Photometric Stellar Variation Due to Extra-Solar Comets

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arXiv:astro-ph/9812381v1 21 Dec 1998 e words: Key of evolution systems. activity, and planetary cometary structure extra-solar of probe would evolution would This consequently time and comets. the extra-solar explore of to allow number large would survey a photometric spatial detect a year. that be per conclude should occultation thus We of it ob- hundreds stars, several the of detect to By thousand possible of accuracy. tens high several at of servation survey photometric large a percents. few by larger blue a the with chromatic in be extinction par- could smaller occultations have these periastrons Never- ticles, small achromatic. with mainly comets if are theless, occultations comet curve to variations the Photometric due cases, occultation. planetary other tri- a some “rounded mimic can most in peculiar However, very In appar- a shape. stars. shows angular” the curve central by light the the detected of find cases, variations be We photometric can tails. ent comets cometary extra-solar extra-solar that by occultations lar Abstract. Accepted / Received A Boulevard 98bis CNRS, Paris, de d’Astrophysique Institut comets. Etangs des extra-solar Lecavelier A. to due variation stellar Photometric nsc ytmi eysotadoems n a to way a find must one and short dust very the is of system lifetime a the such because the in cometary First, is a reasons. latter be two must the for dust If tail of 1997). cloud of des al. this (Lecavelier front et explanation, cloud Lamers correct dusty in 1997, a passage al. or et planet the Etangs a to either of due star on be the observed variations could the 1981 that November of concluded we study 1995), comets al. our et objects brightness, In variations. only stars’ photometric β on the induce Schnei- effect not also 1990; can their are al. by planets et detectable However, (Schneider as- 1996). years many in der several analyzed for photometric problem through well-known pects a planets is extra-solar variation for search The Introduction 1. edopitrqet to requests offprint Send 70.34 51.;0.31 08.16.2) 07.03.1; 05.15.1; 07(03.13.4; are: codes thesaurus later) Your hand by inserted be (will no. manuscript A&A itrspooercvrain Lcvle e Etangs des (Lecavelier variations photometric Pictoris eas siaetenme fdtcin xetdin expected detections of number the estimate also We epromdnmrclsmltoso stel- of simulations numerical performed We cuttos–Cmt lntr systems Planetary – Comets – Occultations .Lcvle e Etangs des Lecavelier A. : 1 .Vidal-Madjar A. , 1 n .Ferlet R. and , ao -51 ai,France Paris, F-75014 rago, ooy ilb anhdi al 02 twl lo a allow accuracy will photometric It a 10 with 2002. few stars seis- 000 early of 30 stellar in about is of launched aim survey be primary (Baglin future will which example near COROT, mology, very an 1997). the as al. in achievable et mission sur- be space will photometric COROT what a of the from take We activity vey. cometary detecting of of presence plan- the by for caused ets. possibly signature and be indirect perturbations, can gravitational an stars al. around as comets et considered of Vidal-Madjar presence ac- 1995, Moreover, cometary Vidal-Madjar 1998). large & show (Ferlet to tivity expected is system planetary rbto fteds utb sue nodrt eval- extra-solar to to order due in variation assumed dis- stellar geometric be photometric must a the dust al.(1997), uate et the Lamers of by tribution out pointed occultation As cometary of model 5. A Sect 2. de- in is be conclusion could The 4. which 3. Sect comets Sect in in given of are curve number tected light the expected of the Estimates give then and tation, photometric a such survey. using comets detecting of probabilities a omrymc oeiprat n h well-known the activity and cometary of important, that case more known much is It formerly age. was given a at system Rosen- 1994, system 1994). Ninkov solar al. the 1962, et in (Dossin bush tails published cometary of been dust polarization have by and light extinction star of of Combes Observations by (1983). discussed al. been have et comets by occultation lar the in (Lecavelier dust 1996). tail of Etangs, cometary cloud des a huge like exactly a direction: and part opposite the star in the edge sharp toward a have pointing cloud should the it curve; spherical, light be observed cannot the cloud by the constrained is because shape Second, continuously. dust