User:Tohline/SSC/BipolytropeGeneralization
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Bipolytrope Generalization
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On 26 August 2014, Tohline finished rewriting the chapter titled "Bipolytrope Generalization" in a very concise manner (go here for this Version2 chapter) then set this chapter aside to provide a collection of older attempts at the derivations. While much of what follows is technically correct, it is overly detailed and cumbersome. Because it likely also contains some misguided steps, we label it in entirety as Work in Progress.
Material that appears after this point in our presentation is under development and therefore
may contain incorrect mathematical equations and/or physical misinterpretations.
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Old Stuff






In addition to the gravitational potential energy, which is naturally written as,



it seems reasonable to write the separate thermal energy contributions as,






where the subscript "i" means "at the interface," and and are dimensionless functions of order unity (all three functions to be determined) akin to the structural form factors used in our examination of isolated polytropes.
While exploring how the freeenergy function varies across parameter space, we choose to hold and fixed. By dimensional analysis, it is therefore reasonable to normalize all energies, length scales, densities and pressures by, respectively,












As is detailed below — first, here, and via an independent derivation, here — quite generally the expression for the normalized free energy is,












where we have introduced the parameter, . After defining the normalized (and dimensionless) configurarion radius, , we can write the normalized free energy of a bipolytrope in the following compact form:



where,









As is further detailed below, the second expression for the coefficient, , ensures that the pressure at the "surface" of the core matches the pressure at the "base" of the envelope; but it should only be employed after an equilibrium radius, , has been identified by locating an extremum in the free energy.
Simplest Bipolytrope
Familiar Setup
As has been shown in an accompanying presentation, for an bipolytrope,









and where (see, for example, in the context of its original definition, or another, separate derivation),






and where (see the associated discussion of relevant mass integrals),
Cleaner Virial Presentation
In an effort to show the similarity in structure among the several energy terms, we have also found it useful to write their expressions in the following forms:









where (see an associated discussion or the original derivation),
and where,









This also means that the three key terms used as shorthand notation in the above expressions for the three energy terms have the following definitions:

Hence, if all the interface pressures are equal — that is, if and, hence also, — then the total thermal energy is,



and the virial is,



The virial should sum to zero in equilibrium, which means,


















Shift to Central Pressure Normalization
Let's rework the definition of in two ways: (1) Normalize to and normalize the pressure to ; (2) shift the referenced pressure from the pressure at the interface to the central pressure , because it is that is directly related to and ; specifically, . Appreciating that, in equilibrium,



the lefthandside of the last expression, above, can be rewritten as,









Hence, the virial equilibrium condition gives,






This result precisely matches the result obtained via the detailed forcebalanced conditions imposed through hydrostatic equilibrium.
Adopting our new variable normalizations and realizing, in particular, that,



the expression alternatively can be rewritten as,















Normalized in this manner, the virial equilibrium (as well as the hydrostatic balance) condition gives,






FreeEnergy Coefficients
Therefore, for an bipolytrope, the coefficients in the normalized freeenergy function are,





















Note that, because in equilibrium, the ratio of coefficients,






The equilibrium condition is,
where,
More General Derivation of FreeEnergy Coefficients B and C
Keep in mind that, generally,





… and, note that … 






where we have introduced the notation,
So, the freeenergy coefficient,









And the freeenergy coefficient,









OLD DERIVATION

NEW DERIVATION


… therefore … 




… and, enforcing in equilibrium … 




… and, also … 



Extrema
Extrema in the free energy occur when,



Also, as stated above, because in equilibrium, the ratio of coefficients,



When put together, these two relations imply,






But the definition of gives,



Hence, extrema occur when,















In what follows, keep in mind that,









OLD DERIVATION

NEW DERIVATION


… hence, as derived in the above table … 




… which, when combined with the condition that identifies extrema, gives … 




These are consistent results because they result in the detailed forcebalance relation, 
Examples
Identification of Local Extrema in Free Energy 












MIN/MAX 










MIN 










MIN 







" 
" 
" 
MAX 










MIN 







" 
" 
" 
MAX 
Free Energy Extrema when: 













Stability 
MIN/MAX 












MIN 
… 
… 
… 




" 
" 
" 


MAX 












MIN 
… 
… 
… 




" 
" 
" 


MAX 












MAX 
… 
… 
… 




" 
" 
" 


MIN 
System should be stable (with free energy minimum) if: 
Solution Strategy
For a given set of freeenergy coefficients, and , along with a choice of the two adiabatic exponents , here's how to determine all of the physical parameters that are detailed in the above example table.
 Step 1: Guess a value of .
 Step 2: Given the pair of parameter values, , determine the interfacedensity ratio, , by finding the appropriate root of the expression that defines the function, . This can be straightforwardly accomplished because, as demonstrated below, the relevant expression can be written as a quadratic function of .
 Step 3: Given the pair of parameter values, , determine the value of the coretototal mass ratio, , from the expression that was obtained from an integration over the mass, namely,



