User:Tohline/SSC/Structure/StahlerMassRadius
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Stahler's MassRadius Relationship for Embedded Polytropes
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Review
In an accompanying chapter that discusses detailed forcebalanced models of embedded (and pressuretruncated) polytropes, we review S. W. Stahler's (1983) pair of parametric relations for the equilibrium mass and equilibrium radius for such systems, namely,






where,
and point out that Stahler (1983) (see his equation B13) explicitly states that the relevant massradius relationship for embedded polytropes is,



In what was intended to be a complementary discussion, our freeenergy analysis of embedded polytropes produced a virial equilibrium expression of the general form,
where,












and,






and,
Structural Form Factors for PressureTruncated Polytropes  


When we went back to compare the massradius relationship that results from our very general virial equilibrium expression to the one published by Stahler for pressuretruncated polytropes, they did not appear to agree. In what follows, we methodically plow through this comparison in considerable detail to uncover whatever discrepancies might exist.
Comparison
First, let's insert the definitions of the coefficients , , and into the virial equilibrium expression, replacing, where necessary, the adiabatic exponent in favor of the polytropic index, using the relation, .






Next, explicitly spelling out as well the definitions of our adopted normalization radius and normalization pressure — recognizing that — and multiply the expression through by .






As has been pointed out in our separate, more general discussion of the virial equilibrium of polytropes, if we multiply this expression through by , set all three structural form factors, , equal to unity, and replace with the notation, , the expression exactly matches the one presented as equation (5) of Whitworth, which reads:
But I like this last version of our derived expression as well because it shows some resemblance to the massradius relationship presented by Stahler and highlighted above: The first term on the lefthandside is a constant times the square of the mass; the third term is a constant times the fourth power of the equilibrium radius; and the middle term shows a crossproduct of the mass and radius (in our case, each is raised to a power other than unity). In an effort to make the comparison with Stahler even clearer, let's rewrite our expression in terms of the mass and equilibrium radius, normalized respectively to and .





















© 2014  2021 by Joel E. Tohline 