Appendix/Ramblings/StrongNuclearForce - Revision history

Jet53man at 17:36, 16 April 2022

2022-04-16T17:36:35Z

← Older revision		Revision as of 13:36, 16 April 2022
Line 153:		Line 153:
	<ul>		<ul>
	<li>		<li>
	Derivation of the [[SSC/FreeFall#Relationship_to_Relativistic_Cosmologies\|Friedmann Equations]] in the context of our discussion of ''Newtonian'' free-fall collapse.		Derivation of the [[SSC/Dynamics/FreeFall#Relationship_to_Relativistic_Cosmologies\|Friedmann Equations]] in the context of our discussion of ''Newtonian'' free-fall collapse.

	<table border="1" cellpadding="10" align="center">		<table border="1" cellpadding="10" align="center">
Line 254:		Line 254:
	Homework set #3 that was assigned to my ASTR4422 class in the spring of 2005 explored how solutions to the ''Newtonian'' free-fall collapse problem can be mapped directly to cosmological models of the expanding universe. The stated objective was to match the "closed universe," <math>~\Omega_0 = 2</math> model presented in Figure 27.4 (p. 1230) of the 1<sup>st</sup> edition of Carroll & Ostlie. (In the spring of 2009, this was assignment #5, and the aim was to match Figure 29.5 from the 2<sup>nd</sup> edition of Carroll & Ostlie.)		Homework set #3 that was assigned to my ASTR4422 class in the spring of 2005 explored how solutions to the ''Newtonian'' free-fall collapse problem can be mapped directly to cosmological models of the expanding universe. The stated objective was to match the "closed universe," <math>~\Omega_0 = 2</math> model presented in Figure 27.4 (p. 1230) of the 1<sup>st</sup> edition of Carroll & Ostlie. (In the spring of 2009, this was assignment #5, and the aim was to match Figure 29.5 from the 2<sup>nd</sup> edition of Carroll & Ostlie.)

	In the free-fall model, the collapse starts from rest at initial radius and density, <math>~r_0</math> and <math>~\rho_0</math>, respectively, in which case — see, for example, our [[SSC/FreeFall#RoleOfIntegrationConstant\|discussion of the role of the integration constant]] —		In the free-fall model, the collapse starts from rest at initial radius and density, <math>~r_0</math> and <math>~\rho_0</math>, respectively, in which case — see, for example, our [[SSC/Dynamics/FreeFall#RoleOfIntegrationConstant\|discussion of the role of the integration constant]] —
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
Line 431:		Line 431:
	</table>		</table>

	As our [[SSC/FreeFall#Parametric\|parametric solution of the Newtonian free-fall problem details]], quite generally we can write,		As our [[SSC/Dynamics/FreeFall#Parametric\|parametric solution of the Newtonian free-fall problem details]], quite generally we can write,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
Line 491:		Line 491:
	</table>		</table>
	</div>		</div>
	This is the form of the equation that has been integrated analytically in our [[~~User:Tohline~~/~~SSC~~/FreeFall#Single_Particle_in_a_Point-Mass_Potential\|separate discussion of Newtonian free-fall collapse]]. Now, in our [http://adsabs.harvard.edu/abs/1983IAUS..100..205T published speculation about a modified force-law to explain flat rotation curves], we proposed (see that publication's equation 1) a gravitational acceleration of the form,		This is the form of the equation that has been integrated analytically in our [[SSC/Dynamics/FreeFall#Single_Particle_in_a_Point-Mass_Potential\|separate discussion of Newtonian free-fall collapse]]. Now, in our [http://adsabs.harvard.edu/abs/1983IAUS..100..205T published speculation about a modified force-law to explain flat rotation curves], we proposed (see that publication's equation 1) a gravitational acceleration of the form,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
Line 508:		Line 508:
	</table>		</table>
	</div>		</div>
	This was intended to represent the modified gravitational acceleration felt by a (massless) test particle moving outside of a point-mass, <math>~M_R</math>. When considering a position ''inside'' of a spherical mass distribution whose radius, <math>~R_2 > R</math>, the first term remains the same because material outside of the location, <math>~R</math>, does not exert a net gravitational acceleration. But the second term cannot be treated that way. Following our [[~~User:Tohline/~~DarkMatter/UniformSphere#General_Derivation_from_Notes_Dated_29_November_1982\|separate discussion of a 1/r force law]], we propose the following acceleration due to such an extended mass source:		This was intended to represent the modified gravitational acceleration felt by a (massless) test particle moving outside of a point-mass, <math>~M_R</math>. When considering a position ''inside'' of a spherical mass distribution whose radius, <math>~R_2 > R</math>, the first term remains the same because material outside of the location, <math>~R</math>, does not exert a net gravitational acceleration. But the second term cannot be treated that way. Following our [[DarkMatter/UniformSphere#General_Derivation_from_Notes_Dated_29_November_1982\|separate discussion of a 1/r force law]], we propose the following acceleration due to such an extended mass source:
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
Line 757:		Line 757:
	=Unresolved Links=		=Unresolved Links=
	<ul>		<ul>
	<li>[[SSC/FreeFall#Relationship_to_Relativistic_Cosmologies\|Friedmann Equations]]</li>		<li>[[SSC/Dynamics/FreeFall#Relationship_to_Relativistic_Cosmologies\|Friedmann Equations]]</li>
	<li>[[SSC/FreeFall#RoleOfIntegrationConstant\|discussion of the role of the integration constant]]</li>		<li>[[SSC/Dynamics/FreeFall#RoleOfIntegrationConstant\|discussion of the role of the integration constant]]</li>
	<li>[[SSC/FreeFall#Parametric\|parametric solution of the Newtonian free-fall problem details]]</li>		<li>[[SSC/Dynamics/FreeFall#Parametric\|parametric solution of the Newtonian free-fall problem details]]</li>
	<li>[[~~User:Tohline~~/~~SSC~~/FreeFall#Single_Particle_in_a_Point-Mass_Potential\|separate discussion of Newtonian free-fall collapse]]</li>		<li>[[SSC/Dynamics/FreeFall#Single_Particle_in_a_Point-Mass_Potential\|separate discussion of Newtonian free-fall collapse]]</li>
	<li>[[~~User:Tohline/~~DarkMatter/UniformSphere#General_Derivation_from_Notes_Dated_29_November_1982\|separate discussion of a 1/r force law]]</li>		<li>[[DarkMatter/UniformSphere#General_Derivation_from_Notes_Dated_29_November_1982\|separate discussion of a 1/r force law]]</li>
	</ul>		</ul>

