## 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
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,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
