OGLE-2016-BLG-1195 AO: Lens, Companion to Lens or Source, or None of the Above?

The most systematically applicable method of measuring host masses for microlensing planets is late-time imaging of the system when the host and source have separated sufficiently to resolve them. The only fundamental requirements of the method are that the host be luminous and that the planet-to-host mass ratio q and Einstein timescale t_E had been adequately measured from the original event. If the host is luminous, the measurement yields the heliocentric lens–source relative proper motion μ _rel,hel and the lens flux (in, e.g., the K band) K_L . By adjusting μ _rel,hel to the geocentric value μ _rel, and combining this with t_E, one obtains the angular Einstein radius θ_E = μ_rel t_E. Then,

$$\begin{eqnarray}&&{\theta }_{{\rm{E}}}\equiv \sqrt{\kappa M{\pi }_{\mathrm{rel}}},\qquad \kappa \equiv \displaystyle \frac{4G}{{c}^{2}\mathrm{au}}\simeq 8.14\,\displaystyle \frac{\mathrm{mas}}{{M}_{\odot }},\end{eqnarray} \tag{ 1 }$$

and,

$$\begin{eqnarray}&&{K}_{L}={M}_{K}({M}_{\mathrm{host}})+5\mathrm{log}\displaystyle \frac{{D}_{L}}{10\,\mathrm{pc}}+{A}_{K}({D}_{L}),\end{eqnarray} \tag{ 2 }$$

providing two relations between the host mass M_host and distance D_L , which can then be solved for both quantities. Then, the mass of the planet m_p = qM_host can be found from the known value of q. Here, ${\pi }_{\mathrm{rel}}\equiv \mathrm{au}({D}_{L}^{-1}-{D}_{S}^{-1})$ is the lens–source relative parallax, M_K (M_host) is the absolute magnitude of the lens, and A_K (D_L ) is its extinction.

Gould (2022) examined a wide range of issues associated with this method, including degeneracies in converting from μ _rel,hel to μ _rel and in combining Equations (1) and (2), uncertainties in various parameters such as the mass–luminosity relation M_K (M_host), the source parallax π_S , the extinction A_K (D_L ), and the light-curve measurement of t_E.

The most vexing issue identified by Gould (2022) was that the "other object" (i.e., other than the source) might not be the lens. Rather it could be either a companion to the lens, a companion to the source, or in rare cases, an ambient star. Gould (2022) showed that such misidentifications do not pose an issue of principle: they can almost always be resolved by additional late-time observations. In all nonpathological cases, if the object is not the lens, the relative proper-motion vector derived from the two late-time observations will not be consistent with uniform motion and lens–source coincidence at the time of the event. From the discrepancy, one can usually distinguish among these cases, and in particular if the object is a companion to the source, then it will hardly move between epochs, thereby permitting the lens to appear subsequently.

The main issue is that, in general, one may not know when to take a second late-time observation. Adaptive optics (AO) observations on large, or extremely large telescopes (ELTs) are expensive, while Gould (2022) estimated that of order 150 mass measurements could be made at AO first light on ELTs from the 2016 to 2022 planet detections by the Korean Microlensing Telescope Network (KMTNet; Kim et al. 2016) alone. Hence, he considered the issue of the false-positive rate to be crucial.

Motivated by these considerations, Gould (2022) distinguished between two cases: those with and without measurements of μ_rel from the event itself. Such measurements require that the normalized source radius ρ = θ_*/θ_E be measured, which in turn requires that the source passes over a caustic or close to a cusp (and that these are covered by the observations). In this case, it is possible to determine θ_E = θ_*/ρ and so μ_rel = θ_E/t_E from the angular source radius θ_*, which can be determined by standard techniques (Yoo et al. 2004).

Gould (2022) estimated that for about 1/4 of microlensing planets, μ_rel cannot be measured from the light curve (i.e., finite-source effects are not detected). For these, he estimated that the false-positive rate (mainly from companions to the lens), would be about 10%. Nominally, the false-positive rate would be exactly the same for events with μ_rel measurements. However, in most cases, one would receive a warning from the fact that this light-curve-based μ_rel was inconsistent with the one derived from the late-time imaging. Gould (2022) estimated that if these cases were excluded (because they could be scheduled for additional late-time observations) then the false-positive rate could be reduced to about 3%. With this precaution, the overall false-positive rate would be about (1/4) × 10% + (3/4) × 3% ∼ 5%. If this is considered to be an acceptable level, then only about (3/4) × 7% ∼ 5% of events would have to be subjected to additional late-time observations. Otherwise, a more aggressive approach would be needed.

Recently, Vandorou et al. (2023) have adopted the opposite approach. They imaged the planetary microlensing event OGLE-2016-BLG-1195 and reported a strong, 3σ, disagreement with the μ_rel measurements from the discovery papers (Bond et al. 2017; Shvartzvald et al. 2017). None of the six previous late-time imaging efforts, OGLE-2005-BLG-071 (Bennett et al. 2020), OGLE-2005-BLG-169 (Batista et al. 2015; Bennett et al. 2015), MOA-2007-BLG-400 (Bhattacharya et al. 2021), MOA-2009-BLG-319 (Terry et al. 2021), OGLE-2012-BLG-0950 (Bhattacharya et al. 2018), and MOA-2013-BLG-220 (Vandorou et al. 2020), had yielded such a strong disagreement. Nevertheless, rather than regarding this disagreement as a warning sign that this was one of the ∼10% expected rate of false positives, they assumed that the "other star" was indeed the lens, that the two original light-curve-based μ_rel measurements were each in error by 3σ, and that these were superseded by their own measurement.

Here, we investigate these issues further. We note that the two original μ_rel measurements were based on completely independent data sets and were consistent with each other at the 1σ level. We therefore combine these two measurements in two ways. First, we use the standard method of combining independent measurements, and, second, we combine the two data sets at the light-curve level. On this basis, we further refine the conflict between the light-curve and late-time-imaging determinations. We then propose a test to determine whether the Vandorou et al. (2023) identification is indeed a false positive.

In their Table 2, Vandorou et al. (2023) reported the offset between the source and another star (which they identified as the lens) to be Δθ = 54.49 ± 2.70 mas at t₀ + Δt, where Δt = 4.12 yr and t₀ is the peak of the event, when the lens and source were separated by ∼15 μas, i.e., far too small to be of interest here. If the "other star" is indeed the lens, then (ignoring for the moment the lens–source relative parallactic motion—see the next paragraph), this would correspond to a heliocentric relative proper motion,

$$\begin{eqnarray}&&{\mu }_{\mathrm{rel},\mathrm{hel}}=\displaystyle \frac{{\rm{\Delta }}\theta }{{\rm{\Delta }}t}=13.22\pm 0.66\,\mathrm{mas}\,{\mathrm{yr}}^{-1}.\end{eqnarray} \tag{ 3 }$$

Before continuing, we note that Vandorou et al. (2023) incorrectly report an error of σ = 0.89 mas yr⁻¹ in their Table 2, evidently by adding in quadrature the errors of the two components of μ _rel,hel. During the 4.12 (equally, 0.12) yr of elapsed time, Earth moved approximately east by about 0.7 au, causing the lens to appear to move west by δ θ = 0.7 π_rel ∼ 90 μas(π_rel/130 μas), where we have normalized to the Shvartzvald et al. (2017) value, which Vandorou et al. (2023) argue, based on their measurement, is way too large. Even if the Shvartzvald et al. (2017) value is correct, and taking account of the direction of the offset, −32°, north through east, this would affect the proper-motion determination by only about 0.01 mas yr⁻¹, substantially more than an order of magnitude below the measurement errors. Hence, this effect can safely be ignored.

