Astrometric Detection of Giant Planets Around Nearby M Dwarfs: The Gaia Potential

Introduction

In the search for an answer to one of the most fundamental questions of Mankind (‘Are we alone?’), the nearest stars, within a few tens of pc from the Sun, provide the most obvious target sample. The fast-developing, highly interdisciplinary field of extrasolar planets has recently witnessed an increase in dedicated experiments aimed at cooler, low-mass M dwarfs, in addition to those focused on stars more like our Sun. There are several important reasons for such a change in perspective, which can be summarized under two main themes: a) the shift in theoretical paradigms in light of new observations, and b) the improved understanding of the observational opportunities for planet detection and characterization provided by these stars.

First, the observational evidence gathered by ultra-high-precision space-borne photometric surveys (e.g., Kepler), although still a matter of debate (Fressin et al. 2013), indicates that the frequency of close-in ($P<50$ d) low-mass planets, i.e. Neptunes and Super-Earths, is an increasing function of decreasing stellar mass (Howard et al. 2012). This result has recently been strengthened by the findings of ground-based radial-velocity (RV) programs carried out with state-of-the-art facilities (e.g., HARPS): Super-Earths with $M_{p}\leq 10$ $M_{\oplus}$ within the Habitable ZoneIn its standard definition, the Habitable Zone corresponds to the range of distances from a given star for which water could be found in liquid form on a planetary surface (Kasting et al. 1993) (HZ) of low-mass stars appear ubiquitous (Bonfils et al. 2013). Very recent analyses of Kepler data have only further corroborated this evidence (Dressing & Charbonneau 2013; Kopparapu 2013). The identification of a rocky, habitable planet is the essential prerequisite to its possible characterization as an actual life-bearing celestial object. It is thus clear why low-mass M dwarfs, seen for long as providers of inhospitable environments for life (Huang 1959; Dole 1964), are now being moved at the center of the stage in the exoplanets arena (Scalo et al. 2007; Tarter et al. 2007, and references therein).

Second, the sample of the nearest ($d<25-30$ pc), relatively bright ($J<9-10$) M dwarfs is amenable to combined studies with a wide array of observational techniques, which can be exploited to the best of their potential providing the opportunity to characterize the architecture of planetary systems across orders of magnitude in mass and orbital separations in a way that’s not readily achievable for Solar analogs. For example, the possibility to reach detection of short-period transiting rocky planets from the ground with modest-size telescopes ($30-50$ cm class) is guaranteed by the small radii of M dwarfs, leading to deep transits ($\Delta m\gtrsim 0.005$ mag) for the case of planets with $2\leq R_{p}\leq 4$ $R_{\oplus}$ (e.g., Charbonneau et al. 2009). In addition, as we have discussed above, the favorable mass ratios allow for detection of rocky, potentially habitable planets with the RV technique, thanks to RV signals with amplitudes (a few m s-1) that are readily detectable with the most precise instruments available to-date. Analogously, at intermediate separations ($\approx 1-4$ AU) high-precision astrometry becomes sensitive to planets in the mass range between Neptune and Jupiter (e.g., Casertano et al. 2008). Finally, the favorable planet-star contrast ratios provided by the low intrinsic luminosity of M dwarfs allows for improved detectability thresholds of giant planets at wide separations ($>5-10$ AU) with direct imaging techniques (e.g., Bowler et al. 2012). For the same reason, atmospheric characterization (via occultation spectroscopy) of transiting close-in Super-Earths can be achieved for this sample (e.g., Tessenyi et al. 2012).

ESA’s Cornerstone mission Gaia, with a present-day launch scheduled for November 2013, will carry out a magnitude limited ($V\leq 20$), all-sky astrometric survey (complemented by onboard photometric and partial spectroscopic information) that is bound to revolutionize our understanding of countless aspects of astronomy and astrophysics within our Milky Way, and beyond (e.g., Perryman et al. 2001). The global impact of Gaia micro-arcsecond-level ($\mu$as) astrometric measurements in the astrophysics of planetary systems has been addressed in the past (e.g., Lattanzi et al. 2000; Sozzetti et al. 2001, 2003; Casertano et al. 2008; Sozzetti 2011). However, those studies only provided general metrics for gauging detectability thresholds as a function of planetary properties (orbital elements, masses), using solar-like stars as the reference primaries. In addition, only brief mentions were made of the potentially huge levels of synergy between Gaia astrometry and other ongoing and planned exoplanet search and characterization programs. The approach adopted to carry out the analysis, particularly at the level of single- and multiple-planets orbital solutions, was still affected by some caveats and simplifying assumptions (e.g., only partial treatment or complete neglection of the problem of identifying adequate starting values for the non-linear fits). Finally, the Gaia astrometric performance, described in those works through a simple Gaussian single-measurement error model, has further evolved. A more realistic error model, which takes into account e.g. the dependence on magnitude, ought to be utilized.

