Commit 624072b3 by Gernot Maier

### shock powered transient paper

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 --- jupyter: jupytext: formats: ipynb,md,py:light text_representation: extension: .md format_name: markdown format_version: '1.3' jupytext_version: 1.13.0 kernelspec: display_name: Python 3 (ipykernel) language: python name: python3 --- # Shock powered transients - [Fang, K. et al (2021)](https://arxiv.org/abs/2007.15742) Step throught this paper to capture the relevant points. python import astropy.units as u import astropy.constants as const import math import matplotlib.pyplot as plt import numpy as np  #### Fang et al (2021) - Figure 1 ## Section 2.1: Shock Dynamics and Thermal Expansion - spherical expanding homologoues ejecta of average velicity $\bar v_{ej}$ - inner layers slower than other layers: $v_{ej}\propto r$ (homologoues velocity profile) ### External medium Density of external medium $n \equiv \rho / m_p$ ($\rho$ = mass densitiy) with a radial profile of $n \propto r^{-k}$ with $k\geq 2$ The external medium is concentrated in a fractional solid angle $f_{\Omega}\leq 1$. For a thin equatorial disk, $f_{\Omega} \sim h/r$. ### Steady wind approximation Assume a steady wind with - mass-loss rate $\dot M$ - wind velocity $v_w$ the density becomes $n \simeq \dot M/(4\pi f_{\Omega}r^2 v_w m_p) = A / (m_p r^2)$ with $A \equiv \dot M/(4\pi f_{\Omega} v_w)$ Assume here $k=2$, otherwise A(r) is a function of radius. #### Typical values From modeling interacting supernovae Smith (2014): - $\dot M \sim 10^{-4} - 1$ M$_{\odot}$ yr$^{-1}$ - $v_w \sim 100 - 1000$ km s$^{-1}$ (note the typo units in the Fang et al paper: A$_*$ should be in g/cm and not g/cm$^2$) python # canonical value A_star = 5.e11 *u.g/u.cm**2 def steady_wind_A(mdot, vw, fomega=1): "mass loss A" return mdot/(4.*math.pi*fomega*vw) v_w = (np.linspace( 100, 1000, 5))*u.km/u.s M_dot = np.logspace( -5., 0., 100)*u.M_sun/u.year plt.gcf().clear() plt.figure(figsize=(6,6)) for z in v_w: A = steady_wind_A(M_dot, z, 1.) label_vw = ("v$_w=$%.0f km s$^{-1}$" % z.value) plt.plot( M_dot, A.to(u.g/u.cm), label=label_vw) plt.xlabel("mass loss rate $\dot M$ (M$_{\odot}$/yr)",fontsize=14) plt.ylabel("A (g/cm)",fontsize=14) plt.yscale('log') plt.xscale('log') plt.legend() plt.title("steady wind loss parameter A") plt.show()  ## Shocks Collision drives two shocks - forward shock into external medium - reverse shock into ejecta Shocks are radiative with rapid cooling of the gas behind both shocks --> gas accumulate in a thin shell which propagates outwards into the external medium. Shocks reach radius $R_{sh}\approx v_{sh}t$ by the time $t$ after the explosion. Shock speed $v_{sh}$ will reach ejecta speed $v_{ej}$ ([Metzer & Pejach (2017)](https://ui.adsabs.harvard.edu/abs/2017MNRAS.471.3200M/abstract) **READ**), which reduces the power of the reverse shock with relation to the forward shock (**???**) $\rightarrow$ emission dominated by forward shock **Kinetic power of the forward shock**: $L_{sh} = \frac{9\pi}{8} f_{\Omega} m_p n_{sh} v_{sh}^3 R_{sh}^2 = \frac{9}{32} \dot M \frac{v_{sh}^3}{v_w} = \frac{9\pi}{8} A f_{\Omega} v_{sh}^3$ - $n_{sh}(R_{sh})$: upstream densitiy ahead of shock **Understand: derived from $1/2 m v^2$?** python  python 