	{{ SGFfooter }}		{{ SGFfooter }}

Jet53man: /* Insert Dependence on (Energy) Density */

2021-07-22T16:54:57Z

Insert Dependence on (Energy) Density

@@ Line 580: / Line 580: @@
    <td align="center"><math>=</math></td>
    <td align="left">
-<math>+ \frac{GM}{R}  - \frac{GM_r}{a_T} \cdot g(\eta) \cdot \ln \biggl(\frac{R}{a_T}\biggr)  \, ,</math>
+<math>+ \frac{GM}{R}  - \frac{GM}{a_T} \cdot g(\eta) \cdot \ln \biggl(\frac{R}{a_T}\biggr)  \, ,</math>
    </td>
 </tr>
@@ Line 642: / Line 642: @@
 </table>
 </td></tr></table>
 We want to kill off the <math>1/r</math> term when the density climbs above some threshold, <math>\rho_H</math>.  Let's try &hellip;

Jet53man: /* Insert Dependence on (Energy) Density */

2021-07-22T16:13:12Z

Insert Dependence on (Energy) Density

@@ Line 574: / Line 574: @@
 The QGP is a regime where the interaction between quarks and gluons is dominated by the ''Coulomb-like'' term in the interaction potential.  The particles interact with one another as though they are not confined; this is the so-called ''asymptotically free'' regime.  Generally speaking, a QGP is achieved in a very high energy-density environment.
-We can mimic this behavior in our modified cosmology by multiplying the logarithmic term in the gravitational potential by a function that varies with the energy-density in such a way that it behaves like an on/off switch.  (More simply, let's have it vary with the mass-density.) That is, let's write,
+We can mimic this behavior in our modified cosmology by multiplying the logarithmic term in the gravitational potential by a function that varies with the energy-density, <math>u,</math> in such a way that it behaves like an on/off switch.  That is, let's write,
 <table border="0" align="center" cellpadding="5">
 <tr>
@@ Line 585: / Line 585: @@
 </table>
 where,
-<div align="center"><math>\eta \equiv \frac{\rho}{\rho_H} \, ,</math></div>
+<div align="center"><math>\eta \equiv \frac{u}{u_{Q}} = \biggl(\frac{R}{R_Q}\biggr)^{-m} \, ,</math></div>
-and try, for example, a density-dependent coefficient of the form,
+and try, for example, an energy-density dependent coefficient of the form,
 <table border="0" align="center" cellpadding="5">
 <tr>
    <td align="right"><math>g(\eta)</math></td>
    <td align="center"><math>\equiv</math></td>
    <td align="left">
-<math>\frac{1}{2} \biggl[ 1 - \tanh(\eta) \biggr]  \, ,</math>
+<math>e^{-\eta}  \, .</math>
    </td>
 </tr>
 </table>
-where,
+(This has the desired ''switch'' behavior:  At high energy-densities, <math>u/u_Q \gg 1,</math> the coefficient goes to zero, while at low densities, <math>u/u_Q \ll 1,</math> the coefficient goes to unity.)
 We want to kill off the <math>1/r</math> term when the density climbs above some threshold, <math>\rho_H</math>.  Let's try &hellip;