In this work, we revisit the topics of planet detection and characterization with Gaia relaxing some of the above assumptions, and focusing on the sample of nearby low-mass M dwarf stars for which Gaia, as one by-product of its all-sky survey, will deliver precision astrometry down to the $V=20$ magnitude limit. The main thrust of this paper is two-fold.

First, we will gauge the Gaia potential for precision astrometry of exoplanets orbiting an actual sample of thousands of known dM stars within $\sim 30$ pc from the Sun (Lépine 2005). We will then express Gaia sensitivity thresholds as a function of system parameters and in view of the latest mission profile, including the most up-to-date astrometric error model. The analysis of the simulations results will also provide insight on the capability of high-precision astrometry to reconstruct the underlying orbital element distributions and occurrence rates of the planetary companions. These results will help in evaluating the expected Gaia recovery rate of actual planet populations around late-type stars.

Second, we will investigate some elements of the synergy between the Gaia data on nearby M dwarfs and other ground-based and space-borne programs for planet detection and characterization, with a particular focus on: a) the potential for Gaia to precisely determine the orbital inclination, which might indicate the existence of transiting long-period planets; b) the ability of Gaia to accurately predict the ephemerides of (transiting and non-transiting) planets around M stars, and c) its potential to help in the precise determination of the emergent flux, for direct imaging and systematic spectroscopic characterization of their atmospheres with dedicated observatories from the ground and in space.

Our paper is organized as follows. In § 2 we describe the adopted simulation setup, and in § 3 we present the statistical and numerical tools used to analyze the simulated datasets. § 4 is devoted to the analysis of the simulation results. Finally, we summarize in § 5 our findings and provide concluding remarks.

Simulation Scheme

The simulation of Gaia observations follows closely the observational scenario described in Casertano et al. (2008). We refer the reader to that source for details. Here we describe and discuss the changes/upgrades made to that setup.

In the representation of the Gaia satellite, the latest nominal Gaia scanning law was utilized, with the two fields of view separated by a basic angle of 106.5 deg, with a spin rate of 60 arcsec s-1, a solar aspect angle between the direction to the Sun and the satellite’s spin axis $\xi=45$ deg, and a precessional period of the spin axis of 63 days. Details on the scanning geometry of the Gaia satellite can be found in, e.g., Lindegren (2010) and Lindegren et al. (2012). The nominal mission duration ($T=5$ yr) was adopted.

The actual list of targets encompasses 3150 low-mass stars (in the approximate range $0.09-0.6$ $M_{\odot}$) within 33 pc from the Sun (Lépine 2005) from the proper-motion limited LSPM-North Catalog (Lépine & Shara 2005). For convenience we are referring to this sample collectively as M dwarfs, even though some of them have estimated masses more compatible with those of late K dwarfs. This subset of the LSPM catalog (dubbed LSPM sub-sample hereafter) is not complete within the identified volume limit, with as much as 32% of stars missing out to 33 pc (Lépine 2005). However, the choice of this catalog over, for example, the more recent Lépine & Gaidos (2011) all-sky catalog was driven by our interest to choose a volume-confined sample (so that distance effects in the detectability of astrometric signals can be more simply taken into account). Using visual and infrared magnitudes available for the sample, we utilized the color-magnitude conversion formulae of Jordi et al. (2010) to obtain $G$-band magnitudes in the Gaia broad-band photometric system. The LSPM sub-sample results to have an average $G\simeq 14.0$ mag. The Delfosse et al. (2000) mass-luminosity relations for low-mass stars were then utilized to obtain mass estimates for all our targets. The LSPM sub-sample results to have an average $M_{\star}\simeq 0.30$ M⊙. In the four panels of Fig. 1 we show the distributions in $G$ mag, distance $d$ Where available, Hipparcos parallaxes are used, photometric distance estimates are otherwise utilized using the Lépine (2005) values. mass $M_{\star}$, and number of Gaia field transits (individual field-of-view crossings) as a function of ecliptic latitude $\beta$ for our LSPM sub-sample. The dependence of the number of Gaia measurements with $\beta$, with the maximum in correspondence of $\beta=\xi$, is a result of the adopted scanning law (see e.g. Lindegren et al. 2012 for details).

