A CORRELATION BETWEEN HOST STAR ACTIVITY AND PLANET MASS FOR CLOSE-IN EXTRASOLAR PLANETS?

In Figure 1, we show the X-ray luminosities of planet-hosting stars with a_p < 0.15 AU from the ROSAT sample as a function of the innermost planet's mass. The data points for HD 41004B and HD 162020 are plotted in red for comparison. We would like to emphasize the special case of 51 Peg, a star not included in the original sample: this star has been observed in several pointed X-ray observations with ROSAT, XMM-Newton, and Chandra, covering different phases of the planetary orbit; it remained undetected in the RASS data and was therefore included by Scharf (2010) only as an upper limit. A detailed analysis of these data has shown (Poppenhäger et al. 2009) the star's X-ray flux to be constant and at a very low level for over 16 years, indicating that the star might be in a Maunder minimum state despite its close-in heavy planet. This system is a significant outlier of the L_X versus M_pl relation presented in Scharf (2010; see Figure 1), although it fulfills the criterion of having a determined phase-averaged X-ray luminosity.

To test for dependencies on the distance d of the stars, we conduct a principal component analysis (PCA; Pearson 1901) on the three logarithmic variables log L_X, log M_p, and log d from the full Scharf (2010) sample. We use the two eigenvectors with the largest eigenvalues as the feature vectors; the third eigenvalue is very small by comparison (0.05 versus 0.95 and 0.14), meaning that one loses very little data variance in this analysis. After reprojecting the data, all three variables show a strong linear trend with respect to each other (see Figure 3). This is a clear indication that the stellar distance is a crucial parameter in this sample that cannot be ignored. It is important to note that in an unbiased sample, L_X and M_p must not depend on stellar distance; if they do, as in this sample, a selection effect is present. This provides an explanation for the apparent dependence of L_X on M_p: the detection limit of L_X increases with increasing distance. The detectability of planets is somewhat intricate, and we investigate dependencies in detail in Section 4.3. In short, a dependency of planetary mass on stellar distance is present; at larger distances, the radial velocity (RV) method favors the detection of heavier planets, and low X-ray luminosities can no longer be detected in RASS, which yields the observed trend of L_X with M_p without having to invoke effects from supposed SPIs.

Also, without performing a PCA on this sample, the dependencies of L_X and M_p on d are revealed by rank correlation tests. We calculate Spearman's ρ, a rank correlation coefficient, which shows a perfect correlation indicated by a value of 1, a perfect anticorrelation by −1, and no correlation by 0. For the full Scharf (2010) sample, we find strong correlations of both L_X and M_p with d, indicated by ρ values of 0.49 and 0.54, respectively, translating to probabilities of 0.6% and 0.2% that such a correlation can be reached by pure chance. This correlation analysis yields the same result as the PCA; the stellar distance is the crucial parameter in this sample that causes the L_X/M_p correlation.

This is also reflected in the behavior of L_X/L_bol where there is no significant correlation with planetary mass for stars with close-in planets (see Figure 2). We also checked this visual result with a Spearman's ρ test while excluding the data from the two incomparable stars. This yields ρ = 0.05, i.e., a very weak positive correlation; the probability that such a ρ value is reached by chance is 87%. This is not surprising: if the trend in L_X is a distance selection effect and not related to the stellar activity level, then the quantity L_X/L_bol, which measures the intrinsic stellar activity level, should be independent from the planetary mass.

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In our further analysis, we use the data presented in Poppenhaeger et al. (2010), which consists of all known planet-hosting stars within a distance of 30 pc from the Sun, with X-ray properties derived from XMM-Newton and ROSAT data. As discussed in detail in Poppenhaeger et al. (2010), we exclude unresolved binary stars and early stars without outer convection zones from our analysis. The errors given are Poissonian errors plus an additional uncertainty of 30% on the X-ray luminosity to account for short-time variations since a large part of our sample consists of pointed XMM-Newton observations. We use the same sample selection criterion on these data as was used in Scharf (2010; planets at a < 0.15 AU). We show the relation between L_X and M_pl in Figure 4, left panel; data from stars that are also present in the sample from Scharf (2010) are plotted as green filled symbols. These stars lie close to a straight line, similar to Figure 1, although the data were collected in single pointings and not averaged over larger portions of the planetary orbit. This shows that the averaging process is not crucial for this kind of analysis; the L_X values derived from XMM-Newton pointings are very similar to the ones from the RASS data. The main difference from Figure 1 is that there are many additional X-ray detections in the lower right corner of the diagram. This is contrary to the assumption that massive, close-in planets cause a lower floor for the X-ray luminosity of their host stars.

