Difference between revisions of "User:Tohline/Appendix/Ramblings/Dyson1893Part1"

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Line 102: Line 102:
   <td align="left">
   <td align="left">
<math>~
<math>~
\frac{4K(\mu)}{R+R_1} \, ,
\frac{2K(k)}{R_1} \, ,
</math>
</math>
   </td>
   </td>
Line 112: Line 112:
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~K(\mu)</math>
<math>~K(k)</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
Line 119: Line 119:
   <td align="left">
   <td align="left">
<math>~
<math>~
\int_0^{\pi/2} d\phi \biggl[1 - \mu^2\sin^2\phi  \bigg]^{-1 / 2}
\int_0^{\pi/2} d\phi \biggl[1 - k^2\sin^2\phi  \bigg]^{-1 / 2}
</math>
</math>
   </td>
   </td>
   <td align="center">&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp;
   <td align="center">&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp;
  <td align="right">
<math>~k</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{R_1^2-R^2}{R_1^2} \biggr]^{1 / 2} \, .
</math>
  </td>
</tr>
</table>
Taking a queue from our [[User:Tohline/2DStructure/ToroidalGreenFunction#Basic_Elements_of_a_Toroidal_Coordinate_System|accompanying discussion of toroidal coordinates]], if we adopt the variable notation,
<div align="center">
<math>~\eta \equiv \ln\biggl(\frac{R_1}{R}\biggr) \, ,</math>
</div>
then we can write,
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\cosh\eta = \frac{1}{2}\biggl[e^\eta + e^{-\eta}\biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{R^2 + R_1^2}{2RR_1} \, ,</math>
  </td>
</tr>
</table>
which implies that,
<!--
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\tanh^2\biggl(\frac{\eta}{2}\biggr) = \frac{\cosh\eta - 1}{\cosh\eta+1}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[\frac{R_1 - R}{R_1 + R}\biggr]^2 \, ,</math>
  </td>
</tr>
</table>
and,
-->
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\biggl[ \frac{2}{\coth\eta +1} \biggr]^{1 / 2} = [1 - e^{-2\eta}]^{1 / 2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ 1 - \biggl(\frac{R}{R_1}\biggr)^2 \biggr]^{1 / 2} = k \, .</math>
  </td>
</tr>
</table>
Now, if we employ the [https://dlmf.nist.gov/19.8#ii ''Descending Landen Transformation'' for the complete elliptic integral of the first kind], we can make the substitution,
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~K(k)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
(1 + \mu)K(\mu) \, ,
</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; where, &nbsp; &nbsp; &nbsp;</td>
   <td align="right">
   <td align="right">
<math>~\mu</math>
<math>~\mu</math>
Line 128: Line 209:
   <td align="center">
   <td align="center">
<math>~\equiv</math>
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\frac{1-\sqrt{1-k^2}}{1+\sqrt{1-k^2}} \, .
</math>
  </td>
</tr>
</table>
But notice that, <math>~\sqrt{1-k^2} = e^{-\eta}</math>, in which case,
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\mu </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1-e^{-\eta}}{1+e^{-\eta}}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1-R/R_1}{1+R/R_1}
</math>
  </td>
  <td align="center">
<math>~=</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
Line 136: Line 250:
</tr>
</tr>
</table>
</table>
Hence, we can write,
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\mathfrak{I}(r,\theta,c) = \frac{2K(k)}{R_1}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{2}{R_1} \biggl[(1+\mu)K(\mu) \biggr] </math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{2K(\mu)}{R_1} \biggl[1+\frac{R_1-R}{R_1+R}  \biggr] </math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{4K(\mu)}{R_1+R} \, .</math>
  </td>
</tr>
</table>
This is the expression for <math>~\mathfrak{I}(r,\theta,c) </math> that was adopted by Dyson at the beginning of his &sect;8.
<table border="1" align="center" cellpadding="8" width="70%">
<tr>
  <th align="center" bgcolor="yellow">
LaTeX mathematical expressions cut-and-pasted directly from
<br />
NIST's ''Digital Library of Mathematical Functions''
  </th>
</tr>
<tr>
  <td align="left">
As a primary point of reference, note that according to [http://dlmf.nist.gov/1.2 &sect;1.2 of NIST's ''Digital Library of Mathematical Functions''], the binomial theorem states that,
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~(a+b)^{n}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
a^{n}+\binom{n}{1}a^{n-1}b+\binom{n}{2}a^{n-2}b^{2}+\dots+\binom{n}{n-1}ab^{n-1}+b^{n},
</math>
  </td>
</tr>
</table>
where, for nonnegative integer values of <math>~k</math> and <math>~n</math> and <math>~k \le n</math>, the notation,
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\binom{n}{k}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{n!}{(n-k)!k!}=\binom{n}{n-k}.
</math>
  </td>
</tr>
</table>
----
'''Our Example:''' &nbsp;Setting <math>~a = 1</math> gives,
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~(1+b)^{n}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
1+\binom{n}{1}b+\binom{n}{2}b^{2}+\binom{n}{3}b^{3}+\binom{n}{4}b^{4}+\dots
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
1+\frac{n!}{(n-1)!}~b + \frac{n!}{(n-2)! 2!}~b^{2}  + \frac{n!}{(n-3)! 3!}~b^{3} + \frac{n!}{(n-4)! 4!}~b^{4} + \dots
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
1+ nb + \biggl[ \frac{n(n-1)}{2!}\biggr] b^{2}  + \biggl[ \frac{n(n-1)(n-2)}{3!} \biggr] b^{3} + \biggl[ \frac{n(n-1)(n-2)(n-3)}{4!} \biggr] b^{4} + \dots
</math>
  </td>
</tr>
</table>
  </td>
</tr>
</table>