The generation of planetary systems proceeded as follows. One planet was generated around each star (assumed not to be orbited by a stellar companion), with mass $M_{p}=1M_{J}$, orbital periods were uniformly distributed in the range $0.01\leq P\leq 15$ yr and eccentricities were uniformly distributed in the range $0.0\leq e\leq 0.6$). The orbital semi-major axis $a_{p}$ was determined using Kepler’s thid law. All other orbital elements (inclination $i$, argument of pericenter $\omega$, ascending node $\Omega$, and epoch of pericenter passage $\tau$) were uniformly distributed within their respective ranges (for the inclination $\cos i$ was uniformly distributed). The resulting astrometric signature induced on the primary was calculated using the standard formula corresponding to the semi-major axis of the orbit of the primary around the barycenter of the system scaled by the distance to the observer: $\varsigma=(M_{p}/M_{\star})\times(a_{p}/d)$. With $a_{p}$ in AU, $d$ in pc, and $M_{p}$ and $M_{\star}$ in M⊙, then $\varsigma$ is evaluated in arcsec. Note that $\varsigma$ corresponds to the true perturbation size only in the case of circular orbits. It is in general only an upper limit to the actual magnitude of the measured perturbation when projection and eccentricity effects are taken into account (e.g., Sozzetti et al. 2003; Reffert & Quirrenbach 2011).

Statistical and Numerical Analysis Tools

The tools utilized in the analysis of the simulated Gaia astrometric data have already been described elsewhere (Casertano et al. 2008). We briefly recall here their main features. First, statistically robust deviations from a single-star model, indicating the presence in the observations residuals of the perturbation due to a companion with a given level of confidence, are identified through the application of a $\chi^{2}$-test or $F$-test (low probabilities of $P(\chi^{2})$ or $P(F)$ signifying likely planet, and unlikely false positive). Then, orbital fits to the data are carried out, using a Markov Chain Monte Carlo (MCMC)-driven global search approach to the identification of good starting guesses for the orbit fitting procedure that combines a period search with a local minimization algorithm (Levenberg-Marquardt). Details on the overall algorithm performance as applied to large datasets of synthetic Gaia observations produced within the context of the Gaia Data Processing and Analysis Consortium (DPAC) http://www.rssd.esa.int/gaia/dpac will be published elsewhere. As described in Casertano et al. (2008), the resulting Gaia observable, the one-dimensional coordinate $\psi$ in the along-scan direction of the instantaneous great circle followed by Gaia at that instant, will then be modeled as $\psi(\alpha,\delta,\mu_{\alpha},\mu_{\delta},\pi,A,B,F,G,P,e,\tau)$, where the five standard astrometric parameters correspond to the actual positions $(\alpha,\delta)$, proper motion components $(\mu_{\alpha},\mu_{\delta})$, and parallax ($\pi$) of each target M dwarf as provided in Lépine (2005), while $A$, $B$, $F$, and $G$ are four of the six Thiele-Innes elements (Green 1985). Planetary masses are derived from the best-fit orbital elements assuming perfect knowledge of the stellar primary mass and utilizing the approximation of the mass-function formula (valid in the limit $M_{p}\ll M_{\star}$):

with $M_{\star}$ in solar-mass units, $P$ in years, $\pi$ and $a_{\star}$ (the semimajor axis of the orbit of the central star around the barycenter) both expressed in arcseconds.

