Measuring the Galactic Distribution of Transiting Planets with WFIRST

2.1.1. Survey Duration and Cadence

2.1.2. Photometric Noise

We consider photometric noise following the standard CCD signal-to-noise equation. We use the values from the science definition team (SDT) report (Spergel et al. 2015) for the photometric zeropoint of the detector, as well as the bias, read noise, gain, dark current, and sky brightness. This report assumes (perhaps conservatively) an error floor in the photometry of 1 mmag, which we add in quadrature to the calculations from the SDT. The SDT estimates of the noise are presented in AB magnitudes.

We compare the estimated noise to that of Gould et al. (2015), who find that, for saturated stars, the precision in a single observation in the W149 bandpass will scale such that

$$\begin{eqnarray}&&\sigma =1.0\times {10}^{(2/15){H}_{\mathrm{Vega}}-2}\,\mathrm{mmag},\end{eqnarray} \tag{ 1 }$$

where ${H}_{\mathrm{Vega}}$ is the apparent H-band magnitude relative to Vega. For reference, ${H}_{\mathrm{AB}}-{H}_{\mathrm{Vega}}=1.39$ mag.

Near the saturation limit of 16.1 mag, the SDT estimate of the precision is very similar to that of Gould et al. (2015). For significantly brighter stars, the Gould et al. (2015) estimate of the noise is markedly lower than the (Spergel et al. 2015) prescription. As saturated stars make up only a small fraction of the stars in the WFIRST field of view, the choice of noise model does not appreciably affect our results. For consistency, we only consider the Spergel et al. (2015) estimate of the noise throughout this analysis, noting that if the Gould et al. (2015) noise estimate is realized, the performance of WFIRST will be improved at the extreme bright end.

We compare both of these relations with the photometric precision of Kepler in Figure 1. To perform a direct comparison to Kepler, we make two corrections. First, we follow the Kepler convention of considering the average noise of observations binned over six hours, the "combined differential photometric precision" or CDPP (Christiansen et al. 2012). Second, the WFIRST bandpass is significantly redder than the Kepler bandpass. As transit searches focus on FGKM stars, with red colors, these stars appear brighter on the WFIRST detector than they would on the Kepler detector. To provide a fair comparison, for the Kepler stars, we use the H-band magnitude of the stars in the Kepler field.

Zoom In Zoom Out Reset image size

The expectation is that WFIRST will achieve a relative precision of 1 part per thousand (ppt) in a single observation of a 15th magnitude star in the W149 bandpass (0.93–2.00 μm). This is equivalent to 200 parts per million (ppm) when binned over six hours, comparable to the precision of Kepler on a star with $r\approx {K}_{p}=15$. However, since a typical G dwarf has an R − H color of 1.1, the same Sun-like star observed with Kepler and WFIRST would be observed at a higher precision with WFIRST, even after accounting for the typical extinction level of ${A}_{H}\approx 0.7$ mag toward the bulge.

In this work, we assume the photometric noise is white, so that there are no correlations between observations. Correlated noise can be the result of spacecraft systematics or stellar p-modes (Gilliland et al. 2010; Campante et al. 2011). The timescale for p-modes is inversely proportional with stellar density: for G dwarfs, the granulation timescale is approximately five minutes; for M dwarfs, 30 s. Like Kepler data, for most stars observations will be spaced widely enough to capture a random phase of p-mode oscillations during each observation. As WFIRST has significantly larger levels of photon noise, the correlated stellar signals will be small by comparison, causing the while instrumental noise to dominate over any red astrophysical effects. Moreover, as WFIRST observes at much redder wavelengths, the complicating effects of stellar spots will be diminished.

We simulate individual transits by injecting a planetary signal into simulated WFIRST data using the prescription for the photometric noise described in the previous section. Every fifteen minutes, starting at a random phase, we collect an observation of the flux from this system: every twelve hours one observation is taken in Z087, while all other observations are in W149. We model the transit light curve with the transit model of Mandel & Agol (2002). We calculate limb darkening coefficients in each bandpass using the online tool developed and described in Eastman et al. (2013), which interpolates the Claret & Bloemen (2011) quadratic limb darkening tables.

Note that unlike the Kepler mission, each data point will consist of a single 52 s observation (290 s at Z087), rather than a series of binned observations over 30 minutes. Each observation will then sample one specific point on the transit light curve as opposed to an integrated measure of the observed flux, meaning morphological light curve distortions due to finite integration time will be minor in the WFIRST data (Kipping 2010).

