A second-order linear differential equation has the form
\[ p(x)\frac{d^2y}{dx^2} + Q(x)\frac{dy}{dx} + R(x)y = G(x). \]
If $G(x)=0$, for all $x$, such equations are called homogeneous linear equations.
If $G(x)\neq 0$, the equation is called nonhomogeneous.
For homogeneous and constant coefficient equations,
\[ ay''+by'+cy=0,\]
the solution is shown below.
Assume the solution has this form: $e^{rx}$. We will get
\[ar^2+br+c=0,\]
\[\Delta=b^2-4ac,\]
\[r_1=\frac{-b+\sqrt{\Delta}}{2a},\]
\[r_2=\frac{-b-\sqrt{\Delta}}{2a}.\]
For $\Delta>0$, the solution is $c_1e^{r_1x}+c_2e^{r_2x}$.
$\Delta=0$, $r_1=r_2$ $c_1e^{r_1x}+c_2xe^{r_2x}$.
$\Delta<0$, $r=\alpha\pm i\beta$ $e^{\alpha x}(c_1\cos(\beta x) + c_2\sin(\beta x))$.
Reference:
https://www.stewartcalculus.com/data/CALCULUS%20Concepts%20and%20Contexts/upfiles/3c3-2ndOrderLinearEqns_Stu.pdf
Interested in particle astrophysics and plasma astrophysics. This blog is my research/private notebook.
Friday, May 17, 2019
Thursday, May 9, 2019
A simple derivation for parker transport equation
The continuity equation in differential form
\begin{equation*}
\frac{\partial f}{\partial t} =-\nabla \cdot \overrightarrow{S.}
\end{equation*}
In a given volume, the rate of change of the number of particles,$\displaystyle \frac{\partial f}{\partial t}$, is given by the numer of particles that across the surface enclosing the volume, assuming no source of particle is present.
$\displaystyle \vec{S}$ is the streaming crrent density of particles across the surface.
\begin{gather*}
\vec{S} \ =\ \vec{V} f\ -K\cdot \nabla f\ \\
\nabla _{p} \cdot \overrightarrow{S_{p}} =\frac{1}{p^{2}}\frac{\partial }{\partial p}\left( p^{2} \langle \dot{p} \rangle f\right) \ ,\ \overrightarrow{S_{p}} =\langle \dot{p} \rangle f\\
\end{gather*}
\begin{equation*}
\frac{\partial f}{\partial t} +\nabla \cdot \left(\vec{V} f-K\cdot \nabla f\right) +\left(\frac{1}{p^{2}}\frac{\partial }{\partial p}\left( p^{2} \langle \dot{p} \rangle f\right)\right) =0
\end{equation*}
note. The streaming is included in $\vec{S}$.
adiabatic cooling resulting the average change in momentum is $\displaystyle \langle \dot{p} \rangle =-\frac{p}{3} \nabla \cdot \vec{V}$.
\begin{gather*}
\frac{\partial f}{\partial t} +f\nabla \cdot \vec{V} +\vec{V} \cdot \nabla f-\frac{( \nabla \cdot \overrightarrow{V)}}{3p^{2}}\left( 3p^{2} f+p^{3}\frac{\partial f}{\partial p}\right)\\
=\frac{\partial f}{\partial t} +f\nabla \cdot \vec{V} +\vec{V} \cdot \nabla f-( \nabla \cdot \overrightarrow{V)}\left( f+\frac{p}{3}\frac{\partial f}{\partial p}\right)\\
=\frac{\partial f}{\partial t} +\vec{V} \cdot \nabla f-\left( \nabla \cdot \vec{V}\right)\frac{p}{3}\frac{\partial f}{\partial p}
\end{gather*}
\begin{array}{l}
note.\ \frac{dT}{dt} =\frac{dp}{dt}\frac{dT}{dp} =\frac{-\nabla \cdot \vec{V}}{3}\left(p*\frac{p}{T+m}\right) =\frac{-( \nabla \cdot \overrightarrow{V)}}{3} T\left(\frac{T+2m}{T+m}\right) .\\
\frac{dE}{dt} =\frac{dp}{dt}\frac{dE}{dp} =\frac{-\left( \nabla \cdot \vec{V}\right)}{3}(pv)
\end{array}
When differential energy density, $U_p=4\pi p^2f$, is convected with velocity $V$, the flux observed is not $VU_p$, but rather $S_p=CVU_p$, where $C=\frac{-1}{3}\frac{\partial lnf}{\partial lnp} = \frac{-1}{3}\frac{p}{f}\frac{\partial f}{\partial p}$ is called the Compton-Getting coefficien.
For $f=Ap^{-\gamma}$, C is $\frac{\gamma}{3}$. if $\gamma$ is 4.7, then the C is $4.7/3\sim 1.57$.
