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Virial Stability of BiPolytropes
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Discussion with Kundan Kadam
The comments provided inside the following box are for Kundan Kadam, providing feedback on a derivation he sent to me ( J. E. Tohline) on Monday, 27 January 2014.
[31 January 2014] I agree with the following derived expressions; they all already appear in my presentation elsewhere on this page: Throughout the envelope …
During the recent December holiday break, I also used the hydrostatic balance relation to derive expressions for the pressure throughout the twocomponent (uniform density core + uniform density envelope) model, as you have done. I agree completely with your derivation of the expressions for — and relationships between — P_{cc}, P_{ci}, and P_{ei}. In addition, I chose to normalize all the pressures to,
Letting an asterisk superscript denote normalized pressures, that is, , my derived expressions are:
and,
First, we can now compare and contrast these newly derived expressions with the analogous expressions that have been derived earlier for a bipolytrope with n_{c} = 5 and n_{e} = 1. For example, at the end of Step #7 in the new derivation, we find the following expressions for the equilibrium radius and total mass of a bipolytrope with (n_{c},n_{e}) = (0,0) expressed in terms of the chosen central density, ρ_{0}, and central pressure, P_{0}, of the configuration:
where the function (note that, for a few days, there was a sign error in this definition of g),
We will also find it useful to combine these two expressions to eliminate direct reference to the central density, ρ_{0}, obtaining,
For comparison, referring back to the "Normalization" discussion and table of "Parameter Values" provided in our example bipolytrope with (n_{c},n_{e}) = (5,1), we have,
where expressions for the various functions, A,η_{s},θ_{i}, are also provided in the table of "Parameter Values." The dimensional normalizations are clearly the same in both types of bipolytropes because, in the latter case, . Second, a comparison between these two types of bipolytropes may help clarify how a discontinuous jump in the mean molecular weight should be handled in the more general virial analysis. Usually ρ_{e} / ρ_{c} is set equal to μ_{e} / μ_{c} at the interface, but if the core and envelope are both treated as having uniform densities, setting the ratio of densities to the ratio of mean molecular weights seems to overconstrain the problem. I haven't figured out yet how to handle this, but maybe this model comparison will help. Third, when conducting the virial analysis, it should now be clear how to evaluate the system's free energy. When the system is treated as being composed of a central uniformdensity spherical core surrounded by a uniformdensity envelope, we have already derived an analytic expression for the total gravitational potential energy, W. The above derivation of the pressure distribution throughout such a configuration — that is, throughout a bipolytrope with (n_{c},n_{e}) = (0,0) — now lets us derive an analytic expression for the thermal energy of the core, S_{core}, and an analytic expression for the thermal energy of the envelope, S_{env}. Whether focusing on the core or the envelope, start with an appreciation that,
Hence,
Then, for either volume segment, the relationship between the internal energy, U, and the thermal energy, S, is,
My initial derivation gives,
where, P_{i} is the pressure at the interface,
and the relationship between the central pressure and the pressure at the interface is,
Items #2 and #3 are the next things I will be working on.

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BiPolytrope Structural Relations
[Following a discussion that Tohline had with Kundan Kadam on 3 July 2013, we have decided to carry out a virial equilibrium and stability analysis of nonrotating bipolytropes.]
We will adopt the following approach:
 Properties of the core
 Uniform density, ρ_{c};
 Polytropic constant, K_{c}, and polytropic index, n_{c};
 Surface of the core at r_{i};
 Properties of the envelope
 Uniform density, ρ_{e};
 Polytropic constant, K_{e}, and polytropic index, n_{e};
 Base of the core at r_{i} and surface at R.
Use the dimensionless radius,
.
Then, ξ_{i} = 1 and .
Expressions for Mass
Inside the core, the expression for the mass interior to any radius, , is,
.
The expression for the mass interior to any position within the envelope, , is,
.
Hence, the mass of the core, the mass of the envelope, and the total mass are, respectively,
;
;
.
Following the work of Schönberg & Chandrasekhar (1942) — see our accompanying discussion — we are seeking equilibrium configurations in the ν − q plane where,
and . 
So we can combine the above expressions to obtain,
or,
.
It is worth noting that exactly the same result arises from an examination of the analytically definable, structural properties of bipolytropes having n_{c} = 5 and n_{e} = 1. That is, the ratio of the average density in the envelope to the average density in the core is,
This is, of course, at it should be. 
It is worth noting that, because , we can write,
and
which is consistent with the above expression for the ratio, The following figure shows how varies with for various choices of the mass density ratio, . It illustrates that, for a given coretototal mass ratio, , the relative location of the interface radius, , can vary between zero and one, but each value of reflects a different ratio of envelopetocore mass density.
Energy Expressions
The gravitational potential energy of the bipolytropic configuration is obtained by integrating over the following differential energy contribution,
.
Hence,
W = W_{core} + W_{env} 











