Difference between revisions of "Entropy of Expanding Universe"
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− | [[Category:Thermodynamics of Universe|(3)]] | + | [[Category:Thermodynamics of Universe|3]] |
+ | |||
+ | __NOTOC__ | ||
+ | |||
+ | <div id="therm23_1"></div> | ||
+ | <div style="border: 1px solid #AAA; padding:5px;"> | ||
+ | === Problem 1 === | ||
+ | Transform the energy conditions for the flat Universe to | ||
+ | conditions for the entropy density (see [http://arxiv.org/abs/1009.4513]) | ||
+ | <div class="NavFrame collapsed"> | ||
+ | <div class="NavHead">solution</div> | ||
+ | <div style="width:100%;" class="NavContent"> | ||
+ | <p style="text-align: left;"></p> | ||
+ | </div> | ||
+ | </div></div> | ||
+ | |||
+ | |||
+ | |||
+ | <div id="therm29"></div> | ||
+ | <div style="border: 1px solid #AAA; padding:5px;"> | ||
+ | === Problem 2 === | ||
+ | Find the entropy density for the photon gas. | ||
+ | <div class="NavFrame collapsed"> | ||
+ | <div class="NavHead">solution</div> | ||
+ | <div style="width:100%;" class="NavContent"> | ||
+ | <p style="text-align: left;">\[ | ||
+ | dS = \frac{dE}{T} + \frac{pdV}{T} = \frac{V}{T}\frac{d\rho}{dT}dT + \frac{\rho+ p}{T}dV \Rightarrow\] \[ \left( \frac{\partial S}{\partial V} | ||
+ | \right)_T = \frac{\rho + p} {T} \Rightarrow | ||
+ | S = \frac{\rho + p}{T}V + f(T).\] | ||
+ | |||
+ | As entropy is proportional to volume, then $f(T) = | ||
+ | 0.$ For the photon gas $p = \rho/3$, therefore | ||
+ | $$ | ||
+ | s = \frac{S}{V} = \frac{4}{3}\frac{\rho}{T} = \frac{4}{3}\alpha | ||
+ | {T^3}. | ||
+ | $$</p> | ||
+ | </div> | ||
+ | </div></div> | ||
+ | |||
+ | |||
+ | |||
+ | <div id="therm30"></div> | ||
+ | <div style="border: 1px solid #AAA; padding:5px;"> | ||
+ | === Problem 3 === | ||
+ | Use the result of the previous problem to derive an alternative proof of the fact that $aT=const$ in the adiabatically expanding Universe. | ||
+ | <div class="NavFrame collapsed"> | ||
+ | <div class="NavHead">solution</div> | ||
+ | <div style="width:100%;" class="NavContent"> | ||
+ | <p style="text-align: left;">$$S = \frac{4}{3}\alpha T^3V = const;\quad V \propto a^3 \Rightarrow aT = const.$$</p> | ||
+ | </div> | ||
+ | </div></div> | ||
+ | |||
+ | |||
+ | |||
+ | <div id="therm31"></div> | ||
+ | <div style="border: 1px solid #AAA; padding:5px;"> | ||
+ | === Problem 4 === | ||
+ | Find the adiabatic index for the CMB. | ||
+ | <div class="NavFrame collapsed"> | ||
+ | <div class="NavHead">solution</div> | ||
+ | <div style="width:100%;" class="NavContent"> | ||
+ | <p style="text-align: left;">$$ | ||
+ | p \propto \rho \propto a^{ - 4};\quad V \propto {a^3} \Rightarrow pV^\gamma = const,\quad \gamma = \frac{4}{3}. | ||
+ | $$</p> | ||
+ | </div> | ||
+ | </div></div> | ||
+ | |||
+ | |||
+ | |||
+ | <div id="therm32"></div> | ||
+ | <div style="border: 1px solid #AAA; padding:5px;"> | ||
+ | === Problem 5 === | ||
+ | Show that the ratio of CMB entropy density to the baryon number density $s_\gamma/n_b$ remains constant during the expansion of the Universe. | ||
+ | <div class="NavFrame collapsed"> | ||
+ | <div class="NavHead">solution</div> | ||
+ | <div style="width:100%;" class="NavContent"> | ||
+ | <p style="text-align: left;">$$s \propto T^3 \propto a^{ - 3};\quad n_b \propto a^{ - 3}.$$ Therefore their ratio remains constant during the expansion of Universe.</p> | ||
+ | </div> | ||
+ | </div></div> | ||
+ | |||
+ | |||
+ | |||