 # --- # jupyter: # jupytext: # formats: ipynb,md,py:light # text_representation: # extension: .py # format_name: light # format_version: '1.5' # jupytext_version: 1.13.0 # kernelspec: # display_name: Python 3 (ipykernel) # language: python # name: python3 # --- # # Shock powered transients # # - [Fang, K. et al (2021)](https://arxiv.org/abs/2007.15742) # # Step throught this paper to capture the relevant points. import astropy.units as u import astropy.constants as const import math import matplotlib.pyplot as plt import numpy as np # #### Fang et al (2021) - Figure 1 # # # ## Section 2.1: Shock Dynamics and Thermal Expansion # # - spherical expanding homologoues ejecta of average velicity $\bar v_{ej}$ # - inner layers slower than other layers: $v_{ej}\propto r$ (homologoues velocity profile) # # ### External medium # # Density of external medium # # $# n \equiv \rho / m_p #$ # ($\rho$ = mass densitiy) # # with a radial profile of # # $# n \propto r^{-k} #$ # with $k\geq 2$ # # The external medium is concentrated in a fractional solid angle $f_{\Omega}\leq 1$. # For a thin equatorial disk, $f_{\Omega} \sim h/r$. # # ### Steady wind approximation # # Assume a steady wind with # # - mass-loss rate $\dot M$ # - wind velocity $v_w$ # # the density becomes # # $# n \simeq \dot M/(4\pi f_{\Omega}r^2 v_w m_p) = A / (m_p r^2) #$ # with # $# A \equiv \dot M/(4\pi f_{\Omega} v_w) #$ # # Assume here $k=2$, otherwise A(r) is a function of radius. # #### Typical values # # From modeling interacting supernovae Smith (2014): # - $\dot M \sim 10^{-4} - 1$ M$_{\odot}$ yr$^{-1}$ # - $v_w \sim 100 - 1000$ km s$^{-1}$ # # (note the typo units in the Fang et al paper: A$_*$ should be in g/cm and not g/cm$^2$) # + # canonical value A_star = 5.e11 *u.g/u.cm**2 def steady_wind_A(mdot, vw, fomega=1): "mass loss A" return mdot/(4.*math.pi*fomega*vw) v_w = (np.linspace( 100, 1000, 5))*u.km/u.s M_dot = np.logspace( -5., 0., 100)*u.M_sun/u.year plt.gcf().clear() plt.figure(figsize=(6,6)) for z in v_w: A = steady_wind_A(M_dot, z, 1.) label_vw = ("v$_w=$%.0f km s$^{-1}$" % z.value) plt.plot( M_dot, A.to(u.g/u.cm), label=label_vw) plt.xlabel("mass loss rate $\dot M$ (M$_{\odot}$/yr)",fontsize=14) plt.ylabel("A (g/cm)",fontsize=14) plt.yscale('log') plt.xscale('log') plt.legend() plt.title("steady wind loss parameter A") plt.show() # - # ## Shocks # # Collision drives two shocks # - forward shock into external medium # - reverse shock into ejecta # # Shocks are radiative with rapid cooling of the gas behind both shocks --> gas accumulate in a thin shell which propagates outwards into the external medium. # # # Shocks reach radius $R_{sh}\approx v_{sh}t$ by the time $t$ after the explosion. # Shock speed $v_{sh}$ will reach ejecta speed $v_{ej}$ ([Metzer & Pejach (2017)](https://ui.adsabs.harvard.edu/abs/2017MNRAS.471.3200M/abstract) **READ**), which reduces the power of the reverse shock with relation to the forward shock (**???**) # # $\rightarrow$ emission dominated by forward shock # **Kinetic power of the forward shock**: # # $# L_{sh} = \frac{9\pi}{8} f_{\Omega} m_p n_{sh} v_{sh}^3 R_{sh}^2 = \frac{9}{32} \dot M \frac{v_{sh}^3}{v_w} = \frac{9\pi}{8} A f_{\Omega} v_{sh}^3 #$ # # - $n_{sh}(R_{sh})$: upstream densitiy ahead of shock # # **Understand: derived from $1/2 m v^2$?**
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