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On the occultations of a binary star by a circum-orbiting dark companion
Planet. Space Sci., Vol. 42, No. I, pp. 539-544, 1994
Pergamon
Copyright 0 1994 Elsevier Science Ltd Printed in Great Britain. All rights reserved 00324633/94 $7.00+0.00 0032-0633(94)00092-l
On the occultations of a binary star by a circum-orbiting dark companion J. Schneider CNRS-UPR
176, Observatoire de Paris, 92195 Meudon, France
Received 12 August 1993 ; revised 6 May 1994; accepted 6 May 1994
.
1. Introduction
The existence of two classes of dark celestial bodiesbrown dwarfs and extrasolar planets (at least orbiting around main sequence stars)-is still an unsolved problem. These are different objects from the point of view of internal structure and formation, but present a large similarity from the point of view of their detection. In this paper we present and discuss a new aspect of the occultation method of searching for a dark companion, also known as the “photometric method”, taking advantage of the precession of orbits, with a particular emphasis on extrasolar planets. A more exhaustive quantitative investigation will be made in a subsequent paper.
Short overview of detection methods
The search for extrasolar planetary systems has recently taken a significant step with the probable discovery of three planets in orbit around the pulsar PSR1257+12 (Wolszczan and Frail, 1994). The next important step would be to detect planets around main sequence stars, Correspondence to : J. Schneider
principally for their relevance to habitability. Among several choices, three methods are generally envisaged for that aim: direct imaging, velocimetry and astrometry. Direct imaging, in which the planet’s image must be extracted from the diffraction peak of the star, requires an angular resolution beyond the scope of telescopes in operation or under construction ; for example it has been shown that the Hubble Space Telescope is unlikely to detect extrasolar planets by that way (Brown and Burrows, 1990). The velocimetric method suffers from possible confusion with stellar activity (Walker et al., 1992). The astrometric method, which tracks the reflex motion of the star due to the gravitational influence of its companion, does not have that disadvantage, but it is not able to observe directly the planet and to give any information on its atmosphere (and perhaps prebiotic activity); presently we can only detect Jupiters. In addition to these methods the occultation method searches for a small drop in luminosity of the star due to a planetary transit. It has its own implemental difficulties but also two advantages : not only can it detect Earth-like habitable planets but it may also in the future allow for spectroscopic studies of the planetary atmosphere (Schneider, 1994).
The occultation method
This method has first been proposed by Struve (1952) and then discussed quantitatively by Rosenblatt (1971), Borucki and Summers (1984) which lead to a proposal for a devoted space mission (Borucki and Koch, 1993) ; a systematic survey of several thousand stars would be necessary. Schneider et al. (1990) have shown that searches for planets in orbit around cool dwarf stars increase both the depth and the probability of occultations. On the other hand, Schneider and Chevreton (1990) have shown that searches around eclipsing binaries increase very significantly the probability of detection events which have very characteristic unambiguous multiple eclipsing patterns. The interest of binary central stars is however
540
suspended to the possibility of having planet formation in binary systems and to the stability of heir orbits. There is a lack of studies for the first point : the only reference is Heppenheimer (1978), who found that planets can indeed form in a binary system, but with too elementary a model. The second point has been addressed more extensively. There are two types of configurations: the inner case where the planet is orbiting around one of the components of the binary and the outer case where the planet is orbiting around both components. All the studies (see for instance Benest (1988), Black (1982), Kubala et al. (1993), Szebehely (1984), and references therein), made either numerically or semi-analytically show that the orbits as stable as soon as its radius is about either one third (inner case) or the triple (outer case) of the separation between components. One may also be concerned with the possible confusion with some features in the star’s light curve due to its activity such as flares or brightness modulation due to the rotation of spots. These features contribute to the background noise, but they cannot mimic occultations. Flares increase the brightness instead of decreasing it. Spot rotation gives a quasi-sinusoidal modulation which furthermore loses its phase after several months because of the finite lifetime of spots, whereas an occultation is perfectly predictable over at least thousands of years. As for the signal to noise ratio S/N, it allows us to detect an occultation as soon as the luminosity drop is larger than N/S. These detectability aspects are studied more extensively in a paper in preparation (Schneider and Doyle, 1994). The combination of both classes of targets4001 dwarf stars and eclipsing binaries-naturally leads to the idea of concentrating searches around eclipsing pairs of dwarf stars of low brightness. But, even in these cases, the chance that the orbital plane of the planet is suitably orientednearly perpendicular to the sky with a deviation better than 1 degree-is not 100% and it is possible that for a given target an occultation never occurs. The example of the solar system shows that the inclination of planetary orbits can vary from 0” to 7”, making the occultation method less attractive. In fact, the orbital plane of a planet is a binary is not stable and precesses around the angular momentum of the binary. It follows from this precession that occultations necessarily occur twice for each revolution of the planet’s orbital plane around its axis of precession. In the remaining part of this paper we investigate the advantages of this effect in the search for dark companions of binary stars. In section 2 the relevant formula is established and a qualitative discussion of the effects of the precession on the eclipses is made. In section 3 the results of numerical simulations is presented. In section 4 consequences are drawn for strategies of future observations with a discussion of some specific cases.
