SUB-SATURN PLANET MOA-2008-BLG-310Lb: LIKELY TO BE IN THE GALACTIC BULGE^*

The finite-source two-point-mass lens magnification calculations are carried out using the improved magnification map technique of Dong et al. (2006, 2009b), which is optimized for high-magnification events. The fitting procedure follows closely that of Dong et al. (2009b). The initial search for two-point-mass lens solutions is conducted over a grid of three parameters: the short-caustic width w, companion/star mass ratio q, and the angle of the source trajectory relative to the companion/star axis α. Since w is a function of q and the companion/star separation d, this is equivalent to also fixing d at various values. The remaining parameters (t₀,u₀,t_E,ρ) are allowed to vary. Two additional parameters for limb darkening are given fixed values (see Figure 2), as will be discussed in Section 4. The source flux f_s and the blended flux f_b are fit independently for each filter and telescope. We use Monte Carlo Markov Chain (MCMC) to minimize χ² with respect to (t₀,u₀,t_E,ρ) at each of the (w, q, α) grid points. There is a well-known degeneracy such that for q ≪ 1, planet/star separations d and d⁻¹ will produce almost identical central caustic structures and consequently indistinguishable light curves for high magnification events such as this (Griest & Safizadeh 1998). We explore a (w, q) grid for each geometry, searching the d ⩾ 1 (in units of the Einstein radius) regime for "wide" solutions and the d < 1 regime for "close" solutions.

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An initial search for two-point-mass lens solutions is conducted over the range of caustic widths −3.5 ⩽ log w ⩽ −1.0 (in units of the Einstein radius), companion mass ratios −5.0 ⩽ log q ⩽ 0, and source trajectory angles 0 ⩽ α ⩽ 2π in the two separate regimes d ⩾ 1 and d < 1. This initial search gives us fairly good estimates of the best-fit parameters and the location of the χ² minima in terms of w and q (and hence also d). For this particular event, however, w and q turn out to be highly correlated. We conduct a refined search over a grid in (d, q) instead of (w, q), and we also allow α to vary as a MCMC variable, rather than discretely. The solid black lines in Figure 3 show Δχ² = 1, 4, 9 contours in the (d, q) plane for the wide (top) and close (bottom) solutions, respectively. For the wide solution, the χ² minimum occurs at d = 1.085 ± 0.003 and q = (3.31 ± 0.26) × 10⁻⁴. The close solution minimum occurs at d = 0.927 ± 0.003 and q = (3.20 ± 0.26) × 10⁻⁴. The mass ratio indicates that the companion to the lens is in fact a planet. As expected, we recover the d ↔ d⁻¹ degeneracy. The wide solution is favored by just Δχ² = 2.06, indicating that the wide/close degeneracy cannot be clearly resolved in this case. The best-fit parameters for both wide and close solutions are recorded in Table 1. Two independent algorithms were used to explore parameter space and both returned essentially identical best fits.

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The wide and close planetary models qualitatively explain several features of the single-lens residuals. The lower panel of Figure 4 shows the extended source at key points in time on its trajectory. The nearly identical central caustics generated by the best-fit wide and close models are both shown. The most prominent features in the residuals, shown in the upper panel of Figure 4, occur as the limb of the source crosses the caustic. The positive and negative spikes that are most evident from the raw data (features 2 and 4 of Figure 4) coincide with the limb of the source entering the strong curved portion of the caustic and exiting the weaker straight segment. Residual patterns like this, characterized by short duration perturbations separated by a relatively flat region, are typical of planetary lens systems affected by strong finite source effects (Griest & Safizadeh 1998; Dong et al. 2009b; Han & Kim 2009). The bottom panel of Figure 1 shows the residuals to the best-fit wide planetary model. The deviations from the point-lens model that initially indicated that the lens was not being accurately modeled are no longer apparent. The planetary model decreases χ² by ∼880 for three additional degrees of freedom, making this a strong detection of a Saturn mass-ratio planet/star system.