We now argue that this measurement is in 4σ disagreement with the μ_rel measurements made by Bond et al. (2017) and Shvartzvald et al. (2017), via the relation,

$$\begin{eqnarray}&&{\mu }_{\mathrm{rel}}=\displaystyle \frac{{\theta }_{* }}{{t}_{* }}=\displaystyle \frac{{\theta }_{* }}{\rho {t}_{{\rm{E}}}},\end{eqnarray} \tag{ 4 }$$

where θ_* is the angular radius of the source, ρ = θ_*/θ_E, and t_* is the source self-crossing time. Substantially different methods are used to measure θ_* and t_*. Hence, we treat them separately. We note that these two measurements are virtually uncorrelated.

2.1.  t_* Measurement

Bond et al. (2017) and Shvartzvald et al. (2017) analyzed disjoint photometric data sets, and each fit these to planetary models, which automatically yielded estimates of t_*. In principle, one might simply take these two measurements and combine them in the standard way to obtain the best overall estimate. However, for several reasons, we adopt a more comprehensive approach.

Our main concern is to check that the overall fits presented in these two papers are consistent. If they were not, it would be evidence that one or the other of these two measurements were dominated by systematics, in which case it would not be appropriate to combine them in the naive way. Second, each group reported multiple models, i.e., two and eight models, respectively. Both groups reported two classes of models, which both labeled "wide" and "close," but which are more accurately called "inner" and "outer" (Gaudi & Gould 1997; Yee et al. 2021). That is, in the inner model, the source passes inside the planetary caustic (over a ridge between the planetary and central caustics), while in the outer model, it passes outside the planetary wing of a central caustic. In both cases, these models (or groups of models) are indistinguishable at the 1σ level. Therefore, we must determine whether these models make essentially identical predictions for t_*. Third, Shvartzvald et al. (2017) simultaneously analyzed Spitzer data and so reported measurements of the microlensing parallax vector π _E. Such measurements are subject to a well-known four-fold degeneracy (Refsdal 1966; Gould 1994), which is the reason that Shvartzvald et al. (2017) reported four times more models. Although we are not directly concerned with the π _E measurements in this section, we must determine whether including π _E in the fit substantially impacts the parameters that we are interested in.

To conduct these investigations, we first reparameterize the fits reported by the two papers in terms of "invariants." That is, one usually expresses the solutions to planetary microlensing events in terms of the seven parameters, (t₀, u₀, t_E, ρ, q, α, s), where u₀ is the impact parameter (normalized to θ_E), α is the angle of the source trajectory relative to the planet–host axis, and s is the planet–host separation (also normalized to θ_E). However, Yee et al. (2012) showed that for planets detected at high magnification the four quantities t_eff ≡ u₀ t_E, t_* ≡ ρ t_E, t_q ≡ qt_E, and f_S t_E, are usually "invariants," i.e., the fractional errors in these quantities are smaller than those naively inferred from the fractional errors of the two factors because these are anticorrelated. Here, f_S is the source flux, although we will not be making use of the f_S t_E invariant until Section 2.2.

Thus, our seven parameters are (t₀, t_eff, t_E, t_*, t_q , α, s). We infer best estimates of the invariant parameters by taking the products of the two factors from the published tables. For the error estimates, we proceed as follows. First, for the cases that asymmetric errors are reported (all from Shvartzvald et al. 2017), we symmetrize them. Second, for an invariant parameter η t_E (i.e., the product of two parameters from the fit, η and t_E), we adopt $\sigma (\eta {t}_{{\rm{E}}})=\eta {t}_{{\rm{E}}}\sqrt{{[\sigma (\eta )/\eta ]}^{2}-{[\sigma ({t}_{{\rm{E}}})/{t}_{{\rm{E}}}]}^{2}}$. Note that Bond et al. (2017) already reported t_*, while Shvartzvald et al. (2017) reported ρ and t_E separately.

Our first step was to check whether the two models (or eight models) were consistent among themselves, i.e., they had essentially the same values and errors (except for s). Half of the eight Shvartzvald et al. (2017) models had negative t_eff and α, but we just considered the absolute values of these quantities for this purpose. We found that the error bars were all the same to within a few percent. Furthermore, we found that the largest difference between models was generally much smaller than the error bars. For Bond et al. (2017), the largest difference was 30% of the error bar (for t₀), while most were of order 20%. For Shvartzvald et al. (2017), the largest difference was 45% of the error bar (for t_E for one solution), while the others were of order 10%–20%. Therefore, for each parameter, we report, in Table 1, the simple average of all solutions (two or eight) from each paper, except that we report the values of s from the two topologies separately. In column 6, we report the difference between the two papers divided by the quadrature sum of their errors. Under the assumption that the results reported from the two papers are not dominated by systematics, we expect these differences to be unit-variance Gaussian distributed. The first six parameters (i.e., excluding s_inner and s_outer) are essentially uncorrelated, implying that χ² is just the sum of the squares of the values in this column, i.e., χ² = 5.7 for six degrees of freedom (dof). The last two rows are essentially uncorrelated with the others, but highly correlated with each other because $\sqrt{{s}_{\mathrm{inner}}{s}_{\mathrm{outer}}}\simeq {s}_{+}^{\dagger }\equiv (\sqrt{4+{u}_{\mathrm{anom}}^{2}}+{u}_{\mathrm{anom}})/2$, where u_anom = t_eff csc α/t_E = 0.64 day/t_E (Gould et al. 2022; Hwang et al. 2022; Ryu et al. 2022). If we add this seventh dof and evaluate it at the mean of their absolute values, i.e., 1.35, then χ² = 7.5 for seven dof. In either case, this test constitutes strong evidence against systematics in either analysis.

Table 1. Combination of Two Fits Comparison

Parameter	B+	σ(B+)	S+	σ(S+)	Δ/σ	Comb.	σ(C)
t0 − 2,457,560	8.7716	0.0020	8.7693	0.0013	0.9432	8.7700	0.0011
teff (day)	0.5225	0.0069	0.5296	0.0046	−0.8605	0.5274	0.0038
tE (day)	10.1850	0.2550	9.9600	0.1100	0.8102	9.9953	0.1010
t* (day)	0.0337	0.0022	0.0286	0.0037	1.1730	0.0324	0.0019
tq (hr)	0.0103	0.0016	0.0133	0.0018	−1.2249	0.0117	0.0012
α (deg)	55.2850	0.2600	55.4988	0.1300	−0.7353	55.4560	0.1163
sinner	1.0698	0.0078	1.0858	0.0078	−1.4505	1.0778	0.0055
souter	0.9957	0.0073	0.9839	0.0072	1.1508	0.9897	0.0051

Note. B+ = Bond et al. (2017); S+ = Shvartzvald et al. (2017); C = Combined.

