REVEALING A UNIVERSAL PLANET–METALLICITY CORRELATION FOR PLANETS OF DIFFERENT SIZES AROUND SOLAR-TYPE STARS

Our method relies on knowing stellar properties for both planet host stars and stars with no planet detections. For the planet host stars, we used stars in Buchhave et al. (2014) whose stellar properties are determined via the Stellar Parameter Classification (SPC) technique (Buchhave et al. 2012). Typical precision for SPC is 50 K, 0.10 dex, and 0.08 dex for T_eff, log(g), and [Fe/H], respectively. We selected the stars in Buchhave et al. (2014) hosting planets with orbital periods less than 100 days, which account for 93% (376/406) of their sample.

Since a large number ($\gt 180,000$) of stars have been observed by the Kepler mission ² , only a tiny fraction of Kepler stars received spectroscopic followup observations; the majority of Kepler stars have only photometrically determined stellar properties (Brown et al. 2011; Huber et al. 2014). Therefore, for stars with no planet detections, we adopted the stellar parameters from the NASA Exoplanet Archive ³ and converted the values into the more representative stellar properties to remove systematic errors (see Section 2.3 for details). We restricted our sample to solar-type dwarf stars ($4800\;{\rm K}\lt {{T}_{{\rm eff}}}\leqslant 6500$ K, log(g) $\geqslant \;4.2$).

Stellar properties for most Kepler stars are from the KIC (Brown et al. 2011). The KIC values are known to have large uncertainties and systematic errors. To investigate the uncertainty and the systematic error, we compared the KIC values with the spectroscopically determined values for the 406 stars in Buchhave et al. (2014) (shown in Figure 1). While we found a large dispersion (∼0.2 dex) of spectroscopically determined metallicities at a given KIC metallicity, there is a positive correlation between the SPC and KIC metallicities, [Fe/H]_SPC = 0.10 + 0.49 × [Fe/H]_KIC. The positive correlation suggests that, although not accurate, the KIC metallicity can be used as a proxy of the stellar metallicity. Dawson & Murray-Clay (2013) used the KIC metallicity to find evidence of planet–planet interactions for giant-planet migration in metal-rich stars. Similarly, for T_eff and log(g), we also found a positive correlation between SPC and KIC values, T_eff,SPC = 1194 + 0.78 × T_eff.KIC, and log(g)_SPC = 1.97 + 0.54 × log(g)_SPC, respectively. The correlations for [Fe/H], T_eff, and log(g) can be used to remove systematic errors, and convert KIC values to values that better represent stellar properties.

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We describe how we converted KIC values to the more representative values as follows. Using the metallicity as an example, we divided KIC metallicity into six bins with a width of 0.2 dex starting at −0.7 dex, and calculated the median value and the standard deviation of the SPC metallicities in each bin (shown as red filled circles and error bars in Figure 1). A more representative metallicity than the KIC metallicity can be calculated by interpolating between adjacent red points. Similarly, a more representative error bar can also be estimated. For T_eff, we used six bins with a width of 417.5 K starting at 4590 K to cover the entire T_eff range. For log(g), we used 10 bins with a width of 0.1 or 0.2 dex starting at 3.2 dex. The width is changed from 0.1 dex to 0.2 dex for bins with log(g) lower than 4.0 dex because of the small number of available data points. We will use the more representative stellar properties for Kepler stars with no planet detections throughout the paper.

We defined two metallicity groups: a metal-poor group with [Fe/H] $\lt -0.05$ and a metal-rich group with [Fe/H] $\gt 0.05$. The metallicity range between −0.05 and 0.05 serves as a buffer zone, limiting the contamination between two metallicity groups. The width of the buffer zone is consistent with the typical error bar quoted in Buchhave et al. (2014). There are ∼53,000 and ∼49,000 stars in the metal-poor and metal-rich groups, respectively. Given the uncertainty of the converted metallicity (∼0.2 dex), we did not further divide the sample along the metallicity dimension. In each metallicity group, we further divided the stars with detected planets into three sub-groups based on planet size: gas-giant planets ($3.9\ {{R}_{E}}\ \lt {{R}_{P}}\ \leqslant \ 22.0\ {{R}_{E}}$), gas-dwarf planets ($1.7\ {{R}_{E}}\ \lt {{R}_{P}}\ \leqslant \ 3.9\ {{R}_{E}}$), and terrestrial planets (${{R}_{P}}\ \leqslant \ 1.7\ {{R}_{E}}$). The scatter plot of planet radius and stellar metallicity is shown in Figure 2.