where R_{0} is an, as yet unspecified, normalization radius, and
I like the form of this expression. The leading term, which scales as R^{ − 1}, encapsulates the behavior of the gravitational potential energy for a given choice of the internal structure, namely, a given choice of ξ_{s}, ν, and density ratio (ρ_{e} / ρ_{c}). Actually, only two internal structural parameters need to be specified — ν and ξ_{s} (or, q). From these two, the expression shown above allows the determination of (ρ_{e} / ρ_{c}).
Sanity Check: Uniform Density Configuration
If ρ_{e} / ρ_{c} = 1, then,
The gravitational potential energy is,

Drawing on expressions developed in our introductory discussion of the virial equation, the internal energy of the bipolytropic configuration is,
U = U_{core} + U_{env} 
= 

where ρ_{0} is an, as yet unspecified, normalization density, and we have allowed for either an isothermal () or an adiabatic () core.
Strategy
Scaling Parameters
We want to vary the total radius, , of the configuration and look for extrema in the free energy, while holding the following parameters fixed: , (or ), , and . So we need to rewrite the expressions for and such that everything is constant except for , or . And, in order to put everything explicitly in terms of the fixed parameters just specified, let's go ahead and define the length, time, and density against which all dimensional quantities will ultimately be scaled.
Chosen Scaling Parameters 


Polytropic Core 
Isothermal Core 










Rescaled Energy Expressions
Isothermal Core
In the case of an isothermal core, the expression for W shown above needs only a slight modification to put it in the appropriate form, namely,


In the case of an isothermal core, the expression for the total internal energy may be rewritten as,




= 


= 

Hence (in the case of an isothermal core),

= 


= 

where,












Polytropic Core
In the case of a polytropic core,

= 

where,









Temperature Across the Interface
In order to ensure that the temperature of the envelope is the same as the temperature of the core when there is a drop in the mean molecular weight at the interface, we need to have,



Note that this reflects the same physical condition as the constraint that is placed on the enthalpy at the interface when we analyze the detailed structure of n_{c} = 5, n_{e} = 1 bipolytropes (see the middle of Table 1 in the accompanying discussion), namely,



In the case of an isothermal core, this implies,









Not Necessarily Useful 










Therefore, the coefficient becomes,



In the case of a polytropic core, this implies,









Therefore, the coefficient becomes,



Pressure Across the Interface
We will relate K_{e} to K_{c} by demanding that initially the pressure is identical in both layers. The relevant algebraic relation will depend on whether the core is isothermal (), or whether it has a finite polytropic index and therefore adjusts adiabatically to compressions or expansions.
Isothermal Core
Guided by the interface conditions presented in Table 2 of our accompanying discussion of the structure of bipolytropes, the condition for pressure balance in the case of an isothermal core should be,
Actually, in the full structural solution,

= 


= 


= 

Adiabatic Core
Guided by the interface conditions presented in Table 2 of our accompanying discussion of the structure of bipolytropes, the condition for pressure balance in the case of polytropic core should be,
Actually, in the full structural solution,

= 


= 


= 


= 

Virial Analysis
Isothermal Core Equilibrium Condition
For a given set of fixed coefficients in the free energy expression, equilibria are identified by setting . Generally,



So, for the case of n_{e} = 3 / 2, the equilibrium radius is given by the condition,















where,



and 



Combined Equilibrium Constraints
But, from the condition on the temperature at the interface we also need,






Hence, the two conditions combined imply,



where,



This, in turn, implies,












But notice, as well, that in order to have real roots of the equilibrium condition, we need . This means,


















Free Energy Expression
To within an additive constant, the free energy may now be written as,
where, and,












Derivatives of Free Energy
Equilibrium Condition
We obtain the equilibrium radius, χ_{E}, when . Hence, the relation governing the equilibrium radius is,

= 


= 


= 

where,









Isothermal Core
In the case of an isothermal core (),

= 



Or, alternatively, via a dimensionless treatment,

= 

Adiabatic Core
In the case of an adiabatice core (),

= 

Stability
At this equilibrium radius, the second derivative of the free energy has the value,