To simulate a realistic estimate of the stellar population in the WFIRST microlensing field, we develop a galactic population generated from the online Besançon models of the galaxy (Robin et al. 2003). To convert the returned apparent magnitudes to near-infrared simulated photometry, we apply the transformations of Bilir et al. (2008). We then apply a correction for interstellar extinction assuming the Cardelli et al. (1989) extinction law with R_v = 2.5, following Nataf et al. (2013). From the derived JHK magnitudes, we approximate the W149 magnitude for each star by assuming W149 = (J + H + K)/3.

We then apply a series of corrections to turn the Beasançon models into a realistic simulation of the stars observed by WFIRST. The Beasançon model outputs the properties and numbers of stars along a given sightline within a certain solid angle. Because each simulated field is not a perfect match to the WFIRST field, we weight each simulated star by the fraction of the simulated field that falls in the WFIRST field. We then apply a correction for the mass function in the bulge. The model assumes stars in the bulge follow the Salpeter IMF (Salpeter 1955). We downweight stars of mass $M\lt 0.5$ M_⊙ by a factor of $0.5/M$, which approximates the IMF of Kroupa (2001). We then apply a uniform correction to all stars to match the overall number of bulge main sequence stars near the WFIRST fields as measured by Calamida et al. (2015). Further details can be found in Penny et al. (2016b).

We simulate two planet populations. First, we simulate planets assuming the occurrence rate is the same as for the Kepler field. We assign planets around solar-type FGK stars following the planet occurrence estimates of Howard et al. (2012). We assign planet radii and orbital periods following the "Cutoff Power-Law Model" of Table 5 of that paper, and bulk occurrence rates for each spectral type following the authors' Table 4. For M dwarfs, we follow the relations of Morton & Swift (2014), specifically their "logflat+exponential" model of the period distribution from their Figure 7 and the radius distribution from their Figure 6. This leads to considerably smaller numbers of giant planets injected around M dwarfs than more massive stars, in line with observations from Kepler.

We also simulate planets taking host star metallicity into account. WFIRST will observe stars at large distances along the galactic metallicity gradient (Rolleston et al. 2000; Pedicelli et al. 2009). Observations suggest the average stellar metallicity changes by −0.05 dex kpc⁻¹ along a line of sight moving radially outward from the center of the galaxy. Thus WFIRST is expected to observe stars at preferentially higher metallicities than the solar neighborhood. Indeed, simulations of the WFIRST field suggest the median G2V dwarf observable by WFIRST with W149 < 19.5 has [Fe/H] = 0.26 (Section 3.2). This is significant because radial velocity surveys have unveiled a correlation between giant planet occurrence and stellar metallicity (Fischer & Valenti 2005; Johnson et al. 2010). Note, however, that the presence of small transiting planets does not appear to be affected by the host star's metallicity (Buchhave & Latham 2015).

To account for this metallicity effect, we weight the planets based on their radii and host star metallicity. Following Johnson et al. (2010), who find planet occurrence scales as ${10}^{1.2[\mathrm{Fe}/{\rm{H}}]}$, we modify the likelihood of all planets with radii larger than 5 ${R}_{\oplus }$ by this factor. For the median star ([Fe/H] = 0.25), this factor increases giant planet occurrence by a factor of two.

After all transits have been simulated, we phase-fold on the known period and measure the significance of the observed transit depth. We use the same $7.1\sigma$ threshold as that used by Kepler to decide whether or not a transit is detected. We also require at least two transits during at least one season to be detected.

The $7.1\sigma$ threshold is not strictly appropriate for these simulations. It was chosen for Kepler so that there would not be more than one false positive with three "transits" out of 100,000 stars. WFIRST will observe a thousand times more stars, which would increase the threshold for detection. At the same time, we have required 2 transits to be detected in a single season. Since WFIRST will have six microlensing seasons, this means there will be ∼12 transits over the course of the mission. Therefore, the effective detection threshold over the mission is higher than $7.1\sigma$. Finally, $7.1\sigma$ was chosen in the absence of correlated noise and systematics, so in practice the true threshold for planet detection in Kepler is higher because of the presence of these effects. Thus, we conclude that $7.1\sigma$ is a reasonable benchmark for the detection of planets, but the true threshold will have to be evaluated once the properties of the data are better understood.

