User:Tohline/SSC/Structure/Lane1870 - Revision history

Tohline: /* Published n = 5/2 Tabulations */

2019-07-19T16:08:41Z

Published n = 5/2 Tabulations

← Older revision		Revision as of 16:08, 19 July 2019
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	<tr><td align="left">		<tr><td align="left">
	<pre>		<pre>
	Horedt 2004 pp. 73-74 [n = 3/2]		Horedt 2004 pp. 73-74 [n = 5/2]
	2.5 <-- n		2.5 <-- n
	23.40646 <-- rho_c/rho_avg		23.40646 <-- rho_c/rho_avg

Tohline: /* Published n = 5/2 Tabulations */

2019-03-28T21:40:03Z

Published n = 5/2 Tabulations

← Older revision		Revision as of 21:40, 28 March 2019
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	<tr><td align="left">		<tr><td align="left">
	<pre>		<pre>
	Horedt 2004 p. 74 [n = 5/2]		Horedt 2004 pp. 73-74 [n = 3/2]
	2.5 <-- n		2.5 <-- n
			23.40646 <-- rho_c/rho_avg

	xi theta -theta' theta^(n+1) "mass" ~~rho_c~~/rho_avg		xi theta -theta' theta^(n+1) "mass" rho/rho_avg
	0.00 1.00000 0.00000 1.0000 0.00000 1.~~0000~~		0.00 1.00000 0.00000 1.0000 0.00000 23.4065
	0.10 0.99834 0.03325 0.9942 0.00033 1.~~0025~~		0.10 0.99834 0.03325 0.9942 0.00033 23.3092
	0.50 0.95960 0.15670 0.8656 0.03917 1.~~0636~~		0.50 0.95960 0.15670 0.8656 0.03917 21.1134
	1.00 0.85194 0.26287 0.5707 0.2629 1.~~2680~~		1.00 0.85194 0.26287 0.5707 0.2629 15.6806
	4.00 0.13768 0.13405 9.684E-04 2.1449 9.~~9463~~		4.00 0.13768 0.13405 9.684E-04 2.1449 1.646E-01
	5.00 0.~~02902~~ 0.08747 4.163E-06 2.1868 19.~~0534~~		5.00 2.9019E-02 0.08747 4.163E-06 2.1868 3.358E-03
	5.300 4.~~2595E~~-03 0.07786 5.044E-09 2.1872 22.~~6891~~		5.300 4.2594E-03 0.07786 5.044E-09 2.1872 2.772E-05
	5.3550 2.1009E-05 0.07627 4.250E-17 2.1872 23.~~4029~~		5.3550 2.1009E-05 0.07627 4.250E-17 2.1872 4.735E-11
	5.35527546 0.00000 7.626491E-02 0.000E+00 2.187200 23.~~4065~~		5.35527546 0.00000 7.626491E-02 0.000E+00 2.187200 0.000E+00
	</pre>		</pre>
	</td></tr>		</td></tr>

