Editing ParabolicDensity/Axisymmetric/Structure/Try1thru7 (section)

===6<sup>th</sup> Try===

====Euler Equation====

From, for example, [[PGE/Euler#in_terms_of_velocity:_2|here]] we can write the,

<div align="center">
<span id="ConservingMomentum:Eulerian"><font color="#770000">'''Eulerian Representation'''</font></span><br />
of the Euler Equation,

{{Template:Math/EQ_Euler02}}
</div>
In  steady-state, we should set <math>\partial\vec{v}/\partial t = 0</math>.  There are various ways of expressing the nonlinear term on the LHS; from [[PGE/Euler#in_terms_of_the_vorticity:|here]], for example, we find,
<div align="center">
<math>
(\vec{v}\cdot\nabla)\vec{v} = \frac{1}{2}\nabla(\vec{v}\cdot\vec{v}) - \vec{v}\times(\nabla\times\vec{v})
= \frac{1}{2}\nabla(v^2) + \vec{\zeta}\times \vec{v} ,
</math>
</div>
where,
<div align="center">
<math>
\vec\zeta \equiv \nabla\times\vec{v}
</math>
</div>
is commonly referred to as the [https://en.wikipedia.org/wiki/Vorticity vorticity].

====Axisymmetric Configurations====

From, for example, [[AxisymmetricConfigurations/PGE#CYLconvectiveOperator|here]], we appreciate that, quite generally, for axisymmetric systems when written in cylindrical coordinates,

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>
(\vec{v} \cdot \nabla )\vec{v} 
</math>
</td>
<td align="center"> 
=
</td>
<td align="left">
<math>
\hat{e}_\varpi \biggl[ v_\varpi \frac{\partial v_\varpi}{\partial\varpi} + v_z \frac{\partial v_\varpi}{\partial z} - \frac{v_\varphi v_\varphi}{\varpi}  \biggr]
+ \hat{e}_\varphi \biggl[ v_\varpi \frac{\partial v_\varphi}{\partial \varpi}  + v_z \frac{\partial v_\varphi}{\partial z} + \frac{v_\varphi v_\varpi}{\varpi}  \biggr]
+ \hat{e}_z \biggl[ v_\varpi \frac{\partial v_z}{\partial\varpi}  + v_z \frac{\partial v_z}{\partial z} \biggr] \, .
</math>
</td>
</tr>
</table>
We seek steady-state configurations for which <math>v_\varpi =0</math>  and <math>v_z = 0</math>, in which case this expression simplifies considerably to,

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>
(\vec{v} \cdot \nabla )\vec{v} 
</math>
  </td>
  <td align="center"> 
<math>=</math>
  </td>
  <td align="left">
<math>
\hat{e}_\varpi \biggl[ - \frac{v_\varphi v_\varphi}{\varpi}  \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center"> 
<math>=</math>
  </td>
  <td align="left">
<math>
\hat{e}_\varpi \biggl[ - \frac{j^2}{\varpi^3}  \biggr]
\, ,
</math>
  </td>
</tr>
</table>
where, in this last expression we have replaced <math>v_\varphi</math> with the specific angular momentum, <math>j \equiv \varpi v_\varphi = (\varpi^2 \dot\varphi)</math>, which is a [[AxisymmetricConfigurations/PGE#Conservation_of_Specific_Angular_Momentum_(CYL.)|conserved quantity in dynamically evolving systems]].  NOTE: &nbsp; Up to this point in our discussion, <math>j</math> can be a function of both coordinates, that is, <math>j = j(\varpi, z)</math>.

As has been highlighted [[AxisymmetricConfigurations/PGE#RelevantCylindricalComponents|here]] for example &#8212; for the axisymmetric configurations under consideration &#8212; the <math>\hat{e}_\varpi</math> and <math>\hat{e}_z</math> components of the Euler equation become, respectively,</span>
<table border="1" align="center" cellpadding="10"><tr><td align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right"><math>{\hat{e}}_\varpi</math>: &nbsp; &nbsp;</td>
  <td align="right">
<math>
- \frac{j^2}{\varpi^3}  
</math>
  </td>
  <td align="center">
=
  </td>
  <td align="left">
<math>
- \biggl[ \frac{1}{\rho}\frac{\partial P}{\partial\varpi} + \frac{\partial \Phi}{\partial\varpi}\biggr]  
</math>
  </td>
</tr>
<tr>
  <td align="right"><math>{\hat{e}}_z</math>: &nbsp; &nbsp;</td>
  <td align="right">
<math>
0 
</math>
  </td>
  <td align="center">
=
  </td>
  <td align="left">
<math>
- \biggl[ \frac{1}{\rho}\frac{\partial P}{\partial z} + \frac{\partial \Phi}{\partial z} \biggr] 
</math>
  </td>
</tr>
</table>
</td></tr></table>