the produce nti ae,w elwt h motn possibility important the with deal we paper, this In nSc ,w ecietemdlo h oeayoccul- cometary the of model the describe we 2, Sect In u h oa ytmi nyoepriua planetary particular one only is system solar the But stel- of observations the system, solar the of case the In β 1 − itrssosta uigtefis 10 first the during that shows Pictoris 4 uigsvrlmnh.Hr,w rdc the predict we Here, months. several during ASTROPHYSICS ASTRONOMY AND 8 er,a years, 2 A. Lecavelier des Etangs et al.: Photometric stellar variation due to extra-solar comets. comets. Then, by taking into account the optical proper- for s = 0.1µm. The ejection velocity is smaller for larger ties of the cometary grains, the photometric variation can grains which have smaller cross section area to mass ratio −1 be estimated. (veject(10µm) 124km s ). We checked that any other realistic values≈ for A and B give similar light curves within 2.1. Distribution of dust in a cometary tail few percents. The evaluation of the distribution of the dust in a 2.2. Stellar parameter cometary tail can be made through particle simulation. The input parameters are the comet orbit, the dust pro- The simulations have been performed with the mass, lu- duction rate, the ejection velocity and the size distribu- minosity and radius of the central star set to solar values tion. We assume a size distribution dn(s) of the form (M∗, L∗ and R∗). Simulation with other parameters could be possible. A larger mass for the central star would induce (1 s /s)m 0 a shorter time scale; a larger luminosity would increase the dn(s)= − n (1) s effect of the radiation pressure on grains; a larger radius as observed in the solar system, where s is the dust size. would decrease the relative extinction (Eq. 6). However, We take s = 0.1µm, n = 4.2, m = n(s s )/s , and the conclusions do not strongly depend on these stellar 0 p − 0 0 sp = 0.5µm (Hanner 1983, Newburn & Spinrad 1985). properties. For instance, adopting the properties of an This distribution starts at s0 = 0.1µm with dn(s0) = 0, A5V star does not change the shape of the light curves. −n peaks at sp = 0.5µm, and has a tail similar to a s By scaling the production rate to the star luminosity, we distribution for large sizes. found a quantitative change by less than a factor of two. Dust sensitivity to radiation pressure is given by the β ratio of the radiation force to the gravitational force. We 2.3. Grain properties take −1 2.3.1. Extinction L∗/M∗ s β =0.2 (2) L /M 1µm The extinction cross-section of a dust grain is 2 ⊙ ⊙ Qext(s, λ)πs , where the extinction efficiency, Qext is where L∗/M∗ is the luminosity-mass ratio of the star. So slightly dependent on the particle size (s) and radiation the orbits of the small grains are more affected by radi- wavelength (λ). If Qsca is the scattering efficiency (see ation pressure than those of large grains. This is a very Sect. 2.3.2) and Qabs is absorption efficiency, Qext(s, λ)= good approximation for particles larger than 0.1µm of any Qsca + Qabs. We take Qabs = 1 if s λ, and Qabs = s/λ realistic composition and for the solar spectrum (Burns et if s < λ, which is a good approximation≥ for optical wave- al. 1979). lengths and grains larger than 0.1µm (Draine & Lee 1984). The dust production rate P is assumed to vary with The total extinction is hence calculated by adding the r, the distance to the star, and is taken to be extinction due to all particles in the line of sight to the − star. The optical depth τ due to the dust is r 2 P = P0 (3) 2 r0 Ngrain/part.Qext(s, λ)πs τ = part. (4) P (see, for example, A’Hearn et al., 1995; Weaver et al., 1997, S Schleicher 1998). The dust production is taken to be zero where S is the projected area of the line of sight. In the beyond 3 AU for a solar luminosity. simulation, one particle represents an optically thin cloud As soon as a grain is produced from a comet nucleus, of several dust grains. The number of physical grains per we assume that it is ejected from the parent body with particle in the simulation, Ngrain/part., is a velocity veject in an arbitrary direction. The grain then follows a trajectory defined by gravitation and radiation 3Mdust dn(s) Ngrain/part. = 3 (5) forces. The ejection velocity depends upon the particle 4πρNpart. Rs dn(s) size. We take veject = √β/(A + B√β) (Sekanina & Lar- R son 1984), which approximates the results of Probstein’s where Mdust is the total dust mass, ρ is the dust density, (1969) two-phase dusty-gas dynamics for the acceleration and Npart. is the number of particles in the simulation. 