Non relativistic small perturbation theory - Revision history

Cosmo All: /* Problem 9 */

2014-02-11T05:39:49Z

‎Problem 9

Cosmo All: /* Problem 6 */

2014-02-11T05:39:07Z

‎Problem 6

Cosmo All: /* Problem 5 */

2014-02-11T05:36:55Z

‎Problem 5

Cosmo All: /* Problem 4 */

2014-02-11T05:36:07Z

‎Problem 4

Cosmo All: /* Problem 3 */

2014-02-11T05:35:13Z

‎Problem 3

Cosmo All: /* Problem 3 */

2014-02-11T05:34:23Z

‎Problem 3

Cosmo All: /* Problem 2 */

2014-02-11T05:33:57Z

‎Problem 2

Cosmo All: /* Problem 2 */

2014-02-11T05:33:30Z

‎Problem 2

Cosmo All at 05:32, 11 February 2014

2014-02-11T05:32:48Z

Cosmo All at 05:31, 11 February 2014

2014-02-11T05:31:26Z

Show changes

@@ Line 374: / Line 374: @@
 === Problem 9 ===
 <p style= "color: #999;font-size: 11px">problem id: per8</p>
-Assuming, that a particular solution to equation from prob. \ref{per6} has the form $\delta _1\left( t \right) \sim H\left( t \right)$, construct the general solution for $\delta (t)$. Consider the flat Universe filled with the substance with $p = w\rho.$
+Assuming, that a particular solution to equation from prob. [[#per6]] has the form $\delta _1\left( t \right) \sim H\left( t \right)$, construct the general solution for $\delta (t)$. Consider the flat Universe filled with the substance with $p = w\rho.$
 <div class="NavFrame collapsed">
    <div class="NavHead">solution</div>
@@ Line 413: / Line 413: @@
 <div id="per9"></div>
 <div style="border: 1px solid #AAA; padding:5px;">
 === Problem 10 ===
 <p style= "color: #999;font-size: 11px">problem id: per9</p>

@@ Line 234: / Line 234: @@
 $$
 \begin{gathered}
-   \tilde{\rho}  = \rho  + \delta \rho \left(\vec x,t \right);\hfill \\
+   \tilde{\rho}  = \rho  + \delta \rho \left(\vec x,t \right);\\
-\vec{\tilde {V}} = \vec V + \delta \vec v;\hfill \\
+\vec{\tilde {V}} = \vec V + \delta \vec v;\\
-\tilde \Phi  = \Phi  + \delta \Phi; \hfill \\
+\tilde \Phi  = \Phi  + \delta \Phi; \\
-   \tilde p = p + \delta p. \hfill \\
+   \tilde p = p + \delta p. \\
 \end{gathered}
 $$
@@ Line 244: / Line 244: @@
 \begin{gathered}
    \frac{\partial \delta \rho}
-{\partial t} + \rho \nabla \delta \vec v + \nabla \left( \delta \rho  \cdot \vec V \right) = 0; \hfill \\
+{\partial t} + \rho \nabla \delta \vec v + \nabla \left( \delta \rho  \cdot \vec V \right) = 0; \\
    \frac{\partial \delta \vec v}
 {\partial t} + \left( \vec V \cdot \nabla \right)\delta \vec v + \left( \delta \vec v \cdot \nabla \right)\vec V + \frac{c_s^2}
-\rho\nabla \delta \rho  + \nabla \delta \Phi  = 0;\hfill \\
+\rho\nabla \delta \rho  + \nabla \delta \Phi  = 0;\\
-   \Delta \delta \Phi  = 4\pi G\delta \rho . \hfill \\
+   \Delta \delta \Phi  = 4\pi G\delta \rho . \\
 \end{gathered}
 $$
@@ Line 263: / Line 263: @@
 Denoting $\delta  = \delta \rho /\rho$, one could rewrite equations in the form
 $$\begin{gathered}
-  \frac{\partial \delta }{\partial t}  + \frac{1}{a}\nabla \delta \vec v = 0;\hfill\\
+  \frac{\partial \delta }{\partial t}  + \frac{1}{a}\nabla \delta \vec v = 0;\\
-    \frac{\partial \delta }{\partial t} + H\delta \vec v + \frac{c_s^2}{a}\nabla \delta  + \frac{1}{a}\nabla \Phi  = 0;\hfill \\
+    \frac{\partial \delta }{\partial t} + H\delta \vec v + \frac{c_s^2}{a}\nabla \delta  + \frac{1}{a}\nabla \Phi  = 0;\\
-\Delta \delta \Phi  = 4\pi G\rho \delta , \hfill \\
+\Delta \delta \Phi  = 4\pi G\rho \delta , \\
 \end{gathered}
 $$
@@ Line 281: / Line 281: @@
 <div id="per7"></div>
 <div style="border: 1px solid #AAA; padding:5px;">
 === Problem 7 ===
 <p style= "color: #999;font-size: 11px">problem id: per7</p>

