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* < 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

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

tdyn 2R3

Gm

12

40R

R

3M

m

12

minutes

tth Gm2

RL3107 m

M

2R

R

L

L years

tnuclear 7 109 m

M

L

L years

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

d ln Rd lnM

If L dyn

If dyn L th

If dyn , th L

AS 3004 Stellar Dynamics

Conservative mass transfer

• Conservative– total mass conserved

– total angular momentum conserved

– consider only Jorb, as Jspin << Jorb

so

– and

Mtot m1 m2 constant

dm1 dm2

Jorb Gm1

2m22a 1 e2 Mtot

12

constant

a constantm1m2 2

AS 3004 Stellar Dynamics

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

0,20,1221

2

22

1

1

2

21

0,20,1

0

3

21

0,20,1

0

2

32

3

4 2

3

mmmm

m

mm

m

mm

mm

P

P

mm

mm

P

P

aPP

aGM tot

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 RL

21

211

121

21

00

121

21

3

21

0,20,1

0

21

3

3

3

mm

mmm

P

P

mmm

mm

P

P

P

P

mmm

mm

mm

mm

P

P

mm

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 L2 point

• Angular momentum not conserved– magnetic braking through stellar wind

– forced to corotate at large distance

– gravitational radiation for close neutron stars

AS 3004 Stellar Dynamics

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:

121

21

221

13

21

22

21

322

31

342

4

ma

mma

mmP

P

mm

ma

mm

aa

GPP

mmG

aP

AS 3004 Stellar Dynamics

• constant linear velocity of star 1 requires– a1 x 2/P = constant

– and as a1 = a m2 / (m1 + m2 ), hence

– the binary period must increase regardless of which star loses mass.

21

1

121

21

23

21

212

1

21

2

2

0 and

constant

constant 2

2

mm

m

P

P

mamma

mmaa

mmG

mm

am

AS 3004 Stellar Dynamics

General case

• assuming circular orbits

• and

• angular momentum change KJ

– alternatively

)(22

)(

)(

2

0,0,0 and,

211

1

2

2

1

1

21

21

211

2

21

2121

21

21

21

mm

m

a

a

m

m

m

m

J

J

mm

ammGJ

Kmm

m

m

m

J

J

KJPamJ

mmmmmm

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

0

332

2

3

2

21

212

21

21

Km

Kmm

mmm

mm

m

P

P

mm

m

a

a

P

P

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

E

E

m

m

a

a

a

a

m

m

E

E

aa

mGm

a

mmGE

mma

mGmE

1

1

1

1

22121

21

21

or ,

22

0 and ,0 if2

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

J

J

mm

m

m

m

a

a

e

ee

mmmm

eGaJ

21

2

1

12

21

21

21

2

12)1(

)1(