2. Precession of the planet orbital plane We assume for simplicity that both stars are in circular orbits and have the same mass M/2. The parameters describing the configuration of the system are: a_, the
J. Schneider : Occultations of a binary star
separation between the two components (sitting on the same orbit since they have equal masses), up and Pp the radius and the period of the planet orbit, assumed to be circular, R* the radius of the stellar components (assumed to be equal), igPthe angle between the binary and planet’s orbits and finally Q the elongation of the ascending node of the planet’s orbit with respect to the intersection of the sky plane and the binary plane. In addition there are two parameters depending on the position of the observer, namely iB and iP, the inclinations of the binary and planetary orbits on the sky. The angle iBPis a constant of motion of the planet, but its orbital plane precesses continuously. To study this precession we use the standard perturbation method. Let H(iep, Q) be the Hamiltonian of a particle orbiting in the binary system. The precession rate is given by the equation (Kaula, 1968)
a=-
l3H
PP
27-4 sin i,,
&P’
(1)
Consider a planet in orbit around both stars. Since the rotation of the binary is much faster than the orbital motion of the planet we can simulate the binary as a ring with the same total mass and radius as the binary. The mean Hamiltonian potential averaged along the planetary orbit is then H=
-~~+~~~(1-3c0s2i,,). P
(2)
4
Using (1) and assuming that the planet mass is negligible, the precession rate is then given by fi=
( )2cosiBp, f.fP
_g P
P
(3)
This results in a precession period for the orbital plane of the planet Ppr given by 2
P”‘=PpF
? (
y. =P
)
cos
1 lap
(4)
If the planet is in orbit around one of the components of the binary the Hamiltonian takes the form H=
-~+~~*(l-3cos2iBp) B as’e,
(5)
which leads, using equation (1) to the precession period
(6) where PB is the orbital period of the binary. According to several authors, numerical simultations show that the condition for the planetary orbits to be stable is asep> 3ap (Harrington, 1977; Black, 1982; Benest, 1988). The precession period is therefore generally much larger than for planets in orbit around both components of the binary and thus is less suited to searching for extrasolar planets ; we will consequently consider only this latter case in the remaining part of this paper, leaving the investigation of inner planetary orbits for a subsequent paper.
(a) : t = 0
(b) :ttP&4
rapidity of the precession: if it is slow, occultations may occur for many orbital periods, if it is fast, occultations may happen for only a few revolutions of the planet. Let us summarize the results of this discussion. (1) If the three following conditions are fulfilled :
(c) : “=Pp,12
(d) : t = 3~~4
Fig. 1, Effect of the precession on the projection of the planet orbit on the sky. The large and the small ellip= represent, respectiveiy, the projections on the sky of the planet and the binary star mbits
Let us discuss the consequences of this precession on the occultations. We introduce two unit vectors I, and tp perpendicular to the orbital planes of the binary and the planet, respectively. The second vector, IP, rotates, under the effect of precession, around 1, with a period P,. Consider the projection EP on the sky plane of the planet orbit. During one precession period it oscillates between two extreme shapes. When the angle between lP and the sky plane, which is equal to n/2-&, reaches its maximum x/2 - iP,mjn,EP is an ellipse with maximum semi-minor axis 6~ = ctpCOS ip,*in ; it then may or may not intersect the binary “track” (generated by the motion on the sky of the two stellar disks along their apparent orbit), depending on the values of the parameters : when bp > br,+ R, (bB is the semi-minor axis of the projection on the sky of the binary orbit) no occultation can take place. If during its rotation around lB the vector IF intersects the sky plane, EP reduces at that very instant to a segment of length a, centered on the binary center of mass ; an occultation is then likely to occur. This happens whenever igP > 7t/2- it+ See Fig. I for a ~sualjzation of the configuration. tether or not an occultation really happens depends in addition on the positions of the two components of the binary and of the planet in their orbits at that moment. Suppose that then the track TB on the sky ia is smaller than 2R&,,; of the two stellar disks along their trajectory is simply connected, otherwise it has a central hole. If in addition (far iB < 2R,/a,,) p, the angle between the semi-major axes of Et, and E,, is smaller than R*/(OSa,,) the trajectory of the planet inescapably encounters at least one of the two stellar disks, whatever their positions on their orbits. If the precession is slow compared to the orbital revolution of the planet an occultation must necessarily occur. It is necessary to explain what is to be meant here by “slow”. Suppose that the projection EP intersects the track TB ; EP is moving under the effect of the precession ; an occultation occurs if that configuration lasts more than the quantity one orbital period Pp Introducing PO= 2R.4asep, this leads to the approximate condition P,,&/(2n) > Pp or ap > (37~/16)(a,,,//3,). It follows that each time the condition iep > 42 - is is satisfied the precession forces an occultation to occur twice during one precession period (cases b and d in Fig. 1) ; in particular if iB = n/2 an occultation is inescapable. A further question then arises : if at some instant along the precession an occultation occzlfs, will it repeat for the subsequent orbital revolutions of the planet? The answer depends on the
an occultation will inevitably occur at least twice during every precession period. If iB = KY’, the maximum value taken by p along the precession is iBP; in that case the condition fi < fi,, is automatically satisfied whenever isr < PO. As an example, for the eclipsing binary CM Dra (ie = 90”, use+, = 3.76R,, R* = 0.25R,), the value of POis 0.13rad = ‘7.5”; for this star occultations occur twice for every precession period whenever up > 34Ro and igP < 7.5”. If the precession would not have been taken into account, occultations would in that case have of iP only for occurred for all o~entat~ons iap < 2R,/a, = 0.8”. (2) If one of the preceding conditions is relaxed, the precession will again force occultations to occur, but only with some statistical frequency p and not for every halfperiod of precession. For given values ofp and P,, at least one occultation will occur for every time interval i PJp. In the next section we investigate numerically the corresponding statistical distributions.
