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Whitworth's (1981) Isothermal Free-Energy Surface
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Initial Temperature Distributions

In an accompanying Wiki page we've discussed in detail (or see the summary page) how to transform back and forth between cgs units and the dimensionless code units that have been adopted by Dominic Marcello in his radiation-hydro simulations of binary mass-transfer. Here we want to probe in more depth what temperature distributions are obtained from the initial polytropic structure once Dominic chooses particular values of the four scaling parameters: \tilde{r}, \tilde{a}, \tilde{g}, and \tilde{c}.

Our derivation of the temperature distribution will center around the following ideas. First, the initial binary model that Dominic obtains from Wes Even's self-consistent-field (SCF) code obeys a polytropic equation of state (EOS), namely,

~P = K_\mathrm{n} \rho^{1+1/n}

with an adopted polytropic index ~n = 3 / 2. Hence, at any point inside either star, the pressure (in code units), Pcode, can be obtained from knowledge of the mass-density (in code units), ρcode, and the polytropic constant, Kcode, via the relation,


[P_\mathrm{total}]_\mathrm{code} = K_\mathrm{code} \rho_\mathrm{code}^{5/3} .

Second, Dominic's models are evolved assuming a more realistic EOS. Specifically, he assumes that the total pressure is given by the expression,

Ptotal = Pgas + Pdeg + Prad,

where mathematical expressions for the ideal gas pressure, Pgas, the electron degeneracy pressure, Pdeg, and the photon radiation pressure, Prad, are provided in an accompanying discussion of analytically prescribed equations of state. (Actually, Dominic is presently ignoring the effects of Pdeg, but because it allows for a more general treatment at some later date, we will assume the more general expression for Ptotal and set Pdeg = 0 near the end of our discussion.)

Various Scalings

Pressure

Now, realizing that pressure has units of energy per unit volume, we conclude that in order to transform between cgs units and code units, Dominic must adopt the relation,


\frac{P_\mathrm{cgs}}{P_\mathrm{code}}

=


\biggl( \frac{m_\mathrm{cgs}}{m_\mathrm{code}} \biggr) \biggl( \frac{\ell_\mathrm{cgs}}{\ell_\mathrm{code}} \biggr)^{-1} \biggl( \frac{t_\mathrm{cgs}}{t_\mathrm{code}} \biggr)^{-2}

 

=


\frac{\Lambda}{ G \bar{\mu}^2} \biggl( \frac{\tilde{g}^3 \tilde{a} }{\tilde{r}^4} \biggr)^{1/2}
\biggl[ \frac{\Lambda}{ c^2 \bar{\mu}^2} \biggl( \frac{\tilde{c}^4 \tilde{g} \tilde{a} }{\tilde{r}^4} \biggr)^{1/2} \biggr]^{-1}
\biggl[ \frac{\Lambda}{ c^3 \bar{\mu}^2} \biggl( \frac{\tilde{c}^6 \tilde{g} \tilde{a} }{\tilde{r}^4} \biggr)^{1/2} \biggr]^{-2}

 

=


\frac{ c^8 a_\mathrm{rad}}{(\Re / \bar{\mu})^4} \biggl( \frac{\tilde{r}^4  }{\tilde{a} \tilde{c}^8} \biggr)

 

=


\frac{ 8\pi^4}{5} \biggl(\frac{m_u}{m_e} \biggr)^4 A_\mathrm{F} \biggl( \frac{\tilde{r}^4 {\bar\mu}^4  }{\tilde{a} \tilde{c}^8} \biggr)

where ~m_e, ~m_u and ~A_\mathrm{F} (the characteristic Fermi pressure) are physical constants defined in our accompanying variables appendix. (Numerical values of these constants can be obtained by scrolling the cursor over the symbols for the constants in this last sentence.) This relation also means that, generally,

\frac{P_\mathrm{cgs}}{A_\mathrm{F}} = \biggl[ \frac{ 8\pi^4}{5} \biggl(\frac{m_u}{m_e} \biggr)^4 \biggl( \frac{\tilde{r}^4 {\bar\mu}^4  }{\tilde{a} \tilde{c}^8} \biggr) \biggr] P_\mathrm{code} ;

and, specifically when Pcgs = Ptotal, we have,


p_\mathrm{total} \equiv \frac{P_\mathrm{total}}{A_\mathrm{F}} = \biggl[ \frac{ 8\pi^4}{5} \biggl(\frac{m_u}{m_e} \biggr)^4 \biggl( \frac{\tilde{r}^4 {\bar\mu}^4  }{\tilde{a} \tilde{c}^8} \biggr) \biggr] K_\mathrm{code} \rho_\mathrm{code}^{5/3} .

