On the dynamics and tidal dissipation rate of the white dwarf in 4U Snezana Prodan¹ & Norman Murray¹ ¹CITA Canadian Institute for Theoretical Astrophysics University of Toronto

Outline 4U and what we know about it Our model How did the system arrive at its current state? Our results Constraint on the tidal dissipation factor Q for white dwarf secondary Conclusions

4U LMXB in NGC 6624 NS+He WD, m wd =( )M ☉ (Rappaport et al. 1987) P ₁ ≈11min (Stella et al. 1987) ‏ Ṗ₁ /P ₁ =(-3.5±1.5)ˉ ⁸ 10yrˉ¹ (Chou &Grindlay 2001)‏ P ₂ ≈170d (luminosity varies by factor ~2) Figure 1. The RXTE/ASM light curve based on 1-day measurements. Solid curve presents the theoretical model of Zdziarski et al. 2007

Superorbital period of 170 days?! P ₂ /P ₁ ≈2x10 ⁴ =>too big for disk precession (Larwood 1998, Wijers& Pringle 1999)‏ Triple hierarchical model proposed by Chou &Grindlay 2001, Zdziarski et al (Z07), Prodan& Murray 2010 (submitted)‏ Third star is WD or M-dwarf on~1 day orbit e oscillations responsible for luminosity variation on a period of 170 days

Our model Kozai mechanism GR periastron precession Rotational bulge Tidal bulge Tidal dissipation Mass transfer Gravitational radiation

Kozai mechanism I and e undergo periodic oscillations->Kozai cycles Kozai cycle are consequence of 1:1 resonance between precession rates of and I ₀ > 39˚.2 Figure 2. Geometry of triple hierarchical system, from Wen 2003

Kozai mechanism e-oscillations-> change MT rate -> luminosity variations 170 day period is set by interactions with third body small increase in e is enough to increase luminosity by factor of ~2 (see Fig 3) Figure 3. orbit-averaged accretion rate as function of e, from Zdziarski et al. 2007

Suppressing effects GR periastron precession tends to reduce amplitude of e variations, increase I 0 Tidal and rotational bulges also promote pericenter precession Tides tend to shrink and circularize the orbit

Gravitational radiation and mass transfer Gravitational radiation tends to shrink and circularize the orbit-> but other effects are stronger Mass transfer expands the orbit

Libration around fixed point 4 ώ : Kozai, tidal bulge, GR, rotational bulge In libration: ώ = 0 Period of libration is determined by initial conditions Figure 4. Phase space portrait for different initial eccentricity.

Period of small oscillations around fixed point Close to i crit period of small oscillations is around 170 days Figure 5. P ₀ vs I ₀ The solid line is analytic solution; the points come from numerical integration of the equations of motion.

Period of small oscillations around fixed point As expected from a out ¯³ dependance of ώ koz, period of small oscillations increases rapidly with a out Figure 6. P ₀ vs a ₂ /a ₁ The solid line is analytic solution; the points come from numerical integration of equations of motion.

Results m c ~ 0.5M ☉ probably wd k ₂ =0.1 (Sirotkin & Kim 2009 (stars with n=1.5))‏ Q=5x10 ⁷ I~ 65˚ 170 days is period of libration Figure 7. e vs t

Detrapping from the resonance Movie 1. Phase space evolution. Orbit evolves from libration to rotation Semimajor axis shrinks => Q~1x10 ⁶

Detrapping from the resonance Figure 8. ω vs t. The action of the separatrix is decreasing and the system is ejected from the resonance Semimajor axis shrinks=> Q~1x10 ⁶

Resonant trapping Movie 2. Phase space evolution. The system is placed in circulation; after about 750yr it gets trapped in libration Semimajor axis expansion => Q~10 ⁸

Resonant trapping Figure 9. ω vs t. The action of the separatrix is increasing and the system is trapped in the resonance Semimajor axis expansion => Q~10 ⁸

Constraint on tidal Q Observations: Ṗ₁ /P ₁ =(-3.5±1.5)10ˉ ⁸ yrˉ¹=> Q/k ₂ ~10 ⁷ for e=0.006 But, triple hierarchical model requires Ṗ₁ /P ₁ >0 => Q/k ₂ ~10 ⁹ Observed negative Ṗ₁ /P ₁ most likely is not intrinsic to the system if we consider 4U to be a hierarchical triple

Constraint on tidal Q ( Ṗ₁ /P ₁ ) obs = ( Ṗ₁ /P ₁ ) Roche + ( Ṗ₁ /P ₁ ) accel + ( Ṗ₁ /P ₁ ) TD ( Ṗ₁ /P ₁ ) obs = (-3.5±1.5)10ˉ ⁸ yrˉ¹ (Chou&Grindlay 2001) ( Ṗ₁ /P ₁ ) Roche > +8.8x10ˉ ⁸ yrˉ¹ (Rappaport et al. 1987) ɑ max /c = 2.5x10ˉ¹ ⁵ sˉ¹ (van der Klis et al. 1993b) => ( Ṗ₁ /P ₁ ) accel = -7.9x10ˉ ⁸ yrˉ¹ |( Ṗ₁ /P ₁ ) TD |=4.5x10ˉ ⁸ yrˉ¹ => Q/k ₂ < 1.5x10 ⁸

Constraint on tidal Q Figure 10. ω vs t. The system is detrapped from the resonance in ~500yrs even though semimajor axis is expanding ( Ṗ₁ /P ₁ ) obs = (-3.5±1.5)10ˉ ⁸ yrˉ¹=> Q/k ₂ =5.5x10 ⁷

Constraint on tidal Q To remain trapped in libration for >10 ⁵ yrs => Q/k ₂ >3x10 ⁹ Figure 11. ω vs t. The system is trapped in the resonance for 10 ⁵ yrs

Conclusions Triple hierarchical model does explain origin of the superorbital period of 170 days, it gives a constraint on Q/k ₂ for the white dwarf secondary (semimajor axis has to expand in order to trap the system in libration) Negative Ṗ₁ /P ₁ can not be reproduced even when we employ both acceleration of the cluster and tidal dissipation.

Thank you

Constraint on tidal Q ( Ṗ₁ /P ₁ ) = ( Ṗ₁ /P ₁ ) Roche + ( Ṗ₁ /P ₁ ) TD >0 => Q/k ₂ >3.4x10 ⁸ Finally 3.4x10 ⁸ <Q/k ₂ <6.7x10 ⁸