Jet53man: /* Steady-State 2nd-Order Tensor Virial Equations */

2021-10-02T16:49:27Z

Steady-State 2nd-Order Tensor Virial Equations

← Older revision		Revision as of 16:49, 2 October 2021
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	=Steady-State 2<sup>nd</sup>-Order Tensor Virial Equations=		=Steady-State 2<sup>nd</sup>-Order Tensor Virial Equations=
			{\| class="2ndOrderTVE" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black"
	==~~Summary~~==		\|-
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	Drawing from our [[~~User:Tohline/~~VE#Virial_Equations_.28Rotating_Frame.29\|accompanying discussion of virial equations as viewed from a rotating frame of reference]], here we employ the 2<sup>nd</sup>-order tensor virial equation (TVE),		<font size="-1">[[H_BookTiledMenu#Ellipsoidal_.26_Ellipsoidal-Like\|<b>Steady-State<br />2<sup>nd</sup>-Order<br />Tensor Virial<br />Equations</b>]]</font>
			\|}
			Drawing from our [[VE#Virial_Equations_.28Rotating_Frame.29\|accompanying discussion of virial equations as viewed from a rotating frame of reference]], here we employ the 2<sup>nd</sup>-order tensor virial equation (TVE),
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	==General Coefficient Expressions==		==General Coefficient Expressions==

	In the context of our discussion of configurations that are triaxial ellipsoids, we begin by adopting the <math>~(\ell, m, s)</math> subscript notation to identify which semi-axis length is the (largest, medium-length, smallest). As has been detailed in an [[~~User:Tohline/~~ThreeDimensionalConfigurations/HomogeneousEllipsoids#Derivation_of_Expressions_for_Ai\|accompanying chapter]], the gravitational potential anywhere inside or on the surface of an homogeneous ellipsoid may be given analytically in terms of the following three coefficient expressions:		In the context of our discussion of configurations that are triaxial ellipsoids, we begin by adopting the <math>~(\ell, m, s)</math> subscript notation to identify which semi-axis length is the (largest, medium-length, smallest). As has been detailed in an [[ThreeDimensionalConfigurations/HomogeneousEllipsoids#Derivation_of_Expressions_for_Ai\|accompanying chapter]], the gravitational potential anywhere inside or on the surface of an homogeneous ellipsoid may be given analytically in terms of the following three coefficient expressions:
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	<tr><td align="center" colspan="3">[ [[~~User:Tohline/~~Appendix/References#EFE\|EFE]], <font color="#00CC00">Chapter 3, §17, Eq. (32)</font> ]</td></tr>		<tr><td align="center" colspan="3">[ [[Appendix/References#EFE\|EFE]], <font color="#00CC00">Chapter 3, §17, Eq. (32)</font> ]</td></tr>
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	===Specific Case of a<sub>1</sub> > a<sub>3</sub> > a<sub>2</sub>===		===Specific Case of a<sub>1</sub> > a<sub>3</sub> > a<sub>2</sub>===

