notebooks updates

notebooks/ShockPoweredTransients-Fang2021.py

@@ -271,7 +271,7 @@ plt.show()
 #
 # Assume here that $L_{pk} \approx L_{sh}$ (all optical emission is shock powered, without any contribution from e.g. radiactive decay).
 
-# ## Calorimetric Technique
+# ## 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.
 #
@@ -300,9 +300,22 @@ plt.show()
 
 # For Novae: $\epsilon_{rel} \sim 0.003 - 0.01$
 
-# **Dynamical tooling time scale:**
+# **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):
+#
+# $k_{es} \approx \sigma_T / m_p \approx 0.38 $ cm$^2$ g$^{-1}$.
+#
+# This is a good approximation for hydrogen-rich ejected; for hydro-poor ejecta, the opacity should be lower.
+
+def t_dyn(n, v_sh, kappa = 0.38 * u.cm**2/u.g):
+    "dynamical time scale (peak time)"
+    return c/v_sh**2/n/(const.m_p*kappa)
+
 
 # **Cooling times scale for radiatively cooled thermal gas**:
 #
@@ -342,3 +355,93 @@ def t_pp(n, sigma_pp=5.e-25*u.cm**2):
     return 1./n/sigma_pp/const.c
 
 
+# **Cooling due to photo pion production:**
+#
+# Threshold for photo pion production:
+#
+# $
+# E_{p,th} \approx (\epsilon_{p\gamma ,th}/\epsilon_{opt}) m_p c^2 = 1.4\times 10^{16} (\epsilon_{opt}/$10 eV$)^{-1}$ eV
+#
+# 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.
+
+# +
+def v_85(v_sh):
+    "convenient shock speed"
+    return v_sh/(30000. * u.km/u.s)
+
+def kappa_03(kappa):
+    "convenient kappa"
+    return kappa / (0.3 *u.cm**2/u.g)
+
+def t_cool_dyn(v_sh, kappa=0.38*u.cm**2/u.g):
+    "ratio cooling to dynamical time scale"
+    
+    return 1.e-3 * kappa_03(kappa) * v_85(v_sh)**3
+
+def t_pp_dyn(v_sh, kappa=0.38*u.cm**2/u.g):
+    "ratio pp cooling to dynamical time scale"
+    return 1.e-3 * kappa_03(kappa) * v_85(v_sh)**2
+
+def t_pp_pgamma(v_sh,e_opt=1*u.eV,f_omega=1.):
+    "ratio of pp to p_gamma cooling"
+    return 4.*v_85(v_sh)**2/e_opt*f_omega
+
+v_sh = (np.logspace( 3, 5.5, 100))*u.km/u.s
+
+
+plt.gcf().clear()
+plt.figure(figsize=(6,6))
+
+r_cool_dyn = t_cool_dyn(v_sh)
+plt.plot(
+    v_sh,
+    r_cool_dyn,
+    label="$t_{cool}/t_{dyn}$")
+
+# H poor environement with lower opacity
+r_cool_dyn_Hpoor = t_cool_dyn(v_sh, 0.5*0.38*u.cm**2/u.g)
+#plt.plot(
+#    v_sh,
+#    r_cool_dyn_Hpoor,
+#    label="$t_{cool}/t_{dyn}$ (H poor)")
+
+r_pp_dyn = t_pp_dyn(v_sh)
+plt.plot(
+    v_sh,
+    r_pp_dyn,
+    label="$t_{pp}/t_{dyn}$")
+
+# depends strongly on f_omega!
+r_pp_pgamma = t_pp_pgamma(v_sh, 1., 1.)
+plt.plot(
+    v_sh,
+    r_pp_pgamma,
+    label="$t_{pp}/t_{p\gamma}$")
+
+plt.xlabel("shock velocity ($km/s$)",fontsize=14)
+plt.ylabel("ratio of cooling time scales",fontsize=14)
+plt.yscale('log')
+plt.xscale('log')
+plt.legend()
+
+plt.title("Ratio of cooling time scales" )
+plt.show()
+# -
+
+# ## Diffusive shock acceleration
+#
+# **Maximum Ion Energy:**
+#
+# Magnetic field strenght near shock (derived using equipartion arguments?):
+#
+# $
+# B_{sh} = \left( 6 \pi \epsilon_B m_p n_{sh} v_{sh}^2 \right)^{1/2}
+# $
+
+