Results

2 Orbit Determination

We show in Fig. 5 the variation of the fractional error on the orbital period as a function of the true simulated value of $P$. As expected, some of the main features of this behaviour already described in Casertano et al. (2008) are recovered. For example, Fig. 5 highlights how Gaia sensitivity decreases significantly both for periods exceeding the mission duration as well as for short-period orbits which are under-sampled (as a direct effect of the scanning law) and translate in very low astrometric signals. On the other hand, well-sampled ($P<T$) orbital periods can be determined with uncertainties of $<10\%$ around the nearest sample ($d<15$ pc, approximately 450 targets). In the same range of periods, the precision improves if a magnitude cut-off ($G\leq 12$, approximately 600 targets) is made, but not to a very significant extent. Bright objects are in fact somewhat affected (in terms of planet detectability and quality of orbit reconstruction) by the presently envisioned gate scheme to avoid saturation on bright stars (see Fig. 2). Instead, at least for the LSPM sub-sample under investigation, the nearest stars ($d<15$ pc) appear to provide the most significant improvement in precision in orbital period determination. The resulting astrometric signatures are typically large enough to allow for good-accuracy orbit reconstruction even for relatively faint objects, for which the per-measurement precision is significantly degraded.

The planetary mass as derived using the mass function approximation will be affected by the uncertainty on $P$, $\pi$, and $a_{\star}$ as obtained from the fitting procedure.We assume here that $M_{\star}$ is perfectly known. While uncertainties in stellar mass at the bottom of the main sequence can easily be on the order of 10-20% (e.g., Boyajian et al. 2012 and references therein), the uncertainties in the model parameters from orbital fits are the dominant source of error when deriving the companion mass in this analysis. Its median for the whole LSPM sub-sample is 1.19 MJ, which reduces to $0.99$ MJ for the sample within 20 pc from the Sun. The two panels of Fig. 6 show how $P$ and $d$ affect the uncertainty on Mp. In particular, planets orbiting stars within $\sim 20$ pc have their masses measured with typical precision of $10\%$, or better, while short- and long-period orbits allow for reduced precision in the derivation of Mp, as expected. For $P>T$, planetary masses are systematically overestimated as a result of the systematic under-estimation of $P$, an effect already shown and discussed in detail by Casertano et al. (2008).

3 Expected Planet Yield

It is worthwhile providing a reference figure of merit on the number of giant planets we can expect Gaia to detect in a given interval of orbital separations, as a way of gauging, in a preliminary fashion, the ability of the survey to reconstruct the underlying orbital elements distributions and occurrence rates in the low-mass star regime. In two recent works, Johnson et al. (2010b) and Bonfils et al. (2013) have provided updated estimates of the fraction of M dwarfs (no distinction in the stellar sub-types given the small-number statistics involved) hosting giant planets within approximately 3 AU. Starting with a northern hemisphere sample observed with HIRES and a southern hemisphere sample observed with HARPS, with different minimum-mass sensitivity thresholds but comparable time baselines, they reach similar conclusions: short-period ($P<100$ days) giants ($M_{p}\sin i>0.3$ $M_{J}$) are quite rare around M dwarfs ($f_{p}<1\%$). At wider separations (roughly, $a<3$ AU), giants orbiting M dwarfs appear to be more frequent: Johnson et al. (2010b) report $f_{p}\simeq 3\pm 1\%$ (corrected for metallicity effects), while Bonfils et al. (2013) obtain $f_{p}\simeq 4^{+5}_{-1}\%$, two estimates which appear consistent with each other, within the error-bars. Note that these values of $f_{p}$ are somewhat higher than those quoted in previous works. For example, Endl et al. (2006) derive an upper limit (at the $1-\sigma$ confidence level) of $f_{p}\simeq 1.3\%$ for giant planets within 1 AU of low-mass stars, while Cumming et al. (2008) infer $f_{p}\simeq 2\%$ for M dwarfs orbited by gas giants within $\sim 3$ AU. It is furthermore worth pointing out how giant planet occurrence rates at intermediate separations ($\sim 3$ AU) around M-dwarf hosts from microlensing surveys (e.g., Gould et al. 2010) appear reasonably in agreement with the above results. Other recent microlensing and high-contrast imaging studies encompassing the range of orbital separations $a<20$ AU for gas giants provide roughly consistent numbers ($f_{p}\simeq 9^{+3}_{-5}\%$ from Cassan et al. (2012) and $f_{p}\simeq 6.5\pm 3.0\%$ from Montet et al. (2013), respectively).

