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Astron. Astrophys. 343, 916-922 (1999)

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4. Probability of detection

From the above modeling, we can evaluate the probability to detect a photometric stellar variation due to an extra-solar comet. We assume a survey of a given number of stars at a given accuracy. We carried out a large number of simulations of cometary occultations in all directions.

4.1. Orbital parameters

The characteristics of the comets are given by their distribution in the parameter space. Concerning the orbital parameters, the periastron density distribution is chosen to be the same as the one observed in the solar system: [FORMULA] with [FORMULA]AU[FORMULA]AU[FORMULA] (A'Hearn et al., 1995). Many sun-grazing comets have been discovered by SOHO with perihelion [FORMULA] AU (Kohl et al. 1997; see also the numerous IAU Circulars). But it is still difficult to infer the distribution for these small periastrons. The above distribution is thus likely biased by an underestimate of the number of comets with small periastron, because such comets are difficult to observe. These star-grazing comets give a larger photometric variation. Therefore, we possibly slightly underestimate the probabilities of detection.

We fixed all the apoastrons at 20 AU, the longitudes of periastron, the ascending nodes and the inclinations are chosen randomly.

4.2. Dust production rate

One of the most important parameter is the dust production rate. Because, it dictates the mass of dust in the tail, and constrains the detectability of the comet. The amplitude of the photometric variation is roughly proportional to that parameter.

We consider that the dust production rate (P) is proportional to the area of the comet's nucleus, and that the distribution of the comets' radii ([FORMULA]) is similar to the distribution observed for the comets, asteroids, and Kuiper belt objects of the solar system. We assume that the number density of objects with radius in the range [FORMULA] to [FORMULA] is [FORMULA]. [FORMULA] is a positive number, typically in the range 3 to 4 (Luu 1995). It is constrained to be [FORMULA] 3-3.5 for the comets' nuclei observed with [FORMULA] km[FORMULA] km[FORMULA] (Fernández 1982, Hughes & Daniels 1982, Brandt et al. 1997) The same distribution is consistent with the observation of the Kuiper belt objects: [FORMULA] with [FORMULA] km[FORMULA] km[FORMULA] (Jewitt 1996), or with theoretical models for the formation of these objects: [FORMULA] for [FORMULA] km (Kenyon & Luu, 1998). With the assumption that [FORMULA], the number of comets with a production rate between P and [FORMULA] is [FORMULA]. We adopt [FORMULA]. Changing [FORMULA] to 3 or 4 would change the probability of detection by less than a factor of two.

Finally, this distribution is normalized by [FORMULA], the number of comets per unit of time passing through the periastron with a production rate larger than [FORMULA]. Hence, we have [FORMULA]. Thus,

[EQUATION]

4.3. Results

Using a large number of various comets, we calculate the probability of detection at a given photometric accuracy. Then, the number of possible detections is simply this probability multiplied by [FORMULA], the duration of the observation [FORMULA] and the number of stars surveyed [FORMULA].

We suppose that the time scale between each measurement is small enough ([FORMULA] hour) that each variation above the detection limit will effectively be detected. As an example, we take [FORMULA] year and [FORMULA], which is the order of magnitude for the future space mission COROT.

The expected number of detections is plotted in Fig. 5. We considered two types of planetary systems. The first one is similar to the solar system with [FORMULA] comets per year with [FORMULA] kg s-1 at [FORMULA] AU. We see that few dozens of comets could be detected at an accuracy of [FORMULA]. We consider it as the pessimistic case.

[FIGURE] Fig. 5. Plot of the number of detections of cometary occultations as a function of the photometric accuracy. Two cases are considered: the "pessimistic case" which is a survey of [FORMULA] sun-like stars, during [FORMULA] year ([FORMULA], [FORMULA] kg s-1), and the "optimistic case" where among the 30000 stars, 1% ([FORMULA]) have a [FORMULA] Pictoris-like activity ([FORMULA], [FORMULA] kg s-1). With an accuracy of [FORMULA], about 10 to 103 detections can be expected. The large difference between the two cases shows that this kind of survey will also give information on the planetary evolution. For comparison, the dotted histogram gives the number of detection of planets assuming that each star has a planetary system like the solar system. With an accuracy larger than [FORMULA], this gives the number of detection of giant planets, and mainly Jupiter-like planets. With an accuracy better than [FORMULA], Earth-like planets will be detected. Each step in the histogram represents the possibility to detect successively the Earth ([FORMULA]), Venus ([FORMULA]), Mars ([FORMULA]) and Mercury ([FORMULA]). We see that accurate photometric survey should detect more comets than planets.

The second type of planetary system is supposed to be a young planetary system with a large cometary activity as during the youth of the solar system. The typical example is the well-known star [FORMULA] Pictoris where comets' infalls are commonly observed through UV and optical spectroscopy (see e.g., Ferlet et al. 1987, Lagrange et al. 1988, Beust et al. 1990, Vidal-Madjar et al. 1994, 1998). For such a planetary system, [FORMULA] comets per year with [FORMULA] kg s-1 at [FORMULA] AU (Beust 1995, Beust et al. 1996). Note that [FORMULA] Pictoris is young but on the main sequence (Crifo et al. 1997), its age is few percents of the age of the solar system. Therefore, in a set of 30 000 stars there should be about [FORMULA] stars with about the same activity as [FORMULA] Pictoris. Thus, few thousands comets could be expected with a survey at [FORMULA] accuracy. We consider it as the optimistic case.

The solar system is likely not exceptional. The bottom-line in Fig. 5 is a good estimate of the lower limit of the expected number of detections. Younger stars may have a higher level of activity, with a larger number of comets; but, although infrared excess have been observed around many main sequence stars, [FORMULA] Pictoris is certainly a very peculiar case (Vidal-Madjar et al. 1998). Thus, the top-line in Fig. 5 gives a good estimate of the upper limit of the expected number of detections.

Note that the major assumption that the dust production rates in the solar system can be extrapolated to large values is realistic. A large rate has effectively been observed in the recent comet Hale-Bopp where it reached of few times [FORMULA] kg s-1 at about 1 AU (Rauer H. et al., 1997, Schleicher et al., 1997, Senay et al. 1997, Weaver et al., 1997).

It is very likely that in the near future a large number of extra-solar comets will be detected through occultations.

For comparison, we also evaluate the probability to detect planets assuming that each star has a planetary system like the solar system. With an accuracy better than [FORMULA], Jupiter can be detected. Below [FORMULA], other giant planets are also detectable. But because of their large orbital periods, their contribution to the number of detections is smaller than the one from Jupiter. With a total probability of [FORMULA] to detect a giant planet in one year, the survey of 30000 stars will give [FORMULA] planets. With an accuracy better than [FORMULA], Earth-like planets start to be detectable. Because of their smaller distance to the star, they have larger contribution to the probability of planet detection. The Earth and Venus can be detected with a probability of [FORMULA] in one year. Mars and Mercury, which are detectable with an accuracy of [FORMULA], give a total probability of [FORMULA]. The comparison with the number of detection of comets shows that accurate photometric surveys should detect more comets than planets.

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© European Southern Observatory (ESO) 1999

Online publication: March 1, 1999
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