Editing SSC/Structure/Lane1870 (section)

==SEGMENT II==

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{{ Lane1870figure }}

'''SEGMENT II''' (beginning on p. 59)
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<font color="darkgreen">
Under the hypothesis that the [http://www.thermopedia.com/content/598/ law of Mariotte] and the law of Poisson prevail throughout the whole mass, and that this mass is in convective equilibrium, we have
<table border="0" cellpadding="5" align="center" width="50%">
<tr>
  <td align="center" colspan="4">'''Lane's Notation'''<br />
----
  </td>
  <td align="center">Modern Notation<br />
----
  </td>
</tr>

<tr>
  <td align="left">&nbsp;</td>
  <td align="right" width="20%">
<math>~\sigma</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
a constant,
  </td>
  <td align="center"><math>~\biggl(\frac{\Re}{\bar\mu}\biggr)</math>&nbsp; is a constant</td>
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<tr>
  <td align="left">(Eq. 1)</td>
  <td align="right" width="20%">
<math>~t</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~t_1 \rho^{k-1} \, ,</math>
  </td>
  <td align="center"><math>~T = T_0 \biggl( \frac{\rho}{\rho_0}\biggr)^{\gamma_g-1} \, ,</math></td>
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<math>~t_1</math> representing the value of <math>~t</math> in the part of the mass where the density is a unit.
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----
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In this last expression, the parameter <math>~k</math> that appears in the exponent <font color="darkgreen">represents the [[SR/IdealGas#gamma_g|ratio of specific heat]] of a gas under constant pressure to its specific heat under constant volume.</font>  Instead of <math>~k</math>, it is more common in modern notation to use, <math>~\gamma_g\equiv c_P/c_V</math>.
<p></p>
----
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<font color="darkgreen">
The theoretical difficulties which, if the supply of solar heat is to be kept up by the potential due to the mutual approach of the parts of the sun's mass consequent on the loss of heat by radiation, come in when we suppose a material departure from these laws of [http://www.thermopedia.com/content/598/ Mariotte] and of Poisson at the extreme temperatures and pressures in the sun's body, or how far such difficulties intervene, will be considered further on.

By means of the constant value of <math>~\sigma</math>, and the value of <math>~t</math> given in (1), the above differential equation is transformed into
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  <td align="center" colspan="3">'''Lane's Notation'''<br />
----
  </td>
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  <td align="right">
<math>~\sigma t_1 d(\rho^k)</math>
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  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \frac{m~R^2}{M~r^2} ~\rho dr </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ k \sigma t_1 \rho^{k-2} d\rho</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \frac{m~R^2}{M~r^2} ~dr </math>
  </td>
</tr>
</table>
the integral of which gives

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  <td align="center" colspan="4">'''Lane's Notation'''<br />
----
  </td>
</tr>

<tr>
  <td align="right">
<math>~- \frac{R^2}{M} ~\int_0^r\frac{m dr}{r^2} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~k \sigma t_1 \int_{\rho_0}^\rho \rho^{k-2} d\rho</math>
  </td>
  <td align="right">&nbsp; &nbsp; &nbsp; </td>
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  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{k \sigma t_1}{k-1}\biggr] \biggl[\rho^{k-1} -  \rho_0^{k-1} \biggr]</math>
  </td>
  <td align="right">&nbsp; &nbsp; &nbsp;</td>
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  <td align="right">
<math>~\Rightarrow ~~~ \biggl[1 - \biggl( \frac{\rho}{\rho_0}\biggr)^{k-1} \biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~+ \biggl[ \frac{k-1}{k \sigma t_1}\biggr] \frac{R^2}{M \rho_0^{k-1}} ~\int_0^r\frac{m dr}{r^2} \, ,</math>
  </td>
  <td align="right">&nbsp; &nbsp; &nbsp; (Eq. 2)</td>
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in which <math>~\rho_0</math> is the value of <math>~\rho</math> at the sun's center.

We have also,

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  <td align="right">
<math>~m </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~4\pi \int_0^r \rho r^2 dr = 4\pi \rho_0 \int_0^r \biggl(\frac{\rho}{\rho_0}\biggr)r^2 dr \, .</math>
  </td>
  <td align="right">&nbsp; &nbsp; &nbsp; (Eq. 3)</td>
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</table>

If now we put

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<math>~r </math>
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<math>~=</math>
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<math>~\biggl[ \frac{k\sigma M t_1}{4(k-1)R^2 \pi \rho_0^{2-k}} \biggr]^{1 / 2} x\, ,</math>
  </td>
  <td align="right">&nbsp; &nbsp; &nbsp; (Eq. 4)</td>
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</table>

we shall have

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  <td align="right">
<math>~m </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~4\pi \rho_0\biggl[ \frac{k\sigma M t_1}{4(k-1)R^2 \pi \rho_0^{2-k}} \biggr]^{3 / 2} \mu \, ,</math>
  </td>
  <td align="right">&nbsp; &nbsp; &nbsp; (Eq. 5)</td>
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</table>

in which

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  <td align="right">
<math>~\mu </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\int_0^x \biggl(\frac{\rho}{\rho_0} \biggr) x^2 dx \, ,</math>
  </td>
  <td align="right">&nbsp; &nbsp; &nbsp; (Eq. 6)</td>
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</table>

and equation (2) becomes

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  <td align="center" colspan="4">'''Lane's Notation'''<br />
----
  </td>
</tr>