Figure 2 shows an example light curve for a Jupiter-sized planet on a 3.0 day orbit around a W149 = 15.0 mag host star with impact parameter b = 0.5. The transit duration is approximately two hours. These transits can be seen by eye, even in the case of single transit events. Over the course of the mission, more than 150 transits of such a hot Jupiter would be observed, leading to approximately 1200 observations during the transit in the W149 bandpass. Moreover, approximately two dozen observations during the transit will be collected in the Z087 bandpass, which might be useful for confirmation of the planetary nature of this signal (Section 4). By fitting transit models and evaluating their likelihood, we measure a transit depth of 0.998 ± 0.002 R_J, assuming perfect knowledge of the stellar host. This planet is detected at $\sim 500\sigma$.

Zoom In Zoom Out Reset image size

Using our simulated light curves we calculate WFIRST's sensitivity to planets as a function of radius and period. We simulate planets with radius and orbital period drawn from log-flat distributions over the ranges $[1,16]$ R_⊕ and $[1,72]$ days, respectively. We then assign an impact parameter for each simulated transiting planet drawn from a uniform distribution over the range $[0,1]$. We assume the host star is a G-dwarf with radius $1{R}_{\odot }$. We then check to see which planets meet our detection threshold. The results are shown in Figure 3. We have chosen to present this calculation in terms of physical parameters rather than transit depth because they are more intuitive. However, since the radius of the host star is fixed, it is trivial to convert to transit depth if desired.

Zoom In Zoom Out Reset image size

We find that, for the brightest stars observed by WFIRST, Neptune-sized planets with orbital periods shorter than one month will be easily detected in a single season of data. The mission will also recover many mini-Neptunes with periods shorter than 20 days, and is likely to recover a small number of planets smaller than 2 ${R}_{\oplus }$ with periods shorter than two days. Over the entire mission, WFIRST will be sensitive to a few Earth-sized planets with orbital periods shorter than two days orbiting the brightest stars. Of the 12 million stars with ${\rm{W}}149\lt 19.5$, the prospects for detecting super-Earths or mini-Neptunes are much lower, but the mission will detect the majority of Neptune-sized planets with periods less than a month and all transiting Jupiter-sized planets in that period range as well. We discuss expected planet yields in Section 3.2.

We use the simulations described in Section 2 to calculate the number of transiting planets that will be detected by WFIRST. We inject planets around the main-sequence dwarf stars brighter than W149 = 21.0. We use the same detection criteria as in Section 3.1, and thus, we limit the range of orbital periods to $P\lt 72\,\mathrm{days}.$

The results are shown in the left-hand panels of Figure 4. Assuming a Kepler-like planet population, we expect WFIRST to detect approximately 13,000 transiting planets orbiting dwarf stars with W149 < 19.5, the majority being giant planets orbiting F and G stars. Similarly, we expect WFIRST to detect 70,000 transiting planets orbiting stars with W149 < 21.0. The mission will also detect approximately 800 planets smaller than Neptune, the majority of which will be orbiting M dwarfs. While large, these numbers are not unexpected given that WFIRST will observe two orders of magnitude more stars than Kepler, which detected several thousand transiting planets.

Zoom In Zoom Out Reset image size

The numbers are even more striking when taking into account the expected metallicity dependence on the occurrence rates of transiting planets (right-hand panels of Figure 4). In this case, we detect more than 150,000 transiting planets over the six seasons of the WFIRST mission. As expected, the number of small planets is unchanged, with the gains made entirely in the population of planets larger than Neptune. WFIRST, by completing this survey, will provide the best assessment of the effects of high metallicity on the population of giant planets, providing clues to the formation and evolution of these systems. With the development of multi-object NIR spectrographs for large telescopes like the VLT, reconnaissance spectroscopy for large numbers of faint stars to measure metallicities will be possible (Cirasuolo et al. 2012). We show the distribution of these planets with respect to the apparent magnitudes of their host stars in the WFIRST bandpass in Figure 5.

