A COMPREHENSIVE STATISTICAL ASSESSMENT OF STAR–PLANET INTERACTION

We selected MS planet-hosting stars from the Exoplanet Orbit Database (EOD; Wright et al. 2011; Han et al. 2014)¹⁰ as of 2013 March. Our sample is limited to stars at distances less than 60 pc, and restricted to RV discovered planets with Msin i < 20 M_Jup (this is deliberately greater than the usual brown dwarf cutoff so as to include all strongly interacting systems, but only HD 162020 has Msin i > 13 M_Jup and a < 0.15 AU). Stars are identified as MS by requiring surface gravity log g > 3.8 and inferred stellar radius R_* < 2 R_☉, which is appropriate for non-evolved stars across the full range of temperatures here considered. (See, e.g., Niedzielski et al. 2015, and references therein for planet-hosting evolved systems.) We require the effective temperature to be between 4000 K and 6500 K. The upper temperature limit excludes early F and hotter stars for which the disappearance of the upper convective zone eliminates dynamo action and leaves them generally X-ray dark. The lower temperature limit excludes M dwarfs, which have efficient dynamos (Kitchatinov & Olemskoy 2011) and coronal activity producing high L_X/L_bol ratios (they are also generally UV active; France et al. 2013). M dwarfs are also less likely to host gas giants and more likely to contain super Earths (e.g., Wu & Lithwick 2013); this is not solely a selection effect, as theoretical models (e.g., Boss 2006) predict that low-mass stars more easily (but certainly not exclusively; see also Johnson et al. 2012) form lower-mass planets.

For multiple-planet systems, we consider only the properties of the planet most relevant for potential magnetic interaction. We rank by M_P/a² in multi-planet systems (see discussion in Section 1); in nearly all cases this is equivalent to choosing the closest-in known planet. The bolometric luminosity is calculated from the B − V color index and the apparent V-band magnitude m_v based on the MS trends tabulated by Bessell et al. (1998); specifically, we take the bolometric correction to be 0.59–2.32(B − V) + 2.60(B − V)² − 0.48(B − V)³. Measurements of R'_HK are from the EOD except for five values for hot Jupiter systems adopted from Knutson et al. (2010)¹¹ and eight values added from Isaacson & Fischer (2010), which updates Wright et al. (2004).¹² Measurements of L_UV/L_bol are taken from Shkolnik (2013).

From the MS parent sample of planet-hosting stars, we select solar analogs as having 5500 K < T_eff < 6000 K and 0.6 < L_bol/L_☉ < 2. Additional criteria were imposed on our new Chandra targets: those 12 stars are non-active (vsin i < 2 km s⁻¹ and $R^{^{\prime }}_{\rm HK}<-4.8$ where known) and have definitively Jovian planets (M_P > 0.1 M_Jup) in low eccentricity orbits (e < 0.3). The activity and mass cuts further reduce detectability bias, and additionally sub-Jovian planets may have distinct magnetic field properties that could blur a statistical signature of interaction in the sample. Low eccentricity establishes that the planet remains at a nearly constant distance from its star, so any interaction is quasi-continuous and X-ray observations do not need to be timed to periastron (but see also Hodgson et al. 2014 for using eccentric systems to test for star–planet interaction).

Targeting low-activity solar analogs reduces detectability bias, but it has both additional advantages and disadvantages for investigating star–planet interaction. The absolute energy produced by planet-induced activity is expected to scale with stellar magnetic field strength (Section 1); however, the fractional increase in X-ray luminosity may be relatively constant, due to L_X correlating with B_* (Pevtsov et al. 2003). In active systems the intrinsic chromospheric or coronal variability, such as from starspots or flares, is generally several times greater than that expected from star–planet interaction (e.g., see discussion of HD 73256 in Shkolnik et al. 2005). Intensive phase-resolved coverage of individual active systems can potentially distinguish stellar from planet-induced variability (Shkolnik et al. 2005, 2008; Miller al. 2012; Scandariato et al. 2013), but for our statistical study that seeks to test planet-induced enhancements in X-ray luminosity by an average factor of a few (Kashyap et al. 2008) it is preferable to mitigate against increased noise. After considering the properties of solar analogs, we broaden the scope of our analysis to the full FGK MS sample that also includes more active stars.

Properties of the 198 FGK MS stars and their relevant planets are provided in Table 1. Figure 1 shows an HR diagram of the full sample, with the cuts selecting solar analogs indicated as dashed lines. Figure 2 shows the semi-major axis and planetary mass for the most relevant planet in each system, as defined above; there is good coverage of the a − M_P plane. Trend lines for constant semi-amplitude K∝ M_P/a^0.5 (which, along with intrinsic stellar noise, determines detection sensitivity) and for expected interaction strength M_P/a² are also shown on Figure 2.