====Strategy====

<font color="red">STEP 1:</font> &nbsp; For the problem being tackled here, we start by recognizing that when considering hydrostatic balance in the <math>\hat{e}_z</math> direction, we have analytically known expressions for both <math>\rho(\varpi, z)</math> and <math>\partial\Phi/\partial z</math>.  This means, therefore, that we can construct an analytical expression for the vertical component of the pressure gradient, specifically,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>
\frac{\partial P}{\partial z} 
</math>
  </td>
  <td align="center">
=
  </td>
  <td align="left">
<math>
- \rho \cdot \frac{\partial \Phi}{\partial z} \, . 
</math>
  </td>
</tr>

<tr>
  <td align="right"><math>\Rightarrow ~~~ \frac{1}{(\pi G\rho_c^2 a_\ell^2)} \cdot \frac{\partial P}{\partial \zeta}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{\rho}{\rho_c} \cdot \frac{\partial}{\partial \zeta} \biggl\{
\frac{\Phi_\mathrm{grav}}{(-\pi G\rho_c a_\ell^2)}
\biggr\}
</math>
  </td>
</tr>
</table>

<font color="red">STEP 2:</font> &nbsp; Because we want the meridional-plane, constant-pressure contours to align with the meridional-plane, constant density contours, we can determine the radial component of the pressure gradient by forcing the slope of the tangent  vector to match the tangent vector of the density contour.

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{\partial P}{\partial \zeta}
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
- \frac{1}{m}\biggl[\frac{\partial P}{\partial \chi}\biggr]
=
-\frac{\chi(1-e^2)}{\zeta} \biggl[\frac{\partial P}{\partial \chi}\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ \frac{1}{(\pi G\rho_c^2 a_\ell^2)} \cdot \frac{\partial P}{\partial \chi}
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
-\frac{\zeta}{\chi(1-e^2)} \biggl[\frac{1}{(\pi G\rho_c^2 a_\ell^2)} \cdot \frac{\partial P}{\partial \zeta}\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
-\frac{\zeta}{\chi(1-e^2)} 
\biggl\{
\frac{\rho}{\rho_c} \cdot \frac{\partial}{\partial \zeta} \biggl[
\frac{\Phi_\mathrm{grav}}{(-\pi G\rho_c a_\ell^2)}
\biggr]
\biggr\} 
\, .
</math>
  </td>
</tr>
</table>

<font color="red">STEP 3:</font> &nbsp; Via the radial component of the hydrostatic balance expression, we can determine analytically the distribution of specific angular momentum.
<table border="0" align="center" cellpadding="8">
<tr>
  <td align="right">
<math>
\frac{j^2}{\varpi^3}  
</math>
  </td>
  <td align="center">
=
  </td>
  <td align="left">
<math>
\biggl[ \frac{1}{\rho}\frac{\partial P}{\partial\varpi} + \frac{\partial \Phi}{\partial\varpi}\biggr]  
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~
\frac{j^2 }{(\pi G \rho_c a_\ell^4)} \cdot \frac{1}{\chi^3}
</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl(\frac{\rho}{\rho_c}\biggr)^{-1} \biggl[\frac{1}{(\pi G \rho_c^2 a_\ell^2)}  \biggr]\frac{\partial P}{\partial \chi}
-
\frac{\partial}{\partial \chi} \biggl\{
\frac{\Phi_\mathrm{grav}}{(-\pi G \rho_c a_\ell^2)}
\biggr\}
</math>
  </td>
</tr>
</table>

<font color="red">STEP 4:</font> &nbsp; From knowledge of both components of <math>\nabla P</math>, see if the expression for the pressure can be ascertained.

