Difference between revisions of "Connection between Temperature and Redshift"

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[[Category:Thermodynamics of Universe|4]]
 
[[Category:Thermodynamics of Universe|4]]
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__NOTOC__
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<div id="therm23"></div>
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<div style="border: 1px solid #AAA; padding:5px;">
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=== Problem 1 ===
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Find the dependence of radiation temperature on the redshift.
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<div class="NavFrame collapsed">
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  <div class="NavHead">solution</div>
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  <div style="width:100%;" class="NavContent">
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    <p style="text-align: left;">
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$$
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T \propto a^{ - 1};\quad T = T_0(1 + z).
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$$
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</p>
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  </div>
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</div></div>
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<div id="therm24"></div>
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<div style="border: 1px solid #AAA; padding:5px;">
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=== Problem 2 ===
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Find the dependence of free non-relativistic gas' temperature on the redshift.
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<div class="NavFrame collapsed">
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  <div class="NavHead">solution</div>
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  <div style="width:100%;" class="NavContent">
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    <p style="text-align: left;">
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$$
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T \propto a^{ - 2};\quad T = T_0(1 + z)^2.
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$$
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</p>
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  </div>
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</div></div>
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<div id="therm25"></div>
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<div style="border: 1px solid #AAA; padding:5px;">
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=== Problem 3 ===
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Estimate the time moment when the recombination started, i.e. when ionized plasma transited to the gas of neutral atoms, and determine the corresponding redshift value. The recombination temperature equals to $T_{rec} \approx 0.3\,eV.$
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<div class="NavFrame collapsed">
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  <div class="NavHead">solution</div>
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  <div style="width:100%;" class="NavContent">
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    <p style="text-align: left;">
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$$
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z_{rec} = \frac{T_{rec}}{T_0} - 1.
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$$
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As $1\mbox{\it eV} \approx 11600\,K$, then
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$$
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z_{rec} \approx \frac{0.3}{2.73}11600 \approx 1300.
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$$
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For a rough estimate assume that the transition $T_{rec} \to T_0$ occurs mainly in the matter-dominated epoch to obtain
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$$
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t_{rec} = t_0\left( \frac{a}{a_0} \right)^{3/2} =
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t_0\left(\frac{T_0}{T_{rec}} \right)^{3/2}\approx
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300\,000\,years.
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$$
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</p>
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  </div>
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</div></div>

Revision as of 21:50, 1 October 2012



Problem 1

Find the dependence of radiation temperature on the redshift.

Problem 2

Find the dependence of free non-relativistic gas' temperature on the redshift.

Problem 3

Estimate the time moment when the recombination started, i.e. when ionized plasma transited to the gas of neutral atoms, and determine the corresponding redshift value. The recombination temperature equals to $T_{rec} \approx 0.3\,eV.$

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