Appendix/Ramblings/OriginOfPlanetaryNebulae - Revision history

Jet53man: /* Commentary */

2023-11-28T00:20:32Z

Commentary

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	===Commentary===		===Commentary===

			====Evolutionary Tracks====
	When we embarked on this investigation, we thought that it would be difficult to quantitatively compare the evolution of our <math>(n_c, n_e) = (5, 1)</math> bipolytrope with more realistic stellar evolutionary tracks — as published, for example, by {{ Iben67full }} — because our toy model does not provide a mechanism for assessing the variation in time of the configuration's surface temperature. In the preceding subsection of this discussion, we have demonstrated that a comparison of evolutions is possible if we focus on an examination of how the star's radius varies as the mass of its core steadily increases.		When we embarked on this investigation, we thought that it would be difficult to quantitatively compare the evolution of our <math>(n_c, n_e) = (5, 1)</math> bipolytrope with more realistic stellar evolutionary tracks — as published, for example, by {{ Iben67full }} — because our toy model does not provide a mechanism for assessing the variation in time of the configuration's surface temperature. In the preceding subsection of this discussion, we have demonstrated that a comparison of evolutions is possible if we focus on an examination of how the star's radius varies as the mass of its core steadily increases.

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	Given that the ''trend'' matches, we will consider this a win! Hopefully a more quantitative match will be obtained when we switch to a bipolytrope that has an isothermal <math>(n_c = \infty)</math> core, instead of an <math>n_c = 5</math> core.		Given that the ''trend'' matches, we will consider this a win! Hopefully a more quantitative match will be obtained when we switch to a bipolytrope that has an isothermal <math>(n_c = \infty)</math> core, instead of an <math>n_c = 5</math> core.

			====Dynamical Instability====

			We have placed one solid-red circular marker on each of the evolutionary tracks that have been drawn from our toy model. In both cases, this marks the point along the track — at a radius less than <math>4 R_\odot</math> — where the <math>(n_c, n_e) = (5, 1)</math> model becomes dynamically unstable. It is tempting to suggest that this indicates that {{ Iben67hereafter }}'s model should also become dynamically unstable before expanding to a radius greater than <math>4 R_\odot</math>. Let's pursue building a bipolytrope that has an isothermal core and see if that system strengthens this argument.

	=Textbook Explanations=		=Textbook Explanations=

Jet53man: /* Commentary */

2023-11-28T00:07:33Z

Commentary

← Older revision		Revision as of 20:07, 27 November 2023
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	===Commentary===		===Commentary===

	When we embarked on this investigation, we thought that it would be difficult to quantitatively compare the evolution of our <math>(n_c, n_e) = (5, 1)</math> bipolytrope with more realistic stellar evolutionary tracks — as published, for example, by {{ Iben67full }} — because our toy model does not provide a mechanism for assessing the variation in time of the configuration's surface temperature. In the preceding subsection of this discussion, we have demonstrated that a comparison of evolutions is possible if we focus on an examination of how the star's radius varies as the mass of its core steadily increases.		When we embarked on this investigation, we thought that it would be difficult to quantitatively compare the evolution of our <math>(n_c, n_e) = (5, 1)</math> bipolytrope with more realistic stellar evolutionary tracks — as published, for example, by {{ Iben67full }} — because our toy model does not provide a mechanism for assessing the variation in time of the configuration's surface temperature. In the preceding subsection of this discussion, we have demonstrated that a comparison of evolutions is possible if we focus on an examination of how the star's radius varies as the mass of its core steadily increases.

			Not surprisingly, we find that the ''trend'' is the same in our toy model as it is in {{ Iben67hereafter }}'s <math>1 M_\odot</math> model:   as the core's mass monotonically increases, the star's radius monotonically increases as well. But there appear to be two significant mismatches. (1) In the earliest stage of its evolution, {{ Iben67hereafter }}'s model exhibits a steeper rise in the mass of the core for a given radius. (2) If we set <math>\mu_e/\mu_c = 1/2</math> — which is the ''expected'' value when considering the fusion of hydrogen into helium — overall, our toy model dumps significantly more mass into the core than does {{ Iben67hereafter }}'s evolutionary model. Dropping the mean-molecular-weight ratio from <math>1 / 2</math> to <math>1 / 4</math> provides a better match, but it is difficult to come up with an astrophysical argument that would justify such a low ratio.

			Given that the ''trend'' matches, we will consider this a win! Hopefully a more quantitative match will be obtained when we switch to a bipolytrope that has an isothermal <math>(n_c = \infty)</math> core, instead of an <math>n_c = 5</math> core.

