Difference between revisions of "User:Tohline/Appendix/Ramblings/Additional Analytically Specified Eigenvectors for Zero-Zero Bipolytropes"
Line 244: | Line 244: | ||
<math>~ | <math>~ | ||
+ ( 4c_0 + 22)\Chi \biggl\{ [\Chi - 2]c_0^2 + [13\Chi -14 -Q (\Chi-2) ]c_0 + [42\Chi -Q (7\Chi-8) -24] \biggr\} | + ( 4c_0 + 22)\Chi \biggl\{ [\Chi - 2]c_0^2 + [13\Chi -14 -Q (\Chi-2) ]c_0 + [42\Chi -Q (7\Chi-8) -24] \biggr\} | ||
</math> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td align="right"> | |||
| |||
</td> | |||
<td align="center"> | |||
<math>~=</math> | |||
</td> | |||
<td align="left"> | |||
<math>~4c_0^3 +[ 26 - 4Q]c_0^2 + [40 - 26Q]c_0 - 40Q | |||
</math> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td align="right"> | |||
| |||
</td> | |||
<td align="center"> | |||
| |||
</td> | |||
<td align="left"> | |||
<math>~ | |||
+ 4\Chi[\Chi - 2]c_0^3 + 4\Chi[13\Chi -14 -Q (\Chi-2) ]c_0^2 + 4\Chi[42\Chi -Q (7\Chi-8) -24]c_0 | |||
</math> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td align="right"> | |||
| |||
</td> | |||
<td align="center"> | |||
| |||
</td> | |||
<td align="left"> | |||
<math>~ | |||
+ 22\Chi[\Chi - 2]c_0^2 + 22\Chi[13\Chi -14 -Q (\Chi-2) ]c_0 + 22\Chi[42\Chi -Q (7\Chi-8) -24] | |||
</math> | |||
</td> | |||
</tr> | |||
</table> | |||
</div> | |||
Using <math>~z</math> in place of <math>~c_0</math>, this can be written in the form of a standard cubic equation. Specifically, | |||
<div align="center"> | |||
<table border="0" cellpadding="5" align="center"> | |||
<tr> | |||
<td align="right"> | |||
<math>~a z^3 + b z^2 + c z + d</math> | |||
</td> | |||
<td align="center"> | |||
<math>~=</math> | |||
</td> | |||
<td align="left"> | |||
<math>~ | |||
0 \, , | |||
</math> | |||
</td> | |||
</tr> | |||
</table> | |||
</div> | |||
where, | |||
<div align="center"> | |||
<table border="0" cellpadding="5" align="center"> | |||
<tr> | |||
<td align="right"> | |||
<math>~a</math> | |||
</td> | |||
<td align="center"> | |||
<math>~\equiv</math> | |||
</td> | |||
<td align="left"> | |||
<math>~ | |||
4 + 4\Chi(\Chi - 2)\, , | |||
</math> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td align="right"> | |||
<math>~b</math> | |||
</td> | |||
<td align="center"> | |||
<math>~\equiv</math> | |||
</td> | |||
<td align="left"> | |||
<math>~ | |||
( 26 - 4Q) + 4\Chi[ 13\Chi -14 -Q (\Chi-2) ] + 22\Chi (\Chi - 2)\, , | |||
</math> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td align="right"> | |||
<math>~c</math> | |||
</td> | |||
<td align="center"> | |||
<math>~\equiv</math> | |||
</td> | |||
<td align="left"> | |||
<math>~ | |||
[40 - 26Q]+ 4\Chi[42\Chi -Q (7\Chi-8) -24] + 22\Chi[13\Chi -14 -Q (\Chi-2) ] \, , | |||
</math> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td align="right"> | |||
<math>~d</math> | |||
</td> | |||
<td align="center"> | |||
<math>~\equiv</math> | |||
</td> | |||
<td align="left"> | |||
<math>~ | |||
- 40Q+ 22\Chi[42\Chi -Q (7\Chi-8) -24] \, . | |||
</math> | </math> | ||
</td> | </td> |
Revision as of 03:46, 19 December 2016
Searching for Additional Eigenvectors of Zero-Zero Bipolytropes
This chapter is an extension of two accompanying discussions: The original discovery and detailed derivation; and the more readable, summary.
