Aps/MaclaurinSpheroidFreeFall - Revision history

Jet53man: /* Oblate Spheroids */

2022-04-09T17:20:43Z

Oblate Spheroids

← Older revision		Revision as of 13:20, 9 April 2022
Line 146:		Line 146:
	These same coefficient expressions may also be found in, for example:  Chapter 3, Eq. (36) of [<b>[[Appendix/References#EFE\|<font color="red">EFE</font>]]</b>]; §4.5, Eqs. (48) & (49) of [<b>[[Appendix/References#T78\|<font color="red">T78</font>]]</b>]; and the first column of Table 2-1 (p. 57) of [<b>[[Appendix/References#BT87\|<font color="red">BT87</font>]]</b>]. Note that, as we have pointed out in a [[Apps/MaclaurinSpheroids#Gravitational_Potential\|separate discussion of Maclaurin Spheroids]], the expressions for <math>A_\ell</math> and <math>A_s</math> have the following values in the limit of a sphere <math>(e=0)</math> or in the limit of an infinitesimally thin disk <math>(e=1)</math>:		These same coefficient expressions may also be found in, for example:  Chapter 3, Eq. (36) of [<b>[[Appendix/References#EFE\|<font color="red">EFE</font>]]</b>]; §4.5, Eqs. (48) & (49) of [<b>[[Appendix/References#T78\|<font color="red">T78</font>]]</b>]; and the first column of Table 2-1 (p. 57) of [<b>[[Appendix/References#BT87\|<font color="red">BT87</font>]]</b>]. Note that, as we have pointed out in a [[Apps/MaclaurinSpheroids#Gravitational_Potential\|separate discussion of Maclaurin Spheroids]], the expressions for <math>A_\ell</math> and <math>A_s</math> have the following values in the limit of a sphere <math>(e=0)</math> or in the limit of an infinitesimally thin disk <math>(e=1)</math>:

	<table align="center" border=1 cellpadding="10">
			<table align="center" border=1 cellpadding="8">
	<tr>		<tr>
	<td colspan="3" align="center">		<td colspan="3" align="center">
	Limiting Values~~<br />~~		'''Table 1:'''  [[Appendix/Ramblings/PowerSeriesExpressions#Maclaurin_Spheroid_Index_Symbols\|Limiting Values]] (for oblate spheroids)
	(for oblate spheroids)
	</td>		</td>
	</tr>		</tr>
Line 158:		Line 158:
	</td>		</td>
	<td align="center">		<td align="center">
	<b><math>e=0</math></b>		<b><math>e \rightarrow 0</math></b>
	</td>		</td>
	<td align="center">		<td align="center">
	<b><math>e=1</math></b>		<b><math>\frac{a_3}{a_1} \rightarrow 0</math></b>
			</td>
			</tr>
			<tr>
			<td align="center">
			<b><math>~\frac{\sin^{-1}e}{e}</math></b>
			</td>
			<td align="center">
			<math>~1 + \frac{e^2}{6} + \mathcal{O}\biggl(e^4\biggr)</math>
			</td>
			<td align="center">
			<math>~\frac{\pi}{2} - \biggl(\frac{a_3}{a_1}\biggr) +\frac{\pi}{4}\biggl(\frac{a_3}{a_1}\biggr)^2
			- \mathcal{O}\biggl(\frac{a_3^3}{a_1^3}\biggr)</math>
	</td>		</td>
	</tr>		</tr>
Line 169:		Line 181:
	</td>		</td>
	<td align="center">		<td align="center">
	<math>\frac{2}{3}</math>		<math>\frac{2}{3}\biggl[1 - \frac{e^2}{5} - \mathcal{O}\biggl(e^4\biggr)\biggr]</math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>0</math>		<math>\frac{\pi}{2} \biggl( \frac{a_3}{a_1}\biggr) - 2\biggl(\frac{a_3}{a_1}\biggr)^2+ \mathcal{O}\biggl(\frac{a_3^3}{a_1^3}\biggr)</math>
	</td>		</td>
	</tr>		</tr>
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	</td>		</td>
	<td align="center">		<td align="center">
	<math>\frac{2}{3}</math>		<math>\frac{2}{3}\biggl[1 + \frac{2e^2}{5} + \mathcal{O}\biggl(e^4\biggr)\biggr]</math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>2</math>		<math> 2 - \pi \biggl( \frac{a_3}{a_1}\biggr) + 4\biggl(\frac{a_3}{a_1}\biggr)^2
			- \mathcal{O}\biggl(\frac{a_3^3}{a_1^3}\biggr)</math>
	</td>		</td>
	</tr>		</tr>

