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Astron. Astrophys. 361, 265-272 (2000)

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4. Activity, rotation, and radial-velocity "jitter"

The basic motivation for computing our [FORMULA] index is to monitor the activity of the stars in the planet-search programme. This can be done essentially in two ways. First, we can try to verify if there is any rotational modulation of the CaII flux. Such a detection would give us the rotational period of the star, and hence any radial-velocity variation with the same periodicity would be highly suspicious (although it would not exclude a planetary explanation). Unfortunately, the obtained precision and the bad time sampling of the data don't permit us to "directly" derive the rotational periods for our stars from rotational modulation of the CaII flux (e.g. Vaughan et al. 1981). An estimate of the rotational periods can therefore only be made using the mean "S" index (Noyes et al. 1984). On the other hand, the fact that we have activity values for such a large sample of stars gives us the possibility of studying the relation between the radial-velocity scatter and the chromospheric activity as expressed by the index [FORMULA].

In Fig. 6 we plot three log-log diagrams of the reduced radial velocity scatter, [FORMULA]'(Vr), against the chromospheric activity expressed by 105 [FORMULA], for 15 F, 98 G and 18 K dwarfs for which we have [FORMULA] values. The [FORMULA]'(Vr) values were computed by subtracting quadratically, for each star, the rms of the mean of the individual photon-noise statistical errors, [FORMULA], from the rms around the mean radial velocity, [FORMULA](Vr). The values of the instrumental long-term errors are not known and thus were not taken into account. This way, [FORMULA]'(Vr) includes unknown systematic instrumental errors as well as dispersion due to activity-related phenomena.

[FIGURE] Fig. 6. Plots of the reduced radial-velocity rms, [FORMULA]'(Vr), as a function of [FORMULA] for F dwarfs (filled triangles), G dwarfs (filled circles) and K dwarfs (filled hexagons) of the CORALIE sample. Down arrows represent stars for which [FORMULA]'(Vr) is lower than [FORMULA]. The solid lines represent the best least-squares linear fit to the points, while the rms of the fit is represented by the dashed lines. Open squares represent stars with known planetary systems when the [FORMULA]'(Vr) was computed without subtracting the orbital solution.

Values of [FORMULA]'(Vr) were computed only for stars with at least 7 CORALIE high-precision radial-velocity measurements. In general, the dispersion of the photon-noise errors around the mean is low, but all points with [FORMULA](Vr) larger than 2 [FORMULA] were eliminated (measures with errors much higher than usual). In order not to underestimate the activity-related scatter, in the cases where we have multiple radial-velocity observations per night, only one (the more precise) was considered. Stars for which linear radial-velocity drifts were found were also excluded from the fits, but some of these may have escaped. For stars with planets, the values of [FORMULA]'(Vr) were computed in the same way after subtraction of the orbital solutions.

First of all, the number of F dwarfs in Fig. 6 is much lower than that of G dwarfs, although they are intrinsically brighter, and thus easier targets for activity determinations. This is in fact another observational bias that occurs because F stars in our sample have usually higher values of [FORMULA] with respect to G dwarfs, and thus they have lower priority in the planet-search programme (the expected radial-velocity "jitter" is higher, e.g. Mayor et al. 1998). This way, often we don't have enough radial-velocity measurements for these stars to put them in the diagram of Fig. 6.

We can easily see from the plots that there is a clear relation between [FORMULA]'(Vr) and activity (expressed by [FORMULA]) for F, G and K dwarfs, confirming the results of Saar et al. (1998). The linear fits hold:


where [FORMULA] [FORMULA] 105 [FORMULA]. The fits have a rms of [FORMULA]=0.17 dex (F dwarfs), [FORMULA]=0.18 dex (G dwarfs), and [FORMULA]=0.19 dex (K dwarfs). 3

These relations clearly show that a trend does exist between [FORMULA]'(Vr) and [FORMULA] (at least for F and G dwarfs), but also that the slopes of the [FORMULA] vs. [FORMULA]'(Vr) relation increase as we go from K to F dwarfs. The zero points for the three relations also decrease slightly with increasing [FORMULA]. These two facts together have an important consequence: they impose different limits on the expected activity induced radial-velocity noise for solar type stars of different spectral types. For example, for an active F dwarf with an activity index [FORMULA]= -4.3 (105 [FORMULA] = 5.0) we expect a value of [FORMULA]'(Vr) between 21 and 45 m s-1 (within 1[FORMULA]). On the other hand, for a K dwarf with the same activity level, we expect a significantly lower radial-velocity "jitter", between 7 and 14 m s-1.

This fact can also be seen in Fig. 7, where we plot the values of [FORMULA]'(Vr) against the [FORMULA] of our programme stars. From the plot we can see that for the upper envelope of the F and early G dwarfs [FORMULA]'(Vr) is in general considerably higher than for late G or K dwarfs. This fact is most probably a real effect and not a sampling bias, since the whole activity range is relatively well covered for all spectral types.

[FIGURE] Fig. 7. Plot of the radial-velocity scatter as a function of spectral type for our programme stars. The size of the dots is proportional to the [FORMULA] of the star. Given the ambiguity in the [FORMULA], the size of the symbols should be seen as a lower value for the rotational velocity of the stars.