| Tiled Menu | Tables of Content | Banner Video | Tohline Home Page | |
In our accompanying summary, we demonstrated how analytically specified eigenvectors can be constructed for the mode labeled, <math>~(\ell, j) = (2,1)</math>. This was done by specifying <math>~\gamma_e</math>, then solving a quartic equation for <math>~q</math>. Shortly after completing this summary chapter, we noticed that an alternate approach may be to specify <math>~q</math>, then solve for <math>~\gamma_e</math>; and this path may be simpler because it may only involve solution of a quadratic equation. If this proves to be the case, then it may also be possible to analytically construct eigenvectors of additional modes. Let's see.
Seek Alternate Solution
According to our accompanying summary discussion, we need to solve the following "matching" expression:
<math>~\frac{14(1+2q^3)^2}{7(1+2q^3)^2 - 5}</math> |
<math>~=</math> |
<math>~ \frac{c_0 + (c_0 + 3)A_{21}q^3 + (c_0 + 6)A_{21}B_{21} q^6}{1 + A_{21}q^3 + A_{21}B_{21}q^6} \, , </math> |
where, recognizing that, <math>~\alpha_e = c_0(c_0+2) \, ,</math>
<math>~A_{21}</math> |
<math>~\equiv</math> |
<math>~\biggl[ \frac{c_0(c_0+5) - (c_0 + 6)(c_0 + 11)}{(c_0 + 3)(c_0+5) - \alpha_e}\biggr] </math> |
|
<math>~=</math> |
<math>~\biggl[ \frac{c_0^2 + 5c_0 - (c_0^2 + 17c_0 + 66)}{(c_0^2 + 8c_0 + 15) - (c_0^2+2c_0)}\biggr] </math> |
|
<math>~=</math> |
<math>~-\biggl( \frac{ 4c_0 + 22}{2c_0 + 5}\biggr) \, ,</math> |
<math>~B_{21}</math> |
<math>~\equiv</math> |
<math>~\biggl[ \frac{(c_0+3)(c_0+8) - (c_0 + 6)(c_0 + 11)}{(c_0 + 6)(c_0+8) - \alpha_e}\biggr] </math> |
|
<math>~=</math> |
<math>~\biggl[ \frac{(c_0^2 +11c_0 + 24) - (c_0^2 + 17c_0 + 66)}{(c_0^2+14c_0+48) - (c_0^2 + 2c_0)}\biggr] </math> |
|
<math>~=</math> |
<math>~-\biggl( \frac{c_0 + 7 }{2c_0+8}\biggr) \, . </math> |
Here, we will assume that <math>~\Chi \equiv q^3</math> is specified, and we seek the corresponding value of <math>~c_0</math>. Given that the LHS of this matching relation is known once <math>~\Chi</math> has been specified, to simply notation, let's also define,
<math>~Q</math> |
<math>~\equiv</math> |
<math>~\frac{14(1+2\Chi)^2}{7(1+2\Chi)^2 - 5} \, .</math> |
Then the matching relation becomes,
<math>~Q</math> |
<math>~=</math> |
<math>~ \frac{c_0 + (c_0 + 3)A_{21}\Chi + (c_0 + 6)A_{21}B_{21} \Chi^2}{1 + A_{21}\Chi + A_{21}B_{21}\Chi^2} </math> |
