IS THE GALACTIC BULGE DEVOID OF PLANETS?

Rather than ask the binary question of whether or not there are planets in the bulge, we can ask are there fewer planets in the bulge than the disk? We have weighted the bulge hosts in our model by an additional factor ${f}_{\mathrm{bulge}}$, the abundance of planets around bulge hosts relative to the abundance around disk hosts, and repeated the AD test at various values of ${f}_{\mathrm{bulge}}$. We plot the results in Figure 2 as the dark blue solid line.⁹ A² rises as ${f}_{\mathrm{bulge}}$ increases, and crosses the p = 0.05 and p = 0.01 lines at ${f}_{\mathrm{bulge}}=0.32$ and $0.54$, respectively. The KS test statistic behaves similarly, but crosses the p = 0.01 line at ${f}_{\mathrm{bulge}}=0.93$. This result suggests that the abundance of planets in the bulge relative to the disk is less than $0.54$ at 3σ. Before we can draw this conclusion however, we must more critically examine our assumptions regarding both the data and our model and also reject any other potential explanations.

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Given the difficulty in measuring the distance of planet hosts, there are a number of ways in which the distribution of distance estimates could be altered in a way that we have not modeled. Relative proper motions are measured routinely in planetary microlensing events, and stellar kinematics are correlated with population, so we can use the distribution of measurements of proper motionas a consistency check on our findings from the previous subsections. Figures 3 and 4 show the results of the same exercise performed with measurements of the relative proper motion.

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The distribution of proper motions should be much more robust to catastrophic errors than the distance distribution, but due to the significant overlap of the distributions of proper motions of bulge and disk stars, it will have less statistical power to disentangle the contributions of each population. Recall also that we are not comparing relative proper motions in the same frame, so it is possible that we are biasing the result.

The lack of statistical power is borne out, with a large range of ${f}_{\mathrm{bulge}}$ producing AD test statistics with large p-values. However, the AD statistic does rise toward smaller values of ${f}_{\mathrm{bulge}}$, reaching $2.704$ at ${f}_{\mathrm{bulge}}=0$, which corresponds to $p=0.039$ and indicates tension with the hypothesis of no bulge planets. However, the distribution of proper motions does not rule out all of the space with ${f}_{\mathrm{bulge}}\lt 0.54$ that is consistent with the distance distribution.

To investigate whether other physical factors could cause the observed distribution of distance estimates, we considered toy models of the planet abundance f as a function of host metallicity [Fe/H], host mass M, and planet semimajor axis a:

$$\begin{eqnarray}&&f\propto {10}^{\alpha [\mathrm{Fe}/{\rm{H}}]},\quad f\propto {M}^{\beta },\quad f\propto {a}^{\gamma },\end{eqnarray} \tag{ 5 }$$

where

$$\begin{eqnarray}&&f=\displaystyle \frac{{d}^{3}N}{d[\mathrm{Fe}/{\rm{H}}]{dMda}},\end{eqnarray} \tag{ 6 }$$

following a long tradition of using power laws to model the distribution of exoplanet abundances (e.g., Stepinski & Black 2000; Lineweaver & Grether 2003; Fischer & Valenti 2005; Johnson et al. 2010). Figure 5 plots the AD and KS test statistics as a function of each parameter α, β, and γ. If a value of α, β, or γ different from our fiducial value of 0 significantly reduces the value of the AD test statistic, then this would indicate that, rather than a different abundance of planets in the disk and bulge, a dependence of planet abundance on intrinsic stellar properties or system architecture may be responsible for the discrepancy that we see between the model and the observational data.

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A value of $\alpha =0.9$ improves the model's match to the distribution of distance estimates, but only to a value of ${A}^{2}=5.263$ with $p=2.2\times {10}^{-3}$. Interestingly, negative values of $\alpha \leqslant -1$ are rejected by the AD test with p ≤ 0.01 for the ${\mu }_{\mathrm{rel}}$ distribution as well as the ${D}_{{\rm{l}}}$ distribution. Were there a strong negative correlation of planet abundance with metallicity, the fraction of stars in the low-metallicity tail of the bulge's broad metallicity distribution would vastly overproduce planets compared to the disk, to the extent that it would even affect the distribution of relative proper motions of planetary microlensing events.