4 by the expanding gas within tens of kilometers from the Npart. is set to a few 10 in order to keep reasonable com- nucleus. The coefficient A and B depend upon many pa- puting time. rameters such as the thermal velocity of the expanding Because the cometary cloud is optically thick but its gas. We used A = B = 1 s km−1 which is a good ap- size is smaller than the size of the star, we mapped the proximation of different values measured for the comets stellar surface through a set of cells in polar coordinates. of the solar system (Sekanina & Larson 1984; Sekanina For each cell i, we calculate the optical depth τi due to the −1 2 1998). This gives an ejection velocity of 585km s particles within this cell of area S ( S = πR∗). The ≈ i i i P A. Lecavelier des Etangs et al.: Photometric stellar variation due to extra-solar comets. 3 ratio of the flux observed through the cloud (Fext) to the In fact, for very peaked forward-scattering function on initial stellar flux (F∗) is very small scattering angle, the finite size of the star must be taken into account (especially when the dust cloud is −(τi) Fext Sie seen superimposed on the star surface). Hence, we mapped = 2 (6) F∗ πR∗ the stellar surface by small arcs centered on the dust par- Xi ticle. We then calculate the total scattering by adding the The number of cells is the best compromise between contribution of each arc. the spatial resolution and the number of particles in the simulation. We take into account that the maximum con- −1 3. Light curves tribution to τi by each particle must be < 10 in order to achieve an accurate result in spite of the quantization The light observed at a given time is the sum of two op- of the extinction. Si and Npart are thus constrained by posite effects: the increase of the brightness through the scattering by particles at small angle from the line of sight S N 3M Q s2dn(s) i part dust ext (7) and the decrease of the brightness through the extinction 10 ≥ 4ρ R s3dn(s) (including absorption) by particles on the line of sight. R The limb darkening is not taken into account in this work, because its effect is negligible. 3.1. First order estimations 3.1.1. Extinction 2.3.2. Scattering From Eqs. 4 and 5, in the case s > λ (Qext = 2), and As already pointed out by Lamers et al. (1997), the main neglecting the inner extinction, we find∼ an upper limit for part of an occulting dust cloud is seen through a very the extinction: small scattering angle. Thus, the total star light forward- F 6M s2dn(s) scattered to the observer can be large because the phase ext = exp dust (10) ∗ 2 R 3 function is strongly peaked to small angles for which the F max − 4πρR∗ s dn(s) diffraction has the dominant contribution. The phase func- R 1 For the particle size distribution given by Eq. 1 tion for the diffraction is (Sect. 2.1), we have ( s2dn(s))/( s3dn(s)) = 0.15µ−1. − 2 2 2 Adopting a dust production of 106 kg s 1 during 100 days, x˜ 1+cos θ J1(˜x sin θ) R R P (θ, x˜)= (8) 13 −3 4π x˜ sin θ we find Mdust 10 kg. Hence (∆F/F∗)max 10 . This order of magnitude≈ is close to the result of more≈ elaborated wherex ˜ =2πs/λ. The phase function has been normalized calculations (Sect. 3.2). We can already conclude that this by P (θ)dΩ = 1 and depends upon the wavelength λ. variation could be detected by a space photometric survey. ExamplesR of such phase functions are shown in Lamers et We note that the photometric variations of β Pic in − al. (1997). 1981 was 5 10 2. The above calculation gives an es- ∼ · The scattered light (Fsca) is evaluated by adding the timate of the lower limit of the needed production rate : − − contribution of each particle in the simulation. P > 5 107 kg s 1. This is consistent with P > 108 kg s 1 obtained∼ · from the estimation of Lamers et∼ al. (1997): 2 8 −1 Fscat Qscaπs P (θ) −(τin+τout) P/∆v > 10 kg km and with an assumed relative ve- = Ngrain/part. e . (9) F∗ r2 partX. locity between the dust and the nucleus of the comet ∆v < 1km s−1. τin is the total extinction along the path from the star to ∼ the scattering grain and τ is from the grain to the ob- out 3.1.2. Scattering server. The grain is at distance r from the star. For s λ, ≥ the scattering efficiency Qsca is assumed to be the diffrac- If we take a comet at r = 1 AU from the star and an tion efficiency for large grains: Qsca = QD = 1 (Pollack & impact parameter on the