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Hence, it is justified to combine them, which is done in columns 7 and 8. In particular, we find,

$$\begin{eqnarray}&&{t}_{* }=0.0324\pm 0.0019\,\mathrm{day},\end{eqnarray} \tag{ 5 }$$

i.e., a 5.9% error.

2.2.  θ_* Measurement

As for the t_* measurement, the two papers relied on completely independent data sets for the measurement of θ_*, which provides a powerful consistency check. On the other hand, both papers employed the same overall method, which is subject to the same systematic errors. Therefore, the two θ_* measurements cannot simply be averaged together as was done for t_*, in Table 1.

The basic method (Yoo et al. 2004) is: first, measure the offset of the source relative to the red clump, ${\rm{\Delta }}{[(V-I),I]={[(V-I),I]}_{S}-[(V-I),I]}_{\mathrm{cl}}$, on a color–magnitude diagram (CMD); second, make use of the "known" dereddened position of the red clump, [(V − I), I]_cl,0 (Bensby et al. 2013; Nataf et al. 2013), to calculate the dereddened source values [(V − I), I]_s,0 = [(V − I), I]_cl,0 + Δ[(V − I), I]; and third, use a color/surface-brightness relation to derive θ_* from [(V − I), I]_s,0. While the first step is a straightforward measurement with (usually) equally straightforward error estimation, the other two steps are subject to systematic errors that are more difficult to quantify.

Bond et al. (2017) and Shvartzvald et al. (2017) found Δ[(V − I), I]_B+2017 = (−0.355 ± 0.021, 3.369 ± 0.018) and Δ[(V − I), I]_S+2017 = (−0.37 ± 0.03, 3.40 ± 0.04), respectively. Before continuing, we note that, of course, each group used its own measurement of I_S from its fit to the data. Because f_S t_E is an invariant, and because the two fits differed by t_E,B+2017/t_E,S+2017 = 1.0226, these I_S values would differ (after calibrating to the same CMD) by 0.024 mag. We put these on the same system by adopting the combined t_E from Table 1, which yields adjusted values Δ[(V − I), I]_B+2017 = (−0.355 ± 0.021, 3.349 ± 0.018) and Δ[(V − I), I]_S+2017 = (−0.37 ± 0.03, 3.404 ± 0.04).

In our view, the Bond et al. (2017) error bars are underestimated. First, they report the error in I_S itself as "0.001," although their reported error in t_E, combined with f_S t_E invariance, implies that it is 0.027, which is substantially larger than their reported total error. In addition, based on our extensive experience, including making all of the CMDs used for the Bensby et al. (2013) calibrations, we do not believe that the clump can be centroided to the precision given by Bond et al. (2017) who quote I_clump = 16.212 ± 0.018 and (V − I)_clump = 2.468 ± 0.007. We note that Bond et al. (2017) also report a separate measurement based on the Optical Gravitational Lensing Experiment (OGLE)-IV data, Δ(V − I) = −0.39 ± 0.03. We then adopt an average of the two values for ΔI and of the three values for Δ(V − I), and we use the Shvartzvald et al. (2017) error bars, which basically reflect the difficulty of centroiding the clump,

$$\begin{eqnarray}&&{\rm{\Delta }}{[(V-I),I]}_{\mathrm{adopted}}=(-0.37\pm 0.03,3.38\pm 0.04).\end{eqnarray} \tag{ 6 }$$

As mentioned above, the next two steps require estimates of the systematic errors. For this line of sight, the "known" position of the clump is [(V − I), I]_cl,0 = (1.06, 14.44). Based on our experience carrying out the Bensby et al. (2013) calibration, we estimate the clump color error as 1.06 ± 0.03. The Nataf et al. (2013) clump magnitude measurement, when combined with various stellar physics arguments, led to an estimate for a Galactocentric distance of R₀ = 8.1 kpc, in remarkable agreement with subsequent direct observations of SgrA*. Therefore, we estimate the systematic error in the clump magnitude as 14.44 ± 0.02, and so find,

$$\begin{eqnarray}&&{[(V-I),I]}_{S,0}=(0.69\pm 0.04,17.82\pm 0.05).\end{eqnarray} \tag{ 7 }$$

As described above, the Yoo et al. (2004) method derives θ_* from the dereddened color and magnitude using a color/surface-brightness relation. Usually, one employs such a relation that is calibrated from angular diameter measurements. However, as the source has almost exactly the color of the Sun, we use its color and absolute magnitude, ${[(V-I),{M}_{I}]}_{\odot }=(0.71,4.10)$, its radius R_⊙ = 695,700 km, as well as the differential color relation $d\mathrm{ln}{\theta }_{* }/d(V-I)=0.966$ quoted by Bond et al. (2017) to obtain θ_* = 0.822 ± 0.037 μas. Finally, we must account for the fact that the source star has an unknown composition and so may have a somewhat different surface brightness from the Sun at the same color. To account for this, we add 2% in quadrature to the error and finally obtain,

$$\begin{eqnarray}&&{\theta }_{* }=0.822\pm 0.041\,\mu \mathrm{as}.\end{eqnarray} \tag{ 8 }$$

For comparison, Bond et al. (2017) derived θ_* = 0.856 ± 0.019 μas, while Shvartzvald et al. (2017) derived θ_* = 0.82 ± 0.07 μas.

Combining Equations (5) and (8) yields,

$$\begin{eqnarray}&&{\mu }_{\mathrm{rel}}=\frac{{\theta }_{* }}{{t}_{* }}=9.27\pm 0.72\,\mathrm{mas}\,{\mathrm{yr}}^{-1}.\end{eqnarray} \tag{ 9 }$$

Ignoring for the moment the difference between heliocentric and geocentric proper motions, ⁷ Equations (3) and (9) differ by 3.95 ± 0.98 mas yr⁻¹, i.e., a 4.0σ discrepancy.

Gould (2022) also mentions contradictions between light-curve-based measurements of the microlens parallax π _E and the lens parameters derived from the AO imaging. The microlens parallax is defined by,

$$\begin{eqnarray}&&{{\boldsymbol{\pi }}}_{{\rm{E}}}\equiv \displaystyle \frac{{\pi }_{\mathrm{rel}}}{{\theta }_{{\rm{E}}}}\,\displaystyle \frac{{{\boldsymbol{\mu }}}_{\mathrm{rel}}}{{\mu }_{\mathrm{rel}}}.\end{eqnarray} \tag{ 10 }$$

In principle, it can be measured from light-curve distortions generated by Earth's annual motion (Gould 1992), but because for most events (and OGLE-2016-BLG-1195, in particular), t_E ≪ 1 yr, this is often impossible, and it is difficult in most other cases.