Jet53man: /* Oblate Spheroids */

2021-11-14T17:58:02Z

Oblate Spheroids

@@ Line 144: / Line 144: @@
 </math>
 </div>
-These same coefficient expressions may also be found in, for example: &nbsp;Chapter 3, Eq. (36) of [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>]; &sect;4.5, Eqs. (48) &amp; (49) of [<b>[[Appendix/References#T78|<font color="red">T78</font>]]</b>]; and the first column of Table 2-1 (p. 57) of [<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>].
+These same coefficient expressions may also be found in, for example: &nbsp;Chapter 3, Eq. (36) of [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>]; &sect;4.5, Eqs. (48) &amp; (49) of [<b>[[Appendix/References#T78|<font color="red">T78</font>]]</b>]; and the first column of Table 2-1 (p. 57) of [<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>].  Note that, as we have pointed out in a [[Apps/MaclaurinSpheroids#Gravitational_Potential|separate discussion of Maclaurin Spheroids]], the expressions for <math>A_\ell</math> and <math>A_s</math> have the following values in the limit of a sphere <math>(e=0)</math> or in the limit of an infinitesimally thin disk <math>(e=1)</math>:
 ===Prolate Spheroids===

Jet53man: /* Discrete Representations */

2021-11-13T23:14:02Z

Discrete Representations

← Older revision		Revision as of 19:14, 13 November 2021
Line 929:		Line 929:
	\biggr] \biggr\}_n \, ,</math>		\biggr] \biggr\}_n \, ,</math>
	</td>		</td>
	<td align="right" width="25%">    <math>(0.~~0000~~)</math></td>		<td align="right" width="25%">    <math>(F_{+1/2} -5.905085\times 10^{-3})</math></td>
	</tr>		</tr>

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	\, .</math>		\, .</math>
	</td>		</td>
	<td align="right" width="25%">    <math>(0.~~0000~~)</math></td>		<td align="right" width="25%">    <math>(K_{+1/2} - 6.695907\times 10^{-3})</math></td>
	</tr>		</tr>
	</table>		</table>
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	<math>R_n + \Delta\cdot F_{n+1/2} \, ,</math>		<math>R_n + \Delta\cdot F_{n+1/2} \, ,</math>
	</td>		</td>
	<td align="right" width="25%">    <math>(0.~~0000~~)</math></td>		<td align="right" width="25%">    <math>(0.99994095)</math></td>
	</tr>		</tr>

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	<math>Z_n + \Delta \cdot K_{n+1/2} \, .</math>		<math>Z_n + \Delta \cdot K_{n+1/2} \, .</math>
	</td>		</td>
	<td align="right" width="25%">    <math>(0.~~0000~~)</math></td>		<td align="right" width="25%">    <math>(0.999933042)</math></td>
	</tr>		</tr>
	</table>		</table>
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	</math>		</math>
	</td>		</td>
	<td align="right" width="25%">    <math>(0.~~0000~~)</math></td>		<td align="right" width="25%">    <math>(1 - 0.809987188)^{1 / 2} = (0.43590459)</math></td>
	</tr>		</tr>
	</table>		</table>