Tohline: /* Published n = 5/2 Tabulations */

2019-03-28T21:21:40Z

Published n = 5/2 Tabulations

← Older revision		Revision as of 21:21, 28 March 2019
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	<tr><td align="left">		<tr><td align="left">
	<pre>		<pre>
	Emden (1907) Tabelle 6 [n = 5/2]		Emden (1907) Tabelle 6 [n = 5/2]
	2.5 <-- n		2.5 <-- n
			24.08 <-- rho_c/rho_avg
	xi theta -theta' theta^(n+1) "mass" ~~rho_c~~/rho_avg
	0.00 1.00000 0.00000 1.0000 0.00000 1.~~0000~~		xi theta -theta' theta^(n+1) "mass" rho/rho_avg
	0.25 0.98971 0.08226 0.9644 0.00514 1.~~0130~~		0.00 1.00000 0.00000 1.0000 0.00000 24.076
	0.50 0.95961 0.15676 0.8656 0.03919 1.~~0632~~		0.25 0.98971 0.08226 0.9644 0.00514 23.462
	0.75 0.91242 0.21798 0.7256 0.12261 1.~~1469~~		0.50 0.95961 0.15676 0.8656 0.03919 21.718
	1.00 0.85196 0.26282 0.5708 0.26282 1.~~2683~~		0.75 0.91242 0.21798 0.7256 0.12261 19.146
	1.25 0.78246 0.29036 0.4238 0.45369 1.~~4350~~		1.00 0.85196 0.26282 0.5708 0.26282 16.130
	1.50 0.70809 0.30213 0.2988 0.67979 1.~~6549~~		1.25 0.78246 0.29036 0.4238 0.45369 13.039
	1.75 0.63246 0.29532 0.2012 0.9044 1.~~9753~~		1.50 0.70809 0.30213 0.2988 0.67979 10.158
	2.00 0.55961 0.28614 0.1311 1.1446 2.~~3299~~		1.75 0.63246 0.29532 0.2012 0.9044 7.659
	2.35 0.46331 0.26290 0.~~0677~~ 1.4516 2.~~9793~~		2.00 0.55961 0.28614 0.1311 1.1446 5.640
	2.50 0.42473 0.25080 0.~~0499~~ 1.5675 3.~~3227~~		2.35 0.46331 0.26290 6.769E-02 1.4516 3.518
	3.00 0.31000 0.20793 0.~~0166~~ 1.8714 4.~~8093~~		2.50 0.42473 0.25080 4.993E-02 1.5675 2.831
	3.5000 0.21752 0.16783 0.~~0048~~ 2.0559 6.~~9515~~		3.00 0.31000 0.20793 1.659E-02 1.8714 1.288
	4.00 0.14300 0.13445 0.~~0011~~ 2.1512 9.~~9169~~		3.5000 0.21752 0.16783 4.800E-03 2.0559 5.313E-01
	4.50 0.08263 0.10813 0.~~0001622~~ 2.1896 13.~~8722 <-~~- ~~Emden's value is 10.539~~		4.00 0.14300 0.13445 1.106E-03 2.1512 1.862E-01
	5.00 0.03384 0.08796 7.129E-06 2.1990 18.~~9480~~		4.50 0.08263 0.10813 1.622E-04 2.1896 4.725E-02
	5.400 0.00128 0.07545 7.503E-11 2.2001 23.~~8569~~		5.00 0.03384 0.08796 7.129E-06 2.1990 5.072E-03
	5.4172 0.00000 0.07500 0.0000 2.2010 24.~~0764~~		5.400 0.00128 0.07545 7.503E-11 2.2001 1.411E-06
			5.4172 0.00000 0.07500 0.0000 2.2010 0.000E+00
	</pre>		</pre>
	</td></tr>		</td></tr>

Tohline: /* Published n = 5/2 Tabulations */

2019-03-28T21:15:39Z

Published n = 5/2 Tabulations

← Older revision		Revision as of 21:15, 28 March 2019
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	<tr>		<tr>
	<td align="left">		<td align="left">
	The column labeled "mass" contains the tabulated quantity, <math>(-\xi^2 \theta^')</math>; and the column labeled "~~rho_c~~/rho_avg" is <~~math~~>~~\xi/(-3\theta^')~~</~~math~~>.		The quantity labeled "rho_c/rho_avg" is <math>[\xi/(-3\theta^')]</math>, evaluated at the surface; the column labeled "mass" contains the tabulated quantity, <math>(-\xi^2 \theta^')</math>; and the column labeled "rho/rho_avg" is θ<sup>n</sup> × rho_c/rho_avg.
	</td>		</td>
	</tr>		</tr>
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	<tr>		<tr>
	<td align="left">		<td align="left">
	The column labeled "mass" contains the tabulated quantity, <math>(-\xi^2 \theta^')</math>; and the column labeled "~~rho_c~~/rho_avg" is <~~math~~>~~\xi/(-3\theta^')~~</~~math~~>.		The quantity labeled "rho_c/rho_avg" is <math>[\xi/(-3\theta^')]</math>, evaluated at the surface; the column labeled "mass" contains the tabulated quantity, <math>(-\xi^2 \theta^')</math>; and the column labeled "rho/rho_avg" is θ<sup>n</sup> × rho_c/rho_avg.
	</td>		</td>
	</tr>		</tr>