line of sight of one half the stellar Cuzzi, 1980). For small particles (s < λ), we used the basic radius, then θ 7′. For the particle size distribution given 4 ≈ ′ approximation Qsca = (s/λ) (van de Hulst, 1957). The by Eq. 1, the phase function is P˜(θ 7 ) 700. Therefore, result is very insensitive to this last assumption because ≈ ≈ 2 F 3M P˜(θ) s dn(s) − forward scattering at small angle is largely dominated by scat dust 4 10 5 (11) diffraction on particles larger than the wavelength. Be- F∗ ≈ 4ρr2 R s3dn(s) ≈ · cause each cloud represented by a particle is very thin, R As a first conclusion, it is clear that both extinction and the total extinction is small (τ , τ < 10−1), we in out and scattered light can be detected by very precise photo- assumed single scattering (no source function). metric survey, although scattering gives photometric vari- 1 The Eq. 12 in Lamers et al. (1997) has a typo-mistake:x ˜ ations an order of magnitude smaller than extinction. Ex- 2 in the first fraction must bex ˜ . tinction appears to be the major process observed when 4 A. Lecavelier des Etangs et al.: Photometric stellar variation due to extra-solar comets. the comet is passing in front of the star. This event can thus be called an occultation.

3.2. Light curves Taking into account the dust spatial distribution and the extinction within the tail, we calculated the light curves of a set of cometary occultations. From a given comet orbit, and dust production rate, we compute the full motion of each dust particle; the result is the variation of the star light as a function of time. Two typical light curves resulting from the simulation are shown in Fig. 1 and 2. The majority of cometary occultations gives light curves with a very particular “rounded triangular” shape (Fig. 1). This occurs when the dense cometary head first occults the star and gives a very fast and sharp brightness decrease. It is then followed by the tail which gives an additional slow decrease. Similarly, the subsequent brightness increase is also sharp when the cometary head is going out of the occulting part. Then the increase slows down and the brightness returns to the normal level when the cometary tail is less and less dense Fig. 1. Plot of the photometric variations during a in front of the star. cometary occultation in red (λ 8000 A,˚ thick solid line) However, in some configurations, the tail can be and in blue (λ 4000 A,˚ thick∼ short-dashed line). The aligned with the line of sight. In these cases the light curves insert is a view∼ from the top when the comet is crossing are more symmetric (Fig. 2). They can mimic planetary the line of sight (Y = 0) at the periastron. The produc- occultations (Fig. 3). Because of the noise, it will be dif- tion rate is 2 106 kg s−1 at 1 AU. The scattered light is ficult from such observations to differentiate between a given by the top· thin lines, the extinction by the bottom comet and a planet. thin lines (long-dashed line for λ 8000 A;˚ dotted line for λ 4000 A).˚ The total variation∼ is plotted with the thick ∼ 3.3. Color signatures lines. This light curve presents the very specific ”rounded triangular” shape observed in the majority of simulations As seen in Sect. 2.3, the light variations show some color of cometary occultations. The difference between the vari- signature due to the optical properties of the grains. Par- ation in the blue and in the red is negligible and less than ticles with a size smaller than the wavelength are less ef- 1%. ficient for extinction. Thus, extinction is smaller at larger wavelengths. The forward scattering is more peaked to the small angles at shorter wavelengths (Lamers et al. 1997). be very difficult to observe. This is beyond the today tech- The scattering is thus larger at smaller wavelengths when nical feasibility. the cometary cloud is occulting the star (a contrario, the However, for comets at small distances from the Sun scattering is larger at larger wavelength when the comet (< 0.5 AU), its appears that the dust size distribution is cloud is seen far from the star; but then the scattered peaked∼ at smaller sizes (Newburn & Spinrad 1985). This light becomes negligible). As a result, there is a balance decrease of ap with the heliocentric distance could be due between the additional scattered light and the extinction; to particle fragmentation. Although it is difficult to guess both are larger at smaller wavelengths. It is difficult to the size distribution for