Jet53man: /* Discrete Representations */

2021-11-13T22:48:14Z

Discrete Representations

← Older revision		Revision as of 18:48, 13 November 2021
Line 870:		Line 870:
	<math>1 + \Delta \cdot F_{+1/2} \, ,</math>		<math>1 + \Delta \cdot F_{+1/2} \, ,</math>
	</td>		</td>
	<td align="right" width="25%">    <math>(0.~~99998524~~)</math></td>		<td align="right" width="25%">    <math>(1.0-1.476226 \times 10^{-5})</math></td>
	</tr>		</tr>

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	<math>1 + \Delta \cdot K_{+1/2} \, .</math>		<math>1 + \Delta \cdot K_{+1/2} \, .</math>
	</td>		</td>
	<td align="right" width="25%">    <math>(0.~~99998326~~)</math></td>		<td align="right" width="25%">    <math>(1.0-1.673924\times 10^{-5})</math></td>
	</tr>		</tr>
	</table>		</table>
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	</math>		</math>
	</td>		</td>
	<td align="right" width="25%">    <math>(0.~~43589357~~)</math></td>		<td align="right" width="25%">    <math>(1 - 0.809996797)^{1 / 2} = (0.435893568)</math></td>
	</tr>		</tr>
	</table>		</table>

Jet53man: /* Discrete Representations */

2021-11-12T22:34:55Z

Discrete Representations

← Older revision		Revision as of 18:34, 12 November 2021
Line 846:		Line 846:
	<td align="left">		<td align="left">
	<math>		<math>
	- \frac{\Delta}{2} \cdot \frac{3\pi^2}{16} \biggl\{ ~~\frac{1}{R^2}~~ \biggl[		- \frac{\Delta}{2} \cdot \frac{3\pi^2}{16} \biggl\{ \biggl[
	\frac{2}{e_0^2} -		\frac{2}{e_0^2} -
	~\frac{2(1-e_0^2)^{1 / 2}}{ e_0^3 } \cdot \sin^{-1}e_0 \biggr] \biggr\}		~\frac{2(1-e_0^2)^{1 / 2}}{ e_0^3 } \cdot \sin^{-1}e_0 \biggr] \biggr\}
	\, .</math>		\, .</math>
	</td>		</td>
	<td align="right" width="25%">    <math>(0.~~0000~~)</math></td>		<td align="right" width="25%">    <math>(-3.347849\times 10^{-3})</math></td>
	</tr>		</tr>
	</table>		</table>
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	<math>1 + \Delta \cdot K_{+1/2} \, .</math>		<math>1 + \Delta \cdot K_{+1/2} \, .</math>
	</td>		</td>
	<td align="right" width="25%">    <math>(0.~~0000~~)</math></td>		<td align="right" width="25%">    <math>(0.99998326)</math></td>
	</tr>		</tr>
	</table>		</table>
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	</math>		</math>
	</td>		</td>
	<td align="right" width="25%">    <math>(0.~~0000~~)</math></td>		<td align="right" width="25%">    <math>(0.43589357)</math></td>
	</tr>		</tr>
	</table>		</table>

Jet53man: /* Discrete Representations */

2021-11-12T21:52:10Z

Discrete Representations

← Older revision		Revision as of 17:52, 12 November 2021
Line 807:		Line 807:
	<font color="red">STEP #0:</font>		<font color="red">STEP #0:</font>
	<ul>		<ul>
	<li>Choose time-independent values of the parameters, <math>(c_0/a_0)</math> and <math>\beta_0</math>, which means that, <math>\sigma_0^2 = 3\pi^2\beta_0/8</math> and <math>e_0 = [1 - (c_0/a_0)^2]^{1 / 2}</math>. Also set the (uniform) integration time step, <math>\Delta</math>; for 201 time steps, for example, set <math>\Delta \approx 0.005</math>. The example values shown below (inside parentheses) assume that <math>(c_0/a_0, \beta_0) = (0.90, 0.0)</math>, which corresponds to one of the model-evolutions presented by {{ LMS65 }}.</li>		<li>Choose time-independent values of the parameters, <math>(c_0/a_0)</math> and <math>\beta_0</math>, which means that, <math>\sigma_0^2 = 3\pi^2\beta_0/8</math> and <math>e_0 = [1 - (c_0/a_0)^2]^{1 / 2}</math>. Also set the (uniform) integration time step, <math>\Delta</math>; for 201 time steps, for example, set <math>\Delta \approx 0.005</math>.<br />
			The example values shown below (inside parentheses) assume that <math>(c_0/a_0, \beta_0) = (0.90, 0.0) ~~\Rightarrow (e_0, \sigma_0^2) = (0.43588989, 0)</math>, which corresponds to one of the model-evolutions presented by {{ LMS65 }}.</li>
	</ul>		</ul>

