Editing SSC/Structure/Lane1870 (section)

==SEGMENT I==

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{{ Lane1870figure }}

'''SEGMENT I''' (beginning on p. 57)
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[Read (by J. Homer Lane at age 49) before the National Academy of Sciences at the session of April 13-16, 1869.]
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Many years have passed since the suggestion was thrown out by Helmholtz, and afterwards by others, that the present volume of the sun is maintained by his internal heat, and may become less in time.  Upon this hypothesis it was proposed to account for the renewal of the heat radiated from the sun, by means of the mechanical power of the sun's mass descending toward his center.  Calculations made by Prof. Pierce, and I believe by others, have shown that this provides a supply of heat far greater than it is possible to attribute to the meteoric theory of Prof. Wm. Thomson, which, I understand, has been abandoned by Prof. Thomson himself as not reconcilable with astronomical facts.  Some years ago the question occurred to me in connection with this theory of Helmholtz whether the entire mass of the sun might not be a mixture of transparent gases, and whether Herschel's clouds might not arise from the precipitation of some of these gases, say carbon, near the surface, with their revaporization when fallen or carried into the hotter subjacent layers of atmosphere beneath; the circulation necessary for the play of this Espian theory being of course maintained by the constant disturbance of equilibrium due to the loss of heat by radiation from the precipitated clouds.  Prof. Espy's theory of storms I first became acquainted with more than twenty years ago from lectures delivered by himself, and, original as I suppose it to be, and well supported as it is in the phenomena of terrestrial meteorology, I have long thought that Prof. Espy's labors deserve a more general recognition than they have received abroad.  It is not surprising, therefore, in a time when the constitution of the sun was exciting so much discussion, that the above suggestions should have occurred to myself before I became aware of the very similar, and in the main identical, views of Prof. Faye, put forth in the [https://en.wikipedia.org/wiki/Comptes_rendus_de_l'Académie_des_Sciences Comptes Rendus].  I sought to determine how far such a supposed constitution of the sun could be made to connect with the laws of the gases as known to us in terrestrial experiments at common temperatures.  Some calculations based upon conjectures of the highest temperature and least density thought supposable at the sun's photosphere led me to the conclusion that it was extremely difficult if not impossible, to make out the connection in a credible manner.  Nevertheless, I mentioned my ideas to Prof. Henry, Secretary of the Smithsonian Institution, when he immediately referred me to a number of the [https://en.wikipedia.org/wiki/Comptes_rendus_de_l'Académie_des_Sciences Comptes Rendus], then recently received, containing Faye's exposition of his theory.  Of course nothing is further from my purpose than to make any kind of claim to any thing in that publication.  After becoming acquainted with his labors I still regarded the theory as seriously lacking, in its physical or mechanical aspect, the direct support of confirmatory observations, and even as being subject to grave difficulty in that direction.  In this attitude I allowed the subject to rest until my friend Dr. Craig, in charge of the Chemical Laboratory of the Surgeon General's office, without any knowledge of Faye's memoir, or of my own suggestions previously made to Prof. Henry and another scientific friend, fell upon the same ideas of the sun's constitution, availing himself, precisely as I had done, of Espy's theory of storms.  Dr. Craig's ideas were communicated to a company of scientific gentlemen early last spring, and soon after, Prof. Newcomb, of the U. S. Naval Observatory, entered into a general survey of the nebular hypothesis.  These communications of Dr. Craig and Prof. Newcomb led me to enter into a renewed examination of the mechanical embarrassment under which I had believed the theory to labor.  Not any longer relying on my first rough estimate based on assumed high temperatures at the photosphere, the question was now inverted.  Assuming the gaseous constitution, and assuming the laws expressed in Poisson's formul&aelig;, known to govern the constitution of gases at common temperatures and densities, what shall we find to be the temperatures and densities corresponding to the observed volume of the sun supposing it were composed of some known gas such as hydrogen, or supposing it to be composed of such a mixture of gases as would be represented by common air.  Pure hydrogen will, of course, give us the lowest temperature of all known substances, under the general hypothesis.

The question was resolved, and the results were communicated in graphical and numerical form in May or June last to two or three scientific friends, but their publication has been delayed by an unavoidable absence of several months from home.