extra-solar comets, it is easy to un- predict what will be the color effect on cometary occulta- derstand that if small particles are more numerous (with tion light curves. s < 0.5µm), the occultation will show a larger color sig- In addition, most of the extinction and scattering due nature. We checked that a comet with a periastron at to the cometary dust is concentrated in the inner coma, 0.3 AU and sp =0.25µm (Newburn & Spinrad 1985) gives where the number of large particles is larger because larger extinction by about 15% in the blue than in the red smaller particles are more efficiently ejected by radiation (Fig. 4). pressure (Sect. 2.1). As a consequence, in the optical, and with the size distribution assumed in Sect 2.1, the light 3.4. Conclusion curves are barely dependent on the wavelength (or color band). The difference between variations in blue and in red Photometric variations due to the stars’ occultation by is smaller than few percents. This color signature would extra-solar comets could be detectable by photometric A. Lecavelier des Etangs et al.: Photometric stellar variation due to extra-solar comets. 5

Fig. 2. Same as Fig.1. Here all the parameters are the Fig. 3. Plot of the photometric variations due to a comet same except that the longitude of the periastron is at similar to the comet of the Fig.2 (histogram). Here the 90◦ from the line of sight. This gives a production rate of result is given as a noisy observation with a photometric 5 105 kg s−1 at 2 AU. The cometary tail is aligned with measurement every hour and an accuracy of 3σ = 10−4. the· line of sight; consequently the light curve presents a For comparison, an occultation by a planet with a radius more symmetrical shape resembling a planetary occulta- of 9 000 km and orbiting at 2 AU is plotted with limb tion. The net color effect is hardly detectable and of the darkening effect taken into account (thick line). The de- order of 1.5%. tected variation is very similar to a planetary occultation, the main difference being extended wings in the cometary case. measurements with an accuracy of 10−3 – 10−4. In most cases, the particular ”rounded triangular”∼ shape of stars at a given accuracy. We carried out a large number the light curve can be an easy diagnostic for the presence of simulations of cometary occultations in all directions. of a comet. However, in some cases, the light curves can mimic an occultation by a more compact object like a clas- sical extra-solar planet (Lecavelier des Etangs et al. 1997). 4.1. Orbital parameters The color signatures of cometary occultations can be too The characteristics of the comets are given by their dis- small to avoid the confusion. The detection of periodicity tribution in the parameter space. Concerning the orbital appears to be critical in the diagnostic of a planetary oc- parameters, the periastron density distribution is chosen cultation. Alternatively, the polarization is another way to to be the same as the one observed in the solar system: discriminate the two phenomena, because the light is scat- dn(q) q0.3dq with q [0.1 AU, 2 AU] (A’Hearn et al., tered by the cometary dust roughly gathered in the same 1995).∝ Many sun-grazing∈ comets have been discovered by plane. But, with a level of at most few percents in polar- SOHO with perihelion q < 0.1 AU (Kohl et al. 1997; see ization of solely the scattered light, this gives < 0.01 % also the numerous IAU Circulars). But it is still difficult polarization in total and would also be difficult to detect. to infer the distribution for these small periastrons. The In cases of planetary and cometary occultations, spectro- above distribution is thus likely biased by an underes- scopic follow-up observations should be planned to allow timate of the number of comets with small periastron, a better analysis of a suspected on-going detection. because such comets are difficult to observe. These star- grazing comets give a larger photometric variation. There- 4. Probability of detection fore, we possibly slightly underestimate the probabilities of detection. From the above modeling, we can evaluate the probability We fixed all the apoastrons at 20 AU, the longitudes to detect a photometric stellar variation due to an extra- of periastron, the ascending nodes and the inclinations are solar comet. We assume a survey of a given number of chosen randomly. 6 A. Lecavelier des Etangs et al.: Photometric stellar variation due to extra-solar comets.

