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(Created page with '__FORCETOC__ =Poisson Equation= {{LSU_HBook_header}} =Drawn from Other Wiki Pages= It is clear, therefore, that Chandrasekhar uses the variable <math>~\vec{u}</math> instead of…')
 
 
(7 intermediate revisions by the same user not shown)
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__FORCETOC__
__FORCETOC__
=Poisson Equation=
=Origin of the Poisson Equation=
 
In deriving the,
<div align="center">
<span id="PGE:Poisson"><font color="#770000">'''Poisson Equation'''</font></span><br />
 
{{User:Tohline/Math/EQ_Poisson01}}
</div>
we will follow closely the presentation found in &sect;2.1 of [<b>[[User:Tohline/Appendix/References#BT87|<font color="red">BT87</font>]]</b>].
 
 
{{LSU_HBook_header}}
{{LSU_HBook_header}}


<font color="#007700">According to Isaac Newton's inverse-square law of gravitation,</font> the acceleration, <math>~\vec{a}(\vec{x})</math>, felt at any point in space, <math>~\vec{x}</math>, due to the gravitational attraction of a distribution of mass, <math>~\rho(\vec{x})</math>, is obtained by integrating over the accelerations exerted by each small mass element, <math>~\rho(\vec{x}^{~'}) d^3x'</math>, as follows:
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\vec{a}(\vec{x})</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\int \biggl[\frac{\vec{x}^{~'} - \vec{x}}{|\vec{x}^{~'} - \vec{x}|^3}\biggr] G\rho(\vec{x}^{~'}) d^3 x' \, ,
</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
[<b>[[User:Tohline/Appendix/References#BT87|<font color="red">BT87</font>]]</b>], p. 31, Eq. (2-2)
  </td>
</tr>
</table>
</div>
where, {{ User:Tohline/Math/C_GravitationalConstant }} is the universal gravitational constant.
==Step 1==
In the astrophysics literature, it is customary to adopt the following definition of the,
<div align="center" id="GravitationalPotential">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="center" colspan="3">
<font color="#770000">'''Scalar Gravitational Potential'''</font>
  </td>
</tr>
<tr>
  <td align="right">
<math>~ \Phi(\vec{x})</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~ -G \int \frac{\rho(\vec{x}^{~'})}{|\vec{x}^{~'} - \vec{x}|} d^3x^' \, .</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
[<b>[[User:Tohline/Appendix/References#BT87|<font color="red">BT87</font>]]</b>], p. 31, Eq. (2-3)<br />
[<b>[[User:Tohline/Appendix/References#EFE|<font color="red">EFE</font>]]</b>], &sect;10, p. 17, Eq. (11)<br />
[<b>[[User:Tohline/Appendix/References#T78|<font color="red">T78</font>]]</b>], &sect;4.2, p. 77, Eq. (12)
  </td>
</tr>
</table>
</div>


=Drawn from Other Wiki Pages=
(Note: &nbsp; As we have detailed in a [[User:Tohline/VE#Setting_the_Stage|separate discussion]], throughout [<b>[[User:Tohline/Appendix/References#EFE|<font color="red">EFE</font>]]</b>] Chandrasekhar adopts a ''different sign convention'' as well as a different variable name to represent the gravitational potential.)  Recognizing that the gradient of the function, <math>~|\vec{x}^{~'} - \vec{x}|^{-1}</math>, with respect to <math>~\vec{x}</math> is,
It is clear, therefore, that Chandrasekhar uses the variable <math>~\vec{u}</math> instead of <math>~\vec{v}</math> to represent the inertial velocity field.  More importantly, he adopts a different variable name ''and a different sign convention'' to represent the gravitational potential, specifically,
<div align="center">
<div align="center">
<table border="0" cellpadding="5" align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~ - \Phi = \mathfrak{B} </math>
<math>~\nabla_x \biggl[ \frac{1}{|\vec{x}^{~'} - \vec{x}|} \biggr]</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
Line 16: Line 81:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~ G \int\limits_V \frac{\rho(\vec{x}^{~'})}{|\vec{x} - \vec{x}^{~'}|} d^3x^' \, .</math>
<math>~
\frac{\vec{x}^{~'} - \vec{x}}{|\vec{x}^{~'} - \vec{x}|^3} \, ,
</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
[<b>[[User:Tohline/Appendix/References#BT87|<font color="red">BT87</font>]]</b>], p. 31, Eq. (2-4)
   </td>
   </td>
</tr>
</tr>
</table>
</table>
</div>
</div>
Hence, care must be taken to ensure that the signs on various mathematical terms are internally consistent when mapping derivations and resulting expressions from [<b>[[User:Tohline/Appendix/References#EFE|<font color="red">EFE</font>]]</b>] into this H_Book.
and given that, in the above expression for the gravitational acceleration, the integration is taken over the volume that is identified by the ''primed'' <math>~(\vec{x}~{'})</math>, rather than the unprimed <math>~(\vec{x})</math>, coordinate system, <font color="#007700">we find that we may write</font> the gravitational acceleration as,
<div align="center">
<table border="0" cellpadding="5" align="center">
 