 Step 4: Given the value of along with the pair of parameter values, , the above expression that defines can be solved to give the relevant value of the dimensionless parameter,
 Step 5: The value of — the coefficient that appears on the righthandside of the above expression that defines — can be determined, given the values of parameter triplet, .
 Step 6: Given the value of and the justdetermined value of the coefficient , the normalized equilibrium radius, that corresponds to the value of that was guessed in Step #1 can be determined from the above definition of , specifically,



 Step 7: But, independent of this guessed value of the condition for virial equilibrium — which identifies extrema in the freeenergy function — gives the following expression for the normalized equilibrium radius:



 Step 8: If , return to Step #1 and guess a different value of . Repeat Steps #1 through #7 until the two independently derived values of the normalized radius match, to a desired level of precision.
 Keep in mind: (A) A graphical representation of the freeenergy function, , can also be used to identify the "correct" value of and, ultimately, the abovedescribed iteration loop should converge on this value. (B) The freeenergy function may exhibit more than one (or, actually, no) extrema, in which case more than one (or no) value of should lead to convergence of the abovedescribed iteration loop.
Material that appears after this point in our presentation is under development and therefore
may contain incorrect mathematical equations and/or physical misinterpretations.
 Go Home 
Detailed Derivations
Dividing the freeenergy expression through by generates,







































We also want to ensure that envelope pressure matches the core pressure at the interface. This means,





















Hence, we can write the normalized (and dimensionless) free energy as,



Keep in mind that, if the envelope and core both have uniform (but different) densities, then , , and
Free Energy and Its Derivatives
Now, the free energy can be written as,









The first derivative of the free energy with respect to radius is, then,



And the second derivative is,









Equilibrium
The radius, , of the equilibrium configuration(s) is determined by setting the first derivative of the free energy to zero. Hence,












This is the familiar statement of virial equilibrium. From it we should always be able to derive the radius of equilibrium configurations.
Stability
To assess the relative stability of an equilibrium configuration, we need to determine the sign of the second derivative of the free energy, evaluated at the equilibrium radius. If the sign of the second derivative is positive, the system is dynamically stable; if the sign is negative, he system is dynamically unstable. Using the above statement of virial equilibrium, that is, setting,






we obtain,












So, if when evaluated at the equilibrium state, the expression inside of the square brackets of this last expression is negative, the equilibrium configuration will be dynamically unstable. We have chosen to write the expression in this particular final form because we generally will be interested in bipolytropes for which the adiabatic exponent of the envelope is greater than and the adiabatic exponent of the core is less than or equal to — that is, . Hence, because the gravitational potential energy, , is intrinsically negative, the system will be dynamically unstable only if the second term (involving ) is greater in magnitude than the first term (involving ).
It is worth noting that, instead of drawing upon and to define the stability condition, we could have used an appropriate combination of and , or the and pair. Also, for example, because the virial equilibrium condition is , it is easy to see that the following inequality also equivalently defines stability:



Examples
(0, 0) Bipolytropes
Review
In an accompanying discussion we have derived analytic expressions describing the equilibrium structure and the stability of bipolytropes in which both the core and the envelope have uniform densities, that is, bipolytropes with . From this work, we find that integrals over the mass and pressure distributions give:












where,















Renormalize
Let's renormalize these energy terms in order to more readily relate them to the generalized expressions derived above.















Also,















Hence,



Given that for the bipolytrope, we can finally write,



and,



Hence the renormalized gravitational potential energy becomes,



and the two, renormalized contributions to the thermal energy become,









Finally, then, we can state that,









Virial Equilibrium and Stability Evaluation
With these expressions in hand, we can deduce the equilibrium radius and relativity stability of bipolytropes using the generalized expressions provided above. For example, from the statement of virial equilibrium we obtain,


















Or, given the above renormalization, this expression can be written as,






And the condition for dynamical stability is,









(5, 1) Bipolytropes
In another accompanying discussion we have derived analytic expressions describing the equilibrium structure of bipolytropes with . Can we perform a similar stability analysis of these configurations? Work in progress!
Best of the Best
One Derivation of Free Energy






Another Derivation of Free Energy
Hence the renormalized gravitational potential energy becomes,



and the two, renormalized contributions to the thermal energy become,

































Finally, then, we can state that,









Note,



We also want to ensure that envelope pressure matches the core pressure at the interface. This means,





















Keep in mind that, if the envelope and core both have uniform (but different) densities, then , , and
Summary
Understanding FreeEnergy Behavior
Step 1: Pick values for the separate coefficients, and of the three terms in the normalized freeenergy expression,



then plot the function, , and identify the value(s) of at which the function has an extremum (or multiple extrema).
Step 2: Note that,















where (see, for example, in the context of its original definition),



and, where,



Also, keep in mind that, if the envelope and core both have uniform (but different) densities, then , , and
Step 3: An analytic evaluation tells us that the following should happen. Using the numerically derived value for , define,
We should then discover that,
Check It



























