Black widow evolution:magnetic braking by an ablated wind
Almog
Yalinewich
Pulsar
Coffee
7.2.20
Mass Loss from a Black Widow
Energy budget
\frac{G M_{\odot}^2}{R_{\odot}} \approx 10^{48} \rm \, erg
L_{\odot} t_H \approx 10^{50} \, \rm erg
Efficiency?
Albedo?
(regressive energy distribution)
Extremely Low Mass Companion
PSR J1719-1438 and PSR J2322-2650
Parker Wind
\frac{\partial}{\partial r} \left(\rho v r^2\right) =0
p = \frac{\rho k T}{\mu}
\rho v \frac{\partial v}{\partial r} + \frac{\partial p}{\partial r} = - \frac{G M \rho}{r^2}
\frac{1}{2} \left(1 - \frac{k T}{\mu v^2} \right ) \frac{\partial v^2}{\partial r} = \frac{2 k T}{\mu r} - \frac{G M}{r^2}
Sonic Point
c = v
R_b \approx \frac{G M}{c^2}
\left(1 - \frac{c^2}{v^2} \right ) \frac{\partial}{\partial r} \frac{v^2}{2} = \frac{2 c^2}{r} - \frac{G M}{r^2}
r \ll R_g
r \gg R_g
Hydrostatic
Coasting
Analytic Solution
\rho v \frac{\partial v}{\partial r} + \frac{\partial p}{\partial r} = - \frac{G M \rho}{r^2}
\rho v \frac{\partial v}{\partial r} + c^2\frac{\partial \rho}{\partial r} = - \frac{G M \rho}{r^2}
\frac{1}{2} v^2 + c^2 \ln \frac{\rho}{\rho_0} + \frac{G M}{R} - \frac{G M}{r} = 0
\rho_b \approx \rho_0 \exp \left(-R_b/R\right)
Complication: Heating & Cooling
\left(1 - \frac{c^2}{v^2} \right ) \frac{\partial}{\partial r} \frac{v^2}{2} = \frac{2 c^2}{r} - \frac{G M}{r^2} + \frac{\partial c^2}{\partial r}
R_b \approx \frac{G M}{c^2} / \left(1+\frac{d \ln c}{d \ln r}\right)
Mass flux
\dot{M} \approx \rho_b R_b^2 c
Radiative Processes
Heating
Inelastic Compton Scattering
Ionisation
H \approx \sigma F n x
\sigma \approx r_e^2
\sigma \approx \lambda r_e \left(\frac{E}{\varepsilon}\right)^3
lines
Inelastic Compton Scattering
x \approx \varepsilon /m_e c^2
Cooling
free free
free bound
C \approx \Lambda n^2
C_{ff} \approx \alpha \frac{m_e c^3}{r_e^4} \left(n r_e^3 \right)^2 \sqrt{\frac{k T}{m_e c^2}}
C_{fb} \approx \alpha^3 \frac{m_e c^3}{r_e^4} \left(n r_e^3 \right)^2 \sqrt{\frac{m_e c^2}{k T}}
lines
Cooling Function
Difficulty in cooling past 10,000 K
Minimum Pressure
p \propto \rho \cdot 10^4 K\propto \rho
High densities
constant temperature
Low densities
ff cooling
T \propto \frac{1}{\rho^2}
p \propto \rho T \propto \frac{1}{\rho}
minimum pressure
p_0
Temperature Scales
k T_g \approx \frac{G m}{R}
Escape temperature
Inverse Compton temperature
k T_{IC} \approx \varepsilon
Characteristic temperature
k T_{ch} \approx H \frac{R}{c_{ch}}
k T_{ch} \approx \left(\sigma^2 F^2 R^2 \mu\right)^{1/3}
Temperature Scales
k T_g \approx \frac{G m}{R}
Escape temperature
Inverse Compton temperature
k T_{IC} \approx \varepsilon
Characteristic temperature
k T_{ch} \approx H \frac{R}{c_{ch}}
k T_{ch} \approx \left(\sigma^2 F^2 R^2 \mu\right)^{1/3}