	=Textbook Explanations=		=Textbook Explanations=

Jet53man: /* Commentary */

2023-11-27T23:43:26Z

Commentary

← Older revision		Revision as of 19:43, 27 November 2023
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	===Commentary===		===Commentary===

	When we embarked on this investigation, we thought that it would be difficult to quantitatively compare the evolution of our <math>(n_c, n_e) = (5, 1)</math> bipolytrope with more realistic stellar evolutionary tracks as published, for example, by {{ Iben67full }}.		When we embarked on this investigation, we thought that it would be difficult to quantitatively compare the evolution of our <math>(n_c, n_e) = (5, 1)</math> bipolytrope with more realistic stellar evolutionary tracks — as published, for example, by {{ Iben67full }} — because our toy model does not provide a mechanism for assessing the variation in time of the configuration's surface temperature. In the preceding subsection of this discussion, we have demonstrated that a comparison of evolutions is possible if we focus on an examination of how the star's radius varies as the mass of its core steadily increases.

	=Textbook Explanations=		=Textbook Explanations=

Jet53man: /* Toy Model */

2023-11-27T23:32:38Z

Toy Model

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	<li><math>\nu</math>:   This is the fraction of the star's total mass that has accumulated in the core by the end of the specified time interval; in our toy model, the total mass is <math>1 M_\odot</math> so <math>\nu</math> is also the core mass expressed in solar masses. Here the value of <math>\nu</math> is obtained by summing over the mass increments, <math>\Delta(M/M_\odot)</math>, that have been burned at all previous evolutionary time intervals, including the present one; for example, at the end of time interval 3 - 4, <math>\nu = 0.0311+0.0322+0.0210 = 0.0842</math>.</li>		<li><math>\nu</math>:   This is the fraction of the star's total mass that has accumulated in the core by the end of the specified time interval; in our toy model, the total mass is <math>1 M_\odot</math> so <math>\nu</math> is also the core mass expressed in solar masses. Here the value of <math>\nu</math> is obtained by summing over the mass increments, <math>\Delta(M/M_\odot)</math>, that have been burned at all previous evolutionary time intervals, including the present one; for example, at the end of time interval 3 - 4, <math>\nu = 0.0311+0.0322+0.0210 = 0.0842</math>.</li>
	</ul>		</ul>

			===Commentary===

			When we embarked on this investigation, we thought that it would be difficult to quantitatively compare the evolution of our <math>(n_c, n_e) = (5, 1)</math> bipolytrope with more realistic stellar evolutionary tracks as published, for example, by {{ Iben67full }}.

	=Textbook Explanations=		=Textbook Explanations=

Jet53man: /* Clayton (1968) */

2023-11-27T23:14:54Z

Clayton (1968)

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	</table>		</table>

	It is via this expression that we have deduced how the radius of ~~Iben~~'s evolving configuration changes over time.		It is via this expression that we have deduced how the radius of {{ Iben67hereafter }}'s evolving configuration changes over time.

	==Hansen & Kawaler (1994)==		==Hansen & Kawaler (1994)==

Jet53man: /* Example Evolution */

2023-11-27T23:12:03Z

Example Evolution

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	===Example Evolution===		===Example Evolution===

	Let's attempt to construct the post-main-sequence evolutionary trajectory for <math>1 M_\odot</math> stars having a couple of different values of the mean-molecular weight jump, <math>\mu_e/\mu_c</math>. This will be for comparison with the evolutionary trajectory published by {{ Iben67 }} [[#IbenFigure\|displayed below]]. Our toy model does not handle radiation, so we cannot plot ~~Luminosity~~ versus effective (surface) temperature, that is, we cannot generate a traditional HR diagram. However, our toy model predicts how the radius of the star should vary with the mass of its core. In the following figure, the solid-green circular markers show how <math>\nu</math> varies with <math>R_\mathrm{surf}/R_\odot</math> for the model sequence in which <math>\mu_e/\mu_c = 1 / 2</math> — note, for example, that among our above numerically evaluated "key relations", <math>(\nu, R_\mathrm{surf}/R_\odot) = (0.40178, 2.548)</math>; this point along the curve is marked by the solid-red circular marker. The solid-yellow circular markers show the same information for ~~the~~ model sequence in which <math>\mu_e/\mu_c = 1 / 4</math>.		Let's attempt to construct the post-main-sequence evolutionary trajectory for <math>1 M_\odot</math> stars having a couple of different values of the mean-molecular weight jump, <math>\mu_e/\mu_c</math>. This will be for comparison with the <math>1 M_\odot</math> evolutionary trajectory published by {{ Iben67 }}; hereafter, {{ Iben67hereafter }}. A remake of this published trajectory is [[#IbenFigure\|displayed below]].

