OGLE-2018-BLG-1269Lb: A Jovian Planet With A Bright, $I=16$ Host

The normalized source radius $\rho$ is precisely measured (see Table 1). This implies that we can measure $\theta_{\rm E}=\theta_{*}/\rho$ provided that we can estimate the angular source radius $\theta_{*}$. The Einstein radius is related to the lens mass $M$ and the relative parallax $\pi_{\rm rel}$ by

Then, we can use the measured $\theta_{\rm E}$ to constrain the lens properties. Hence, we first estimate $\theta_{*}$ by following the approach of Yoo et al. (2004).

Based on the KMTC03 pyDIA reduction calibrated to the OGLE-III catalog (Szymański et al., 2011), we build a $(V-I,I)$ color-magnitude diagram (CMD) with stars centered on the event location (see Figure 5). We next find the source position of $(V-I,I)_{\rm S}=(2.40\pm 0.02,19.42\pm 0.01)$ from the best-fit model. We also estimate the giant clump (GC) centroid as $(V-I,I)_{\rm GC}=(2.82\pm 0.05,16.34\pm 0.07)$, which yields an offset

where $(V-I,I)_{0,{\rm GC}}=(1.06,14.36)$ is the intrinsic GC centroid (Bensby et al., 2013; Nataf et al., 2013). Using this offset, we obtain the dereddened source position as $(V-I,I)_{0,{\rm S}}=(V-I,I)_{\rm S}-\Delta(V-I,I)=(0.64\pm 0.05,17.44\pm 0.07)$. This suggests that the source is either a late F or an early G dwarf.

We then apply $(V-I,I)_{0,{\rm S}}$ to the $VIK$ relation (Bessell & Brett, 1988) and $(V-K)/\theta_{*}$ relation (Kervella et al., 2004) to derive

where we add $5\%$ error in quadrature to $\theta_{*}$ to account for the uncertainty of $(V-I,I)_{0,{\rm GC}}$ and the color/surface-brightness conversion of the Galactic-bulge population relative to locally calibrated stars. With the measured $\rho$, we obtain

The geocentric relative lens-source proper motion is then

The unusually large values of $\theta_{\rm E}$ and $\mu_{\rm rel}$ suggest that the lens lies in the Galactic disk. From the definition of $\theta_{\rm E}$ (Equation (4)),

Thus, given the lens flux constraint, that will be discussed in the following subsection, the lens must be $D_{\rm L}={\rm au}/(\pi_{\rm rel}+\pi_{\rm S})\lesssim 3~{}{\rm kpc}$ (unless the lens is black hole). We note that the measured $\mu_{\rm rel}$ is also consistent with the typical values of disk lenses.

Gaia data (Gaia Collaboration et al., 2016, 2018; Luri et al., 2018) will play a critical role in the derivation of the lens physical characteristics, in several different respects. As has become relatively common in recent years, we will make use of the Gaia proper-motion measurement of the “baseline object”. However, in contrast to most events with such a measurement, in this case, the baseline object is strongly dominated by the lens (or at least a stellar component of the lens system). Thus, the Gaia parallax measurement is also relevant¹¹1By contrast, Gaia parallax measurements of microlens sources, which are nearly all in the bulge, are of no practical use.. Moreover, we will in this paper, for the first time, make use of the Gaia position measurement of the baseline object. That is, we will use the full position, parallax, proper motion (PPPM) Gaia solution at various points in the analysis. Hence, we introduce all of these measurements here, together with some context and cautions on their use.