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In the following, the term "source" refers to the star that was microlensed, while the term "target" refers to all the light that is aligned with the source in the NACO images. We calibrate the NACO H-band magnitude of the target via the route IRSF-to-2MASS and find H_NACO,calib = 17.47 ± 0.05. Then using the IRSF-to-CTIO transformation, we convert the measured NACO flux into the instrumental CTIO system, H_target,CTIO = 21.29 ± 0.05. We stress that this indirect road NACO-to-IRSF-to-CTIO is actually the most accurate one to estimate the magnitude in the instrumental CTIO system.

We also carry out the following independent check. We measure aperture fluxes f_i on the NACO image for stars 1, 2, 3, and the target listed in Table 3. For the target, we correct the result for contaminating flux from star 3 and from another much fainter nearby star. Then using the same stars i = 1, 2, 3 on the IRSF image, we obtain an estimate of the target magnitude on the IRSF system: H_{target;i,IRSF} = H_i,IRSF + 2.5log(f_i/f_target). We take the average of these three estimates (whose standard error of the mean is only 0.012 mag), and then apply the previously derived conversion from IRSF to CTIO. We find H_target,CTIO = 21.27.

As a further sanity check, we apply a similar procedure to compare the NACO and CTIO images directly. As stated at the outset, we expect that this will be less accurate both because there are only two viable comparison stars (1 and 2) and because the CTIO flux measurements are less accurate than those of IRSF. Nevertheless, we find a similar result: H_target,CTIO = 21.32, although with substantially worse precision.

We finally adopt H_target,CTIO = 21.28 ± 0.05, where the error bar reflects our estimate of the systematic error. Clearly, our two primary methods of estimating this quantity agree much more closely than this, but there still could be systematic effects common to both. We regard 0.05 mag as a conservative overestimate of the error.

Inserting the H-band CTIO observations into the planetary model, we obtain the unmagnified source flux, H_source,CTIO = 21.55 ± 0.05 on the instrumental CTIO system. The error bar accounts for the uncertainty in the fit by allowing all parameters, including parallax, to vary freely. However, for the purpose of determining the blend on the NACO image, we are more interested in the magnified flux from the source at t_NACO, the time the image was taken. The magnification is determined by the separation between the source and lens at t_NACO, u_NACO = (t_NACO − t₀)/t_E in units of the Einstein radius. The unmagnified source flux is anti-correlated with the Einstein crossing time t_E, so the dispersion in the magnified flux is slightly smaller than the dispersion in the unmagnified flux. The uncertainty in the magnified flux is related to the model uncertainty in the unmagnified flux (f_s) by

$$\begin{equation} {{\sigma (Af_s)\over A\sigma (f_s)}} = 1 + {{d\ln A}\over {d\ln u}} \end{equation} \tag{ 9 }$$

which in the point-lens approximation (generally valid on the wings on the light curve) translates to³⁸

$$\begin{equation} {{\sigma (Af_s)\over A\sigma (f_s)}} = 3 -{2 A^2}[1 -(1-A^{-2})^{3/2}]. \end{equation} \tag{ 10 }$$

In our case, the analytical result ${{\sigma (Af_s)\over A\sigma (f_s)}} = 0.77$ is very close the result calculated using MCMC data.

Figure 7 shows probability distributions for the magnified source flux constructed from the MCMC chains. Without parallax (black histogram), the close and wide solutions give the same source magnitude at the time of the NACO image, H_{magnified,CTIO} = 21.45 ± 0.04, while the best-fit solution with unconstrained parallax (gray) gives a flux ∼2% brighter. As noted in Section 6, the best-fit parallax implies a planetary lens mass and is likely a spurious detection. If we constrain parallax so that the lens mass is at least 0.08 M_☉ (π_E ⩽ 0.25), the best-fit magnified source flux is identical to the case of no parallax. Thus, our best estimate of the flux strictly from the source is H_{magnified,CTIO} = 21.45 ± 0.04, while the light aligned with the event on the NACO image is H_target,CTIO = 21.28 ± 0.05. We consider 21.28 + 0.05 = 21.33 to be a robust lower limit on the amount of light detected in the NACO image. Therefore, excess light unrelated to the source is detected at the 3σ level.