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Table 1. Sample Properties

Name	Teff	Lbol	R'HK	RMS	Msin i	a	Per	K	MP/a2	1/a	LX	$\frac{L_{\rm X}}{L_{\rm bol}}$	Obsa
	(K)	(erg s1)		(m s−1)	(MJup)	(AU)	(days)	(m s−1)	($\frac{M_{\rm Jup}}{ {\rm AU}^{2}}$)	(AU−1)	(erg s−1)
HD 142 b	6248	34.04	−4.92	12.00	1.31	1.04	350.30	33.90	1.20	0.96	<27.90	< −6.14
HD 1237 b	5536	33.38	−4.44	19.20	3.37	0.49	133.71	167.00	13.79	2.02	28.75 ± 0.16	−4.63	B
HD 1461 b	5765	33.60	−5.02	3.43	0.02	0.06	5.77	2.70	5.95	15.74	<27.81	< −5.79
HIP 2247 b	4714	32.98	−4.79	4.10	5.12	1.34	655.60	173.30	2.86	0.75	<28.18	< −4.80
HD 2638 b	5192	33.21	−99.00	3.30	0.48	0.04	3.44	67.40	251.39	22.95	<28.47	< −4.74
HD 3651 b	5220	33.31	−5.02	6.30	0.23	0.29	62.22	15.90	2.64	3.39	27.12 ± 0.18	−6.19	H
HD 4113 b	5688	33.66	−99.00	8.40	1.65	1.27	526.62	97.10	1.02	0.79	<28.37	< −5.29
HD 4208 b	5600	33.42	−4.95	3.40	0.81	1.65	828.00	19.06	0.30	0.60	<28.10	< −5.32
HD 4308 b	5695	33.58	−99.00	1.30	0.05	0.12	15.56	4.07	3.36	8.39	26.66 ± 0.14	−6.92	C
HD 5388 b	6297	34.22	−4.98	3.33	1.97	1.76	777.00	41.70	0.63	0.57	<28.54	< −5.68
HD 6434 b	5835	33.65	−4.90	10.60	0.40	0.14	22.00	34.20	19.68	7.04	<26.60	< −7.05	C
HIP 5158 b	4962	32.86	−4.80	2.47	1.43	0.89	345.72	57.00	1.81	1.13	<28.31	< −4.55
HD 6718 b	5746	33.61	−4.97	1.79	1.56	3.55	2496.00	24.10	0.12	0.28	<28.56	< −5.05
HD 7199 b	5386	33.46	−4.80	2.63	0.30	1.36	615.00	7.76	0.16	0.73	<28.19	< −5.27
HD 7449 b	6024	33.67	−4.85	3.81	1.31	2.34	1275.00	41.59	0.24	0.43	<28.25	< −5.42
HD 7924 b	5177	33.14	−4.89	2.78	0.03	0.06	5.40	3.87	9.08	17.66	27.27 ± 0.26	−5.87	F
HD 8535 b	6136	33.85	−4.95	2.49	0.68	2.44	1313.00	11.80	0.11	0.41	<28.52	< −5.33
HD 8574 b	6049	33.95	−5.09	14.20	1.81	0.76	227.00	58.30	3.15	1.32	<28.38	< −5.57
HD 9446 b	5793	33.55	−4.50	15.10	0.70	0.19	30.05	46.60	19.52	5.29	28.49 ± 0.28	−5.06	F
$\upsilon$ And b	6212	34.12	−4.98	13.66	0.67	0.06	4.62	68.21	189.78	16.84	27.66 ± 0.12	−6.46	O

Notes. Ordered by R.A. The 23 bolded names make up the solar analogs subsample (the two italicized names are excluded due to high R'_HK values). ^aObservatory: no entry is RASS limit; B/F is RASS Bright/Faint source; P/H is ROSAT pointed PSPC/HRI detection; X is XMM-Newton detection or limit; C is Chandra detection or limit; O is other as explained in the text.

Only a portion of this table is shown here to demonstrate its form and content. A machine-readable version of the full table is available.

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Chandra archival coverage of solar analogs includes observations of 51 Peg (ObsID 10825; PI: Schmitt; Poppenhäger et al. 2009), HD 4308 (ObsID 12339; PI: Schmitt), and ρ CrB (ObsID 12396; PI: Saar; Saar & Testa 2012). We reprocessed these data as described below. XMM-Newton observations targeting planet-hosting stars within d < 30 pc (Kashyap et al. 2008; Poppenhäger et al. 2010) include an five additional solar analogs (47 UMa, HD 190360, HD 217107, 16 Cyg B, and HD 70642; 51 Peg and HD 4308 also have XMM-Newton coverage), for which we take L_X measurements from Poppenhäger et al. (2010). For completeness we also include complementary ROSAT coverage of five solar analogs at d < 30 pc¹³ (HD 39091, HD 82943, HD 147513, HD 150706, HD 210277), calculating L_X from RASS bright or faint source catalog net count rates. However, two (HD 147513 and HD 150706) of these five solar analogs with ROSAT coverage are quite active (likely because they are young), with $\log {R^{^{\prime }}_{\rm HK}}>-4.5$, and they are set aside for this analysis as they do not provide a proper point of comparison with true solar analogs; note that they both have only distant Jupiter-mass planets known, so their activity cannot be related to star–planet interaction. To this archival data we add our 12 new Chandra observations (PI: Miller, ObsIDs 13658–13669) to construct a sample of 23 solar analogs with sensitive X-ray coverage.

The new and archival Chandra observations of solar analogs are detailed in Table 2. These observations were taken with the ACIS-S array, with the target positioned at the aim point of the S3 chip; the soft-band sensitivity of this back-illuminated chip is helpful for detecting coronal emission (Poppenhäger et al. 2009). All observations were taken using Very Faint telemetry to optimize background removal. The data were reduced using the CIAO software package, version 4.5, using standard techniques that are briefly described below.