Fig. 4. Same as Fig.1. Here all the parameters are the Fig. 5. Plot of the number of detections of cometary oc- same except that the periastron is at 0.3 AU and the par- cultations as a function of the photometric accuracy. Two ticle size distribution is taken with ap = 0.25µm. The cases are considered: the “pessimistic case” which is a sur- 7 −1 production rate corresponds to 2 10 kg s at 0.3 AU. vey of nstar = 30000 sun-like stars, during Tobs = 1 year · 3 −1 Because there are more particles smaller than the wave- (ncomet = 100, P0 = 10 kg s ), and the “optimistic length, we see a dipper occultation in the blue by about case” where among the 30000 stars, 1% (nstar = 300) have 6 −1 15%. a β Pictoris-like activity (ncomet = 100, P0 = 10 kg s ). With an accuracy of 10−4, about 10 to 103 detections can be expected. The large diﬀerence between the two cases 4.2. Dust production rate shows that this kind of survey will also give information on the planetary evolution. One of the most important parameter is the dust produc- For comparison, the dotted histogram gives the number of tion rate. Because, it dictates the mass of dust in the tail, detection of planets assuming that each star has a plane- and constrains the detectability of the comet. The ampli- tary system like the solar system. With an accuracy larger tude of the photometric variation is roughly proportional than 10−4, this gives the number of detection of giant to that parameter. planets, and mainly Jupiter-like planets. With an accu- We consider that the dust production rate (P ) is pro- racy better than 10−4, Earth-like planets will be detected. portional to the area of the comet’s nucleus, and that Each step in the histogram represents the possibility to the distribution of the comets’ radii (Rc) is similar to detect successively the Earth (8 10−5), Venus (7 10−5), the distribution observed for the comets, asteroids, and Mars (2 10−5) and Mercury (10· −5). We see that· accu- Kuiper belt objects of the solar system. We assume that rate photometric· survey should detect more comets than the number density of objects with radius in the range Rc planets. −γ to Rc + dRc is dn(Rc) Rc dRc. γ is a positive number, typically in the range 3∝ to 4 (Luu 1995). It is constrained to be 3 – 3.5 for the comets’ nuclei observed with Finally, this distribution is normalized by ncomet, the R [0.∼1km, 100km] (Fern´andez 1982, Hughes & Daniels number of comets per unit of time passing through the ∈ periastron with a production rate larger than P . Hence, 1982, Brandt et al. 1997) The same distribution is consis- ∞ 0 tent with the observation of the Kuiper belt objects: γ 3 we have ncomet dn(P ). Thus, ∝ P0 with R [100 km, 400 km] (Jewitt 1996), or with theoret-∼ R ∈ (1−γ)/2 ical models for the formation of these objects: γ 4 for ncomet P0 . (12) R < 200 km (Kenyon & Luu, 1998). With the assumption∼ ∝ ∼ 2 that P Rc , the number of comets with a production 4.3. Results rate between∝ P and P + dP is dn(P ) P −(γ+1)/2dP . We adopt γ = 3.5. Changing γ to 3 or 4∝ would change the Using a large number of various comets, we calculate the probability of detection by less than a factor of two. probability of detection at a given photometric accuracy. A. Lecavelier des Etangs et al.: Photometric stellar variation due to extra-solar comets. 7

Then, the number of possible detections is simply this tions is smaller than the one from Jupiter. With a total probability multiplied by n , the duration of the ob- probability of 10−4 to detect a giant planet in one year, comet ∼ servation Tobs and the number of stars surveyed nstar. the survey of 30000 stars will give 3 planets. With an − ∼ We suppose that the time scale between each measure- accuracy better than 10 4, Earth-like planets start to be ment is small enough (< 1 hour) that each variation above detectable. Because of their smaller distance to the star, the detection limit will∼ effectively be detected. As an ex- they have larger contribution to the probability of planet ample, we take Tobs = 1 year and nstar = 30000, which detection. The Earth and Venus can be detected with a is the order of magnitude for the future space mission probability of 10−2 in one year. Mars and Mercury, ∼ − COROT. which are detectable with an accuracy of 10 5, give a − ∼ The expected number of detections is plotted in Fig. 5. total probability of 3 10 2. The comparison with the ∼ · We considered two types of planetary systems. The first number of detection of comets shows that accurate pho- one is similar to