Influence of cosmological expansion on local systems - Revision history

Cosmo All at 22:04, 12 November 2012

2012-11-12T22:04:01Z

Cosmo All: /* Problem 4. */

2012-07-23T20:37:54Z

‎Problem 4.

Cosmo All: /* Problem 4. */

2012-07-23T20:36:35Z

‎Problem 4.

Cosmo All: /* Problem 2. */

2012-07-23T20:35:01Z

‎Problem 2.

Cosmo All: /* Problem 3. */

2012-07-23T20:34:15Z

‎Problem 3.

Cosmo All: /* Problem 1. */

2012-07-23T20:33:10Z

‎Problem 1.

Cosmo All at 20:32, 23 July 2012

2012-07-23T20:32:49Z

Cosmo All at 20:21, 23 July 2012

2012-07-23T20:21:50Z

Show changes

Cosmo All at 19:34, 23 July 2012

2012-07-23T19:34:48Z

Show changes

Cosmo All at 16:34, 23 July 2012

2012-07-23T16:34:39Z

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-[[Category:Dynamics of the Universe in the Big Bang Model|7]]
+[[Category:Dynamics of the Universe in the Big Bang Model|8]]
 __NOTOC__
 <p style="text-align: left;">Does the expansion of space mean that everything in it is stretched? Galaxies? Atoms? A shallow answer to this question is: "bounded" systems do not take part in the expansion. However, if space is stretched, then how can these systems not experience some, at least minimal, extension? Should bounded systems be stretched less intensively? The following several problems attempt to clarify the question by the example of a simple model: a classical atom, which consists of a negatively charged electron with negligible mass, rotating around a positively charged massive nucleus.</p>

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 \[\ddot r =  - \frac{4r_0^3}{3t^4}\omega _0^2.\]
-Using $r_0 =1.5\cdot 10^{11}${ m} (distance from Earth to Sun), ${\omega _0}= 2\pi /T_0 \simeq 2 \cdot 10^{ - 7}s^{- 1}$ ($T_0$ is the Earth's year) and $t$ the age of the Universe, we obtain $\ddot r \simeq  - 3.17 \cdot 10^{ - 47}${m/s}${}^2$. This quantity should be compared to gravitational acceleration of the Earth caused by the Sun $g = GM_\odot/r_0^2 \simeq 6 \cdot 10^{ - 3}$m/s${}^2$, which is 44 orders of magnitude larger.</p>
+Using $r_0 =1.5\cdot 10^{11}$m (distance from Earth to Sun), ${\omega _0}= 2\pi /T_0 \simeq 2 \cdot 10^{ - 7}s^{- 1}$ ($T_0$ is the Earth's year) and $t$ the age of the Universe, we obtain $\ddot r \simeq  - 3.17 \cdot 10^{ - 47}$m/s$^2$. This quantity should be compared to gravitational acceleration of the Earth caused by the Sun $g = GM_\odot/r_0^2 \simeq 6 \cdot 10^{ - 3}$m/s$^2$, which is 44 orders of magnitude larger.</p>
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 \[\ddot r =  - \frac{4r_0^3}{3t^4}\omega _0^2.\]
-Using $r_0 =1.5\cdot 10^{11}${\it m} (distance from Earth to Sun), ${\omega _0}= 2\pi /T_0 \simeq 2 \cdot 10^{ - 7}s^{- 1}$ ($T_0$ is the Earth's year) and $t$ the age of the Universe, we obtain $\ddot r \simeq  - 3.17 \cdot 10^{ - 47}${\it m/s}${}^2$. This quantity should be compared to gravitational acceleration of the Earth caused by the Sun $g = GM_\odot/r_0^2 \simeq 6 \cdot 10^{ - 3}${\it m/s}${}^2$, which is 44 orders of magnitude larger.</p>
+Using $r_0 =1.5\cdot 10^{11}${ m} (distance from Earth to Sun), ${\omega _0}= 2\pi /T_0 \simeq 2 \cdot 10^{ - 7}s^{- 1}$ ($T_0$ is the Earth's year) and $t$ the age of the Universe, we obtain $\ddot r \simeq  - 3.17 \cdot 10^{ - 47}${m/s}${}^2$. This quantity should be compared to gravitational acceleration of the Earth caused by the Sun $g = GM_\odot/r_0^2 \simeq 6 \cdot 10^{ - 3}$m/s${}^2$, which is 44 orders of magnitude larger.</p>
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 where $C$ is the constant of electrostatic interaction. Now we have to take into account the effect of expansion. According to $R = a(t)r$, if a point takes part in the cosmological expansion only, i.e has constant comoving coordinates $r,\theta,\varphi$, its radial acceleration is
 \[\left. {\ddot R} \right|_{expansion}
-	= R\frac{{\ddot a}}{a}.\]
+	= R\frac{\ddot a}{a}.\]
 We can treat this terms as a radial force acting on a unit mass and add it to the equation of motion, leading to
-\[\ddot R =   \frac{L^2}{R^3} -
+\[\ddot R =   \frac{L^2}{R^3} -	\frac{C}{R^2} + R\frac{\ddot a}{a}.\]
-	\frac{C}{R^2} + R\frac{\ddot a}{a}.\]
 The solution of this equation $R\big(r(t)\big)$ for some given $a(t)$ allows us to find the radial coordinate of the electron at the given time. Using the integral of motion $L\equiv {R^2}\dot \varphi$, we can find the angular coordinate $\varphi (t)$. Thus we can obtain full description of the orbit both in physical $R,\theta ,\varphi$ and in the comoving $r,\theta ,\varphi$ frames.</p>
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 === Problem 3. ===
 Write the effective potential for the electron for the case of exponential expansion $a(t) = e^{\beta t}$ and use it to analyze the dynamics for the case $L^2=C$, where $C$ is the constant of electrostatic interaction.