We have used a numerical model for ~~~l~tions developed in a previous work (Schneider and Chevreton, 1990) to which we have added the precession of the orbital plane of the planet. We have used the approximation for the precession period given in the preceding section. The model is applied for different configurations characterized by the following parameters : iB the binary inclination on the sky asepthe binary separation M the binary total mass iap the angle between the binary and the planet’s planes 9 the planet semi-major axis. From these parameters one deduces the precession period given by the expression (3). For each configuration we simulated the motion, projected on the sky, of the binary star and the planet, along 100 periods of precession. We have in each case investigated the occurrence of occultations and have in particular determined : the proportion f = N,,,,,/N,, of the planet’s orbits which, in the course of the precession, intersect the track TB on the sky of the binary ; the statistical frequency pYwhen the projections Ep and
J. Schneider : Occuitations of a binary star
542 Table 1. ascp = 3.8, A4 = 0.5 M,,
up = 12R,
IBP
ap= 15R,
Ppr = 372 d Texp (days)
P
f
(deg)
0.5
1
1 2 3 4 5 7 10 20
1
1 1
0 0
0.52 0.36 0.3 0.27 0.21 0.18 0.14
0.89 0.86 0.76 0.74 0.81 0.72 0.5
90 120 130 135 146 152 160
R, = 0.25 R,, igp = 0”
.f 1
1
0
0.9 0.4 0.28 0.23 0.2 0.17 0.14 0.11
0.91 0.9 0.86 0.82 0.8 0.76 0.71 0.54
41 242 290 310 324 336 348
b3P
s
ap = 15R,
0 0 0.04 0.27 0.32 0.27 0.15 0.11
1 2 3 4 5 7 10 20
P,, = 503 d P
TeXp (days)
0 0 1 0.78 0.88 0.81 0.73 0.54
up = 20Ro
.f
238 182 170 201 213 223
0 0 0.02 0.23 0.25
0.15 0.12 0.09
TB intersect, of occurrences of at least one occultation during one period of precession. For a given precession period Ppr the ratio f leads to a time interval fPpr during which occultations can occur in a precession period ; therefore TeXp= (I -f )P,,/2 is at any instant the maximum time interval after which the precession will force an occultation to occur ; we will call Texp the expectation time for occultations. On the other hand, the statistical frequency p leads to a mean time interval 0.5P,,/p after which one occultation must statistically occur. The results are given in Tables l-3 for different sets of parameters.
Table 3. a,, = 50R,, fBP
(de& 0.5 1 2 3 4 :: 10 20
M = 2Mn, R, = lR,, is = 0”
up = 150R, f P 1 0.49 0.28 0.21 0.18 0.16 0.15 0.14 0.11
1 0.92 0.83 0.82 0.8 0.77 0.68 0.52 0.32
P,, = 4880 d Texp Wv4 0 1240 1760 1930 2000 2050 2075 2100 2170
up = 30R,
P,, = P
f 0.85 0.36 0.2 0.14 0.12 0.1 0.09 0.07 0.06
0.97 0.93 0.88 0.85 0.8 0.8 0.75 0.7 0.5
9189 d Terp(days) 690 2940 3680 3950 4100 4130 4200 4250 4330
R, = 0.6 R,, iBp= 4”
Table 2. asep = 3.8, M = lM,,
(deg)
Ppr = 812 d TeXp (days)
P
Ppr= 1378d P
TeXp (days)
0 0
1 0.83
1 0.73 0.66 0.55
670 530 515 585 600 630
f
up = 30R,
0 0 0 0.18 0.15 0.1 0.08 0.06
Ppr= 5700 d P
Texp (days)
0 0 0
0.94 0.93 8:E3 0.5
2320 2410 25.50 2610 2680
4. Discussion In this section we discuss the consequences of the preceding results in view of different stategies for observations.
Discovery of planets
The occultation method allows the detection of inner planets not accessible to the velocimetric and the astrometric methods. The reason for that is that these methods, being linearly sensitive to the planet mass, are presently limited to Jovian masses and that Jupiters close to the central stars are excluded by the evaporation of their hydrogen. As has already been pointed out in a preceding paper, the time characteristics of the occultations can give an estimate of the period Pp; in the case of multiple occultations this estimate can be fairly good (Schneider and Chevreton, 1990). Let us recall the argument: for a single star, the star/planet configuration is characterized by two unknowns, up and b, the impact parameter (closest distance on the sky) of the planet’s projected trajectory to the star; only one quantity, the duration D,, = (P,/n)~~/ up of the occultation, is measurable. The problem is mathematically underdetermined and only an approximate estimate Pp > FcD&~GMR-$/~ can be given. For the occultation of a binary star there are three unknowns, ap, b (relative to the center of the
J. Schneider : Occultations of a binary star
u a
u
b
Ll c
u
d
Fig. 2. Schematic representation of an example of a multi-occultations pattern for a planet in orbit at a, = 50 R, around a binary star with a separation of 3.5 R, between the two components. The two stars have the same radius R, = 0.25R,. In
the cases a, b and d the planet transit is for one of the components of the binary ; in case c the transit is for the other component. The longest occultation (b) corresponds to the case where the planet and the star go in the same direction. In the three other cases the planet and the star go in opposite directions on the sky plane binary) and the angte 6 defined in section 2. But since the central star is binary and moving rapidly, it leads to multiple occultations as shown by an example in Fig. 2. There are then four (or more) measurable quantities : the durations D,,,, and DOcc,zof each occultation related to the diameters of the two stars, their temporal separation Tsep,related to the separation between the two stars projected on the sky, and the time of the first occultation compared to the time of occultation of the binary. The problem is mathematically overdete~ined and it is possible to give a good estimate of the orbital period of the planet which is important to predict the epoch of occurrence of the next occultation. If we relax the hypothesis of a circular orbit for the planet there is a fourth unknown, namely ep the eccentricity, so now the problem is well determined. Suppose that an occultation or a multi-occultation is detected at some instant t,. From its temporal characteristics an orbital period for the planet Pp can be estimated. Because of the precession it is possible that at the time to+ Pp no occultation occurs. An absence of occultation at to + Pp would therefore not rule out the existence of an occultation at t,. Even if an occultation or a multi-occultation repeats at t, + Pp, the effect of precession would be that the characteristics would not remain identical to those at t, : a change in the temporal characteristics of a drop in luminosity of the star does not imply that it is not caused by an occultation. A negative result of a photometric monitoring over a period P around a binary would not imply the nonexistence of a planet with an orbital period P or smaller. A continuous monitoring over a time larger than the expectation time T,,(P) corresponding to P, as defined in section 3, would be necessary to draw final conclusions on the presence or absence of a planet with an orbital period P. The effect of precession can also be used to extend the sample of stars to monitor. An occultation can take place whenever the inclination & is sufficiently close to 7c/2 (we have seen in sections 2 and 3 that there is no analytical formula for the upper limit of Pp in case of a binary star). During the precession ir covers the interval [ie+ r&&- iBp]. Even if iB is not close to 90”, ip can approach 90” insofar as igp, the angle between the binary and planetary orbital planes, can take values significantly different from zero. If we take the solar system as a comparison point, it leads us to admit values up to 10” (from Mercury orbital inclination) or 20” (from the inclination of the Moon’s orbit) for the angle isp. The same range of
543 values can therefore be admitted for 190”- &I. It implies that the search for occultation should not be restricted to eclipsing binaries but should be extended to spectroscopic binaries with high inclinations (for binaries for which iB can be determined).