Table 2. Chandra Observations of Solar Analogs

Name	ObsID	HJD	ϕ	Exp	Cts	Rate	Flux	Dist	LX	mv	Lbol	$\frac{L_{\rm X}}{L_{\rm bol}}$
HD 4308	12339a	2455622.92	0.097	14.88	20.68	1.39	5.49	22.06	26.66 ± 0.14	6.55	33.58	−6.92
HD 6434	13662	2456202.55	0.430	8.97	<3.12	<0.35	<1.38	41.37	<26.60	7.72	33.65	< −7.05
HD 28185	13664	2456202.67	0.092	8.68	<3.17	<0.37	<1.46	42.34	<26.64	7.80	33.67	< −7.03
HD 30177	13668	2456117.61	0.483	14.88	<3.31	<0.22	<0.87	52.83	<26.62	8.41	33.62	< −7.00
HD 49674	13661	2456263.06	0.960	11.94	42.71	3.58	14.13	44.23	27.67 ± 0.11	8.10	33.58	−5.91
HD 102117	13663	2456196.61	0.991	7.00	4.93	0.70	2.76	39.70	26.87 ± 0.27	7.47	33.73	−6.86
HD 107148	13665	2456248.46	0.704	9.57	6.91	0.72	2.84	51.20	27.10 ± 0.22	8.01	33.74	−6.64
HD 134987	13666	2455937.42	0.385	4.99	2.92b	0.58	2.29	26.21	26.43 ± 0.39	6.47	33.77	−7.34
ρ CrB	12396a	2455944.05	0.644	9.86	5.88	0.60	2.37	17.24	26.07 ± 0.24	5.39	33.82	−7.75
HD 168746	13660	2456226.63	0.590	9.96	<3.17	<0.32	<1.26	42.73	<26.59	7.95	33.60	< −7.01
HD 178911B	13659	2455933.36	0.914	9.94	20.88	2.10	8.29	42.59	27.41 ± 0.11	7.97	33.60	−6.19
HD 187123	13658	2455970.76	0.465	9.76	<3.15	<0.32	<1.26	48.26	<26.70	7.83	33.75	< −7.05
HD 188015	13667	2456171.39	0.872	12.92	19.73	1.53	6.04	57.01	27.52 ± 0.14	8.24	33.74	−6.22
HD 204313	13669	2456222.07	0.786	9.66	<3.12	<0.32	<1.26	47.37	<26.69	7.99	33.67	< −6.98
51 Peg	10825a	2454806.96	0.746	4.92	7.94	1.61	6.35	15.61	26.42 ± 0.21	5.45	33.72	−7.30

Notes. Ordered by R.A. Column details: HJD is the heliocentric Julian Date at the beginning of the observation; ϕ is the radial-velocity orbital phase relative to our line of sight; Exp is the effective exposure in ks; Cts is the net counts within a 2'' aperture centered on the object coordinates at the time of the observation, or else the 95% confidence upper limit; Rate is the net counts per ks; Flux is the unabsorbed 0.2–2 keV flux for a mekal model with log T = 7; Dist is the distance in parsecs calculated from Hipparcos parallax; L_X is the 0.2–2 keV X-ray luminosity in erg s⁻¹ (expressed as a logarithm). ^aFrom archival Chandra data; see Section 2.2. All other Chandra observations are from our Cycle 13 program. ^bHD 134987 is formally a detection at >90% confidence with our conservative methodology but is found by wavdetect; see Section 2.2.

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The data were reprocessed with the EDSER subpixel optimization applied, with Very Faint particle background cleaning applied, with the charge transfer inefficiency correction applied, with time-dependent gain correction applied, and using the most recently available CALDB calibration files (including the updated ACIS contamination model that accounts for condensation on the optical blocking filters). The standard grades of 02346 were retained. Observations were checked for background flaring; point sources were identified on the S3 chip using wavdetect and removed for these purposes, then the deflare script was run on a lightcurve binned to 200 s and any identified intervals of high background were filtered out of the level 2 event file. A 0.8 keV exposure map was used to determine the effective exposure times at the source positions, and to excise edge regions (<1.5%). A 0.15–2 keV image of the S3 chip was created for each observation and used for all subsequent analysis. The coronal emission from these solar-type stars is not expected to extend to harder energies (we verified that at most a few percent of the source counts have energies >2 keV), and insufficient counts are available to justify subdividing this energy band.

The targeted stars are optically bright, with a median v = 7.8. For effective temperatures of 5500–6000 K, these magnitudes are expected to result in approximately one photoelectron per pixel per standard 3.2 s frame exposure registering for the S3 chip near the aimpoint. Each excess photoelectron shifts the bias level by 3.4 eV, slightly altering the observed X-ray energy for a given event. However, for these observations the total number of X-ray counts is too low to conduct spectral analysis, so this slight contamination has no impact on the derived fluxes (i.e., subarray binning was not required). The optical photoelectrons cannot themselves register as an X-ray event, since the low energy cutoff we use of 0.15 keV is ∼44 photoelectrons, corresponding to stars 4 mag brighter.

The 0.15–2 keV images of the targets are provided in Figure 3. Because the targets are relatively nearby, it is necessary to take proper motions into account. The J2000 coordinates were adjusted to the epoch of the Chandra observations using the proper motions measured by Hipparcos (van Leeuwen 2007). The red and green circles show r = 2'' apertures at the J2000 and updated positions, respectively. Source detection is assessed through running wavdetect at a significance threshold of 10⁻⁶ over the S3 chip and separately by calculating whether the number of observed counts is greater than could arise from background fluctuations at 95% confidence (per the Bayesian-derived tables given in Kraft et al. 1991). Five sources are clearly detected: HD 188015, HD 107148, HD 102117, HD 49674, and HD 178911B. Five sources are clearly not detected: HD 204313, HD 28185, HD 6434, HD 168746, and HD 187123. One source is borderline: HD 134987 has three counts and is formally a detection at 90%–95% confidence using the criteria of Kraft et al. (1991), but those three counts are angularly concentrated within ∼12 (the on-axis PSF) and it is confirmed as a detection by wavdetect. One source is confused: HD 30177 has a clear detection but that source is ∼4'' distant from the expected target location; we conservatively consider this to be an unrelated object, most plausibly a wide-orbit companion.