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We adopted the following Monte Carlo method to estimate the fraction of stars with detected planets. For each Kepler star with no planet detection, we assigned the star with stellar properties (T_eff, log(g), and [Fe/H]) based on the converted, more representative values. The values are then perturbed assuming Gaussian distributions with standard deviations that are equal to the more representative error bars as shown in Figure 1. For each Kepler planet host star in our sample, we perturbed the measurements of stellar properties (T_eff, log(g), and [Fe/H]) and planet properties (R_P ) with the reported measurement uncertainties. With this set of simulated data, we counted the number of stars with detected planet candidates (N_P ) and the number of searched stars (N_S ) in each sub-region as shown in Figure 2. We then calculated the ratio of N_P and N_S , a ratio which we refer to as the fraction of stars with detected planets. We repeated this procedure 1000 times and calculated the median values and the standard deviations for N_P and N_S . Since N_P and N_S were counted in each iteration, we considered counting errors which were assumed to follow Poisson distribution. Therefore, the final uncertainties of N_P and N_S are the summation in quadrature of the standard deviation and the Poisson noise. This approach takes into account both the statistical uncertainty and the uncertainties of stellar and planetary properties.

The fractions of stars with detected planets are given in Table 1. The reported values and uncertainties are the mode and the 68% credible interval of the ${{N}_{P}}/{{N}_{S}}$ distribution. We emphasize that measurements of the planet occurrence rates for planets of different sizes require further information such as detection sensitivity and survey incompleteness. Thus, the numbers that we list should not be construed to be the estimates of the true planet occurrence rates. However, the ratio of ${{N}_{P}}/{{N}_{S}}$ between two metallicity groups is a measure of the ratio of the planet occurrence rates of two metallicity groups, i.e., the relative planet occurrence rate of metal-rich to metal-poor stars. This is true only if the detection sensitivity and survey incompleteness affect the two metallicity groups in the same way.

Table 1. Comparison of the Fractions of Stars with Detected Planets for Two Metallicity Groups in Different Planet Radius Ranges

	Metal-poor			Metal-rich			$\frac{{{({{N}_{P}}/{{N}_{S}})}_{{\rm rich}}}}{{{({{N}_{P}}/{{N}_{S}})}_{{\rm poor}}}}$
Planet Radius	NP	NS	${{N}_{P}}/{{N}_{S}}$	NP	NS	${{N}_{P}}/{{N}_{S}}$
${{R}_{P}}\leqslant 1.7{{R}_{E}}$	51.0 ± 8.1	53289 ± 287	9.6 ± 1.5 × 10−4	82.0 ± 10.2	49366 ± 284	16.6 ± 2.1 × 10−4	$1.72_{-0.17}^{+0.19}$
$1.7{{R}_{E}}\leqslant {{R}_{P}}\leqslant 3.9{{R}_{E}}$	36.0 ± 7.1	53289 ± 287	6.8 ± 1.3 × 10−4	68.0 ± 9.5	49366 ± 284	13.8 ± 1.9 × 10−4	$2.03_{-0.26}^{+0.29}$
$3.9{{R}_{E}}\leqslant {{R}_{P}}\leqslant 22.0{{R}_{E}}$	3.0 ± 2.1	53289 ± 287	5.6 ± 4.0 × 10−5	24.0 ± 5.4	49366 ± 284	48.6 ± 10.9 × 10−5	$9.30_{-3.04}^{+5.62}$