Photometric conjrmation ofdetectedplanets
The unquestionable discovery of occultations by an (until yet) unknown planet would require a photometric monitoring over a period larger than its orbital period. But the detection of occultations would also be useful in the case of a planet already discovered by another method. It would serve to determine, or to measure with higher precision, some parameters of the planet, such as the inclination of its orbit with respect to the binary’s plane, its size (through the amount of the drop in brightness of the star) and perhaps the composition of its atmosphere (Schneider, 1994). Suppose a Jupiter is detected by some method around a nearly edge-on binary. The subsequent detection of the transits would be rather easy. From the astrometric or velocimetric data the time tc of next conjunction of the planet and the central star (epoch of zero difference between the mean value and the actual value either for the position or for the velocity of the star) would be easy to estimate (up to an additional factor Pp/2). At tc an occultation may or may not occur, depending on the value of the impact parameter b. The effect of the precession would be that in a time shorter than Pprec/2= Pp(ap/asep)2/2 an occultation would necessarily occur. For example a Jupiter orbiting at a distance of lOOR, a binary dM seen exactly edge-on with a separation of 30R, would produce an occultation every 3.3 years, or less, during a period of 0.7 year for igp = 5” around well estimated conjunction epochs tc. A monitoring would be necessary only near the epochs tc + nPp.
Photometric search for brown dwarfs
The detection of occultations to Jupiter-sized planets is restricted to the range up > R,(T,/200K)’ where T* is the temperature of the star. Closer orbits would lead to hydrogen evaporation of the Jupiter. Therefore the detection of planets in close orbits, for which the probability of detection is larger, is restricted to small planets which induce a smaller drop in brightness. These limitations do not hold for brown dwarfs. This class of objects can therefore be in principle searched for by the same occultation method, although it is not clear if compact triple systems can exist.
5. Conclusion We have shown that the precession of the orbital plane of a planet in revolution around a binary star forces occultations of the binary by the planet for nearly edge-on binaries. It significantly increases the probability of photometric detection of planets and can be used to extend the
544
sample of stars for the photometric search of extrasolar planets. It changes the temporal signature of the multioccultations. Since the occultation method allows for the determination of some parameters of the planet or its orbit otherwise unreachable, this method must be viewed in some cases as a complement to astrometric and velocimetric searches for extrasolar planets. The results of the present study must consequently be borne in mind before analysing observations or planning photometric monitorings.
References Benest, D., Planetary orbits in the elliptic restricted problem. Celest. Mech. 43,47, 1988. Black, D. C., A simple criterion for determining the dynamical stability of three-body systems. Astron. J. 263, 854, 1982. Borucki, W. and Summers, A. L., The photometric method of detecting other planetary systems. Icarus 58, 121, 1984. Borucki, W. and Koch, D., Multiplex approach of the photometric detection of planets, in Proceedings of the Planetary Systems : Formation and Detection ’ Conference in press, 1994. Brown, R. A. and Burrows, C. J., On the feasibility of detecting extrasolar planets by reflected starlight using the Hubble Space Telescope. Zcarus 87,484, 1990.
J. Schneider : quotations
of a binary star
Harrington, R. S., Planetary orbits in binary stars. Astron. J. 82, 753, 1977. Kaula, W. M., An Introduction to Planetary Physics. p. 166. Wiley, New York, 1968. Kubala A., Black, D. and Szebebely, V., Stability of outer planetary orbits around binary stars : a comparison of Hill’s and Laplace’s stability criteria. Celest. Mech. 56, 51, 1993. Rosenblatt, F., A two color photometric method for detection of extrasolar planetary systems. Icarus 14,71, 1971. Schneider J., On the search for 0, in the atmosphere of extrasolar planets, in Proceedings of the Planetary Systems : Formation and Detection’ Conference. Astrophys. Space Sci., in press, 1994. Schneider, J. and Chevreton, M., The photometric search for Earth-sized extrasolar planets by occultation in binary systems. Astron. Astrophys. 232, 251, 1990. Schneider, J., Chevreton, M. and Martin, E. L., New efforts in the search for extrasolar planets, in ‘24’hESLAB Symposium’ (edited by B. Battrick), ESA SP-315, 67, 1990. Schneider, J. and Doyle, L., On the feasability of the photometric search search of extrasolar planets in eclipsing binaries, 1994. Szebehely V., Review of concepts of stability. Celest. Mech. 34, 49, 1984. Struve O., Proposal for a project of high precision stellar radial velocity work. The Observatory 72, 199, 1952. Walker, G. A. H. et al., y Cephei : rotation or planetary companion? Astrophys. J. Lett. 396, L91, I992 Wolszczan, A. and D. A. Frail, Con~rmation of earth-mass planets orbiting the millisecond Pulsar PSR B1257 + 12. Scierzce, 1994.