Nevertheless, OGLE-2016-BLG-1195 was observed by Spitzer from solar orbit (Yee et al. 2015), and such observations can in principle measure π _E even for very short events (Refsdal 1966; Gould 1994). However, such satellite-parallax measurements are only straightforward if the satellite observations cover both the rising and falling sides of the light curve. Because of constraints on Spitzer operations, observations could not be immediately triggered. In particular, observations of OGLE-2016-BLG-1195 did not begin until 1.6 days after the ground-based peak. Moreover, the total flux variation was only about 2.5 flux units, whereas in several other cases, it has been shown that Spitzer light curves show systematics at the level of 0.5–1 flux unit. Hence, in our view, results derived from cases with few-flux-unit variations must be treated cautiously, but can still provide valuable information. For example, Spitzer observations of Kojima-1 (Zang et al. 2020) covered only the extreme falling wing of the light curve, with a flux variation of only 5 units, yet it delivered precise parallax information that has been independently confirmed by other techniques (Dong et al. 2019; Fukui et al. 2019).

However, rather than seeing the contradiction with previous π _E measurements as a reason for caution and deeper investigation of their own results, Vandorou et al. (2023) took this contradiction as "proof" that Spitzer π _E measurements derived from low flux variations are unreliable. By contrast, in our view, such a conclusion would only be appropriate if the AO results were independently confirmed in some other aspect, such as agreeing with the very precise, and multiple confirmed measurements of μ_rel. In fact, the radical disagreement of the AO-based and light-curve-based proper-motion measurements implies that the AO observations cannot be used to cast doubt on the Spitzer π _E. Thus, in the following section, we will retain an open mind regarding the Spitzer π _E measurement.

In general, when a planet/host mass measurement is made based on a single late-time AO observation, one should always consider the possibility that the "other star" is not the lens, but rather is a companion to either the lens or source or is an ambient star. Moreover, as pointed out by Gould (2022), these possibilities must be taken even more seriously when the AO measurement appears to contradict previously known facts about the event. The contradiction that is expected to be most frequent is between the AO-based μ _rel,hel and the light-curve-based μ_rel. In the present case, this contradiction is quite severe. Moreover, as just discussed in Section 3, it is augmented by a conflict between the AO measurement and the Spitzer π _E. See Figure 3 of Vandorou et al. (2023).

4.1. Companion to the Lens (Host)

The "other star" could be a companion to the lens (host). As we will soon show, such a companion must be separated from the lens by at least tens of θ_E (so tens of astronomical units), and therefore it will be moving with nearly identical proper motion to the lens. Thus, it can be robustly predicted that a second epoch, which could be taken "now" (in 2023) will show a vector displacement from its 2020 position of about (2023 – 2020)μ_rel ≃ 28 mas. Such an observation would rule out a source companion (which would hardly move) and would render the explanation of an ambient star extremely unlikely. Hence, it would confirm the "lens-companion" hypothesis. Furthermore, by extrapolating the companion proper motion back to t₀ (in 2016) one would find the separation between the "lens" (planet host) and its companion in arcseconds.

However, such a measurement would not, in itself, tell us the mass or distance of the host or the planet. That is, the lens could be anywhere along the line of sight, and at each possible distance, the lens would have a mass that is consistent with the measured ${\theta }_{{\rm{E}}}=\sqrt{\kappa M{\pi }_{\mathrm{rel}}}$, and the companion would have a mass consistent with its measured flux. Nevertheless, such a measurement would tell us the future positions as functions of time of the lens relative to both the source and the companion, which would permit an informed decision on when the lens could be imaged (possibly with more advanced instruments). Even if the lens were dark, one could still determine the distance of the companion (and so the lens system) by multicolor, or possibly spectral observations of the companion. Combining this distance with the θ_E measurement would then yield the host (and planet) mass. Thus, the first step is simply to obtain another AO epoch, which could be done immediately.

Next, is the lens-companion hypothesis consistent with the Spitzer π _E measurement being correct? Recall first that there were eight such π _E solutions. However, these come in four (inner/outer) pairs, whose π _E are nearly identical. Moreover, the scalar amplitudes are very similar among these four solutions: π_E = (0.437, 0.473, 0.482, 0.430), leading to very similar π_rel = (0.111, 0.120, 0.122, 0.109) mas. Hence, all solutions are at similar distances D_L = 4.2 kpc, and they also have similar lens masses M_host ∼ 0.07 M_⊙. At this distance, the measured flux of the companion K = 19.96, together with the extinction A_K = 0.24 (Gonzalez et al. 2012) and the mass–luminosity relation of Benedict et al. (2016), yields a companion mass M_comp = 0.4 M_⊙, hence, a mass ratio ⁸ Q = M_comp/M_host ≃ 6.

In principle, such a massive companion could be ruled out because it might predict a light-curve distortion near the peak of this relatively high-magnification event that is not seen. To determine whether this is the case, we predict the position of the lens relative to the source, using the measured μ_rel and the four values of π _rel, to obtain the corresponding μ _rel = μ_rel π _E/π_E, and finally convert to heliocentric,

$$\begin{eqnarray}&&{{\boldsymbol{\mu }}}_{\mathrm{rel},\mathrm{hel}}={{\boldsymbol{\mu }}}_{\mathrm{rel}}+\displaystyle \frac{{\pi }_{\mathrm{rel}}}{\mathrm{au}}{{\boldsymbol{v}}}_{\oplus ,\perp },\end{eqnarray} \tag{ 11 }$$

where v _⊕,⊥(N, E) = (−0.76, +28.94) km s⁻¹ is the velocity of Earth, projected on the sky at t₀.

These models are illustrated in Figure 1. The green arrows represent the four heliocentric proper motions, propagated over the 4.12 yr between t₀ and the Keck observations. The blue point is the measured position of the "other star" in 2020 relative to the source, which is at the origin. In these models the "other star" is assumed to be a companion to the host. The black points then represent the companion position relative to the source (and so the host) at t₀. The red points represent the position of the host relative to the source in 2020.

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Note that for two of the solutions, the companion was separated from the source at t₀ by 65 or 70 mas, so if there had been an AO observation at that time, such models could have been confirmed or ruled out. However, to the best of our knowledge, no such observations were taken. There were AO observations taken in 2018 by Vandorou et al. (2023), when the companion would have been separated from the source by about 57 or 60 mas for these two models. These observations were of lower quality than the 2020 observations, so it is not clear whether they could have detected the companion at the position predicted by these two models (i.e., models 3 and 4 in Figure 1). In any case, there are no constraints on models 1 and 2.

Note that models (1, 2, 3, 4) predict separations between the lens companion and the source of about (56, 60, 76, 78) mas in mid 2023. Hence, all would be detectable under good conditions, while models 3 and 4 (the two solutions that could not have been probed by the 2018 observations) would be detectable even under moderately good conditions.

For all four solutions, the companion is separated from the lens by at least Δθ_comp > 27 mas, which corresponds to s_comp = Δθ_comp/θ_E > 106. This would induce a Chang & Refsdal (1979, 1984) caustic of radius $w=2Q/{s}_{\mathrm{comp}}^{2}\sim {10}^{-3}$, which is about 50 times smaller than the closest passage of the source, i.e., u₀ ∼ 0.05. Hence, the companion would not have induced any noticeable effect on the light curve near peak.

4.2. Companion to the Source

4.3. Ambient Star

There are two additional issues that impact the plausibility of the identification of the "other star" as the lens. While neither appears to be as severe as the 4σ discrepancy in the proper-motion measurement, both do need to be considered.