Density

In a similar manner we recognize that the density transformation must be governed by the relation ... (express this in terms of χ3 so that it is obvious how to introduce ρcode into the quartic equation, below).


\frac{\rho_\mathrm{cgs}}{\rho_\mathrm{code}}

=


\biggl( \frac{m_\mathrm{cgs}}{m_\mathrm{code}} \biggr) \biggl( \frac{\ell_\mathrm{cgs}}{\ell_\mathrm{code}} \biggr)^{-3}

 

=


\frac{\Lambda}{ G \bar{\mu}^2} \biggl( \frac{\tilde{g}^3 \tilde{a} }{\tilde{r}^4} \biggr)^{1/2}
\biggl[ \frac{\Lambda}{ c^2 \bar{\mu}^2} \biggl( \frac{\tilde{c}^4 \tilde{g} \tilde{a} }{\tilde{r}^4} \biggr)^{1/2} \biggr]^{-3}

 

=


\frac{ c^6 a_\mathrm{rad}}{(\Re / \bar{\mu})^4} \biggl( \frac{\tilde{r}^4  }{\tilde{a} \tilde{c}^6} \biggr)

 

=


\frac{ \pi^4}{5} \biggl(\frac{m_u}{m_e} \biggr)^4 \biggl( \frac{m_e}{m_p} \biggr) \frac{B_\mathrm{F}}{\mu_e} \biggl( \frac{\tilde{r}^4 {\bar\mu}^4  }{\tilde{a} \tilde{c}^6} \biggr)

Hence,

\chi^3 \equiv \frac{\rho_\mathrm{cgs}}{B_\mathrm{F}} = \frac{ \pi^4}{5} \biggl(\frac{m_u}{m_e} \biggr)^4 \biggl( \frac{m_e}{m_p} \biggr) \frac{1}{\mu_e} \biggl( \frac{\tilde{r}^4 {\bar\mu}^4  }{\tilde{a} \tilde{c}^6} \biggr) \rho_\mathrm{code}

Temperature

Also, the temperature scaling can be rewritten as follows.


\frac{T_\mathrm{cgs}}{T_\mathrm{code}}

=


\frac{c^2}{ (\Re/ \bar{\mu})} \biggl(\frac{\tilde{r} }{\tilde{c}^2}\biggr)

 

=


\biggl(\frac{m_u}{m_e} \biggr) T_e \biggl(\frac{\tilde{r} \bar\mu }{\tilde{c}^2}\biggr)

Hence,


\frac{T_\mathrm{cgs}}{T_e} = \biggl(\frac{m_u}{m_e} \biggr) \biggl(\frac{\tilde{r} \bar\mu }{\tilde{c}^2}\biggr)  T_\mathrm{code} .

Total Pressure Relation

In an accompanying page of our Wiki-based H_Book, we show that, when normalized to ~A_\mathrm{F}, the analytic expression for the dimensionless total pressure takes the form,

~p_\mathrm{total} = \biggl(\frac{\mu_e m_p}{\bar{\mu} m_u} \biggr) 8 \chi^3 \frac{T}{T_e} + F(\chi) + 
\frac{8\pi^4}{15} \biggl( \frac{T}{T_e} \biggr)^4

Based on the above-derived relations, this can now be rewritten in terms of the variables used in Dominic's simulations as follows:

\biggl[ \frac{ 8\pi^4}{5} \biggl(\frac{m_u}{m_e} \biggr)^4 \biggl( \frac{\tilde{r}^4 {\bar\mu}^4  }{\tilde{a} \tilde{c}^8} \biggr) \biggr] K_\mathrm{code} \rho_\mathrm{code}^{5/3}

=

\biggl(\frac{\mu_e m_p}{\bar\mu m_u}\biggr) 8 \biggl[\frac{ \pi^4}{5} \biggl(\frac{m_u}{m_e} \biggr)^4 \biggl( \frac{m_e}{m_p} \biggr) \frac{1}{\mu_e} \biggl( \frac{\tilde{r}^4 {\bar\mu}^4  }{\tilde{a} \tilde{c}^6} \biggr) \rho_\mathrm{code} \biggr] \biggl(\frac{m_u}{m_e} \biggr) \biggl(\frac{\tilde{r} \bar\mu }{\tilde{c}^2}\biggr)  T_\mathrm{code}

 

 