Pergamon
Copyright 0 1994 Elsevier Science Ltd Printed in Great Britain. All rights reserved 00324633/94 $7.00+0.00 0032-0633(94)00092-l
On the occultations of a binary star by a circum-orbiting dark companion J. Schneider CNRS-UPR
176, Observatoire de Paris, 92195 Meudon, France
Received 12 August 1993 ; revised 6 May 1994; accepted 6 May 1994
.
1. Introduction
The existence of two classes of dark celestial bodiesbrown dwarfs and extrasolar planets (at least orbiting around main sequence stars)-is still an unsolved problem. These are different objects from the point of view of internal structure and formation, but present a large similarity from the point of view of their detection. In this paper we present and discuss a new aspect of the occultation method of searching for a dark companion, also known as the “photometric method”, taking advantage of the precession of orbits, with a particular emphasis on extrasolar planets. A more exhaustive quantitative investigation will be made in a subsequent paper.
Short overview of detection methods
The search for extrasolar planetary systems has recently taken a significant step with the probable discovery of three planets in orbit around the pulsar PSR1257+12 (Wolszczan and Frail, 1994). The next important step would be to detect planets around main sequence stars, Correspondence to : J. Schneider
principally for their relevance to habitability. Among several choices, three methods are generally envisaged for that aim: direct imaging, velocimetry and astrometry. Direct imaging, in which the planet’s image must be extracted from the diffraction peak of the star, requires an angular resolution beyond the scope of telescopes in operation or under construction ; for example it has been shown that the Hubble Space Telescope is unlikely to detect extrasolar planets by that way (Brown and Burrows, 1990). The velocimetric method suffers from possible confusion with stellar activity (Walker et al., 1992). The astrometric method, which tracks the reflex motion of the star due to the gravitational influence of its companion, does not have that disadvantage, but it is not able to observe directly the planet and to give any information on its atmosphere (and perhaps prebiotic activity); presently we can only detect Jupiters. In addition to these methods the occultation method searches for a small drop in luminosity of the star due to a planetary transit. It has its own implemental difficulties but also two advantages : not only can it detect Earth-like habitable planets but it may also in the future allow for spectroscopic studies of the planetary atmosphere (Schneider, 1994).
The occultation method
This method has first been proposed by Struve (1952) and then discussed quantitatively by Rosenblatt (1971), Borucki and Summers (1984) which lead to a proposal for a devoted space mission (Borucki and Koch, 1993) ; a systematic survey of several thousand stars would be necessary. Schneider et al. (1990) have shown that searches for planets in orbit around cool dwarf stars increase both the depth and the probability of occultations. On the other hand, Schneider and Chevreton (1990) have shown that searches around eclipsing binaries increase very significantly the probability of detection events which have very characteristic unambiguous multiple eclipsing patterns. The interest of binary central stars is however
540
suspended to the possibility of having planet formation in binary systems and to the stability of heir orbits. There is a lack of studies for the first point : the only reference is Heppenheimer (1978), who found that planets can indeed form in a binary system, but with too elementary a model. The second point has been addressed more extensively. There are two types of configurations: the inner case where the planet is orbiting around one of the components of the binary and the outer case where the planet is orbiting around both components. All the studies (see for instance Benest (1988), Black (1982), Kubala et al. (1993), Szebehely (1984), and references therein), made either numerically or semi-analytically show that the orbits as stable as soon as its radius is about either one third (inner case) or the triple (outer case) of the separation between components. One may also be concerned with the possible confusion with some features in the star’s light curve due to its activity such as flares or brightness modulation due to the rotation of spots. These features contribute to the background noise, but they cannot mimic occultations. Flares increase the brightness instead of decreasing it. Spot rotation gives a quasi-sinusoidal modulation which furthermore loses its phase after several months because of the finite lifetime of spots, whereas an occultation is perfectly predictable over at least thousands of years. As for the signal to noise ratio S/N, it allows us to detect an occultation as soon as the luminosity drop is larger than N/S. These detectability aspects are studied more extensively in a paper in preparation (Schneider and Doyle, 1994). The combination of both classes of targets4001 dwarf stars and eclipsing binaries-naturally leads to the idea of concentrating searches around eclipsing pairs of dwarf stars of low brightness. But, even in these cases, the chance that the orbital plane of the planet is suitably orientednearly perpendicular to the sky with a deviation better than 1 degree-is not 100% and it is possible that for a given target an occultation never occurs. The example of the solar system shows that the inclination of planetary orbits can vary from 0” to 7”, making the occultation method less attractive. In fact, the orbital plane of a planet is a binary is not stable and precesses around the angular momentum of the binary. It follows from this precession that occultations necessarily occur twice for each revolution of the planet’s orbital plane around its axis of precession. In the remaining part of this paper we investigate the advantages of this effect in the search for dark companions of binary stars. In section 2 the relevant formula is established and a qualitative discussion of the effects of the precession on the eclipses is made. In section 3 the results of numerical simulations is presented. In section 4 consequences are drawn for strategies of future observations with a discussion of some specific cases.
