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Astron. Astrophys. 328, 311-320 (1997)

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6. A model for the occultation by a planet

In this section we develop the two phases of this phenomenon. We shall analyse the sphere of influence, and finally we will conclude on the parameters which can be derived from a model of planetary occultation.

6.1. A dust-free sphere of influence

The explanation of the brightening of [FORMULA]  Pic by the passage of a cleared out zone in the dust ring in front of the star requires two properties. (a) The Hill sphere must be cleared of dust. This is necessary to reduce the extinction before and after the occultation. (b) The dust disk must be seen edge-on and produce an extinction larger than 0.06 mag. within a radial extent about equal to the radius of the cleared region around the planet. This is necessary to explain the amplitude of the variation.

To show the effect of the Hill sphere, we carried out the dynamical evolution of particles in resonance 1:1, i.e. with the same orbital period as the planet. Numerical simulations have been done with a model developed on the CM5 massively parallel computer. The program calculates the spatial distribution of dust at the equilibrium between injection by large bodies and destruction by collision (Lecavelier des Etangs et al. 1996b). The particles are assumed to be produced by parent bodies in a similar manner as described by Lecavelier des Etangs et al. (1996a), i.e. following the relation between the eccentricity of the orbits of dust grains and the [FORMULA] ratio of radiation forces to gravitational forces: [FORMULA].

The initial and final stage of such a model are represented in Fig. 5. We see that particles in 1:1 resonance are trapped around the [FORMULA] and [FORMULA] Lagrangian points. The Poynting-Robertson (PR) effect perturbs the orbits of dust particles around the star, so that they evolve into horseshoe orbits which continuously expand in size, until they reach close encounters with the planet (see also Liou & Zook 1995a and 1995b). Then these particles are accreted or ejected into very eccentric orbits. Therefore, at the steady state between particle injection and destruction, the influence sphere of the planet is clear of dust. Moreover, the regions around the Lagrangian points have large over-densities. We can thus explain not only the brightness increase during few days before and after JD 4918 by the cleared influence sphere, but also the brightness increase on long time scale which were observed before, i.e. the decrease of about 0.01 magnitude from 1979 to 1982. This variation can be interpreted by the slow decrease of the extinction by matter gathered around one of the Lagrange points: this region of over-density is rotating at the same speed as the planet.

[FIGURE] Fig. 5. Spatial distribution of dust in resonance 1:1 with the planet. The initial conditions are given at the left, with the azimuthal distribution at bottom. The steady state is given at the right. The influence sphere is represented by a circle around the planet. We see that at the steady state this region is free of dust, and that there is accumulation of matter around the two [FORMULA] and [FORMULA] Lagrangian points. These over-densities are located at [FORMULA] from the planet.

At a first glance, it seems that the three different variations are simultaneously explained: the slow brightening from 1979 to 1982 by the over-densities of the dust ring in a Lagrangian point, the variations before and after JD 4918 by the hole around the influence sphere of the planet and the dip in the lightcurve on JD 4918 by the planet occultation. However, the second requirement is a strong hypothesis: the extinction by dust or bodies in 1:1 resonance must be relatively large. This can be consistent with the infrared observations if the dust ring is flat and has an opening angle smaller than the outer part of the disk. In that case, the extinction can be relatively large in a small solid angle covered by this ring, and the total absorption can be small and consistent with the total infrared emission. Indeed, there is no reason to believe that the inner part of the disk at few astronomical units has an opening angle as large as the opening angle of the outer part, simply because the opening angle of the outer disk is due to the inclination distribution of the dust parent bodies (Lecavelier des Etangs et al. 1996a). Alternatively, as suggested in Sect.  3.1, there could be a flat ring of large particles in which dissipative collisions play the same role as the Poynting-Robertson drag. In conclusion, all the observed variations can be explained if a flat ring of particles is orbiting with a planet, except for its sphere of influence which would be relatively clear of dust.

6.2. Constraints from the occultation light curve

The dip in the lightcurve on JD 4918 might be due to occultation by the planet, independent of the explanation of the brightening. We compare the observations with detailed calculations. During the phase when the projected planet disk partially occults the stellar disk, the star light must quickly decrease (hundredths of magnitude per hour). The observed duration of this partial occultation phase is of the order of hours. Moreover, during the occultation, the planet hides different regions of the stellar surface along its path from limb to limb, depending on the impact parameter. Thus, due to the stellar limb-darkening, the light curve must present a very well defined shape.