@@ Line 181: / Line 181: @@
      <p style="text-align: left;">a) $$
 \begin{gathered}
-\pi G{\rho _0} - c_S^2{k_J}^2 > 0,\hfill \\
+\pi G{\rho _0} - c_S^2{k_J}^2 > 0,\\
-\omega _1 =  - \omega _2 = \omega \operatorname{Im}\delta _0 = 0,\hfill \\
+\omega _1 =  - \omega _2 = \omega \operatorname{Im}\delta _0 = 0,\\
   \rho = \rho _0\operatorname{Re}(1 + \delta_0e^{\omega _{1,2}t+i\vec k\vec x})
-= \rho _0(1 + \delta _0e^{\omega _{1,2}t}\cos \vec k\vec x), \hfill \\
+= \rho _0(1 + \delta _0e^{\omega _{1,2}t}\cos \vec k\vec x), \\
 \vec v = \operatorname{Re}(i\frac{\vec k}
 {k}e^{i\vec k\vec x}\frac{\omega }
 {k}\delta _0e^{\omega _{1,2}t}) =  - \frac{\vec k\omega}
-{k}e^{\omega _{1,2}t}\sin \vec k\vec x .\hfill \\
+{k}e^{\omega _{1,2}t}\sin \vec k\vec x .\\
 \end{gathered}
 $$
@@ Line 194: / Line 194: @@
 b)\[
 \begin{gathered}
-   \omega  =  \pm i\sqrt {c_S^2k^2 - 4\pi G\rho _0},  \hfill \\
+   \omega  =  \pm i\sqrt {c_S^2k^2 - 4\pi G\rho _0},  \\
-   \rho  = \rho _0\left[ 1 - \delta _0\cos \left( \vec k\vec x - \left| \omega  \right|t \right) \right],\hfill\\
+   \rho  = \rho _0\left[ 1 - \delta _0\cos \left( \vec k\vec x - \left| \omega  \right|t \right) \right],\\
 \vec v = \frac{\vec k\omega }
-{k}\cos \left( \vec k\vec x - \left| \omega  \right|t \right),\hfill \\
+{k}\cos \left( \vec k\vec x - \left| \omega  \right|t \right),\\
-   \left| \omega  \right| = \sqrt {c_S^2k^2 - 4\pi G\rho _0}.  \hfill \\
+   \left| \omega  \right| = \sqrt {c_S^2k^2 - 4\pi G\rho _0}.  \\
 \end{gathered}
 \]
@@ Line 212: / Line 212: @@
 $$
 \begin{gathered}
-\rho =\rho _0\left[1 -\delta _0\cos \left(\vec k\vec x \right)\right], \hfill \\
+\rho =\rho _0\left[1 -\delta _0\cos \left(\vec k\vec x \right)\right], \\
 \vec v = \frac{\vec k\omega }
-{k}\cos \left( \vec k\vec x\right). \hfill \\
+{k}\cos \left( \vec k\vec x\right). \\
 \end{gathered}
 $$
@@ Line 224: / Line 224: @@
 <div id="per6"></div>
 <div style="border: 1px solid #AAA; padding:5px;">
 === Problem 6 ===
 <p style= "color: #999;font-size: 11px">problem id: per6</p>