Gaia reports PPPM values (at epoch 2015.5) of (R.A.,Decl.)_J2000 = (17:58:46.4171136073, $-27$:37:04.543560775) $\pm(0.16,0.14)\,{\rm mas}$,

and

Gaia also reports all $10$ correlation coefficients, but the only one of interest for our purposes is the one associated with the last equation, $0.31$. Before continuing, we note that Gaia parallaxes have a color-dependent zero-point error. For relatively red stars (due to intrinsic color or reddening), the shift is measured to be $\pi_{\rm shift}=0.055\,{\rm mas}$ (Zinn et al., 2019). Hence, we correct the baseline-object parallax to be

While the Gaia PPPM catalog is by far the best large-scale astrometric database ever constructed, its performance in the crowded fields of the Galactic bulge is not at the same level as in high-latitude fields, nor even as in the other parts of the Galactic plane. For example, Hirao et al. (2020) found that the Gaia proper-motion measurement of the baseline-object of OGLE-2017-BLG-0406 was spurious. This itself shows that Gaia measurements in crowded fields must be treated with caution.

However, Hirao et al. (2020) also showed, based on generally more precise (and likely, more accurate) OGLE proper motions of stars in the same field, that the reported Gaia proper motions of most stars are very reliable. In particular, after Hirao et al. (2020) eliminated stars with $\sigma(\pi)/\pi<-2$ (which included OGLE-2017-BLG-0406S itself) and those with $\sigma(\mu_{\rm north})>0.6\,{\rm mas\,yr^{-1}}$ or $\sigma(\mu_{\rm east})>0.6\,{\rm mas\,yr^{-1}}$, that only 1–2% of Gaia proper motions were $>3\,\sigma$ outliers. However, the Gaia proper-motion errors had to be renormalized by a factor $2.2$ to enforce $\chi^{2}$/dof = $1$. Although the exact reason for this renormalization is not known, it is likely that bulge-field crowding is a major contributing cause. In particular, the Gaia mirror has an axis ratio of three, meaning that the Gaia point spread function (PSF) has the same ratio. Hence, as Gaia observes a field at various random orientations, light from faint ambient stars can enter the elongated PSF “aperture”, leading to “random” shifts in the astrometric centroid. This can lead to “excess noise” relative to the photon-based error estimates, and this “excess noise” is tabulated as the “astrometric excess noise sig (AENS)” parameter. In so far as this “excess noise” is truly “random”, it just degrades the measurement, which is reflected in the reported uncertainties. However, because it is likely due to real stars, whose positions change very little, and because the observing pattern is also not truly “random”, this “excess noise” can lead to systematic errors that are larger than the random errors.

In their study of Gaia proper-motion errors, Hirao et al. (2020) considered stars with AENS$<10$, so their results strictly apply to such stars. They did not notice any trends in behavior with AENS, and (though not specifically reported), they also did not notice any trends for AENS at a few times this level.

For OGLE-2018-BLG-1269, Gaia reports AENS=10.4. We therefore apply the Hirao et al. (2020) error renormalization to the above Gaia measurements. Although Hirao et al. (2020) only studied proper-motion errors (because this is the only quantity for which OGLE measurements are superior to Gaia), we apply this renormalization to all PPPM quantities.

For the position measurement, the formal errors $(\sim 0.15\,{\rm mas})$ are so small that they play no practical role, even after renormalization. So we ignore these errors. For the parallax measurement, the renormalized error is $\sim 45\%$ of the measured value. Hence, its role is mainly qualitative confirmation that the lens is nearby. The renormalized proper-motion errors ($\sim 0.7\,{\rm mas\,yr^{-1}}$) are still relatively small, and this measurement will play a crucial role at several points.

For the Bayesian analysis, we will incorporate the Gaia astrometric measurements in addition to the usual microlensing-parameter measurements.