Allowing non-zero values of either β and γ does not significantly reduce A² (though implausibly strong negative correlations do slightly increase A²), which implies that these parameters do not significantly affect the distribution of distance estimates. Note, however, that our model has some shortcomings for both these parameters. First, for both the disk and bulge H14 do not simulate events with host stars more massive than $1\,{M}_{\odot }$. This cutoff is reasonable for the older bulge, where stars more massive would have evolved off the main sequence, but for the younger disk more massive main-sequence stars will still be present. If there were a dependence of planet abundance on host mass, the relatively longer lever arm of the mass distribution in the disk could have an effect on the distance distribution. Second, H14 only explores a coarse grid of semimajor axis, so it is not clear whether a smooth distribution of semimajor axis would change the result. Because the Einstein ring radius ${r}_{{\rm{E}}}$ depends on ${D}_{{\rm{l}}}$ and the sensitivity of microlensing to planets is strongly peaked around separations of $1{r}_{{\rm{E}}}$, a dependence of planet abundance on semimajor axis would affect the distance distribution of planet hosts.

Given that radial velocity surveys have found a correlation between both host mass and metallicity, we can ask how such a correlation would change our findings from Section 5.1. With no correlation with mass or metallicity (α = β = 0), we found 95% limits of $0.02\leqslant {f}_{\mathrm{bulge}}\leqslant 0.32$ from AD tests on the distributions of ${D}_{{\rm{l}}}$ and ${\mu }_{\mathrm{rel}}$. Weighting by the trend of Johnson et al. (2010) between mass, metallicity, and planet abundance (α = 1.2, β = 1.0) produces a qualitatively similar model distribution to our fiducial model, but the number of disk hosts at distances of 3–5 kpc is enhanced at the expense of bulge planets between 6 and 9 kpc. This enhancement of planets in the inner disk is due to the radial metallicity gradient from APOGEE rising above the mean metallicity of the bulge in this region. The AD and KS statistics as a function of ${f}_{\mathrm{bulge}}$, including the mass and metallicity correlations, are plotted in Figures 2 and 4 with lighter shades. The 95% confidence limits in this case are $0.06\leqslant {f}_{\mathrm{bulge}}\leqslant 0.32$. The 1% upper limits on ${f}_{\mathrm{bulge}}$ increase from $0.54$ in the no-correlation case to $0.7$ with the observed correlations. Note that ${f}_{\mathrm{bulge}}$ has a slightly different interpretation in the two scenarios; with no mass or metallicity correlations, ${f}_{\mathrm{bulge}}$ can be interpreted as the ratio of planets per star in the bulge relative to that in the disk, but with the correlations, ${f}_{\mathrm{bulge}}$ is only the contribution to this ratio that has not been accounted for already by mass and metallicity correlations.

It is certain that our model is inaccurate at some level. The Galactic model of Han & Gould (2003) used by H14 is relatively old now in comparison to many new results on the structure and kinematics of the bulge and disk over the intervening time (e.g., Clarkson et al. 2008; McWilliam & Zoccali 2010; Nataf et al. 2010; Wegg & Gerhard 2013). Additionally, the disk of Han & Gould (2003) does not have a central hole, so one might consider disk stars near the Galactic center to belong to the bulge. But the effect of a lack of a disk hole might be cancelled by not also accounting for stars that were born in the disk and later swept up by the bar. Despite these problems, it is difficult to imagine how the various inaccuracies in the model could combine to form an error of at least a factor of three (for our 95% confidence limits accounting for the mass and metallicity correlations of Johnson et al. 2010) in the relative microlensing event rates between the disk and bulge that would be necessary to explain the disagreement between model and data.