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 \[V(R) = \frac{L^2}{2R^2} - \frac{C}{R}
 	- \frac{1}{2}\beta ^2R^2.\]
-<gallery widths=400px heights=300px>
+<gallery widths=600px heights=500px>
 File:3_9.jpg|The effective potential for $a = e^{\beta t},\;\beta=0.01$ with different values of $L$.
 </gallery>
-At small $L$ (or, equivalently, large $\beta$), the potential energy decreases monotonically  with $R$. Radial forces push the electron into the region in which the term related to the expansion of the Universe $\beta ^2R^2$ is dominating. As $L$ increases, the effective potential acquires a minimum (see figure \ref{fig39}) in the neighborhood of $L=1$, in which the electron is captured. Thus our model demonstrates the effect "all or nothing": the electron is either captured in the minimum $(R=const)$, or joins the Hubble flow ($r=const$). It should be stressed, that the chosen exponential form of expansion is unessential. In any case the centrifugal and electrostatic forces decrease with distance, while the cosmological term increases.</p>
+At small $L$ (or, equivalently, large $\beta$), the potential energy decreases monotonically  with $R$. Radial forces push the electron into the region in which the term related to the expansion of the Universe $\beta ^2R^2$ is dominating. As $L$ increases, the effective potential acquires a minimum (see figure) in the neighborhood of $L=1$, in which the electron is captured. Thus our model demonstrates the effect "all or nothing": the electron is either captured in the minimum $(R=const)$, or joins the Hubble flow ($r=const$). It should be stressed, that the chosen exponential form of expansion is unessential. In any case the centrifugal and electrostatic forces decrease with distance, while the cosmological term increases.</p>
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 === Problem 4. ===
 Why does the Solar system not expand despite of expansion of all the Universe? Give quantitative arguments.

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-     <p style="text-align: left;">If $R=const$, it means there is no expansion. If $r=const$, it means the atom is copmletely ``involved'' into the cosmological expansion (they say, joins the Hubble flow). Simultaneous alteration of both coordinates corresponds to an intermediate situation. The question whether the intermediate situation can be realized should be considered separately.</p>
+     <p style="text-align: left;">If $R=const$, it means there is no expansion. If $r=const$, it means the atom is copmletely "involved" into the cosmological expansion (they say, joins the Hubble flow). Simultaneous alteration of both coordinates corresponds to an intermediate situation. The question whether the intermediate situation can be realized should be considered separately.</p>
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 === Problem 2. ===
 Derive the equation of motion for the atom's electron accounting for the cosmological expansion.

← Older revision		Revision as of 16:34, 23 July 2012
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−	[[Category:Dynamics of the Universe in the Big Bang Model]]	+	[[Category:Dynamics of the Universe in the Big Bang Model\|7]]