# ## 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, $f_{\Omega} \sim h/r$.
#
# assumed f_omega in the following
f_omega=1.
# ### 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
defsteady_wind_A(mdot,vw,fomega=1):
"mass loss A"
returnmdot/(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))
forzinv_w:
A=steady_wind_A(M_dot,z,f_omega)
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 ($f_{\Omega}=%.2f$)"%f_omega)
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
# with $n_{sh}(R_{sh})$: upstream density ahead of shock
#
# +
defkinetic_power(A,v_sh,fomega=1):
"kinetic power of forward shock"
return9.*math.pi/8*A*fomega*v_sh**3
v_sh=(np.logspace(3,5,3))*u.km/u.s
A=np.logspace(11.,18.,100)*u.g/u.cm
plt.gcf().clear()
plt.figure(figsize=(8,8))
forzinv_sh:
l_kin=kinetic_power(A,z,f_omega)
label_vw=("v$_{sh}=$%.0f km s$^{-1}$"%z.value)
plt.plot(
A,
l_kin.to(u.erg/u.s),
label=label_vw)
plt.xlabel("steady wind loss parameter A ($g/cm$)",fontsize=14)
plt.ylabel("$L_{sh} (erg/s)$",fontsize=14)
plt.yscale('log')
plt.xscale('log')
plt.legend()
plt.title("Kinetic power of forward shock ($f_{\Omega}=%.2f$)"%f_omega)
plt.show()
# -
# **Shock heating:**
#
# Gas immediatly behind the shock is heated to: $kT \simeq \frac{3}{16} \mu m_p v_{sh}^2 \approx 11 v_{8.5}^2$ keV.
#
# with $v_{sh}=v_{sh}/(3000 km s^{-1}$.
# Mean molecular weight of the fully ionized gas (solar composition): $\mu=0.62$
#
# (would be $\mu=2$ if it would be hydrogen-poor gas)
mu=0.62
# +
defshock_temperature(v_sh,mu):
"gas temperature behind the shock"
return3./16.*mu*const.m_p*v_sh**2
v_sh=(np.logspace(3,5,100))*u.km/u.s
T_sh=shock_temperature(v_sh,mu)
plt.gcf().clear()
plt.figure(figsize=(6,6))
plt.plot(
v_sh,
T_sh.to(u.keV))
plt.xlabel("shock velocity ($km/s$)",fontsize=14)
plt.ylabel("shock temperature ($keV$)",fontsize=14)
plt.yscale('log')
plt.xscale('log')
plt.title("Temperature of gas behind shock ($\mu=%.2f$)"%mu)
plt.show()
# -
# Large photoelectric opacity of the external medium: most of $L_{sh}$ is absorped and reprocessed via continuum and line emission into the optical wavelengths.
#
# Time delay of optical emission, as reprocessed emission from nearby the shock must propagate through the column of external medium: $\Sigma = \int_{R_sh}^{\infty} ndr \sim n_{sh} R_{sh}$.
#
# Optical diffussion time scale:
#
# $t_{diff} \approx \tau_{opt} \cdot (R_{sh}/c)$
#
# with the optical depth $\tau_{opt} \equiv \sum \sigma_{opt}$ ($\sigma_{opt}$ is the effective cross section at visual wavelengths).
#
# Adiabatic losses can be neglected as long as $t_{diff} \leq t_{dyn}$ with the expansion time scale of the shocked gas $t_{dyn} = R_{sh}/v_{sh}$:
#
# $
# \tau_{opt} \lesssim c/v_{sh} \ \ \ (3)
# $
# This condition is satisfied at times later than $t$:
#
# $
# t \gtrsim t_{pk} \approx \frac{c}{v_{sh}^2 n_{sh}} \sigma_{opt} = \frac{\dot M \kappa_{opt}}{4\pi f_{\Omega}c v_w} = \frac{A \kappa_{opt}}{c} \ \ (4)
# $
#
# with the optical opacity $\kappa_{opt}\equiv \sigma_{opt}/m_p$ (opacity: cross section for absorbing photons per units mass material; units [m$^2$/kg]).
#
# Label $t_{pk}$ is introduced here, as the critical time often corresponds to the peak of the light curve.
#
# Any diffusion through the ejecta and corrections to $t_{diff}$ due its non-spherical geometry are ignored in this step.
# +
defpeak_time(a,kappa):
"peak time"
returna*kappa/const.c
kappa_es=0.38*u.cm**2/u.g
t_pk=peak_time(A,kappa_es)
plt.gcf().clear()
plt.figure(figsize=(6,6))
plt.plot(
A,
t_pk.to(u.s))
plt.xlabel("steady wind loss parameter A ($g/cm$)",fontsize=14)
plt.ylabel("peak time $t_{pk}$ ($s$)",fontsize=14)
plt.yscale('log')
plt.xscale('log')
plt.title("Peak time ($\kappa=%.2f$ cm$^2$/g )"%kappa_es.value)
plt.show()
# -
# **Shock velocity is function of peak luminosity and peak time**:
# Assume here that $L_{pk} \approx L_{sh}$ (all optical emission is shock powered, without any contribution from e.g. radiactive decay).
# ## Calorimetry
#
# Conditions on the optical depth are very similar to the required conditions for particle acceleration and that the shock discontinuity is mediated by collisionless plasma. At times earlier than $t_{pk}$, the trapped radiations thickens the shock transition to macroscopic scale and does not allow any particle injection.
#
# 1. no efficient particle acceleration before $t_{pk}$
# 2. fixed fraction $\epsilon_{rel}$ of the shock power $L_{sh}$ is placed into relatistic particles
#
# Total energy of relativistic particles $E_{rel}$ is proportional to the fraction of radiated optical fluence $f_{sh}$:
#
# $
# E_{rel} \approx f_{sh} \epsilon_{rel} E_{opt}
# $
#
# $\epsilon_{rel} E_{opt}$ is an upper limit to the energy of accelerated particles, as $f_{sh}<1$.
#
# The total optical energy of shock-powered transients places an upper bound on the total gamma-ray and neutrino emission, as the relativistic particles cool quickly (**calorimetric method**).
#
# ## Coooling time scales
#
# (skip discussions on cooling time scales, **revisit later**)
#
# Shocks are generally radiative ($t_{cool} \ll t_{dyn}$) for $v_{sh} \lesssim 10,000-30,000$ km s$^{-1}$.
#
#
#
# For Novae: $\epsilon_{rel} \sim 0.003 - 0.01$
# **Shock dynamical time scale:**
#
# $t_{dyn} \sim R_{sh} / v_{sh}$
#
# Assume in the following $t_{dyn} \sim t_{pk}$.
#
# Approximation of opacity in fully ionized gas (electron scattering):
# with $\epsilon_{p\gamma ,th}=(m_{\pi}+m_{\pi}^2/m_p)c^2 \approx 150$ MeV
# **Cooling time scale summary**
#
# All ratios at $t_{pk}$.
#
# Ratios $t_{cool}/t_{dyn}$ and $t_{pp}/t_{dyn}$ show that both thermal and non-thermal particles cool efficienctly around $t_{pk}$ and that a calometric approach can be considered for shock-powered transients.