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From direct examination of the NACO H image, we find the density of stars within 0.5 mag of the H-band magnitude of the excess light to be $0.94\,\rm arcsec^{-2}$. Hence, the prior probability that such a random star lies buried under the NACO image is 5.1%.

The source is a main-sequence G star. From Table 7 of Duquennoy & Mayor (1991), we find that 9.4% of their sample (of 164 stars) have companions within the mass-ratio range of 0.57 < q < 0.76, i.e., the mass range corresponding to within ±0.5 mag of the observed excess flux. However, from Figure 5 of Duquennoy & Mayor (1991), 22% of companions lie outside ∼1000 AU, the size of the NACO PSF projected on the source plane. A further 3% have orbits shorter than ∼3 days, which would have given rise to observable "xallarap" signals in the microlensing light curve. Hence, if bulge G stars are like the local sample, 0.094 × 0.75 = 7.1% of them have companions within 0.5 mag of the observed excess flux, and lie at separations where they would not have been detected. This is comparable to the corresponding value for ambient stars.

If the lens had a companion, it would induce shear on the lens's gravitational field, which would generate a small Chang & Refsdal (1979, 1984) caustic at the center of magnification of the lensing system. This would in turn produce spikes in the light curve at HJD' 4656.36 and 4656.44, when the lens center-of-magnification crosses the limb of the source. The residuals to the planetary model (Figure 1) strongly limit any such spikes. To put this constraint on a quantitative basis, we fit the light curve to models that have two additional parameters, ϕ, the angle between the planet axis and the binary-companion axis, and w_com, the width of the caustic induced by the companion,

$$\begin{equation} w_{\rm com} = 4 {q_{\rm com}\over d_{\rm com}^2}, \end{equation} \tag{ 11 }$$

where (q_com, d_com) are the mass ratio and separation of the companion. We hold (w_com, ϕ) at a grid of fixed values and minimize χ² with respect to all other parameters.³⁹

This search reveals an improvement of Δχ² = −7.3 for two additional degrees of freedom, with a best fit of (log w_com, ϕ) = (−3.28, 40°). This improvement is too small to claim a detection, since it could occur by chance with probability 2.6%, and could also be due to low-level systematics. Thus, while this test raises the tantalizing possibility that the excess light is due to a companion to the lens, we mainly focus on the 3 σ upper limits to shear from a companion with w_com < 1 × 10⁻³ over almost all angles. See Figure 8. By comparison, w_planet = 5 × 10⁻³ (see Figure 4).

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Equation (11) can also be written as

$$\begin{equation*} w_{\rm com} = {4\kappa M_{\rm com} \pi _{\rm rel}\over \Delta \theta _{\rm com}^2}, \end{equation*}$$

where M_com is the companion mass, and Δθ_com is its angular separation. If the lens is in the bulge and the excess light is due to its companion, then (see Section 8.4) M_com ∼ 0.6 M_☉, and so w_com < 1 × 10⁻³ implies

$$\begin{equation} \Delta \theta _{\rm com} > 16\,{\rm mas}{\pi _{\rm rel}\over 37\,\rm \mu as} = 16\,{\rm mas}\biggl ({M\over 0.08\,M_\odot }\biggr)^{-1}. \end{equation} \tag{ 12 }$$

For foreground-disk lenses, the limit on Δθ_com continues to grow, but much more slowly, to 70 mas for $(M,\pi _{\rm rel}) = (0.003\,M_\odot,1\,\rm mas)$.

Hence, in contrast to source companions, which are permitted over 4 decades of separation, companions to the lens are restricted to 1–2 decades for bulge lenses and a somewhat narrower range for disk lenses.

We stress that the excess light could be due to any of the three options above. The excess light could also be due to the lens. As we show below, this requires the lens star to be relatively massive and so quite close to the source. Such small source–lens distances are generally disfavored by phase space and kinematic factors. However, as mentioned previously, evaluating the prior probability of this scenario requires adopting a specific assumption of frequency of planets as a function of host mass, which is poorly constrained.

