4 How to explain it?

The problem concerning the significance of the high-metallicity trend observed for stars with planets being "solved'', our enquiry is turned to the cause of these differences and to the implications this result might have for planetary system formation and evolution scenarios.

4.1 Comparing the distributions

One very interesting feature of the distribution of stars with planets is well illustrated in Fig. 2. The shape of this distribution is well described by a quadratic rising with [Fe/H] (up to [Fe/H] $\sim$ +0.35 - dashed curve in the diagram). In other words, the distribution is rising with Z (the mass fraction of heavy elements) up to a value of 0.04, falling then abruptly. This sharp cut-off in the distribution of stars with planets is probably due to the fact that we may be looking at a limit on the metallicity of the stars in the solar neighbourhood.

Clearly, the rise is expected if we consider that the probability of forming planets is proportional to the metallicity content. In one sense, an increase in the number of dust particles (N) in the young disc will increase the probability of a shock by N². The rate of formation of planetesimals, and thus the probability of creating a planet core with sufficient mass to accrete gas from the disc before it disappears, will increase quadratically with Z (considering that there is a linear relation between these two variables).

But this model is probably too simple. Other, more complicated mechanisms intervene, however, such as the fact that an increase in [Fe/H] will also probably increase the opacity of the disc, changing its physical conditions (e.g. Schmidt et al. 1997), such as the accretion rate or its vertical structure. The current result may be used in exactly this way in constraining these mechanisms.

On the other hand, the distribution of [Fe/H] in a volume-limited sample of stars (with and without planet hosts) apparently decreases with increasing [Fe/H] for [Fe/H] > 0.0 (e.g. Favata et al. 1997). Taking this effect into account would result in an even steeper rise of the relative frequency of stars with planets with the metallicity.

This is exactly what we can see from Fig. 3. In panels (a) and (c) we present a comparison between the [Fe/H] distribution of stars with planets (shaded bars) and the same distribution for two volume-limited samples of dwarfs (empty bars). The volume-limited distribution in panel (a) was taken from this paper (objects presented in Table 1 plus the five stars with planets making part of our volume limited sample - see Sect. 2.1 for details). For panel (c), the distribution represents the metallicities of about 1000 stars in the CORALIE planet-search sample as determined from a calibration of the CORALIE cross-correlation function surface (Santos et al., in preparation). Since these two field star distributions represent the actual distribution of metallicities in the solar neighborhood (within statistical errors), we can use them to estimate the "real'' percentage of stars with planets for each metallicity bin that we would find if the metallicity distribution of field stars was "flat'' - panels (b) and (d). As we can see from these plots, there is a steep rise of the percentage of stars with planets with [Fe/H] (we will concentrate on panel (d) given that it presents better statistics than panel (b)). Actually, the plot shows that more than 75% of the stars with planets would have ${\rm [Fe/H]}>+0.2$ dex. But even more striking is the fact that $\sim$85% of the surface of the distribution falls in the region of ${\rm [Fe/H]}>0.0$. Although this is a preliminary result, the plots leave no doubts about the strong significance of the increase in the frequency of known planetary systems found with the metallicity content of their host stars.

We note that for the low metallicity tail of the distribution, this result is based on relatively poor statistics. The small relative number of stars with planets found with low [Fe/H] values is, however, perfectely consistent with recent studies that failed to find any planetary host companions amid the stars in the metal-poor globular cluster 47 Tuc (Gilliland et al. 2000).

4.2 A simple "pollution'' model

We tried to determine whether it would be possible to obtain a metallicity distribution for the stars without planets similar to that of stars with extrasolar planets by simply adding material to their convective envelopes. To make this simple model, we computed the convective zone mass for each star using Eqs. (5) and (6) of Murray et al. (2001) - metallicity-dependent relations for M< 1.2 $M_{\odot}$; only stars in this mass range were used, since the comparison sample is not complete for higher masses. We are supposing that the "pollution'', if significant, has taken place after the star reaches the ZAMS. As discussed in Paper I, if we consider the pollution to occur in a pre-main-sequence phase, higher quantities of material would be needed, since the convection envelope masses are bigger (D'Antona & Mazzitelli 1994). In fact, recent results show that gas-disc lifetimes can be higher than previously thought (up to 20 Myr - Thi et al. 2001), and thus we can probably expect significant pollution when the star is already on the main sequence.

Two models were then constructed: the first was built on the supposition that the quantity of material falling into the star is the same for all the objects. A second, and possibly more realistic model, considered the pollution to be proportional to Z. The quantity of "added material'' was chosen so that the distributions (stars with and without planets) peak at about the same value of [Fe/H].

The results are shown in Fig. 4 (open bars), compared with the results obtained for stars with planets within the same mass regime. As we can see, the shape of the planet sample, particularly the sharp cutoff at high metallicity, is not correctly obtained for any of the models. The resulting samples are more symmetric, but most of all, in both models there are always a few points that end up with extremely high metallicities (>0.8 dex). This is particularly unpleasant, since the inclusion of higher-mass stars (such as some of the observed exoplanet hosts) would even boost some bins to higher values, given that the mass of the convection zone is supposed to be lower. As discussed by Murray et al. (2001), however, Li and Be observations suggest that the total "mixing'' mass increases after 1.2 $M_{\odot}$, eventually reducing this problem. Note also that the strong cut-off in [Fe/H] for the planet host star sample remains, even though this time we have not included stars with M > 1.2 $M_{\odot}$. This shape is not at all reproduced by our computations.