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

Revision as of 20:05, 17 September 2018

Dyson (1893a) Part I: Some Details

This chapter provides some derivation details relevant to our accompanying discussion of Dyson's analysis of the gravitational potential exterior to an anchor ring.

Whitworth's (1981) Isothermal Free-Energy Surface
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Overview

In his pioneering work, F. W. Dyson (1893a, Philosophical Transactions of the Royal Society of London. A., 184, 43 - 95) and (1893b, Philosophical Transactions of the Royal Society of London. A., 184, 1041 - 1106) used analytic techniques to determine the approximate equilibrium structure of axisymmetric, uniformly rotating, incompressible tori. C.-Y. Wong (1974, ApJ, 190, 675 - 694) extended Dyson's work, using numerical techniques to obtain more accurate — but still approximate — equilibrium structures for incompressible tori having solid body rotation. Since then, Y. Eriguchi & D. Sugimoto (1981, Progress of Theoretical Physics, 65, 1870 - 1875) and I. Hachisu, J. E. Tohline & Y. Eriguchi (1987, ApJ, 323, 592 - 613) have mapped out the full sequence of Dyson-Wong tori, beginning from a bifurcation point on the Maclaurin spheroid sequence.

External Potential

On p. 59, at the end of §6 of Dyson (1893a), we find the following expression for the potential at point "P", anywhere exterior to an anchor ring:

<math>~\frac{\pi V(r,\theta)}{M}</math>

<math>~=</math>

<math>~ \mathfrak{I}(r,\theta,c) ~+~ \frac{1}{2^3}\biggl(\frac{a^2}{c}\biggr) \frac{d}{dc} \biggl[ \mathfrak{I}(r,\theta,c)\biggr] ~-~ \frac{1}{2^6\cdot 3}\biggl(\frac{a^2}{c}\biggr)^2 \frac{d^2}{dc^2} \biggl[ \mathfrak{I}(r,\theta,c)\biggr] ~+~\cdots </math>

 

 

<math>~ ~+~(-1)^{n+1} \frac{2}{2n+2} \biggl[ \frac{1\cdot 3\cdot 5 \cdots (2n-3)}{2^2\cdot 4^2\cdot 6^2\cdots(2n)^2} \biggr] \biggl(\frac{a^2}{c}\biggr)^n \frac{d^n}{dc^n} \biggl[ \mathfrak{I}(r,\theta,c)\biggr] ~+~ \cdots </math>

where (see beginning of §8 on p. 61),

<math>~\mathfrak{I}(r,\theta,c)</math>

<math>~\equiv</math>

<math>~ \int_0^\pi d\phi \biggl[r^2 - 2cr\sin\theta \cos\phi +c^2\biggr]^{-1 / 2} </math>

 

<math>~=</math>

<math>~ 2\int_0^{\pi/2} d\phi \biggl[ R_1^2 - (R_1^2-R^2)\sin^2\phi \biggr]^{-1 / 2} </math>

 

<math>~=</math>

<math>~ \frac{2}{R_1}\int_0^{\pi/2} d\phi \biggl[ 1 - \biggl( \frac{R_1^2-R^2}{R_1^2}\biggr) \sin^2\phi \biggr]^{-1 / 2} </math>

 