2. Precession of the planet orbital plane We assume for simplicity that both stars are in circular orbits and have the same mass M/2. The parameters describing the configuration of the system are: a_, the
J. Schneider : Occultations of a binary star
separation between the two components (sitting on the same orbit since they have equal masses), up and Pp the radius and the period of the planet orbit, assumed to be circular, R* the radius of the stellar components (assumed to be equal), igPthe angle between the binary and planet’s orbits and finally Q the elongation of the ascending node of the planet’s orbit with respect to the intersection of the sky plane and the binary plane. In addition there are two parameters depending on the position of the observer, namely iB and iP, the inclinations of the binary and planetary orbits on the sky. The angle iBPis a constant of motion of the planet, but its orbital plane precesses continuously. To study this precession we use the standard perturbation method. Let H(iep, Q) be the Hamiltonian of a particle orbiting in the binary system. The precession rate is given by the equation (Kaula, 1968)
a=-
l3H
PP
27-4 sin i,,
&P’
(1)
Consider a planet in orbit around both stars. Since the rotation of the binary is much faster than the orbital motion of the planet we can simulate the binary as a ring with the same total mass and radius as the binary. The mean Hamiltonian potential averaged along the planetary orbit is then H=
-~~+~~~(1-3c0s2i,,). P
(2)
4
Using (1) and assuming that the planet mass is negligible, the precession rate is then given by fi=
( )2cosiBp, f.fP
_g P
P
(3)
This results in a precession period for the orbital plane of the planet Ppr given by 2
P”‘=PpF
? (
y. =P
)
cos
1 lap
(4)
If the planet is in orbit around one of the components of the binary the Hamiltonian takes the form H=
-~+~~*(l-3cos2iBp) B as’e,
(5)
which leads, using equation (1) to the precession period
(6) where PB is the orbital period of the binary. According to several authors, numerical simultations show that the condition for the planetary orbits to be stable is asep> 3ap (Harrington, 1977; Black, 1982; Benest, 1988). The precession period is therefore generally much larger than for planets in orbit around both components of the binary and thus is less suited to searching for extrasolar planets ; we will consequently consider only this latter case in the remaining part of this paper, leaving the investigation of inner planetary orbits for a subsequent paper.
(a) : t = 0
(b) :ttP&4
rapidity of the precession: if it is slow, occultations may occur for many orbital periods, if it is fast, occultations may happen for only a few revolutions of the planet. Let us summarize the results of this discussion. (1) If the three following conditions are fulfilled :
(c) : “=Pp,12
(d) : t = 3~~4
Fig. 1, Effect of the precession on the projection of the planet orbit on the sky. The large and the small ellip= represent, respectiveiy, the projections on the sky of the planet and the binary star mbits
Let us discuss the consequences of this precession on the occultations. We introduce two unit vectors I, and tp perpendicular to the orbital planes of the binary and the planet, respectively. The second vector, IP, rotates, under the effect of precession, around 1, with a period P,. Consider the projection EP on the sky plane of the planet orbit. During one precession period it oscillates between two extreme shapes. When the angle between lP and the sky plane, which is equal to n/2-&, reaches its maximum x/2 - iP,mjn,EP is an ellipse with maximum semi-minor axis 6~ = ctpCOS ip,*in ; it then may or may not intersect the binary “track” (generated by the motion on the sky of the two stellar disks along their apparent orbit), depending on the values of the parameters : when bp > br,+ R, (bB is the semi-minor axis of the projection on the sky of the binary orbit) no occultation can take place. If during its rotation around lB the vector IF intersects the sky plane, EP reduces at that very instant to a segment of length a, centered on the binary center of mass ; an occultation is then likely to occur. This happens whenever igP > 7t/2- it+ See Fig. I for a ~sualjzation of the configuration. tether or not an occultation really happens depends in addition on the positions of the two components of the binary and of the planet in their orbits at that moment. Suppose that then the track TB on the sky ia is smaller than 2R&,,; of the two stellar disks along their trajectory is simply connected, otherwise it has a central hole. If in addition (far iB < 2R,/a,,) p, the angle between the semi-major axes of Et, and E,, is smaller than R*/(OSa,,) the trajectory of the planet inescapably encounters at least one of the two stellar disks, whatever their positions on their orbits. If the precession is slow compared to the orbital revolution of the planet an occultation must necessarily occur. It is necessary to explain what is to be meant here by “slow”. Suppose that the projection EP intersects the track TB ; EP is moving under the effect of the precession ; an occultation occurs if that configuration lasts more than the quantity one orbital period Pp Introducing PO= 2R.4asep, this leads to the approximate condition P,,&/(2n) > Pp or ap > (37~/16)(a,,,//3,). It follows that each time the condition iep > 42 - is is satisfied the precession forces an occultation to occur twice during one precession period (cases b and d in Fig. 1) ; in particular if iB = n/2 an occultation is inescapable. A further question then arises : if at some instant along the precession an occultation occzlfs, will it repeat for the subsequent orbital revolutions of the planet? The answer depends on the
an occultation will inevitably occur at least twice during every precession period. If iB = KY’, the maximum value taken by p along the precession is iBP; in that case the condition fi < fi,, is automatically satisfied whenever isr < PO. As an example, for the eclipsing binary CM Dra (ie = 90”, use+, = 3.76R,, R* = 0.25R,), the value of POis 0.13rad = ‘7.5”; for this star occultations occur twice for every precession period whenever up > 34Ro and igP < 7.5”. If the precession would not have been taken into account, occultations would in that case have of iP only for occurred for all o~entat~ons iap < 2R,/a, = 0.8”. (2) If one of the preceding conditions is relaxed, the precession will again force occultations to occur, but only with some statistical frequency p and not for every halfperiod of precession. For given values ofp and P,, at least one occultation will occur for every time interval i PJp. In the next section we investigate numerically the corresponding statistical distributions.
