OGLE-2018-BLG-1269Lb: A Jovian Planet with a Bright I = 16 Host

The normalized source radius ρ 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 θ_*. The Einstein radius is related to the lens mass M and the relative parallax π_rel by

$$\begin{eqnarray}&&{\theta }_{{\rm{E}}}\equiv \sqrt{\kappa M{\pi }_{\mathrm{rel}}};{\pi }_{\mathrm{rel}}=\mathrm{au}\left(\displaystyle \frac{1}{{D}_{{\rm{L}}}}-\displaystyle \frac{1}{{D}_{{\rm{S}}}}\right).\end{eqnarray} \tag{ 3 }$$

Then, we can use the measured θ_E to constrain the lens properties. Hence, we first estimate θ_* 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 ${(V-I,I)}_{\mathrm{GC}}=(2.82\pm 0.05,16.34\pm 0.07)$, which yields an offset

$$\begin{eqnarray}\begin{array}{rcl}{\rm{\Delta }}(V-I,I) & = & {\left(V-I,I\right)}_{\mathrm{GC}}-{\left(V-I,I\right)}_{0,\mathrm{GC}}\\ & = & (1.76\pm 0.05,1.98\pm 0.07),\end{array}\end{eqnarray} \tag{ 4 }$$

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

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

$$\begin{eqnarray}&&{\theta }_{* }=0.948\pm 0.068\,\mu \mathrm{as},\end{eqnarray} \tag{ 5 }$$

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

$$\begin{eqnarray}&&{\theta }_{{\rm{E}}}=1.602\pm 0.118\,\mathrm{mas}.\end{eqnarray} \tag{ 6 }$$

The geocentric relative lens–source proper motion is then

$$\begin{eqnarray}&&{\mu }_{\mathrm{rel}}=\displaystyle \frac{{\theta }_{{\rm{E}}}}{{t}_{{\rm{E}}}}=8.29\pm 0.61\,\mathrm{mas}\,{\mathrm{yr}}^{-1}.\end{eqnarray} \tag{ 7 }$$

The unusually large values of θ_E and μ_rel suggest that the lens lies in the Galactic disk. From the definition of θ_E (Equation (3)),

$$\begin{eqnarray}&&{\pi }_{\mathrm{rel}}=0.22\,\mathrm{mas}{\left(\displaystyle \frac{{\theta }_{{\rm{E}}}}{1.6\mathrm{mas}}\right)}^{2}{\left(\displaystyle \frac{M}{1.4{M}_{\odot }}\right)}^{-1}.\end{eqnarray} \tag{ 8 }$$

Thus, given the lens flux constraint, which will be discussed in the following subsection, the lens must be ${D}_{{\rm{L}}}\,=\mathrm{au}/({\pi }_{\mathrm{rel}}+{\pi }_{{\rm{S}}})\lesssim 3\,\mathrm{kpc}$ (unless the lens is a black hole). We note that the measured μ_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 relevant.³² Moreover, in this paper, for the first time, we will 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) $\pm (0.16,0.14)\ \mathrm{mas}$,

$$\begin{eqnarray}&&{\pi }_{{\rm{G}}}=0.73\pm 0.18\,\mathrm{mas},\end{eqnarray} \tag{ 9 }$$

and

$$\begin{eqnarray}&&{{\boldsymbol{\mu }}}_{{\rm{G}}}(N,E)=(-1.24\pm 0.26,-1.58\pm 0.31)\,\mathrm{mas}\,{\mathrm{yr}}^{-1}.\end{eqnarray} \tag{ 10 }$$

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 ${\pi }_{\mathrm{shift}}=0.055\,\mathrm{mas}$ (Zinn et al. 2019). Hence, we correct the baseline object parallax to be

$$\begin{eqnarray}&&{\pi }_{\mathrm{base}}=0.78\pm 0.18\,\mathrm{mas}.\end{eqnarray} \tag{ 11 }$$

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, or 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 σ(π)/π < −2 (which included OGLE-2017-BLG-0406S itself) and $\sigma ({\mu }_{{\rm{n}}{\rm{o}}{\rm{r}}{\rm{t}}{\rm{h}}})\gt 0.6$ or $\sigma ({\mu }_{\mathrm{east}})\gt 0.6\,\mathrm{mas}\,{\mathrm{yr}}^{-1}$, only 1%–2% of Gaia proper motions were >3σ outliers. However, the Gaia proper-motion errors had to be renormalized by a factor of 2.2 to enforce χ²/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 3, 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. Insofar 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 $(\sim 0.15\,\mathrm{mas})$ 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 ∼45% of the measured value. Hence, its role is mainly qualitative confirmation that the lens is nearby. The renormalized proper-motion errors ($\sim 0.7\,\mathrm{mas}\,{\mathrm{yr}}^{-1}$) are still relatively small, and this measurement will play a crucial role at several points.