Finally, while theoretical arguments based on the core-accretion model of giant planet formation (e.g., Laughlin et al. 2004; Ida & Lin 2005; Alibert et al. 2011) clearly predict the existence of a trend of decreasing $f_{p}$ with decreasing $M_{\star}$, as observed (Johnson et al. 2010b), the predicted planet fractions in a given stellar mass range do not necessarily agree with the observations. For example, Kennedy & Kenyon (2008) predict $f_{p}\sim 1\%$ within $\sim 2-3$ AU of $M_{\star}<0.5$ M⊙ M dwarfs, a value somewhat lower than the observed fraction. Any discrepancy could point to either insufficient depth in the analysis of the observational data or to the necessity to further the theoretical understanding of planet formation processes in the low-mass star regime. However, at present any attempt to study fine structure details in the comparison between theory and observations is severely hampered by small-number statistics. In this respect, Gaia high-precision astrometry of thousands of nearby M dwarfs will likely help to shed light into the matter, as this unbiased sample screened for giant planets by Gaia will contribute to significantly reduce the uncertainties on the occurrence rate estimates. For example, based on the above simulation results we can infer how precisely a value of $f_{p}=3\%$ for Jupiter-mass companions within 3 AU around M0-M9 stars could be determined by using the number of detections ($n\simeq 2600$) and the number of stars for which an astrometric detection was possible ($N\simeq 9\times 10^{4}$), the latter derived based on the detection efficiency estimates presented in § 4.1 using the Besancon galaxy model. Using this non-parametric description (see e.g., Cumming et al. 2008) and by adopting the standard Poisson uncertainty limits $(1/n+1/N)^{1/2}$ (e.g., Burgasser et al. 2003), we then obtain $f_{p}=3.00\pm 0.06$, an improvement by a factor $\sim 15$ with respect to, e.g., the Johnson et al. (2010b) estimates.

4 Measuring Transiting Systems Configurations

The class of transiting planets is of particular importance, as the simultaneous determination of their masses (via Doppler measurements) and radii (via transit photometry) provides the means to estimate their densities, a fundamental proxy for understanding their interior compositions (e.g., Charbonneau et al. 2007, and references therein). Furthermore, if the primaries are sufficiently bright, transiting planets can be further characterized using the techniques of transmission and occultation spectroscopy to determine the chemistry and dynamics of their atmospheres (e.g., Seager & Deming 2010).

On the one hand, detection of planetary transits is normally achieved via investigation of photometric lightcurves. The general prospects for transiting short-period (giant) planet detection with Gaia using its onboard photometry have recently been revisited by Dzigan & Zucker (2012). On the other hand, Gaia high-precision astrometry, by measuring directly the inclination angle of an orbit (unlike Doppler spectroscopy), can in principle allow to uncover the existence of a possibly transiting planet at wider orbital separations (typically $a>0.5$ AU). We focus here on gauging the potential of Gaia to identify astrometrically extrasolar planets in orbits compatible with transit configurations.

We show in Fig. 8 the fractional error in $i$ as a function of $i$ itself as determined in the simulations. The three cases correspond to the full LSPM sub-sample within 33 pc from the Sun, stars within 15 pc and with planets with $P<T$, and stars with $G<12$ and with planets with $P<T$ (i.e., using the same selection criteria of § 4.2 and the additional constraint of well-sampled orbits). The overall trend confirms the findings of Sozzetti et al. (2001), with the Gaia astrometric observations becoming less sensitive to the inclination itself as we move towards a quasi-face-on configuration ($i\approx 0$ deg), and a corresponding increase of the fractional error on this parameter. The immediate conclusion is that $i\simeq 90$ deg could be determined with uncertainties of just a few degrees for Jupiter-mass companions on well-sampled orbits with $P<T$ around the nearest or brightest M dwarfs. Wider-separation ($a>0.3$ AU) systems with close to edge-on configurations, indicating the presence of a planet that might transit and/or be occulted by its primary, would then become very interesting targets for follow-up photometry, to ascertain whether the prediction is verified or not.

The possibility to study a sample of transiting cold (i.e., long-period) giant planets around nearby low-mass stars is certainly intriguing, for systematic comparison with their strongly irradiated, short-period counterparts. While their typical transit depths (significantly exceeding 0.01 mag) would not pose a challenge even for modest-precision photometric systems, ground-based transit searches lack sufficient sensitivity at long periods due to the impossibility to guarantee continuous coverage over extended time baselines, a necessary prerequisite given that the infrequent transits make it difficult to build enough signal-to-noise ratio. Space-borne instruments can fulfill the requirement of uninterrupted photometric coverage, and indeed Kepler has identified transiting giant planet candidates on $\sim 1$ AU orbits (Fressin et al. 2013). However, these orbit F-G-K dwarfs at hundreds of pc from the Sun, with typical infrared magnitudes of $J\approx 12-13$ mag. If any such objects were detected around low-mass stars in the solar neighborhood (tens of pc), as they would orbit much brighter primaries at infrared wavelengths, they would then constitute prime targets for atmospheric characterization via transit and occultation spectroscopy with future ground-based and particularly space-borne instrumentation.