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At a given photometric precision, it is more difficult to detect a transiting planet around a larger star, so stellar radius affects the planet detection sensitivity. To investigate whether detection sensitivity is the same for stars from two metallicity groups, we checked the dependence of stellar radius on metallicity. The top panels in Figure 3 show stars with spectroscopic followup observations from Buchhave et al. (2014). We limited the comparison for stars with $4800\;{\rm K}\lt {{T}_{{\rm eff}}}\leqslant 6500$ K and log(g) $\geqslant \;4.2$, which is consistent with our sample selection criteria. We found that metal-rich stars are ∼20% larger than metal-poor stars. We performed a two-sided K-S test, with a p value of $1.8\times {{10}^{-5}}$, and a maximum difference of the cumulative distribution of 0.29. A two-sided Kuiper test, which is more sensitive to tails of distributions, shows a similar result with a p value of 7 × 10⁻⁴. Next, we checked if the stellar radius dependence on metallicity is a systematic for the SPC method. We collected stars with spectroscopically or asteroseismically determined stellar properties from Huber et al. (2014), but excluded stars from Buchhave et al. (2012). These stars consist of a sample whose stellar properties are determined free of the systematics (if any) of the SPC method. With this sample of 182 stars, we again found that metal-rich stars are larger; the median radius of the metal-rich sample is ∼5% larger than that for the metal-poor sample. The K-S test gives a p value of 0.006. However, the Kuiper test is inconclusive (p = 0.089 $\geqslant \;0.01$). Finally, we checked all stars whose radii are measured with asteroseismic and interferometric methods (bottom panels in Figure 3, Boyajian et al. 2012, 2013; von Braun et al. 2014; Chaplin et al. 2014). In this case, we found that metal-rich stars are ∼5% larger than metal-poor stars, but neither the K-S test (p = 0.14) nor the Kuiper test (p = 0.41) is significant. However, the maximum difference between the two cumulative distributions is 0.24, similar to the two previous cases. The smaller sample size (N = 88) may explain the inconclusive tests.

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In a transiting planet survey like the Kepler mission, planet detection is more difficult around larger stars for a given planet size, so there is a bias against planet detection for metal-rich stars whose radii are larger than metal-poor stars. This bias may not be an issue for large planets, whose signals can be detected regardless of the 5%–20% stellar radius difference. However, the detection bias gets stronger for terrestrial planets and small gas-dwarf planets; the transiting signals of these planets may be only marginally detected given the Kepler precision. Therefore, the ratio of the fraction of stars with planets for metal-rich stars to that for metal-poor stars reflects a lower limit of the relative planet occurrence rate of metal-rich stars to metal-poor stars.

We found a universal planet–metallicity correlation: not only gas-giant planets but also gas-dwarf planets and terrestrial planets occur more frequently in metal-rich stars (Figure 4). The dependence of the planet occurrence rate on metallicity decreases with decreasing planet size. The planet occurrence rates of gas-giant planets, gas-dwarf planets, and terrestrial planets are $9.30_{-3.04}^{+5.62}$, $2.03_{-0.26}^{+0.29}$, and $1.72_{-0.17}^{+0.19}$ times higher for metal-rich stars than for metal-poor stars, respectively. Given the detection bias against metal-rich stars (Section 2.5), these values are the lower limits for the planet–metallicity correlation: the dependence of the planet occurrence rate on metallicity may be stronger.

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We further divided the sample into two with different effective temperature ranges, one with $4800\;{\rm K}\lt {{T}_{{\rm eff}}}\leqslant 5650{\rm K}$, and the other one with $5650\;{\rm K}\lt {{T}_{{\rm eff}}}\leqslant 6500$ K. The dividing T_eff was selected to match the mean T_eff of the sample. For the sub-sample with lower T_eff, we found that the relative planet occurrence rates of metal-rich to metal-poor stars are $10.44_{-5.19}^{+1.97}$, $1.51_{-0.26}^{+0.31}$, and $1.34_{-0.18}^{+0.25}$ for gas-giant planets, gas-dwarf planets, and terrestrial planets, respectively. The relative planet occurrence rates in the sub-sample with higher T_eff are $7.24_{-2.51}^{+7.86}$, $2.39_{-0.36}^{+0.48}$, and $1.94_{-0.22}^{+0.26}$ for gas-giant planets, gas-dwarf planets, and terrestrial planets, respectively. The relative planet occurrence rates for gas-giant planets are similar between two T_eff sub-samples, but they are different for smaller planets (Figure 4). This may suggest that, for smaller planets, the metallicity dependence of the planet occurrence rate is weaker for stars with lower T_eff than for stars with higher T_eff. However, the hypothesis replies heavily on more observations in the future to confirm or refute.