Tohline: /* Published n = 3/2 Tabulations */

2019-03-28T21:15:20Z

Published n = 3/2 Tabulations

← Older revision		Revision as of 21:15, 28 March 2019
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	<tr>		<tr>
	<td align="left">		<td align="left">
	The column labeled "mass" contains the tabulated quantity, <math>(-\xi^2 \theta^')</math>; and the column labeled "~~rho_c~~/rho_avg" is <~~math~~>~~\xi/(-3\theta^')~~</~~math~~>.		The quantity labeled "rho_c/rho_avg" is <math>[\xi/(-3\theta^')]</math>, evaluated at the surface; the column labeled "mass" contains the tabulated quantity, <math>(-\xi^2 \theta^')</math>; and the column labeled "rho/rho_avg" is θ<sup>n</sup> × rho_c/rho_avg.
	</td>		</td>
	</tr>		</tr>
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	<tr>		<tr>
	<td align="left">		<td align="left">
	The column labeled "mass" contains the tabulated quantity, <math>(-\xi^2 \theta^')</math>; and the column labeled "~~rho_c~~/rho_avg" is <~~math~~>~~\xi/(-3\theta^')~~</~~math~~>.		The quantity labeled "rho_c/rho_avg" is <math>[\xi/(-3\theta^')]</math>, evaluated at the surface; the column labeled "mass" contains the tabulated quantity, <math>(-\xi^2 \theta^')</math>; and the column labeled "rho/rho_avg" is θ<sup>n</sup> × rho_c/rho_avg.
	</td>		</td>
	</tr>		</tr>

Tohline: /* Published n = 3/2 Tabulations */

2019-03-28T21:09:04Z

Published n = 3/2 Tabulations

← Older revision		Revision as of 21:09, 28 March 2019
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	Horedt 2004 pp. 73-74 [n = 3/2]		Horedt 2004 pp. 73-74 [n = 3/2]
	1.5 <-- n		1.5 <-- n
			5.990704011 <-- rho_c/rho_avg
	xi theta -theta' theta^(n+1) "mass" ~~rho_c~~/rho_avg
	0.00 1.00000 0.00000 1.0000 0.00000 1.~~0000~~		xi theta -theta' theta^(n+1) "mass" rho/rho_avg
	0.10 0.99833 0.03328 0.9958 0.00033 1.~~0015~~		0.00 1.00000 0.00000 1.0000 0.00000 5.990704
	0.50 0.95910 0.16054 0.9009 0.04014 1.~~0381~~		0.10 0.99833 0.03328 0.9958 0.00033 5.9757
	1.00 0.84517 0.28726 0.6567 0.2873 1.~~1604~~		0.50 0.95910 0.16054 0.9009 0.04014 5.6270
	3.00 0.15886 0.28425 1.006E-02 2.5583 3.~~5180~~		1.00 0.84517 0.28726 0.6567 0.2873 4.6547
	3.60 1.1091E-02 0.20939 1.295E-05 2.7137 5.~~7309~~		3.00 0.15886 0.28425 1.006E-02 2.5583 3.793E-01
	3.650 7.6392E-04 0.20372 1.613E-08 2.7141 5.~~9723~~		3.60 1.1091E-02 0.20939 1.295E-05 2.7137 6.997E-03
	3.65375374 0.00000 2.033013E-01 0.0000 2.714055 ~~5.9907~~		3.650 7.6392E-04 0.20372 1.613E-08 2.7141 1.265E-04
			3.65375374 0.00000 2.033013E-01 0.0000 2.714055 0
	</pre>		</pre>
	</td></tr>		</td></tr>

