Mass transfer in binary systems

1. Mass transfer in binary systems • Mass transfer occurs when • star expands to fill Roche-lobe • due to stellar evolution • orbit, and thus Roche-lobe, shrinks till R* < RL • due to orbital evolution: magnetic braking, gravitational radiation etc • Three cases • Case A: Mass transfer while donor is on main sequence • Case B: donor star is in (or evolving to) Red Giant phase • Case C: SuperGiant phase • Mass transfer changes mass ratios • changes Roche-lobe sizes • can drive further mass transfer

2. Timescales • Dynamical timescale • timescale for star to establish hydrostatic equilibrium • Thermal timescale • timescale for star to establish thermal equilibrium • Nuclear timescale • timescale of energy source of star • ie main sequence lifetime

3. Star reacts to mass loss, • expands/contracts • Roche-lobe also expands or contracts • Define • then the star will transfer mass on a dynamical timescale • star reacts more to change in Roche-lobe • then hydrostatic equilirbium easily maintained • star transfers mass on thermal timescale • stable on thermal timescale • mass transfer due to stellar evolution, nuclear timescale

4. Conservative mass transfer • Conservative • total mass conserved • total angular momentum conserved • consider only Jorb, as Jspin << Jorb so • and

5. Cons. mass transfer cont. • As • if we have initial values, m1,0 , m2,0 , P0 • then • then the relative change in the Period due to mass transfer is

6. But for conservative mass transfer • then • and thus • If more massive star transfers mass to companion • then P decreases, and a decreases • until masses become equal, minimum a, P • if mass transfer 1 to 2 continues, orbit expands, as do RL

7. Non-conservative mass transfer • General case • What happens if either mass, or angular moment are not conserved • mass not conserved when • mass transfer / loss through stellar wind • rapid Roche-lobe overflow • receiving star can’t accrete all mass / angular momentum • mass loss through L2 point • Angular momentum not conserved • magnetic braking through stellar wind • forced to corotate at large distance • gravitational radiation for close neutron stars

8. Stellar Wind • wind from primary escapes system • dm1/dt < 0, and dm2/dt = 0, • total angular momentum decreases with mass loss • specific angular momentum constant • mass losing star has constant orbital velocity • From Kepler’s third law: • differentiating:

9. constant linear velocity of star 1 requires • a1 x 2p/P = constant • and as a1 = a m2 / (m1 + m2 ), hence • the binary period must increase regardless of which star loses mass.

10. General case • assuming circular orbits • and • angular momentum change KJ • alternatively

11. from Kepler’s 3rd law • Combining the three equations yields: • General case of mass transfer in circular orbits • reduces to earlier results when • K can include effects of magnetic braking or gravitational radiation, etc

12. Catastrophic mass loss • From a nova or a supernova type event • can drive eccentricity into the system • Total Energy of the system is • the rate of change of the size of orbit

13. Total angular momentum of (eccentric) orbit is • differentiating to give • can drive significant eccentricity into system • possibly disrupting it such that e>0