We caution, however, that we are using very simple models to make this calculation. The inclusion of more parameters, such as a time- or disc-mass-dependent accretion (which might be related to the stellar mass), or a mass-dependent size of the central disc hole (e.g. Udry et al. 2001), might modify the results. In any case, we have shown that a simple "pollution'' model cannot explain the observations.

Figure 5: A plot of [Fe/H] as a function of mass for planet host stars (filled circles) and "comparison'' stars (open squares). The line represents the upper limit for the field star [Fe/H]. See text for more details.

4.3 The mass dependence

It has been suggested (Laughlin & Adams 1997) that if the pollution scenario is correct one would be able to see a trend of [Fe/H] with stellar mass, since the high-mass stars have tiny convective envelopes, and thus a higher probability of having a noticeable increase in [Fe/H] (here we have chosen stellar mass and not the mass of the convective shell, since if the pollution scenario is correct, the original mass for the convection zone for the stars with planets would not be correctly calculated). In Fig. 5 we present such a plot.

There are three features in the plot that deserve a comment. First, the region for $M\ge$ 1.2 $M_{\odot}$ has a very low number of stars without planets. This is because in a volume-limited sample of stars, there are very few dwarfs in this mass regime (the IMF decreases with mass). However, these stars are bright, and thus easier targets for planet searches, which explains the high number of planetary candidates in this region. On the other hand, for masses $M\le$ 0.8 $M_{\odot}$, we see the opposite effect: stars are fainter, and thus difficult targets for planet searches, but are more numerous in a volume-limited sample. There are probably, however, two more biases in the diagram. In fact, the comparison sample was cut in (B-V). For a given temperature, a higher-metallicity star also has higher mass. Thus, stars in the upper part of the diagram (more metal-rich) are moved to the righthand side relative to stars in the lower part. For a fixed $T_{\rm eff}$, 0.2 dex in metallicity imply $\sim$0.05 $M_{\odot}$. On the other hand, if we take a given volume-limited sample of dwarfs, lower-mass dwarfs tend to be older than high-mass dwarfs; in proportion there are thus fewer metal-rich dwarfs in the "high''-mass region of the plot. These facts might explain the lack of stars with planetary companions presenting $M\ge$ 1.2 $M_{\odot}$ and low [Fe/H]. In any case, the most striking thing in the diagram is the fact that all comparison stars have [Fe/H] $\le$ 0.17 dex, while about 55% of stars with planets have metallicity above that value. Furthermore, contrary to former studies (Laughlin 2000) this diagram does not suggest that the metallicity of the upper envelope of stars with planets is increasing with stellar mass. This point renders unlikely the possibility that the high metallicity of these stars is due to pollution. As discussed above, however, for stellar masses higher than $\sim$1.2 $M_{\odot}$, the total "mixing'' mass may increase. This may explain to some extent the lower value of the upper envelope of [Fe/H] for masses above this limit, but it is not certain that it is enough to explain the difference. An extension of our comparison sample for stars with M> 1.2 $M_{\odot}$ would be very useful.

As mentioned, our results and interpretations are opposed to those presented by Laughlin (2000), who used photometric indices to compute the metallicities for a sample of stars with planets, and for a large volume-limited sample of dwarfs used as "comparison''. We believe that there are two important reasons for the difference. First, the trend he finds in [Fe/H] vs. stellar mass is mainly due to the few "low''-[Fe/H] planet host stars having $M< 1~M_{\odot}$. On the other hand, in his plot there are a few very high metallicity "comparison'' stars in the region of $M< 1.1~M_{\odot}$. If we believe in the results presented in this paper, the probability that these stars harbour planets is very high. His sample might in fact be biased by the presence of planet host stars in his "comparison'' sample.

 

 

Table 4: Planetary and orbital parameters used in Figs. 6 and 7. Stars with multiple companions are not included. The values were compiled from the literature.