Fortunately, this precisely matches our earlier derivation, which states that,



Playing With One Example
By setting,






2.5 
1.0 
2.0 
a plot of versus exhibits the following, two extrema:
extremum 







MIN 
1.1824 
− 0.611367 

0.66891 
1.3378 
1.0693 
1.0694 
MAX 
9.6722 
+ 0.508104 

0.004313 
0.008625 
2.4786 
2.4786 
The last two columns of this table confirm the internal consistency of the relationships presented in Step 3, above. But what does this mean in terms of the values of , , and the related ratio of densities at the interface, ?
Let's assume that what we're trying to display and examine is the behavior of the freeenergy surface for a fixed value of the ratio of densities at the interface. Once the value of has been specified, it is clear that the value of (and, hence, also ) is set because has also been specified. But our specification of along with also forces a particular value of . It is unlikely that these two values of will be the same. In reality, once and have both been specified, they force a particular pair. How do we (easily) figure out what this pair is?
Let's begin by rewriting the expressions for and in terms of just and the ratio, , keeping in mind that, for the case of a uniformdensity core (of density, ) and a uniformdensity envelope (of density, ),



hence,



and 



Putting the expression for in the desired form is simple because only appears as a leading factor. Specifically, we have,






The expression for can be written in the form,












Generally speaking, the equilibrium radius, , which appears in the expression for , is not known ahead of time. Indeed, as is illustrated in our simple example immediately above, the normal path is to pick values for the coefficients, , , and , and determine the equilibrium radius by looking for extrema in the freeenergy function. And because is not known ahead of time, it isn't clear how to (easily) figure out what pair of physical parameter values, , give selfconsistent values for the coefficient pair, .
Because we are using a uniform density core and uniform density envelope as our base model, however, we do know the analytic solution for . As stated above, it is,









Combining this expression with the one for gives us the desired result — although, strictly speaking, it is cheating! We can now methodically choose pairs and map them into the corresponding values of and . And, via an analogous "cheat," the choice of also gives us the selfconsistent value of . In this manner, we should be able to map out the freeenergy surface for any desired set of physical parameters.
Second Example
Explain Logic
The figure presented here, on the right, shows a plot of the free energy, as a function of the dimensionless radius, , where,



and, where we have used the parameter values,






0.201707 
0.0896 
0.002484 
Directly from this plot we deduce that this freeenergy function exhibits a minimum at and that, at this equilibrium radius, the configuration has a freeenergy value, . Via the steps described below, we demonstrate that this identified equilibrium radius is appropriate for an bipolytrope (with the justspecified core and envelope adiabatic indexes) that has the following physical properties:
 Fractional core mass, ;
 Coreenvelope interface located at ;
 Density jump at the coreenvelope interface, .
Step 1: Because the ratio, , is a linear function of the density ratio, , the full definition of the freeenergy coefficient, , can be restructured into a quadratic equation that gives the density ratio for any choice of the parameter pair, . Specifically,






and this can be written in the form,



where,









Hence,



(For our physical problem it appears as though only the positive root is relevant.) For the purposes of this example, we set and examined a range of values of to find a physically interesting value for the density ratio. We picked:




















Step 2: Next, we chose the parameter pair,
and determined the following parameter values from the known analytic solution:
















Construction Multiple Curves to Define a FreeEnergy Surface
Okay. Now that we have the hang of this, let's construct a sequence of curves that represent physical evolution at a fixed interfacedensity ratio, , but for steadily increasing coretototal mass ratio, . Specifically, we choose,
From the known analytic solution, here are parameters defining several different equilibrium models:
Identification of Local Minimum in Free Energy 






































Here we are examining the behavior of the freeenergy function for bipolytropic models having , , and a density ratio at the coreenvelope interface of . The figure shown here, on the right, displays the three separate freeenergy curves, — where, is the normalized configuration radius — that correspond to the three values of given in the first column of the above table. Along each curve, the local freeenergy minimum corresponds to the the equilibrium radius, , recorded in column 6 of the above table. 
Each of the freeenergy curves shown above has been entirely defined by our specification of the three coefficients in the freeenergy function, , and . In each case, the values of these three coefficients was judiciously chosen to produce a curve with a local minimum at the correct value of corresponding to an equilibrium configuration having the desired model parameters. Upon plotting these three curves, we noticed that two of the curves — curves for and — also display a local maximum. Presumably, these maxima also identify equilibrium configurations, albeit unstable ones. From a careful inspection of the plotted curves, we have identified the value of that corresponds to the two newly discovered (unstable) equilibrium models; these values are recorded in the table that immediately follows this paragraph. By construction, we also know what values of , and are associated with these two identified equilibria; these values also have been recorded in the table. But it is not immediately obvious what the values are of the model parameters that correspond to these two equilibrium models.
Subsequently Identified Local Energy Maxima 




























Related Discussions
 Newer, Version2 of Bipolytrope Generalization derivations.
 Analytic solution with and .
 Analytic solution with and .
© 2014  2020 by Joel E. Tohline 