====Implication====

Hence,
<table border="0" align="center" cellpadding="8">

<tr>
  <td align="right">
<math>
\frac{j^2 }{(\pi G \rho_c a_\ell^4)} \cdot \frac{1}{\chi^3}
</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
-\frac{\zeta}{\chi(1-e^2)} 
\cdot
\frac{\partial}{\partial \zeta} \biggl[
\frac{\Phi_\mathrm{grav}}{(-\pi G\rho_c a_\ell^2)}
\biggr]
-
\frac{\partial}{\partial \chi} \biggl\{
\frac{\Phi_\mathrm{grav}}{(-\pi G \rho_c a_\ell^2)}
\biggr\}
\, .
</math>
  </td>
</tr>
</table>

Now, given that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{ \Phi_\mathrm{grav}(\varpi,z)}{(-\pi G\rho_c a_\ell^2)} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{1}{2} I_\mathrm{BT}  
- A_\ell \chi^2  - A_s \zeta^2  
 + \frac{1}{2}\biggl[(A_{s s} a_\ell^2) \zeta^4
+ 2(A_{\ell s}a_\ell^2 )\chi^2 \zeta^2
+ (A_{\ell \ell} a_\ell^2)  \chi^4 \biggr]
\, ,
</math>
  </td>
</tr>
</table>
we see that the pair of partial derivative expressions are:
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{\partial}{\partial \zeta} \biggl[
\frac{\Phi_\mathrm{grav}}{(-\pi G\rho_c a_\ell^2)}
\biggr]
 </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
2\biggl[(A_{s s} a_\ell^2) \zeta^3
+ (A_{\ell s}a_\ell^2 )\chi^2 \zeta
- A_s \zeta
\biggr]
\, ;
</math>
  </td>

<tr>
  <td align="right">
<math>\frac{\partial}{\partial \chi} \biggl[
\frac{\Phi_\mathrm{grav}}{(-\pi G\rho_c a_\ell^2)}
\biggr]
 </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
2\biggl[
(A_{\ell \ell} a_\ell^2)  \chi^3 + (A_{\ell s}a_\ell^2 )\chi \zeta^2 - A_\ell \chi\biggr]
\, .
</math>
  </td>
</tr>
</table>

As a result we find,

<table border="0" align="center" cellpadding="8">

<tr>
  <td align="right">
<math>
\frac{j^2 (1-e^2)}{(\pi G \rho_c a_\ell^4)} \cdot \frac{1}{\chi^2}
</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
-2\zeta 
\biggl[(A_{s s} a_\ell^2) \zeta^3
+ (A_{\ell s}a_\ell^2 )\chi^2 \zeta
- A_s \zeta
\biggr]
-
2\chi(1-e^2)
\biggl[
(A_{\ell \ell} a_\ell^2)  \chi^3 + (A_{\ell s}a_\ell^2 )\chi \zeta^2 - A_\ell \chi\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
\Rightarrow ~~~ \frac{j^2 (1-e^2)}{(2\pi G \rho_c a_\ell^4)} \cdot \frac{1}{\chi^2}
</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl[-(A_{s s} a_\ell^2) \zeta^4
- (A_{\ell s}a_\ell^2 )\chi^2 \zeta^2
+ A_s \zeta^2
\biggr]
+
(1-e^2)
\biggl[
-(A_{\ell \ell} a_\ell^2)  \chi^4 - (A_{\ell s}a_\ell^2 )\chi^2 \zeta^2 + A_\ell \chi^2\biggr]
</math>
  </td>
</tr>
</table>