+ | <div id="ter_6n"></div> | ||
+ | <div style="border: 1px solid #AAA; padding:5px;"> | ||
+ | === Problem 6 === | ||
+ | Estimate the current entropy density of the Universe. | ||
+ | <div class="NavFrame collapsed"> | ||
+ | <div class="NavHead">solution</div> | ||
+ | <div style="width:100%;" class="NavContent"> | ||
+ | <p style="text-align: left;">The current entropy of the Universe is determined by the CMB photons. A rough estimate neglects the contribution of other particles, then one obtains | ||
+ | \[s = 2\frac{2\pi ^2}{45}T_0^3.\] | ||
+ | For $T_0 \approx 2.725K$ | ||
+ | \[s \approx 1.5 \cdot 10^3 \, cm^{-3}.\]</p> | ||
+ | </div> | ||
+ | </div></div> | ||
+ | |||
+ | |||
+ | |||
+ | <div id="ter_7n"></div> | ||
+ | <div style="border: 1px solid #AAA; padding:5px;"> | ||
+ | === Problem 7 === | ||
+ | Estimate the entropy of the observable part of the Universe. | ||
+ | <div class="NavFrame collapsed"> | ||
+ | <div class="NavHead">solution</div> | ||
+ | <div style="width:100%;" class="NavContent"> | ||
+ | <p style="text-align: left;">Size of the observable part of the Universe is $l_{H0} \sim 10^{28} \mbox{cm}$ and therefore | ||
+ | \[S = \frac{4\pi}{3}sl_{H0}^3 \sim 10^{88}.\]</p> | ||
+ | </div> | ||
+ | </div></div> | ||
+ | |||
+ | |||
+ | |||
+ | <div id="ter_10"></div> | ||
+ | <div style="border: 1px solid #AAA; padding:5px;"> | ||
+ | |||
+ | === Problem 7 === | ||
+ | Why is the expansion of the Universe described by the Friedman equations adiabatic? | ||
+ | <div class="NavFrame collapsed"> | ||
+ | <div class="NavHead">solution</div> | ||
+ | <div style="width:100%;" class="NavContent"> | ||
+ | <p style="text-align: left;">It is because the heat flow is absent in the homogeneous and isotropic Universe.</p> | ||
+ | </div> | ||
+ | </div></div> | ||
+ | |||
+ | |||
+ | |||
+ | <div id="ter_11"></div> | ||
+ | <div style="border: 1px solid #AAA; padding:5px;"> | ||
+ | === Problem 8 === | ||
+ | Show that entropy is conserved during the expansion of the Universe described by the Friedman equations. | ||
+ | <div class="NavFrame collapsed"> | ||
+ | <div class="NavHead">solution</div> | ||
+ | <div style="width:100%;" class="NavContent"> | ||
+ | <p style="text-align: left;">The first law of thermodynamics | ||
+ | \[ | ||
+ | dE + pdV = TdS | ||
+ | \] | ||
+ | can be transformed into the form | ||
+ | \[ | ||
+ | a^3 \left[\frac{d\rho}{dt} + 3H(\rho + p)\right] =T\frac{dS}{dt}.\] | ||
+ | Then with the conservation equation | ||
+ | \[\frac{d\rho}{dt} + 3H(\rho + p) = 0,\] which follows from the Friedmann equation, one obtains \[ \frac{dS}{dt} = 0 \Rightarrow S = const.\]</p> | ||
+ | </div> | ||
+ | </div></div> | ||
+ | |||
+ | |||
+ | |||
+ | <div id="ter_12"></div> | ||
+ | <div style="border: 1px solid #AAA; padding:5px;"> | ||
+ | === Problem 9 === | ||
+ | Show that the entropy density behaves as $s\propto a^{-3}$. | ||
+ | <div class="NavFrame collapsed"> | ||
+ | <div class="NavHead">solution</div> | ||
+ | <div style="width:100%;" class="NavContent"> | ||
+ | <p style="text-align: left;">\[S = const \Rightarrow sV = const \Rightarrow sV_0 a^3 = const \Rightarrow sa^3 = const \Rightarrow s \propto a^{ - 3}.\]</p> | ||
+ | </div> | ||
+ | </div></div> | ||
+ | |||
+ | |||
+ | |||
+ | <div id="ter_13"></div> | ||
+ | <div style="border: 1px solid #AAA; padding:5px;"> | ||
+ | === Problem 10 === | ||
+ | Using only thermodynamical considerations, show that if the energy density of some component is $\rho=const$ | ||