the solar system with ncomet = 100 comets tometric surveys should detect more comets than planets. per year with P P = 103 kg s−1 at r = 1 AU. We see ≥ 0 0 that few dozens of comets could be detected at an accuracy 5. Conclusion of 10−4. We consider it as the pessimistic case. The second type of planetary system is supposed to be We performed detailed numerical simulations of stars’ oc- a young planetary system with a large cometary activity as cultations by extra-solar comets. We extracted the appar- during the youth of the solar system. The typical example ent photometric variations of the central stars due to these is the well-known star β Pictoris where comets’ infalls are putative comets. We have shown that: commonly observed through UV and optical spectroscopy 1) Extra-solar comets can be detected through pho- (see e.g., Ferlet et al. 1987, Lagrange et al. 1988, Beust tometric variations due to occultation by dusty tails. et al. 1990, Vidal-Madjar et al. 1994, 1998). For such a In many cases, the light curve shows a very particular planetary system, ncomet = 100 comets per year with P “rounded triangular” shape. However, in some remaining 6 −1 ≥ P0 = 10 kg s at r0 = 1 AU (Beust 1995, Beust et cases, the curve can mimic a planetary occultation. al. 1996). Note that β Pictoris is young but on the main 2) The photometric variations due to cometary occul- sequence (Crifo et al. 1997), its age is few percents of the tations are mainly achromatic. This property will allow age of the solar system. Therefore, in a set of 30000 stars to discriminate the occultations by comets from intrinsic there should be about nstar 300 stars with about the stellar variations. However, the confusion with planetary same activity as β Pictoris. Thus,≈ few thousands comets occultations cannot be efficiently removed by color mea- could be expected with a survey at 10−4 accuracy. We surements in the optical. consider it as the optimistic case. 3) The number of detections which can be expected The solar system is likely not exceptional. The bottom- from a large photometric survey of several tens of thou- − line in Fig. 5 is a good estimate of the lower limit of the sand of stars at high accuracy (10 4) is of the order of expected number of detections. Younger stars may have a several hundreds of occultations per year. higher level of activity, with a larger number of comets; These detections will allow to explore the evolution but, although infrared excess have been observed around of the cometary activity through the correlation with the many main sequence stars, β Pictoris is certainly a very stellar age. Indeed, we know from the solar system ex- peculiar case (Vidal-Madjar et al. 1998). Thus, the top- ploration that the cometary activity significantly changed line in Fig. 5 gives a good estimate of the upper limit of with time. Moreover, comets are believed to be the primi- the expected number of detections. tive bricks of the planetary formation, and planets per- Note that the major assumption that the dust pro- turbation are needed to push them toward the central duction rates in the solar system can be extrapolated to star. It is thus clear that the detection and analysis of large values is realistic. A large rate has effectively been the cometary activity around nearby stars will give im- observed in the recent comet Hale-Bopp where it reached portant information on the structure and evolution of the of few times 105 kg s−1 at about 1 AU (Rauer H. et al., planetary systems. 1997, Schleicher et al., 1997, Senay et al. 1997, Weaver et al., 1997). Acknowledgements. We would like to express our gratitude to Alain Leger for fruitful discussions. We also thank the referee It is very likely that in the near future a large number Henny Lamers for his useful comments which improved the of extra-solar comets will be detected through occulta- paper. tions. We warmly thank Remi Cabanac for his critical reading of the For comparison, we also evaluate the probability to de- paper. tect planets assuming that each star has a planetary system like the solar system. With an accuracy better than 10−2, Jupiter can be detected. Below 10−3, other giant References planets are also detectable. But because of their large or- A’Hearn M.F., Millis R.L., Schleicher D.G., Osip D.J., Birch bital periods, their contribution to the number of detec- R.V., 1995, Icarus 118, 223 8 A. Lecavelier des Etangs et al.: Photometric stellar variation due to extra-solar comets.

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