For the planet’s disc to occult the stellar disc, the orbital inclination must satisfy:

with $R_{\star}$ and $R_{p}$ the stellar and planetary radii, respectively. In general, the geometric transit probability can be expressed as (e.g., Barnes 2007):

Based on the above considerations, for fractional uncertainties on the inclination angle of 10%, 5%, and 2%, Gaia could detect 255, 85, and 10 systems, respectively, formally compatible with transiting configurations within the $1\sigma$ error-bars. On the one hand, using the $f_{p}=3\%$ estimate the expectation is that only 40 systems in the BGM sample would actually have $i$ above the critical value for transits to occur in practice. If the sample of actually transiting systems were entirely composed of systems with sufficiently high values of astrometric signals for which Gaia could deliver orbits with $i$ determined within 10% accuracy, then we find that the sample of candidate transiting planets identified by Gaia would encompass $\sim 85\%$ of false positives. On the one hand, tightening the requirements on the precision with which $i$ can be determined might allow to select a smaller sample of candidates with fewer false positives. On the other hand, it might well happen that a candidate system in transit with $i$ measured less precisely is in fact transiting, while one with $i$ more accurately measured in fact is not. This will depend in practice on the actual shapes of the period and mass distributions, on the details of planet frequency as a function of spectral sub-type, and distance from the Sun of the actual M dwarf sample that will be observed by Gaia.

Possibly transiting giants planets uncovered astrometrically by Gaia will have to be confirmed by means of follow-up photometric observations, that could readily be carried out from the ground even with modest-size telescopes. In perspective, any experiment designed for this purpose will also have to keep the above caveats into consideration. For example, such studies would benefit from the availability of additional Doppler measurements aimed at improving the accuracy of the orbital solutions and the corresponding transit ephemeris predictions. It will also be important to dentify the correct balance between size of the candidate sample and expectations of false positive rates, as such issues could have a significant impact on follow-up programs to verify the actual transiting nature of the detected systems. Finally, note that Gaia-detected intermediate-separation giants on orbits compatible with transit configurations might also help revisit the photometric light-curve databases of existing (e.g., MEarth, Nutzman & Charbonneau 2008; APACHE, Giacobbe et al. 2012; Sozzetti et al. 2013) and upcoming (NGTS, Wheatley et al. 2013) ground-based surveys focusing on late-type dwarfs as well as those of other successful programs, such as Super-WASP, HATNet, and HATSouth, looking for missed or uncategorized transit events.

5 Predicting Giant Planets’ Location and Brightness

As shown by Benedict et al. (2006) using HST/FGS measurements of the $\varepsilon$ Eridani system, astrometry, by determining the full orbital geometry and mass of a planetary companion, can have significant value for future direct-imaging programs and for the interpretation of emergent flux measurements. For example, by determining the times, angular separation and position angle at periastron and apoastron passage, it will be possible to predict where and when a planet will be at its brightest (this is also relevant for eccentric planets which can undergo orders of magnitude of variation in apparent brightness along the orbit), thus a) crucially helping in the optimization of direct imaging observations and b) resolving at least in part important model degeneracies in predictions of an exoplanet apparent brightness in reflected host star light as functions of orbit geometry, companion mass, system age, orbital phase, cloud cover, scattering mechanisms, and degree of polarization (e.g., Sudarsky et al. 2005; Burrows et al. 2004; Madhusudhan & Burrows 2012).

