Changes between Version 3 and Version 4 of u/erica/CoreCollapseBlog


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Timestamp:
03/15/18 18:18:34 (7 years ago)
Author:
Erica Kaminski
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  • u/erica/CoreCollapseBlog

    v3 v4  
     1'''3/15/18'''
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     6'''3/5/18'''
     7
     8The HSE module takes as input:
     9
     10        -Pout (ambient pressure that the HSE module integrates inward from
     11
     12        -Starting density profile that the HSE module interpolates between and uses when solving the discretized HSE equation in spherical coords
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     15
     16'''2/26/18'''
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     18The Lane Emden equation non-dimensionalizes the HSE equation in an attempt to find analytic solutions. However, analytic solutions only exist for certain values of the polytropic index. Thus, if we are looking for numerical solutions, we can instead just start with the equation of HSE in spherical coordinates and discretize it:
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     20$P_{i+1} = P_i - \frac{G M_i,{enc}}{r_i^2} \Delta r_i \rho_i $
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     22The order of accuracy of this discretization depends on the way M_enc is calculated (see Jonathans description on the wiki). Since there might be some undesirable error depending on how M_enc is calculated, might be worthwhile to see how Liu’s approximate analytic solution for a gamma=5/3 sphere compares to the numerically integrated solution found by this method.
     23
     24What this discretization gives us then is (for an arbitrary gamma law EOS), the pressure at each zone of a spherical mass distribution in HSE. However, note this method requires a boundary condition (the starting pressure, $P_i$) and the density profile ($\rho_i$). As written, the algorithm takes the starting pressure to be the external pressure, and using the density profile integrates this equation inwards to the center of the sphere. Eventually then, the inner pressure should match the one from the profiles Chris sent me, if that thing is in or near HSE as he says it is.
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     26Note, if we were to chose to integrate in the opposite direction then we would instead have to define a cut-off radius to the integration, which would yield the outer pressure of a sphere in HSE given a particular density profile and specified inner pressure boundary condition.
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     28The algorithm that performs the above described operation in AstroBEAR is called the ‘HSE self-gravity profiles object’. This object interpolates (using cubic spline) the density profile of the polytrope for arbitrary resolution given a finite, discrete set of density profile points. Using this interpolated function, the pressure at each cell center is then calculated using the above function (M_enc is calculated inside this object).
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     30
    131'''2/7/18'''
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    333Fryer sent progenitor data. The density, pressure, temperature, and velocity profiles of this progenitor look like:
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    5 [[Image()]]
     35[[Image(progenitor_profiles.png, 100%)]]
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    737The goal is to get these profiles initialized on an Eulerian grid in astrobear and to make sure the solution can be numerically stable over multiple dynamical times. The question of stability is interesting and something we will turn to next.
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    9 The pressure and density profiles of a polytropic sphere in hydrostatic equilibrium (HSE) are given by the Lane-Emden equation:
     39The pressure and density profiles of a polytropic sphere in hydrostatic equilibrium (HSE) are given by the Lane-Emden equation. This equation is derived by combining the equation of HSE with Poisson’s equation for gravity and the pressure/density relationship of a polytrope:
    1040
    11 This equation is derived by combining the equation of HSE with Poisson’s equation for gravity
    12 and the pressure/density relationship of a polytrope:
    13 
    14 
     41$P=K\rho^{1+1/n}$
    1542
    1643where  is the polytropic index. Note the polytropic index  is related to the adiabatic index () through the equation:
     
    2047Exact analytic solutions of the Lane-Emden equation exist only for certain values of . These are:
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    22  =  (exact solutions exist)
     49$n = 0,1,5 ~(exact ~solutions~ exist)$
    2350
    24 However, for other values of the Lane-Emden must be numerically integrated.
     51However, for other values of $n$ the Lane-Emden must be numerically integrated.
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    26 In astrobear, I have written a numerical solver for  which corresponds to the Isothermal limit (). In principle, we would be able to initialize an isothermal sphere on the grid using the inner and outer density of the sphere and its radius, but given the density contrast here is >> than the critical value of 14.1, this sphere would be highly gravitationally unstable — I imagine its evolution would be close to that of the singular isothermal sphere.
     53In astrobear, I have written a solver for $n=\infty$ which corresponds to the Isothermal limit ($\gamma=1$). In principle, we would be able to initialize an isothermal sphere on the grid using the inner and outer density of the sphere and its radius, but given the density contrast here is >> than the critical value of 14.1, this sphere would be highly gravitationally unstable — I imagine its evolution would be close to that of the singular isothermal sphere.
    2754
    2855Thus, we need an alternative solution to modeling the progenitor.
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    30 I could easily modify the algorithm I have for setting up a polytropic sphere in HSE for (which corresponds to an adiabatic sphere of ).
     57I could easily modify the algorithm I have for setting up a polytropic sphere in HSE for $n=1.5$ (which corresponds to an adiabatic sphere of $\gamma=5/3$). This would add a new approximate analytic solution to the module for this new value of $n$. However, first will explore using the code's prebuilt routines for numerically integrating the equation of HSE.
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