Cosmo All at 01:30, 23 May 2014

2014-05-23T01:30:38Z

Cosmo All: Created page with "10

=== Problem 1 ===
problem id: ..."

2014-05-23T01:07:24Z

Created page with "10 <div id="Siv_1"></div> <div style="border: 1px solid #AAA; padding:5px;"> === Problem 1 === <p style= "color: #999;font-size: 11px">problem id: ..."

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[[Category:Cosmo warm-up|10]]

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=== Problem 1 ===
<p style= "color: #999;font-size: 11px">problem id: Siv_1</p>
(after C.Sivaram, Dark Energy may link the numbers of Rees, arXiv: 0710.4993)
Given $\Lambda$-dominated Universe, the requirement that for various large scale structures (held together by self gravity) to form a variety of length scales, their gravitational self energy density should at least match the ambient vacuum energy repulsion, as was shown to imply [16. C Sivaram, Astr. Spc. Sci, 219, 135; IJTP, 33, 2407, 1994, 17. C Sivaram, Mod. Phys. Lett., 34, 2463, 1999]
a scale invariant mass-radius relationship to the form (for the various structures):
\[\frac M{R^2}\approx\sqrt\Lambda\frac{c^2} G.\]
This equation predicts a universality of $M/R^2$ for a large variety of structures. Check this statement for such structures as a galaxy, a globular cluster, a galaxy cluster.
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<div class="NavHead">solution</div>
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<p style="text-align: left;">For a typical spiral galaxy $M_{gal}\approx10^{12} M_\odot$, $R\approx30kpc$, for globular clusters, $M\approx10^{6} M_\odot$, $R\approx100pc$, for galaxy clusters, $M_C\approx10^{16} M_\odot$, $R_C\approx3Mpc$, so for all these structures the equation holds.</p>
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=== Problem 2 ===
<p style= "color: #999;font-size: 11px">problem id: Siv_2</p>
(after C.Sivaram, Scaling Relations for self-Similar Structures and the Cosmological Constant, arXiv: 0801.1218)
In recent papers [13. Sivaram, C.: 1993a, Mod. Phys. Lett. 8,321.; 14. Sivaram, C.: 1993b, Astrophys. Spc. Sci. 207, 317.; 15. Sivaram, C.: 1993c, Astron. Astrophys. 275, 37.; 16. Sivaram, C.: 1994a, Astrophysics. Spc. Sci., 215, 185.; 17. Sivaram, C.: 1994b. Astrphysics .Spc .Sci., 215,191.; 18. Sivaram, C.: 1994c. Int. J. Theor. Phys. 33, 2407.], it was pointed out that the surface gravities of a whole hierarchy
of astronomical objects (i.e. globular clusters, galaxies, clusters, super clusters,
GMC's etc.) are more or less given by a universal value $a_0\approx cH_0\approx 10^{-8} cm\ s^{-2}$ a o ƒ° cHo ƒ° 10-8 cms-2. Thus
\[a=\frac{GM}{R^2}\approx a_0\]
for all these objects, $M$ being their typical mass and $R$ their typical radius. Also interestingly enough it was also pointed out [4. Sivaram, C.: 1982, Astrophysics. Spc. Sci. 88,507.; 5. Sivaram, C.: 1982, Amer. J. Phy. 50, 279.; 6. Sivaram, C.: 1983, Amer. J. Phys. 51, 277.; 7. Sivaram, C.: 1983, Phys. Lett. 60B, 181.] that the gravitational self energy of a typical elementary particle (hadron) was shown to be
\[E_G\approx\frac{Gm^3 c}{\hbar}\approx\hbar H_0\]
implying the same surface gravity value for the particle
\[a_h=\frac{GM}{r^2}\approx \frac{Gm^3 c}{\hbar}\times\frac c\hbar\approx cH_0\approx a_0.\]
Calculate actual value of the ratio
\[\frac M{R^2}\approx\sqrt\Lambda\frac{c^2} G\sim1\]
for such examples as a galaxy, whole Universe, globular cluster, a GMC, a supercluster, nuclei, an electron, Solar system, planetary nebula.
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<p style="text-align: left;">\begin{tabular}{|c|c|}
\hline
$M/R^2$ ($g/cm^2$) & Object \\
\hline
$10^{45}/(3\times10^{22})^2\sim1$ & Galaxy \\
$10^{56}/(10^{28})^2\sim1$ & Universe \\
$10^{38}/(10^{19})^2\sim1$ & globular cluster, GMC, etc. \\
$few\times10^{48}/(few\times10^{24})^2\sim1$ & supercluster \\
$10^{-23}/(10^{-12})^2\sim1$ & nuclei \\
$10^{-27}/(10^{-13})^2\sim1$ & electron \\
$10^{33}/(10^{16})^2\sim1$ & Solar system, planetary nebula\\
\hline
\end{tabular}</p>
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Play with Numbers after Sivaram - Revision history

Cosmo All at 01:30, 23 May 2014

Cosmo All: Created page with "10 === Problem 1 === problem id: ..."

Cosmo All: Created page with "10

=== Problem 1 ===
problem id: ..."