Tohline: /* Published n = 3/2 Tabulations */

2019-03-28T21:01:42Z

Published n = 3/2 Tabulations

← Older revision		Revision as of 21:01, 28 March 2019
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	Emden (1907) Tabelle 4 [n = 3/2]		Emden (1907) Tabelle 4 [n = 3/2]
	1.5 <-- n		1.5 <-- n
			6.000360963 <-- rho_c/rho_avg
	xi theta -theta' theta^(n+1) "mass" ~~rho_c~~/rho_avg
	0.00 1.00000 0.00000 1.0000 0.00000 1.~~0000~~		xi theta -theta' theta^(n+1) "mass" rho/rho_avg
	0.25 0.98966 0.08268 0.9744 0.00517 1.~~0079~~		0.00 1.00000 0.00000 1.0000 0.00000 6.0004
	0.50 0.95911 0.16057 0.9009 0.04014 1.~~0380~~		0.25 0.98966 0.08268 0.9744 0.00517 5.9075
	0.75 0.91008 0.22988 0.7901 0.12931 1.~~0875~~		0.50 0.95911 0.16057 0.9009 0.04014 5.6361
	1.00 0.84516 0.28727 0.6567 0.28727 1.~~1603~~		0.75 0.91008 0.22988 0.7901 0.12931 5.2095
	1.25 0.76761 0.33061 0.5162 0.51658 1.~~2603~~		1.00 0.84516 0.28727 0.6567 0.28727 4.6621
	1.50 0.68132 0.35752 0.3832 0.80442 1.~~3985~~		1.25 0.76761 0.33061 0.5162 0.51658 4.0354
	1.75 0.58994 0.37168 0.2673 1.1383 1.~~5695~~		1.50 0.68132 0.35752 0.3832 0.80442 3.3745
	2.00 0.49670 0.37209 0.1739 1.4884 1.~~7917~~		1.75 0.58994 0.37168 0.2673 1.1383 2.7189
	2.25 0.40477 0.36119 0.1042 1.8285 2.~~0765~~		2.00 0.49670 0.37209 0.1739 1.4884 2.1005
	2.50 0.31678 0.34120 0.0565 2.1325 2.~~4424~~		2.25 0.40477 0.36119 0.1042 1.8285 1.5452
	2.75 0.23468 0.34750 0.0267 2.6280 2.~~6379~~		2.50 0.31678 0.34120 0.0565 2.1325 1.0698
	2.8085 0.21617 0.30788 0.0217 2.4285 3.~~0407~~		2.75 0.23468 0.34750 0.0267 2.6280 6.8217E-01
	3.00 0.15972 0.28442 0.0102 2.5598 3.~~5159~~		2.8085 0.21617 0.30788 0.0217 2.4285 6.0307E-01
	3.25 0.09258 0.25261 0.0026 2.6682 4.~~2886~~		3.00 0.15972 0.28442 0.0102 2.5598 3.8302E-01
	3.50 0.03335 0.22147 0.0002 2.7130 5.~~2678~~		3.25 0.09258 0.25261 0.0026 2.6682 1.6903E-01
	3.625 0.00659 0.20680 0.0000 2.7175 5.~~8430~~		3.50 0.03335 0.22147 0.0002 2.7130 3.6544E-02
	3.64 0.00350 0.20511 0.0000 2.7176 5.~~9155~~		3.625 0.00659 0.20680 0.0000 2.7175 3.2100E-03
	3.6571 0.00000 0.20316 0.0000 2.7171 6.~~0004~~		3.64 0.00350 0.20511 0.0000 2.7176 1.2425E-03
			3.6571 0.00000 0.20316 0.0000 2.7171 0.0000E+00
	</pre>		</pre>
	</td></tr>		</td></tr>