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	\biggr] \biggr\} \, ,</math>		\biggr] \biggr\} \, ,</math>
	</td>		</td>
	<td align="right" width="25%">    <math>(0.~~0000~~)</math></td>		<td align="right" width="25%">    <math>(-2.952452 \times 10^{-3})</math></td>
	</tr>		</tr>

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	<math>1 + \Delta \cdot F_{+1/2} \, ,</math>		<math>1 + \Delta \cdot F_{+1/2} \, ,</math>
	</td>		</td>
	<td align="right" width="25%">    <math>(0.~~0000~~)</math></td>		<td align="right" width="25%">    <math>(0.99998524)</math></td>
	</tr>		</tr>

Jet53man: /* Discrete Representations */

2021-11-12T21:37:23Z

Discrete Representations

← Older revision		Revision as of 17:37, 12 November 2021
Line 807:		Line 807:
	<font color="red">STEP #0:</font>		<font color="red">STEP #0:</font>
	<ul>		<ul>
	<li>Choose time-independent values of the parameters, <math>(c_0/a_0)</math> and <math>\beta_0</math>, which means that, <math>\sigma_0^2 = 3\pi^2\beta_0/8</math> and <math>e_0 = [1 - (c_0/a_0)^2]^{1 / 2}</math>. Also set the (uniform) integration time step, <math>\Delta</math>; for ~~200~~ time steps, for example, set <math>\Delta \approx 0.005</math>. The example values shown below (inside parentheses) assume that <math>(c_0/a_0, \beta_0) = (0.90, 0.0)</math>, which corresponds to one of the model-evolutions presented by {{ LMS65 }}.</li>		<li>Choose time-independent values of the parameters, <math>(c_0/a_0)</math> and <math>\beta_0</math>, which means that, <math>\sigma_0^2 = 3\pi^2\beta_0/8</math> and <math>e_0 = [1 - (c_0/a_0)^2]^{1 / 2}</math>. Also set the (uniform) integration time step, <math>\Delta</math>; for 201 time steps, for example, set <math>\Delta \approx 0.005</math>. The example values shown below (inside parentheses) assume that <math>(c_0/a_0, \beta_0) = (0.90, 0.0)</math>, which corresponds to one of the model-evolutions presented by {{ LMS65 }}.</li>
	</ul>		</ul>

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	<li>Use the first and third discrete equations to solve for the "R" and "Z" velocities at time step <math>n=0</math>, namely,</li>		<li>Use the first and third discrete equations to solve for the "R" and "Z" velocities at time step <math>n=0</math>, namely,</li>
	</ul>		</ul>
	<table border="0" cellpadding="3" align="center">		<table border="0" width="80%" cellpadding="3" align="center">

	<tr>		<tr>
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	\biggr] \biggr\} \, ,</math>		\biggr] \biggr\} \, ,</math>
	</td>		</td>
			<td align="right" width="25%">    <math>(0.0000)</math></td>
	</tr>		</tr>

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	\, .</math>		\, .</math>
	</td>		</td>
			<td align="right" width="25%">    <math>(0.0000)</math></td>
	</tr>		</tr>
	</table>		</table>
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	</ul>		</ul>

