S H I N E: April 2020

Wednesday, April 22, 2020

Evolution of SNR

http://www.astro.uu.se/~hoefner/astro/teach/apd_files/apd_SNR_dyn.pdf
Size distribution of supernova remnants and the interstellarmedium: the case of M 33
https://www.astro.umd.edu/~richard/ASTR680/supernova%20remnants.pdf
https://heasarc.gsfc.nasa.gov/docs/objects/snrs/snrstext.html

TERMUX INSTALL MATPLOTLIB error: ft2build.h / fthread.h

error: can not find ft2build.h

ft2build.h in the below directory: /data/data/com.termux/files/usr/include/freetype2

solution:

Reference:

https://www.reddit.com/r/termux/comments/bhiinx/problem_installing_matplotlib/

https://github.com/matplotlib/matplotlib/issues/3029/

Monday, April 20, 2020

From pitch angle diffusion to spatial diffusion coefficient

\[ dp/dt = qv/c \times (B_0 + \delta B) \]
\[dp_{\parallel}/dt = q/c (v_{\perp} \times \delta B) \]
\[p_{\perp} = \mu p \]
\[ d\mu/dt = q/(pc)v(1-\mu^2)^{1/2} \delta B \cos(\Omega t -kx +\psi) \]
\[ \delta B(x,t) = \delta B \cos(\Omega t -kx +\psi) \]

Averaging over random phase of the waves $\psi$,
\[ \int_0^{2\pi} \cos^2\psi d\psi = \pi\]
\[\cos(a+b) = \cos(a)\cos(b) - \sin(a)\sin(b) \]
in the frame in which the wave is at rest, we can write $x=\mu v t$,
\[ \langle \delta \mu (t')\delta \mu (t'') \rangle_{\psi} = \frac{q^2v^2(1-\mu^2)(\delta B)^2}{2c^2p^2} \cos[(\Omega -kv\mu)(t-t'')] \].

Integrating over time,
\[ \langle \delta \mu (t')\delta \mu (t'') \rangle_{t} = \frac{q^2v^2(1-\mu^2)(\delta B)^2}{2c^2p^2} \int dt' \int dt'' \cos[(\Omega -kv\mu)(t-t'')] \\
= \frac{q^2 v(1-\mu^2)(\delta B)^2}{c^2 p^2 \mu} \delta(k-\Omega/(v\mu)) \delta t \].

Chromebook linux beta

Open Chrome OS developer shell Ctrl + Alt + T
vmc list
vmc start termina

check for default Linux container
lxc list

If the penguin container is listed, you can try manually starting it
logout
vmc container termina penguin

destroy a virtual machine

vmc destroy vmc_name

Reference:
https://support.google.com/chromebook/thread/6127686?hl=en

Sunday, April 12, 2020

Integration to sum

The definite integral of a continuous function $f$ over the interval $[a,b]$, denoted by $\int_a^b f(x) dx$, is the limit of a Riemann sum as the number of subdivisions approach infinity.
\[ \int_a^b f(x)dx = \lim_{n \to \infty} \sum_{i=1}^n \delta x \cdot f(x_i) \]
where $\delta x = (b-a)/n$, and $x_i = a+\delta x \cdot i$.

Example
\[\int_{\pi}^{2\pi} cos(x)dx \]
\[\delta x = (b-a)/n = (2\pi-\pi)/n = pi/n\]
\[x_i = a + \delta x \cdot i = \pi + \pi/n \cdot i \]
Therefore
\[\int_{\pi}^{2\pi} \cos(x)dx = \lim_{n\to \infty}\sum_{i=1}^{n} \frac{\pi}{n}\cdot \cos(\pi+\frac{\pi i}{n}) \]

Reference:
https://www.khanacademy.org/math/ap-calculus-ab/ab-integration-new/ab-6-3/a/definite-integral-as-the-limit-of-a-riemann-sum