We first ask whether the host is consistent with being the primary contributor to the blended light? If it is, then there must be a star simultaneously consistent with the microlensing properties and the blended light. For this analysis, we use the measured Einstein radius $\theta_{\rm E}$ and the adopted source parallax $\pi_{\rm S}$ (Equation (2)) to map a set of model isochrones to the calibrated CMD. Given $\theta_{\rm E}$ and $\pi_{\rm S}$, we can take a star with given mass $M_{\rm iso}$, intrinsic color $(V-I)_{0,{\rm iso}}$, and absolute magnitude $M_{I,{\rm iso}}$, to estimate the distance to the star $D_{\rm iso}$. We next find the dereddened $I$- and $V$-band magnitudes by $I_{0,{\rm iso}}$ = $M_{I,{\rm iso}}+5{\rm log}(D_{{\rm iso}}/{\rm pc})-5$ and $V_{0,{\rm iso}}$ = $I_{0,{\rm iso}}+(V-I)_{0,{\rm iso}}$. We then find the position of the star $(V-I,I)_{\rm iso}$ in the calibrated CMD using the partial extinction model (Equation (23)). Finally, we build an “observed” isochrone with $[M,D,(V-I),I]_{\rm iso}$ from all stars listed in the model isochrone. For the three cases of ${\rm[Fe/H]}=(-0.5,0.0,+0.3)$, we then construct “observed” isochrones with different ages, and compare them to the blended light to estimate the blend mass $M_{\rm b}$ and distance $D_{\rm b}$.

We find that two “observed” isochrones can match the blended light (see Figure 9). That is, the two curves for $(\rm{[Fe/H]},{\rm age})=(0.0,6\,{\rm Gyr})$ and $(+0.3,4.5\,{\rm Gyr})$ pass through the blend position with the offset of $(5.5,8.5)\times 10^{-3}$, respectively. The estimated mass and distance to the blend are $(M_{\rm b},D_{\rm b})=(1.16\,M_{\odot},2.49\,{\rm kpc})$ for $(\rm{[Fe/H]},{\rm age})=(0.0,6\,{\rm Gyr})$ and $(M_{\rm b},D_{\rm b})=(1.38\,M_{\odot},2.80\,{\rm kpc})$ for $(\rm{[Fe/H]},{\rm age})=(+0.3,4.5\,{\rm Gyr})$. These estimates imply that for typical disk populations with $0\leq{\rm[Fe/H]}\leq 0.3$, $M_{\rm b}$ and $D_{\rm b}$ are in the range of $1.16\leq M_{\rm b}/M_{\odot}\leq 1.38$ and $2.49\leq D_{\rm b}/{\rm kpc}\leq 2.80$, respectively. These ranges show remarkable agreement with the prediction from the Bayesian analysis (see Figure 7). This implies that the host is consistent with causing the blended light, which is then a subgiant (or possibly late turnoff) star.

We still must consider the possibility that the blended light is primarily due to a stellar companion to the host, rather than the host itself. That is, it is due to a star that does not directly enter into the microlensing event but is gravitationally bound to the host. This alternative explanation for the blended light can be conceptually divided into two cases: either the host contributes very little light to the blend, or the host and its brighter stellar companion both contribute significantly to the observed blended light. When we quantitatively evaluate the probability that the host dominates the blended light, we will treat these two alternative cases as a single case. However, in the qualitative treatment that follows immediately below, we make a conceptual distinction between them.

These six classes of systems can contribute the three cases of events as follows. Class (A) can contribute only to events of case (1). Class (B) cannot contribute to any events with a OGLE-208-BLG-1269 type light curve because companions in this period range would have given rise to recognizable signals in the light curve ($P<10^{4}\,$days) or would violate the $Gaia$-based source-blend separation measurement ($P<10^{5.3}\,$days).

Class (C) is excluded for cases (2) and (3) because the centroid of light would be displaced from the the host (and therefore the host) by more than $12$ mas. However, it is permitted for case (1) because the light from the companion would not significantly displace the light centroid.

Class (D) can contribute to events of case (1) but not of case (2) or (3). That is, the light contributed by the stellar companion would not be enough to qualitatively alter the photometric appearance of the combined light relative to an isolated turnoff/subgiant star, so case (1) is compatible. However, the mass of the host for case (2) is too low to be compatible with microlensing constraints (see Table 2). Therefore, case (2) is excluded. And case (3) is also excluded because a $Q<0.5$ companion cannot contribute significantly.