Jet53man: /* Insert Dependence on (Energy) Density */

2021-07-22T15:19:34Z

Insert Dependence on (Energy) Density

@@ Line 574: / Line 574: @@
 The QGP is a regime where the interaction between quarks and gluons is dominated by the ''Coulomb-like'' term in the interaction potential.  The particles interact with one another as though they are not confined; this is the so-called ''asymptotically free'' regime.  Generally speaking, a QGP is achieved in a very high energy-density environment.
-We can mimic this behavior in our modified cosmology by assuming that the coefficient on the <math>1/r</math> term in the gravitational acceleration varies with the energy-density of the fluid.  (More simply, let's have it vary with the ''mass''-dentiy.)  We want to kill off the <math>1/r</math> term when the density climbs above some threshold, <math>\rho_H</math>.  Let's try &hellip;
+We can mimic this behavior in our modified cosmology by multiplying the logarithmic term in the gravitational potential by a function that varies with the energy-density in such a way that it behaves like an on/off switch.  (More simply, let's have it vary with the mass-density.) That is, let's write,
 <table border="0" align="center" cellpadding="5">
 <tr>

Jet53man at 00:16, 22 July 2021

2021-07-22T00:16:25Z

← Older revision		Revision as of 20:16, 21 July 2021
Line 609:		Line 609:
	* [https://books.google.com/books/about/Theoretical_Nuclear_and_Subnuclear_Physi.html?id=mfphXc8b-2IC Walecka, John Dirk], ''Theoretical Nuclear and Subnuclear Physics'', World Scientific (2004)		* [https://books.google.com/books/about/Theoretical_Nuclear_and_Subnuclear_Physi.html?id=mfphXc8b-2IC Walecka, John Dirk], ''Theoretical Nuclear and Subnuclear Physics'', World Scientific (2004)

			=Unresolved Links=
			<ul>
			<li>[[SSC/FreeFall#Relationship_to_Relativistic_Cosmologies\|Friedmann Equations]]</li>
			<li>[[SSC/FreeFall#RoleOfIntegrationConstant\|discussion of the role of the integration constant]]</li>
			<li>[[SSC/FreeFall#Parametric\|parametric solution of the Newtonian free-fall problem details]]</li>
			<li>[[User:Tohline/SSC/FreeFall#Single_Particle_in_a_Point-Mass_Potential\|separate discussion of Newtonian free-fall collapse]]</li>
			<li>[[User:Tohline/DarkMatter/UniformSphere#General_Derivation_from_Notes_Dated_29_November_1982\|separate discussion of a 1/r force law]]</li>
			</ul>

	{{ SGFfooter }}		{{ SGFfooter }}

Jet53man: /* ASTR4422 Class Notes */

2021-07-22T00:10:57Z

ASTR4422 Class Notes

← Older revision		Revision as of 20:10, 21 July 2021
Line 254:		Line 254:
	Homework set #3 that was assigned to my ASTR4422 class in the spring of 2005 explored how solutions to the ''Newtonian'' free-fall collapse problem can be mapped directly to cosmological models of the expanding universe. The stated objective was to match the "closed universe," <math>~\Omega_0 = 2</math> model presented in Figure 27.4 (p. 1230) of the 1<sup>st</sup> edition of Carroll & Ostlie. (In the spring of 2009, this was assignment #5, and the aim was to match Figure 29.5 from the 2<sup>nd</sup> edition of Carroll & Ostlie.)		Homework set #3 that was assigned to my ASTR4422 class in the spring of 2005 explored how solutions to the ''Newtonian'' free-fall collapse problem can be mapped directly to cosmological models of the expanding universe. The stated objective was to match the "closed universe," <math>~\Omega_0 = 2</math> model presented in Figure 27.4 (p. 1230) of the 1<sup>st</sup> edition of Carroll & Ostlie. (In the spring of 2009, this was assignment #5, and the aim was to match Figure 29.5 from the 2<sup>nd</sup> edition of Carroll & Ostlie.)

	In the free-fall model, the collapse starts from rest at initial radius and density, <math>~r_0</math> and <math>~\rho_0</math>, respectively, in which case — see, for example, our [[~~User:Tohline/~~SSC/FreeFall#RoleOfIntegrationConstant\|discussion of the role of the integration constant]] —		In the free-fall model, the collapse starts from rest at initial radius and density, <math>~r_0</math> and <math>~\rho_0</math>, respectively, in which case — see, for example, our [[SSC/FreeFall#RoleOfIntegrationConstant\|discussion of the role of the integration constant]] —
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
Line 431:		Line 431:
	</table>		</table>

	As our [[~~User:Tohline/~~SSC/FreeFall#Parametric\|parametric solution of the Newtonian free-fall problem details]], quite generally we can write,		As our [[SSC/FreeFall#Parametric\|parametric solution of the Newtonian free-fall problem details]], quite generally we can write,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

Jet53man: /* Standard Presentation */

2021-07-22T00:09:40Z

Standard Presentation

← Older revision		Revision as of 20:09, 21 July 2021
Line 153:		Line 153:
	<ul>		<ul>
	<li>		<li>
	Derivation of the [[~~User:Tohline/~~SSC/FreeFall#Relationship_to_Relativistic_Cosmologies\|Friedmann Equations]] in the context of our discussion of ''Newtonian'' free-fall collapse.		Derivation of the [[SSC/FreeFall#Relationship_to_Relativistic_Cosmologies\|Friedmann Equations]] in the context of our discussion of ''Newtonian'' free-fall collapse.