@@ Line 135: / Line 135: @@
 === Problem 4 ===
 <p style= "color: #999;font-size: 11px">problem id: per4</p>
-Consider time dependent adiabatic perturbations and find the characterictic scale of instability (so-called Jeans instability).
+Consider time dependent adiabatic perturbations and find the characteristic scale of instability (so-called Jeans instability).
 <div class="NavFrame collapsed">
    <div class="NavHead">solution</div>
@@ Line 142: / Line 142: @@
 $$
 \begin{gathered}
-   \omega \delta _0 =  - ikw_0, \hfill \\
+   \omega \delta _0 =  - ikw_0, \\
-   \omega \vec w_0 =  - i\vec kf_0 - i\vec kc_S^2\delta _0, \hfill \\
+   \omega \vec w_0 =  - i\vec kf_0 - i\vec kc_S^2\delta _0, \\
-    - k^2f_0 = 4\pi G\rho _0\delta _0,\hfill \\
+    - k^2f_0 = 4\pi G\rho _0\delta _0,\\
-   \omega \sigma _0 = 0. \hfill \\
+   \omega \sigma _0 = 0. \\
 \end{gathered}
 $$
@@ Line 155: / Line 155: @@
 $$
 \begin{gathered}
-\pi G\rho _0 - c_S^2k_J^2 = 0, \hfill \\
+\pi G\rho _0 - c_S^2k_J^2 = 0, \\
   k_J = \frac{1}
-{c_S}\sqrt {4\pi G\rho _0},\hfill \\
+{c_S}\sqrt {4\pi G\rho _0},\\
    \lambda _J = \frac{2\pi}
 {k_J} = c_S\sqrt {\frac{\pi }
-{G\rho _0}} . \hfill \\
+{G\rho _0}} . \\
 \end{gathered}
 $$</p>
@@ Line 169: / Line 169: @@
 <div id="per5"></div>
 <div style="border: 1px solid #AAA; padding:5px;">
 === Problem 5 ===
 <p style= "color: #999;font-size: 11px">problem id: per5</p>

@@ Line 116: / Line 116: @@
 \end{gathered}
 $$
-The equations for perturbations in this case (see. problem \ref{per2}) have the form:
+The equations for perturbations in this case (see. problem [[#per2]]) have the form:
 $$
 \omega \delta _0 =  - ikw_0;

@@ Line 110: / Line 110: @@
 $$
 \begin{gathered}
-   \delta  = \delta _0e^{\omega t}, \hfill \\
+   \delta  = \delta _0e^{\omega t}, \\
-  \vec w =\vec w_0e^{\omega t}, \hfill \\
+  \vec w =\vec w_0e^{\omega t}, \\
-  f = f_0e^{\omega t},\hfill \\
+  f = f_0e^{\omega t},\\
-  \sigma  = \sigma _0e^{\omega t}. \hfill \\
+  \sigma  = \sigma _0e^{\omega t}. \\
 \end{gathered}
 $$
@@ Line 132: / Line 132: @@
 <div id="per4"></div>
 <div style="border: 1px solid #AAA; padding:5px;">
 === Problem 4 ===
 <p style= "color: #999;font-size: 11px">problem id: per4</p>

@@ Line 51: / Line 51: @@
    \frac{\partial \vec v}
 {\partial t} + \left(\vec v\nabla \right)\vec v + \frac{1}
-{\rho }\nabla P + \nabla \Phi  = 0, \hfill \\
+{\rho }\nabla P + \nabla \Phi  = 0, \\
    \Delta \Phi  = \nabla  \cdot \left(\nabla \Phi\right) = 4\pi G\rho , \\
    \frac{\partial S}
@@ Line 78: / Line 78: @@
 where $c_S^2$ is adiabatic speed of sound.
-Substituting these expressions into hydrodynamic equations, we obtain the foolowing system:
+Substituting these expressions into hydrodynamic equations, we obtain the following system:
 $$
 \begin{gathered}
    \frac{d\delta}
-{dt} + i\vec k\vec w = 0; \hfill \\
+{dt} + i\vec k\vec w = 0; \\
   \frac{d\vec w}
 {dt} + i\vec kf + i\vec kc_S^2\delta+i\vec kh^2\rho _0^{-1}\sigma  =0; \\
 {k^2}f =  - 4\pi {\rho _0}\delta ; \\
 \frac{d\sigma }
-{dt} = 0. \hfill \\
+{dt} = 0. \\
 \end{gathered}
 $$</p>