Under the assumption that the NACO blended light is due to the lens, we can estimate the lens mass using its inferred instrumental flux H_lens,CTIO = 23.38 ± 0.41, the measured Einstein radius $\theta _{\rm E}\equiv \sqrt{\kappa M_{\rm L} \pi _{\rm rel}} = 0.155\pm 0.011$, and an assumed range of mass–luminosity relations, which depend on the lens's unknown metallicity. While our actual calculation is fully self-consistent, the basic result can be understood intuitively as follows. For any possible mass consistent with the observed flux, the lens–source distance will be quite small, D_LS ∼ π_relD²_S/AU = 300 pc(M_L/0.7 M_☉)⁻¹ relative to D_S, and the range of possible values due to different candidate masses is even smaller. Hence, we can just adopt D_LS = 300 pc. For fixed D_LS the lens lies, on average, D_LS/2 or 0.04 mag in front of the local bulge density peak (defined by the clump giants). At Galactic latitude b = −4.56, the dispersion of distance moduli of bulge sources is (5/ln 10)sin b/0.6 = 0.29 mag for an adopted bulge flattening of 0.6. We adopt an absolute magnitude for the clump of $M_{H,\rm clump} = -1.41\pm 0.05$, and so from the observed H_clump,CTIO = 17.35, we infer

$$\begin{eqnarray*} M_{H,\rm lens} &=& M_{H,\rm clump} + (H_{\rm lens,CTIO} - H_{\rm clump,CTIO}) + (0.04 \pm 0.29) \nonumber\\ &=& 4.66 \pm 0.50. \end{eqnarray*}$$

Then using six isochrones generated by the Dartmouth Stellar Evolution Database (Dotter et al. 2008) with [Fe/H] ranging from −0.5 to 0.5 and age ranging from 5 Gyr to 10 Gyr, we estimate the lens mass to be M_L = 0.67 ± 0.14 M_☉.

Assuming the excess light is indeed due to the lens and using the resulting estimate of the lens mass, we can now estimate the properties of the planetary companion. The planet mass is m_p = 74 ± 17 M_⊕, roughly 80% the mass of Saturn. Taking account of the uncertainties in both the distance to the lens and the angular Einstein radius, as well as the wide/close degeneracy, the projected separation between the planet and host star is 1.25 ± 0.10 AU.

Note that if the blended light is not due to the lens, then the lens must be fainter than this light and so (unless it is a remnant) also of lower mass. The planet would then be of proportionately lower mass as well.

For completeness, we also remark on the possibility that the host is a remnant. A white-dwarf host is at least 5 times less likely than a luminous lens simply because the space densities of white dwarfs and main sequence stars per unit mass are similar over this mass range (Gould 2000a), but for the former the excess light must be "explained" by a companion to the source or lens or by an ambient star, while for the later, the excess light is directly associated with the lens. In addition, since stars lose of order half their mass in the course of becoming white dwarfs, their planets migrate outward by the same factor. Since the planet is currently at 1.25 AU in projection, it would have had difficulty surviving the asymptotic giant branch phase of evolution unless (with somewhat low probability) its physical separation were substantially larger than its projected separation.

Even 2 decades ago, pulsar timing experiments would have been sensitive to Saturn-mass planets in Earth-like orbits around more than 300 pulsars (Bailes et al. 1993, Figure 1), and it is likely that many more have been searched to this level today. Hence, the probability that the host is a neutron star is negligibly small.

Microlensing event MOA-2008-BLG-310 is one of only two published high magnification events to date for which the source is as large or larger than the central caustic. It bears many similarities to the other, MOA-2007-BLG-400 (Dong et al. 2009b). Like that earlier event, the planetary perturbations in the light curve are not immediately apparent, having been smoothed out by finite-source effects. We find that a Saturn mass ratio planet/star model is nevertheless favored over the single-lens model by a significant reduction in χ² (Δχ² ∼ 880 for an additional 3 degrees of freedom). Using VLT NACO (together with IRSF) photometry, we definitively detect excess light blended with the source that is due to the lens, or a companion to the lens or the source, or an unassociated star. Regardless of the origin of this excess light, however, it places an upper limit on the lens flux and so on its mass. The planet's Saturn-like mass ratio therefore implies that it has a sub-Saturn mass.