<tr>
  <td align="right">
<math>~\vec{a}(\vec{x})</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\int G\rho(\vec{x}^{~'})  \nabla_x \biggl[ \frac{1}{|\vec{x}^{~'} - \vec{x}|} \biggr]d^3 x'
</math>
  </td>
</tr>
 
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \nabla_x \biggl\{ G \int \biggl[ \frac{\rho(\vec{x}^{~'}) }{|\vec{x}^{~'} - \vec{x}|} \biggr]d^3 x'\biggr\}</math>
  </td>
</tr>


&hellip; which <font color="#007700">expresses simply the conservation of the angular momentum of the system</font>. The symmetric part of the tensor expression gives what is generally referred to as (see [<b>[[User:Tohline/Appendix/References#EFE|<font color="red">EFE</font>]]</b>] for details) the,
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-\nabla_x \Phi \, .</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
[<b>[[User:Tohline/Appendix/References#BT87|<font color="red">BT87</font>]]</b>], p. 31, Eq. (2-5)
  </td>
</tr>
</table>
</div>
 
==Step 2==
Next, we realize that the divergence of the gravitational acceleration takes the form,
<div align="center">
<div align="center">
<span id="PGE:TVE"><font color="#770000">'''Tensor Virial Equation'''</font></span><br />
<table border="0" cellpadding="5" align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\nabla_x \cdot \vec{a}(\vec{x})</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\nabla_x \cdot \int \biggl[\frac{\vec{x}^{~'} - \vec{x}}{|\vec{x}^{~'} - \vec{x}|^3}\biggr] G\rho(\vec{x}^{~'}) d^3 x'
</math>
  </td>
</tr>
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~\frac{1}{2} \frac{d^2 I_{ij}}{dt^2}</math>
&nbsp;
   </td>
   </td>
   <td align="center">
   <td align="center">
Line 35: Line 168:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~2 \mathfrak{T}_{ij} + \mathfrak{W}_{ij} + \delta_{ij}\Pi \, ,</math>
<math>~
\int G\rho(\vec{x}^{~'}) \biggl\{ \nabla_x \cdot \biggl[\frac{\vec{x}^{~'} - \vec{x}}{|\vec{x}^{~'} - \vec{x}|^3}\biggr] \biggr\} d^3 x' \, .
</math>
   </td>
   </td>
</tr>
</tr>
<tr>
<tr>
   <td align="center" colspan="3">
   <td align="center" colspan="3">
[<b>[[User:Tohline/Appendix/References#EFE|<font color="red">EFE</font>]]</b>], p. 23, Eq. (51)<br />
[<b>[[User:Tohline/Appendix/References#BT87|<font color="red">BT87</font>]]</b>], p. 31, Eq. (2-6)
[<b>[[User:Tohline/Appendix/References#BT87|<font color="red">BT87</font>]]</b>], p. 213, Eq. (4-78)
   </td>
   </td>
</tr>
</tr>
</table>
</table>
</div>
</div>
Examining the expression inside the curly braces, we find that,
<div align="center">
<table border="0" cellpadding="5" align="center">


<tr>
  <td align="right">
<math>~\nabla_x \cdot \biggl[\frac{\vec{x}^{~'} - \vec{x}}{|\vec{x}^{~'} - \vec{x}|^3}\biggr] </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \frac{3}{|\vec{x}^{~'} - \vec{x}|^3}
+ 3 \biggl[ \frac{ (\vec{x}^{~'} - \vec{x}) \cdot (\vec{x}^{~'} - \vec{x}) }{|\vec{x}^{~'} - \vec{x}|^5}\biggr]
</math>
  </td>
</tr>
</table>
</div>
(Note: &nbsp; Ostensibly, this last expression is the same as equation 2-7 of [<b>[[User:Tohline/Appendix/References#BT87|<font color="red">BT87</font>]]</b>], but apparently there is a typesetting error in the BT87 publication.  As printed, the denominator of the first term on the right-hand side is <math>~|\vec{x}^{~'} - \vec{x}|^1</math>, whereas it should be <math>~|\vec{x}^{~'} - \vec{x}|^3</math> as written here.)  <font color="#007700">When <math>~(\vec{x}^{~'} - \vec{x}) \ne 0</math>, we may cancel the factor <math>~|\vec{x}^{~'} - \vec{x}|^2</math> from top and bottom of the last term in this equation to conclude that</font>,
<div align="center">
<table border="0" cellpadding="5" align="center">