We have used a numerical model for ~~~l~tions developed in a previous work (Schneider and Chevreton, 1990) to which we have added the precession of the orbital plane of the planet. We have used the approximation for the precession period given in the preceding section. The model is applied for different configurations characterized by the following parameters : iB the binary inclination on the sky asepthe binary separation M the binary total mass iap the angle between the binary and the planet’s planes 9 the planet semi-major axis. From these parameters one deduces the precession period given by the expression (3). For each configuration we simulated the motion, projected on the sky, of the binary star and the planet, along 100 periods of precession. We have in each case investigated the occurrence of occultations and have in particular determined : the proportion f = N,,,,,/N,, of the planet’s orbits which, in the course of the precession, intersect the track TB on the sky of the binary ; the statistical frequency pYwhen the projections Ep and
J. Schneider : Occuitations of a binary star
542 Table 1. ascp = 3.8, A4 = 0.5 M,,
up = 12R,
IBP
ap= 15R,
Ppr = 372 d Texp (days)
P
f
(deg)
0.5
1
1 2 3 4 5 7 10 20
1
1 1
0 0
0.52 0.36 0.3 0.27 0.21 0.18 0.14
0.89 0.86 0.76 0.74 0.81 0.72 0.5
90 120 130 135 146 152 160
R, = 0.25 R,, igp = 0”
.f 1
1
0
0.9 0.4 0.28 0.23 0.2 0.17 0.14 0.11
0.91 0.9 0.86 0.82 0.8 0.76 0.71 0.54
41 242 290 310 324 336 348
b3P
s
ap = 15R,
0 0 0.04 0.27 0.32 0.27 0.15 0.11
1 2 3 4 5 7 10 20
P,, = 503 d P
TeXp (days)
0 0 1 0.78 0.88 0.81 0.73 0.54
up = 20Ro
.f
238 182 170 201 213 223
0 0 0.02 0.23 0.25
0.15 0.12 0.09
TB intersect, of occurrences of at least one occultation during one period of precession. For a given precession period Ppr the ratio f leads to a time interval fPpr during which occultations can occur in a precession period ; therefore TeXp= (I -f )P,,/2 is at any instant the maximum time interval after which the precession will force an occultation to occur ; we will call Texp the expectation time for occultations. On the other hand, the statistical frequency p leads to a mean time interval 0.5P,,/p after which one occultation must statistically occur. The results are given in Tables l-3 for different sets of parameters.
Table 3. a,, = 50R,, fBP
(de& 0.5 1 2 3 4 :: 10 20
M = 2Mn, R, = lR,, is = 0”
up = 150R, f P 1 0.49 0.28 0.21 0.18 0.16 0.15 0.14 0.11
1 0.92 0.83 0.82 0.8 0.77 0.68 0.52 0.32
P,, = 4880 d Texp Wv4 0 1240 1760 1930 2000 2050 2075 2100 2170
up = 30R,
P,, = P
f 0.85 0.36 0.2 0.14 0.12 0.1 0.09 0.07 0.06
0.97 0.93 0.88 0.85 0.8 0.8 0.75 0.7 0.5
9189 d Terp(days) 690 2940 3680 3950 4100 4130 4200 4250 4330
R, = 0.6 R,, iBp= 4”
Table 2. asep = 3.8, M = lM,,
(deg)
Ppr = 812 d TeXp (days)
P
Ppr= 1378d P
TeXp (days)
0 0
1 0.83
1 0.73 0.66 0.55
670 530 515 585 600 630
f
up = 30R,
0 0 0 0.18 0.15 0.1 0.08 0.06
Ppr= 5700 d P
Texp (days)
0 0 0
0.94 0.93 8:E3 0.5
2320 2410 25.50 2610 2680
4. Discussion In this section we discuss the consequences of the preceding results in view of different stategies for observations.
Discovery of planets
The occultation method allows the detection of inner planets not accessible to the velocimetric and the astrometric methods. The reason for that is that these methods, being linearly sensitive to the planet mass, are presently limited to Jovian masses and that Jupiters close to the central stars are excluded by the evaporation of their hydrogen. As has already been pointed out in a preceding paper, the time characteristics of the occultations can give an estimate of the period Pp; in the case of multiple occultations this estimate can be fairly good (Schneider and Chevreton, 1990). Let us recall the argument: for a single star, the star/planet configuration is characterized by two unknowns, up and b, the impact parameter (closest distance on the sky) of the planet’s projected trajectory to the star; only one quantity, the duration D,, = (P,/n)~~/ up of the occultation, is measurable. The problem is mathematically underdetermined and only an approximate estimate Pp > FcD&~GMR-$/~ can be given. For the occultation of a binary star there are three unknowns, ap, b (relative to the center of the
J. Schneider : Occultations of a binary star
u a
u
b
Ll c
u
d
Fig. 2. Schematic representation of an example of a multi-occultations pattern for a planet in orbit at a, = 50 R, around a binary star with a separation of 3.5 R, between the two components. The two stars have the same radius R, = 0.25R,. In
the cases a, b and d the planet transit is for one of the components of the binary ; in case c the transit is for the other component. The longest occultation (b) corresponds to the case where the planet and the star go in the same direction. In the three other cases the planet and the star go in opposite directions on the sky plane binary) and the angte 6 defined in section 2. But since the central star is binary and moving rapidly, it leads to multiple occultations as shown by an example in Fig. 2. There are then four (or more) measurable quantities : the durations D,,,, and DOcc,zof each occultation related to the diameters of the two stars, their temporal separation Tsep,related to the separation between the two stars projected on the sky, and the time of the first occultation compared to the time of occultation of the binary. The problem is mathematically overdete~ined and it is possible to give a good estimate of the orbital period of the planet which is important to predict the epoch of occurrence of the next occultation. If we relax the hypothesis of a circular orbit for the planet there is a fourth unknown, namely ep the eccentricity, so now the problem is well determined. Suppose that an occultation or a multi-occultation is detected at some instant t,. From its temporal characteristics an orbital period for the planet Pp can be estimated. Because of the precession it is possible that at the time to+ Pp no occultation occurs. An absence of occultation at to + Pp would therefore not rule out the existence of an occultation at t,. Even if an occultation or a multi-occultation repeats at t, + Pp, the effect of precession would be that the characteristics would not remain identical to those at t, : a change in the temporal characteristics of a drop in luminosity of the star does not imply that it is not caused by an occultation. A negative result of a photometric monitoring over a period P around a binary would not imply the nonexistence of a planet with an orbital period P or smaller. A continuous monitoring over a time larger than the expectation time T,,(P) corresponding to P, as defined in section 3, would be necessary to draw final conclusions on the presence or absence of a planet with an orbital period P. The effect of precession can also be used to extend the sample of stars to monitor. An occultation can take place whenever the inclination & is sufficiently close to 7c/2 (we have seen in sections 2 and 3 that there is no analytical formula for the upper limit of Pp in case of a binary star). During the precession ir covers the interval [ie+ r&&- iBp]. Even if iB is not close to 90”, ip can approach 90” insofar as igp, the angle between the binary and planetary orbital planes, can take values significantly different from zero. If we take the solar system as a comparison point, it leads us to admit values up to 10” (from Mercury orbital inclination) or 20” (from the inclination of the Moon’s orbit) for the angle isp. The same range of
543 values can therefore be admitted for 190”- &I. It implies that the search for occultation should not be restricted to eclipsing binaries but should be extended to spectroscopic binaries with high inclinations (for binaries for which iB can be determined).