The first element of the synergy between Gaia astrometry and future direct-imaging projects consists in being able to quantify the accuracy with which it will be possible to predict where to look around a given star, based on the companion mass and orbital parameters determination. We show in Fig. 9 the average rates of degradation $\Delta\varrho$ and $\Delta\vartheta$ in the estimated orbital separation $\varrho$ and position angle $\vartheta$ of the planet (expressed in mas yr-1 and deg yr-1, respectively) as a function of the orbital period. On average, the knowledge of the planet’s ephemeris will degrade at rates of $\Delta\varrho<1$ mas yr-1 and $\Delta\vartheta<2$ deg yr-1, for orbits with $P<T$. These numbers are over an order of magnitude smaller than the degradation levels attained by present-day ephemerides predictions based on mas-level precision HST/FGS astrometry (Benedict et al. 2006). In the present sample of intermediate-separation giant planets around M dwarfs (see Table 1), one could then conclude that at least two objects, GJ 832b and GJ 433c, with typical separations of $0.7^{\prime\prime}$ and $0.4^{\prime\prime}$, respectively, represent prime candidates for such an investigation when Gaia data will become available, particularly if combined with existing radial-velocity datasets (thus improving the accuracy of the ephemeris predictions). In this regime of orbital separations instruments such as SPHERE on the VLT (Kasper et al. 2012) and particularly PCS on the E-ELT (see https://www.eso.org/sci/facilities/eelt/instrumentation/) are in fact expected to achieve very high contrast ratios.

A second element of the above mentioned synergy relates to the effectiveness with which a precise knowledge of the companion mass and orbital geometry of the system from astrometry can be used to predict accurate times of optimal visibility for direct imaging and eventually help discriminate between different atmospheric compositions. Using as an illustrative example that of an isotropically, perfectly reflecting Lambertian surface (e.g., Burrows 2005; Madhusudhan & Burrows 2012), the left panel of Figure 10 shows how the error in the planetary phase function $\Phi(\lambda)$ varies as a function of the error in the derived phase angle $\lambda$ based on the orbital parameters determined for each of the 1-MJ companions around the LSPM M-dwarf sub-sample. In the Figure, for each target in the LSPM sub-sample the errors $\Delta\lambda$ and $\Delta\Phi(\lambda)$ are defined as the difference between the orbit-averaged derived value and the orbit-averaged true value of each quantity. For both $\lambda$ and $\Phi(\lambda)$ the median differences are very close to zero, indicating that there is little bias in both estimates. Based on the distribution of the absolute differences for the two quantities, the median uncertainty on the phase results to be $\Delta\lambda\sim 7$ deg, and that on the phase function $\Delta\Phi(\lambda)\sim 0.05$.

The rms uncertainty on the phase-averaged phase function is representative of the quality with which this quantity could be determined based on Gaia astrometry (assuming a specific atmospheric model), but not at all phases. The right panel of Figure 10 shows how the error in the phase function varies with $\lambda$. In the plot, each point corresponds to the median of the absolute differences between the derived and the true value of $\Phi(\lambda)$ in each 10-deg bin in $\lambda$ (considering only the subsets of the LSPM sample with planets whose orbits have been sampled in any given interval in phase angle). Note that, depending e.g., on orbit geometry and details on the atmospheric scattering properties the value of $\Phi(\lambda)$ can vary by a factor of 2 or more at a given phase angle (Sudarsky et al. 2005; Madhusudhan & Burrows 2012) for intermediate-separation giant planets. Based on the result shown in the right panel of Figure 10, then for a value of the phase angle of say $\lambda=70$ deg, it would be possible to distinguish, on average, between the phase function of a Lambert sphere ($\Phi(\lambda)\sim 0.5$) and that of Jupiter ($\Phi(\lambda)\sim 0.2$) at the $2-\sigma$ level (see e.g. Figure 3 of Sudarsky et al. 2005). However, a difference between $\Phi(\lambda)\sim 0.9$ for a Lambert sphere and $\Phi(\lambda)\sim 0.6$ for Jupiter at $\lambda=30$ deg might still fall within the typical errors.