Zoom In Zoom Out Reset image size

3.3.1. Single Transit Events

3.3.2. Planets Around Evolved Stars

If WFIRST observes transits from multiple planets around a single star, this by itself significantly increases the probability that the transits are indeed due to real planets rather than astrophysical false positives. The initial Kepler data release contained 444 multiple-candidate systems from a total of ∼1600 candidate systems (see Lissauer et al. 2011, 2012). Thirty-eight of those systems have $r\gt 3{R}_{\oplus }$ and $P\lt 72$ days, making them easily detectable by WFIRST. It would not be surprising for WFIRST to discover more than one thousand planetary systems with multiple transiting planets over the course of its mission.

If there are multiple planets in a system, this opens the possibility of measuring TTVs. TTVs are the deviation of the time of the transit center from a linear ephemeris and are caused by dynamical interactions between the various bodies in the system (Agol et al. 2005; Holman & Murray 2005; Lithwick & Wu 2012). The exact nature of any TTV curve depends on the architecture of any particular TTV system: two planetary systems with identical planets but different orbital eccentricities, arguments of periapse, or longitudes of ascending node would exhibit different TTV signals. They are the most straightforward way for WFIRST to directly confirm the planetary nature of transit signals.

Kepler has had enormous success at measuring TTV signals. They have been used to confirm the planetary nature of transiting signals (Holman et al. 2010; Fabrycky et al. 2012; Ford et al. 2012; Xie 2013), to detect the presence of non-transiting planets (Ballard et al. 2011; Nesvorný et al. 2012, 2013), and to infer masses and eccentricities of planetary systems (Hadden & Lithwick 2014; Jontof-Hutter et al. 2015, 2016). Based on data from the Kepler mission, TTV signals have been analyzed around more than 2,500 KOIs (Holczer et al. 2015, 2016). They have detected 260 KOIs with TTVs on timescales $\gt 100$ days, i.e., likely to be due to a companion rather than an astrophysical false positive. Of these, 163 have TTV amplitudes larger than 15 minutes (see below). Based on the number of planets we expect WFIRST to be able to detect and the timing precision we expect the mission to achieve on individual transits 4.2, hundreds of systems with observable TTVs should be detected over the observing campaign.

We should note that WFIRST TTVs will have a few differences with respect to Kepler TTVs. The longer time baseline (5 years as opposed to 4) will enable the possible detection of transit timing signals with longer periods, such as those due to the Roemer delay from a hierarchical binary star or the orbital evolution of a giant planet in a short orbit (Ragozzine & Wolf 2009; Maciejewski et al. 2016). However, the large gaps between seasons result in degeneracies in the TTV solutions. At the same time, WFIRST TTVs will be less severely affected by starspots. Starspots complicate the measurement of TTVs by distorting the light curve both in and out of transit. Since WFIRST will observe in the near-IR, where the effects of starspots are significantly minimized due to their lower contrast, this reduces the possibility of significant starspot-induced timing errors.

To better understand the detection of TTVs with WFIRST, we simulate transit events in order to estimate the precision to which we will be able to measure transit times. We model our benchmark system after Kepler-9b and Kepler-9c, the first planets confirmed via TTVs (Holman et al. 2010). We use orbital periods for the two planets of 19.2 and 38.9 days. Kepler has shown that less massive planets more often exhibit TTVs than more massive planets (Mazeh et al. 2013), and yet a transit must be detected in order to measure a TTV. Thus, we simulate planets near the bottom of our detectability contours in order to understand a typical TTV signal seen by WFIRST. We assume the two planets are mini-Neptunes with masses of 10 ${M}_{\oplus }$ and radii of 3 ${R}_{\oplus }$. These are significantly smaller than the real Kepler-9 planets leading to smaller TTVs and larger uncertainty in the measured time of transit center. We simulate transits of these planets orbiting a Sun-like star with W149 = 15.0, so that the photometric precision on each data point is 1 part per thousand, assigning impact parameters at random.

First, we consider the precision with which WFIRST can measure the times of individual transits. We focus on the precision for the inner planet, because more transits will be observed over the course of the WFIRST mission. To begin, we fit a transit model to simulated transits for the inner planet, fixing the limb darkening to that predicted by Claret & Bloemen (2011) for a Sun-like star in the H-band but allowing the transit parameters to vary. After simulating many transits, we find a median uncertainty in the measured transit time for each individual transit of 28 minutes. Over the course of a single season, several transits will be observed. If we phase-fold over all observed transits inside an observing season, we find a median uncertainty in the average transit time of the folded transit of 15 minutes.