The plot of Fig. 7 also suggests that to higher [FORMULA] stars (larger symbols) corresponds generally a higher [FORMULA]'(Vr) for the same [FORMULA]. The values of [FORMULA] go from [FORMULA]0 to 15.4 km s-1, and were obtained from the width of the CORAVEL cross-correlation dip (Benz & Mayor 1984). A higher [FORMULA] is indeed expected to induce more radial-velocity "jitter" for the same activity level and spectral type, since it will introduce more important line asymmetries (Mayor et al. 1998; Saar et al. 1998). But some dispersion in this relation is expected, because for a given mass the activity level is physically related with the rotational period (Noyes et al. 1984; Patten & Simon 1996) and not with [FORMULA], the projected component of the rotational velocity.

It is interesting to verify that the F and early G dwarfs in the plot of Fig. 7 have in general higher values of [FORMULA] with respect to the late G and K dwarfs. This may indicate that "bluer" dwarfs need higher rotational velocities to produce the same chromospheric activity level (we remember that our samples of F and G dwarfs have a similar distribution of [FORMULA] values). Such a fact is also expected, because the depth of the convective layer increases as we go from F to K dwarfs, also increasing the convection overturn time and the dynamo effect (Noyes et al. 1984).

The higher radial-velocity scatter of the F stars is thus likely to be explained by two separate mechanisms acting simultaneously. On the one hand, it might be related with the higher "intrinsic" [FORMULA] for a given activity level. On the other hand, for solar-type stars, the convective velocities decrease with increasing [FORMULA], making convective inhomogeneities less important (Gray 1984; Saar & Donahue 1997). Since convection movements introduce an extra velocity field (Saar & Donahue 1997), to stars of different spectral types but of the same activity level and [FORMULA] will correspond different radial-velocity perturbations, being larger for F dwarfs (Saar et al. 1998).

The obtained relations between [FORMULA]'(Vr) and [FORMULA] can be particularly useful when dealing with extra-solar planet search programmes: together with the knowledge of the projected rotational velocity ([FORMULA]) of the star, they can be used to distinguish, to a first approximation, activity-related scatter from orbital radial-velocity variations. For example, a periodic radial-velocity variation with a rms of 40 m s-1 for a K dwarf would be, even if the star is chromospherically very active, an indication that we are much probably dealing with a planetary system. Such a limit may however not be comfortable for an active G or F dwarf, for which much higher activity-related "jitter" is expected (especially when the [FORMULA] is high).

To better illustrate this fact we can take the case of the two extra-solar planet parent stars HD 130322, (Udry et al. 2000) and HD 192263 (Santos et al. 2000a). Both are early K dwarfs that have high chromospheric activity levels (with [FORMULA] of -4.39 and -4.35, respectively). However, the orbital radial-velocity signal is much higher than the expected activity-related scatter. It is interesting to point out, however, that in both cases the radial-velocity rms around the orbital solution is relatively high (about 15 m s-1 for HD 130322 and 13 m s-1 for HD 192263), as expected according to the fits of Fig. 6.

The future addition of more data to the plots of Fig. 6 and Fig. 7 is probably necessary to better constrain and clarify the validity and precision of these relations. This is particularly noticeable in the cases of F and K dwarfs, for which we don't have many points yet. For G dwarfs, however, the present results are perfectly consistent with the ones presented in Santos et al. (2000b), even though the number of points has almost doubled.

In all cases (except F dwarfs) stars with known planetary systems are all considerably (more than 2 [FORMULA]) above the fits when compared to "single" stars. On the other side, there are two objects (HD 55720 and HD 73322, late G and K dwarfs, respectively) that are positioned more than 2 [FORMULA] below the fit. The reason for this deviation may have to do with a statistical bias caused by the relatively low number of points used to compute the values of [FORMULA]'(Vr), or to a projection effect (star seen pole-on). However, we cannot exclude the presence of some physical process possibly responsible for the observed high values of chromospheric activity, but not able to produce changes in the observed radial-velocities. For example, observations of the Sun showed that the presence of chromospheric "plages" 4 is not necessarily connected to the existence of a spot group (Howard 1996). We can thus imagine that these two chromospherically active stars may lack the photospheric features capable of producing important radial-velocity variations.

Since the instrumental long-term errors have not been subtracted, the physical meaning of all these quantities has not, however, a straightforward interpretation. For example, the exponents of the relations are probably under-evaluated while the zero points are over-estimated. Thus, for a given chromospheric-activity level (expressed by [FORMULA]), the value of the computed [FORMULA]'(Vr) is probably higher than the "real" value. This can be further stressed if we imagine that our sample may still contain some stars with unknown planetary companions, inducing radial-velocity variations (still treated as "noise"). This lead us to conclude that the results must be considered as upper-limits for the expected [FORMULA]'(Vr). Qualitatively, the fact that the instrumental errors have not been taken into account does not have any serious implications, and the main conclusions would remain the same.

As an observational test we have computed the value expected for the radial-velocity rms of the Sun, according to our relations. If we consider that the instrumental long-term errors in radial velocity amount to about 7 m s-1 (Queloz et al. 2000b), and taking a value of [FORMULA] = -4.94 (Noyes et al. 1984), we obtain from Eq. 3, [FORMULA](Vr)[FORMULA] 5 m s-1 (subtracting quadratically the errors). This is compatible with the results obtained by McMillan et al. (1993), that showed that the long term radial-velocity "jitter" of the Sun is lower than [FORMULA]4 m s-1. The small excess might be related to the existence of undetected planetary companions amid the stars in the plots.

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Online publication: September 5, 2000