+ | than the state equation for that component reads $p=-\rho$. | ||
+ | <div class="NavFrame collapsed"> | ||
+ | <div class="NavHead">solution</div> | ||
+ | <div style="width:100%;" class="NavContent"> | ||
+ | <p style="text-align: left;">By definition, | ||
+ | \[ | ||
+ | p = - \left( \frac{\partial E}{\partial V} \right)_S. | ||
+ | \] | ||
+ | If $\rho = const$, then \[\frac{\partial E}{\partial V} = \rho\Rightarrow p = - \rho.\]</p> | ||
+ | </div> | ||
+ | </div></div> | ||
+ | |||
+ | |||
+ | |||
+ | <div id="ter_14"></div> | ||
+ | <div style="border: 1px solid #AAA; padding:5px;"> | ||
+ | === Problem 11 === | ||
+ | Show that the product $aT$ is an approximate invariant in the Universe dominated by relativistic particles. | ||
+ | <div class="NavFrame collapsed"> | ||
+ | <div class="NavHead">solution</div> | ||
+ | <div style="width:100%;" class="NavContent"> | ||
+ | <p style="text-align: left;">Consider a volume $V$, filled by photons and let it expands with the same rate as the whole Universe, i.e. $V \propto a^3 (t)$. For radiation $\rho = \alpha T^4 $, $p = \rho/3.$ The first law of thermodynamics applied to the expanding Universe with no heat flow reads | ||
+ | \[ | ||
+ | \frac{dE}{dt} = - p\frac{dV}{dt}. | ||
+ | \] | ||
+ | As $E = \rho V$, then | ||
+ | \[ | ||
+ | \frac{1}{T}\frac{dT}{dt} = - \frac{1}{3V}\frac{dV}{dt},\] or equivalently \[ \frac{d}{dt}(\ln T) = - \frac{d}{dt}(\ln a)\Rightarrow aT = const. | ||
+ | \]</p> | ||
+ | </div> | ||
+ | </div></div> | ||
+ | |||
+ | |||
+ | |||
+ | |||
+ | |||
+ | <div id="ter_15"></div> | ||
+ | <div style="border: 1px solid #AAA; padding:5px;"> | ||
+ | === Problem 12 === | ||
+ | Show that for a system of particles in thermal equilibrium, | ||
+ | \[\frac{dp}{dT} =\frac{1}{T} \left(\rho +p\right)\] | ||
+ | <div class="NavFrame collapsed"> | ||
+ | <div class="NavHead">solution</div> | ||
+ | <div style="width:100%;" class="NavContent"> | ||
+ | <p style="text-align: left;">The entropy of a system of particles in thermal equilibrium, is a function $S=S(V,T)$ such that | ||
+ | \[dS=\frac{1}{T} \left(dE+pdV\right)=\frac{1}{T} \left[d(\rho V)+pdV\right]=\frac{1}{T} \left[V\frac{d\rho }{dT} dT+(p+\rho )dV\right]\] | ||
+ | Hence | ||
+ | \[\frac{\partial S}{\partial V} =\frac{1}{T} \left(\rho +p\right),\quad \frac{\partial S}{\partial V} =\frac{\partial S}{\partial T} =\frac{V}{T} \frac{d\rho }{dT} \] | ||
+ | These derivatives must satisfy | ||
+ | \[\frac{\partial }{\partial T} \left[\frac{1}{T} \left(\rho +p\right)\right]=\frac{\partial }{\partial V} \left(\frac{V}{T} \frac{d\rho }{dT} \right)\] | ||
+ | Finally obtain | ||
+ | \[\frac{dp}{dT} =\frac{1}{T} \left(\rho +p\right)\]</p> | ||
+ | </div> | ||
+ | </div></div> | ||
+ | |||
+ | |||
+ | |||
+ | |||
+ | <div id="ter_16"></div> | ||
+ | <div style="border: 1px solid #AAA; padding:5px;"> | ||
+ | === Problem 12 === | ||
+ | Show that for the substance with the equation $p=w\rho$ \[T\propto \rho ^{\frac{w}{w+1} } \] | ||
+ | <div class="NavFrame collapsed"> | ||
+ | <div class="NavHead">solution</div> | ||
+ | <div style="width:100%;" class="NavContent"> | ||
+ | <p style="text-align: left;">\[\begin{array}{l} {\frac{dp}{dT} =\frac{1}{T} \left(\rho +p\right)\to \frac{d\rho }{dT} =\frac{1+w}{w} \frac{\rho }{T} ,} \\ {\frac{w}{w+1} \frac{d\rho }{\rho } =\frac{dT}{T} ,} \\ {T\propto \rho ^{\frac{w}{w+1} } } \end{array}\] </p> | ||
+ | </div> | ||