Given these expectations for the measurement precision, we then consider if WFIRST data can be used to identify interacting systems with TTVs. We use TTVFast (Deck et al. 2014) to integrate our test system as a dynamically interacting planetary system over a simulated WFIRST campaign. The simulated deviations of the transit times are shown as the gray Xs in Figure 6. We then simulate observations of these systems over hypothetical WFIRST seasons. For each observed transit, we add a random offset drawn from a uniform distribution on the range (25 minutes, 40 minutes), similar to the predicted scatter on measurements of the times of individual transits, and assign an uncertainty on the observed time of transit equal to this value (red points in Figure 6). The average time of transit for each particular season is also shown (black points). We purposefully schedule the WFIRST seasons to coincide with the smallest observed TTV signal to simulate a worst-case scenario.

Zoom In Zoom Out Reset image size

Fitting only the transit times of the inner planet, we find that a dynamically interacting planet model fit the data considerably better than a linear ephemeris (${\rm{\Delta }}{\chi }^{2}=64$). In this case, these planets would be easily confirmed via WFIRST observations. It is very difficult to contrive a set of observations of this planetary system that would not have detectable TTVs with WFIRST. However, the inferred masses of the transiting planets are a function of the unknown eccentricity: a pair of 10 ${M}_{\oplus }$ planets or a pair of 25 ${M}_{\oplus }$ planets can both explain the observed TTVs.

Given this simulation, we conclude that WFIRST TTVs can be used to confirm planets, but will not be robust for measuring their masses. However, for the brightest stars, it may be possible to identify particular transits that would be useful for precise determination of planet masses. These transits could then be targeted for observations from other facilities in order to measure their TTVs. Furthermore, based on demographics alone, this method will be best for confirming smaller planets, which are more likely to have companions to produce a TTV signal. In contrast to small planets, most giant planets are most often detected in isolation, without an additional transiting companion (Steffen et al. 2012). However, the detection of TTVs requires the existence of a second planet. So far, there is only one hot Jupiter system with detected TTVs induced by the presence of an additional planet (Becker et al. 2015). We expect only a few of the hot Jupiters detected by WFIRST will be confirmed by this method.

Although Kepler shows that hot Jupiters are unlikely to exhibit TTVs, they can be confirmed by observations of their secondary eclipses. The depth of the secondary eclipse yields a measurement of the brightness temperature, and thus the flux in that bandpass (e.g., Charbonneau et al. 2005). Previous experience from Kepler shows that with Kepler data alone it is difficult to confirm planets via secondary eclipses. While Kepler detected thousands of planets, it was only able to confirm planetary systems via detection of their phase curves and secondary eclipses for a handful of these (e.g. Esteves et al. 2013; Quintana et al. 2013; Angerhausen et al. 2015). Because the Kepler bandpass spans approximately 0.4–0.9 μm, near the peak of a typical stellar spectrum but far bluer than the typical planetary spectrum, only the hottest, largest planets are detectable by their own emission. However, WFIRST, with its primary bandpass spanning 0.927–2.000 μm, will be significantly more effective at observing planetary emission directly.

The bottom panels of Figure 2 show the secondary eclipses of a typical hot Jupiter, which has an eclipse depth equivalent to the transit of a $3{R}_{\oplus }$ planet. As a more extreme example, WASP-12 b's secondary eclipse depth integrated accros the WFIRST bandpass is nearly 2 parts per thousand (Croll et al. 2011; Stevenson et al. 2014a), matching the transit depth of a 4.9 ${R}_{\oplus }$ planet. Figure 3 shows that we expect detections of secondary eclipses of analogous planets to be detected around stars as faint as ${\rm{W}}149=21.0$, by the end of the mission. Even smaller, cooler planets will be detected in secondary eclipse around the million stars with W149 brighter than 15.0.

To determine the feasibility of observing secondary eclipses with WFIRST, we model the secondary eclipses of each transiting planet injected in Section 3.2. We estimate the relative flux of each planet and its host star in the WFIRST bandpass assuming the planet radiates as a perfect blackbody. This decision is a simplification of the nonthermal physics of exoplanet atmospheres. By comparing to theoretical spectra from the BT-Settl spectral library of Baraffe et al. (2015) as well as actual near-IR observations of hot Jupiters, we find it is too optimistic in the observed planet flux by approximately a factor of two, so we divide all estimated fluxes by that factor to account for this approximation.

