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

Herbig Haro 30

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

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