1. AS 3004 Stellar Dynamics 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 * < R L –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. AS 3004 Stellar Dynamics 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. AS 3004 Stellar Dynamics 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. AS 3004 Stellar Dynamics Conservative mass transfer Conservative –total mass conserved –total angular momentum conserved –consider only J orb, as J spin << J orb so –and

5. AS 3004 Stellar Dynamics Cons. mass transfer cont. As –if we have initial values, m 1,0, m 2,0, P 0 –then then the relative change in the Period due to mass transfer is

6. AS 3004 Stellar Dynamics 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 R L

7. AS 3004 Stellar Dynamics 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 L 2 point Angular momentum not conserved –magnetic braking through stellar wind –forced to corotate at large distance –gravitational radiation for close neutron stars

8. AS 3004 Stellar Dynamics Stellar Wind wind from primary escapes system –dm 1 /dt < 0, and dm 2 /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. AS 3004 Stellar Dynamics constant linear velocity of star 1 requires –a 1 x 2  /P = constant –and as a 1 = a m 2 / (m 1 + m 2 ), hence –the binary period must increase regardless of which star loses mass.

10. AS 3004 Stellar Dynamics General case assuming circular orbits and angular momentum change KJ –alternatively

11. AS 3004 Stellar Dynamics 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. AS 3004 Stellar Dynamics 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. AS 3004 Stellar Dynamics Total angular momentum of (eccentric) orbit is –differentiating to give can drive significant eccentricity into system –possibly disrupting it such that e>0