+ | </div></div> | ||
+ | |||
+ | |||
+ | |||
+ | |||
+ | <div id="ter_17"></div> | ||
+ | <div style="border: 1px solid #AAA; padding:5px;"> | ||
+ | === Problem 13 === | ||
+ | Show that for the substance with the equation $p=w\rho $ | ||
+ | \[T\propto a^{-3w} \] | ||
+ | <div class="NavFrame collapsed"> | ||
+ | <div class="NavHead">solution</div> | ||
+ | <div style="width:100%;" class="NavContent"> | ||
+ | <p style="text-align: left;">\[T\propto \rho ^{\frac{w}{w+1} } ,\quad \rho \propto a^{-3(1+w)} ,\to T\propto a^{-3w} \] </p> | ||
+ | </div> | ||
+ | </div></div> | ||
+ | |||
+ | |||
+ | |||
+ | |||
+ | |||
+ | |||
+ | <div id="ter_18"></div> | ||
+ | <div style="border: 1px solid #AAA; padding:5px;"> | ||
+ | === Problem 14 === | ||
+ | Obtain a generalization of Stefan-Boltzmann law for cosmological fluid, described by generalized polytropic equation of state $p=w\rho +k\rho ^{1+1/n} $. | ||
+ | <div class="NavFrame collapsed"> | ||
+ | <div class="NavHead">solution</div> | ||
+ | <div style="width:100%;" class="NavContent"> | ||
+ | <p style="text-align: left;">For the polytropic equation of state the thermodynamical equation | ||
+ | \[\frac{dp}{dT} =\frac{1}{T} \left(\rho +p\right)\] | ||
+ | becomes | ||
+ | \[\left(w+k(1+1/n)\rho ^{1/n} \right)\frac{d\rho }{dt} =\frac{1}{T} \left(w+1+k\rho ^{1/n} \right)\rho \] | ||
+ | This equation can be integrated into | ||
+ | \[T=T_{*} \left[1\pm \left(\rho /\rho _{*} \right)^{1/n} \right]^{\left(w+n+1\right)/\left(w+1\right)} \left(\rho /\rho _{*} \right)^{w/\left(w+1\right)} \] | ||
+ | where $T_{*} $is a constant of integration, the upper sign corresponds to$k>0$, and the lower sign corresponds to$k<0$. This relation can be viewed as a generalization of Stefan-Boltzmann law.</p> | ||
+ | </div> | ||
+ | </div></div> | ||
+ | |||
+ | |||
+ | |||
+ | |||
+ | |||
+ | |||
+ | |||
+ | <div id="ter_19"></div> | ||
+ | <div style="border: 1px solid #AAA; padding:5px;"> | ||
+ | === Problem 15 === | ||
+ | Show that the velocity of sound for cosmological fluid described by generalized polytropic equation of state $p=w\rho +k\rho ^{1+1/n} $ vanishes at the point where the temperature is extremum. | ||
+ | <div class="NavFrame collapsed"> | ||
+ | <div class="NavHead">solution</div> | ||
+ | <div style="width:100%;" class="NavContent"> | ||
+ | <p style="text-align: left;">For the generalized polytropic equation of state the velocity of sound is given by | ||
+ | \[\begin{array}{l} {c_{s}^{2} =\frac{dp}{d\rho } =w\pm \left(w+1\right)\frac{n+1}{n} \left(\frac{\rho }{\rho _{*} } \right)^{1/n} ,} \\ {\rho _{*} =\left[\frac{w+1}{\left|k\right|} \right]^{n} } \end{array}\] | ||
+ | The upper sign corresponds to$k>0$, and the lower sign corresponds to$k<0$. As we have seen in the previous problem | ||
+ | \[T=T_{*} \left[1\pm \left(\rho /\rho _{*} \right)^{1/n} \right]^{\left(w+n+1\right)/\left(w+1\right)} \left(\rho /\rho _{*} \right)^{w/\left(w+1\right)} \] | ||
+ | The extremum of temperature (when it exists) is located at | ||
+ | \[\rho _{e} =\rho _{*} \left[\mp \frac{wn}{\left(w+1\right)(n+1)} \right]^{n} \] | ||
+ | It is easy to see that the velocity of sound vanishes at the point where the temperature is extremum.</p> | ||
+ | </div> | ||
+ | </div></div> |
Latest revision as of 23:29, 8 January 2013
Problem 1
Transform the energy conditions for the flat Universe to conditions for the entropy density (see [1])
Problem 2
Find the entropy density for the photon gas.