<math>~\Rightarrow~~~ 0</math> |
<math>~=</math> |
<math>~[c_0 + (c_0 + 3)A_{21}\Chi + (c_0 + 6)A_{21}B_{21} \Chi^2 ] - Q[1 + A_{21}\Chi + A_{21}B_{21}\Chi^2 ] </math> |
|
<math>~=</math> |
<math>~\biggl[c_0 - (c_0 + 3)\biggl( \frac{ 4c_0 + 22}{2c_0 + 5}\biggr)\Chi + (c_0 + 6)\biggl( \frac{ 4c_0 + 22}{2c_0 + 5}\biggr)\biggl( \frac{c_0 + 7 }{2c_0+8}\biggr) \Chi^2 \biggr] - Q\biggl[1 - \biggl( \frac{ 4c_0 + 22}{2c_0 + 5}\biggr)\Chi + \biggl( \frac{ 4c_0 + 22}{2c_0 + 5}\biggr)\biggl( \frac{c_0 + 7 }{2c_0+8}\biggr)\Chi^2 \biggr] </math> |
|
<math>~=</math> |
<math>~\biggl[c_0(2c_0+5)(2c_0+8) - (c_0 + 3)(2c_0+8)( 4c_0 + 22)\Chi + (c_0 + 6)( 4c_0 + 22)(c_0 + 7 ) \Chi^2 \biggr] </math> |
|
|
<math>~ - Q\biggl[(2c_0+5)(2c_0+8) - (2c_0+8)( 4c_0 + 22)\Chi + ( 4c_0 + 22)( c_0 + 7 )\Chi^2 \biggr] </math> |
|
<math>~=</math> |
<math>~\biggl\{c_0(4c_0^2 + 26c_0 + 40) + ( 4c_0 + 22)\Chi [(c_0^2 + 13c_0 + 42 ) \Chi- (2c_0^2 +14c_0 +24)] \biggr\} </math> |
|
|
<math>~ - Q\biggl\{ (4c_0^2 + 26c_0 + 40) + ( 4c_0 + 22)\Chi [c_0(\Chi-2) + (7\Chi-8) ] \biggr\} </math> |
|
<math>~=</math> |
<math>~4c_0^3 +[ 26 - 4Q]c_0^2 + [40 - 26Q]c_0 - 40Q </math> |
|
|
<math>~ + ( 4c_0 + 22)\Chi \biggl\{ [\Chi - 2]c_0^2 + [13\Chi -14 -Q (\Chi-2) ]c_0 + [42\Chi -Q (7\Chi-8) -24] \biggr\} </math> |
|
<math>~=</math> |
<math>~4c_0^3 +[ 26 - 4Q]c_0^2 + [40 - 26Q]c_0 - 40Q </math> |
|
|
<math>~ + 4\Chi[\Chi - 2]c_0^3 + 4\Chi[13\Chi -14 -Q (\Chi-2) ]c_0^2 + 4\Chi[42\Chi -Q (7\Chi-8) -24]c_0 </math> |
|
|
<math>~ + 22\Chi[\Chi - 2]c_0^2 + 22\Chi[13\Chi -14 -Q (\Chi-2) ]c_0 + 22\Chi[42\Chi -Q (7\Chi-8) -24] </math> |
Using <math>~z</math> in place of <math>~c_0</math>, this can be written in the form of a standard cubic equation. Specifically,
<math>~a z^3 + b z^2 + c z + d</math> |
<math>~=</math> |
<math>~ 0 \, , </math> |
where,
<math>~a</math> |
<math>~\equiv</math> |
<math>~ 4 + 4\Chi(\Chi - 2)\, , </math> |
<math>~b</math> |
<math>~\equiv</math> |
<math>~ ( 26 - 4Q) + 4\Chi[ 13\Chi -14 -Q (\Chi-2) ] + 22\Chi (\Chi - 2)\, , </math> |
<math>~c</math> |
<math>~\equiv</math> |
<math>~ [40 - 26Q]+ 4\Chi[42\Chi -Q (7\Chi-8) -24] + 22\Chi[13\Chi -14 -Q (\Chi-2) ] \, , </math> |
<math>~d</math> |
<math>~\equiv</math> |
<math>~ - 40Q+ 22\Chi[42\Chi -Q (7\Chi-8) -24] \, . </math> |
Related Discussions
© 2014 - 2021 by Joel E. Tohline |