5 Discussion: On the nature of the long-term oscillation

The long-term well behaved sinusoidal modulations observed in our sample is not known to occur in other types of blue variable stars. Here we propose as a possible cause for the long-term oscillation the precession of an elliptical disc around a blue star in a semi-detached binary system. The origin of the disc ellipticity and precession could be the tidal interaction between the disc and the Roche-lobe filling secondary star, in a similar way that occurs in short-orbital period dwarf novae of the SU UMa class during superoutburst (e.g. Patterson 2001).

 

 

Table 1: The double-periodic blue stars. OGLE name and MACHO identification are given for every star, along with their photometric period and error. The half amplitude of the long-term variation is also given for the OGLE I band and the MACHO b band, along with the epoch (HJD-2 400 000) for the faint-to-bright mean level crossing for the MACHO b band. The half amplitude of the MACHO b-r light curve is also given, along with their difference in phase with the b curve, considering the epoch for positive to negative mean b-r level crossing as reference. Dashes indicate non-detected or marginal colour variations. A note indicates the appearance of the short-term variability: double-wave (dw), single wave (sw) or eclipsing (e).

Star	MACHO ID	P1 (d)	P2 (d)	AI	Ab	Eb	Ab-r	$\Delta\Phi$	Note
OGLE00451755-7323436	212.15675.158	5.178(5)	171(15)	0.12	0.11	50350.2(8)	0.02	-0.35	dw
OGLE00474820-7319061	212.15847.466	5.497(7)	177(10)	0.12	0.10	50329.3(7)	-	-	dw
OGLE00553643-7313019	211.16304.169	5.092(5)	176(17)	0.08	0.04	50238(1)	0.02	0.53	dw
OGLE05025323-6909493	1.3686.53	8.025(18)	255(30)	0.17	-	-	-	-	dw
OGLE05040378-6917508	1.3805.130	6.223(7)	207(20)	0.11	-	-	-	-	dw
OGLE05060009-6855025	1.4174.42	3.849(7)	230(22)	0.19	-	-	-	-	sw
OGLE05101621-6854290	79.4900.185	4.301(6)	319(34)	0.06	0.04	50344(2)	0.02	0.53	sw
OGLE05115466-6846369	2.5144.4555	9.138(2)	361(29)	0.26	-	-	-	-	e
OGLE05142677-6910559	79.5501.400	6.515(2)	224(14)	0.13	0.08	50430(2)	0.04	-0.43	e
OGLE05143758-6852259	79.5506.139	5.372(6)	185(11)	0.14	0.10	50237.3(5)	0.03	0.55	dw?
OGLE05152654-6923257	79.5740.5092	6.292(8)	205(15)	0.04	0.04	50275.3(6)	-	-	dw
OGLE05155332-6925581	79.5739.5807	7.2835(16)	188(11)	0.11	0.08	50254(1)	0.03	0.54	e
OGLE05171401-6936374	78.5979.58	8.309(4)	311(21)	0.13	0.08	50345(1)	0.03	0.48	e
OGLE05194110-6931171	78.6343.81	6.9044(10)	226(13)	0.11	0.06	50247(2)	0.02	0.59	e
OGLE05195898-6917013	80.6468.83	2.410(10)	140(5)	0.06	0.04	50267.6(8)	-	-	sw
OGLE05203325-6910146	80.6469.95	5.737(5)	182(10)	0.04	0.05	50357.1(8)	-	-	dw
OGLE05260516-6954534	77.7426.140	3.632(3)	233(15)	0.07	0.07	50425.7(7)	-	-	sw
OGLE05274332-6950556	77.7669.1013	7.320(11)	227(20)	0.05	0.03	50280(1)	0.02	0.54	dw
OGLE05285370-6952194	77.7911.26	15.854(31)	620(70)	0.07	0.06	50388(2)	0.01	0.67	dw
OGLE05294913-6949103	77.8033.140	7.184(8)	258(20)	0.10	0.09	50382.2(6)	0.01	-0.38	dw
OGLE05295881-6934075	77.8036.5142	5.597(4)	179(8)	0.12	-	-	-	-	dw
OGLE05313130-7012584	7.8269.36	9.231(21)	960(176)	0.04	0.03	51226(4)	-	-	sw
OGLE05333926-6956229	81.8636.51	7.863(15)	257(18)	0.03	0.01	50335(2)	0.01	0.52	dw
OGLE05371342-7010580	11.9237.2121	10.913(23)	421(40)	0.12	-	-	-	-	dw
OGLE05390681-7027487	11.9475.96	6.967(12)	276(15)	0.10	0.01	50335(2)	0.01	0.52	dw
OGLE05390992-7019262	11.9477.138	6.632(8)	198(15)	0.10	0.05	50301(1)	0.02	0.56	dw
OGLE05391746-7044019	11.9592.22	7.151(8)	219(20)	0.06	0.04	50401.0(8)	0.01	-0.49	dw
OGLE05410217-7011043	76.9842.2444	7.352(13)	264(26)	0.11	0.11	50395(2)	-	-	dw
OGLE05410942-7002215	76.9844.110	6.586(9)	245(23)	0.12	0.09	50510(2)	-	-	dw
OGLE05435003-7057431	15.10314.144	5.012(5)	173(12)	0.12	0.09	50250.7(7)	-	-	dw

The 3:1 resonance between a disc particle orbiting the primary and the binary system occurs at  $R_{\rm disc} \approx 0.46a$, where a is the binary separation. Elliptical orbits at this radius will experience a dynamical apsidal advance with a period given by:

$$P_{\rm p} = \frac{\sqrt{1+q}}{0.37q}(R_{\rm disc}/0.46a)^{-2.3}P_{\rm o}$$ (3)

(Murray 2000) where q is the ratio between the mass of the secondary star and the primary star and $P_{\rm o}$ the orbital period. If we interpret P₁ as the orbital period and P₂ as the precession period, then the above equation imposes a strong correlation between both periodicities, as effectively seen in Fig. 3. The observed slope of $P_{2}/P_{1}\approx 35$ should imply a mass ratio $q \approx 0.08$ for $R_{\rm disc} = 0.46 a$. There are few Algols with q < 0.1 (e.g. Richards & Albright 1999), and they do not show the long-term oscillations described in this paper. However it is exciting that only in these low q systems two important conditions are fulfilled: the primary radius is smaller than the stream circularization radius, making possible the formation of an accretion disc (e.g. Richards & Albright 1999) and the 3:1 resonance radius is below the tidal truncation radius, so the disc could in principle grown beyond the 3:1 resonance and may eventually start to precess (Fig. 4). The secondary eclipse, in this view, could be the eclipse of the disc.

We have examined an alternative explanation suggested by the referee, namely, nodal precession of a tilted disc. However, this phenomenon seems to appear in X-ray binaries with very different mass ratios (and different $P_{\rm p}/P_{\rm o}$values, e.g.  Larwood 1998) and not to be confined to the low mass ratio region, as suggested by the strong concentration of $P_{\rm p}/P_{\rm o}$ values around 35.

Acknowledgements

We thank the referee J. Thorstensen, and also D. Sasselov, for interesting comments on a first version of this paper. REM acknowledges support by Grant Fondecyt 1030707 and DI UdeC 202.011.030-1.0. MPD thanks CNPq support under grant # 301029. This paper utilizes public domain data originally obtained by the OGLE and MACHO Projects.