From VistrailsWiki
< User:Tohline
Revision as of 21:27, 29 September 2014 by Tohline (talk | contribs) (Begin separate chapter discussion Staher's (1983) mass-radius relationship for n=5 embedded polytropes)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to navigation Jump to search

Stahler's Mass-Radius Relationship for Embedded Polytropes

Whitworth's (1981) Isothermal Free-Energy Surface
|   Tiled Menu   |   Tables of Content   |  Banner Video   |  Tohline Home Page   |

In an accompanying chapter that discusses detailed force-balanced models of embedded (and pressure-truncated) polytropes, we review S. W. Stahler's (1983) pair of parametric relations for the equilibrium mass and equilibrium radius for such systems, namely,

<math> ~M </math>


<math> M_\mathrm{SWS} \biggl( \frac{n^3}{4\pi} \biggr)^{1/2} \biggl\{ \theta_n^{(n-3)/2} \xi^2 \biggl| \frac{d\theta_n}{d\xi} \biggr| \biggr\}_{\xi_e} </math>

<math> ~R_\mathrm{eq} </math>


<math> R_\mathrm{SWS} \biggl( \frac{n}{4\pi} \biggr)^{1/2} \biggl\{ \xi \theta_n^{(n-1)/2} \biggr\}_{\xi_e} </math>


<math>M_\mathrm{SWS} = \biggl( \frac{n+1}{nG} \biggr)^{3/2} K_n^{2n/(n+1)} P_\mathrm{e}^{(3-n)/[2(n+1)]} \, ,</math>

<math> R_\mathrm{SWS} = \biggl( \frac{n+1}{nG} \biggr)^{1/2} K_n^{n/(n+1)} P_\mathrm{e}^{(1-n)/[2(n+1)]} \, , </math>

and point out that Stahler (1983) (see his equation B13) explicitly states that the relevant mass-radius relationship for <math>~n = 5</math> embedded polytropes is,

<math> \biggl( \frac{M}{M_\mathrm{SWS}} \biggr)^2 - 5 \biggl( \frac{M}{M_\mathrm{SWS}} \biggr)\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{SWS}} \biggr) + \frac{20\pi}{3} \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{SWS}} \biggr)^4 </math>


<math> ~0 \, . </math>

Whitworth's (1981) Isothermal Free-Energy Surface

© 2014 - 2021 by Joel E. Tohline
|   H_Book Home   |   YouTube   |
Appendices: | Equations | Variables | References | Ramblings | Images | myphys.lsu | ADS |
Recommended citation:   Tohline, Joel E. (2021), The Structure, Stability, & Dynamics of Self-Gravitating Fluids, a (MediaWiki-based) publication,