Next, regarding <font color="red">STEP 4</font>,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right"><math>\frac{1}{(\pi G\rho_c^2 a_\ell^2)} \cdot \frac{\partial P}{\partial \zeta}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{\rho}{\rho_c} \cdot \frac{\partial}{\partial \zeta} \biggl\{
\frac{\Phi_\mathrm{grav}}{(-\pi G\rho_c a_\ell^2)}
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
2\biggl[1 - \chi^2 - \zeta^2(1-e^2)^{-1} \biggr] \biggl[(A_{s s} a_\ell^2) \zeta^3
+ (A_{\ell s}a_\ell^2 )\chi^2 \zeta
- A_s \zeta
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
2\biggl[(A_{s s} a_\ell^2) \zeta^3
+ (A_{\ell s}a_\ell^2 )\chi^2 \zeta
- A_s \zeta
\biggr]
-
2\biggl[(A_{s s} a_\ell^2) \chi^2 \zeta^3
+ (A_{\ell s}a_\ell^2 )\chi^4 \zeta
- A_s \chi^2\zeta
\biggr]
-
2\biggl[(A_{s s} a_\ell^2) \zeta^5(1-e^2)^{-1}
+ (A_{\ell s}a_\ell^2 )\chi^2 \zeta^3(1-e^2)^{-1}
- A_s \zeta^3(1-e^2)^{-1}
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
2 \biggl\{
(A_{s s} a_\ell^2) \zeta^3
+ (A_{\ell s}a_\ell^2 )\chi^2 \zeta
- A_s \zeta
-(A_{s s} a_\ell^2) \chi^2 \zeta^3
- (A_{\ell s}a_\ell^2 )\chi^4 \zeta
+ A_s \chi^2\zeta
-(A_{s s} a_\ell^2) \zeta^5(1-e^2)^{-1}
- (A_{\ell s}a_\ell^2 )\chi^2 \zeta^3(1-e^2)^{-1}
+ A_s \zeta^3(1-e^2)^{-1}
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
2 \biggl\{
\biggl[ (A_{\ell s}a_\ell^2 )\chi^2 - A_s - (A_{\ell s}a_\ell^2 )\chi^4 + A_s \chi^2\biggr]\zeta

+ \biggl[ (A_{s s} a_\ell^2) -(A_{s s} a_\ell^2) \chi^2 - (A_{\ell s}a_\ell^2 )\chi^2 (1-e^2)^{-1} + A_s (1-e^2)^{-1}\biggr] \zeta^3
+ \biggl[-(A_{s s} a_\ell^2) (1-e^2)^{-1} \biggr]\zeta^5
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right"><math>\Rightarrow ~~~ \frac{12 P}{(2\pi G\rho_c^2 a_\ell^2)} </math></td>
  <td align="center"><math>\sim</math></td>
  <td align="left">
<math>
6\biggl[ (A_{\ell s}a_\ell^2 )\chi^2 - A_s - (A_{\ell s}a_\ell^2 )\chi^4 + A_s \chi^2\biggr]\zeta^2

+ 3\biggl[ (A_{s s} a_\ell^2) -(A_{s s} a_\ell^2) \chi^2 - (A_{\ell s}a_\ell^2 )\chi^2 (1-e^2)^{-1} + A_s (1-e^2)^{-1}\biggr] \zeta^4
+ 2\biggl[-(A_{s s} a_\ell^2) (1-e^2)^{-1} \biggr]\zeta^6
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
6(A_{\ell s}a_\ell^2 )\zeta^2\chi^2 - 6A_s\zeta^2 - 6(A_{\ell s}a_\ell^2 )\zeta^2\chi^4 + 6A_s \zeta^2 \chi^2

+ 3(A_{s s} a_\ell^2)\zeta^4 -3(A_{s s} a_\ell^2) \zeta^4\chi^2 - 3(A_{\ell s}a_\ell^2 )\zeta^4\chi^2 (1-e^2)^{-1} + 3A_s \zeta^4(1-e^2)^{-1} 
- 2(A_{s s} a_\ell^2) \zeta^6(1-e^2)^{-1} 
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
+ \biggl[ 3(A_{s s} a_\ell^2)\zeta^4  + 3A_s \zeta^4(1-e^2)^{-1} - 6A_s\zeta^2 - 2(A_{s s} a_\ell^2) \zeta^6(1-e^2)^{-1}\biggr] 
+ \biggl[ 6(A_{\ell s}a_\ell^2 )\zeta^2 + 6A_s \zeta^2 - 3(A_{s s} a_\ell^2) \zeta^4 - 3(A_{\ell s}a_\ell^2 )\zeta^4 (1-e^2)^{-1}\biggr]\chi^2 
+ \biggl[ - 6(A_{\ell s}a_\ell^2 )\zeta^2 \biggr] \chi^4 
\, ;
</math>
  </td>
</tr>
</table>