			Our toy model does not handle radiation, so we cannot plot luminosity versus effective (surface) temperature, that is, we cannot generate a traditional HR diagram. However, our toy model predicts how the radius of the star should vary with the mass of its core. In the following figure, the solid-green circular markers show how the fractional mass, <math>\nu</math>, varies with <math>R_\mathrm{surf}/R_\odot</math> for the model sequence in which <math>\mu_e/\mu_c = 1 / 2</math> — note, for example, that among our above numerically evaluated "key relations", <math>(\nu, R_\mathrm{surf}/R_\odot) = (0.40178, 2.548)</math>; this point along the curve is marked by the solid-red circular marker. The solid-yellow circular markers show the same information for our toy model sequence in which <math>\mu_e/\mu_c = 1 / 4</math>.

	<table border="0" align="center" cellpadding="5">		<table border="0" align="center" cellpadding="5">
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	</table>		</table>

	The parameter values displayed in the table shown here — immediately on the right of our figure — were determined as follows, referencing the data from [[#IbenFigure\|~~Iben67a~~ presented below]]:		The parameter values displayed in the table shown here — immediately on the right of our figure — were determined as follows, referencing the data from [[#IbenFigure\|Iben67 presented below]]:
	<ul>		<ul>
	<li>Interval: —The points displayed along the <math>1 M_\odot</math> evolutionary trajectory in Figure 1 of ~~Iben67a~~ are numbered in chronological order, from 1 to 13 (except points 8 and 9 are not included). Here, our parameter values reference the ''time intervals'' between adjacent points; for example, our table refers to interval "1 - 2," interval "2 - 3," etc.</li>		<li>Interval: —The points displayed along the <math>1 M_\odot</math> evolutionary trajectory in Figure 1 of {{ Iben67hereafter }} are numbered in chronological order, from 1 to 13 (except points 8 and 9 are not included). Here, our parameter values reference the ''time intervals'' between adjacent points; for example, our table refers to interval "1 - 2," interval "2 - 3," etc.</li>
	<li><math>(\Delta t)_9</math>:   This refers to the amount of evolutionary time that separates the two points of an identified interval. For example, Table 1 of ~~Iben67a~~ states that the star's age at "point 1" along its evolutionary trajectory is <math>0.0506\times 10^{9}~\mathrm{yrs}</math> and at "point 2" its age is <math>3.8209\times 10^{9}~\mathrm{yrs}</math>; hence the interval of time between points 1 and 2 is <math>(\Delta t)_9 = (3.8209 - 0.0506)\times 10^{9}~\mathrm{yrs}= 3.770 \times 10^{9}~\mathrm{yrs}</math>.</li>		<li><math>(\Delta t)_9</math>:   This refers to the amount of evolutionary time that separates the two points of an identified interval. For example, Table 1 of {{ Iben67hereafter }} states that the star's age at "point 1" along its evolutionary trajectory is <math>0.0506\times 10^{9}~\mathrm{yrs}</math> and at "point 2" its age is <math>3.8209\times 10^{9}~\mathrm{yrs}</math>; hence the interval of time between points 1 and 2 is <math>(\Delta t)_9 = (3.8209 - 0.0506)\times 10^{9}~\mathrm{yrs}= 3.770 \times 10^{9}~\mathrm{yrs}</math>.</li>
	<li><math>[\log_{10}(L/L_\odot)]_\mathrm{avg}</math>:  We have performed a linear interpolation ''in log<sub>10</sub>'' of the luminosity listed for each point along the evolutionary trajectory in order to obtain an ''average'' value of the luminosity across each interval. For example, the star's luminosity at "point 1" is approximately <math>\log(L/L_\odot) \approx -0.140</math> while at "point 2" it is <math>\log(L/L_\odot) \approx 0.0</math>. Hence, while evolving across the 1 - 2 interval, we assume that its luminosity is constant and has the value, <math>[\log_{10}(L/L_\odot)]_\mathrm{avg} = \tfrac{1}{2}[0.0 + (- 0.140)] = -0.07</math>.</li>		<li><math>[\log_{10}(L/L_\odot)]_\mathrm{avg}</math>:  We have performed a linear interpolation ''in log<sub>10</sub>'' of the luminosity listed for each point along the evolutionary trajectory in order to obtain an ''average'' value of the luminosity across each interval. For example, the star's luminosity at "point 1" is approximately <math>\log(L/L_\odot) \approx -0.140</math> while at "point 2" it is <math>\log(L/L_\odot) \approx 0.0</math>. Hence, while evolving across the 1 - 2 interval, we assume that its luminosity is constant and has the value, <math>[\log_{10}(L/L_\odot)]_\mathrm{avg} = \tfrac{1}{2}[0.0 + (- 0.140)] = -0.07</math>.</li>
	<li><math>[\log_{10}(R/R_\odot)]_\mathrm{avg}</math>:  The average stellar radius over an interval is obtained in the same way. For example, while evolving across the 1 - 2 interval, we assume that its radius is constant and has the value, <math>[\log_{10}(R/R_\odot)]_\mathrm{avg} = \tfrac{1}{2}[-0.073 + (- 0.027)] = -0.05</math>.</li>		<li><math>[\log_{10}(R/R_\odot)]_\mathrm{avg}</math>:  The average stellar radius over an interval is obtained in the same way. For example, while evolving across the 1 - 2 interval, we assume that its radius is constant and has the value, <math>[\log_{10}(R/R_\odot)]_\mathrm{avg} = \tfrac{1}{2}[-0.073 + (- 0.027)] = -0.05</math>.</li>
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	For example, for interval 1 - 2, <math>\Delta M/M_\odot \approx (0.0097 \times 3.770 \times 10^{-0.070}) = 0.0311</math>.		For example, for interval 1 - 2, <math>\Delta M/M_\odot \approx (0.0097 \times 3.770 \times 10^{-0.070}) = 0.0311</math>.
	</li>		</li>
	<li><math>\nu</math>:   This is fraction of the star's total mass that has accumulated in the core; in our toy model, the total mass is <math>1 M_\odot</math> so <math>\nu</math> is also the core mass expressed in solar masses. Here the value of <math>\nu</math> is obtained by summing over the mass increments, <math>\Delta(M/M_\odot)</math>, that have been burned at all previous evolutionary time intervals, including the present one; for example, at the end of time interval 3 - 4, <math>\nu = 0.0311+0.0322+0.0210 = 0.0842</math>.</li>		<li><math>\nu</math>:   This is the fraction of the star's total mass that has accumulated in the core by the end of the specified time interval; in our toy model, the total mass is <math>1 M_\odot</math> so <math>\nu</math> is also the core mass expressed in solar masses. Here the value of <math>\nu</math> is obtained by summing over the mass increments, <math>\Delta(M/M_\odot)</math>, that have been burned at all previous evolutionary time intervals, including the present one; for example, at the end of time interval 3 - 4, <math>\nu = 0.0311+0.0322+0.0210 = 0.0842</math>.</li>
	</ul>		</ul>