= 


= 

which, when combined with the condition for equilibrium gives,

= 


= 


= 

The equilibrium configuration is stable as long as this second derivative is positive.
Isothermal Core
Hence, for a bipolytrope with an isothermal core (), the configuration is stable as long as,
Adiabatic Core
In the adiabatic case (), the configuration is stable as long as,
Examples
Isothermal Core with n_{e} = 3 / 2
Consider the case examined by Schönberg & Chandrasekhar (1942), that is, the case of an isothermal core and an envelope with n_{e} = 3 / 2. The equilibrium radius is given by the expression,

= 


= 


= 


= 

And the system is stable when,
A couple of physical attributes are now clear:
 Physical configurations only exist for .
 For each value of , there are two equilibrium configurations, given by the roots of the quadratic equation for χ_{E}; the "negative" branch is stable but the "positive" branch is unstable.
Note that,
and,
But, this last expression must be less than or equal to unity, which implies,
This doesn't seem to have the correct behavior, for example, the smaller values of ν should be the stable ones, so there must be a mistake in the derivation.
Adiabatic Core with n_{c} = 5 and n_{e} = 1
Consider the case with an analytical structure derived by Eagleton, Faulkner, and Cannon (1998, MNRAS, 298, 831), that is, the case of an adiabatic core having n_{c} = 5 and an envelope with n_{e} = 1. The equilibrium radius is,
Old and Probably Irrelevant Discussion
Summary Expressions (New)
In the above derivations, we have adopted the notation,
Now, guided by the earlier discussion of pressurebounded isothermal spheres, we choose the following normalization energy and radius:
and
Also, by analogy, it is useful to define the dimensionless parameter,
(It is worth noting that if we set n_{e} = − 1, the dimensionless parameter Π_{I} becomes identical to the parameter Π as defined in the context of our discussion of the BonnorEbert sphere. But in order to complete the analogy with the BonnorEbert sphere discussion, we would also need to change the sign on the last term in the above expression for the free energy because in the earlier discussion the external pressure was an external, confining condition whereas here it is included as an internal energy of the system.)
Relevant Expressions for Isothermal Core 


















ν 




[On 8 November 2013, J. E. Tohline wrote: I just confirmed that the simpler expression for the normalized total free energy, , matches the more complicated version. I don't like the result because the third term in the free energy  the one contributed by the internal energy of the envelope  is independent of the radius of the configuration, χ; it works out this way because my expression for Π_{I} has a radial dependence that exactly cancels out the explicit radial dependence that appears in the more complicated expression. But maybe it's okay after all because this expression is intended to show how the free energy varies across the (q,ν) plane, and the effect of Π_{I} appears implicitly through the specification of χ, or visa versa.]
Summary Expressions (Old)
In the above derivations, we have adopted the notation,
Now, guided by the dimensional aspects of the various coefficients in the free energy expression, we choose the following normalization energy and radius:
and
When combined with the expression for ρ_{norm}, these become,
and
So, the primary scales are determined after specifying two parameters: M_{tot} and K_{e}. We also obtain,
Relevant Expressions for Isothermal Core 











3κ_{I}ν 


Subsequently, we will also find it useful to have expressions for the following coefficient ratios:

= 


= 


= 


= 


= 


= 


= 


= 


= 


= 


= 


= 

Derivatives of Free Energy
Equilibrium Condition
We obtain the equilibrium radius, χ_{E}, when . Hence, the relation governing the equilibrium radius is,

= 


= 


= 

where,
Stability
At this equilibrium radius, the second derivative of the free energy has the value,

= 


= 

which, when combined with the condition for equilibrium gives,

= 


= 


= 

Finally, the equilibrium configuration is stable as long as this second derivative is positive. Hence, for a bipolytrope with an isothermal core (), the configuration is stable as long as,
In the adiabatic case (), the configuration is stable as long as,
Examples
Isothermal Core with n = 3 / 2 Envelope
When the core is isothermal and n_{e} = 3 / 2, the equilibrium condition is:
At the same time, the condition for stability is,
Isothermal Core with n = 1 Envelope
When the core is isothermal and n_{e} = 1, the equilibrium condition is:
(We need to solve this cubic equation.)
At the same time, the condition for stability is,
Old (and probably incorrect) cases
Envelope with n = 3 / 2
If we choose an n_{e} = 3 / 2 envelope, we obtain stability for,
In this case, the equilibrium radius condition is,
Envelope with n = 1
If, instead, we choose an n_{e} = 1 envelope, we obtain stability for,
In this case, the equilibrium radius condition is,
© 2014  2019 by Joel E. Tohline 