The adiabatic rate of change of momentum $\dot p = −(1/3)p\nabla\cdot V$ referred to above, is actually the rate of change in the non-inertial frame. The rate in the stationary frame is $\dot p=−(1/3)pV \cdot (\nabla f/f)$.
Reference
Radiation Environments and Damage in Silicon Semiconductors Silicon chapter4
Cosmic-Ray Modulation Equations
\begin{equation*}
\frac{\partial f}{\partial t} =-\nabla \cdot \overrightarrow{S.}
\end{equation*}
In a given volume, the rate of change of the number of particles,$\displaystyle \frac{\partial f}{\partial t}$, is given by the numer of particles that across the surface enclosing the volume, assuming no source of particle is present.
$\displaystyle \vec{S}$ is the streaming crrent density of particles across the surface.
\begin{gather*}
\vec{S} \ =\ \vec{V} f\ -K\cdot \nabla f\ \\
\nabla _{p} \cdot \overrightarrow{S_{p}} =\frac{1}{p^{2}}\frac{\partial }{\partial p}\left( p^{2} \langle \dot{p} \rangle f\right) \ ,\ \overrightarrow{S_{p}} =\langle \dot{p} \rangle f\\
\end{gather*}
\begin{equation*}
\frac{\partial f}{\partial t} +\nabla \cdot \left(\vec{V} f-K\cdot \nabla f\right) +\left(\frac{1}{p^{2}}\frac{\partial }{\partial p}\left( p^{2} \langle \dot{p} \rangle f\right)\right) =0
\end{equation*}
note. The streaming is included in $\vec{S}$.
adiabatic cooling resulting the average change in momentum is $\displaystyle \langle \dot{p} \rangle =-\frac{p}{3} \nabla \cdot \vec{V}$.
\begin{gather*}
\frac{\partial f}{\partial t} +f\nabla \cdot \vec{V} +\vec{V} \cdot \nabla f-\frac{( \nabla \cdot \overrightarrow{V)}}{3p^{2}}\left( 3p^{2} f+p^{3}\frac{\partial f}{\partial p}\right)\\
=\frac{\partial f}{\partial t} +f\nabla \cdot \vec{V} +\vec{V} \cdot \nabla f-( \nabla \cdot \overrightarrow{V)}\left( f+\frac{p}{3}\frac{\partial f}{\partial p}\right)\\
=\frac{\partial f}{\partial t} +\vec{V} \cdot \nabla f-\left( \nabla \cdot \vec{V}\right)\frac{p}{3}\frac{\partial f}{\partial p}
\end{gather*}
\begin{array}{l}
note.\ \frac{dT}{dt} =\frac{dp}{dt}\frac{dT}{dp} =\frac{-\nabla \cdot \vec{V}}{3}\left(p*\frac{p}{T+m}\right) =\frac{-( \nabla \cdot \overrightarrow{V)}}{3} T\left(\frac{T+2m}{T+m}\right) .\\
\frac{dE}{dt} =\frac{dp}{dt}\frac{dE}{dp} =\frac{-\left( \nabla \cdot \vec{V}\right)}{3}(pv)
\end{array}
When differential energy density, $U_p=4\pi p^2f$, is convected with velocity $V$, the flux observed is not $VU_p$, but rather $S_p=CVU_p$, where $C=\frac{-1}{3}\frac{\partial lnf}{\partial lnp} = \frac{-1}{3}\frac{p}{f}\frac{\partial f}{\partial p}$ is called the Compton-Getting coefficien.
For $f=Ap^{-\gamma}$, C is $\frac{\gamma}{3}$. if $\gamma$ is 4.7, then the C is $4.7/3\sim 1.57$.
The adiabatic rate of change of momentum $\dot p = −(1/3)p\nabla\cdot V$ referred to above, is actually the rate of change in the non-inertial frame. The rate in the stationary frame is $\dot p=−(1/3)pV \cdot (\nabla f/f)$.