	<table border="1" cellpadding="10" align="center">		<table border="1" cellpadding="10" align="center">

Jet53man: Created page with "FORCETOC =Radial Dependence of the Strong Nuclear Force= ==Wikipedia as a Resource== [https://en.wikipedia.org/wiki/Quark–gluon_pl..."

2021-07-22T00:06:04Z

Created page with "__FORCETOC__  =Radial Dependence of the Strong Nuclear Force= ==Wikipedia as a Resource== [https://en.wikipedia.org/wiki/Quark–gluon_pl..."

New page

__FORCETOC__

=Radial Dependence of the Strong Nuclear Force=

==Wikipedia as a Resource==
[https://en.wikipedia.org/wiki/Quark–gluon_plasma QGP]:  
<ul>
<li>
"QGP (quark-gluon plasma) is a state of matter in which the elementary particles that make up the hadrons of baryonic matter are freed of their strong attraction for one another under extremely high energy densities."
</li>
<li>
"In normal matter quarks are ''confined''; in the QGP (quark-gluon plasma) quarks are ''deconfined''."
</li>
<li>
"In classical QCD quarks are the ''fermionic'' components of hadrons (mesons and baryons) while the gluons are considered the boson components of such particles. The gluons are the force carriers, or bosons, of the QCD color force, while the quarks by themselves are their fermionic matter counterparts."
<ul>
<li>Electrons (spin 1/2 particles) and (as a composite particle) protons are fermions; they obey Fermi-Dirac statistics.</li>
<li>According to the Standard Model of Particle Physics, photons (spin 1 particles) are one of only 5 elementary bosons; they obey Bose-Einstein statistics.</li>
</ul>
</li>
</ul>

[https://en.wikipedia.org/wiki/Color_confinement Color confinement]: 
<ul>
<li>
"[This] phenomenon can be understood qualitatively by noting that the force-carrying [bosonic] gluons of QCD have color charge [as well as do the fermionic quarks], unlike the photons of QED. Whereas the electric field between electrically charged particles decreases rapidly as those particles are separated, the gluon field between a pair of color charges forms a narrow flux tube (or string) between them. Because of this behavior of the gluon field, the strong force between the particles is constant regardless of their separation."
</li>
</ul>

[https://en.wikipedia.org/wiki/Strong_interaction Strong interaction]:  
<ul>
<li>
"Unlike all other forces … the strong force does not diminish in strength with increasing distance between pairs of quarks. After a limiting distance (about the size of a hadron) has been reached, it remains at a strength of about 10,000 newtons, no matter how much farther the distance between the quarks." Hence, the effective potential has a term that is linear in r.
</li>
</ul>

==Tidbits==
From an [https://physics.stackexchange.com/questions/8452/is-there-an-equation-for-the-strong-nuclear-force online chat]:
<ul>
<li>
From the study of the spectrum of quarkonium (bound system of quark and antiquark) and the comparison with positronium one finds as potential for the strong force,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~V(r)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \frac{4}{3} \cdot \frac{\alpha_s(r) \hbar c}{r} + kr \, ,
</math>
</td>
</tr>
</table>
</div>
where, the constant <math>~k</math> determines the field energy per unit length and is called string tension. For short distances this resembles the Coulomb law, while for large distances the <math>~kr</math> factor dominates (confinement). It is important to note that the coupling <math>~\alpha_s</math> also depends on the distance between the quarks.

This formula is valid and in agreement with theoretical predictions only for the quarkonium system and its typical energies and distances. For example charmonium: <math>~r \approx 0.4~\mathrm{fm}</math>.
</li>
<ul>
<li>
Of course, the "breaking of the flux tube" has no classical or semi-classical analogue, making this formulation better for hand waving than calculation.
</li>
<li>
This is fine for the quark-qark interaction, but people reading this answer should be careful not to interpret it as a nucleon-nucleon interaction.
</li>
</ul>
<li>
At the level of quantum hadron dynamics (i.e., the level of nuclear physics, not the level of particle physics where the real strong force lives) one can talk about a Yukawa potential of the form,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~V(r)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \frac{g^2}{4\pi c^2} \cdot \frac{e^{-mr}}{r} \, ,
</math>
</td>
</tr>
</table>
</div>
where <math>~m</math> is roughly the pion mass and <math>~g</math> is an effective coupling constant. To get the force related to this you would take the derivative in <math>~r</math>.