It is possible that the simulations of H14 do not accurately reflect the process by which most of the planets were found, and therefore alter the distribution of distances. The simplest way in which such an error may manifest itself is by an additional dependence of planet detection efficiency on the parameters of the microlensing events. We can again use toy power-law models to test how a term unaccounted for in the detection efficiency behaves. The event timescale ${t}_{{\rm{E}}}$ and impact parameter ${u}_{0}$ are the most likely culprits, but we also test s and q, and a derived parameter $| s-1/s|$, which describes the approximate position of the planetary caustics and has a minimum at s = 1 where microlensing is most sensitive; the results are shown in Figure 6.

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An additional weighting of ${t}_{{\rm{E}}}^{\sim 1}$ improves the match for the ${D}_{{\rm{l}}}$ distribution (p = 0.023), but at the cost of increased tension with the ${\mu }_{\mathrm{rel}}$ distribution (p = 0.013), and would not completely solve the problem. Such a weighting is a potentially plausible explanation for the lack of planets at large ${D}_{{\rm{l}}}$ because bulge–bulge lensing events tend to have shorter timescales, but longer timescales allow for more time to organize follow-up observations of high-magnification events, which the survey-only simulations of H14 would not capture. Applying a ${t}_{{\rm{E}}}^{+1}$ weighting also brings the median of the distribution of timescales of model events to 39.5 days (instead of 23.5 days for the unweighted distribution), which is more in line with the median timescale of the sample, which is 40 days.

Despite high-magnification events being over-represented in the observed sample relative to the simulations, there is no improvement offered by an inverse scaling with the impact parameter $| {u}_{0}|$. A power law in s does not provide any improvement. A strong negative power of $\,| s-1/s{| }^{-1\to -2}$ does provide an improvement, which would imply that there was a selection effect that favored planets closer to the Einstein ring. This might be a plausible explanation if many putative discoveries of planets in high-magnification events are not published because of potential degeneracy with stellar mass binaries; alternatively, planets with large $| s-1/s|$ may be missed because follow-up observations tend to cover only the higher-magnification portions of light curves. Finally, an inverse power law with mass ratio ${q}^{\lesssim -0.5}$ could explain the differences between the data and the model, but this seems implausible because we already account for the measured planetary mass function of Cassan et al. (2012), and this would require that the mass function be roughly twice as steep. Furthermore, a slope twice as steep as that of Cassan et al. (2012) is inconsistent by more than 3σ with the constraints on the slope of the planetary mass function reported by Clanton & Gaudi (2016), which were derived from a joint analysis of results from microlensing, radial velocity, and direct imaging surveys. Finally, we also tried increasing the Δχ² threshold for the model and found no significant change in the test statistics until small-number statistics in the model became an issue.

It is also possible that the distance estimates we use are the cause of the disagreement between the data and the model. As mentioned in Section 2, the authors who compute Bayesian estimates of the distance and mass would caution us not to use them for statistical studies. However, we should also cast a critical eye upon the parallax measurements that are generally thought to be accurate.

The constraints that shape the posterior distribution of the Bayesian distance estimate come primarily from the measurements of the angular Einstein radius and proper motion that are routine for most planetary microlensing events.¹⁰ ${\theta }_{{\rm{E}}}$ and ${\mu }_{\mathrm{rel}}$ are derived from measurements of the crossing time of the source radius t_*, ${t}_{{\rm{E}}}$, and the source color. t_* can essentially be "read off" from the duration of finite-source effects in the light curve. Though there is some sensitivity to the choice of limb darkening parameters, it should be determined very robustly given sufficient coverage over the finite-source features. ${t}_{{\rm{E}}}$ and the source color can be smoothly degenerate with blending, which means that uncertainties should account for this degeneracy; however, discrete degeneracies between different models in the q–s plane can cause catastrophic errors in ${\mu }_{\mathrm{rel}}$ and ${\theta }_{{\rm{E}}}$. Bayesian distance estimates can incorporate the multimodal posteriors that such a discrete degeneracy would produce, but our analysis cannot. Some events in the observed sample are affected by such a discrete degeneracy (e.g., MOA-2007-BLG-192, Bennett et al. 2008), but they are in the minority.

