1. The Minimum Mass Ratio for Contact Close Binary
Bojan Arbutina
University of Belgrade, Serbia
The Minimum Mass Ratio for Contact Close Binary
Systems of W Ursae Majoris-type
Stellar Mergers workshop, Lorentz center
Leiden, 21 Sept Oct 2009

2. CBs of W UMa-type Roche model: O B A F G K M
The Minimum Mass Ratio for Contact CBS of W UMa-type
CBs of W UMa-type
contact systems
Roche model:
spectral type: late F-K
common convective envelope, nearly equal temperatures (although q =M2/M1 ~ 0.5)
two sub-types: A and W
primary components seems to be normal MS stars, secondaries are oversized for their ZAMS masses, and can be found left from the main-sequence (see e.g. Hilditch 2001)
O B A F G K M
critical equipotential surfaces (Roche lobes):
- degree of contact (overcontact degree):

3. Dynamical evolution driven presumably by angular momentum loss (AML)
The Minimum Mass Ratio for Contact CBS of W UMa-type
Dynamical evolution
driven presumably by angular momentum loss (AML)
magnetic activity, starspots, magnetized stellar wind
secular, tidal or Darwin instability
tidal forces circulization and synchronization
if the timescale for the synchronization is smaller that the AML timescale, system will remain synchronized and orbit will shrink until, at some critical separation, the instability sets in
- rotational and orbital angular momentum become comparable
instability condition: d Jtot = 0 (Jorb = 3 Jspin)
MERGER!
(Rasio 1995, Rasio & Shapiro 1995)
W UMa blue stragglers, FK Com
a significant number of W UMa-type binary systems among blue stragglers in open and globular clusters
(Kaluzny & Shara 1988).
Sir George Howard Darwin ( )

4. The minimum mass ratio for W UMa-type CBs
The Minimum Mass Ratio for Contact CBS of W UMa-type
The minimum mass ratio for W UMa-type CBs
(Eggleton 1983, Yakut & Eggleton 2005)
- qmin =
AW UMa, q = 0.075
(Paczynski 1964,
Rucinski 1992,
Pribulla & Rucinski 2008)
- critical separation (Rasio 1995)
- k is dimensionless gyration radius which depends on the density distribution (for homogenous sphere k2 = 2/5)
polytrope
– disagreement between theory and observations – there are systems with the mass ratio smaller than qmin observed !
polytrope
Sun:

5. - “spherical symmetry”, r R volume radius, see Eggleton (2006)
The Minimum Mass Ratio for Contact CBS of W UMa-type
- contribution of the rotational AM of the secondary (Li & Zhang 2006, Arbutina 2007)
- qmin =
deformation of the primary due to rotation and companion – nonzero quadrupole moment – “apsidal motion constant”
- qmin = ( )
- structure of the primary (k depends on the central condensation, or )
- “spherical symmetry”, r R volume radius, see Eggleton (2006)

6. - instability condition:
The Minimum Mass Ratio for Contact CBS of W UMa-type
- instability condition:

7. - significantly lower minimum mass ratio (Arbutina 2009) :
The Minimum Mass Ratio for Contact CBS of W UMa-type
- significantly lower minimum mass ratio (Arbutina 2009) :
qmin =
contact CBs of W UMa-type with an extremely low mass ratio

8. The Minimum Mass Ratio for Contact CBS of W UMa-type

9. Interesting systems - AW UMa Pribulla & Rucinski (2008) find higher
The Minimum Mass Ratio for Contact CBS of W UMa-type
Interesting systems
- AW UMa
Pribulla & Rucinski (2008) find higher
mass ratio q = 0.1 and suggest that
AW UMa may not be a contact binary?
- qmin could be slightly higher if contribution from the secondary is taken into account, but it
could be lower if the star is more evolved (more centrally condensed than n = 3 polytrope)
- differential rotation (Hilditch 2001) - Yakut & Eggleton (2005) proposed it as a possible
mechanism for thermal energy transfer from the primary to the secondary component in contact
binaries, which leads to the equalization of temperatures in the common envelope.
- unstable merger?
V857 Her, q = 0.065? (Qian et al. 2006)
- spectroscopic mass ratio, Pribulla et al. (2009)