HD	Star	$M~\sin{i}$	Period	a	e
number		( $M_{\rm Jup}$)	(days)	(AU)
1237	GJ 3021	3.51	133.7	0.494	0.51
6434	HD 6434	0.48	22.1	0.15	0.295
10697	109 Psc	6.60	1072	2.12	0.12
12661	HD 12661	2.93	264.0	0.789	0.33
13445	Gl 86	3.77	15.8	0.11	0.04
16141	HD 16141	0.23	75.8	0.35	0.28
17051	$\iota$ Hor	2.36	311.3	0.93	0.22
19994	HD 19994	1.88	454.2	1.23	0.2
22049	$\epsilon$ Eri	0.84	2518	3.4	0.6
28185	HD 28185	5.6	385	1.0	0.06
37124	HD 37124	1.04	154.8	0.55	0.31
38529	HD 38529	0.81	14.3	0.129	0.27
46375	HD 46375	0.26	3.02	0.041	0.02
52265	HD 52265	1.08	119.2	0.5	0.35
75289	HD 75289	0.44	3.48	0.04	0.065
75732A	55 Cnc	0.82	14.66	0.12	0.03
80606	HD 80606	3.51	111.8	0.44	0.93
89744	HD 89744	7.54	265.0	0.91	0.7
92788	HD 92788	3.77	340.8	0.97	0.36
95128	47 UMa	2.57	1084	2.1	0.13
108147	HD 108147	0.35	11.05	0.10	0.57
117176	70 Vir	7.75	116.7	0.48	0.4
120136	$\tau$ Boo	4.29	3.31	0.047	0.02
121504	HD 121504	0.89	64.62	0.32	0.13
130322	HD 130322	1.04	10.72	0.088	0.044
134987	23 Lib	1.57	259.6	0.81	0.24
143761	$\rho$ CrB	1.15	39.6	0.23	0.07
145675	14 Her	5.65	1650	2.84	0.37
168746	HD 168746	0.25	6.41	0.066	0.00
169830	HD 169830	3.04	229.9	0.82	0.35
177830	HD 177830	1.26	391.6	1.1	0.41
186427	16 Cyg B	1.75	804.4	1.61	0.67
187123	HD 187123	0.60	3.10	0.042	0.01
190228	HD 190228	5.03	1161	2.3	0.5
192263	HD 192263	0.75	24.13	0.15	0.00
209458	HD 209458	0.68	3.52	0.047	0.00
210277	HD 210277	1.31	435.6	1.09	0.34
217014	51 Peg	0.47	4.23	0.05	0.00
217107	HD 217107	1.34	7.11	0.071	0.14
222582	HD 222582	5.44	575.9	1.35	0.71

If we concentrate on the mass interval for which we have both stars with planets and comparison stars (region with 1.2  $M_{\odot} > M > 0.7$ $M_{\odot}$), we obtain a mean [Fe/H] = +0.13 for the star with planet sample, and -0.08 for the comparison star sample. This makes a difference of about 0.21 dex, for a mean stellar mass of 0.95 $M_{\odot}$. To pollute the convective envelope of a 0.95 solar mass star containing about 0.04 $M_{\odot}$ in the convection zone (D'Antona & Mazzitelli 1994) in order to obtain the observed difference, one would need about 10 earth masses of pure iron. Considering the composition of C1 chondrites, this would mean $\sim$5 times more in silicate material, a value that seems excessively high. The addition of giant planets, also very rich in gas, would need even more material. For example, the addition of Jupiter to the present-day Sun would only be able to increase its metallicity by $\sim$0.05 dex, while adding two jupiters would increase 0.08 dex (e.g. Gonzalez 1998). We are supposing here that Jupiter has a rocky core, which is not necessarily true (Guillot 1999).

Figure 6: Left: distribution of [Fe/H] for stars with planets orbiting with semi-major axes lower than 0.1 AU (shaded histogram), and with semi-major axes greater than 0.1 AU (open histogram). The vertical lines denote stars with more than one planetary-mass companion. Right: cumulative fraction of the [Fe/H] distribution for the long (thin line) and short (thick line) period systems. Althought some trend is suggested regarding a possible higher metallicity for the stars with short period planets, a Kolmogorov-Smirnov test gives a probability of $\sim$0.7, which is not significant.

Murray et al. (2001) have reported evidence that all stars in the solar neighbourhood have accreted about 0.4 Earth masses of iron, suggesting that "terrestrial-type material is common around solar type stars''. This seems to be particularly evident in the objects with M> 1.3-1.4 $M_{\odot}$, which is in accordance with expectations (Laughlin et al. 1997). If confirmed, and as discussed by these authors, this result further stresses the difficulty of obtaining the [Fe/H] differences we observe considering a pollution model: 0.4 Earth masses of iron are definitely not able to explain the observed differences between stars with planets and stars "without'' planets.

4.4 Using evolved stars as a test

One other interesting note can be added by looking at the "star-with-planet'' sample. If the pollution scenario were correct, we would expect evolved stars (sub-giants) having planetary companions to have lower metallicities than their dwarf counterparts, since the convection envelope increases in mass as the star evolves off the main sequence (Sackmann et al. 1993), diluting its metallicity content.

There are eight stars in Tables 2 and 3 having spectroscopic $\log{g}$ values below $\sim$4.10 dex (probably already slightly evolving away from the main sequence): HD 10697 (3.96), HD  38529 (4.01), HD 117176 (3.90), HD 143761 (4.10), HD 168443 (4.10), HD 169830 (4.04), HD 177830 (3.32) and HD 190228 (4.02). If we compute the mean [Fe/H] we obtain a value of +0.08 $\pm$ 0.25 (with values going from -0.29 to +0.39), only slightly lower than the value of +0.15 $\pm$ 0.22 for the sample comprising only stars with $\log g > 4.10$ (values going from -0.55 to +0.50). As mentioned also by Gonzalez et al. (2001), this is even more striking when we notice that the higher-metallicity star in this "low $\log{g}$'' sample (HD 177830) is that having the lower surface gravity.