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  <td align="right">
<math>~1 - \biggl( \frac{\rho}{\rho_0}\biggr)^{k-1} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\int_0^\xi \frac{\mu dx}{x^2} \, .</math>
  </td>
  <td align="right">&nbsp; &nbsp; &nbsp; (Eq. 7)</td>
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Drawing upon ''[[SSCpt2/SolutionStrategies#Technique_2|Technique 2]]'' from our discussion of various solution strategies, in a [[SSC/Structure/Polytropes#Lane-Emden_Equation|separate chapter]] we have derived what is now commonly referred to as the,
<div align="center" id="LaneEmden">
<font color="maroon">'''Polytropic Lane-Emden Equation'''</font>

{{ Math/EQ_SSLaneEmden01 }}
</div>
where, given that <math>~n = (\gamma_\mathrm{g}-1)^{-1}</math> and <math>~K_n = P_c \rho_c^{-(n+1)/n}</math>, we have,

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  <td align="center" colspan="3">'''Modern Notation'''<br />
----
  </td>
  <td align="center">&nbsp; &nbsp; &nbsp; </td>
  <td align="center" colspan="3">'''Lane's Notation'''<br />
----
  </td>
</tr>

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  <td align="right">
<math>~\Theta_H</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{\rho}{\rho_c} \biggr)^{1/n}</math>
  </td>
  <td align="center">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; </td>
  <td align="right">
<math>~\Theta_H</math>
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  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{\rho}{\rho_0} \biggr)^{k-1}</math>
  </td>
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</table>

and the dimensionless radial coordinate, <math>~\xi</math>, is defined via the relation,
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="center" colspan="3">'''Modern Notation'''<br />
----
  </td>
  <td align="center">&nbsp; &nbsp; &nbsp; </td>
  <td align="center" colspan="3">'''Lane's Notation'''<br />
----
  </td>
</tr>

<tr>
  <td align="right">
<math>~r</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[\frac{(n+1)K_n}{4\pi G} \cdot \rho_c^{(1-n)/n} \biggr]^{1/2} \xi</math>
  </td>
  <td align="center">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; </td>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
&nbsp;
  </td>
</tr>

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  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
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<math>~\biggl[\frac{\gamma_\mathrm{g}}{4\pi G(\gamma_\mathrm{g}-1)} \cdot \frac{P_c}{\rho_c^2} \biggr]^{1/2} \xi</math>
  </td>
  <td align="center">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; </td>
  <td align="right">
<math>~r</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[\frac{k}{4\pi G(k-1)} \biggl(\frac{GM_\oplus}{R_\oplus^2}\biggr)\frac{\sigma t_c}{\rho_c} \biggr]^{1/2} \xi</math>
  </td>
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  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
&nbsp;
  </td>
  <td align="center">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; </td>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[\frac{k}{4\pi (k-1)} \biggl(\frac{M_\oplus}{R_\oplus^2}\biggr)\frac{\sigma t_1 \rho_c^{k-1}}{\rho_c} \biggr]^{1/2} \xi</math>
  </td>
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</table>
Comparing this last expression with Lane's equation (4), it is therefore clear that Lane's dimensionless radial function, <math>~x</math>, is identical to the now more commonly used dimensionless radial function, <math>~\xi</math>.  Note, as well, that Integrating the ''Polytropic Lane-Emden Equation'' once gives,
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  <td align="right">
<math>~\xi^2 \frac{d\Theta_H}{d\xi}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \int_0^\xi \xi^2 \Theta_H^n d\xi</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \xi^2 \frac{d}{d\xi}\biggl( \frac{\rho}{\rho_0}  \biggr)^{k-1}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \int_0^\xi \xi^2 \biggl( \frac{\rho}{\rho_0}  \biggr) d\xi</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \mu_\mathrm{Lane} \, ,</math>
  </td>
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</table>
where the function, <math>~\mu_\mathrm{Lane}</math>, is as defined by Lane's equation (6); and integrating a second time gives,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\int_{\rho_0}^\rho d\biggl( \frac{\rho}{\rho_0}  \biggr)^{k-1}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \int_0^\xi \frac{\mu_\mathrm{Lane}}{\xi^2} d\xi</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ 1 - \biggl( \frac{\rho}{\rho_0}  \biggr)^{k-1}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~+ \int_0^\xi \frac{\mu_\mathrm{Lane}}{\xi^2} d\xi \, ,</math>
  </td>
</tr>
</table>
which is identically Lane's equation (7).  It is therefore clear that Lane's equation (7) provides a mathematical model of the same stellar-interiors problem as &#8212; and is simply an ''integrated'' form of &#8212; the 2<sup>nd</sup>-order ODE that now customarily carries his name (along with Emden's).