This is a semi-classical approximation, but it is good enough that [https://books.google.com/books/about/Theoretical_Nuclear_and_Subnuclear_Physi.html?id=mfphXc8b-2IC Walecka] used it briefly in his book.
</li>
<li>
The nuclear force is now understood as a residual effect of the even more powerful strong force, or strong interaction, which is the attractive force that binds particles called quarks together, to form the nucleons themselves. This more powerful force is mediated by particles called gluons. Gluons hold quarks together with a force like that of electric charge but of far greater power.

Marek is talking of the strong force that binds the quarks within the protons and neutrons. There are charges, called colored charges on the quarks, but protons and neutrons are color neutral. Nuclei are bound by the interplay between the residual strong force, the part that is not shielded by the color neutrality of the nucleons, and the electro magnetic force due to the charge of the protons. That also cannot be simply described. Various potentials are used to calculate nuclear interactions.
</li>
</ul>

From the arXiv preprint of a review article by [https://physics.stackexchange.com/questions/8452/is-there-an-equation-for-the-strong-nuclear-force A. Deur, S. J. Brodsky, & G. F. de Téramond (2020)] titled, ''The QCD Running Coupling'':
<ul>
<li>
(middle of p. 10) "We illustrate this behavior" — that is, "… the scale dependence of the coupling" — "for the coupling that arises in the static case of heavy sources and which provides a simple physical picture. Historically, and in the case of linear theories with massless force carriers, a force coupling constant is a universal coefficient that links the force to the 'charges' of two bodies (''e.g.,'' the electric charge for electricity or the mass for gravity) divided by the distance dependence <math>1/r^2</math>."
</li>
<li>
(middle of p. 10, continued) "In QFT (quantum field theory) … for weak enough forces, the first Born approximation dominates higher order contributions and the <math>1/q^2</math> propagator in momentum yields the familiar <math>1/r^2</math> factor in coordinate space. However, higher orders do contribute and deviations from the <math>1/r^2</math> law thus occur. This extra r-dependence is folded in the coupling which then acquires a scale dependence."
</li>
</ul>

==Pointers from Richard Imlay ''circa'' 1983==
When I asked Richard Imlay (high-energy experimentalist at LSU) for a reference to high-energy physics articles in which quark-quark interactions have been expressed in terms of a radially dependent (e.g., logarithmic ) potential, he pointed me to the following:
* Quigg, C. & Rosner, J. L. [http://adsabs.harvard.edu/abs/1977PhLB...71..153Q (1977), Physics Letters, 71B, pp. 153-157], ''Quarkonium level spacings''
* Tuts, P. Michael [https://books.google.com/books/about/Proceedings_of_the_1983_International_Sy.html?id=ktnvAAAAMAAJ (1983), Proceedings of the 1983 International Symposium on Lepton and Photon Interactions at High Energies, edited by D. G. Cassel & D. Kreinick (Cornell University, Ithaca, 1983), pp. 284-327], ''Experimental results in heavy quarkonia''

<div align="center">
<table border="0" align="center" cellpadding="3" width="85%">
<tr>
<th align="center" colspan="2">Figure 1 from Tuts (1983)</th>
</tr>
<tr>
<td align="center" rowspan="2">
[[File:ImlayFig1 cropped.png|center|300px|Figure 1 from Tuts (1983)]]
</td>
<td align="left">
References Cited in Figure Caption:
<ol start="46">
<li>[<font color="orange">'''curve 3'''</font>] E. Eichten, E. ''et al.'' [http://adsabs.harvard.edu/abs/1980PhRvD..21..203E (1980), Phys. Rev. D21, pp. 203-233], ''Charmonium: Comparison with experiment''</li>
<li>[<font color="orange">'''curve 2'''</font>] W. Buchmuller, G. Grunberg, & S.-H. H. Tye [http://adsabs.harvard.edu/abs/1980PhRvL..45..103B (1980), PRL, 45, pp. 103-106], ''Regge slope and the Λ parameter in quantum chromodynamics: An empirical approach via quarkonia''; <br />
W. Buchmuller & S.-H. H. Tye [http://adsabs.harvard.edu/abs/1981PhRvD..24..132B (1981), Phys. Rev. D24, pp. 132-156], ''Quarkonia and quantum chromodynamics''</li>
<li>[<font color="orange">'''curve 1'''</font>] A. Martin [http://adsabs.harvard.edu/abs/1980PhLB...93..338M (1980), Physics Letters B93, pp. 338-342]; ''A fit of upsilon and charmonium spectra''<br />
A. Martin [http://adsabs.harvard.edu/abs/1981PhLB..100..511M (1981) Physics Letters B100, pp. 511-514], ''A simultaneous fit of <math>~b\bar{b}</math>, <math>~c\bar{c}</math>, <math>~s\bar{s}</math> (bcs Pairs) and <math>~c\bar{s}</math> spectra''</li>
<li>[<font color="orange">'''curve 4'''</font>] G. Bhanot & S. Rudaz [http://adsabs.harvard.edu/abs/1978PhLB...78..119B (1978), Physics Letters B78, pp. 119-124], ''A new potential for quarkonium''</li>
</ol>
</td>
</tr>
<tr>
<td align="center"> </td>
</tr>
</table>
</div>