According to our model, the distribution of distances as a function of ${\theta }_{{\rm{E}}}$ is monotonic, and roughly linear in $\mathrm{log}({\theta }_{{\rm{E}}}/\mathrm{mas})$ with a slope of ∼10 kpc dex⁻¹ and an rms scatter of ∼1.3 kpc in each 0.1 dex bin (see also Figure 7 below); the dominant contributor to this scatter is the mass distribution of the hosts. Other Galactic models used as priors in Bayesian distance estimates will likely have similar properties. Often ${\theta }_{{\rm{E}}}$ measurements are limited by a systematic error in the conversion of the source color to an angular source size (e.g., Yoo et al. 2004) with a magnitude of ∼0.05–0.1 in $\mathrm{ln}({\theta }_{{\rm{E}}}/\mathrm{mas})$. This implies that, with only a ${\theta }_{{\rm{E}}}$ measurement, it should be possible to localize the distance of most lenses to within ∼2 kpc or less. Therefore, we conclude that while the Bayesian estimates of mass are likely to be unreliable, and are likely to conform to our prior on the mass, the Bayesian distance estimates are likely to be much more useful.

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The precision of the Bayesian distance estimates is certainly enough to conduct the present study. Even if one were suspicious of the exact form of the distance distribution that Bayesian estimates use as priors, with such uncertainties a more crude test that asks whether the number of hosts with distances less than 5 kpc (beyond which bulge hosts should begin to contribute) is consistent with the model assuming binomial statistics, should be robust. When this test is performed, the model predicts $9.0\pm 2.5$ hosts closer than 5 kpc, and the model weighted by the trends in mass and metallicity abundance of Johnson et al. (2010) predicts $9.4\pm 2.6$ hosts. There are $17$ hosts closer than 5 kpc in our sample, implying that our fiducial model and the model weighted by the trends in mass and metallicity are inconsistent with the data at 3.2σ and 3.0σ respectively.

The final ingredient to examine is the parallax distance measurements. As can be seen in Figure 1, all of the $12$ hosts in events with parallax measurements are within 5 kpc, which is not surprising given that there is a strong bias toward nearby lenses as discussed in Section 2. Of the $12$ hosts, four are between 4 and 5 kpc, five are closer than 2 kpc, and three are closer than 1 kpc. By itself, this may not be too surprising given the observational biases. However, if in contrast to the rest of the paper we assume that the fiducial model is accurate, the number of hosts at extremely small distances (<2 kpc) is extremely surprising. For the number of hosts in our sample, our model would predict $0.26$ hosts in the 0–1 kpc bin and $1.24$ in the 1–2 kpc bin, compared to the three hosts in each (including the Bayesian estimated hosts). In other words, if all the hosts in the first two bins are in the correct bins, the model would predict that the sample would contain $61.9$ planet hosts, twice the number that are actually in the sample! Moving four or five out of the six events with ${D}_{{\rm{l}}}\lt 2\,\mathrm{kpc}$ to the 6–8 kpc range (where there is the largest deficit of hosts relative to the model) would remove all tension with the model.

So, could the parallax distance estimates be wrong? As described in Section 2, these require both parallax and finite-source measurements, but as discussed earlier in this section, the finite-source measurements, which yield ${\theta }_{{\rm{E}}}$, should be fairly robust. Until recently, the only way to measure the microlensing parallax ${\pi }_{{\rm{E}}}$ was via orbital parallax.¹¹ Orbital parallax is an often subtle effect that arises from the departure of the apparent trajectory of the source from linear motion caused by the Earth's acceleration in its orbit. ${\pi }_{{\rm{E}}}$ is actually a vector quantity ${{\boldsymbol{\pi }}}_{{\rm{E}}}$, with magnitude ${\pi }_{{\rm{E}}}={\pi }_{\mathrm{rel}}/{\theta }_{{\rm{E}}}$ and the same direction as the vector of relative proper motion. Often only the so-called parallel component of the vector (in the direction parallel to the direction of the Earth's acceleration, which causes an asymmetric distortion to the light curve) is well measured, with the perpendicular component (which causes a symmetric distortion) being degenerate with the impact parameter and blending. In addition, orbital parallax is also degenerate with the orbital motions of the lens (Batista et al. 2011; Skowron et al. 2011) and the source (xallarap, e.g., Poindexter et al. 2005), which can cause similar apparent changes to the source trajectory. The magnitude of the parallax distortions and their duration (covering the whole magnified portion of the light curve) can also make them vulnerable to corruption by long-term systematic trends in photometry.

