Editing Apps/Wong1973Potential (section)

====Contribution from Individual Modes====

It is instructive to examine how large a contribution to the ''composite'' potential, <math>\Phi_\mathrm{W}</math>, is made by each term in the series summation.  Letting <math>\Phi_{\mathrm{W}0}</math> represent the dimensionless amplitude of the zeroth-order "mode", we have,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\Phi_{\mathrm{W}0} (\eta,\theta) \equiv \biggl( \frac{a}{GM} \biggr) \Phi_\mathrm{W}(\eta,\theta)\biggr|_{n=0}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
-D_0
(\cosh\eta - \cos\theta)^{1 / 2} ~A_0(\cosh\eta) \, .
</math>
  </td>
</tr>
</table>
The 2D contour plot generated by this function is displayed &#8212; along with the label, n = 0 &#8212; in the upper-right corner of the central panel of Figure 3.  For comparison and reference, the ''composite'' 2D contour plot that appears in the bottom-right panel of Figure 1 has been redisplayed in the upper-left corner of the same Figure 3 panel.  Immediately we appreciate that the zeroth-order contribution to the potential, <math>\Phi_{\mathrm{W}0}</math>, on its own plays a dominant role in determining the overall structure of the composite potential well.  Notice that this component depends on the ''polar angle'', <math>\theta</math>, only through the coefficient, <math>(\cosh\eta - \cos\theta)^{1 / 2}</math>, which it shares in common with all other terms in the series.


<table align="center" border="1" cellpadding="10">
<tr><th align="center" colspan="3">Figure 3: &nbsp; Contribution from Various ''Polar Angle'' "Modes"</th></tr>
<tr>
  <td align="center">n = 1 (magnification 2)</td>
<td align="center" rowspan="2" bgcolor="#D0FFFF">
[[File:Montage04.png|350px|Mode Ensemble]]
</td>
  <td align="center">n = 3 (magnification 100)</td>
</tr>
<tr>
<td align="center" bgcolor="#D0FFFF">
[[File:MovieWongN2.gif|300px|Contribution to potential by mode n = 1 (magnified by 2)]]
</td>
<td align="center" bgcolor="#D0FFFF">
[[File:MovieWongN4b.gif|300px|Contribution to potential by mode n = 3 (magnified by 100)]]
</td>
</tr>
</table>


The second (n = 1) term in the series is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\Phi_{\mathrm{W}1}(\eta,\theta) \equiv \biggl( \frac{a}{GM} \biggr) \Phi_\mathrm{W}(\eta,\theta)\biggr|_{n=1}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
-2D_0
(\cosh\eta - \cos\theta)^{1 / 2} ~\cos(\theta) A_1(\cosh\eta) \, .
</math>
  </td>
</tr>
</table>
The 2D contour plot generated by this function is displayed &#8212; along with the label, n = 1 &#8212; in the left-hand column of the central panel of Figure 3.  By analogy with Figure 2, above, a three-dimensional ''animated'' depiction of this same potential contour surface is displayed in the left-hand panel of Figure 3.  Note, however, that the overall amplitude has been multiplied by a factor of two &#8212; that is, the warped 3D surface has been generated by the function, <math>2\Phi_{\mathrm{W}1}</math> &#8212; in order to more vividly illustrate the structural modification to the potential that arises when this term is added to the leading, zeroth-order term.

In a directly analogous fashion, 2D contour plots showing the behavior of <math>\Phi_{\mathrm{W}2}</math>, <math>\Phi_{\mathrm{W}3}</math>, and <math>\Phi_{\mathrm{W}4}</math>, have been generated and displayed &#8212; along with the corresponding labels, n = 2, 3, &amp; 4 &#8212; in the central panel of Figure 3.  A 3D animated depiction of the warped, n = 3 surface is presented in the right-hand panel of Figure 3; note that, in this instance, the overall amplitude has been multiplied by a factor of one-hundred &#8212; that is, the warped 3D surface has been generated by the function, <math>100\Phi_{\mathrm{W}3}</math>.

A similarly vivid 3D display of the structure arising from the n = 2 and n = 4 terms in the series would have required, respectively, "magnification" of <math>\Phi_{\mathrm{W}2}</math> by approximately a factor of fifteen and "magnification" of <math>\Phi_{\mathrm{W}4}</math> by a factor of five-hundred; successively larger magnifications would be required for terms associated with successively higher ''polar angle'' indices, n.  We conclude, therefore, that a ''composite'' potential containing only the first four terms in the series represents the ''exact'' potential to an accuracy better than one percent.