Additionally, in this sample there is no significant correlation in the relation between the X-ray activity indicator L_X/L_bol versus planetary mass (see Figure 4, right panel); testing for rank correlation yields ρ = 0.003, i.e., practically no correlation at all. This is also true for stars with far-out planets, for which no SPI-related effects are expected (Figure 5). The only significant correlation present in the whole sample is the one between X-ray luminosity and the product of planetary mass and inverse semimajor axis. For the intrinsic X-ray activity measured by L_X/L_bol no such correlation is present. As discussed in Poppenhaeger et al. (2010), the L_X correlation is equally strong in a subsample of stars with small, far-out planets as well as in stars with heavy, close-in planets. Poppenhaeger et al. (2010) conclude that the correlation is caused by selection effects from planet detection; if it was caused by SPI, it should be strong in the second subsample and weak in the first subsample.

For the sake of completeness, we also conducted a PCA for this sample. Here we find for the reprojected data that there is a strong linear trend of M_p with d, but no apparent trends of L_X with d or M_p. This is due to the fact that we also use data from pointed observations in our sample, where observations of more distant targets usually have longer exposure times. This prevents the correlation of L_X with d which is present in the sample of Scharf (2010), and therefore also no correlation between L_X and M_p is present here.

In the above section we used a volume-limited sample of planet-hosting stars in which most of the stars have been detected in X-rays. In that sense, the sample is complete with regard to X-ray flux. However, the sample is most probably not complete regarding the detection of planets. Therefore, selection effects induced by planetary detection methods need to be analyzed carefully (see also O'Toole et al. 2009; Hartman 2010). The portion of planets that are detected by transits is quickly growing since the start of the CoRoT and Kepler observations, but for stars in the solar neighborhood, the dominant detection mechanism still is the RV method. We now provide an investigation of such selection effects in the basic planetary and stellar parameters that are present the sample used above.

The RV detectability of a planet depends on several properties: first, on the brightness of the star itself and therefore the quality of the signal in which one searches for RV variations; and second, on the stellar mass M_*, the planetary mass M_p, and the planetary period P. Other influences such as eccentricity of the planetary orbit are ignored here. Specifically, the RV semi-amplitude is proportional to P^−1/3 × M_p × M^−2/3_*. Thus, it should be easier to detect low-mass planets around low-mass stars for a given (fixed) sensitivity.

The first half of our hypothesis is that the RV detectability decreases as the stellar distance d increases since the apparent brightness of the star decreases. This means that at larger distances, planets should be found around intrinsically brighter stars. On the main sequence, this implies earlier and therefore more massive stars. We test this with Spearman's ρ for the sample from Poppenhaeger et al. (2010); we use the complete sample, not just the subsample with close-in planets, since we aim to analyze selection effects for all detected planets. We find that indeed the stellar mass is strongly positively correlated with distance (ρ = 0.36, P_false < 1%).

The second half of our hypothesis is that because of the dependencies of the RV amplitude on stellar and planetary parameters, mostly massive planets in close orbits will be found around massive stars, as the RV amplitude will be too small to be detected otherwise. Accordingly, we find that stellar mass and planetary mass are positively correlated (ρ = 0.46, P_false < 0.1%); the correlation between stellar mass and the quantity P^−1/3 × M_p is even stronger with ρ = 0.53. We also show the correlation of M_* and P^−1/3 × M_p in Figure 6. However, for a linear graphical representation, one has to choose specific functions (for example, logarithms) of the parameters to be plotted. This may not be the real relation of the two quantities; furthermore, we have already shown that there is an additional dependence of the RV detectability on stellar mass and distance which may also influence other correlations. Therefore, this plot in which a linear trend is indicated does not directly represent the true extent of the correlation as demonstrated by the rank correlation analysis. Indeed, it is advantageous to use rank correlations here since they do not rely on any specific parameterization of the quantities in question.

The above analysis confirms our detectability considerations. At larger distances, planets are detected around brighter, more massive stars, and these planets are more massive and in closer orbits in order to produce a strong and therefore detectable RV signal.

In our PCA calculations in Section 4.1 (Figure 3), the planets in wider orbits, depicted by open symbols, show a larger scatter around the various correlations than the close-in planets. When we analyze only the systems with wide-orbit planets (a ⩾ 0.15 AU) for rank correlations, we find that the correlations between planetary mass and stellar mass as well as between stellar mass and P^−1/3 × M_p are of comparable strength as in the full sample. However, the correlation between distance and stellar mass is weaker (P_false ≈ 10%). One could speculate that in the outer-planet sample the RV signal is weaker due to the larger distance between star and planet, and this might favor detections around brighter stars, yielding less "leverage" for the correlation (the mean stellar mass is actually larger for the outer-planet sample). However, we do not have a compelling and easy explanation for this.

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In summary, we deduce that the correlation of L_X and M_p is a combined selection effect of X-ray flux limits in the RASS and planet detectability by the RV method. In the sample that is not X-ray-flux-limited, the correlation of planetary mass and X-ray activity is consequently not present, as demonstrated in Figure 4.

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