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X-ray counts were extracted from the indicated 2'' apertures with the (nearly negligible) background estimated from nearby source-free regions. Net count rates were converted to unabsorbed fluxes using PIMMS¹⁴ for a fixed plasma/MEKAL model with three components at temperatures of 1 MK, 3 MK, and 10 MK (log T = 6.0, 6.5, and 7.0 or 0.09, 0.27, and 0.86 keV, respectively), with relative flux normalizations of 1:1:1. Solar abundances and an N_H = 10¹⁸ cm⁻² were assumed. This provides energy conversion factors between net rates and fluxes similar to those adopted by previous studies of star–planet interaction; for example, while we use 0.2–2 keV fluxes throughout, the ROSAT PSPC and HRI rates would translate into 0.1–4.5 keV fluxes through conversion by a factor of 5.6 × 10⁻¹² and 2.9 × 10⁻¹¹, comparable to the factors of 6.5 × 10⁻¹² and 2.8 × 10⁻¹¹ adopted by Kashyap et al. (2008). This model also approximately reproduces the relative counts observed within the 0.2–0.45, 0.45–0.75, and 0.75–2.0 keV bands in XMM-Newton observations of solar-type stars (Poppenhäger et al. 2010). The conversion factor from Chandra/ACIS-S (Cycle 13) net count rates to unabsorbed flux (both over 0.2–2 keV) is then 5.59 × 10⁻¹². Minor modifications in these parameters would alter the resulting fluxes by less than the statistical errors. X-ray luminosities are calculated for the Hipparcos distances (van Leeuwen 2007), as for the bolometric luminosities. X-ray luminosities are expressed as logarithms throughout and given in units of erg s⁻¹. We add an uncertainty of 20% to the statistical errors on the X-ray luminosities derived from the Chandra observations to account for inaccuracies in counts-to-flux conversion from using a fixed spectral model.

X-ray luminosities for the remainder of the full FGK MS sample are recalculated from the net count rates given in Poppenhäger et al. (2010)¹⁵ for stars observed by XMM-Newton (including those from Kashyap et al. 2008), or else from ROSAT coverage. For objects lacking Chandra or XMM-Newton L_X values, we searched the RASS bright and faint source catalogs as well as source catalogs generated from pointed PSPC and HRI observations, prioritizing the latter. Net 0.1–2.5 keV ROSAT count rates were converted to 0.2–2 keV fluxes by factors of 3.95 × 10⁻¹² and 2.02 × 10⁻¹¹ for the PSPC and HRI detectors, respectively, while the energy conversion factor for XMM-Newton observations with the pn detector and the thick/medium/thin optical blocking filter is 3.90/2.66/2.43 × 10⁻¹². The XMM-Newton conversion factors were calculated using XSPEC and for the fixed spectral model provide good agreement with the luminosities calculated by Poppenhäger et al. (2010), with a median offset of 0.03 dex and a scatter of 0.10 dex.¹⁶ Approximate L_X upper limits were estimated for undetected stars from the typical RASS faint source catalog limit of 10⁻¹³ erg s⁻¹ cm⁻². In two cases we use literature values for L_X that are based on the median of multiple high-quality observations: $\upsilon$ And has Chandra coverage described in Poppenhäger et al. (2011), while HD 179949 has XMM-Newton coverage described in Scandariato et al. (2013).

We tested for a statistically significant correlation within the 23 solar analogs with X-ray coverage using the Kendall τ statistic implemented within ASURV¹⁷ (Feigelson & Nelson 1985), which provides more accurate results than a Spearman correlation test for small sample sizes. This is a non-parametric ranking test that does not assume any particular functional relationship between the variables. X-ray upper limits are accounted for, but measurement errors are not; all points are equally weighted. We considered four different potential interaction scalings: L_X and L_X/L_bol as activity indicators, versus M_P/a² and 1/a as interaction-strength proxies (Figure 4). The values of τ are 0.28 and 0.21 for L_X and L_X/L_bol as a function of M_P/a², and 0.42 and 0.35 as a function of 1/a. These correspond to probabilities for no correlation of 78%, 83%, 67%, and 73%; the null hypothesis cannot be rejected, and by this metric the solar analogs show no significant evidence for increased X-ray luminosity in hot Jupiter systems.

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We next tested for a statistically significant correlation by assessing whether a positive slope is present in a linear fit (using logarithmic quantities). We use the Bayesian linear regression code of Kelly (2007), which accounts for both measurement errors and upper limits. The resulting best-fit values are reported as the median of 10000 draws from the posterior distribution, with uncertainties corresponding to 1σ given as the standard deviation. The preferred slopes are near zero, specifically β = −0.05 ± 0.23, −0.06 ± 0.24 and β = 0.01 ± 0.29, 0.02 ± 0.30 for L_X, L_X/L_bol as a function of M_P/a² and 1/a. Kashyap et al. (2008) report an enhancement by a factor of ∼4 in X-ray emission for close-in planets, of which they estimate ∼2$^{+2}_{-0.7}$ is due to star–planet interaction, with the remainder attributable to selection effects. The separation between the putative weakly interacting and strongly interacting systems is ∼1.7 dex in M_P/a²; a slope of 0.3 would then produce an increase of 0.5 dex (or ∼3) in X-ray luminosity, so we take the probability of β > 0.3 as the likelihood that interaction is present. The close-in and distant systems in 1/a are separated by ∼1.0 dex, so here we consider β > 0.5 to provide positive evidence of interaction. The posterior distribution of β is shown in Figure 5; in all cases, slopes above these cutoff values are excluded at ≳ 94% confidence.¹⁸ The insets show 500 draws from the posterior distribution for the intercepts and slopes. The linear regression slopes likewise and independently do not produce significant evidence for star–planet interaction in the solar analogs.