<tr>
  <td align="right">
<math>~\nabla_x \cdot \biggl[\frac{\vec{x}^{~'} - \vec{x}}{|\vec{x}^{~'} - \vec{x}|^3}\biggr] = 0</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; when, &nbsp; &nbsp; &nbsp;
  </td>
  <td align="left">
<math>~
(\vec{x}^{~'} \ne \vec{x}) \, .
</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
[<b>[[User:Tohline/Appendix/References#BT87|<font color="red">BT87</font>]]</b>], p. 31, Eq. (2-8)
  </td>
</tr>
</table>
</div>
<font color="#007700">Therefore, any contribution to the integral must come from the point <math>~\vec{x}^{~'} = \vec{x}</math>, and we may restrict the volume of integration to a small sphere &hellip; centered on this point.  Since</font>, for a sufficiently small sphere, <font color="#007700">the density will be almost constant through this volume, we can take <math>~\rho(\vec{x}~{'}) = \rho(\vec{x})</math> out of the integral.</font>  Via the divergence theorem (for details, see appendix 1.B &#8212; specifically, equation 1B-42 &#8212; of [<b>[[User:Tohline/Appendix/References#BT87|<font color="red">BT87</font>]]</b>]), the remaining volume integral may be converted into a surface integral over the small volume centered on the point <math>~\vec{x}^{~'} = \vec{x}</math> and, in turn, this surface integral may be written in terms of an integral over the solid angle, <math>~d^2\Omega</math>, to give:
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\nabla_x \cdot \vec{a}(\vec{x})</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-G\rho(\vec{x}) \int d^2\Omega
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-4\pi G\rho(\vec{x}) \, .
</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
[<b>[[User:Tohline/Appendix/References#BT87|<font color="red">BT87</font>]]</b>], p. 32, Eq. (2-9b)
  </td>
</tr>
</table>
</div>
==Step 3==
Finally, combining the results of ''Step 1'' and ''Step 2'' gives the desired,
<div align="center">
<span id="PGE:Poisson"><font color="#770000">'''Poisson Equation'''</font></span><br />
{{User:Tohline/Math/EQ_Poisson01}}
</div>
which serves as one of the [[User:Tohline/PGE#Principal_Governing_Equations|principal governing equations]] in our examination of the '''Structure, Stability, &amp; Dynamics of Self-Gravitating Fluids'''.


=See Also=
=See Also=

Latest revision as of 01:38, 6 July 2017

Origin of the Poisson Equation

In deriving the,

Poisson Equation

LSU Key.png

<math>\nabla^2 \Phi = 4\pi G \rho</math>

we will follow closely the presentation found in §2.1 of [BT87].


Whitworth's (1981) Isothermal Free-Energy Surface
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According to Isaac Newton's inverse-square law of gravitation, the acceleration, <math>~\vec{a}(\vec{x})</math>, felt at any point in space, <math>~\vec{x}</math>, due to the gravitational attraction of a distribution of mass, <math>~\rho(\vec{x})</math>, is obtained by integrating over the accelerations exerted by each small mass element, <math>~\rho(\vec{x}^{~'}) d^3x'</math>, as follows:

<math>~\vec{a}(\vec{x})</math>

<math>~=</math>

<math>~ \int \biggl[\frac{\vec{x}^{~'} - \vec{x}}{|\vec{x}^{~'} - \vec{x}|^3}\biggr] G\rho(\vec{x}^{~'}) d^3 x' \, , </math>

[BT87], p. 31, Eq. (2-2)

where, <math>~G</math> is the universal gravitational constant.

Step 1

In the astrophysics literature, it is customary to adopt the following definition of the,

Scalar Gravitational Potential

<math>~ \Phi(\vec{x})</math>

<math>~\equiv</math>

<math>~ -G \int \frac{\rho(\vec{x}^{~'})}{|\vec{x}^{~'} - \vec{x}|} d^3x^' \, .</math>

[BT87], p. 31, Eq. (2-3)
[EFE], §10, p. 17, Eq. (11)
[T78], §4.2, p. 77, Eq. (12)

(Note:   As we have detailed in a separate discussion, throughout [EFE] Chandrasekhar adopts a different sign convention as well as a different variable name to represent the gravitational potential.) Recognizing that the gradient of the function, <math>~|\vec{x}^{~'} - \vec{x}|^{-1}</math>, with respect to <math>~\vec{x}</math> is,

<math>~\nabla_x \biggl[ \frac{1}{|\vec{x}^{~'} - \vec{x}|} \biggr]</math>

<math>~=</math>

<math>~ \frac{\vec{x}^{~'} - \vec{x}}{|\vec{x}^{~'} - \vec{x}|^3} \, , </math>

[BT87], p. 31, Eq. (2-4)

and given that, in the above expression for the gravitational acceleration, the integration is taken over the volume that is identified by the primed <math>~(\vec{x}~{'})</math>, rather than the unprimed <math>~(\vec{x})</math>, coordinate system, we find that we may write the gravitational acceleration as,

