Common Envelope
COBRA Discussion Group
Almog Yalinewich
8.3.19
A picture is worth 1000 words
Algol Paradox
Less massive but more evolved
Mass transfer
V471 Tau
Eclipsing Binary
V471 Tau
Very short period (12h), sma
\approx 3 R_{\odot}
V471 Tau
K type
white dwarf
Planetary Nebula
NGC 6720, The Ring Nebula
NGC 6543, the Cat's Eye Nebula
Planetary Nebula
V1309 Sco
Bipolar Outflow?
Theoretical Analysis
Energy Conservation
m_c
m_s
m_s
\frac{G \left(m_c+m_e\right) m_e}{\lambda R}
m_e
R
a_i
a_f
\alpha \left[\frac{G m_s m_c}{a_f} - \frac{G m_s m_c}{a_i}\right]
Addition Energy Sources
Accretion onto Companion
Companion Spin
Recombination
Energy sink - binding energy to companion
+fusion
Recombination
13.6 \, \rm eV
1 \, M_{\odot}
1 \, \rm au
9 \, \rm eV
Technically, the envelope is unbound
\alpha \nless 1
Calibration
Mass ratio
Conservation of Angular Momentum
\frac{\Delta J}{J_i} \approx \gamma \frac{m_e}{m_e+m_c+m_s}
\frac{a_f}{a_i} = \left[\frac{m_c+m_e}{m_c} \frac{M_s+m_c +m_e}{m_s+m_c}\right]^2 \left[1-\gamma \frac{m_e}{m_e+m_c+m_s}\right]^2
a_i \approx R
Calibration
Mass transfer Stability
J = \frac{M_d M_r}{M_r +M_d} \sqrt{G \left(M_d + M_r\right) a}
\frac{\Delta a}{a} = 2 \Delta M \left[\frac{1}{M_d} - \frac{1}{M_r}\right]
= \frac{\left(M_d-\Delta M\right) \left(M_r+\Delta M\right)}{M_r +M_d} \sqrt{G \left(M_d + M_r\right) \left(a+\Delta a\right)}
Angular momentum
after mass transfer
M_d > M_r \Rightarrow \Delta a<0
M_d < M_r \Rightarrow \Delta a>0
Stable
Unstable
Stable Mass Transfer
M_d < \textcolor{blue}{M_r}
Unstable Mass Transfer
M_d > \textcolor{blue}{M_r}
Mnemonic
Progressive
unstable
Regressive
stable
insert snide political comment
Statistics
Bimodal White Dwarf Binary Period Distribution
Eccentricity-Period Distribution
Eccentricity-Period Distribution
Eccentricity-Period Distribution - General
Period Mass Ratio Relation
Tight
binaries
favour
large q
Alpha Prescription
Alpha Prescription
Gamma Prescription
Simulations
Difficulties
Inherently 3D
Self Gravity
Multiscale
Radiation
Length Scales
200 R_{\odot}
3 R_{\odot}
Timescales
Rapid braking
Slow braking
high eccentricity
low eccentricity
Accretion Rate
v_k \approx \sqrt{\frac{G M_c}{R}}
R
R_b \approx \frac{G M_c}{v_k^2} \approx R
Super Eddington Accretion
\dot{M} \approx M \sqrt{\frac{G M}{R^3}} \approx 1 \, \frac{M_{\odot}}{\rm year}
Without radiation
\dot{M}_e \approx 10^{-9} \frac{M}{M_{\odot}} \frac{M_{\odot}}{\rm year}
Eddington limit
Jet formation?
Adiabatic Simulations
Adiabatic Simulations
Very Little Mass Accretion
Loss of Mass through L2
Loss of Mass through L2
Loss of Mass through L2
Radiation in CE of a Massive Star
Adiabatic
Radiative
Tenuous atmosphere
Different from low mass stars
Radiation in CE of a Massive Star
Envelope retained
More Eccentric
Larger separation
Jets
Jets
Common Envelope
By almog yalinewich