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We also tested the significance of our result by conducting simulations, to assess whether 23 objects is sufficient to identify or exclude interaction. For each trial, we fixed the values of M_P/a² at those for the observed solar analogs, and then generated L_X/L_bol for two hypothetical distributions: first, a normal distribution about −7 with scatter 0.7 dex (matching the preferred value fit to the data) and no dependence upon M_P/a², and second, a normal distribution about the line L_X/L_bol = −7 + 0.3 × (log M_P/a² − 0.6), again with 0.7 dex scatter. For the second model the slope is selected such that the increase in L_X/L_bol is 0.5 dex over the 1.7 orders of magnitude separating hypothetically weakly and strongly interacting systems, which matches the ∼2 (4) factor of increased X-ray activity found by Kashyap et al. (2008) after (prior) to controlling for selection effects. In only 7% of cases with no input interaction is the best-fit slope fit to the random realizations greater than 0.3 (i.e., ∼7% Type I error). In only 4% of cases with a significant input interaction is the best-fit slope less than zero (i.e., ∼4% Type II error). These simulations indicate that our sample of solar analogs is sufficiently large in size and dynamic range to guard against randomly pathological distributions and verifies that the results of the correlation and linear regression tests are secure.

Finally, we consider the orbital phase at which the Chandra observations of solar analogs were conducted. It is possible that magnetic star–planet interaction could occur preferentially near a particular phase. For example, a reconnection-induced hotspot slightly leading the sub-planetary point on the stellar surface would cross the line of sight prior to the planet, at a phase slightly less than unity. Based on Ca ii H and K variability in HD 179949 and $\upsilon$ And, Shkolnik et al. (2008) identified ϕ ∼ 0.8 as an orbital phase at which star–planet interaction is preferentially manifested. On the other hand, Pillitteri et al. (2010, 2011, 2014a) find X-ray activity in HD 189733 near ϕ ∼ 0.5 (i.e., when the planet is behind the star). A planet-induced coronal hotspot being dragged across the stellar surface should be visible over ∼0.5 of the planetary orbit, with the projected surface area peaking along the line of sight. We determined the heliocentric Julian date at the midpoint of the Chandra observation, and converted this to the line-of-sight orbital phase using the orbital parameters in the EOD (derived from fitting RV measurements). Figure 6 shows ϕ versus L_X/L_bol for solar analogs hosting close-in (red) and distant (blue) systems. For context, we also plot X-ray luminosities from single-pointing non-RASS coverage of FGK MS stars, generally from XMM-Newton observations. We also show the XMM-Newton monitoring campaign on HD 179949 from Scandariato et al. (2013, their Table 6); for extensive X-ray coverage and discussion of $\upsilon$ And, HD 179949, or HD 189733 we refer the reader to Poppenhäger et al. (2011), Saar et al. (2008) plus Scandariato et al. (2013), and Poppenhäger et al. (2013) plus Pillitteri et al. (2010, 2011, 2014a), respectively. Taking both detection and limits into account, there are no particular phases at which the close-in systems show systematic X-ray enhancements (or large positive-only scatter). While the number of points is not large enough to draw definitive conclusions, there is no evidence from these observations of a preferred phase at which planet-induced enhancements routinely occur in hot Jupiter systems.

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For the full MS sample, there are sufficient stars to compare L_X and L_X/L_bol within four discrete bins of M_P/a² and 1/a. The Kaplan Meier mean is computed within ASURV, taking X-ray upper limits into account, and the X-ray luminosities are centered to L_X − 28 and L_X/L_bol + 6. The results are plotted as large crosses in Figure 7, with the horizontal bars indicating bin boundaries and vertical bars indicating the error on the mean within that bin. While the uncertainties are large, L_X increases by 0.56 ± 0.25 dex from the lowest to the highest M_P/a² bin (with means, as logarithms, of −0.53 and 2.56 M_Jup AU⁻², respectively, a separation of ∼3 dex), and L_X/L_bol similarly increases by 0.60 ± 0.29 across this same interval. The difference is less pronounced for 1/a as a function of X-ray luminosity; from the most distant to the most close-in bins (with means, as logarithms, of −0.32 to 1.36 AU⁻¹, respectively, a separation of ∼1.7 dex), the increase is 0.31 ± 0.27 and 0.57 ± 0.29 for L_X and L_X/L_bol, respectively.

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Testing the MS sample for a correlation with ASURV, the Kendall τ values are 2.37 and 1.91 for L_X and L_X/L_bol as a function of M_P/a², which correspond to probabilities for no correlation of 1.8% and 5.6%; the null hypothesis is rejected at ≳ 2σ, confirming the increase in X-ray luminosity for hot Jupiters in the full sample, in contrast to the results for the solar analogs alone. On the other hand, the Kendall τ values are 1.09 and 1.55 for L_X and L_X/L_bol as a function of 1/a, which correspond to probabilities for no correlation of 28% and 12%; here the null hypothesis cannot be rejected.

Qualitatively similar results are obtained for fitting L_X or L_X/L_bol as a function of M_P/a² or 1/a (again with all quantities expressed as logarithms and centered). The best-fit linear relations (calculated with the Bayesian IDL routine of Kelly (2007) are overplotted in Figure 7 as dashed black lines. This methodology is completely independent of the mean values found within bins with ASURV, but it may be observed that there is good agreement in the trends. Specifically, the preferred slopes, with 1σ errors, are 0.19 ± 0.08 and 0.20 ± 0.08 for L_X and L_X/L_bol as a function of M_P/a², and 0.11 ± 0.13 and 0.22 ± 0.15 as a function of 1/a, with corresponding likelihood over the full posterior for the slope to be greater than zero of 99.6%, 99.3%, 79.8%, and 93.7%. (These preferred slopes are still significantly less than the values of 0.3 and 0.5 that would correspond to a typical increase in X-ray emission by a factor of ∼3 for M_P/a² and 1/a, respectively.)