<math>~\vec{a}(\vec{x})</math>

<math>~=</math>

<math>~\int G\rho(\vec{x}^{~'}) \nabla_x \biggl[ \frac{1}{|\vec{x}^{~'} - \vec{x}|} \biggr]d^3 x' </math>

 

<math>~=</math>

<math>~ \nabla_x \biggl\{ G \int \biggl[ \frac{\rho(\vec{x}^{~'}) }{|\vec{x}^{~'} - \vec{x}|} \biggr]d^3 x'\biggr\}</math>

 

<math>~=</math>

<math>~-\nabla_x \Phi \, .</math>

[BT87], p. 31, Eq. (2-5)

Step 2

Next, we realize that the divergence of the gravitational acceleration takes the form,

<math>~\nabla_x \cdot \vec{a}(\vec{x})</math>

<math>~=</math>

<math>~ \nabla_x \cdot \int \biggl[\frac{\vec{x}^{~'} - \vec{x}}{|\vec{x}^{~'} - \vec{x}|^3}\biggr] G\rho(\vec{x}^{~'}) d^3 x' </math>

 

<math>~=</math>

<math>~ \int G\rho(\vec{x}^{~'}) \biggl\{ \nabla_x \cdot \biggl[\frac{\vec{x}^{~'} - \vec{x}}{|\vec{x}^{~'} - \vec{x}|^3}\biggr] \biggr\} d^3 x' \, . </math>

[BT87], p. 31, Eq. (2-6)

Examining the expression inside the curly braces, we find that,

<math>~\nabla_x \cdot \biggl[\frac{\vec{x}^{~'} - \vec{x}}{|\vec{x}^{~'} - \vec{x}|^3}\biggr] </math>

<math>~=</math>

<math>~ - \frac{3}{|\vec{x}^{~'} - \vec{x}|^3} + 3 \biggl[ \frac{ (\vec{x}^{~'} - \vec{x}) \cdot (\vec{x}^{~'} - \vec{x}) }{|\vec{x}^{~'} - \vec{x}|^5}\biggr] </math>

(Note:   Ostensibly, this last expression is the same as equation 2-7 of [BT87], but apparently there is a typesetting error in the BT87 publication. As printed, the denominator of the first term on the right-hand side is <math>~|\vec{x}^{~'} - \vec{x}|^1</math>, whereas it should be <math>~|\vec{x}^{~'} - \vec{x}|^3</math> as written here.) When <math>~(\vec{x}^{~'} - \vec{x}) \ne 0</math>, we may cancel the factor <math>~|\vec{x}^{~'} - \vec{x}|^2</math> from top and bottom of the last term in this equation to conclude that,

<math>~\nabla_x \cdot \biggl[\frac{\vec{x}^{~'} - \vec{x}}{|\vec{x}^{~'} - \vec{x}|^3}\biggr] = 0</math>

      when,      

<math>~ (\vec{x}^{~'} \ne \vec{x}) \, . </math>

[BT87], p. 31, Eq. (2-8)

Therefore, any contribution to the integral must come from the point <math>~\vec{x}^{~'} = \vec{x}</math>, and we may restrict the volume of integration to a small sphere … centered on this point. Since, for a sufficiently small sphere, the density will be almost constant through this volume, we can take <math>~\rho(\vec{x}~{'}) = \rho(\vec{x})</math> out of the integral. Via the divergence theorem (for details, see appendix 1.B — specifically, equation 1B-42 — of [BT87]), the remaining volume integral may be converted into a surface integral over the small volume centered on the point <math>~\vec{x}^{~'} = \vec{x}</math> and, in turn, this surface integral may be written in terms of an integral over the solid angle, <math>~d^2\Omega</math>, to give:

<math>~\nabla_x \cdot \vec{a}(\vec{x})</math>

<math>~=</math>

<math>~ -G\rho(\vec{x}) \int d^2\Omega </math>

 

<math>~=</math>

<math>~ -4\pi G\rho(\vec{x}) \, . </math>

[BT87], p. 32, Eq. (2-9b)

Step 3

Finally, combining the results of Step 1 and Step 2 gives the desired,

Poisson Equation

LSU Key.png

<math>\nabla^2 \Phi = 4\pi G \rho</math>

which serves as one of the principal governing equations in our examination of the Structure, Stability, & Dynamics of Self-Gravitating Fluids.

See Also

Whitworth's (1981) Isothermal Free-Energy Surface

© 2014 - 2021 by Joel E. Tohline
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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