We assume circular orbits for all planets, so that the duration of the secondary eclipse is identical to the transit duration, and assume no limb darkening or spatial variations in the received flux from the planet itself. We attempt to detect each secondary eclipse in the same manner we detect each transit, declaring the eclipse detected if it is observed at $7.1\sigma$, but making no a priori assumptions about the time of secondary in our search.

By the end of the mission, of the planets detected in our simulations in Section 3.2, we expect to detect approximately 1600 planets in secondary eclipse if the planet occurrence rate is the same as the Kepler field, and 2900 planets if the planet occurrence rate scales with metallicity in the same way as it does in the solar neighborhood. As shown in Figure 7, these eclipses are predominantly around G and K stars at a few kpc. F stars are too luminous relative to the planets for the secondary eclipses to be regularly detected, while giant planets are too rare around M dwarfs to detect their secondaries in significant numbers. WFIRST will detect more secondary eclipses than have been detected for all known exoplanets to date. For many of these systems, WFIRST could plausibly detect phase curves as well, as we discuss in Section 5.3.3.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

If secondary eclipses are to be used to confirm planetary systems, observers will need to be able to separate true secondary eclipses from false positive events. It is unlikely that a background binary would be observed with a period as close to that of a transiting system to mimic a secondary eclipse signal, especially with a five-year time baseline. The only conceivable false positive scenarios involve a single eclipsing binary system masquerading as a transit and secondary eclipse. As a planetary secondary eclipse depth will represent the eclipse of a ∼1000 K body, most stellar-stellar secondary eclipses will be too deep to mimic a stellar-planetary secondary eclipse. The most plausible false positive case is then a significantly blended stellar binary, blended either because of a bright background star, or because the binary itself is in the background of the target star.

As we discuss in Section 5.1, because of the small pixel scale for WFIRST the rates of both of these events should be fairly low, similar to Kepler (Morton & Johnson 2011). Even better, in many cases stellar-stellar secondary eclipses should be discernible from stellar-planetary eclipses as the durations of ingress and egress should be much longer for stellar-sized objects than for planetary-sized objects. Additionally, high-resolution reconnaissance spectroscopy of these systems should enable us to detect spectral signatures of blended stars in each system if they exist. Therefore, the vast majority of candidate planetary secondary eclipse detections should be real events, with a relatively small minority being astrophysical false positives.

Blended light from multiple stars in the PSF or in a given pixel crates two problems for reliably detecting transiting planets. First, the extra light can dilute a real planetary transit causing it to appear shallower than otherwise expected. Dilution could then lower the SNR of observed transit signals, complicating the detection of small planets orbiting faint stars. Second, the dilution can be so severe that a background eclipsing binary mimics a transiting planet signal.

The Kepler mission had smaller pixels on the detector relative to wide-field, ground-based transit searches, making the stellar density per pixel lower (Morton & Johnson 2011). As a consequence, it had a significantly lower rate of false positive transit signals than previous transit search missions from the ground.

The Kepler Input Catalog contains approximately 4.5 million stars brigher than K_p = 21 that were observable by the detector's 94.6 million pixel detector (Haas et al. 2010), meaning on average there were 4.8 stars for every 100 pixels. WFIRST has much smaller pixels than Kepler (011 versus 4''), but the campaign fields are significantly more crowded, containing approximately 300 million stars brighter than W149 = 28 (see Figure 6 of Gould et al. 2015, for an estimate of the stellar density as a function of magnitude). As a result, the field will contain approximately 10 stars per 100 pixels within this limit, for a crowding rate approximately a factor of two larger than Kepler. The Kepler limit is five magnitudes fainter than the limit for the faintest stars in the exoplanet survey (K_p = 16), where here the WFIRST limit is seven magnitudes fainter than the faintest planet hosts considered. Many of these potential blended stars will immediately be able to be ruled out as causes of false positive events, if their contribution to the light curve is less than the observed transit depth. The same techniques used to identify blends for Kepler and ground-based transit surveys should continue to be relevant for WFIRST. In particular, observations of centroid shifts of the photocenter of light during transit events and differences in the depth of alternating transits should provide information about false positives.

Transits of a dark object across the face of a star should be, to first order, achromatic. False positive events caused by eclipsing binaries, where multiple objects are self-luminous, will have wavelength-dependent depth variations as different portions of the stellar SEDs are sampled at different bandpasses. Multiband photometry can then be used to separate transiting planets from background eclipsing binary events.