The significance by which the preferred slope exceeds zero is strongly dependent upon a handful of extreme systems. Specifically, if the six systems (HIP 14810, HD 73256, tau Boo, HD 162020, HD 179949, and HD 189733) with M_P/a² > 450 M_Jup AU⁻² were to be omitted, preferred slopes for all four fits would be consistent with zero (0.06 ± 0.09, 0.04 ± 0.09, −0.10 ± 0.14, 0.03 ± 0.15). HIP 14810 has only a loose X-ray limit, but the other five extreme systems are X-ray luminous and also chromospherically active, with $R_{\rm HK}^{^{\prime }}\gtrsim -4.7$. For completeness, we also fit linear relations to the extreme systems only (solid cyan lines in Figure 7), but because there are only six points the uncertainties in the slopes (dotted cyan lines) are quite large and permit slopes of zero, preventing us from drawing any definitive conclusions.

An older MS star will have lower activity than an otherwise similar but younger counterpart, independent of planetary properties. Because the inferred signatures of star–planet interaction upon the chromospheric emission cores in the Ca ii H and K lines is at the few percent level (Shkolnik et al. 2008), the value of $R^{^{\prime }}_{\rm HK}$ for a given star should be nearly or entirely independent of planetary properties (e.g., Canto Martins et al. 2011; see further discussion in Section 5). We search for excess X-ray luminosity beyond that related to the intrinsic stellar activity by removing the dependence of L_X/L_bol upon $R^{^{\prime }}_{\rm HK}$, as parameterized for MS stars by Mamajek & Hillenbrand (2008). The resulting preferred slopes are consistent with zero at the ≲ 1σ level (0.03 ± 0.06, 0.03 ± 0.06, 0.09 ± 0.12, and 0.09 ± 0.11). The preferred intercepts are modestly negative (−0.23, −0.24, −0.27, −0.26, all ±0.09), perhaps reflective of the (selected) lower activity of RV targeted stars compared to those in the field (see also discussion in Shkolnik 2013). If only the most chromospherically active stars are considered, there are 31 stars with $R^{^{\prime }}_{\rm HK}>-4.8$, of which 16 have X-ray detections, and the preferred slopes for this subset against L_X or L_X/L_bol are consistent with zero at the 1σ level.

An alternative method of testing whether a potential correlation between two variables could be related to a third variable is given by the Kendall partial tau test. This has been implemented for data including censoring by Akritas & Siebert (1996) and we use their methodology to test whether the correlation between L_X and M_P/a² remains significant when controlling for $R^{^{\prime }}_{\rm HK}$. Kelly et al. (2007) identify instances in which this test can produce misleading results, but our sample is not subject to the strong multiple variable correlations that can be potentially problematic, and we use this test only as an additional check. For reference, with the third variable also equal to M_P/a² but summed with a random Gaussian of standard deviation 0.001, τ_{1, 3} = 0.9998 and τ_{12, 3} = 5.6 × 10⁻³, with σ = 7.5 × 10⁻³, properly identifying the third variable as relevant; in contrast, controlling for a uniformly zero third variable produces τ_{1, 3} = 0.015 and τ_{12, 3} = 0.053, with σ = 0.026, properly rejecting the third variable as relevant. Controlling for $R^{^{\prime }}_{\rm HK}$, τ_{1, 2} = 0.038, τ_{1, 3} = 0.058, τ_{2, 3} = 0.12, and τ_{12, 3} = 0.031, with σ = 0.023. This indicates that the null hypothesis cannot be rejected and so the influence of $R^{^{\prime }}_{\rm HK}$ upon the L_X versus M_P/a² correlation cannot be ruled out, consistent with the regression results. For completeness, we verified that a partial correlation with either distance or stellar temperature is rejected. These results are consistent with and reinforce those obtained from the two-variable correlation and linear regression tests.

While magnetic star–planet interaction might provide one explanation for these results, at least for the most extreme systems, it is necessary to consider also selection biases as well as planet and stellar evolution effects.

The full sample of MS FGK planet-hosting stars is susceptible to selection biases; in particular, weakly interacting systems may only be detectable around inactive stars, resulting in a deficit of systems with low-mass, distant planets and high L_X values and inducing a spurious correlation that must be removed prior to evaluating potential star–planet interaction signatures. This incompleteness is apparent in a diagram of the velocity semi-amplitude versus the residual "jitter" noise (Figure 8). Here K is primarily influenced by the planetary properties (the range of stellar masses is less than that in period and planetary mass) while RMS reflects the intrinsic activity of the parent star. There is a dearth of points at low K and high RMS values due to sensitivity limitations in radial velocity searches.

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This bias results in relatively fewer systems at low M_P/a² and high L_X values. We confirm that a significant selection bias is present in our data through testing the scaling of the velocity semi-amplitude K with distance, and find that a positive correlation is present at >3σ whereas in a complete sample no dependence would be expected. (Excluding the six most extreme systems with M_P/a² > 450 M_Jup AU⁻² does not remove the correlation with K).