Our measurement of the angular Einstein radius θ_E constrains a combination of the lens mass and distance. We thereby conclude that if the lens is a star, then it must be in the bulge.

We are not able to resolve the close/wide degeneracy in the geometry of the planetary system. However, the separate solutions for the lens/star separation d differ only by a factor of 1.17. The d ↔ 1/d degeneracy is not as severe in this case because the planet is located very close to the Einstein radius.

Figure 9 shows the mass versus equilibrium temperature for planets that have been detected orbiting main-sequence stars via radial velocity, transits, direct imaging, astrometry, and microlensing. The position of MOA-2008-BLG-310Lb is shown under two assumptions. The hexagon symbol indicates its position assuming that the excess flux is due to the lens. We then obtain a host mass of M_L = 0.67 ± 0.14 M_☉ and so a planet mass of 0.23 ± 0.05 M_jup. The "tail" extending toward lower masses and colder temperatures assumes that the excess light is not from the lens (and so is due to a companion to the source or lens). The path of this tail is determined by the measurement of θ_E = 0.155 mas, which constrains the product of the lens mass and lens–source relative parallax to be θ²_E/κ = Mπ_rel = 3 M_☉ μas.

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All possibilities for the lens identification are interesting. In particular, the host might be a brown dwarf or free-floating planet in the foreground disk. In this case, the Saturn-mass-ratio companion would actually be a moon. Future observations with the HST or AO could distinguish among these three possibilities. First, of course, mere detection of the light from the host would confirm it is a star and therefore that this is indeed a bulge planetary system, the first unambiguous such detection.

If the excess light is due to the lens, then Figure 9 shows that this detection is beginning to probe a new part of parameter space. The previous eight planets detected via microlensing span a relatively wide range of mass from a few Earth masses to several Jupiter masses, but are largely located at relatively cold equilibrium temperatures of ∼40–80 K, similar to the outer planets of our solar system. In contrast, the radial velocity and transit methods are generally only sensitive to sub-Saturn mass planets with relatively warm equilibrium temperatures of ≳300 K. Therefore, little is currently known about the demographics of sub-Saturn mass planets with equilibrium temperatures between 100 and 300 K. MOA-2008-BLG-310Lb would make this the first microlensing planet to fall on the Snow Line.

Of course, if the excess light is not due to the lens, then we have no hard information on the lens mass. However, the "tail" in Figure 9 indicates an interesting possibility: that MOA-2008-BLG-310LB is a "cold Neptune" orbiting a low mass star, similar to several other such detections.

What are the prospects for characterizing the host and its planet? As we discussed in Section 6, the event contains essentially no parallax information. Hence, the only path toward measuring the lens mass and distance is direct detection of the host (or possibly its companion). For either the wide or close solution, the geocentric proper motion is μ_geo = θ_E/t_E = 5.1 mas yr^-1. The heliocentric and geocentric proper motions differ by

$$\begin{equation} |{\mbox{\boldmath $\mu $}}_{\rm hel} - {\mbox{\boldmath $\mu $}}_{\rm geo}| = |{{\bf v}}_{\oplus,\perp }|\pi _{\rm rel} = {\theta _{\rm E}^2 v_{\oplus,\perp }\over \kappa M} = 0.018\,{M_\odot \over M}\,{\rm mas\,yr^{-1}}, \end{equation} \tag{ 13 }$$

where v_⊕,⊥ = 28 km s⁻¹ is the velocity of the Earth projected on the plane of the sky at the peak of the event. Hence, if the lens is luminous (M ≳ 0.08 M_☉), then the heliocentric and geocentric proper motions are essentially identical, and so the magnitude of the heliocentric lens–source relative proper motion is well determined.