Jet53man: /* Example Evolution */

2023-11-27T19:45:20Z

Example Evolution

← Older revision		Revision as of 15:45, 27 November 2023
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	===Example Evolution===		===Example Evolution===

	Let's attempt to construct the post-main-sequence evolutionary trajectory for <math>1 M_\odot</math> stars having a couple of different values of the mean-molecular weight jump, <math>\mu_e/\mu_c</math>. This will be for comparison with [[#IbenFigure\|~~Iben's (1967a) evolution~~]] ~~displayed below~~. Our toy model does not handle radiation, so we cannot plot Luminosity versus effective (surface) temperature, that is, we cannot generate a traditional HR diagram. However, our toy model predicts how the radius of the star should vary with the mass of its core. In the following figure, the solid-green circular markers show how <math>\nu</math> varies with <math>R_\mathrm{surf}/R_\odot</math> for the model sequence in which <math>\mu_e/\mu_c = 1 / 2</math> — note, for example, that among our above numerically evaluated "key relations", <math>(\nu, R_\mathrm{surf}/R_\odot) = (0.40178, 2.548)</math>; this point along the curve is marked by the solid-red circular marker. The solid-yellow circular markers show the same information for the model sequence in which <math>\mu_e/\mu_c = 1 / 4</math>.		Let's attempt to construct the post-main-sequence evolutionary trajectory for <math>1 M_\odot</math> stars having a couple of different values of the mean-molecular weight jump, <math>\mu_e/\mu_c</math>. This will be for comparison with the evolutionary trajectory published by {{ Iben67 }} [[#IbenFigure\|displayed below]]. Our toy model does not handle radiation, so we cannot plot Luminosity versus effective (surface) temperature, that is, we cannot generate a traditional HR diagram. However, our toy model predicts how the radius of the star should vary with the mass of its core. In the following figure, the solid-green circular markers show how <math>\nu</math> varies with <math>R_\mathrm{surf}/R_\odot</math> for the model sequence in which <math>\mu_e/\mu_c = 1 / 2</math> — note, for example, that among our above numerically evaluated "key relations", <math>(\nu, R_\mathrm{surf}/R_\odot) = (0.40178, 2.548)</math>; this point along the curve is marked by the solid-red circular marker. The solid-yellow circular markers show the same information for the model sequence in which <math>\mu_e/\mu_c = 1 / 4</math>.