The events with ${D}_{{\rm{l}}}\lt 2\,\mathrm{kpc}$ are OGLE-2006-BLG-109, MOA-2007-BLG-192, MOA-2010-BLG-328, OGLE-2012-BLG-0563, OGLE-2013-BLG-0341, and OGLE-2013-BLG-0723; we will use abbreviated names (see the notes for Table 1) for events in our sample from here on. We have highlighted them on ${\theta }_{{\rm{E}}}$–${\mu }_{\mathrm{rel}}$, ${\theta }_{{\rm{E}}}$–${D}_{{\rm{l}}}$, and ${\mu }_{\mathrm{rel}}$–${D}_{{\rm{l}}}$ diagrams in Figure 7, which visually compares our host sample to our model. Model points were drawn with replacement from the model, and displaced randomly from their actual values (in order to better visualize point densities) by $\mathrm{log}({\theta }_{{\rm{E}}}/\mathrm{mas})\pm 0.1$, $\mathrm{log}({\mu }_{\mathrm{rel}}/\mathrm{mas}\,{\mathrm{yr}}^{-1})\pm 0.1$, and ${D}_{{\rm{l}}}\pm 0.25\,\mathrm{kpc}$, where the ± sign indicates the range of a uniform distribution.

Recalling that ${\theta }_{{\rm{E}}}$ and ${\mu }_{\mathrm{rel}}$ should be relatively robustly measured, in the ${\mu }_{\mathrm{rel}}$–${\theta }_{{\rm{E}}}$ panel of Figure 7 the nearby hosts do not look to be especially unusual; they cluster toward larger ${\theta }_{{\rm{E}}}$, but are generally within the cloud of model points, with O06-109 (labeled a) being the most extreme. However, in the ${D}_{{\rm{l}}}$–${\theta }_{{\rm{E}}}$ and ${D}_{{\rm{l}}}$–${\mu }_{\mathrm{rel}}$ plots, at least three of the nearby hosts (b, c, and f, or M07-192, M10-328, and O13-0723, respectively) stand out from the cloud of model points; these three hosts are the ones within 1 kpc. O06-109 lies at the edge of the model cloud in the ${D}_{{\rm{l}}}$–${\mu }_{\mathrm{rel}}$ plot and O13-0341 lies at the edge in the ${D}_{{\rm{l}}}$–${\theta }_{{\rm{E}}}$ plot. O12-0563, being a Bayesian distance estimate, unsurprisingly lies within the cloud of points.

The majority of the scatter in ${\theta }_{{\rm{E}}}$ in the model is caused by the range of lens masses. This can be seen by comparing the ${\theta }_{{\rm{E}}}$ scatter in the ${D}_{{\rm{l}}}$–${\theta }_{{\rm{E}}}$ plot between the fiducial red and blue points and the black points, which are the same realization with ${\theta }_{{\rm{E}}}$ plotted as if every star had the same mass of $1\,{M}_{\odot }$. From this it can easily be seen that the small distance estimates imply extremely small estimates of host mass. The host masses in our model range from 0.08 to $1\,{M}_{\odot }$. The authors of papers on M07-192, M10-328, and O13-0723 estimate host masses of 0.084, 0.11, and $0.031\,{M}_{\odot }$, respectively (Kubas et al. 2012; Furusawa et al. 2013; Udalski et al. 2015a). These events have relatively normal angular Einstein radii and relative proper motions, so the low mass and distance estimates are being driven entirely by the parallax measurements. Given that at least some of the events closer than 2 kpc occupy a region of parameter space that our model predicts should be empty, it is worth looking at each one individually in more detail to see if we can plausibly move them to a larger distance and bring them in line with our model's expectations.