This known proper motion can then serve as an anchor point for the interpretation of future high-resolution images, which could in principle directly detect the lens or demonstrate unequivocally that it is not luminous and so is a sub-stellar object in the foreground disk. However, as we now show, such unambiguous results actually require that new images, with FWHM ∼50 mas, be obtained "immediately," i.e., before the lens and source have separated significantly.

Suppose, by contrast, the first epoch consisted solely of the 132 mas FWHM images already in hand, and that second epoch AO images were obtained 10 years later, which (unlike the first epoch) did reach the diffraction limit of 50 mas. The lens will then be 50 mas from the source, and so separately resolved if it is luminous. But if such a star were observed at 50 mas, how could we be certain it was the lens? If the excess light were due to an ambient star or to a companion to the lens or the source, then this object could also happen to be 50 mas from the source at this latter epoch. For the ambient-star and source–companion cases this is obvious. For the lens–companion case, the shear limits discussed in Section 8.3 do place some constraints on future lens–companion positions, but as we will make clear further below, these still allow it to be 50 mas from the source after 10 years.

On the other hand, suppose that nothing was detected in such 10 year post-event images. In this case, we would know that the lens was not in the bulge, but we would still not be able to determine whether the excess light had been due to an ambient star, or companions to the lens or source. And it would be of substantial interest to do so because, in this case, a lens companion would be the only clue to the distance (and so mass) of the lens.

Let us consider now how the situation would change if new images were obtained immediately with 50 mas resolution, either from HST or using AO. Such images would either resolve out the excess flux, or restrict it to 50 mas radius (smaller in the case of HST as discussed below). If it were resolved, then the appearance of a "new" star 10 years later would have to be due to the lens or its companion. (In principle, such a "new" star could be an ambient star that had been hidden at the time of the first epoch, but the probability of this is reduced by (50/132)² = 0.14 and is further reduced by the chance that it would happen to be very close to 50 mas from the source at the second epoch.) The proper motion of the excess light relative to the source would tell us whether it was an ambient star, a companion to the source or lens, which in the last case would give the direction of proper motion, thereby confirming that the "new" star was either the lens or a second (and very close) companion to the lens. These last two possibilities could not be strictly differentiated. However, as discussed in Section 8.3, companions cannot be too close because of the limits on shear, so the second companion could potentially be strictly ruled out depending on the analysis of the other stars in the image.

On the other hand, if the first epoch image did not resolve out the excess flux, then, as argued above, the ambient star hypothesis would be so much less likely that it could be ignored. Appearance of a "new" star in the second epoch would then be either the lens or a very close companion to the lens. Again, the strong constraints on the shear would translate into very strong constraints on the lens–companion scenario.

If the first epoch were carried out with HST then these constraints could be tightened further. Color-dependent centroid shifts (between say V and I) can be detected for star separations down to about 15 mas (assuming an I-band flux ratio of 11%), which (as outlined in Section 8.3) is quite close to the minimum lens–companion separation, unless the lens is in bulge.

In brief, immediate observations with HST, with follow-up 3 (HST) to 10 (AO) years hence, could unambiguously distinguish between a bulge and disk lens, and if the former, give a good measurement of the lens mass and distance. Immediate AO observations, if they achieved 50 mas, would significantly constrain the possible options, but would not yield an absolutely air-tight case.

We note that the calibrated source magnitude is (V, I, H)_source = (20.76, 19.28, 17.73) ± 0.05 and the H-band magnitude of the blend is H_blend = 19.56 ± 0.41. If the blend is in the bulge, then (V, I)_blend ∼ (24.0, 21.9).

The procedures just outlined are challenging but alternative routes to secure detection of bulge planets are, if anything, more difficult. Gaudi (2000) discussed the prospects for detecting transiting planets in the bulge and Sahu et al. (2006) reported the detection of 16 candidate bulge planets from a transit survey carried out with the HST. Two of these were bright enough for radial-velocity follow-up, one of which showed variations consistent with a planet with mass m = 10 M_Jupiter and the other showed upper limits m < 4 M_Jupiter. The stars are so bright, however, that their inferred masses indicate that at least one (and possibly both) probably lie in the foreground disk. Nevertheless, this technique could in principle be pushed harder, particularly when larger telescopes come on line. Even then, however, lower-mass planets, m ≲ M_Jupiter will probably only be accessible with microlensing.