5.1. Galactic Kinematics

5.2. Apparent Source Star Is Too Bright

As discussed in Section 2, the first response to an apparent conflict between the light-curve-based measurement of μ_rel and the AO-based measurement of μ _rel,hel should be a review of the published literature to determine how secure this conflict actually is. We carried such a review and found the conflict to be at the 4.0σ level. The strength of this conflict provided the context for investigating other possible explanations for the detection of the "other star" in Sections 4 and 5.

However, another possibility is that the original light-curve analysis was incorrect and that a corrected value of μ_rel might be more consistent with the AO-based μ _rel,hel. In Section 2, we argued that the original analyses were likely to be robust because there had been two such analyses that used independent data sets and independent codes, and we measured the difference between these as having χ²/dof = 7.5/7.

Nevertheless, there are several reasons for pursuing this course. The most important is that the analysis of even seemingly "simple" planetary events like OGLE-2016-BLG-1195 has progressed substantially over the intervening 7 yr and continues to do so. In particular, new degeneracies have been discovered and new methods for finding degenerate solutions are being developed. As we will review below, one of these degeneracies can, in principle, lead to dramatic changes in t_* and therefore in μ_rel = θ_*/t_*. Second, while seemingly unlikely, it remains possible that mistakes were made in the original analyses. Third, the disagreement between the two measurements of t_* was at the 1.2σ level, with the weighted average strongly dominated by the slower-μ_rel result of Bond et al. (2017). Hence, if there were an error in that analysis, this could, by itself, significantly reduce the tension. Finally, there have recently been important improvements to the KMT data reductions, and these could in principle also change the result.

Moreover, Vandorou et al. (2020) charted a path of such light-curve reanalyses when they found that their AO observations conflicted with the upper limit on lens light derived from the original light-curve analysis by Yee et al. (2014) of MOA-2013-BLG-220. ⁹

While we mainly report in this section on the results using rereduced KMT data, we note that we first checked on the result of combining all the light curves (i.e., OGLE, MOA, and KMT) from the published literature (Bond et al. 2017; Shvartzvald et al. 2017). We found that the best-fit values of all eight parameters listed Table 1 were well within the 1σ ranges shown in the final two columns of that table. By far, the largest "discrepancy" was for t₀, which differed by ∼0.5σ. In addition, we found that while the central value of t_eff was almost identical to that of Table 1, its error bar was only half as big.

As stated above, to conduct our reanalysis, we first rereduced the KMT data using an improved pipeline (H. Yang et al. 2023, in preparation) that is now routinely applied to essentially all new published KMT events. Inspection of the images showed that the KMTC01 data are affected by a "spike" from a neighboring bright star. Hence, we eliminated these from the analysis. Because KMTC does not cover the anomaly and because (by favorable chance) the event lies in two other high-cadence fields (BLG41 and BLG42), the elimination of these data is not expected to have a major effect.

The results of our analysis are given in Table 2, where we show both the standard parameters and the three invariant parameters (t_eff, t_*, t_q ) in order to enable comparison with Table 1.

Table 2. Static 2L1S Models for OGLE-2016-BLG-1195 (OGLE+MOA+KMT-new)

Parameters	Central		Resonant
	Central Inner	Central Outer	Resonant Inner	Resonant Outer
χ2/dof	29,461.4/30,470	29,459.6/30,470	29,487.3/30,470	29,486.6/30,470
t0 − 7560	8.7707 ± 0.0010	8.7706 ± 0.0009	8.7707 ± 0.0009	8.7707 ± 0.0009
u0	0.0528 ± 0.0007	0.0528 ± 0.0007	0.0530 ± 0.0007	0.0531 ± 0.0007
tE (days)	9.91 ± 0.11	9.91 ± 0.11	9.89 ± 0.11	9.88 ± 0.10
ρ (10−3)	3.20 ± 0.18	3.20 ± 0.17	3.52 ± 0.09	3.56 ± 0.10
α (degree)	55.25 ± 0.11	55.23 ± 0.11	55.28 ± 0.11	55.31 ± 0.11
s	1.0763 ± 0.0049	0.9911 ± 0.0044	1.0402 ± 0.0006	1.0254 ± 0.0007
q (10−5)	4.62 ± 0.44	4.60 ± 0.43	2.75 ± 0.13	2.77 ± 0.15
$\mathrm{log}q$	−4.337 ± 0.042	−4.339 ± 0.041	−4.561 ± 0.021	−4.559 ± 0.023
teff (days)	0.5237 ± 0.0015	0.5235 ± 0.0015	0.5241 ± 0.0016	0.5246 ± 0.0016
t* (days)	0.0317 ± 0.0017	0.0317 ± 0.0016	0.0348 ± 0.0009	0.0352 ± 0.0009
tq (hr)	0.0110 ± 0.0010	0.0110 ± 0.0010	0.0065 ± 0.0003	0.0066 ± 0.0003
fS, OGLE	0.2304 ± 0.0030	0.2304 ± 0.0030	0.2311 ± 0.0031	0.2315 ± 0.0029

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There is one major and one minor feature of note about the new analysis. The major feature is that there are four solutions, including the two solutions found in the original papers (Bond et al. 2017; Shvartzvald et al. 2017), plus two additional solutions that were identified in our new grid search (Figure 2). In the original solutions, the bump peaking at 7569.13 in Figure 3 is a major-image anomaly generated by the source crossing a ridge that extends from the central (or resonant) caustic in the direction of the planet. In the additional solutions, the source crosses two closely spaced caustics in the planetary wing of a resonant caustic. See Figure 4. This degeneracy was discovered by Ryu et al. (2022) and was dubbed the "central-resonant" degeneracy by Yang et al. (2022).

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Of particular relevance in the current context is that such a difference in morphologies can significantly impact t_*. This occurred for both of the cases in Ryu et al. (2022; i.e., KMT-2021-BLG-1391 and KMT-2021-BLG-0253), as well as for one of the two cases in Yang et al. (2022; i.e., KMT-2021-BLG-1689 but not KMT-2021-BLG-0171). As Ryu et al. (2022) recount, this degeneracy was discovered by a systematic effort to identify symmetries among multiple solutions. They hypothesized that because the degeneracy was found in two out of four randomly chosen events, it could be common and therefore should be checked for in archival events with bump-type anomalies at relatively high magnification. OGLE-2016-BLG-1195 is indeed a typical example of such an anomaly.

Table 2 shows that the additional solutions (labeled "resonant") indeed have significantly different values of t_* from the original solutions. Specifically, the values are t_* = 0.0317 ± 0.0017 days (central) versus t_* = 0.0350 ± 0.0009 days (resonant). This difference is far smaller than the factor ∼ 2 difference found by Ryu et al. (2022) for their two cases. Moreover, in the present context, the new solutions have larger t_* (so smaller μ_rel), which would only strengthen the tension, not relieve it. Finally, the resonant solutions are disfavored in the combined analysis by Δχ² = 27, which would normally lead to their being reported but discounted in the final assessment of the planet's characteristics. Nevertheless, because the χ² difference is divided between the two independent data sets (see Figure 3), it would have been substantially smaller in the original analyses had these additional solutions been recognized, and it could not have been so decisively rejected. ¹⁰ In particular, if the resonant solution had been recognized, it could not have been ruled out based solely on the Δχ² ∼ 7 difference from an analysis restricted to OGLE and MOA data. Nevertheless, this modest Δχ² combined with the phase-space arguments of Yang et al. (2022), would (in a more modern context) probably be enough to reject this solution.