	<table border="0" cellpadding="3" align="center">		<table border="0" width="80%" cellpadding="3" align="center">

	<tr>		<tr>
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	<math>1 + \Delta \cdot F_{+1/2} \, ,</math>		<math>1 + \Delta \cdot F_{+1/2} \, ,</math>
	</td>		</td>
			<td align="right" width="25%">    <math>(0.0000)</math></td>
	</tr>		</tr>

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	<math>1 + \Delta \cdot K_{+1/2} \, .</math>		<math>1 + \Delta \cdot K_{+1/2} \, .</math>
	</td>		</td>
			<td align="right" width="25%">    <math>(0.0000)</math></td>
	</tr>		</tr>
	</table>		</table>
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	<li>Determine the eccentricity at this advanced time.</li>		<li>Determine the eccentricity at this advanced time.</li>
	</ul>		</ul>
	<table border="0" cellpadding="5" align="center">		<table border="0" width="80%" cellpadding="5" align="center">

	<tr>		<tr>
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	</math>		</math>
	</td>		</td>
			<td align="right" width="25%">    <math>(0.0000)</math></td>
	</tr>		</tr>
	</table>		</table>
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	<li>Given knowledge of the various variable values at time-step, <math>(n-1/2)</math> and <math>n</math>, use the first and third discrete evolution relations to determine <math>F_{n+1/2}</math> and <math>K_{n+1/2}</math>; specifically,</li>		<li>Given knowledge of the various variable values at time-step, <math>(n-1/2)</math> and <math>n</math>, use the first and third discrete evolution relations to determine <math>F_{n+1/2}</math> and <math>K_{n+1/2}</math>; specifically,</li>
	</ul>		</ul>
	<table border="0" cellpadding="3" align="center">		<table border="0" width="80%" cellpadding="3" align="center">

	<tr>		<tr>
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	\biggr] \biggr\}_n \, ,</math>		\biggr] \biggr\}_n \, ,</math>
	</td>		</td>
			<td align="right" width="25%">    <math>(0.0000)</math></td>
	</tr>		</tr>

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	\, .</math>		\, .</math>
	</td>		</td>
			<td align="right" width="25%">    <math>(0.0000)</math></td>
	</tr>		</tr>
	</table>		</table>
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	<li>Similarly, from the second and fourth discrete relations, determine the advanced coordinate positions.</li>		<li>Similarly, from the second and fourth discrete relations, determine the advanced coordinate positions.</li>
	</ul>		</ul>
	<table border="0" cellpadding="3" align="center">		<table border="0" width="80%" cellpadding="3" align="center">

	<tr>		<tr>
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	<math>R_n + \Delta\cdot F_{n+1/2} \, ,</math>		<math>R_n + \Delta\cdot F_{n+1/2} \, ,</math>
	</td>		</td>
			<td align="right" width="25%">    <math>(0.0000)</math></td>
	</tr>		</tr>

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	<math>Z_n + \Delta \cdot K_{n+1/2} \, .</math>		<math>Z_n + \Delta \cdot K_{n+1/2} \, .</math>
	</td>		</td>
			<td align="right" width="25%">    <math>(0.0000)</math></td>
	</tr>		</tr>
	</table>		</table>
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	<li>Determine the eccentricity at this advanced time.</li>		<li>Determine the eccentricity at this advanced time.</li>
	</ul>		</ul>
	<table border="0" cellpadding="5" align="center">		<table border="0" width="80%" cellpadding="5" align="center">

	<tr>		<tr>
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	</math>		</math>
	</td>		</td>
			<td align="right" width="25%">    <math>(0.0000)</math></td>
	</tr>		</tr>
	</table>		</table>