To understand the likelihood of the above result being produced by a random sample, we repeated the Monte Carlo simulations in Section 2.4. Instead of using the metallicities from Buchhave et al. (2014), we randomly assigned metallicities to planet host stars following the more representative metallicity distribution of the overall Kepler stellar population. In only 1 out of 1000 trials we observed a value higher than 1.72, which is the relative planet occurrence rate for terrestrial planets. We did not get a higher value than the reported relative planet occurrence rate for gas-dwarf and gas-giant planets. The result indicates that it is highly unlikely that the universal planet–metallicity correlation is mimicked by a random sample. For a sample of planet host stars with randomized metallicities, the relative planet occurrence rates of metal-rich to metal-poor stars are $1.07_{-0.40}^{+0.50}$, $1.06_{-0.18}^{+0.20}$, and $1.06_{-0.19}^{+0.21}$ for gas-giant planets, gas-dwarf planets, and terrestrial planets, respectively. All the values are consistent with unity, which is what we expect from a randomized sample with no planet–metallicity correlation.

The positive correlation of the gas-giant planet occurrence rate with stellar metallicity is consistent with previous results based on the Kepler data (Schlaufman & Laughlin 2011; Buchhave et al. 2012; Everett et al. 2013). The median [Fe/H] for Kepler metal-poor and metal-rich stars in our sample is −0.14 dex and 0.23 dex, respectively. Our result indicates that a change of 0.37 dex in [Fe/H] results in a change of $9.30_{-3.04}^{+5.62}$ times in the gas-giant planet occurrence rate. In comparison, an exponential law dependence with a power ∼2.0 (Fischer & Valenti 2005; Udry & Santos 2007) predicts that the boost of the planet occurrence rate due to metal enhancement is ∼5.5. Considering the uncertainties of the metallicity measurement and the exponential power, our result for gas-giant planets is consistent with previous studies.

For terrestrial planets, we found that the planet occurrence rate for metal-poor stars is lower than that for metal-rich stars at 4.2 σ level. In comparison, Buchhave et al. (2012) found that the average metallicity for stars hosting planets smaller than 2 R_E is comparable to the solar value, which is a necessary condition for a null-correlation between plant occurrence and stellar metallicity. However, their finding is not sufficient to conclude the null-correlation without knowing the overall metallicity distribution of Kepler stars. Buchhave et al. (2014) measured metallicities for a larger sample of planet host stars. For stars with terrestrial planets (${{R}_{P}}\ \leqslant \ 1.7\ {{R}_{E}}$), the median metallicity is 0.04 dex. Among 207 stars with terrestrial planets, 122 have metallicities above the solar value and 85 have metallicities equal to or lower than the solar value. As the sample size increases, the median metallicity for stars with terrestrial planets becomes higher than the solar value although it is still consistent with the solar value.

We investigated whether the metallicity distribution of the overall Kepler stars is statistically identical to that for stars hosting terrestrial planet. Assuming the more representative metallicities for Kepler stars based on the comparison shown in Figure 1, we compared the metallicity distribution of Kepler stars to that for stars with terrestrial planets. The K-S test (p = 0.005 $\leqslant \;0.01$) rejects the null hypothesis that the metallicity distribution for terrestrial planet host stars is drawn from the overall Kepler stars. The Kuiper test is inconclusive (p = 0.05 $\geqslant 0.01$) although hinting the same result as the K-S test. This test shows the importance of taking into consideration the metallicity distributions of both planet host stars and the overall stellar sample. From the difference in metallicity distribution, we went through the analyses described in this paper, and found that terrestrial planets are $1.72_{-0.17}^{+0.18}$ times more abundant around metal-rich stars than metal-poor stars.

The finding of a positive correlation with metallicity for small planets was also mentioned by Schlaufman & Laughlin (2011), but they found this correlation for late-K dwarfs, which we did not consider in our sample. Schlaufman & Laughlin (2011) did not find a significant planet–metallicity correlation for earlier spectral type. However, they used $g-r$ colors as a proxy for stellar metallicity, which is not as precise as spectroscopically determined values, and is prone to contamination of giant stars (Mann et al. 2012).