The minor feature is that the best-fit value of t_* is slightly smaller than the one in Table 1 and also has a smaller error bar: t_* = 0.0317 ± 0.0017 days versus the previous value of t_* = 0.0324 ± 0.0019 days. While this change is small compared to the errors, it still must be taken into account to make the best estimate of the discrepancy between the light-curve-based and AO-based proper-motion measurements.

The net result of this investigation is that the light-curve-based proper motion increased from μ_rel = 9.27 ± 0.69 mas yr⁻¹ (Equation (9)) to μ_rel = 9.52 ± 0.69 mas yr⁻¹. Comparing to Equation (3) (and ignoring the correction from heliocentric to geocentric), this implies a conflict of 3.9σ rather than 4.0σ. Thus our light-curve reanalysis confirms the strong tension between the light-curve-based and AO-based proper-motion measurements.

High-resolution imaging of the hosts of microlensing planets in which the host star is resolved from the source, which was pioneered by Batista et al. (2015) and Bennett et al. (2015) for Keck-AO and Hubble Space Telescope (HST) imaging, respectively, has immense scientific prospects. As demonstrated by Gould (2022), this approach can yield planetary and host masses for a statistically complete sample of well over 100 planets, beginning at first AO light on ELTs, probably about 2030. There is no other technique that can do this, in particular covering such a wide range of host and planet masses, projected separations, and Galactic environments.

However, as also shown by Gould (2022), only very fragmentary results are possible prior to ELT observations because only a small subset of the full sample will be accessible to present-day instruments. Hence, the main scientific return of high-resolution observations today is technical in nature, in effect, establishing and refining the viability of the technique. A good example is provided by Vandorou et al. (2023), who were able to detect and measure the 16:1 flux ratio of a neighbor at 54 mas, i.e., inside the Keck-AO FWHM. This, together with earlier related achievements by the same group, greatly increases confidence in the method, a confidence that will be sorely needed to gain the necessary observing time on ELTs in a highly competitive environment.

Nevertheless, as also systematically analyzed by Gould (2022), the technical issues arising from this method are not restricted to imaging technology: false positives and false negatives must be driven down to an acceptable level. Hence, the techniques for identifying these must not only be cataloged in theory, they must also be tested in practice.

In this paper, we have shown that the host identification reported by Vandorou et al. (2023) raises three different red flags. The most striking of these, and also the one that Gould (2022) argued would be the most common indication of a false positive, is the strong conflict between the AO-based proper-motion measurement (Equation (3)) and the one derived from the published light-curve analyses (Equation (9)). The second red flag (argued by Gould 2022 to be less common) is the conflict with a previous microlens-parallax measurement. In addition to these two known red flags, we also found that the Vandorou et al. (2023) host identification leads to a solution that is kinematically disfavored relative to the one derived from the Spitzer microlens-parallax measurement.

None of these red flags proves that this host identification is incorrect. Together, however, they do imply that the result must be more deeply investigated, and if these investigations do not lead to clear rejection, then the identification must be confirmed by directly measuring the proper motion of the putative host relative to the source by additional late-time observations.

Prompted by these three red flags, we have conducted such an investigation based on existing data. We noted that the source appears to be "too bright" relative to the flux values implied by the light-curve analysis, something that was also previously noted by the Vandorou et al. (2023) authors but not reported in their published paper. We considered various explanations for this discrepancy, including one in which Vandorou et al. (2023) may have identified the wrong "star" (or stellar asterism) on which to conduct their analysis. We then found preliminary evidence in favor of this hypothesis. ¹¹

An important conclusion of Vandorou et al. (2023) was that by "disproving" the Spitzer parallax solution, they helped to demonstrate the general unreliability of Spitzer parallax measurements. While we do not agree that the Spitzer measurement has in fact been "disproved," we do agree that testing of Spitzer parallax measurements by high-resolution imaging is extremely important. Gould (2022) cataloged 12 planetary events with Spitzer parallaxes (his Table 3), of which five have giant sources and lens–source relative proper-motion measurements (his Figure 4). Because giant sources can require much longer wait times, these potential targets may have to wait until well after ELT AO first light. In such cases, the Spitzer-based mass measurements will be essential to including these planets in statistical studies. Hence, testing the reliability of Spitzer-based parallaxes on events with fainter sources, e.g., OGLE-2016-BLG-1195, is crucial for establishing the conditions under which these giant-source planetary events can be included. We note that new Spitzer planets are still being discovered (so not yet cataloged by Gould 2022), including OGLE-2019-BLG-0679 (Jung et al. 2023) and OGLE-2017-BLG-1275 (Ryu et al. 2023), with the former having a giant source. Thus, one of the technical goals of current high-resolution studies should definitely be to test Spitzer parallaxes when feasible, in particular focusing on those that, like OGLE-2016-BLG-1195, do not have giant-star sources.

This research has made use of the Keck Observatory Archive (KOA), which is operated by the W. M. Keck Observatory and the NASA Exoplanet Science Institute (NExScI), under contract with the National Aeronautics and Space Administration. J.C.Y. acknowledges support from US NSF grant No. AST-2108414. Y.S. acknowledges support from BSF grant No. 2020740. J.Z. and W.Z. acknowledge support by the National Science Foundation of China (grant No. 12133005). W.Z. acknowledges the support from the Harvard-Smithsonian Center for Astrophysics through a CfA Fellowship. This work was authored by employees of Caltech/IPAC under Contract No. 80GSFC21R0032 with the National Aeronautics and Space Administration.

We measure the position of the source relative to the frame of field stars in its neighborhood. Such a measurement can potentially have two different applications. First, it allows us to identify the star (or asterism) on the Keck-AO images that contains the source and therefore should be the target of detailed analysis. Second, it could also permit a measurement of the source's proper motion relative to this frame. Then, in turn, the frame proper motion could be put on an absolute scale using the Gaia sources within it. Combining such a measurement of μ _S,hel with a measurement of μ _rel,hel from the imaging would yield μ _L,hel = μ _rel,hel + μ _S,hel, which could help clarify the nature of the host.

These two potential applications have very different accuracy requirements. The surface density of stars (or asterisms) that are bright enough to contain the K ∼ 17.25 source is relatively low. Hence, barring a pathological pileup of such stars, an accuracy of a few tens of milliarcseconds should be adequate. In fact, there are only two "stars" in the broad neighborhood of the source that are sufficiently bright to contain the source, and these are separated by ∼175 mas. See Figure 1 of Vandorou et al. (2023). Hence, few-tens-of-milliarcseconds accuracy is indeed enough. On the other hand, in order to constrain μ _L significantly, the accuracy of μ _S should be ≲σ_μ ∼ 3 mas yr⁻¹, i.e., the known dispersion of bulge stars. Hence, the position measurement should have an accuracy ≲ σ_μ Δt ∼ 12 mas yr⁻¹, which is more demanding.