Tohline: /* SEGMENT III */

2019-03-27T22:50:55Z

SEGMENT III

← Older revision		Revision as of 22:50, 27 March 2019
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	</td>		</td>
	<td align="center">		<td align="center">
	<math>~\frac{3M_\odot}{4\pi R_\odot^3} \cdot \frac{(x')^3}{3\mu^'} = \rho_\odot \cdot \frac{(x')^3}{3\mu^'} \, .</math>		<math>~\frac{3M_\odot}{4\pi R_\odot^3} \cdot \frac{(x')^3}{3\mu^'} = \bar\rho_\odot \cdot \frac{(x')^3}{3\mu^'} \, .</math>
	</td>		</td>
	</tr>		</tr>
	</table>		</table>

	Lane concludes (see near the bottom of his p. 63) that, <math>~\rho_0 = 28.16</math> for the model with <math>~k = 1.4</math>, and (see near the top of his p. 64) that, <math>~\rho_0 = 7.11</math> for the model with <math>~k = 1\tfrac{2}{3}</math>. This implies that he adopted a value of the Sun's mean density of, <math>~\rho_\odot = 1.20</math> — apparently measured, in some manner we have as yet been unable to decipher, in units of the specific gravity of the Earth. These calculated values — 28.16 and 7.11, respectively — of the central specific gravity are the only ''quantitative'' labels that appear in Lane's diagram that shows the variation with radius of the "Absolute density" in his pair of models (see further elaboration, below).		Lane concludes (see near the bottom of his p. 63) that, <math>~\rho_0 = 28.16</math> for the model with <math>~k = 1.4</math>, and (see near the top of his p. 64) that, <math>~\rho_0 = 7.11</math> for the model with <math>~k = 1\tfrac{2}{3}</math>. This implies that he adopted a value of the Sun's mean density of, <math>~\bar\rho_\odot = 1.20</math> — apparently measured, in some manner we have as yet been unable to decipher, in units of the specific gravity of the Earth. These calculated values — 28.16 and 7.11, respectively — of the central specific gravity are the only ''quantitative'' labels that appear in Lane's diagram that shows the variation with radius of the "Absolute density" in his pair of models (see further elaboration, below).
	</td>		</td>
	</tr>		</tr>

Tohline: /* SEGMENT III */

2019-03-27T22:48:07Z

SEGMENT III

← Older revision		Revision as of 22:48, 27 March 2019
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	<tr>		<tr>
	<td align="left">		<td align="left">
	<font color="red">'''NOTE:'''</font> In his equation (14), Lane states that the central density is given by the expression,		<font color="red">'''NOTE:'''</font>  In his equation (14), [https://archive.org/details/mobot31753002152772/page/56 Lane (1870)] states that the central density is given by the expression,
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
			<tr>
			<td align="center" colspan="2">      </td>
			<td align="center" colspan="1">'''Lane's Notation'''<br />
			----
			</td>
			<td align="center" colspan="2">      </td>
			<td align="center" colspan="1">'''Modern Notation'''<br />
			----
			</td>
			</tr>
	<tr>		<tr>
	<td align="right">		<td align="right">
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	<math>~=</math>		<math>~=</math>
	</td>		</td>
	<td align="~~left~~">		<td align="center">
	<math>~\frac{m' (x')^3}{4\pi \mu^' (r')^3}</math>		<math>~\frac{m' (x')^3}{4\pi \mu^' (r')^3}</math>
			</td>
			<td align="center">      </td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="center">
			<math>~\frac{3M_\odot}{4\pi R_\odot^3} \cdot \frac{(x')^3}{3\mu^'} = \rho_\odot \cdot \frac{(x')^3}{3\mu^'} \, .</math>
	</td>		</td>
	</tr>		</tr>
	</table>		</table>