Photometric conjrmation ofdetectedplanets
The unquestionable discovery of occultations by an (until yet) unknown planet would require a photometric monitoring over a period larger than its orbital period. But the detection of occultations would also be useful in the case of a planet already discovered by another method. It would serve to determine, or to measure with higher precision, some parameters of the planet, such as the inclination of its orbit with respect to the binary’s plane, its size (through the amount of the drop in brightness of the star) and perhaps the composition of its atmosphere (Schneider, 1994). Suppose a Jupiter is detected by some method around a nearly edge-on binary. The subsequent detection of the transits would be rather easy. From the astrometric or velocimetric data the time tc of next conjunction of the planet and the central star (epoch of zero difference between the mean value and the actual value either for the position or for the velocity of the star) would be easy to estimate (up to an additional factor Pp/2). At tc an occultation may or may not occur, depending on the value of the impact parameter b. The effect of the precession would be that in a time shorter than Pprec/2= Pp(ap/asep)2/2 an occultation would necessarily occur. For example a Jupiter orbiting at a distance of lOOR, a binary dM seen exactly edge-on with a separation of 30R, would produce an occultation every 3.3 years, or less, during a period of 0.7 year for igp = 5” around well estimated conjunction epochs tc. A monitoring would be necessary only near the epochs tc + nPp.
Photometric search for brown dwarfs
The detection of occultations to Jupiter-sized planets is restricted to the range up > R,(T,/200K)’ where T* is the temperature of the star. Closer orbits would lead to hydrogen evaporation of the Jupiter. Therefore the detection of planets in close orbits, for which the probability of detection is larger, is restricted to small planets which induce a smaller drop in brightness. These limitations do not hold for brown dwarfs. This class of objects can therefore be in principle searched for by the same occultation method, although it is not clear if compact triple systems can exist.
5. Conclusion We have shown that the precession of the orbital plane of a planet in revolution around a binary star forces occultations of the binary by the planet for nearly edge-on binaries. It significantly increases the probability of photometric detection of planets and can be used to extend the
544
sample of stars for the photometric search of extrasolar planets. It changes the temporal signature of the multioccultations. Since the occultation method allows for the determination of some parameters of the planet or its orbit otherwise unreachable, this method must be viewed in some cases as a complement to astrometric and velocimetric searches for extrasolar planets. The results of the present study must consequently be borne in mind before analysing observations or planning photometric monitorings.
References Benest, D., Planetary orbits in the elliptic restricted problem. Celest. Mech. 43,47, 1988. Black, D. C., A simple criterion for determining the dynamical stability of three-body systems. Astron. J. 263, 854, 1982. Borucki, W. and Summers, A. L., The photometric method of detecting other planetary systems. Icarus 58, 121, 1984. Borucki, W. and Koch, D., Multiplex approach of the photometric detection of planets, in Proceedings of the Planetary Systems : Formation and Detection ’ Conference in press, 1994. Brown, R. A. and Burrows, C. J., On the feasibility of detecting extrasolar planets by reflected starlight using the Hubble Space Telescope. Zcarus 87,484, 1990.
J. Schneider : quotations
of a binary star
Harrington, R. S., Planetary orbits in binary stars. Astron. J. 82, 753, 1977. Kaula, W. M., An Introduction to Planetary Physics. p. 166. Wiley, New York, 1968. Kubala A., Black, D. and Szebebely, V., Stability of outer planetary orbits around binary stars : a comparison of Hill’s and Laplace’s stability criteria. Celest. Mech. 56, 51, 1993. Rosenblatt, F., A two color photometric method for detection of extrasolar planetary systems. Icarus 14,71, 1971. Schneider J., On the search for 0, in the atmosphere of extrasolar planets, in Proceedings of the Planetary Systems : Formation and Detection’ Conference. Astrophys. Space Sci., in press, 1994. Schneider, J. and Chevreton, M., The photometric search for Earth-sized extrasolar planets by occultation in binary systems. Astron. Astrophys. 232, 251, 1990. Schneider, J., Chevreton, M. and Martin, E. L., New efforts in the search for extrasolar planets, in ‘24’hESLAB Symposium’ (edited by B. Battrick), ESA SP-315, 67, 1990. Schneider, J. and Doyle, L., On the feasability of the photometric search search of extrasolar planets in eclipsing binaries, 1994. Szebehely V., Review of concepts of stability. Celest. Mech. 34, 49, 1984. Struve O., Proposal for a project of high precision stellar radial velocity work. The Observatory 72, 199, 1952. Walker, G. A. H. et al., y Cephei : rotation or planetary companion? Astrophys. J. Lett. 396, L91, I992 Wolszczan, A. and D. A. Frail, Con~rmation of earth-mass planets orbiting the millisecond Pulsar PSR B1257 + 12. Scierzce, 1994.