In the WFIRST mission, one data point will be collected every 12 hr in the Z087 filter, or one data point for every 47 obtained in W149. For the example hot Jupiter transiting a Sun-like star with a three-day period, only 24 data points will be obtained during the transit event in Z087 over the entire mission, approximately one data point for every six transits. The situation will be even worse for planets with longer orbital periods, or those with higher impact parameters and shorter transit durations.

We can assume that the transit ephemeris and orbital parameters are known from the W149 photometry used to detect planetary transit signals. Therefore, we only need to fit three parameters in the Z087 transit model: two to describe the limb darkening and one to describe the transit depth. For this case, fitting the Z087 photometry we measure a transit depth to a precision of 3.7%. Therefore, an 11% difference in transit depth between Z087 and W149 is the minimum detectable difference at $3\sigma$ confidence using data from the entire mission. This is sufficient to rule out many, but not all, stellar false positives.

For example, a false positive M7 dwarf with a temperature of 2900 K and a radius equal to Jupiter's has a flux density smaller than the Sun by a factor of 5.7 in the W149 filter and 11.6 in the Z087 filter, leading to a 9% change in the observed transit depth between the two filters. A moderate increase in the cadence of Z087 observations would be required in order to detect these depth variations to identify false positives. However, as long as the orbit is aligned such that secondary eclipses are observable from Earth, this star would induce a 2 ppt secondary eclipse, easily detectable with WFIRST photometry.

While Z087 photometry may be useful at the current cadence in extreme cases, secondary eclipse photometry will be much more significant, as long as the companion's orbit is aligned such that secondary eclipses are visible. An increased rate of Z087-band photometry, perhaps as often as once every three hours, would provide more opportunities to separate transiting hot Jupiters from self-luminous brown dwarfs or giant planets.

Finally, we note that validation via wavelength dependent transit depth can be complicated by the effects of starspots. This is true both in the case where the planet crosses starspots, affecting the light curve shape, and where starspots are located at different latitudes, affecting the transit depth and out-of-transit flux. Due to the nature of the W149 bandpass, we expect spots to have a minimal effect on the observed light curve. They will be more prevalent in the Z087 photometry, but still diminished relative to the Kepler bandpass.

Although a transit is the most obvious signal in a light curve of a planet orbiting a star, the companion planet affects the observed light curve throughout its orbit. Phase curve variations are the sum of three separate effects: thermal emission, reflected light from the host star, relativistic Doppler beaming, and ellipsoidal variations. These variations have been discussed in previous work as a method to measure planetary masses (Faigler & Mazeh 2011; Shporer et al. 2011; Mislis et al. 2012), to detect new transiting objects (Faigler et al. 2015), and to understand the atmospheres of transiting planets (Knutson et al. 2007; Faigler & Mazeh 2015).

5.3.1. Thermal Emission

5.3.2. Reflected Light

In addition to thermal emission, there is also an observable signal in reflected light from the host star, which often dominates the phase curve signal in Kepler data (Shporer & Hu 2015). For WFIRST we do not expect this signal to be significant. The reflected light signal has amplitude

$$\begin{eqnarray}&&\displaystyle \frac{{F}_{R}}{{F}_{0}}={A}_{\lambda }{\left(\displaystyle \frac{2a}{{R}_{p}}\right)}^{2},\end{eqnarray} \tag{ 2 }$$

where F_R is the amplitude of the signal, F₀ the flux from the star, A_λ the albedo, a the orbital semimajor axis and R_p the planet radius.

While albedos are ∼0.1 in the optical, averaged across W149 the albedo for giant planets is typically $\sim {10}^{-3}$. For the orbital parameters of WASP-43 b, we would then expect the amplitude of the signal to be $\sim {10}^{-7}$, well below what is observable. Typical albedos in the Z087 bandpass will be on the order of 10⁻², so the amplitude of the signal will be an order of magnitude larger, but as the number of observations in this bandpass will be a factor of 50 lower, negating much of this benefit and making systematics harder to identify and mitigate.