Restricting consideration to a portion of the K-RMS plane with complete coverage, defined here as 20 < K ⩽ 500 m s⁻¹ and 2 < RMS ⩽ 15 m s⁻¹ (the solid square in Figure 8), produces a subsample of 110 systems, of which 30 have X-ray detections, for which no correlation (slope 0.017 ± 0.028) is now present between K and distance. Four of the five X-ray detected extreme systems with M_P/a² > 450 M_Jup AU⁻² are included in this subset; Tau Boo is excluded with RMS > 15 m s⁻¹. This subsample does show a significant dependence of L_X or L_X/L_bol upon M_P/a² (≳ 99% probability that the slope >0) and the preferred slopes are similar to those found for the full sample (albeit less tightly constrained), which suggests that the RV-sensitivity selection bias is not the sole driver associating enhanced X-ray emission with hot Jupiter systems. If instead the subsample is restricted to 20 < K ⩽ 200 m s⁻¹ and 2 < RMS ⩽ 15 m s⁻¹ (the dotted line in Figure 8; 95 systems, of which 24 have X-ray detections), four of the five extreme systems are now excluded as having K > 200 m s⁻¹ (HD 179949 is retained). Here again there is no correlation between K and distance, but there is still suggestive evidence for a correlation between L_X or L_X/L_bol and M_P/a² (≳ 91% probability that the slope >0, dropping to 80% if HD 179949 is excluded). Removing the dependence of K upon distance through selection of a K-RMS complete subsample does not eliminate the trend of increasing L_X or L_X/L_bol toward greater M_P/a², unless the most extreme systems are all deliberately excluded from the subsample.

We emphasize that the identification of a selection bias within a sample does not necessarily indicate that any correlation with planetary properties is due to that bias. In particular, since K scales with M_P/a^0.5, most proxies for interaction strength, including the M_P/a² preferred here, will correlate with K.

Planet-hosting stars are sometimes present in binary systems (∼10%–20%; Raghavan et al. 2006; Roell et al. 2012). Binarity increases intrinsic X-ray activity even in long-period systems, perhaps related to initial formation (Pye et al. 1994). In addition, any unresolved X-ray emitting secondary companions would inflate the apparent X-ray luminosity of the planet-hosting primary; this is more of a concern for the XMM-Newton and particularly ROSAT observations, which lack the angular resolution of Chandra, but note that 25% contamination would only increase L_X by 0.1 dex. Among our new Chandra targets, HD 188015 and HD 178911B are in wide binary¹⁹ star systems (Raghavan et al. 2006) with projected component separations of 16'' and 13'' (corresponding to 790 and 680 AU), respectively. Both companions are detected in the Chandra observations (Figure 3). The dynamics and evolution of planetary systems are sensitive to binarity; for example, Kaib et al. (2013) demonstrate that outer exoplanets within wide binaries can be destabilized as the system responds to Milky Way tidal forces and passing stars, an effect that could artificially link (surviving) hot Jupiter systems with binary (more active) stars.

We consider whether binarity corresponds with increased X-ray activity in stars flagged as binary in the EOD (since these stars possess sensitive RV measurements required to discover planets, this assessment should also be complete with respect to close stellar companions). The full sample of 198 planet-hosting MS stars contains 36 known binary systems. The Kaplan Meier mean values of L_X − 28 are −0.94 ± 0.09 and −0.58 ± 0.18 for single stars (46/162 detected) and binary (16/36 detected) systems, respectively, calculated within ASURV. The mean values of L_X/L_bol + 6 are −0.56 ± 0.10 and −0.30 ± 0.17. While the distributions of L_X are marginally distinct (probability of 4.2% using the Peto & Prentice test), the distributions of L_X/L_bol are not inconsistent (Peto & Prentice probability 25%). The binary systems also have, on average, modestly more extreme planets, with mean log M_P/a² = 0.92 and mean log 1/a = 0.37 versus 0.35 and 0.21 for the single stars. Among the most extreme hot Jupiter systems, τ Boo and $\upsilon$ And are known binaries (Butler et al. 1997), as is HD 189733 (Roell et al. 2012).

In Section 3.2 we found that a handful of extreme systems with M_P/a² > 450 M_Jup/AU² drive the correlation with L_X or L_X/L_bol in the FGK MS sample. Extreme systems are easier to detect in RV searches (see Section 4.1), so a tendency toward higher X-ray luminosities within this bin is not a selection bias.²⁰ Compared to the solar analogs for which no evidence of magnetic star–planet interaction is found, these extreme systems have similar masses but smaller semi-major axes. In addition, their shorter orbital periods produce faster rotation rates (for tidal locking at a < 0.15 AU; Bodenheimer et al. 2001), which would also increase the interaction energy. The stellar magnetic fields are also likely to be higher, based on the R'_HK values; for example, HD 189733 has a directly measured value of about 30 G (Fares et al. 2010), although for HD 179949 it is only a few Gauss (Fares et al. 2012), comparable to the Sun. We briefly highlight the properties of the X-ray-detected extreme systems and summarize previous phase-resolved studies that searched for or identified apparent star–planet interaction.

HD 73256 is a solar-temperature (T_eff = 5600 K) but active star, with $R_{\rm HK}^{^{\prime }}=-4.5$ and L_X/L_bol = −4.9. Shkolnik et al. (2005) find variation of the Ca ii H and K cores to be modulated with the stellar rotation period (with a flare observed near orbital phase 0.03), but note that the amplitude of the stellar variability could dilute a planet-induced signature. Shkolnik et al. (2008) retain it as a candidate for observable interaction.