\[ dS = \frac{dE}{T} + \frac{pdV}{T} = \frac{V}{T}\frac{d\rho}{dT}dT + \frac{\rho+ p}{T}dV \Rightarrow\] \[ \left( \frac{\partial S}{\partial V} \right)_T = \frac{\rho + p} {T} \Rightarrow S = \frac{\rho + p}{T}V + f(T).\] As entropy is proportional to volume, then $f(T) = 0.$ For the photon gas $p = \rho/3$, therefore $$ s = \frac{S}{V} = \frac{4}{3}\frac{\rho}{T} = \frac{4}{3}\alpha {T^3}. $$
Problem 3
Use the result of the previous problem to derive an alternative proof of the fact that $aT=const$ in the adiabatically expanding Universe.
$$S = \frac{4}{3}\alpha T^3V = const;\quad V \propto a^3 \Rightarrow aT = const.$$
Problem 4
Find the adiabatic index for the CMB.
$$ p \propto \rho \propto a^{ - 4};\quad V \propto {a^3} \Rightarrow pV^\gamma = const,\quad \gamma = \frac{4}{3}. $$
Problem 5
Show that the ratio of CMB entropy density to the baryon number density $s_\gamma/n_b$ remains constant during the expansion of the Universe.
$$s \propto T^3 \propto a^{ - 3};\quad n_b \propto a^{ - 3}.$$ Therefore their ratio remains constant during the expansion of Universe.
Problem 6
Estimate the current entropy density of the Universe.
The current entropy of the Universe is determined by the CMB photons. A rough estimate neglects the contribution of other particles, then one obtains \[s = 2\frac{2\pi ^2}{45}T_0^3.\] For $T_0 \approx 2.725K$ \[s \approx 1.5 \cdot 10^3 \, cm^{-3}.\]
Problem 7
Estimate the entropy of the observable part of the Universe.
Size of the observable part of the Universe is $l_{H0} \sim 10^{28} \mbox{cm}$ and therefore \[S = \frac{4\pi}{3}sl_{H0}^3 \sim 10^{88}.\]
Problem 7
Why is the expansion of the Universe described by the Friedman equations adiabatic?
It is because the heat flow is absent in the homogeneous and isotropic Universe.
Problem 8
Show that entropy is conserved during the expansion of the Universe described by the Friedman equations.
The first law of thermodynamics \[ dE + pdV = TdS \] can be transformed into the form \[ a^3 \left[\frac{d\rho}{dt} + 3H(\rho + p)\right] =T\frac{dS}{dt}.\] Then with the conservation equation \[\frac{d\rho}{dt} + 3H(\rho + p) = 0,\] which follows from the Friedmann equation, one obtains \[ \frac{dS}{dt} = 0 \Rightarrow S = const.\]
Problem 9
Show that the entropy density behaves as $s\propto a^{-3}$.
\[S = const \Rightarrow sV = const \Rightarrow sV_0 a^3 = const \Rightarrow sa^3 = const \Rightarrow s \propto a^{ - 3}.\]
Problem 10
Using only thermodynamical considerations, show that if the energy density of some component is $\rho=const$ than the state equation for that component reads $p=-\rho$.
By definition, \[ p = - \left( \frac{\partial E}{\partial V} \right)_S. \] If $\rho = const$, then \[\frac{\partial E}{\partial V} = \rho\Rightarrow p = - \rho.\]
Problem 11
Show that the product $aT$ is an approximate invariant in the Universe dominated by relativistic particles.