O06-109 (a) was an extremely complex event with five distinct features from two planets (Gaudi et al. 2008). Parallax and finite-source effects combined to give mass and distance estimates of $0.51\,{M}_{\odot }$ and 1.51 kpc. The event also had observable terrestrial parallax (with ${\rm{\Delta }}{\chi }^{2}\sim 12$, Bennett et al. 2010), and H = 17.2 mag of blended light coincident with the source in AO imaging matches essentially exactly that which is expected from the parallax solution. Given that terrestrial parallax cannot be mimicked by xallarap or orbital motion and the bright and unmistakable blended light, we conclude that there is very little chance that the O06-109 distance estimate is significantly in error.

M07-192 (b) was a high-magnification event with poor coverage over its peak, and only four data points cover the planetary deviation. As such, there were many models (16) that could fit the data well, but these all had similar planetary mass ratios, with the only significant difference being two classes of models with ${\theta }_{{\rm{E}}}$ differing by a factor of ∼2 (Bennett et al. 2008); even if the solution with the smaller ${\theta }_{{\rm{E}}}$ were chosen, the distance to the host would be less than 2 kpc, but the relative proper motion would be reduced to 2–3 mas yr⁻¹. Bennett et al. (2008) found xallarap models that fit the data, but argued that these had a low probability relative to the parallax model. From AO imaging, Kubas et al. (2012) find that there is a K_s = 19.2 star blended with the source, but do not measure its color very precisely. Kubas et al. (2012) do not question the parallax measurement of Bennett et al. (2008) and so interpret the blended light as being due to a 0.08 ${M}_{\odot }$ star nearby, but the K_s magnitude they measure is compatible with a 0.7 ${M}_{\odot }$ star at ∼7 kpc, and would only cause some tension with their 0.37 mag $J-{K}_{{\rm{s}}}$ error bar. Bennett et al. (2008) estimate that the a priori probability of the parallax signal being caused by xallarap is 0.32%, accounting for the distribution of binary periods, but not accounting for the increased stellar density in the bulge relative to the nearby disk. They guess that the increased density in the bulge would increase this probability by a factor of about 7, but do not rigorously compute this number. Given the difficulty in computing such a probability, it might be reasonable to assume that it might actually be significantly larger.

M10-328 (c) was a low-magnification event with a ∼5 day long planetary deviation that was poorly covered due to bad weather, except for the exit of the caustic over a cusp (Furusawa et al. 2013). The authors find a xallarap model that fits better than the parallax model by Δχ² ∼ 5, but argue that the parallax interpretation is most likely because an unconstrained xallarap fit with a period fixed to 1 yr yields an orbital orientation that is coincident with the Earth's relative to the event's line of sight. We agree that this would seem to rule out xallarap as an impostor for parallax. Additionally, the large angular Einstein radius of the event would imply that, if the host were in the bulge, it would be $\sim 1\,{M}_{\odot }$, and therefore bright unless it were a stellar remnant. It therefore seems very unlikely that we could plausibly place this host in the bulge. The only remaining hope of doing so is that the poor coverage of the planetary anomaly and the combination of a cusp crossing with orbital motion on the single exit of the caustic result in a larger than reported uncertainty on the angular Einstein radius or a misestimation of it.

O12-0563 (d) was a high-magnification event that was densely covered over the peak (Fukui et al. 2015). Surprisingly, despite the source not crossing the central caustic, the authors claim that they are able to measure ${\theta }_{{\rm{E}}}$ to be 1.36 mas to a precision of ∼10%. While formally the authors measure parallax, they argue that the detection is likely due to systematics, and so we do not count it among the events with parallax detections. AO imaging shows blended light of K_s = 17.7. The authors do not consider a xallarap model. The large ${\theta }_{{\rm{E}}}=1.36$ mas of the event would by itself exclude the possibility of a bulge lens, unless it were a neutron star or black hole. Even if ${\theta }_{{\rm{E}}}$ were overestimated, it would need to be severely overestimated to allow us to move the lens into the bulge.