The steps toward making this measurement are basically standard but still require some description. The first step is to measure the source position on the reference frame of the field stars in the pixel-coordinate frame of the camera. For this purpose, we start by making pyDIA (Albrow 2017) reductions of the KMTC01 images. The reference image is formed by stacking many good images. It is aligned to and then subtracted from a series of images in which the source is magnified. The resulting difference images then basically consist of an isolated point-spread function (PSF), whose position is easy to measure. ¹²

Next we repeat this procedure on three additional data sets: KMTC41, KMTS01, and KMTA01. We align the images by cross-matching relatively bright, I < 17.5 stars, making a two-dimensional linear (six parameter) transformation, and iteratively rejecting outliers. Then, for each cross-matched star (and also for the microlensed source position), we have four separate measurements, all in the KMTC01 reference system. From these we find the mean position, scatter (standard deviation), and standard error of the mean (s.e.m.). In particular, for the microlensed source, we find transformed positions (153.0735, 150.3169), (153.0633, 150.2900), (153.1118, 150.3588), and (153.1165, 150.3539) for KMTC01, KMTC41, KMTS01, and KMTA01, respectively. This yields a mean and s.e.m. of,

$$\begin{eqnarray}&&\begin{array}{l}{(X,Y)}_{S,\mathrm{KMTC}01}=(153.0913,150.3299)\\ \quad \pm \ (0.0134,0.0168)\ \mathrm{pixels},\end{array}\end{eqnarray} \tag{ A1 }$$

corresponding to errors of (5.4, 6.7) mas. ¹³ In Table 3, we list the pixel positions and instrumental magnitudes and colors of the 20 brightest stars lying in the region of the Keck OSIRIS image. We also list the pseudo-K-band magnitude, K_pseudo ≡ I − (V − I) as a rough indicator of the relative brightness expected in the K band. ¹⁴ These indicators are used in Figure 5 to color the points that represent these 20 stars, which should enable easy comparison to the Keck K-band images.

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Using these, together with the microlensed source position in Equation (A1), it will be possible for any reader to make his/her own estimate of the source position on the Keck image. In addition, we provide the original KMTNet pyDIA CMD and star list files to enable complete reproduction of the analysis, if desired.

Next we proceed to make our own such measurements for the NIRC2 (2018) and OSIRIS (2020) epochs.

First, we retrieved the Keck images via the Keck Observatory Archive (KOA ¹⁵ ). For the NIRC2 2018 epoch we use 13 good images. For the OSIRIS 2020 epoch we use 23 good images. For each image (in each epoch) we construct an astrometric catalog using PSF fitting for the centroids of all sources detected in the vicinity of the event, using the software package described in Ofek (2014) and E. O. Ofek et al. (2023, in preparation) and specifically the astrometric tools used in Ofek (2019). The (23 or 13) catalogs are then aligned using third-order polynomials. The mean position and standard deviation is then measured for each entry. We then set the zero-point of this catalog as the measured position of the "northwest star," which we will find below is the closest to the microlensed source. Note that the standard errors of the mean of this star are (0.10, 0.06) mas and (0.44, 2.23) mas, for OSIRIS and NIRC2, respectively, which are small compared to the other errors in the problem. We express these offsets in arcseconds. Note that in the KMT system, the first coordinate (in pixels) increases to the west, while in the Keck system, the first coordinate (in arcseconds) increases to the east.

Next, we identify a restricted subset of the KMT stars that have Gaia counterparts with proper-motion measurements. We exclude from consideration any star with KMT position errors (s.e.m.) greater than 10 mas. In fact, all but one of these excluded stars have Gaia entries but without proper-motion measurements, likely due to crowding and/or unresolved sources. For each such star, we find (if present) the OSIRIS and NIRC2 counterparts. Table 4 lists the resulting 13 stars. Column 1 is a cross reference to Table 3. Columns 2 and 3 give the Gaia proper motions as well as their errors (in the second row). Column 4 gives the Gaia RUWE indicator. Columns (5, 6) and (7, 8) give the OSIRIS and NIRC2 positions, respectively, with their standard deviations shown in the second row. Note that for four of the 2 × 13 = 26 cases, there is no measurement (hence, no entry). Thus, there are potentially 10 cross-matches for OSIRIS and 12 for NIRC2. However, we find that the brightest three NIRC2 stars are seriously saturated, with bagel-morphology images. Corresponding to this, their standard deviations are dramatically larger than those of the fainter NIRC2 stars. Hence, we exclude these, leaving nine NIRC2 stars.

Next we propagate the KMT positions forward using the Gaia proper motions by 2.10 and 4.12 yr for NIRC2 and OSIRIS, respectively. For this purpose we subtract an estimate of the mean proper motion of the bulge, μ _bulge(E, N) = (−3.00, −5.20) mas yr⁻¹, from each Gaia proper motion. That is, ultimately we will be measuring the offset between the positions of Keck stars in the 2018/2020 bulge frame from the position of the microlensed source in the 2016 bulge frame.

We multiply the reported Gaia errors by 1.5 to take account of the general difficulty of proper-motion measurements in the bulge and then add these in quadrature to the KMT position errors. We also include the s.e.m.s of the Keck measurements, but these are generally too small to matter.

We then carry out a second-order (quadratic) transformation from the KMT frame to the Keck frames and thus derive the positional offsets of the "northwest star" from the nominal position of the microlensed source. We find offsets ("northwest star" minus microlensed source) of Δ(E, N)₂₀₁₈ = (−26.8, −1.6) ± (1.2, 2.2) mas with χ²/dof = 90/6 and Δ(E, N)₂₀₂₀ = (−46.3, −4.5) ± (1.9, 3.2) mas with χ²/dof = 108/8, for NIRC2 and OSIRIS, respectively.

Clearly, our formalism has not captured all sources of error. One potential cause of additional errors is that the KMT and Keck images have substantially different resolutions and bandpasses, so that each can be affected in different ways by stars that are uncataloged, either because they are too faint or are buried within the PSFs of cataloged stars. There could be others. We adopt the simple expedient of renormalizing the final error bars by $\sqrt{{\chi }^{2}/\mathrm{dof}}$ to obtain,

$$\begin{eqnarray}&&\begin{array}{c}{\rm{\Delta }}{\left(E,N\right)}_{2018}=(-26.8,-1.6)\pm (4.6,8.5)\,{\rm{mas}},\\ {\rm{\Delta }}{\left(E,N\right)}_{2020}=(-46.3,-4.5)\pm (7.0,11.8)\,{\rm{mas}}.\end{array}\end{eqnarray} \tag{ A2 }$$

To get a further handle on the errors, we repeat these calculations by eliminating either the largest or two-largest outliers. In seven of the eight cases, (two epochs) × (two eliminations) × (two components), we find that the changes are <10 mas, while in one case (NIRC2, two eliminations, decl.), the change is 24 mas. Hence, we estimate that the true errors are of order 15 mas. Nevertheless, for purposes of display in Figure 6, we show the formal errors of Equation (A2). This figure also shows the microlensed source position and the position of the "southeast star," i.e., the one identified by Vandorou et al. (2023) as the microlensed source. Recall from the discussion above that the Keck star positions are in the bulge frame at their respective epochs, while the microlensed source position is in the bulge frame from 2016. Hence, one possible reason that the microlensed source is displaced from the northwest star is that the latter moved toward the west in the intervening years. However, the two Keck positions are actually consistent within the errors. Hence it is also possible that the northwest star has moved very little in the bulge frame, while the microlensed source has remained within the Keck PSF of this bright star.