	<table border="0" align="center" cellpadding="5">		<table border="0" align="center" cellpadding="5">
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	</td>		</td>
	<td align="center">		<td align="center">
	<table border="1" align="center" cellpadding="5">		<table border="1" align="center" cellpadding="8">

	<tr>		<tr>
	<td align="center" colspan="6">Parameters values deduced from~~<br />~~Figure 1 and Table 1 of ~~[https://ui.adsabs.harvard.edu/abs/1967ApJ...147..624I/abstract Iben67a]~~</td>		<td align="center" colspan="6">Parameters values deduced from Figure 1 and Table 1 of {{ Iben67figure }}</td>
	</tr>		</tr>

Jet53man: /* Zero-Age Main Sequence (ZAMS) Configuration */

2023-11-27T19:41:02Z

Zero-Age Main Sequence (ZAMS) Configuration

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	----		----

	In the context of our examination of ~~the~~ [[#IbenFigure\|~~Iben (1967) evolutionary track,~~ below]], we can write quite generally that,		In the context of our examination of a <math>1 M_\odot</math> evolutionary track published by {{ Iben67full }} — [[#IbenFigure\|see below]] — we can write quite generally that,

	<table border="0" align="center" cellpadding="8">		<table border="0" align="center" cellpadding="8">

Jet53man: /* Clayton (1968) */

2023-11-27T19:36:52Z

Clayton (1968)

← Older revision		Revision as of 15:36, 27 November 2023
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	----		----

	<span id="IbenFigure">Figure 6-17 (p. 492)</span> in Clayton (1968) displays a reproduction of Figure 1 from ~~[https://ui.adsabs.harvard.edu/abs/1967ApJ...147..624I/abstract Iben (1967a, ApJ, Vol. 147, pp. 624 - 649)]~~. It displays evolutionary tracks of lower-main-sequence population I stars of mass <math>1 M_\odot , 1.25 M_\odot ,</math> and <math>1.5 M_\odot</math>. Using a pen and ruler to draw scale lines across the figure, the following table catalogues the data from which the <math>1 M_\odot</math> track in the figure was produced.		<span id="IbenFigure">Figure 6-17 (p. 492)</span> in Clayton (1968) displays a reproduction of Figure 1 from {{ Iben67full }}. It displays evolutionary tracks of lower-main-sequence population I stars of mass <math>1 M_\odot , 1.25 M_\odot ,</math> and <math>1.5 M_\odot</math>. Using a pen and ruler to draw scale lines across the figure, the following table catalogues the data from which the <math>1 M_\odot</math> track in the figure was produced.

	<table border="0" align="center" cellpadding="0">		<table border="0" align="center" cellpadding="0">

Jet53man: /* Example Evolution */

2023-11-26T22:09:49Z

Example Evolution

← Older revision		Revision as of 18:09, 26 November 2023
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	<td align="center">0.06</td>		<td align="center">0.06</td>
	<td align="center">-0.003</td>		<td align="center">-0.003</td>
	<td align="center">0.~~0232~~</td>		<td align="center">0.0322</td>
	<td align="center">0.0633</td>		<td align="center">0.0633</td>
	</tr>		</tr>
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	For example, for interval 1 - 2, <math>\Delta M/M_\odot \approx (0.0097 \times 3.770 \times 10^{-0.070}) = 0.0311</math>.		For example, for interval 1 - 2, <math>\Delta M/M_\odot \approx (0.0097 \times 3.770 \times 10^{-0.070}) = 0.0311</math>.
	</li>		</li>
			<li><math>\nu</math>:   This is fraction of the star's total mass that has accumulated in the core; in our toy model, the total mass is <math>1 M_\odot</math> so <math>\nu</math> is also the core mass expressed in solar masses. Here the value of <math>\nu</math> is obtained by summing over the mass increments, <math>\Delta(M/M_\odot)</math>, that have been burned at all previous evolutionary time intervals, including the present one; for example, at the end of time interval 3 - 4, <math>\nu = 0.0311+0.0322+0.0210 = 0.0842</math>.</li>
	</ul>		</ul>