and,

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{1}{(\pi G\rho_c^2 a_\ell^2)} \cdot \frac{\partial P}{\partial \chi}
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
-\frac{\zeta}{\chi(1-e^2)} 
\biggl\{
\frac{\rho}{\rho_c} \cdot \frac{\partial}{\partial \zeta} \biggl\{
\frac{\Phi_\mathrm{grav}}{(-\pi G\rho_c a_\ell^2)}
\biggr\}
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
-\frac{\zeta}{\chi(1-e^2)} 
\biggl\{
2\biggl[(A_{s s} a_\ell^2) \zeta^3
+ (A_{\ell s}a_\ell^2 )\chi^2 \zeta
- A_s \zeta
\biggr]
-
2\biggl[(A_{s s} a_\ell^2) \chi^2 \zeta^3
+ (A_{\ell s}a_\ell^2 )\chi^4 \zeta
- A_s \chi^2\zeta
\biggr]
-
2\biggl[(A_{s s} a_\ell^2) \zeta^5(1-e^2)^{-1}
+ (A_{\ell s}a_\ell^2 )\chi^2 \zeta^3(1-e^2)^{-1}
- A_s \zeta^3(1-e^2)^{-1}
\biggr]
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
-\frac{1}{(1-e^2)} 
\biggl\{
2\biggl[(A_{s s} a_\ell^2) \chi^{-1}\zeta^4
+ (A_{\ell s}a_\ell^2 )\chi \zeta^2
- A_s \chi^{-1}\zeta^2
\biggr]
-
2\biggl[(A_{s s} a_\ell^2) \chi \zeta^4
+ (A_{\ell s}a_\ell^2 )\chi^3 \zeta^2
- A_s \chi\zeta^2
\biggr]
-
2\biggl[(A_{s s} a_\ell^2) \chi^{-1}\zeta^6(1-e^2)^{-1}
+ (A_{\ell s}a_\ell^2 )\chi \zeta^4(1-e^2)^{-1}
- A_s \chi^{-1}\zeta^4(1-e^2)^{-1}
\biggr]
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{2}{(1-e^2)} 
\biggl\{
-(A_{s s} a_\ell^2) \chi^{-1}\zeta^4
- (A_{\ell s}a_\ell^2 )\chi \zeta^2
+ A_s \chi^{-1}\zeta^2
+
(A_{s s} a_\ell^2) \chi \zeta^4
+ (A_{\ell s}a_\ell^2 )\chi^3 \zeta^2
- A_s \chi\zeta^2
+
(A_{s s} a_\ell^2) \chi^{-1}\zeta^6(1-e^2)^{-1}
+ (A_{\ell s}a_\ell^2 )\chi \zeta^4(1-e^2)^{-1}
- A_s \chi^{-1}\zeta^4(1-e^2)^{-1}
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{2}{(1-e^2)} 
\biggl\{
\biggl[A_s \zeta^2  -(A_{s s} a_\ell^2) \zeta^4 - A_s \zeta^4(1-e^2)^{-1} + (A_{s s} a_\ell^2) \zeta^6(1-e^2)^{-1} \biggr]\chi^{-1}
+ \biggl[
- (A_{\ell s}a_\ell^2 )\zeta^2
- A_s \zeta^2
+(A_{s s} a_\ell^2) \zeta^4
+ (A_{\ell s}a_\ell^2 )\zeta^4(1-e^2)^{-1}
\biggr]\chi
+ \biggl[(A_{\ell s}a_\ell^2 )\zeta^2 \biggr]\chi^3 
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math> \Rightarrow ~~~
\frac{(1-e^2)P}{(2\pi G\rho_c^2 a_\ell^2)} 
</math>
  </td>
  <td align="center">
<math>\sim</math>
  </td>
  <td align="left">
<math>
\biggl[A_s \zeta^2  -(A_{s s} a_\ell^2) \zeta^4 - A_s \zeta^4(1-e^2)^{-1} + (A_{s s} a_\ell^2) \zeta^6(1-e^2)^{-1} \biggr]\ln(\chi)
+ \frac{1}{2}\biggl[- (A_{\ell s}a_\ell^2 )\zeta^2- A_s \zeta^2+(A_{s s} a_\ell^2) \zeta^4+ (A_{\ell s}a_\ell^2 )\zeta^4(1-e^2)^{-1}\biggr]\chi^2
+ \frac{1}{4}\biggl[(A_{\ell s}a_\ell^2 )\zeta^2 \biggr]\chi^4 
\, .
</math>
  </td>
</tr>
</table>