Difference between revisions of "User:Tohline/Appendix/Ramblings/Photosphere"

From VistrailsWiki
Jump to navigation Jump to search
(→‎More Careful Derivation: Continue derivation (2))
(→‎More Careful Derivation: Derivation continued (3))
Line 8: Line 8:


===More Careful Derivation===
===More Careful Derivation===
As the accretion stream from the less massive white dwarf impacts the surface of the accretor supersonically, it will heat the accreted material to a temperature,  
As the accretion stream from the less massive white dwarf impacts the surface of the accretor supersonically, it will heat the accreted material to a post-shock temperature,  
<div align="center">
<div align="center">
<math>T_{sh} \sim \frac{GM_a m_p}{kR_a}</math> ,
<math>T_{sh} \approx \frac{GM_a m_p}{kR_a}</math> ,
</div>
</div>
where <math>M_a</math> and <math>R_a</math> are the mass and radius, respectively, of the accretor.  Assuming the post-shock material is optically thick to photon radiation, we should ask to what "photospheric" radius, <math>R_{ph}</math>, the envelope of the accretor will have to swell in order for the star (in steady state) to be able to radiate all of the accretion energy?  It seems that it will need,
where <math>M_a</math> and <math>R_a</math> are the mass and radius, respectively, of the accretor.  Assuming the post-shock material is optically thick to photon radiation, we should ask to what "photospheric" radius, <math>R_{ph}</math>, the envelope of the accretor will have to swell in order for the star (in steady state) to be able to radiate all of the accretion energy?  It seems that it will need,
Line 30: Line 30:
</math>
</math>
</div>
</div>
 
Replacing <math>T_{sh}</math> by the approximate expression shown above gives,
<div align="center">
<math>
\biggl[ \frac{R_{ph}}{R_a} \biggr]^2 \approx \frac{GM_a \dot{M}}{\pi ac R_a^3} \biggl[  \frac{kR_a}{GM_a m_p} \biggr]^4
= \frac{R_a}{\pi ac G^3 M_a^3} \biggl[  \frac{k}{ m_p} \biggr]^4 \dot{M} \, .
</math>
</div>


&nbsp;<br />
&nbsp;<br />
{{LSU_HBook_footer}}
{{LSU_HBook_footer}}

Revision as of 16:42, 6 May 2011

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

Locating the Photosphere of Stably Accreting Double White Dwarf Binaries

Context

At our regularly scheduled astrophysics group meeting on Monday, 2 May 2011, Juhan Frank and I started debating the answer to the following question: What should the photospheric radius be of the common envelope that surrounds a stably accreting, double white dwarf (DWD) binary? That is, does an accreting DWD binary that is destined to be an AM CVn system look like a single bloated star? The various mathematical relations that we think are relevant to this question were sketched on the whiteboard in room 218 Johnston Hall (CCT). Here is a photo of that whiteboard discussion and derivation.

May2011WhiteBoard.JPG


More Careful Derivation

As the accretion stream from the less massive white dwarf impacts the surface of the accretor supersonically, it will heat the accreted material to a post-shock temperature,

<math>T_{sh} \approx \frac{GM_a m_p}{kR_a}</math> ,

where <math>M_a</math> and <math>R_a</math> are the mass and radius, respectively, of the accretor. Assuming the post-shock material is optically thick to photon radiation, we should ask to what "photospheric" radius, <math>R_{ph}</math>, the envelope of the accretor will have to swell in order for the star (in steady state) to be able to radiate all of the accretion energy? It seems that it will need,

<math> 4\pi R_{ph}^2 \sigma T_{sh}^4 = L_{acc} </math>

<math> \Rightarrow~~~~~\biggl[ \frac{R_{ph}}{R_a} \biggr]^2 = \frac{L_{acc}}{\pi ac R_a^2 T^4_{sh}} = \frac{GM_a \dot{M}}{\pi ac R_a^3 T^4_{sh}} \, , </math>

where we've set <math>4\sigma = ac</math> and,

<math> L_{acc} = \frac{GM_a \dot{M}}{R_a} \, . </math>

Replacing <math>T_{sh}</math> by the approximate expression shown above gives,

<math> \biggl[ \frac{R_{ph}}{R_a} \biggr]^2 \approx \frac{GM_a \dot{M}}{\pi ac R_a^3} \biggl[ \frac{kR_a}{GM_a m_p} \biggr]^4 = \frac{R_a}{\pi ac G^3 M_a^3} \biggl[ \frac{k}{ m_p} \biggr]^4 \dot{M} \, . </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) Vistrails.org publication, https://www.vistrails.org/index.php/User:Tohline/citation