There are three other planets detected by microlensing for which the distances are neither measured nor strongly constrained, OGLE-2005-BLG-169Lb (Gould et al. 2006), OGLE-2005-BLG-390Lb (Beaulieu et al. 2006), and MOA-2007-BLG-400Lb (Dong et al. 2009b). In addition, the distance to MOA-2003-BLG-53/OGLE-2003-BLG-235 is not precisely enough measured to determine unambiguously whether it is in the inner disk or the outer bulge. In all four cases, both θ_E and μ are measured, so we estimate the minimum lens mass that would allow the lens–planet system to be in the bulge and the time that must elapse before definitive imaging observations can be undertaken. As mentioned below, all of these events have large proper motions, μ ≳ 7 mas yr^-1, which generally favor disk lenses.

For OGLE-2005-BLG-169Lb, θ_E = 1.00 ± 0.22 mas and μ = 7–10 mas yr^-1. Even adopting the 1σ lower limit on θ_E, then π_rel>75 μas for stellar hosts with M ⩽ M_☉. Thus, for bulge sources at D_s = 8 kpc, the lens distance is no more than 5 kpc. Hence, the lens is almost certainly in the disk. Measurements to confirm this relatively secure conclusion could be made as early as seven years after the event, i.e., 2012.

For MOA-2007-BLG-400Lb, θ_E = 0.32 ± 0.02 mas and μ = 8.2 ± 0.5 mas yr^-1. Adopting D_s = 8 kpc, the lens would only lie within 2 kpc of the source provided that M ≳ 0.30 M_☉. Thus, this is a reasonable, but not particularly strong candidate for a bulge lens. The source is a moderately bright subgiant, so for 10 m class telescopes it is perhaps best to wait for the separation to reach 70 mas, which will require about nine years, i.e., in 2016.

For OGLE-2005-BLG-390Lb, θ_E = 0.21 ± 0.03 mas and μ = 6.8 ± 1.0 mas yr^-1. It is therefore the best previous candidate for a bulge lens since θ²_E, which is the product of the mass and relative parallax, is only a factor 1.6 times larger than for MOA-2008-BLG-310Lb. This means that if it were at the bottom of the main sequence, it would lie about 3 kpc in front of the source and therefore most likely lie in the disk, but if it had significantly larger mass it would be in the bulge. However, in this case, the source is a G4 III giant with I₀ = 14.25, which implies M_H ∼ −0.85. A lens close to the bottom of the main sequence has M_H ∼ 11 and so (even accounting for its closer distance) would appear 25,000 times fainter than the source. While this is an extreme case, it would appear prudent to wait for the lens to move three FWHM away from the source, which for 10 m class telescopes would require about 20 years, i.e., 2025. If larger telescopes with AO come on line before that, it will of course be possible to make the measurement sooner.

Finally, MOA-2003-BLG-53/OGLE-2003-BLG-235 has a proper motion of $3.3\pm 0.4\,\rm mas\,yr^{-1}$, which was sufficient to measure the color-dependent centroid shift from HST observations taken just 1.78 yr after the event, but only at the ∼3 σ level (Bennett et al. 2006). Based on this measurement, the lens distance is estimated to be D_L = 5.8^+0.6_−0.7 kpc, which reflects a roughly 30% error in the lens–source relative parallax π_rel. Thus, this planetary system could be in the inner disk or the outer bulge. The color-dependent centroid shift could certainly be measured more accurately today, but this would not dramatically decrease the uncertainty in D_L, which is fundamentally limited by the 25% uncertainty in θ²_E = κMπ_rel, and so in π_rel. Thus, pending spectra of this I ∼ 21 lens after it is fully separated from the source, it will be difficult to prove whether or not this planet is in the bulge.