Tau Boo is a hot (T_eff = 6400 K) and nearby (d = 15.6 pc) star that is moderately active, with $R_{\rm HK}^{^{\prime }}=-4.7$ and L_X/L_bol = −5.1. (A more recent Chandra observation resolves the secondary and finds a somewhat lower X-ray value for the primary; Poppenhäger & Wolk 2014a). Because the stellar rotation period is identical or nearly so to the planetary orbital period, identification of persistent interaction requires long-epoch studies. Walker et al. (2008) do indeed find that starspot activity is concentrated near a fixed orbital phase. However, as noted by Shkolnik et al. (2008), tidal locking of the star to the planet would produce low relative velocities between field lines, limiting the energy available in magnetic interactions.

HD 162020 is a cool (T_eff = 4800 K) star that is categorized as pre-MS in SIMBAD. Poppenhäger & Schmitt (2011) consequently exclude it when examining the correlation found by Scharf (2010). It is extremely X-ray luminous, with L_X = 29.1 erg s⁻¹ and L_X/L_bol = −3.8, and also has a very massive hot Jupiter, with Msin i = 15.2 M_Jup. There is no measurement of R'_HK available in the literature.

HD 179949 is a hot (T_eff = 6200 K) star that is moderately active, with $R_{\rm HK}^{^{\prime }}=-4.6$ and L_X/L_bol = −5.3. It is the first identified candidate to display star–planet interaction (Shkolnik et al. 2003) and has been extensively investigated since, with possible Ca ii H and K variability phased with the planet at some epochs but not others (Shkolnik et al. 2008), and potential X-ray star–planet synchronicity identified by Saar et al. (2008). It is now clear that the magnetic field configuration is complex (Fares et al. 2012) and so a simple hotspot model might not be applicable. A recent large XMM-Newton plus optical study of HD 179949 did not find significant evidence of star–planet interaction in either X-rays or Ca ii H and K emission (Scandariato et al. 2013).

HD 189733 is a cool (T_eff = 5000 K) star that hosts a transiting hot Jupiter. As such, it is among the most observed exoplanet systems to date, with numerous deep campaigns at optical (ground-based and HST and X-ray (Chandra and XMM-Newton) wavelengths. The star is active, with $R_{\rm HK}^{^{\prime }}=-4.5$ and L_X/L_bol = −4.8. Pillitteri et al. (2010, 2011, 2014a) find cases of X-ray flaring near orbital phase ϕ ∼ 0.5 (i.e., when the planet is behind the star). Shkolnik et al. (2008) find Ca ii H and K emission variability to phase with the star but suggest a residual signature of interaction near ϕ ∼ 0.8; on the other hand, Fares et al. (2010) find no evidence of magnetospheric interactions. Poppenhäger et al. (2013; also Pillitteri et al. 2014a) find that the companion is not X-ray bright and so the system age is likely older than would be inferred from the activity of the primary.

Based on phase-resolved studies of the most promising candidates, in particular HD 179949, the evidence for star–planet interaction is mixed; at best it seems the phenomenon is observable at select epochs. We reiterate that these extreme systems are not X-ray luminous relative to their chromospheric activity (Figure 9), which has been suggested to be the case for star–planet interaction (e.g., Kashyap et al. 2008). It is therefore possible that the most extreme systems are more likely to be active, but for reasons unrelated to direct magnetic interaction.

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Extreme systems are susceptible to cumulative tidal evolution. A close-in gas giant can potentially spin-up its host star, temporarily halting or reversing the usual decline in rotation and dynamo activity with MS age, while gradually shrinking its orbit until infall (Jackson et al. 2009; Debes & Jackson 2010; see also Lanza & Shkolnik 2014 for multi-planet possibilities). There is some observational support for activity rejuvenation in binary systems; for CoRoT-2 and HD 189733, the hot Jupiter hosting primary is active while the companion is X-ray quiescent (Schröter et al. 2011; Pillitteri et al. 2011; Poppenhäger & Wolk 2014a, 2014b). In these cases M/a³ is much larger for the planet than for the companion star, and the lack of simultaneous activity rejuvenation in the companion implies the tidal influence of the planet is more relevant. In addition, τ Boo is apparently tidally locked to the orbit of its hot Jupiter (Shkolnik et al. 2008) at a rapid rotation period of only ∼3.2 days, strong evidence for planetary spin-up. At the same time, extreme systems are tidally unstable on timescales that depend sensitively on the orbital semi-major axis, eccentricity, planetary composition, and stellar size. For example, WASP-18 has a 10 M_Jup planet in a 0.94 day orbit and an estimated age of ∼700 Myr (Hellier et al. 2009; Pillitteri et al. 2014b); the remaining lifetime of WASP-18b is likely quite short, only ∼50 Myr (Hellier et al. 2009).²¹ The relative paucity of hot Jupiter systems with stellar ages above a few gigayears and semi-major axes less than 0.05 AU is cited by Jackson et al. (2009) as reflecting tidal destruction on gigayear timescales. Subsequent age-rotation-activity evolution (unaffected by any remaining outer planets) would then decrease X-ray luminosity by 1–2 orders of magnitude over the MS lifetime (Ribas et al. 2005). If in fact extreme systems do not survive beyond a few gigayears, then they are primarily present around younger stars that are both inherently more active and susceptible to cumulative tidal spin-up activity rejuvenation.

This interpretation is completely consistent with both the lack of star–planet interaction in the solar analogs (which do not feature the most extreme systems, and are screened to exclude active and likely young stars) and the trend toward greater X-ray luminosities with M_P/a² in the full FGK MS sample. It may be that selection biases (acting primarily at low M_P/a²) and cumulative tidal influences (relevant primarily at high M_P/a²) combine to mimic the statistical signature of star–planet interaction. Of course, these effects do not rule out magnetic recombination events producing enhanced activity,²² but they do suggest that this type of interaction is not required to explain the statistical trends in X-ray studies.