O13-0341 (e) is a triple-lens event with a very small planetary signal (the dip) that by itself might be overlooked (Gould et al. 2014). However, the well-covered crossing of a large, binary-star central-caustic is also better fit if a planet with the same properties as would cause the dip is included in the model. A low-amplitude bump during an earlier observing season identifies the binary as a wide binary, and so the planet orbits one of the stars. The parallax signal in the light curve is strong. The authors do not mention whether they considered a xallarap model, but the large proper motion and large angular Einstein radius place the event in a region of parameter space that is almost entirely occupied by disk stars.

O13-0723 (f) is a second triple-lens planetary event discovered in 2013. It also seems to have a strong parallax signal (Udalski et al. 2015a). Again, the planet orbits only one of the binary stars, but in this case the argument for this scenario over a circumbinary planet is based upon an argument that the parameters of the circumbinary model require fine tuning. The authors do not mention whether they considered a xallarap model, but unlike O13-0341, the location of the event in the ${\mu }_{\mathrm{rel}}$–${\theta }_{{\rm{E}}}$ plane is occupied by both bulge and disk stars and such a model should be considered.¹²

While both O13-0341 and O13-0723 seem to have strong parallax signals, it is unsettling, to say the least, that the only two planets discovered in binary-star microlensing events with caustic crossings also have extremely small estimates of host distance (and mass).¹³ There have been other binary microlensing events with small estimates of mass and distance, e.g., OGLE-2009-BLG-151/MOA-2009-BLG-232 (0.39 kpc, $0.025\,{M}_{\odot }$ total mass), OGLE-2011-BLG-0420 (1.99 kpc, $0.034\,{M}_{\odot }$, Choi et al. 2013), OGLE-2012-BLG-0358 (1.73 kpc, $0.024\,{M}_{\odot }$, Han et al. 2013a), OGLE-2013-BLG-0578 (1.16 kpc, $0.16\,{M}_{\odot }$, Park et al. 2015), and MOA-2011-BLG-149 (1.07 kpc, $0.16\,{M}_{\odot }$, Shin et al. 2012). For these binaries it is easy to dismiss the unusually small masses and distances as a publication bias: were they further away, parallax would not be measurable, which would mean their low mass would go unnoticed, and there would be nothing interesting to write a paper about. However, this bias is less prominent for planetary microlensing events because all planetary discoveries are publishable, though there could be a delay in publishing "less interesting" ones without parallax measurements, which would bias planets in binary detections to smaller distances. It is perhaps more likely that planets in binaries would suffer a bias that works in the opposite direction, due to the possibility that parallax effects, in addition to the triple-lens nature, would make the events more difficult to model and hence delay publication.

One might think that there is another possible bias that could result in us detecting planets in binary systems only when they are nearby. That is, if the binary star prevents the formation or survival of planets on wider orbits, then it will only be possible to detect the surviving, closer-in planets in events with small physical Einstein ring radii ${r}_{{\rm{E}}}$. However, ${r}_{{\rm{E}}}\propto \sqrt{{D}_{{\rm{l}}}({D}_{{\rm{s}}}-{D}_{{\rm{l}}})}$, and so there are lenses with small physical Einstein ring radii near their sources in the bulge as well as near us, the observer; there should be many more planets discovered in binary systems in the bulge than in the nearby disk. We are therefore forced to ask the question: is there something going wrong with the modeling of parallax in triple-lens microlensing events (and maybe in binary-star events too)? Given the complexity of triple-lens models it seems plausible that something could be going wrong, but it is difficult to guess at what this might be, especially in the case of O13-0341 where the parallax, angular Einstein radius, and relative proper motion seem to tell a consistent story.

