User:Tohline/SSC/Structure/PolytropesASIDE1
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ASIDE: Whitworth's Scaling
In his study of the "global gravitational stability [of] onedimensional polytropes," Whitworth (1981, MNRAS, 195, 967) normalizes (or "references") various derived mathematical expressions for configuration radii, R, and for the pressure exerted by an external bounding medium, P_{ex}, to quantities he refers to as, respectively, R_{rf} and P_{rf}. The paragraph from his paper in which these two reference quantities are defined is shown here:
In order to map Whitworth's terminology to ours, we note, first, that he uses M_{0} to represent the spherical configuration's total mass, which we refer to simply as M; and his parameter η is related to our via the relation,
Hence, Whitworth writes the polytropic equation of state as,
whereas, using our standard notation, this same key relation is written as,
;
and his parameter K_{η} is identical to our .
According to the second (bottom) expression identified by the red outlined box drawn above,
and inverting the expression inside the green outlined box gives,
Hence,
or, gathering all factors of P_{rf} to the lefthand side,
Analogously, according to the first (top) expression identified inside the red outlined box,
Examples  

η = 1 + 1 / n 
P_{rf} 
R_{rf} 
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© 2014  2019 by Joel E. Tohline 