Premising that the unit of density shall correspond to a unit of mass in the cube of the unit of length, the unit of force to the force of terrestrial gravity in the unit of mass, and the unit of pressure or elasticity in the gas to the unit of force on a surface equal to the square of the unit of length:
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  <td align="center" colspan="3">'''Lane's Notation'''<br />
----
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  <td align="center">Modern Notation<br />
----
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Let &nbsp;<math>~r</math>
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<math>~=</math>
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the distance of an element of the sun's mass from the sun's center,
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  <td align="center"><math>~r</math></td>
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<math>~t</math>
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<math>~=</math>
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the temperature of the element,
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  <td align="center">{{ Math/VAR_Temperature01 }}</td>
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<math>~\sigma t</math>
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<math>~=</math>
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its atmospheric [https://en.wikipedia.org/wiki/Subtangent subtangent], referred to the force of gravity at the earth's surface, or height of the column of homogeneous gas, whose terrestrial gravitating force would equal its elasticity,
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  <td align="center"><math>~\frac{R_\oplus^2}{GM_\oplus}\biggl( \frac{\Re}{\bar\mu} \biggr) T</math></td>
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<math>~\rho</math>
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<math>~=</math>
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its density, or mass of its unit volume
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  <td align="center"><math>~\rho</math></td>
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<font size="+1">?</font>
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<math>~=</math>
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force of terrestrial gravity in its unit volume,
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  <td align="center"><font size="+1">?</font></td>
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<math>~\rho\sigma t</math>
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<math>~=</math>
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its elasticity, or elastic force per unit surface,
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  <td align="center"><math>\biggl( \frac{R_\oplus^2}{GM_\oplus} \biggr)</math> {{ Math/VAR_Pressure01 }}</td>
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<math>~R</math>
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<math>~=</math>
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the earth's radius,
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  <td align="center"><math>~R_\oplus</math></td>
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<math>~M</math>
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<math>~=</math>
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the earth's mass,
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  <td align="center"><math>~M_\oplus</math></td>
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<math>~m</math>
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<math>~=</math>
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the mass of the part of the sun's body contained in the concentric sphere whose radius is <math>~r</math>,
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  <td align="center"><math>~M_r</math></td>
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<math>~\frac{M~r^2}{m~R^2} ~\sigma t</math>
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<math>~=</math>
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the [https://en.wikipedia.org/wiki/Subtangent subtangent] of the gas under its actual gravitating force in the sun.
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  <td align="center"><math>~\frac{r^2}{GM_r}\biggl( \frac{\Re}{\bar\mu} \biggr) T</math></td>
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The condition of equilibrium between the gravitating force of a thin horizontal layer of gas whose thickness is <math>~dr</math>, and the difference of elastic force between its lower and upper surfaces is expressed by the equation,
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  <td align="center" colspan="3">'''Lane's Notation'''<br />
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  <td align="center">&nbsp; &nbsp; &nbsp; </td>
  <td align="center" colspan="3">Modern Notation<br />
----
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<math>~d(\rho \sigma t)</math>
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<math>~=</math>
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<math>~- \frac{m~R^2}{M~r^2} ~\rho dr \, .</math>
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  <td align="center">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; </td>
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<math>~\biggl( \frac{R_\oplus^2}{GM_\oplus} \biggr)dP</math>
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<math>~=</math>
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<math>~- \biggl( \frac{R_\oplus^2}{GM_\oplus} \biggr)\frac{GM_r}{r^2} ~\rho dr \, .</math>
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We see that, drawing upon his background in meteorology, Lane is suggesting that the radial dependence of various internal properties of the Sun is governed by the ''key'' integro-differential equation that is now commonly referred to as a statement of,
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<font color="maroon">'''Hydrostatic Balance'''</font>

{{ Math/EQ_SShydrostaticBalance01 }}
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where,
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{{ Math/EQ_SSmassConservation01 }}
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Our derivation of this mathematical statement of hydrostatic balance has been tagged, ''[[SSCpt2/SolutionStrategies#Technique_1|Technique 1]]'', in our accompanying discussion of various [[SSCpt2/SolutionStrategies#Solution_Strategies|solution strategies]].