The positive planet–metallicity correlation for terrestrial and gas-dwarf planets is seemingly at odds with other previous studies from Doppler surveys (Udry & Santos 2007; Sousa 2008; Bouchy et al. 2009; Mayor et al. 2011). However, there are several possible explanations for the disagreement. First, small sample size may cause the insignificant planet–metallicity correlation. Mayor et al. (2011) studied 23 Doppler planets less massive than 30 M_⨁. They found that the median of their host stars metallicities matches with the median metallicities of the overall stellar sample. To investigate the effect of a small sample size, we downsized our sample to mimic their analysis. We randomly selected 822 Kepler stars to simulate their overall stellar sample, and 23 planet host stars (${{R}_{P}}\ \leqslant$ 3.9 R_E ) to simulate their low-mass planet host star sample, Then, we compared the metallicity distributions of these two samples using the K-S test and the Kuiper test. We found that only in 7.3% (for the K-S test) and 2.5% (for the Kuiper test) of trials the p values are smaller than 0.01. Therefore, the non-detection of a planet–metallicity correlation in Mayor et al. (2011) may be attributed to a small sample size for low-mass planet host stars.

Second, the weaker dependence of the planet occurrence rate on metallicity for stars with lower effective temperatures may be another possible explanation. If Mayor et al. (2011) sample is dominated by stars with ${{T}_{{\rm eff}}}\leqslant 5650$ K, then the planet–metallicity correlation is not as significant as it is for stars with higher T_eff. The relative planet occurrence rate for these lower T_eff stars is less than 2σ away from unity, which is the value corresponding to no planet–metallicity correlation (see Figure 4). To find the fraction of stars with ${{T}_{{\rm eff}}}\leqslant 5650$ K, we checked the HARPS FKG star sample (Sousa et al. 2011), 55% of stars in the sample have T_eff lower than 5650 K. In comparison, 37.5% of the Kepler solar-type stars have T_eff lower than 5650 K. The relative portion of stars with ${{T}_{{\rm eff}}}\leqslant 5650$ K is thus higher for the HARPS sample than for the Kepler sample. We therefore expect a less metallicity dependence of planet occurrence rate for the HARPS sample. We conducted a similar simulation to test small size effect on stars with lower T_eff, only 4.9% (for the K-S test) and 1.8% (for the Kuiper test) of trials returned p values that indicate significant difference (p ≤ 0.01). Therefore, the difference in sample size and T_eff distribution may explain the discrepancy of the planet–metallicity correlation for small planets between the Kepler survey and previous Doppler surveys.

With a sample of 406 Kepler planet host stars taken from Buchhave et al. (2014), we compared the stellar properties from the KIC and those from the SPC analysis. From the comparison, we derived a method to convert the KIC values for stellar properties to more representative values. Based on the more representative stellar properties, we divided stars into metal-poor and metal-rich groups, and counted the number of stars (N_S ) and the number of planet host stars (N_P ) that belong to these two metallicity groups. Next, we calculated the fraction of stars with planets (${{N}_{P}}/{{N}_{S}}$). We used the ratio of the fractions of stars with planets for metal-rich to metal-poor stars as an estimation of the relative planet occurrence rate, a quantity evaluating the planet occurrence rate change due to metal enhancement. We found a universal planet–metallicity correlation for solar-type stars ($4800\;{\rm K}\lt {{T}_{{\rm eff}}}\leqslant 6500$ K, log(g) $\geqslant \;4.2$): planets of all sizes are more abundant around metal-rich stars than they are around metal-poor stars. The relative planet occurrence rates for metal-rich stars to metal-poor stars are $9.30_{-3.04}^{+5.62}$, $2.03_{-0.26}^{+0.29}$, and $1.72_{-0.17}^{+0.18}$ for gas-giant planets ($3.9\ {{R}_{E}}\ \lt {{R}_{P}}\ \leqslant \ 22.0\ {{R}_{E}}$), gas-dwarf planets ($1.7\ {{R}_{E}}\ \lt {{R}_{P}}\ \leqslant \ 3.9\ {{R}_{E}}$), and terrestrial planets (${{R}_{P}}\ \leqslant \ 1.7\ {{R}_{E}}$), respectively. Since we found that metal-rich stars are 5%–20% larger than metal-poor stars, there is a detection bias against planets around metal-rich stars, so the relative planet occurrence rates reported in this paper are the lower limits.