Jet53man: /* Discrete Representations */

2021-11-12T21:26:42Z

Discrete Representations

← Older revision		Revision as of 17:26, 12 November 2021
Line 807:		Line 807:
	<font color="red">STEP #0:</font>		<font color="red">STEP #0:</font>
	<ul>		<ul>
	<li>Choose time-independent values of the parameters, <math>(c_0/a_0)</math> and <math>\beta_0</math>, which means that, <math>\sigma_0^2 = 3\pi^2\beta_0/8</math> and <math>e_0 = [1 - (c_0/a_0)^2]^{1 / 2}</math>. Also set the (uniform) integration time step, <math>\Delta</math>; for 200 time steps, for example, set <math>\Delta \approx 0.005</math>. The example values (~~shown~~ inside parentheses) ~~below~~ assume that <math>(c_0/a_0, \beta_0) = (0.90, 0.0)</math>, which corresponds to one of the model-evolutions presented by {{ LMS65 }}.</li>		<li>Choose time-independent values of the parameters, <math>(c_0/a_0)</math> and <math>\beta_0</math>, which means that, <math>\sigma_0^2 = 3\pi^2\beta_0/8</math> and <math>e_0 = [1 - (c_0/a_0)^2]^{1 / 2}</math>. Also set the (uniform) integration time step, <math>\Delta</math>; for 200 time steps, for example, set <math>\Delta \approx 0.005</math>. The example values shown below (inside parentheses) assume that <math>(c_0/a_0, \beta_0) = (0.90, 0.0)</math>, which corresponds to one of the model-evolutions presented by {{ LMS65 }}.</li>
	</ul>		</ul>

Jet53man: /* Discrete Representations */

2021-11-12T21:24:17Z

Discrete Representations

← Older revision		Revision as of 17:24, 12 November 2021
Line 807:		Line 807:
	<font color="red">STEP #0:</font>		<font color="red">STEP #0:</font>
	<ul>		<ul>
	<li>Choose time-independent values of the parameters, <math>(c_0/a_0)</math> and <math>\beta_0</math>, which means that, <math>\sigma_0^2 = 3\pi^2\beta_0/8</math> and <math>e_0 = [1 - (c_0/a_0)^2]^{1 / 2}</math>. Also set the (uniform) integration time step, <math>\Delta</math>; for 200 time steps, for example, set <math>\Delta \approx 0.005</math>.</li>		<li>Choose time-independent values of the parameters, <math>(c_0/a_0)</math> and <math>\beta_0</math>, which means that, <math>\sigma_0^2 = 3\pi^2\beta_0/8</math> and <math>e_0 = [1 - (c_0/a_0)^2]^{1 / 2}</math>. Also set the (uniform) integration time step, <math>\Delta</math>; for 200 time steps, for example, set <math>\Delta \approx 0.005</math>. The example values (shown inside parentheses) below assume that <math>(c_0/a_0, \beta_0) = (0.90, 0.0)</math>, which corresponds to one of the model-evolutions presented by {{ LMS65 }}.</li>
	</ul>		</ul>

Jet53man: /* Discrete Representations */

2021-11-12T21:14:40Z

Discrete Representations

@@ Line 807: / Line 807: @@
 <font color="red">STEP #0:</font>
 <ul>
-   <li>Choose time-independent values of the parameters, <math>\sigma_0^2</math> and <math>e_0^2</math>, and set the integration time step, <math>\Delta</math></li>
+   <li>Choose time-independent values of the parameters, <math>(c_0/a_0)</math> and <math>\beta_0</math>, which means that, <math>\sigma_0^2 = 3\pi^2\beta_0/8</math> and <math>e_0 = [1 - (c_0/a_0)^2]^{1 / 2}</math>.  Also set the (uniform) integration time step, <math>\Delta</math>; for 200 time steps, for example, set <math>\Delta \approx 0.005</math>.</li>
 </ul>
@@ Line 878: / Line 878: @@
    <td align=left">
 <math>1 + \Delta \cdot K_{+1/2} \, .</math>
    </td>
 </tr>
@@ Line 948: / Line 967: @@
    <td align=left">
 <math>Z_n + \Delta \cdot K_{n+1/2} \, .</math>
    </td>
 </tr>