Reference
Radiation Environments and Damage in Silicon Semiconductors Silicon chapter4
Cosmic-Ray Modulation Equations
My dissertation materials
1. energy equipartition to determine the magnetic field
2. derive the parker equation and adiabatic energy loss
https://indico.cern.ch/event/656460/contributions/2679912/attachments/1686296/2711748/TPE-handout.pdf
ON THE EQUATION OF TRANSPORT FOR COSMIC-RAY PARTICLES IN THE INTERPLANETARY REGION
ON THE EQUATION OF TRANSPORT FOR COSMIC-RAY PARTICLES IN THE INTERPLANETARY REGION
adiabatic
cooling
3. different description on the parker eq: distribution function vs number density
| Cosmic Rays in the Interplanetary Radiation Environments and Damage in Silicon Semiconductors Silicon |
4. Cosmic ray astrophysics (history and general review)
5.Cosmic-ray acceleration and transport, and diffuse galactic gamma-ray emission
http://adsabs.harvard.edu/abs/1983SSRv...36....3V
6. introduction to turbulence
http://www.astronomy.ohio-state.edu/~ryden/ast825/ch7.pdf
7. review of cosmic ray physics
8.Magnetic fields in supernova remnants and pulsar-wind nebulae
https://arxiv.org/pdf/1104.4047.pdf
9. particle trajectory in magnetic field
Revisiting the Equipartition Assumption in Star-forming Galaxies
10. MF review
Practical Modeling of Large-Scale Galactic MagneticFields: Status and Prospects
11. electron spectrum in SNR
Analytical solutions for energy spectra of electrons accelerated by nonrelativistic shock-waves in shell type supernova remnants
COSMIC-RAY ELECTRON EVOLUTION IN THE SUPERNOVA REMNANT RX J1713.7–3946
Evolution of cosmic ray electron spectra in magnetohydrodynamical simulations
12. sun magnetic field 11 evolution
http://www.astrophys-space-sci-trans.net/4/19/2008/astra-4-19-2008.pdf
13. More detailed Monte Carlo models of shock acceleration that include the nonlinear e†ects of the accelerated particles on the shock itself (Ellison et al. 1995, 1996) produce somewhat di†erent results. For the same overall compression ratio, the particle spectra at nonrelativistic energies tend to have spectral indices steeper than that given by the simple model discussed above (2.5), while at ultrarelativistic energies, the spectra Ñatten to indices somewhat less than discussed above (1.5).
https://iopscience.iop.org/article/10.1086/304894/pdf
14.
Radio Properties of Pulsar Wind Nebulae
https://link.springer.com/chapter/10.1007%2F978-3-319-63031-1_1
15. radio index in snr
https://arxiv.org/pdf/1907.00966.pdf
15. introduction to CR
http://astro.tsinghua.edu.cn/~xbai/teaching/PlasmaCourse2016/Ay253_2016_12_CosmicRays.pdf
16. CR escape
https://arxiv.org/pdf/1801.08890.pdf
17. Protheroe (2004) performed a detailed study of the shape of the
cutoff for particles experiencing diffusive shock acceleration
18. Diffusive Shock Acceleration and Magnetic Field Amplification
15. radio index in snr
https://arxiv.org/pdf/1907.00966.pdf
15. introduction to CR
http://astro.tsinghua.edu.cn/~xbai/teaching/PlasmaCourse2016/Ay253_2016_12_CosmicRays.pdf
16. CR escape
https://arxiv.org/pdf/1801.08890.pdf
18. Diffusive Shock Acceleration and Magnetic Field Amplification
Wednesday, May 8, 2019
Solutions to problems when plot with matplotlib
1. can not find 'tkinter' in matplotlib
import matplotlib matplotlib.use('agg')
import matplotlib matplotlib.use('agg')
Plot contour/scatter figure by matplotlib
This is my code.
import matplotlib.pyplot as plt
import matplotlib.tri as tri
import numpy as np
x,y,z = np.loadtxt('GRID007.dat',unpack=True)
#plt.tricontourf(x,y,z,np.linspace(int(min(z)),120,30,endpoint=True))
#plt.colorbar(ticks=range(int(min(z)),130,10))
''' plt.tricontourf(x,y,z,[min(z),37,74,111])
'''
plt.scatter(x,y,c=z,vmax=120)
plt.plot(1.07204,0.762,color='red',linestyle="None",marker='*',label=r'best fit ($\chi^2=100$')
#plt.legend()
#plt.colorbar(ticks=[min(z),111,222,333])
plt.colorbar()
plt.savefig(filename='scatter007.pdf',format='pdf',dpi=300)
plt.show()
Google search tricks
1) when lots of word are missing
If it’s the lengthier half of the phrase you can’t remember rather than a single key word, try writing out the first and last words and putting “AROUND + (the approximate number of missing words)“ between them. For example: amazon around(5) best
2)Google searches may vary depending on your search history and location. If you don't want these factors to alter your results, try the following things:
Do your search as normal. On the results page, add this code to the end of the URL: &pws=0
If it’s the lengthier half of the phrase you can’t remember rather than a single key word, try writing out the first and last words and putting “AROUND + (the approximate number of missing words)“ between them. For example: amazon around(5) best
2)Google searches may vary depending on your search history and location. If you don't want these factors to alter your results, try the following things:
Do your search as normal. On the results page, add this code to the end of the URL: &pws=0
3)search by date
before:2019-3-2
after:2019-1-5
example: google around(4) easier before:2019-02-02 after:2019-1-1
4)add in the url : &num=3
&filter=0
4)add in the url : &num=3
&filter=0
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