Additionally, my handwritten notes, ''circa'' 1983, point to:
* S. M. Alladin & K. S. V. S. Narasimhan [http://adsabs.harvard.edu/abs/1982PhR....92..339A (1982), Physics Reports, 92 (#6), pp. 339 - 397], ''Gravitational interactions between galaxies'' — in 2018, this does not now seem relevant.
* J. Gasser & H. Leutwyler [http://adsabs.harvard.edu/abs/1982PhR....87...77G (1982), Physics Reports, 87, Issue 3, pp. 77 - 169], ''Quark masses''
* G. Altarelli [http://adsabs.harvard.edu/abs/1982PhR....81....1A (1982), Physics Reports, 81, Issue 1, pp. 1 - 129]. ''Partons in quantum chromodynamics''

=Cosmologies=

==Standard Presentation==
<ul>
<li>
Derivation of the [[User:Tohline/SSC/FreeFall#Relationship_to_Relativistic_Cosmologies|Friedmann Equations]] in the context of our discussion of ''Newtonian'' free-fall collapse.

<table border="1" cellpadding="10" align="center">
<tr><th align="center">
Newtonian Description of Pressure-Free Collapse
</th></tr>
<tr><td align="left">

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

<tr>
<td align="right">
<math>~\biggl( \frac{\dot{R}}{R} \biggr)^2</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{8}{3}\pi G \rho - \frac{k(R_i, v_i)}{R^2} \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\frac{\ddot{R}}{R}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- \frac{4}{3}\pi G \rho \, ,</math>
</td>
</tr>

<tr>
<td align="right">
where,     <math>~k(R_i,v_i)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{8}{3}\pi G \rho_i R_i^2 - v_i^2 \, .</math>
</td>
</tr>
</table>
</div>

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

</li>
<li>
[http://www.astro.caltech.edu/~george/ay21/readings/Friemanetal_DE_ARAA.pdf Frieman, Turner & Huterer (2008, ARAA, 46, 385 - 432)] provide an excellent, very readable review of dark matter and dark energy in the context of various cosmologies; see also, chapter 29 of [https://www.scribd.com/doc/301615425/An-Introduction-to-Modern-Astrophysics Carroll & Ostlie (2007, 2<sup>nd</sup> Edition)]. Their equations (2) and (3) are written in the following table — with factors of <math>~c^2</math> inserted to explicitly clarify how the dimensional units are the same for every term in each equation.

<table border="1" cellpadding="10" align="center">
<tr><th align="center">
Friedmann equations:<br />
''Field equations of GR applied to the FRW metric''
</th></tr>
<tr><td align="left">

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

<tr>
<td align="right">
<math>~H^2 = \biggl( \frac{\dot{a}}{a} \biggr)^2</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{8}{3}\pi G \rho - \frac{k}{a^2} + \frac{\Lambda c^2}{3}\, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\frac{\ddot{a}}{a}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- \frac{4}{3}\pi G \biggl[\rho + \frac{3p}{c^2} \biggr] + \frac{\Lambda c^2}{3} \, .</math>
</td>
</tr>
</table>
</div>

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

</li>
</ul>

==ASTR4422 Class Notes==
Homework set #3 that was assigned to my ASTR4422 class in the spring of 2005 explored how solutions to the ''Newtonian'' free-fall collapse problem can be mapped directly to cosmological models of the expanding universe. The stated objective was to match the "closed universe," <math>~\Omega_0 = 2</math> model presented in Figure 27.4 (p. 1230) of the 1<sup>st</sup> edition of Carroll & Ostlie. (In the spring of 2009, this was assignment #5, and the aim was to match Figure 29.5 from the 2<sup>nd</sup> edition of Carroll & Ostlie.)