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Figure 7 shows the Keck images from the two epochs (NIRC2 2018 and OSIRIS 2020) with the (2016 bulge frame) position of the microlensed source superposed as a blue circle.

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Currently, the balance of evidence favors that the microlensed source is associated with the northwest star, rather than the southeast star that was identified by Vandorou et al. (2023). However, in our view, it would be premature to claim this as a fact. First, the evidence that we have presented must be weighed against the evidence (based on completely independent microlensing survey images) that led Vandorou et al. (2023) to the conclusion that their southeast-star identification was correct. Second, as we have discussed, the nonlinear transformation between the KMT and Keck frames are limited by the number of stars. In fact, we lacked enough stars even to make the standard third-order (cubic) transformation. One way to resolve this would be to take an HST image in the I band. This would require only a linear transformation. Moreover, it would allow direct comparison with the microlensing photometry (also in the I band), and so permit a more precise estimate of the excess flux.

However, perhaps the simplest approach would be to wait several more years and then take a Keck or HST image. For example, in 2025, the source and lens will be separated by ∼90 mas. If the Spitzer parallax is basically correct, then the host (near the star/BD boundary) will probably still not be visible. But if the host has a more typical mass, as argued by Vandorou et al. (2023), then it will be separately resolved in these images, probably near the northwest star, but likely visible in either case.

For the present, the main result of this appendix is to reinforce the need for a cautious approach to the identification of the host of OGLE-2016-BLG-1195.

The goal of difference-image astrometry is to locate the source position on the seeing-limited reference image, with the ultimate aim of comparing this reference-plus-source image to late-time high-resolution images. In the simplified presentation above, we implicitly assumed that nothing had moved between the epochs of the images entering the reference image and those of the magnified images. However, for a variety of reasons, including just convenience, the two may in principle be separated by some interval Δt, which could be either positive or negative. For completeness, we note that this could, in principle, lead to difficulties in aligning the reference and magnified images due to random motions relative to the bulk motion of the frame, but in practice this is almost never the case. Hence, we ignore this issue here. We further note that in most cases, the astrometric error of the source position is dominated by scintillation noise, rather than photon noise. However, this scintillation noise is automatically accounted for in our approach (which is often adopted) of estimating the astrometric errors from the scatter of multiple measurements.

After the magnified images (for simplicity, just called "image" I ) are photometrically and astrometrically aligned to the reference image R , and then R is convolved to match the PSF of I , a difference image D = I − R is formed by subtracting the second from the first. In the approximation that nothing has moved between the construction of R and I , D = f_S (A − 1) P _S , where f_S is the source flux, A is the magnification, and P _S is a normalized PSF whose centroid is at the location of the source θ _S . Then, in this approximation,

$$\begin{eqnarray}&&{{\boldsymbol{\theta }}}_{S}={\boldsymbol{c}}({{\boldsymbol{P}}}_{S})=\frac{{\boldsymbol{c}}({{\boldsymbol{D}}}_{S})}{(A-1){f}_{S}},\end{eqnarray} \tag{ B1 }$$

where c is the centroid operator that, in effect, just sums over the pixel values in P _S (of D ). ¹⁶

However, if either the microlensed source or the blended light, f_B , lying within the PSF (or both) have moved, then the subtraction operation will yield P _diff = ( I − R )/[f_S (A − 1)], whose centroid is displaced from θ _S by,

$$\begin{eqnarray}&&{\rm{\Delta }}{\boldsymbol{\theta }}={\boldsymbol{c}}({{\boldsymbol{P}}}_{\mathrm{diff}})-{{\boldsymbol{\theta }}}_{S}=\frac{{\boldsymbol{c}}({\boldsymbol{I}}-{\boldsymbol{R}})}{(A-1){f}_{S}}-{{\boldsymbol{\theta }}}_{S}{\rm{.}}\end{eqnarray} \tag{ B2 }$$

In general, the blended light can take complex forms. However, as long as both the source and the components of the blended light remain well within the FWHM of the PSF, one can ignore these complexities. This is the case we consider here. It is not difficult, in principle, to consider blends that, e.g., straddle the PSF. However, such a more general treatment would take us too far afield. Then, Equation (B2) can be evaluated,

$$\begin{eqnarray}&&{\rm{\Delta }}{\boldsymbol{\theta }}=\displaystyle \frac{({f}_{S}A{{\boldsymbol{\theta }}}_{S,1}+{f}_{B}{{\boldsymbol{\theta }}}_{B,1})-({f}_{S}{{\boldsymbol{\theta }}}_{S,0}+{f}_{B}{{\boldsymbol{\theta }}}_{B,0})}{(A-1){f}_{S}}-{{\boldsymbol{\theta }}}_{S,0},\end{eqnarray} \tag{ B3 }$$

where the subscripts "0" and "1" refer to the reference epoch and the magnified epoch, respectively. Taking note that θ _S,1 = θ _S,0 + μ _S Δt and θ _B,1 = θ _B,0 + μ _B Δt, Equation (B3) simplifies to,

$$\begin{eqnarray}&&{\rm{\Delta }}{\boldsymbol{\theta }}=\left[\displaystyle \frac{A}{A-1}{{\boldsymbol{\mu }}}_{S}+\displaystyle \frac{{f}_{B}}{(A-1){f}_{S}}{{\boldsymbol{\mu }}}_{B}\right]{\rm{\Delta }}t.\end{eqnarray} \tag{ B4 }$$

The first point to note about Equation (B4) is that if Δt is small, then this effect is negligible. This is the case for our measurement because the reference image is constructed from epochs of the same year as the event and on either side of the peak. Specifically, the mean epoch of the reference image is displaced from t₀ by Δt = −0.03 yr. However, high-quality reference images are often used to carry out photometry for many years. This is feasible because the photometric error induced (for an isolated source and in the approximation of a Gaussian PSF) is just $\delta I=\mathrm{log}{(32)(\delta \theta /\mathrm{FWHM})}^{2}$, which is tiny provided that the source has moved δ θ = μ_S Δt ≪ FWHM.

However, the astrometric effects can be much larger. For example, for very-low-magnification events, which includes some of the important class of giant-source free-floating planet (FFP) candidates (e.g., OGLE-2012-BLG-1323; Mróz et al. 2019), it is possible for A/(A − 1) ≳ 10, so that even with typical μ_S ∼ 4 mas yr⁻¹ and a modest time offset from the reference image, Δt ∼ 5 yr, the error in the source position could be Δθ_S ≳ 200 mas, which could dramatically impact the science interpretation. See, e.g., Gould (2014). Similarly, such typical motions of a giant-star blend that was a factor 10 brighter than the magnified source could create similar artificial offsets. For example, such a blend could lie at 200 mas, so well inside the seeing-limited PSF, but easily resolved in AO images of next-generation ELTs.

In most cases, these problems can be avoided simply by constructing reference images from the same year as the peak magnification of the event. However, this may be difficult in some cases, while in others, the specific application might require even greater care to assure that Δt is as close to zero as possible.