tex] - stellar mass, radius

    Definite integral dr with respect to r from 0 to R:

    [tex]\inline{ dP = dr \int_{r = 0(P = P_c)}^{r = R(P = 0) \rightarrow R} - \frac{Gm \rho}{r^2} \left( 1+ \frac{P}{ \rho c^2} \right) \left( 1+ \frac{4 \pi r^3 P}{mc^2} \right) \left( 1- \frac{2Gm}{c^2r} \right)^{-1}}[/tex]

    Definite integral dr with respect to r from 0 to infinity:
    [tex]\inline{ dP = dr \int_{r = 0(P = P_c)}^{r = \infty(P = 0)} - \frac{Gm \rho}{r^2} \left( 1+ \frac{P}{ \rho c^2} \right) \left( 1+ \frac{4 \pi r^3 P}{mc^2} \right) \left( 1- \frac{2Gm}{c^2r} \right)^{-1}}[/tex]

    [tex]\inline{ dP = -Gm \left( \frac{P}{c^2p} +1 \right) p \left( \frac{ \left( 32 \pi G^3 m^3 P + c^8 p \right) \log \left( c^2 r - 2Gm \right)}{2c^6Gmp} - \frac{c^2 \log \left( r \right)}{2Gm} + \frac{ 4 \pi c^2 P r^2 + 16 \pi GmPr}{2c^4P} \right)}[/tex]

    Definite integral dP with respect to P from 0 to infinity:
    [tex]\inline{ dr = dP \int_{P = 0}^{P = \infty} - \frac{r^2}{Gm \rho} \left( 1+ \frac{P}{ \rho c^2} \right)^{-1} \left( 1+ \frac{4 \pi r^3 P}{mc^2} \right)^{-1} \left( 1- \frac{2Gm}{c^2r} \right)}[/tex]

    [tex]\inline{ dr = \frac{ \left( 1- \frac{ 2Gm}{ c^2r} \right) r^2 \left( \frac{ c^2p \log \left( 4 \pi P r^3 +c^2 P \right) }{ 4 \pi r^3 -1} - \frac{ c^2 \log \left( P + c^2p \right) p}{ 4 \pi r^3 -1} \right) }{Gmp}}[/tex]

    Does anyone know how to demonstrate and perform the first integration listed above, with given conditions to solve an equasion solution from 0 to R, for the Oppenheimer-Volkov Mass Limit and neutron stars configurations?

    The best two papers I could locate are incomplete and missing some keys and also the solutions to the final steps in performing the correct integrations.

    Note that the Tolman-Oppenheimer-Volkov equasion (TOV) in this paper may contain a mass placement error in the numerator, instead of denominator, listed in the second set of brackets which does not match the equasions listed in previous papers:
    www.jlab.org/HYP2003/talks/Bombaci.pdf

    Equasional mass error?
    [tex]\left(1+ \frac{4 \pi r^3P(r)}{c^2} m(r) \right)[/tex] ???

    According to this paper, The solutions of the TOV equasions depend parametrically on the central density:
    [tex]\rho_c = \rho (r=0)[/tex]

    Boundary conditions:
    [tex]m(r=0)=0[/tex]
    [tex]P(r=R)=P_{sur}[/tex]
    R - stellar radius

    TOV solutions:
    [tex]P = P(r, \rho_c)[/tex]
    [tex]m = m(r, \rho_c)[/tex]

    Note that this paper is still incomplete, and the solutions listed are still only conditional solutions. A complete paper would contain the complete formulated equasion with proofs for the TOV mass limit solution estimate.

    Note that the more massive a neutron star is, the smaller its radius. Therefore certain physical density geometries cannot be effectively applied.

    Reference:
    www.itkp.uni-bonn.de/~metsch/STAR/star_pdf.pdf
    www.ganil.fr/snns/tina/talks/talk_haensel.pdf
    www.jlab.org/HYP2003/talks/Bombaci.pdf

     
    Last edited: Jul 22, 2004
  2. jcsd
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