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--- courses:ast301:1 [2024/02/17 23:35] – [7. Instability and pulsation] asad
+++ courses:ast301:1 [2024/03/07 01:12] (current) – [4.2 Dynamical timescale] asad
@@ Line 113: / Line 113: @@
 where we assume $N(N-1)\approx N^2$ because $N$ is large. The factor $N(N-1)/2$ comes from the **combination** of two items from a total of $N$ items: $_NC_2=N!/[2!(N-2)!]=N(N-1)/2$. If total mass $M=Nm$ then
-$$ M \langle v_i^2 \rangle_{av} - G\frac{M^2}{2} \langle r_{ij}^{-1} \rangle_{av} $$
+$$ M \langle v_i^2 \rangle_{av} - G\frac{M^2}{2} \langle r_{ij}^{-1} \rangle_{av} = 0$$
 and hence the **virial mass** of a cluster
@@ Line 153: / Line 153: @@
 The time needed to fall by a distance $R$ is found from the constant-force expression $s=at^2/2$, i. e. the dynamical timescale
-$$ \tau_{in} = \sqrt{\frac{R}{a}} = \sqrt{R^3}{GM} $$
+$$ \tau_{in} = \sqrt{\frac{R}{a}} = \sqrt{\frac{R^3}{GM}} $$
 where density $\rho=M/R^3$ giving rise to
@@ Line 360: / Line 360: @@
 If the oscillation amplitude is high, we call the stars //pulsating variables// that include quasi-periodic //cepheid variables// and RR Lyrae variables which are used to measure the distance of galaxies. Let us deal with the basic thermodynamics controlling these two types of variable stars.
-{{:courses:ast301:starcarnot.webp?nolink|}}
+{{:courses:ast301:starcarnot.webp?nolink&750|}}
 The oscillation of these stars can be explained using the 4-step Carnot cycle used for describing the changes of state of a volume of gas. During the cycle, the gas absorbs heat and does some work on the surroundings. On the P-V plot above, work $W=\int P dV$ is the area under a curve with certain limits. Work is positive for movement to the right (upper path, isothermal expansion), negative for movement to the left (lower path, isothermal compression). The lower path yields lower negative work due to lower pressure compared to the the positive work for the upper path. So the net work is positive, which is always the case for clockwise cycle. If the cycle was counterclockwise, work would be done on the gas, i. e. work would be negative.
@@ Line 370: / Line 370: @@
 $$ \delta Q = dU + \delta W $$
-where the $d$ and $\delta$ make it clear that $U$ is a //state variable// (internal property of a system) while $Q$ and $W$ are not. Pressure $P$, volume $V$, temperature $T$ and entropy $S$ are also state variables. $\delta Q$ is positive when gas absorbs heat, $\delta W$ is positive when gas works on the surroundings. Over a complete cycle
+where the $d$ and $\delta$ make it clear that $U$ is a //state variable// (internal property of a system) while $Q$ and $W$ are not. Pressure $P$, volume $V$, temperature $T$ and entropy $S$ are also state variables. $\delta Q$ is positive when gas absorbs heat, $\delta W$ is positive when gas works on the surroundings. Over a complete cycle, the loop integral
 $$ \oint dU = 0 \Rightarrow W = + \oint \delta Q $$
@@ Line 388: / Line 388: @@
 which can be plugged in the equation of entropy to get
+$$ W = \oint \delta Q \approx \oint \frac{\Delta T(t)}{T_0} \delta Q $$
+which means $W$ and $\Delta T$ have the same sign, heat is absorbed (work on the surrounding) when temperature increases and discharged (work on the gas) when temperature decreases. This is similar to a **heat engine**, for example, an internal combustion engine. In both cases, heat is introduced during an expansion at high temperature ($T_2$) and dumped during a compression at low temperature ($T_1$).
+The ultimate **condition for pulsation** is achieved when we integrate the above equation over the whole mass of the star:
+$$ W \approx \int_M \oint_Q \frac{\Delta T(t,m)}{T_0(m)} \delta Q(m) \ dm > 0. $$
+In a heat engine, the operative mechanism is the burning of fuel. In a star, the operative mechanism is the **valving of heat** by the changing opacity in a star's ionization transition zone as shown below.
+{{:courses:ast301:pulsation.webp?nolink&700|}}
+Inward of the transition zone gas is hot and ionized, outward the gas is cold and neutral. For H and He, the transition zone is close to the surface. The transition zone itself is a mix of hot and cold gas.
+(a) When a star compresses (due to a breakdown of equilibrium), temperature in the transition zone increases leading to more ionization. The number of free electrons increases leading to a higher **opacity**, meaning obstruction to photons. Photons are trapped by the free electrons via scattering. The trapped outward radiation provides heat which is absorbed at a high temperature a required by a heat engine.
+(b) The trapped heat causes the star to expand and cool. This leads to recombination for many atoms, so ionization and number of free electrons decrease. The highway of photons is open again and the photons carry away the trapped heat. Heat is released at a low temperature.
+Due to higher volume and lower temperature, pressure decreases ($P \propto T/V$) and the outer layers of the star collapse due to self-gravity leading to a compression which takes us back to (a).
+RR Lyrae and Cepheid variables can be used to measure large distance because they are very luminous (300 to 30,000 times higher than the sun) and their luminosity is related to their period.