Hot Wind
T_g < T_{IC} < T_{ch}
Sonic point is extremely close to the surface
\frac{R_b}{R} - 1 \ll 1
Mass loss rate
\rho_b \approx \frac{p_0}{c_{IC}^2} \approx \frac{\mu p_0}{k T_{IC}}
\dot{M} \approx R^2 \rho_b c_{IC}
Intermediate Wind
T_g < T_{ch} < T_{IC}
Sonic point is withing a few stellar radii
R_b \approx R
Mass loss rate
\rho_b \approx \frac{p_0}{c_{ch}^2} \approx \frac{\mu p_0}{k T_{ch}}
\dot{M} \approx R^2 \rho_b c_{ch}
Cold Wind
T_{ch} < T_g < T_{IC}
Sonic point is much larger than stellar radius
R_b \gg R
speed of sound
c_s \approx v_g
\dot{M} \approx \frac{p_0}{c_{ch}} \frac{T_{ch}^2}{T_g^2}
velocity
k T_g \approx H \frac{R}{v}
mass loss
Roche Lobe Overflow
Gas escapes through narrow angle
\theta_g \approx \sqrt{\frac{T_{ch}}{T_g}}
Net mass loss
\dot{m} \approx R^2 \theta_g^2 \frac{p_0}{c_{ch}}
Mass Loss
Application
Characteristic temperature from Compton heating
k T_{ch} \approx \mu^{1/3} \left(\sigma_T F R\right)^{1/3}
Efficiency
\frac{G m \dot{m}}{R} \approx - \eta L_{\gamma} \left(\frac{R}{a}\right)^2
\eta \approx 2.2 \cdot 10^{-4} \left(\frac{L_{\gamma}}{L_{\odot}}\right)^{1/3} \left(\frac{m}{10^{-2} M_{\odot}}\right)^{1/9} \left(\frac{P_{\rm orb}}{1 \rm \, h}\right)^{-2/9}
Evaporation time
t_{\rm evap} \approx m/\dot{m} \approx
\approx 27 \, {\rm Gyr} \left(\frac{L_{\gamma}}{L_{\odot}}\right)^{-4/3} \left(\frac{m}{10^{-2} M_{\odot}}\right)^{8/9} \left(\frac{P_{\rm orb}}{1 \rm \, h}\right)^{-4/9}
Comparison to Observations
Magnetic Braking
Overview
Synchronised
Lagging
Magnetic breaking
compensation from orbital angular momentum
Net result: Depletion of orbital angular momentum
Alfven Radius
Split monopole magnetic field
B \approx B_0 \left(\frac{R_{NS}}{r}\right)^2
equilibrium between magnetic and ram pressure
B_0^2 R_A^2 \left(\frac{R}{R_A}\right)^4 \approx \dot{m} v \left(R_A\right)
Sun Spin Down
R_A \approx R \sqrt{\frac{B_{\odot}^2 R_{\odot}^2}{\dot{M}_{\odot} v_{\odot}}}
Matter is co - rotating up to the Alfven radius, so mass loss required for factor of two spin - down is
\frac{\Delta M}{M} \approx \frac{R^2}{R_A^2}
Spin Down Time
t_{\rm mag} \approx \frac{m}{\dot{m}} \frac{a^2}{R_A^2} \approx
\approx 31 \left(\frac{L_{\gamma}}{L_{\odot}}\right)^{-\frac{4}{9}} \left(\frac{P_{\rm orb}}{1 \, \rm h}\right)^{-\frac{34}{27}} \left(\frac{m}{10^2 M_{\odot}}\right)^{\frac{2}{27}} \left(\frac{B_0}{10^2 \, \rm G}\right)^{-\frac{4}{3}}
Evaporation time determined by
\min \left(t_{\rm evap}, t_{\rm mag}\right)
Comparison to Observations
Conclusion
Evaporation with fixed orbital parameters is too slow to explain observations
mass loss is regulated by magnetic braking
questions?
Backup slides
black widow magnetic breaking
By almog yalinewich
