User:Tohline/SSC/Virial/FormFactors

Structural Form Factors

As has been defined in a companion, introductory discussion, three key dimensionless structural form factors are:

<math>~\mathfrak{f}_M </math>	<math>~\equiv</math>	<math>~ \int_0^1 3\biggl[ \frac{\rho(x)}{\rho_c}\biggr] x^2 dx = \biggl( \frac{\bar\rho}{\rho_c} \biggr)_\mathrm{eq} \, ,</math>
<math>~\mathfrak{f}_W</math>	<math>~\equiv</math>	<math>~ 3\cdot 5 \int_0^1 \biggl\{ \int_0^x \biggl[ \frac{\rho(x)}{\rho_c}\biggr] x^2 dx \biggr\} \biggl[ \frac{\rho(x)}{\rho_c}\biggr] x dx\, ,</math>
<math>~\mathfrak{f}_A</math>	<math>~\equiv</math>	<math>~ \int_0^1 3\biggl[ \frac{P(x)}{P_c}\biggr] x^2 dx = \biggl( \frac{\bar{P}}{P_c} \biggr)_\mathrm{eq} \, ,</math>

where, <math>~x \equiv r/R_\mathrm{limit}</math>.

One Detailed Example (n = 5)

Here we derive detailed expressions for the above subset of structural form factors in the case of spherically symmetric configurations that obey an <math>~n=5</math> polytropic equation of state. The hope is that this will illustrate, in a clear and helpful manner, how the task of calculating form factors is to be carried out, in practice; and, in particular, to provide one nontrivial example for which analytic expressions are derivable. This should simplify the task of debugging numerical algorithms that are designed to calculate structural form factors for more general cases that cannot be derived analytically. The limits of integration will be specified in a general enough fashion that the resulting expressions can be applied, not only to the structures of isolated polytropes, but to pressure-truncated polytropes that are embedded in a hot, tenuous external medium and to the cores of bipolytropes.

Key Foundations

We use the following normalizations, as drawn from our more general introductory discussion:

Adopted Normalizations <math>~(n=5; ~\gamma=6/5)</math>
<math>~R_\mathrm{norm}</math> <math>~\equiv</math> <math>~\biggl( \frac{G}{K} \biggr)^{5/2} M_\mathrm{tot}^{2} </math> <math>~P_\mathrm{norm}</math> <math>~\equiv</math> <math>~\biggl( \frac{K^{10}}{G^{9} M_\mathrm{tot}^{6}} \biggr) </math> <math>~E_\mathrm{norm}</math> <math>~\equiv</math> <math>~ P_\mathrm{norm} R_\mathrm{norm}^3 = \biggl( \frac{K^5}{G^3} \biggr)^{1/2} </math> <math>~\rho_\mathrm{norm}</math> <math>~\equiv</math> <math>~\frac{3M_\mathrm{tot}}{4\pi R_\mathrm{norm}^3} = \frac{3}{4\pi} \biggl( \frac{K}{G} \biggr)^{15/2} M_\mathrm{tot}^{-5} </math> <math>~c^2_\mathrm{norm}</math> <math>~\equiv</math> <math>~\frac{P_\mathrm{norm}}{\rho_\mathrm{norm}} = \frac{4\pi}{3} \biggl( \frac{K^5}{G^3} \biggr)^{1/2} M_\mathrm{tot}^{-1} </math>
Note that the following relations also hold: <math>~E_\mathrm{norm} = P_\mathrm{norm} R_\mathrm{norm}^3 = \frac{G M_\mathrm{tot}^2}{ R_\mathrm{norm}} = \biggl( \frac{3}{4\pi} \biggr) M_\mathrm{tot} c_\mathrm{norm}^2</math>

© 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) Vistrails.org publication, https://www.vistrails.org/index.php/User:Tohline/citation