It is possible to test the predictions of the parallax models to some degree. They predict the direction of the relative source–lens proper motion, and after ∼10 years the lens and source may separate enough for this direction to be measured. An expedited method to test the validity of the parallax models would be to model only the ground-based data for upcoming binary microlensing events that are observed by Spitzer (e.g., Udalski et al. 2015c; Zhu et al. 2015b) or K2 Campaign 9 (Howell et al. 2014; Henderson et al. 2015), and then test the prediction of orbital parallax with the much more robust Spitzer or K2 satellite measurements of parallax. Udalski et al. (2015c) performed this test for a planetary event and found that the orbital and Spitzer-measured satellite parallaxes agreed, but the binary event modeled by Zhu et al. (2015b) did not have a detectable signature of orbital parallax, and alas Spitzer did not observe the past events whose parallax solutions we are worried about.

While we can offer no convincing evidence to do so, potentially two of the six hosts with ${D}_{{\rm{l}}}\lt 2\,\mathrm{kpc}$ (M07-192 and O13-0723) could plausibly be moved to larger distances if xallarap is mimicking parallax in either event or if there were errors in the parallax modeling of O13-0723. It is worth noting that Poindexter et al. (2005) found that five out of 22 single-lens events with parallax-like signals that they modeled were more likely to have been caused by xallarap. We might therefore expect that ∼20% of our $12$ parallax events (i.e., two or three) would actually be caused by xallarap. In fact, O07-368 should also be added to our parallax-like sample, because it has a reasonable parallax solution, but Sumi et al. (2010) find that it is better fit and more likely caused by xallarap. If the parallax solution of O07-368 had been believed blindly without considering the possibility of xallarap, the event would also have been in our subsample with suspiciously small distance estimates of less than 2 kpc. If we remove M07-192 and O13-0723 from our sample, AD tests like those we performed in Section 5.1 have p-values greater than 0.01 for ${f}_{\mathrm{bulge}}\lt 0.96$, and so would go most of the way toward explaining the discrepancy between the model and the observed data.

We have found that various factors can help to bring the model distribution toward consistency with the null hypothesis, namely that the planet abundance in the bulge is no different from that in the disk, but do not go all the way. We now investigate whether combining all the factors can make the model consistent. We assume that planet abundance is a function of both host mass and metallicity with the same dependence as found by radial velocity surveys (Johnson et al. 2010, $\propto M{10}^{1.2[\mathrm{Fe}/{\rm{H}}]}$). We also assume that the simulations of H14 do not fully capture the increased sensitivity to planets in longer-timescale events for planets detected by follow-up observations, and so weight the simulated planet detections by ${t}_{{\rm{E}}}^{1}$. Finally, we remove the event O13-0723 from our sample of distance estimates, due to a non-planetary model being found that better explains the data on the event (Han et al. 2016), which was reported after this paper was submitted.

Figure 8 shows the AD statistic as a function of ${f}_{\mathrm{bulge}}$ for the model with all of the combined corrections, for the distributions of both distance and relative proper motion. For ${f}_{\mathrm{bulge}}=1$, the model is much more consistent with both the ${D}_{{\rm{l}}}$ and ${\mu }_{\mathrm{rel}}$ distributions, with p > 0.05 in both cases. The full range of ${f}_{\mathrm{bulge}}$ allowed with a threshold of p > 0.01 for both ${D}_{{\rm{l}}}$ and ${\mu }_{\mathrm{rel}}$ distributions is ${f}_{\mathrm{bulge}}\lesssim 1.7$. This might suggest that the puzzle has been solved. However, there is still some tension with the ${f}_{\mathrm{bulge}}=1$ scenario, with significantly smaller values of ${f}_{\mathrm{bulge}}$ providing better fits to the ${D}_{{\rm{l}}}$ and ${\mu }_{\mathrm{rel}}$ distributions. We expect that further events with erroneous parallax measurements implying very small distance estimates are likely to be the cause of this remaining tension.

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