	When we discuss configurations in which <math>~a_1 > a_3 > a_2 > 0</math> — these are usually referred to in [[~~User:Tohline/~~Appendix/References#EFE\|EFE]] as prolate S-Type Riemann ellipsoids — we must instead adopt the associations, <math>~(A_1, a_1) \leftrightarrow (A_\ell, a_\ell)</math>, <math>~(A_2, a_2) \leftrightarrow (A_s, a_s)</math>, and <math>~(A_3, a_3) \leftrightarrow (A_m, a_m)</math>. This means that the coefficients, <math>~A_1</math>, <math>~A_2</math>, and <math>~A_3</math> are defined by the expressions,		When we discuss configurations in which <math>~a_1 > a_3 > a_2 > 0</math> — these are usually referred to in [[Appendix/References#EFE\|EFE]] as prolate S-Type Riemann ellipsoids — we must instead adopt the associations, <math>~(A_1, a_1) \leftrightarrow (A_\ell, a_\ell)</math>, <math>~(A_2, a_2) \leftrightarrow (A_s, a_s)</math>, and <math>~(A_3, a_3) \leftrightarrow (A_m, a_m)</math>. This means that the coefficients, <math>~A_1</math>, <math>~A_2</math>, and <math>~A_3</math> are defined by the expressions,
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	<tr><td align="center" colspan="3">[ [[~~User:Tohline/~~Appendix/References#EFE\|EFE]], <font color="#00CC00">Chapter 7, §48d, footnote to Table VII (p. 143)</font> ]</td></tr>		<tr><td align="center" colspan="3">[ [[Appendix/References#EFE\|EFE]], <font color="#00CC00">Chapter 7, §48d, footnote to Table VII (p. 143)</font> ]</td></tr>
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	<tr><td align="center" colspan="3">[ [[~~User:Tohline/~~Appendix/References#EFE\|EFE]], <font color="#00CC00">Chapter 7, §47, Eq. (1)</font> ]</td></tr>		<tr><td align="center" colspan="3">[ [[Appendix/References#EFE\|EFE]], <font color="#00CC00">Chapter 7, §47, Eq. (1)</font> ]</td></tr>
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	==Equilibrium Expressions==		==Equilibrium Expressions==
	[<b>[[~~User:Tohline/~~Appendix/References#EFE\|<font color="red">EFE</font>]]</b> §11(b), p. 22] <font color="#007700">Under conditions of a stationary state, [the tensor virial equation] gives,</font>		[<b>[[Appendix/References#EFE\|<font color="red">EFE</font>]]</b> §11(b), p. 22] <font color="#007700">Under conditions of a stationary state, [the tensor virial equation] gives,</font>
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	<tr><td align="center" colspan="3">[ [[~~User:Tohline/~~Appendix/References#EFE\|EFE]], <font color="#00CC00">Chapter 2, §12, Eq. (64)</font> ]</td></tr>		<tr><td align="center" colspan="3">[ [[Appendix/References#EFE\|EFE]], <font color="#00CC00">Chapter 2, §12, Eq. (64)</font> ]</td></tr>
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	<tr><td align="center" colspan="3">[ [[~~User:Tohline/~~Appendix/References#EFE\|EFE]], <font color="#00CC00">Chapter 3, §22, Eq. (128)</font> ]</td></tr>		<tr><td align="center" colspan="3">[ [[Appendix/References#EFE\|EFE]], <font color="#00CC00">Chapter 3, §22, Eq. (128)</font> ]</td></tr>
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	<font color="#007700">where</font>		<font color="#007700">where</font>
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	<tr><td align="center" colspan="3">[ [[~~User:Tohline/~~Appendix/References#EFE\|EFE]], <font color="#00CC00">Chapter 3, §22, Eq. (129)</font> ]</td></tr>		<tr><td align="center" colspan="3">[ [[Appendix/References#EFE\|EFE]], <font color="#00CC00">Chapter 3, §22, Eq. (129)</font> ]</td></tr>
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	<tr><td align="center" colspan="4">[ [[~~User:Tohline/~~Appendix/References#EFE\|EFE]], <font color="#00CC00">Chapter 7, §47, Eq. (3)</font> ]</td></tr>		<tr><td align="center" colspan="4">[ [[Appendix/References#EFE\|EFE]], <font color="#00CC00">Chapter 7, §47, Eq. (3)</font> ]</td></tr>
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	whereas for <math>~i=3</math> and <math>~j=2</math>,		whereas for <math>~i=3</math> and <math>~j=2</math>,
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	<tr><td align="center" colspan="4">[ [[~~User:Tohline/~~Appendix/References#EFE\|EFE]], <font color="#00CC00">Chapter 7, §47, Eq. (4)</font> ]</td></tr>		<tr><td align="center" colspan="4">[ [[Appendix/References#EFE\|EFE]], <font color="#00CC00">Chapter 7, §47, Eq. (4)</font> ]</td></tr>
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	<td align="right">      if    <math>~i = j \, .</math>		<td align="right">      if    <math>~i = j \, .</math>
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	<tr><td align="center" colspan="4">[ [[~~User:Tohline/~~Appendix/References#EFE\|EFE]], <font color="#00CC00">Chapter 7, §47, Eq. (5)</font> ]</td></tr>		<tr><td align="center" colspan="4">[ [[Appendix/References#EFE\|EFE]], <font color="#00CC00">Chapter 7, §47, Eq. (5)</font> ]</td></tr>
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	This has allowed us to set to zero one of the integrals in each of these last two expressions. In what follows, we will benefit from recognizing, as well, that,		This has allowed us to set to zero one of the integrals in each of these last two expressions. In what follows, we will benefit from recognizing, as well, that,
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	<tr><td align="center" colspan="3">[ [[~~User:Tohline/~~Appendix/References#EFE\|EFE]], <font color="#00CC00">Chapter 7, §47, Eq. (6)</font> ]</td></tr>		<tr><td align="center" colspan="3">[ [[Appendix/References#EFE\|EFE]], <font color="#00CC00">Chapter 7, §47, Eq. (6)</font> ]</td></tr>
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	and subtracting the pair gives,		and subtracting the pair gives,
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	<tr><td align="center" colspan="3">[ [[~~User:Tohline/~~Appendix/References#EFE\|EFE]], <font color="#00CC00">Chapter 7, §47, Eq. (7)</font> ]</td></tr>		<tr><td align="center" colspan="3">[ [[Appendix/References#EFE\|EFE]], <font color="#00CC00">Chapter 7, §47, Eq. (7)</font> ]</td></tr>
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Jet53man: /* Steady-State 2nd-Order Tensor Virial Equations */

174.64.14.12: Created page with "__FORCETOC__ =Steady-State 2nd-Order Tensor Virial Equations= ==Summary== Drawing from our User:Tohline/VE#Virial_Equatio..."

174.64.14.12: Created page with "FORCETOC =Steady-State 2^nd-Order Tensor Virial Equations= ==Summary== Drawing from our User:Tohline/VE#Virial_Equatio..."