We chose a metallicity buffer zone with a width of 0.1 dex, which is comparable to the reported error from Buchhave et al. (2014). However, the error for the stars without spectroscopic observations may be higher than 0.1 dex. We changed the width of the buffer zone to investigate how the width changes our results. For a buffer zone width of 0.15 dex, the relative planet occurrence rates for metal-rich stars to metal-poor stars are $12.13_{-4.21}^{+12.46}$, $2.20_{-0.29}^{+0.31}$, and $1.81_{-0.19}^{+0.21}$ for gas-giant planets, gas-dwarf planets, and terrestrial planets, respectively. For a buffer zone width of 0.20 dex, the relative planet occurrence rates for metal-rich stars to metal-poor stars are $13.91_{-5.07}^{+11.45}$, $2.41_{-0.33}^{+0.43}$, and $1.92_{-0.21}^{+0.22}$ for gas-giant planets, gas-dwarf planets, and terrestrial planets, respectively. Widening the buffer zone has two effects. First, the widening increases the metallicity difference between the metal-rich and metal-poor sample and thus magnifies the planet–metallicity correlation. Second, the widening decreases the number of available data points and thus increases the uncertainty. As the buffer zone width increases, a higher relative planet occurrence rate is observed for each type of planet, although the numbers overlap within error bars. The increasing relative planet occurrence rate with increasing buffer zone width is another piece of evidence for the universal planet–metallicity correlation. Widening the buffer zone leads to two samples that are more separated by metallicity, which reveals a stronger metallicity dependence of planet occurrence rate, i.e., a higher relative planet occurrence rate.

In Section 3.1, we reported a weaker dependence of the small planet occurrence rate on metallicity for stars with lower effective temperatures ($4800\;{\rm K}\lt {{T}_{{\rm eff}}}\leqslant 5650$ K). The relative planet occurrence rates of metal-rich to metal-poor stars are $1.51_{-0.26}^{+0.31}$ and $1.34_{-0.18}^{+0.25}$ for gas-dwarf planets and terrestrial planets, respectively. In comparison, the relative planet occurrence rates for stars with higher T_eff ($5650\;{\rm K}\lt {{T}_{{\rm eff}}}\leqslant 6500$ K) are $2.39_{-0.36}^{+0.48}$ and $1.94_{-0.22}^{+0.26}$ for gas-dwarf planets and terrestrial planets, respectively. This may be explained by a competition effect between the formation of planetesimals and the dispersal of dust materials in disk. Dust in a protoplanetary disk is dispersed with a time scale of several Myr (Haisch et al. 2001; Mamajek 2009). Observations of transition disks demonstrate that the central region of the circumstellar disk clears before the outer regions (e.g., Isella et al. 2009), and imaging surveys of the brightest disks in the nearby Ophiuchus star forming region demonstrate that central clearings are very common in even the most massive disks at ages of a few Myr (Andrews et al. 2009, 2010). The observed properties of transitional disks are consistent with clearing due to the dynamical influence of a low-mass companion embedded in the disk (e.g., Andrews et al. 2011), although photo-evaporation is likely to play a role in disk dispersal, possibly at later stages (e.g., Alexander et al. 2013). There is also some evidence that the dust disks around intermediate mass stars evolve more rapidly than those around solar-type stars (e.g., Yasui et al. 2014). Therefore, planetesimal formation may be cut off at an earlier time in the disks around more massive stars. In order to rapidly form larger planetesimals that are immune to the effects of photo-evaporation and the reduced accretion flow across the gap formed by low-mass companions embedded in the disk, systems with higher metallicity are favored in forming close-in terrestrial and gas-dwarf planets around more massive stars. By comparison, the clearing process may take place slightly later for less massive stars, such that planetesimals can afford to form more slowly. Therefore, metallicity dependence for close-in terrestrial and gas-dwarf planets around less massive stars is not as strong as it is for more massive stars.