In the free-fall model, the collapse starts from rest at initial radius and density, <math>~r_0</math> and <math>~\rho_0</math>, respectively, in which case — see, for example, our [[User:Tohline/SSC/FreeFall#RoleOfIntegrationConstant|discussion of the role of the integration constant]] —
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~k_i</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{2G}{r_i} \biggl[ \frac{4}{3} \pi \rho_i r_i^3 \biggr] \, .</math>
</td>
</tr>
</table>
</div>
Hence, we have,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~H^2 = \biggl( \frac{\dot{R}}{R} \biggr)^2</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{8}{3}\pi G \rho - \frac{2G}{r_i} \biggl[ \frac{4}{3} \pi \rho_i r_i^3 \biggr] \frac{1}{R^2}</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{8}{3}\pi G \rho_i \biggl[ \frac{\rho}{\rho_i} - \frac{r_i^2}{R^2} \biggr] </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{8}{3}\pi G \rho_i \biggl[ \biggl(\frac{r_i}{R}\biggr)^3 - \biggl(\frac{r_i}{R}\biggr)^2 \biggr] </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{8}{3}\pi G \rho_i \biggl[ \sec^6\zeta - \sec^4\zeta \biggr] \, .</math>
</td>
</tr>
</table>
Now, adopting the terminologies, <math>~\Omega \equiv \rho/\rho_\mathrm{crit}</math> and, for any <math>~H</math>, <math>~\rho_\mathrm{crit} \equiv 3H^2/(8\pi G) ~~\Rightarrow ~~ H^2 = 8\pi G \rho/(3\Omega)</math>, we have,

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

<tr>
<td align="right">
<math>~\frac{8\pi G \rho}{3\Omega}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{8}{3}\pi G \rho_i \biggl[ \sec^6\zeta - \sec^4\zeta \biggr]</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~\frac{1}{\Omega}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{\rho_i}{\rho} \biggl[ \sec^6\zeta - \sec^4\zeta \biggr] = 1 - \cos^2\zeta \, .</math>
</td>
</tr>
</table>
Hence, if in the present epoch [denoted by subscript 0], <math>~\Omega = \Omega_0 = 2</math> (as in the Carroll & Ostlie figure that we're trying to match), then in our "free-fall" model, the present epoch occurs at the dimensionless time given by,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~1 - \cos^2\zeta_0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{1}{2}</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \cos^2\zeta_0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{1}{2}</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \zeta_0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{\pi}{4} \, .</math>
</td>
</tr>
</table>
</div>
This, in turn, implies that,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~H_0^2</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{8}{3}\pi G \rho_i \biggl[ \sec^6\zeta_0 - \sec^4\zeta_0 \biggr] </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{8}{3}\pi G \rho_i \biggl[ 2^3 - 2^2\biggr] = \frac{32}{3}\pi G \rho_i </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ \frac{1}{\tau_\mathrm{ff}^2} \biggl[\frac{3\pi}{32G\rho_i}\biggr] \frac{32}{3}\pi G \rho_i = \biggl(\frac{\pi}{\tau_\mathrm{ff}} \biggr)^2 \, .</math>
</td>
</tr>
</table>

As our [[User:Tohline/SSC/FreeFall#Parametric|parametric solution of the Newtonian free-fall problem details]], quite generally we can write,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~t</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{2\tau_\mathrm{ff}}{\pi} \biggl[ \zeta + \frac{1}{2}\sin(2\zeta)\biggr]</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{2}{\pi} \biggl[\frac{3\pi}{32G\rho_i} \biggr]^{1 / 2} \biggl[ \zeta + \frac{1}{2}\sin(2\zeta)\biggr]</math>
</td>
</tr>
</table>
</div>

==With Logarithmic Potential Included==
Let's return to the ''Newtonian'' expression for the acceleration equation and replace the time-dependent density, <math>~\rho</math>, with the time-independent mass, that is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{\ddot{R}}{R}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- \frac{4}{3} ~\pi G\rho = - \frac{GM_R}{R^3} </math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \ddot{R}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- \frac{GM_R}{R^2} \, .</math>
</td>
</tr>
</table>
</div>
This is the form of the equation that has been integrated analytically in our [[User:Tohline/SSC/FreeFall#Single_Particle_in_a_Point-Mass_Potential|separate discussion of Newtonian free-fall collapse]]. Now, in our [http://adsabs.harvard.edu/abs/1983IAUS..100..205T published speculation about a modified force-law to explain flat rotation curves], we proposed (see that publication's equation 1) a gravitational acceleration of the form,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\ddot{R}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- \frac{GM_R}{R^2} \biggl[1 + \frac{R}{a_\mathrm{T}}\biggr] \, .</math>
</td>
</tr>
</table>
</div>
This was intended to represent the modified gravitational acceleration felt by a (massless) test particle moving outside of a point-mass, <math>~M_R</math>. When considering a position ''inside'' of a spherical mass distribution whose radius, <math>~R_2 > R</math>, the first term remains the same because material outside of the location, <math>~R</math>, does not exert a net gravitational acceleration. But the second term cannot be treated that way. Following our [[User:Tohline/DarkMatter/UniformSphere#General_Derivation_from_Notes_Dated_29_November_1982|separate discussion of a 1/r force law]], we propose the following acceleration due to such an extended mass source:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\ddot{R}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \frac{G}{R^2} \biggl[\frac{4}{3}\pi \rho R^3\biggr]
- \frac{G}{a_T} \biggl[ \frac{4}{3}\pi\rho R_2\biggr] R \biggl\{
1 - 3 \sum_{n=1}^{\infty} \biggl( \frac{R}{R_2} \biggr)^{2n} \biggl[(2n-1)(2n+1)(2n+3) \biggr]^{-1}
\biggr\} \, .
</math>
</td>
</tr>
</table>
</div>
Furthermore, let's equate <math>~R_2</math> with the "size of the universe," namely, <math>~ct</math>; and let's again define the mass inside of the Lagrangian <math>~R</math> as <math>~M_R</math>. Then we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\ddot{R}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \frac{GM_R}{R^2}
- \frac{GM_R }{R^2} \biggl( \frac{R_2}{a_T}\biggr) \biggl\{
1 - 3 \sum_{n=1}^{\infty} \biggl( \frac{R}{R_2} \biggr)^{2n} \biggl[(2n-1)(2n+1)(2n+3) \biggr]^{-1}
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \frac{GM_R}{R^2}
- \frac{GM_R }{R^2} \biggl( \frac{ct}{a_T}\biggr) \biggl\{
1 - 3 \sum_{n=1}^{\infty} \biggl( \frac{R}{ct} \biggr)^{2n} \biggl[(2n-1)(2n+1)(2n+3) \biggr]^{-1}
\biggr\} \, .
</math>
</td>
</tr>
</table>
</div>