We carried out calculations of a planet occultation taking into account the limb-darkening and the partial occultation phase. The input parameters of such a model are [FORMULA] and b, respectively the radius of the planet and the impact parameter (both in unit of stellar radius), d the distance to the star, [FORMULA] the magnitude of the star through the disk just before and after the occultation, and [FORMULA] the time of central occultation.

There are five parameters and we have only five measurements! However if the data would not present a sharp decrease followed by a longer slow one, they would never be compatible with such a model. Indeed, due to the limb-darkening of the stellar photosphere, any solution must present a slow decrease after the sharp one produced when the planet starts to pass in front of the star. This is exactly the behavior of the observed light variations.

Moreover, the five measurements of JD 4918 are only compatible with the set of solutions given by reasonable values of these parameters. The best fit is found for [FORMULA] =0.225, [FORMULA], d =5 AU, [FORMULA] =JD 4919.04 and [FORMULA] =-0.064 (Fig. 6).

[FIGURE] Fig. 6. Variations of a star brightness due to an occultation by a planet as a function of the time assuming a planet distance of 5 AU. b is the impact parameter in stellar radius ([FORMULA]). The planet radius is [FORMULA]. The limb-darkening and partial occultation are taken into account. The [FORMULA]  Pic magnitude measurements has been plotted assuming a differential brightness of 0.064 mag between the mean magnitude of [FORMULA]  Pic and the magnitude just before the occultation.

For a set of values for [FORMULA] and d, we have computed the squared difference between the measurements and the best fits, the three other parameters being relaxed. We have plotted the [FORMULA] isoprobabilities of such a model with 3 degrees of freedom (Fig. 7). It can be seen that [FORMULA] must be greater than 0.2 stellar radius, which is derived from the relation between the planet size and magnitude variation. This size is much larger than the maximal possible size of cold gaseous planets whose radius can never be larger than 1.5 times the radius of Jupiter or 0.1  [FORMULA] (Saumon et al. 1996). There are however three solutions to that problem: either the planet is not gaseous, which is not expected, or it is a hydrogen dominated warm (young) gaseous planet with large radius in a youthful system, finally this putative planet can be surrounded by large rings which can contribute to the amplitude and duration of the photometric variation.

[FIGURE] Fig. 7. Plot of the isoprobability of [FORMULA] of the squared differences between the data and the model as a function of given [FORMULA] and d. The dotted contour gives the best fit with [FORMULA] =0.225 and d =5 AU. The solid line contours represent the domain compatible with the data at a level of respectively 33%, 50%, 66% and 95%. Thus the planet radius must be greater than 0.2 stellar radius. A strong correlation is found between the possible distance to the star and planet radius. Dashed lines represent places where the time of central occultation, [FORMULA], must be JD 4918.8 and JD 4919.1. If the variations before and after JD 4918 are actually symmetrical, then we must have [FORMULA] (Lamers et al. 1997), and the region outside the zone given by these two limits must not be considered. Dot-short dashed line represents places where the final time of occultation [FORMULA] must be JD 4919.56 which is the date when the first measurements of the next day was obtained. If at that time the occultation is already finished, then the zone at the right of this line is not compatible with the data. Thus, the distance between the star and the putative planet must be less than 8 AU.

More importantly, d must be less than 8 AU in order to explain the short duration of the event and the fast variation on JD 4918. An eccentric orbit (not considered here) could slightly change this constrain. This gives a maximum period of 19 years, and the best fit with d =5 AU gives a most probable period of 9 years. A continuous survey of [FORMULA]  Pic photometric variations is thus recommended.

6.3. U band variations

As it can be seen in Fig. 2, the variations do not show color signatures except for the two first measurements of JD 4198 when the brightness of [FORMULA]  Pic was at its maximum and the U brightness increase was possibly 25% to 50% greater than for the other bands. If real (and not produced by atmospheric extinction), this phenomenon could be explained in the context of the occultation model. The extinction by dust in shorter wavelengths is expected to be larger than in longer wavelengths. Thus, a decrease of extinction must produce a larger increase of brightness in U band, and could explain the non-linearity in the color variation. However, as the planet occultation is achromatic, then the planet must be surrounded by material presenting the same larger extinction in the U band (dust rings?).

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© European Southern Observatory (ESO) 1997

Online publication: March 24, 1998