Consider a volume $V$, filled by photons and let it expands with the same rate as the whole Universe, i.e. $V \propto a^3 (t)$. For radiation $\rho = \alpha T^4 $, $p = \rho/3.$ The first law of thermodynamics applied to the expanding Universe with no heat flow reads \[ \frac{dE}{dt} = - p\frac{dV}{dt}. \] As $E = \rho V$, then \[ \frac{1}{T}\frac{dT}{dt} = - \frac{1}{3V}\frac{dV}{dt},\] or equivalently \[ \frac{d}{dt}(\ln T) = - \frac{d}{dt}(\ln a)\Rightarrow aT = const. \]
Problem 12
Show that for a system of particles in thermal equilibrium, \[\frac{dp}{dT} =\frac{1}{T} \left(\rho +p\right)\]
The entropy of a system of particles in thermal equilibrium, is a function $S=S(V,T)$ such that \[dS=\frac{1}{T} \left(dE+pdV\right)=\frac{1}{T} \left[d(\rho V)+pdV\right]=\frac{1}{T} \left[V\frac{d\rho }{dT} dT+(p+\rho )dV\right]\] Hence \[\frac{\partial S}{\partial V} =\frac{1}{T} \left(\rho +p\right),\quad \frac{\partial S}{\partial V} =\frac{\partial S}{\partial T} =\frac{V}{T} \frac{d\rho }{dT} \] These derivatives must satisfy \[\frac{\partial }{\partial T} \left[\frac{1}{T} \left(\rho +p\right)\right]=\frac{\partial }{\partial V} \left(\frac{V}{T} \frac{d\rho }{dT} \right)\] Finally obtain \[\frac{dp}{dT} =\frac{1}{T} \left(\rho +p\right)\]
Problem 12
Show that for the substance with the equation $p=w\rho$ \[T\propto \rho ^{\frac{w}{w+1} } \]
\[\begin{array}{l} {\frac{dp}{dT} =\frac{1}{T} \left(\rho +p\right)\to \frac{d\rho }{dT} =\frac{1+w}{w} \frac{\rho }{T} ,} \\ {\frac{w}{w+1} \frac{d\rho }{\rho } =\frac{dT}{T} ,} \\ {T\propto \rho ^{\frac{w}{w+1} } } \end{array}\]
Problem 13
Show that for the substance with the equation $p=w\rho $ \[T\propto a^{-3w} \]
\[T\propto \rho ^{\frac{w}{w+1} } ,\quad \rho \propto a^{-3(1+w)} ,\to T\propto a^{-3w} \]
Problem 14
Obtain a generalization of Stefan-Boltzmann law for cosmological fluid, described by generalized polytropic equation of state $p=w\rho +k\rho ^{1+1/n} $.
For the polytropic equation of state the thermodynamical equation \[\frac{dp}{dT} =\frac{1}{T} \left(\rho +p\right)\] becomes \[\left(w+k(1+1/n)\rho ^{1/n} \right)\frac{d\rho }{dt} =\frac{1}{T} \left(w+1+k\rho ^{1/n} \right)\rho \] This equation can be integrated into \[T=T_{*} \left[1\pm \left(\rho /\rho _{*} \right)^{1/n} \right]^{\left(w+n+1\right)/\left(w+1\right)} \left(\rho /\rho _{*} \right)^{w/\left(w+1\right)} \] where $T_{*} $is a constant of integration, the upper sign corresponds to$k>0$, and the lower sign corresponds to$k<0$. This relation can be viewed as a generalization of Stefan-Boltzmann law.
Problem 15
Show that the velocity of sound for cosmological fluid described by generalized polytropic equation of state $p=w\rho +k\rho ^{1+1/n} $ vanishes at the point where the temperature is extremum.
For the generalized polytropic equation of state the velocity of sound is given by \[\begin{array}{l} {c_{s}^{2} =\frac{dp}{d\rho } =w\pm \left(w+1\right)\frac{n+1}{n} \left(\frac{\rho }{\rho _{*} } \right)^{1/n} ,} \\ {\rho _{*} =\left[\frac{w+1}{\left|k\right|} \right]^{n} } \end{array}\] The upper sign corresponds to$k>0$, and the lower sign corresponds to$k<0$. As we have seen in the previous problem \[T=T_{*} \left[1\pm \left(\rho /\rho _{*} \right)^{1/n} \right]^{\left(w+n+1\right)/\left(w+1\right)} \left(\rho /\rho _{*} \right)^{w/\left(w+1\right)} \] The extremum of temperature (when it exists) is located at \[\rho _{e} =\rho _{*} \left[\mp \frac{wn}{\left(w+1\right)(n+1)} \right]^{n} \] It is easy to see that the velocity of sound vanishes at the point where the temperature is extremum.