==Insert Dependence on (Energy) Density==

The QGP is a regime where the interaction between quarks and gluons is dominated by the ''Coulomb-like'' term in the interaction potential. The particles interact with one another as though they are not confined; this is the so-called ''asymptotically free'' regime. Generally speaking, a QGP is achieved in a very high energy-density environment.

We can mimic this behavior in our modified cosmology by assuming that the coefficient on the <math>1/r</math> term in the gravitational acceleration varies with the energy-density of the fluid. (More simply, let's have it vary with the ''mass''-dentiy.) We want to kill off the <math>1/r</math> term when the density climbs above some threshold, <math>\rho_H</math>. Let's try …
<table border="0" align="center" cellpadding="5">
<tr>
<td align="right"><math>\ddot{R}</math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math>- \frac{GM}{R^2} \biggl\{ 1 + \biggl[\frac{\rho}{\rho_H} - 1\biggr]^{-2} \frac{R}{a_T} \biggr\} \, .</math>
</td>
</tr>
</table>
Note that the <math>~R-</math>dependent potential from which this expression for the acceleration is derived is,
<table border="0" align="center" cellpadding="5">
<tr>
<td align="right"><math>~\Phi(R)</math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math>+ \frac{GM}{R} - \frac{GM_r}{a_T} \biggl[\frac{\rho}{\rho_H} - 1\biggr]^{-2} \ln \biggl(\frac{R}{a_T}\biggr) \, .</math>
</td>
</tr>
</table>

This expression for the gravitational acceleration has the desired properties:
<ul>
<li>
In the early universe, when <math>\rho/\rho_H \gg 1</math>, the density-dependent coefficient of the second (confining) term goes to zero; we have an ''asymptotically free regime'' in which a Coulomb-like potential dominates throughout the universe.
</li>
<li>
As the universe expands, the density will steadily drop. For <math>\rho/\rho_H \ll 1</math>, the density-dependent coefficient of the confining term approaches unity and we retrieve our originally proposed, modified cosmology; that is, the potential is dominated by a logarithmic term for all distances greater than <math>~a_T</math>.
</li>
</ul>

=Potentially Useful References=
* Wikipedia — [https://en.wikipedia.org/wiki/Nuclear_binding_energy#Semiempirical_formula_for_nuclear_binding_energy Semiempirical Formula for the Nuclear Binding Energy]
* [https://books.google.com/books/about/Theoretical_Nuclear_and_Subnuclear_Physi.html?id=mfphXc8b-2IC Walecka, John Dirk], ''Theoretical Nuclear and Subnuclear Physics'', World Scientific (2004)

{{ SGFfooter }}

Appendix/Ramblings/StrongNuclearForce - Revision history

Jet53man at 17:36, 16 April 2022

Jet53man: /* Insert Dependence on (Energy) Density */

Jet53man: /* Insert Dependence on (Energy) Density */

Jet53man: /* Insert Dependence on (Energy) Density */

Jet53man at 00:16, 22 July 2021

Jet53man: /* ASTR4422 Class Notes */

Jet53man: /* Standard Presentation */

Jet53man: Created page with "__FORCETOC__ =Radial Dependence of the Strong Nuclear Force= ==Wikipedia as a Resource== [https://en.wikipedia.org/wiki/Quark–gluon_pl..."

Jet53man: Created page with "FORCETOC =Radial Dependence of the Strong Nuclear Force= ==Wikipedia as a Resource== [https://en.wikipedia.org/wiki/Quark–gluon_pl..."