			Lane concludes (see near the bottom of his p. 63) that, <math>~\rho_0 = 28.16</math> for the model with <math>~k = 1.4</math>, and (see near the top of his p. 64) that, <math>~\rho_0 = 7.11</math> for the model with <math>~k = 1\tfrac{2}{3}</math>. This implies that he adopted a value of the Sun's mean density of, <math>~\rho_\odot = 1.20</math> — apparently measured, in some manner we have as yet been unable to decipher, in units of the specific gravity of the Earth. These calculated values — 28.16 and 7.11, respectively — of the central specific gravity are the only ''quantitative'' labels that appear in Lane's diagram that shows the variation with radius of the "Absolute density" in his pair of models (see further elaboration, below).
	</td>		</td>
	</tr>		</tr>
	</table>		</table>


	[https://archive.org/details/mobot31753002152772/page/56 Lane (1870)] did not publish a table of numbers to detail how the density varied with radius throughout the interiors of his pair of polytropic models. But both [https://books.google.com/books?id=MiDQAAAAMAAJ&printsec=frontcover#v=onepage&q&f=true Emden (1907)] and [http://adsabs.harvard.edu/abs/2004ASSL..306.....H Horedt (2004)] did. From the tabulated data provided by these two researchers — see below — we have constructed curves showing how the dimensionless density (abscissa),		[https://archive.org/details/mobot31753002152772/page/56 Lane (1870)] did not publish a table of numbers to detail how the density varied with radius throughout the interiors of his pair of polytropic models. But both [https://books.google.com/books?id=MiDQAAAAMAAJ&printsec=frontcover#v=onepage&q&f=true Emden (1907)] and [http://adsabs.harvard.edu/abs/2004ASSL..306.....H Horedt (2004)] did. From the tabulated data provided by these two researchers — see below — we have constructed curves showing how the dimensionless density (abscissa),

Tohline: /* SEGMENT III */

2019-03-27T22:10:28Z

SEGMENT III

← Older revision		Revision as of 22:10, 27 March 2019
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	</table>		</table>
	The value of this density ratio, <math>~\rho_c/\bar\rho</math>, as determined for both models from the dimensionless masses and radii calculated by all three researchers has also been included for comparison in this table.		The value of this density ratio, <math>~\rho_c/\bar\rho</math>, as determined for both models from the dimensionless masses and radii calculated by all three researchers has also been included for comparison in this table.

			<table border="1" width="80%" align="center" cellpadding="10">
			<tr>
			<td align="left">
			<font color="red">'''NOTE:'''</font> In his equation (14), Lane states that the central density is given by the expression,
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\rho_0</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\frac{m' (x')^3}{4\pi \mu^' (r')^3}</math>
			</td>
			</tr>
			</table>

			</td>
			</tr>
			</table>

	[https://archive.org/details/mobot31753002152772/page/56 Lane (1870)] did not publish a table of numbers to detail how the density varied with radius throughout the interiors of his pair of polytropic models. But both [https://books.google.com/books?id=MiDQAAAAMAAJ&printsec=frontcover#v=onepage&q&f=true Emden (1907)] and [http://adsabs.harvard.edu/abs/2004ASSL..306.....H Horedt (2004)] did. From the tabulated data provided by these two researchers — see below — we have constructed curves showing how the dimensionless density (abscissa),		[https://archive.org/details/mobot31753002152772/page/56 Lane (1870)] did not publish a table of numbers to detail how the density varied with radius throughout the interiors of his pair of polytropic models. But both [https://books.google.com/books?id=MiDQAAAAMAAJ&printsec=frontcover#v=onepage&q&f=true Emden (1907)] and [http://adsabs.harvard.edu/abs/2004ASSL..306.....H Horedt (2004)] did. From the tabulated data provided by these two researchers — see below — we have constructed curves showing how the dimensionless density (abscissa),