+ F(\chi) + \frac{8\pi^4}{15} \biggl[ \biggl(\frac{m_u}{m_e} \biggr) \biggl(\frac{\tilde{r} \bar\mu }{\tilde{c}^2}\biggr)  T_\mathrm{code}\biggr]^4

\Rightarrow ~~~~~~\biggl( \frac{\tilde{r}^4 {\bar\mu}^4  }{\tilde{a} \tilde{c}^8} \biggr) K_\mathrm{code} \rho_\mathrm{code}^{5/3} - \biggl[ \frac{5}{ 8\pi^4} \biggl(\frac{m_e}{m_u} \biggr)^4 \biggr]F(\chi)

=

\biggl(\frac{\mu_e}{\bar\mu}\biggr) \biggl[ \frac{1}{\mu_e} \biggl( \frac{\tilde{r}^4 {\bar\mu}^4  }{\tilde{a} \tilde{c}^6} \biggr) \rho_\mathrm{code} \biggr]  \biggl(\frac{\tilde{r} \bar\mu }{\tilde{c}^2}\biggr)  T_\mathrm{code} + \frac{1}{3} \biggr[ \biggl(\frac{\tilde{r} \bar\mu }{\tilde{c}^2}\biggr)  T_\mathrm{code}\biggr]^4

\Rightarrow ~~~~~~ K_\mathrm{code} \rho_\mathrm{code}^{5/3} - \biggl( \frac{\tilde{a} \tilde{c}^8}{\tilde{r}^4 {\bar\mu}^4  } \biggr)\biggl[ \frac{5}{ 8\pi^4} \biggl(\frac{m_e}{m_u} \biggr)^4 \biggr]F(\chi)

=

  \tilde{r} \rho_\mathrm{code} T_\mathrm{code} + \frac{\tilde{a}}{3} T_\mathrm{code}^4 .

EOS Quartic Solution

We can view this last expression as having the form,


a_4 T_\mathrm{code}^4 + a_1 T_\mathrm{code} - a_0 = 0 ,

where,

a4


\equiv


\frac{\tilde{a}}{3} ,

a1


\equiv


\tilde{r}\rho_\mathrm{code} ,

a0


\equiv


K_\mathrm{code} \rho_\mathrm{code}^{5/3} - \biggl( \frac{\tilde{a} \tilde{c}^8}{\tilde{r}^4 {\bar\mu}^4  } \biggr)\biggl[ \frac{5}{ 8\pi^4} \biggl(\frac{m_e}{m_u} \biggr)^4 \biggr]F(\chi) ,

that is, it is a quartic equation describing the relationship between Tcode and ρcode. Following our accompanying H_Book Wiki discussion, the solution to this quartic equation is,


T_\mathrm{code} = \theta \mathcal{K}(\lambda) ,

where,

θ


\equiv


\biggl[ \frac{a_1}{4 a_4} \biggr]^{1/3}

λ


\equiv


\biggl[ \frac{256~ a_0^3 a_4}{27 a_1^4} \biggr]^{1/3} ,


\mathcal{K}(\phi(\lambda))


\equiv


\phi^{-1/3} \biggl[ (\phi - 1)^{1/2} - 1 \biggr] ,

φ


\equiv


2^{3/2} \biggl[ 1 + (1 + \lambda^3)^{1/2} \biggr]^{1/2}
\biggl\{ \biggl[ 1 + (1 + \lambda^3)^{1/2} \biggr]^{2/3} - \lambda \biggr\}^{-3/2} .


Application to Dominic Marcello's Rad-Hydro Models

Temperature

Currently Dominic is ignoring the effects of electron degeneracy pressure, so in applying the above T(ρ) solution to his models we can set F(χ) = 0 in the definition of a0. Doing this, we find that,

θ

=


\biggl[ \frac{3 \tilde{r}\rho_\mathrm{code}}{4 \tilde{a}} \biggr]^{1/3} ,

λ

=


\biggl[ \biggl(\frac{256~ \tilde{a} }{81~ \tilde{r}^4} \biggr)K_\mathrm{code}^3 \rho_\mathrm{code}  \biggr]^{1/3} .

For purposes of discussion, we will define ρ1 as the value of ρcode when λ = 1, that is,

\rho_1 \equiv  \frac{81~ \tilde{r}^4}{256~ \tilde{a} K_\mathrm{code}^3} .

