Difference between revisions of "User:Tohline/SSC/Structure/BonnorEbert"

From VistrailsWiki
Jump to navigation Jump to search
(Begin discussion of Bonnor-Ebert Sphere)
(→‎Governing Relations: Correct the links to the Bonnor and Ebert papers)
Line 36: Line 36:
<math>\frac{1}{r^2} \frac{d}{dr}\biggl( r^2 \frac{d\ln\rho}{dr} \biggr) =-  \frac{4\pi G}{c_s^2} \rho</math> .
<math>\frac{1}{r^2} \frac{d}{dr}\biggl( r^2 \frac{d\ln\rho}{dr} \biggr) =-  \frac{4\pi G}{c_s^2} \rho</math> .
</div>
</div>
In their studies of pressure-bounded isothermal spheres, [http://http://adsabs.harvard.edu/abs/1955ZA.....37..217E Ebert] (1955, ZA, 37, 217) and [http://http://adsabs.harvard.edu/abs/1956MNRAS.116..351B Bonnor] (1956, MNRAS, 116, 351) both started with this governing ODE, but developed its solution in different ways.  Here we present both developments while highlighting transformations between the two.
In their studies of pressure-bounded isothermal spheres, [http://adsabs.harvard.edu/abs/1955ZA.....37..217E Ebert] (1955, ZA, 37, 217) and [http://adsabs.harvard.edu/abs/1956MNRAS.116..351B Bonnor] (1956, MNRAS, 116, 351) both started with this governing ODE, but developed its solution in different ways.  Here we present both developments while highlighting transformations between the two.
 


=Related Wikipedia Discussions=
=Related Wikipedia Discussions=

Revision as of 01:24, 28 October 2012

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

Isothermal Sphere (structure)

Here we supplement the simplified set of principal governing equations with an isothermal equation of state, that is, <math>~P</math> is related to <math>~\rho</math> through the relation,

<math>P = c_s^2 \rho \, ,</math>

where, <math>c_s</math> is the isothermal sound speed. Comparing this <math>~P</math>-<math>~\rho</math> relationship to

Form A
of the Ideal Gas Equation of State,

LSU Key.png

<math>~P_\mathrm{gas} = \frac{\Re}{\bar{\mu}} \rho T</math>

we see that,

<math>c_s^2 = \frac{\Re T}{\bar{\mu}} = \frac{k T}{m_u \bar{\mu}} \, ,</math>

where, <math>~\Re</math>, <math>~k</math>, <math>~m_u</math>, and <math>~\bar{\mu}</math> are all defined in the accompanying variables appendix. It will be useful to note that, for an isothermal gas, <math>~H</math> is related to <math>~\rho</math> via the expression,

<math> dH = \frac{dP}{\rho} = c_s^2 d\ln\rho \, . </math>

Governing Relations

Adopting solution technique #2, we need to solve the following second-order ODE relating the two unknown functions, <math>~\rho</math> and <math>~H</math>:

<math>\frac{1}{r^2} \frac{d}{dr}\biggl( r^2 \frac{dH}{dr} \biggr) =- 4\pi G \rho</math> .

Using the <math>~H</math>-<math>~\rho</math> relationship for an isothermal gas presented above, this can be rewritten entirely in terms of the density as,

<math>\frac{1}{r^2} \frac{d}{dr}\biggl( r^2 \frac{d\ln\rho}{dr} \biggr) =- \frac{4\pi G}{c_s^2} \rho</math> .

In their studies of pressure-bounded isothermal spheres, Ebert (1955, ZA, 37, 217) and Bonnor (1956, MNRAS, 116, 351) both started with this governing ODE, but developed its solution in different ways. Here we present both developments while highlighting transformations between the two.

Related Wikipedia Discussions


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