<math>~=</math>

<math>~ \frac{2K(k)}{R_1} \, , </math>

and, where furthermore,

<math>~K(k)</math>

<math>~=</math>

<math>~ \int_0^{\pi/2} d\phi \biggl[1 - k^2\sin^2\phi \bigg]^{-1 / 2} </math>

      and      

<math>~k</math>

<math>~\equiv</math>

<math>~ \biggl[ \frac{R_1^2-R^2}{R_1^2} \biggr]^{1 / 2} \, . </math>

Taking a queue from our accompanying discussion of toroidal coordinates, if we adopt the variable notation,

<math>~\eta \equiv \ln\biggl(\frac{R_1}{R}\biggr) \, ,</math>

then we can write,

<math>~\cosh\eta = \frac{1}{2}\biggl[e^\eta + e^{-\eta}\biggr]</math>

<math>~=</math>

<math>~\frac{R^2 + R_1^2}{2RR_1} \, ,</math>

which implies that,

<math>~\biggl[ \frac{2}{\coth\eta +1} \biggr]^{1 / 2} = [1 - e^{-2\eta}]^{1 / 2}</math>

<math>~=</math>

<math>~\biggl[ 1 - \biggl(\frac{R}{R_1}\biggr)^2 \biggr]^{1 / 2} = k \, .</math>

Now, if we employ the Descending Landen Transformation for the complete elliptic integral of the first kind, we can make the substitution,

<math>~K(k)</math>

<math>~=</math>

<math>~ (1 + \mu)K(\mu) \, , </math>

      where,      

<math>~\mu</math>

<math>~\equiv</math>

<math>~ \frac{1-\sqrt{1-k^2}}{1+\sqrt{1-k^2}} \, . </math>

But notice that, <math>~\sqrt{1-k^2} = e^{-\eta}</math>, in which case,

<math>~\mu </math>

<math>~=</math>

<math>~ \frac{1-e^{-\eta}}{1+e^{-\eta}} </math>

<math>~=</math>

<math>~ \frac{1-R/R_1}{1+R/R_1} </math>

<math>~=</math>

<math>~ \frac{R_1-R}{R_1+R} \, . </math>

Hence, we can write,

<math>~\mathfrak{I}(r,\theta,c) = \frac{2K(k)}{R_1}</math>

<math>~=</math>

<math>~ \frac{2}{R_1} \biggl[(1+\mu)K(\mu) \biggr] </math>

 

<math>~=</math>

<math>~\frac{2K(\mu)}{R_1} \biggl[1+\frac{R_1-R}{R_1+R} \biggr] </math>

 

<math>~=</math>

<math>~\frac{4K(\mu)}{R_1+R} \, .</math>

This is the expression for <math>~\mathfrak{I}(r,\theta,c) </math> that was adopted by Dyson at the beginning of his §8.

LaTeX mathematical expressions cut-and-pasted directly from
NIST's Digital Library of Mathematical Functions

As a primary point of reference, note that according to §1.2 of NIST's Digital Library of Mathematical Functions, the binomial theorem states that,

<math>~(a+b)^{n}</math>

<math>~=</math>

<math>~ a^{n}+\binom{n}{1}a^{n-1}b+\binom{n}{2}a^{n-2}b^{2}+\dots+\binom{n}{n-1}ab^{n-1}+b^{n}, </math>

where, for nonnegative integer values of <math>~k</math> and <math>~n</math> and <math>~k \le n</math>, the notation,

<math>~\binom{n}{k}</math>

<math>~=</math>

<math>~ \frac{n!}{(n-k)!k!}=\binom{n}{n-k}. </math>


Our Example:  Setting <math>~a = 1</math> gives,

<math>~(1+b)^{n}</math>

<math>~=</math>

<math>~ 1+\binom{n}{1}b+\binom{n}{2}b^{2}+\binom{n}{3}b^{3}+\binom{n}{4}b^{4}+\dots </math>

 

<math>~=</math>

<math>~ 1+\frac{n!}{(n-1)!}~b + \frac{n!}{(n-2)! 2!}~b^{2} + \frac{n!}{(n-3)! 3!}~b^{3} + \frac{n!}{(n-4)! 4!}~b^{4} + \dots </math>

 

<math>~=</math>

<math>~ 1+ nb + \biggl[ \frac{n(n-1)}{2!}\biggr] b^{2} + \biggl[ \frac{n(n-1)(n-2)}{3!} \biggr] b^{3} + \biggl[ \frac{n(n-1)(n-2)(n-3)}{4!} \biggr] b^{4} + \dots </math>



 

Whitworth's (1981) Isothermal Free-Energy Surface

© 2014 - 2021 by Joel E. Tohline
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