Class (E) can contribute to either case (1) or case (2). Because the two stars in the lens system must be on the same isochrone, in the class (E) mass-ratio range, the more massive star must be above the turnoff and the less massive one must be below the turnoff. Hence, they differ by at least one magnitude, which implies that they contribute substantially differently to the total light of the blend. From the lower-right panels of Figure 7, it is clear that over most of this mass-ratio range, the lower-mass star would have a similar color to the blend. Hence, the brightness of the higher-mass star would be reduced by $\sim 0.1$ to $\sim 0.5$ mag, while its color would hardly be altered, relative to the blend. Thus, its position on the CMD would be essentially the same as that of the blend. In particular, it would be projected against the same green contours, and, in fact, slightly closer to the yellow contours.

Because class (F) systems can contribute only to case (3), we can now qualitatively evaluate the relative likelihood of case (1) (blended light from “turnoff/subgiant star” is dominated by the host, i.e., classes (A), (C), (D), and part of class (E)) and case (2) (blended light from “turnoff/subgiant star” is dominated by a companion to the host, i.e., part of class (E)). Then we will return to case (3).

For class (E), in which there are two stars in the system, i.e., higher- and lower-mass stars, the overall probability of lensing is higher than for a single star by $\sqrt{1+Q}$ because there are two well-separated lenses that could give rise to the event. And the relative probability of the lower-mass star giving rise to the event is $\sqrt{Q}$. Therefore, relative to single-host case, the absolute lensing probabilities of two stars scale as $1$ and $\sqrt{Q}$, respectively. We can then approximate the lower-mass events of class (E) by $\sqrt{Q}\sim\sqrt{0.7}\sim 0.84$. Then, the probability for case (2) relative to case (1) can be directly evaluated: $p_{2}/p_{1}=(0.05\times 0.84)/(0.36+0.28+0.10+0.05)=0.05$.

Naively, event case (3) appears highly disfavored because only system class (F) contributes to it, and this comprises only $1\%$ of all systems. In fact, however, this case requires close examination for proper evaluation.

We first consider the very special subcase that the host and its companion are identical. Then their colors would be the same as the blend, but their magnitudes would be $0.75$ mag fainter than the blend. In principle, this might have put them on the main sequence. In this case, the low relative probability of such binary systems ($1\%$) would have been counterbalanced by the fact that main-sequence stars are far more common than turnoff/subgiants of the same color. In fact, however, Figure 9 shows that this position ($0.75$ mag below the blend) is not inhabited by stars on any or the fairly broad range of isochrones that we have displayed.

If we consider the broader case of approximately (rather than exactly) equal masses for the two components ($Q\sim 0.9$), we see that essentially the same (above) argument applies to the case. The less massive star will be fainter and bluer than the blend, while the more massive star will be fainter and redder. The upper panel of Figure 9 shows that at ${\rm[Fe/H]}=+0.3$, it is possible for a star to exist on, e.g., the $10$-Gyr isochrone that is $0.3$ mag fainter and somewhat redder than the blend. However, this position invalidates the main advantage of event cases that was just mentioned above: the lens (or its companion) remains a subgiant and is not on the more populous main sequence. Hence, the probability of this solution is very low. Using the same procedure as above, we derive $p_{3}/p_{1}=(0.01\times 1.9)/(0.36+0.28+0.10+0.05)=0.02$.

We now conduct a quantitative analysis aimed at both testing the qualitative ideas presented above and deriving a more precise quantitative result. Our starting point is to draw random events from the same Galactic model used for the Bayesian analysis described above and to weight each event by the same $(t_{\rm E},\theta_{\rm E},\pi_{\rm L},\mbox{\boldmath$\pi$}_{\rm E})$ priors. However, for each simulated event, we either “accept” or “reject” it according to whether the combined light from the host and some companion drawn from the same isochrone is compatible with the blended light. That is, each simulated event has a corresponding $I$-band magnitude and $V-I$ color; if there exists a companion along the same isochrone for which the combined light is compatible with the blend, the event is “accepted”. The entire isochrone is reddened in the same manner as was done for the case that the blend is dominated by the host light. We consider the same $(3\times 5=15)$ isochrones that were analyzed for the host=blend case, i.e., case (1). We note that after investigating these separate-isochrone cases, we must still combine them to obtain an overall relative probability of case (1) versus cases (2)+(3). This step will also require incorporating information about binary frequency.