Summary and Conclusions

In this work we report results from a detailed numerical experiment designed to assess the potential of ESA’s Cornerstone mission Gaia to detect and characterize astrometrically giant planetary companions to our closest neighbors, the reservoir of cool low-mass M dwarfs within $\sim 30$ pc from the Sun. The paper was motivated by the need to revisit and update Gaia’s planet detection potential now that we are within a few months from launch, relaxing some of the caveats and simplifying assumptions of previous analyses (e.g., using an up-to-date Gaia error model and employing an actual list of stars in input). A second aim of this work was to begin shedding light on some of the potentially relevant synergies between Gaia astrometry and other ongoing and planned planet detection and characterization programs, both from the ground and in space. The results obtained in this work have been specifically tailored to a sample of nearby, low-mass M dwarfs. The main findings in this experiment can be summarized as follows:

for detected giant planets with periods in the range $0.2-5$ yr (i.e., with accurately determined masses and orbits), inclination angles corresponding to quasi-edge-on configurations will be determined with enough precision (a few percent) so that it will be possible to identify candidate transiting planets in a regime of orbital separations which is inaccessible from the ground and only marginally probed from space by dedicated transit discovery missions such as CoRoT and Kepler. Based on the BGM sample results, Gaia might be able to measure accurately the orbits of 10 potentially transiting intermediate-separation giants around nearby M dwarfs. Considering inclination angles determined with 10% accuracy, the sample of ‘astrometric’ candidate long-period transiting planets might encompass more than 250 systems. However, the majority of these candidates ($\sim 85\%$) would be likely false positives. Ground-based monitoring campaigns will be instrumental in unveiling the true nature of the systems.;

for well-sampled orbits ($P<T$), the uncertainties on planetary ephemerides, separation $\varrho$ and position angle $\vartheta$, will degrade at typical rates of $\Delta\varrho<1$ mas yr-1 and $\Delta\vartheta<2$ deg yr-1, respectively. These are over an order of magnitude smaller than the degradation levels attained by present-day ephemerides predictions based on mas-level precision astrometry;

Planetary phases will be measured with typical uncertainties $\Delta\lambda$ of several degrees, resulting (under the assumption of simple purely scattering atmospheres) in phase-averaged errors on the phase function $\Delta\Phi(\lambda)\approx 0.05$, and expected phase-averaged uncertainties in the determination of the emergent flux of well-measured, intermediate-separation ($0.3<a<2.0$ AU) giant planets of $\sim 20\%$. The combination of detailed models of giant exoplanets’ systems and reliable ephemerides from Gaia astrometry could then greatly help both in the selection of good targets for direct-imaging instruments and for the physical interpretation of positive observational results.

Our findings constitute a first step in the characterization of the full impact of Gaia astrometry in the realm of exoplanets orbiting low-mass stars. Indeed, several important issues will be worthy of future investigations, particularly now that the launch of Gaia is looming very close. For example, it would be valuable to provide an assessment of the effectiveness of the combination of Gaia data with high-precision RVs for the sample of objects listed in Table 1, and an extension of this study to multiple-systems configurations (such as the GJ 876 system) also ought to be carried out. We have assumed all stars in the Lépine (2005) LSPM sub-sample used here to be single, but this is not likely to be a realistic approximation, and the problem of astrometric planet detection in the presence of orbital motion induced by a distant companion star (e.g., Sozzetti 2005) will have to be tackled eventually. It might also be worthwhile to investigate to which extent accurate orbital solutions indicating potentially transiting intermediate-separation giant planets could allow to infer precise transit times for successive photometric follow-up. The fine details of the synergy resulting by the actual combination of Gaia and direct imaging devices data for improving the interpretation of observables in reflected light (phase curves, geometric albedos, polarization parameters) of extrasolar planets in terms of the underlying scattering mechanisms and in turn chemical and thermal properties of their atmospheres have also been left largely unexplored. Nevertheless, the results presented here help to quantify the actual relevance of the Gaia observations of the large sample of nearby M dwarfs in a synergetic effort to optimize the planning and interpretation of follow-up/characterization measurements of the discovered systems by means of transit photometry, and upcoming and planned ground-based as well as space-borne observatories for direct imaging (e.g., VLT/SPHERE, E-ELT/PCS) and simultaneous multi-wavelength spectroscopy (e.g., EChO, JWST).

Acknowledgments

We thank U. Abbas, D. Busonero, and A. Spagna for helpful discussions. This research has made use of the VizieR catalogue access tool, CDS, Strasbourg, France, and of NASA’s Astrophysics Data System. We gratefully acknowledge partial support from the European Science Foundation (ESF) within the Gaia Research for European Astronomy Training Research Network Programme. This work has been funded in part by ASI under contract to INAF I/058/10/0 (Gaia Mission - The Italian Participation to DPAC). An anonymous referee provided a thorough, critical review, and very valuable comments and suggestions that significantly improved an earlier version of the manuscript.

References