As the accompanying discussion points out, the limiting behavior of the quartic solution is as follows:

For \rho_\mathrm{code} \ll \rho_1

    \cdots    

T_\mathrm{code} \approx \frac{a_0}{a_1}

=

\biggl( \frac{K_\mathrm{code}}{\tilde{r}} \biggr) \rho_\mathrm{code}^{2/3}

For \rho_\mathrm{code} \gg \rho_1

    \cdots    

T_\mathrm{code} \approx \biggl( \frac{a_0}{a_4} \biggr)^{1/4}

=


\biggl[ \biggl( \frac{3K_\mathrm{code}}{\tilde{a}} \biggr)\rho_\mathrm{code}^{5/3} \biggr]^{1/4}

Now let's plug in numerical values for the two stars in Dominic's Case A, q0 = 0.7 model evolution, as drawn from the accompanying properties table.

Case A:   \tilde{g} = 1; \tilde{c} = 198; \tilde{r} = 0.44; \tilde{a} = 0.044; \bar\mu = 4/3;

Star

Kcode

θ

λ

ρ1

Tcode (for \rho_\mathrm{code} \ll \rho_1)

Prad / Pgas (for \rho_\mathrm{code} \ll \rho_1)

Accretor

0.2571

1.957 \rho_\mathrm{code}^{1/3}

0.3980\rho_\mathrm{code}^{1/3}

15.86

0.5843~\rho_\mathrm{code}^{2/3}

6.65\times 10^{-3}~\rho_\mathrm{code}

Donor

0.2364

1.957 \rho_\mathrm{code}^{1/3}

0.3660\rho_\mathrm{code}^{1/3}

20.4

0.5373~\rho_\mathrm{code}^{2/3}

5.17\times 10^{-3}~\rho_\mathrm{code}


Other Physical Variables

The ratio of radiation pressure to gas pressure (see the last column of the above table) is calculated via the relation,


\frac{1}{\Gamma} = \frac{P_\mathrm{rad}}{P_\mathrm{gas}} = \biggl( \frac{\tilde{a}}{3\tilde{r}} \biggr) \frac{T_\mathrm{code}^3}{\rho_\mathrm{code}} .

Also note that,


\beta \equiv \frac{P_\mathrm{gas}}{P_\mathrm{total}} = \frac{1}{1+P_\mathrm{rad}/P_\mathrm{gas}} .

In order to avoid establishing stellar structures that are convectively unstable, Dominic also needs to choose an evolutionary ratio of specific heats, γ, such that its value is everywhere greater than a critical value, γc, established at the center of the accretor. From Equation (131) in Chapter II of Chandrasekhar (1967) we see that γc depends on each star's central value of β — that is, it depends on βc — and on each star's structural \Gamma_1 \equiv d\ln P/d\ln \rho (which is 5 / 3 for our two ~n = 3 / 2 polytropic stars) in the following way:


\gamma_c = \biggl[ \frac{12(1-\beta_c)(\Gamma_1 - \beta_c) -\beta_c(\Gamma_1 - \beta_c) - (4-3\beta_c)^2}{12(1-\beta_c)(\Gamma_1 - \beta_c)- (4-3\beta_c)^2}  \biggr].

Plugging \rho_\mathrm{code}^\mathrm{max} into these expressions lets us tabulate various properties at the center of both stars.

Central Stellar Values

 

Approximations

From Quartic Solution

Star

\rho^\mathrm{max}_\mathrm{code}

T_\mathrm{code}^\mathrm{max}

\frac{P_\mathrm{rad}}{P_\mathrm{gas}}\biggr|_c

βc

T_\mathrm{code}^\mathrm{max}

\frac{P_\mathrm{rad}}{P_\mathrm{gas}}\biggr|_c

βc

γc

T_\mathrm{cgs}^\mathrm{max}

Accretor

1.0000

0.5843

6.65 \times 10^{-3}

0.99339

0.5805

6.522\times 10^{-3}

0.99352

1.67765

9.39\times 10^{7}~\mathrm{K}

Donor

0.6077

0.3854

3.14\times 10^{-3}

0.99687

0.3842

3.111\times 10^{-3}

0.99690

1.67188

6.22\times 10^{7}~\mathrm{K}

The central values of Prad / Pgas obtained via the quartic solution match exactly the values that Dominic read straight from the rad-hydrocode, namely, 6.5\times 10^{-3} (for the accretor) and 3.1\times 10^{-3} (for the donor). (See email from Dominic to Joel dated 8/4/2010.) In this email, Dominic also stated that he chose γ = 1.67114094; I'm not quite sure how he derived this value.


 

Whitworth's (1981) Isothermal Free-Energy Surface

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