Figure 10 shows separate $(1,2,3)\sigma$ contours, in the lens mass-distance plane, for the all [accepted+rejected] (black, dark grey, grey) and [accepted-only] (red, yellow, green) simulated events. We first focus on the five ${\rm[Fe/H]}=-0.5$ isochrones. These show that the accepted contours lie well away from the contours for all trials in each of the panels. This implies that a very small fraction are accepted. Numerically, we find that the $6$-, $8$-, and $10$-Gyr isochrones have the highest rate of acceptance: about $0.2\%$ for each (see Table 3). To the extent that these do not overlap (which is partial), they would add constructively. Thus, these three isochrones contribute about $0.6\%$. The other two isochrones contribute negligibly.

We next focus on the ${\rm[Fe/H]}=+0.3$ isochrones. Again, the oldest three isochrones contribute the most. However, such old, very metal rich stars are very rare within a few kpc of the Sun. Hence, we ignore these. The two youngest isochrones together contribute $<1\%$.

Lastly, we examine the solar-metallicity isochrones. These contribute $(1.4\%,3.0\%,3.7\%)$ for the $(6,8,10)$ Gyr isochrones, respectively. However, $10$-Gyr solar-metallicity stars are extremely rare, and $8$-Gyr solar-metallicity stars are fairly rare, so we make an overall estimate of $3\%$ for solar metallicity stars. We note that the two youngest isochrones contribute negligibly.

Next, we examine Figure 11, which shows where hosts (red, yellow, green) and stellar companions (black, magenta, cyan) lie on the theoretical isochrones for all $15$ isochrone cases. We note that only “accepted” events are shown. The most important feature of these diagrams is that the stellar companion tracks are almost all confined to the subgiant branch. This confirms the basic logic of the approach that we outlined in the enumeration in Section 5.2, i.e., of considering the relative probability of systems that contain a subgiant-branch star. Recall that if the stellar companion were on the main-sequence for case (3), but it were on the subgiant branch for case (2), then we would need to take account of the fact that main-sequence stars are more common than subgiants.

Now, the companion is actually on the main sequence for the top two $2$-Gyr isochrones, and it is on the turnoff for the metal rich $4$-Gyr isochrone. However, recall from Figure 10 that the former contributes negligibly and the latter contributes $<1\%$. Even if this percentage were augmented by a factor $\sim 5$ due to slower evolution on the turnoff, its contribution would still be small.

Thus, considering that both ${\rm[Fe/H]}=0.0$ and ${\rm[Fe/H]}=+0.3$ can contribute to case (1) as discussed in Section 5.1, while $<5\%$ of stellar populations at these metallicities can contribute to cases (2)+(3), we estimate that from this quantitative analysis alone, the probability for cases (2)+(3) relative to case (1) is $p_{a}<5\%$.

We now must take account of the fact that Classes (A), (C), (D), and (E) can contribute to case (1), while Classes (E) and (F) can contribute to cases (2)+(3). This contributes a relative probability of $p_{b}/(1-p_{b})=(0.05\times 0.84+0.01\times 1.9)/(0.36+0.28+0.10+0.05)=0.077$, i.e., $p_{b}=7\%$. Therefore, the “total probability” that the host dominates the blend light is $(1-p_{a}{\times}p_{b})>99.6\%$.

Finally, we ask why the quantitative analysis gave much more certainty (“$>99.6\%$”) that the host dominates the blended light than the qualitative analysis? The primary reason is that in the qualitative analysis, we implicitly assumed that, for most cases, there would be some isochrone that could provide the “extra light” from a turnoff/subgiant star that could be added to the host to make the observed blended light. However, Figure 10 shows that this is not the case.