@@ Line 48: / Line 48: @@
 \begin{gathered}
    \frac{\partial \rho}
-{\partial t} + \nabla  \cdot \left(\rho \vec v \right) = 0, \hfill \\
+{\partial t} + \nabla  \cdot \left(\rho \vec v \right) = 0, \\
    \frac{\partial \vec v}
 {\partial t} + \left(\vec v\nabla \right)\vec v + \frac{1}
 {\rho }\nabla P + \nabla \Phi  = 0, \hfill \\
-   \Delta \Phi  = \nabla  \cdot \left(\nabla \Phi\right) = 4\pi G\rho , \hfill \\
+   \Delta \Phi  = \nabla  \cdot \left(\nabla \Phi\right) = 4\pi G\rho , \\
    \frac{\partial S}
-{\partial t} + \left(\vec v\nabla \right)S = 0, \hfill \\
+{\partial t} + \left(\vec v\nabla \right)S = 0, \\
 \end{gathered}
 $$
@@ Line 61: / Line 61: @@
 Perturbed solution could be expressed as follows:
 $$\begin{gathered}
-\rho (\vec x,t) = \rho _0\left[ 1 + \delta (t)e^{i\vec k\vec x}\right];\hfill \\
+\rho (\vec x,t) = \rho _0\left[ 1 + \delta (t)e^{i\vec k\vec x}\right];\\
-\vec v(\vec x,t) = \vec w(t)e^{i\vec k\vec x};\hfill \\
+\vec v(\vec x,t) = \vec w(t)e^{i\vec k\vec x};\\
-   p (\vec x,t) = p_0 + f(t)e^{i\vec k\vec x};\hfill \\
+   p (\vec x,t) = p_0 + f(t)e^{i\vec k\vec x};\\
-   S = S_0 + \sigma (t)e^{i\vec k\vec x};\hfill \\
+   S = S_0 + \sigma (t)e^{i\vec k\vec x};\\
    p = p_0 + \frac{\partial p}
 {\partial \rho}\left( \rho  - \rho _0 \right) + \frac{\partial p}
-{\partial S}\left( S - S_0\right) = p_0+ c_S^2\rho _0\delta e^{i\vec k\vec x} + h^2\sigma e^{i\vec k\vec x}. \hfill \\
+{\partial S}\left( S - S_0\right) = p_0+ c_S^2\rho _0\delta e^{i\vec k\vec x} + h^2\sigma e^{i\vec k\vec x}. \\
 \end{gathered}
 $$
@@ Line 84: / Line 84: @@
 {dt} + i\vec k\vec w = 0; \hfill \\
   \frac{d\vec w}
-{dt} + i\vec kf + i\vec kc_S^2\delta+i\vec kh^2\rho _0^{-1}\sigma  =0; \hfill \\
+{dt} + i\vec kf + i\vec kc_S^2\delta+i\vec kh^2\rho _0^{-1}\sigma  =0; \\
-{k^2}f =  - 4\pi {\rho _0}\delta ; \hfill \\
+{k^2}f =  - 4\pi {\rho _0}\delta ; \\
 \frac{d\sigma }
 {dt} = 0. \hfill \\
@@ Line 96: / Line 96: @@
 <div id="per3"></div>
 <div style="border: 1px solid #AAA; padding:5px;">
 === Problem 3 ===
 <p style= "color: #999;font-size: 11px">problem id: per3</p>