5.3.3. Doppler Beaming

As a planet and host star orbit their mutual center of mass, the flux emitted from the star is beamed toward the direction of travel due to the changing the velocity of the host star. A consequence of special relativity, the signal is observable at the non-relativistic speeds at which stars move during their orbits. To first order, the amplitude of the beaming signal is

$$\begin{eqnarray}&&\displaystyle \frac{{F}_{{\rm{D}}}}{{F}_{0}}=(3-\alpha )\displaystyle \frac{{K}_{s}}{c},\end{eqnarray} \tag{ 3 }$$

where F_D is the amplitude of the signal, F₀ the flux from the stationary star, α the shape of the SED at the observed wavelength, K_s the Doppler semiamplitude of the star, and c the speed of light (Loeb & Gaudi 2003). The SED is relevant because, as the star's velocity is modulated, the Doppler shift affects what features of the stellar spectrum fall in our bandpass. For most stars, the W149 filter will fall on the Rayleigh–Jeans tail of the SED, where $\alpha =2$.

For a typical hot Jupiter (${K}_{s}\sim 150$ m s⁻¹), the beaming amplitude will be ∼0.5 parts per million, well below the sensitivity of WFIRST. However, this effect will be useful for detecting more massive objects of similar radii masquerading as hot Jupiters, such as brown dwarfs or very low mass stars. A 50 ${M}_{\mathrm{Jup}}$ object with a three-day period would exhibit a 25 ppm signal. In Section 3.1 we determined that we can measure a transit depth to a precision of 40 ppm. That transit event has a duration of 1.5 hr, whereas the beaming signal occurs throughout the orbit. This implies that beaming will be measured in many of these false positive scenarios. However, the smooth, sinusoidal signature of the beaming signals may be harder to distinguish from instrumental noise than the sharp box-shaped signature of a transit. Moreover, if the secondary is luminous in the WFIRST bandpass, it will exhibit its own Dopper beaming signal which will destructively interfere with the primary signal, reducing the magnitude of the observable effect (Shporer et al. 2010).

5.3.4. Ellipsoidal Variations

Ellipsoidal variations are an achromatic phenomenon caused by changes in the sky-projected shape of a star as a planet orbits, affecting the star's gravitational potential. The signal has twice the frequency of the planet's orbit. Following Loeb & Gaudi (2003), to first order the magnitude of the signal is

$$\begin{eqnarray}&&\displaystyle \frac{{F}_{E}}{{F}_{0}}\sim \beta \displaystyle \frac{{M}_{p}}{{M}_{s}}{\left(\displaystyle \frac{a}{{R}_{s}}\right)}^{-3},\end{eqnarray} \tag{ 4 }$$

Here, β is a term which depends on the nature of gravity darkening for the host star. For Sun-like stars, this value is approximately 0.45. M_p/M_s is the mass ratio between the planet and star and a/R_s is the reduced orbital semimajor axis.

In general, the signal is of a similar magnitude to the Doppler beaming signal, and only likely to be useful in separating brown dwarfs from planets: transiting planets will only be notable by a nondetection of their ellipsoidal variations.

McDonald et al. (2014) note that in the case of Euclid, a color-dependence in observed ellipsoidal variations would be a signature of a background eclipsing binary, as the signal would be achromatic but the relative flux between the foreground and background target would vary between the two bandpasses. The same is true here, although with the cadence of Z087 observations we do not expect this effect to be detectable. In any cases where such an effect would be detectable, variations in the eclipse depth between the bandpasses would also be detectable, likely at a much higher significance. Unlike the Doppler beaming case, because the signal occurs at twice the orbital period, additional signals from a luminous secondary would constructively interfere with the signal from the primary, making the signal even easier to detect.

Ground-based adaptive optics (AO) imaging is typically used to rule out false positive blends of nearby star systems and to understand the level of dilution in transit light curves (Law et al. 2014). In principle, the transiting planet candidates discovered by WFIRST can be followed up by adaptive optics systems on 10 m telescopes, or upcoming 30 m class telescopes that are expected to be built before the WFIRST launch date. These observations may not provide much leverage over the WFIRST data themselves. The diffraction limit of a 30 m telescope in K-band is ∼20 mas. While considerably smaller than the WFIRST pixel scale of 011 pixel⁻¹, this still corresponds to a projected separation of $\approx 20\,\mathrm{au}$ for a Sun-like star with W149 = 14.5, at a distance of $\approx 1\,\mathrm{kpc}$. The diffraction limit also corresponds to a projected separation of 200 au for a Sun-like star at 1 kpc with W149 = 19.5, meaning many bound binary companions will be unresolved even when operating a thirty-meter telescope at the diffraction limit.