4.3.1. Gaia Baseline Object Is $\lt 20\,\mathrm{mas}$ from Source

Gaia astrometry is generally given for epoch 2015.50, whereas the event peaked at 2018.61. In order to compare the position of the source (at 2018.61) with the position of the Gaia baseline object at the same time, we first propagate the positions of all Gaia stars (including the baseline object) forward in time by Δt = 2018.61–2015.50 = 3.11 yr. That is, for each star in the field, i, we calculate

$$\begin{eqnarray}&&{{\boldsymbol{\theta }}}_{i,\mathrm{Gaia}}^{2018.61}={{\boldsymbol{\theta }}}_{i,\mathrm{Gaia}}^{2015.5}+{{\boldsymbol{\mu }}}_{i,\mathrm{Gaia}}{\rm{\Delta }}t,\end{eqnarray} \tag{ 12 }$$

where ${{\boldsymbol{\mu }}}_{i,\mathrm{Gaia}}$ is the proper motion of the Gaia object. We then cross-match the Gaia and KMTNet pyDIA catalogs within a 2' square, excluding entries that fail a relatively forgiving cut on an empirical $(G-I)/(V-I)$ color–color relation and with a 1'' astrometric cut (to allow for optical distortion of the KMTNet camera). Next, we fit for a transformation from Gaia to KMTNet pyDIA coordinates using all matches obtained from the previous step, except the baseline object, by minimizing the unrenormalized χ²,

$$\begin{eqnarray}&&{\chi }^{2}=\displaystyle \sum _{i}{\left[{{\boldsymbol{\theta }}}_{i,\mathrm{KMTNet}}^{2018.61}-{T}_{n}({{\boldsymbol{\theta }}}_{i,\mathrm{Gaia}}^{2018.61})\right]}^{2},\end{eqnarray} \tag{ 13 }$$

where T_n is a nth-order polynomial transformation, i.e., 6, 12, and 20 parameters for n = (1, 2, 3).

We find that the results vary very little, but the n = 3 polynomial fit is slightly better than the others. We recursively eliminate outliers, of which there are 46 objects³³ for 487 original matches. The scatter is 19 mas, which is almost an order of magnitude larger than the typical formal propagated errors in ${{\boldsymbol{\theta }}}_{i,\mathrm{Gaia}}^{2018.61}$. Although the Gaia errors are probably somewhat underestimated in crowded fields (Hirao et al. 2020), it is still the case that this scatter is completely dominated by the errors of KMTNet pyDIA astrometry.

We then apply the resulting transformation to the propagated Gaia blend position ${{\boldsymbol{\theta }}}_{{\rm{b}},\mathrm{Gaia}}^{2018.61}$ and subtract this from the pyDIA source position that is derived from difference images:

$$\begin{eqnarray}\begin{array}{rcl}{\rm{\Delta }}{\boldsymbol{\theta }}(N,E) & = & {{\boldsymbol{\theta }}}_{{\rm{S}},\mathrm{KMTNet}}^{2018.61}-{T}_{3}({{\boldsymbol{\theta }}}_{{\rm{b}},\mathrm{Gaia}}^{2018.61})\\ & = & (6.9,9.2)\pm (4.4,4.4)\ \mathrm{mas}.\end{array}\end{eqnarray} \tag{ 14 }$$

The error comes from three sources added in quadrature: error in the transformation coefficients (1.0 mas), error in the propagated value of ${{\boldsymbol{\theta }}}_{{\rm{b}},\mathrm{Gaia}}^{2018.61}$ (2.0 mas), and error in the pyDIA measurement of ${{\boldsymbol{\theta }}}_{{\rm{S}},\mathrm{KMTNet}}^{2018.61}$ (4.0 mas).

That is, the Gaia baseline object lies 11.5 ± 4.4 mas from the source at the time of the event. We repeat this exercise using OGLE data and obtain 18.5 ± 4.4 mas, which confirms that the source and the Gaia baseline object are very close.

4.3.2. Probability of Chance Superposition Is p = 3 × 10⁻⁶

4.3.3. Blended Light Is Due to the Lens System

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

4.4.1. Inputs from Gaia

To do so, we first note that for the three parameters ${\boldsymbol{X}}=(\pi ,{\boldsymbol{\mu }})$ measured by Gaia, the observed (baseline object) quantities are related to the underlying physical (source and lens) quantities³⁴ by (Ryu et al. 2019)

$$\begin{eqnarray}&&{{\boldsymbol{X}}}_{\mathrm{base}}=(1-\eta ){{\boldsymbol{X}}}_{{\rm{L}}}+\eta {{\boldsymbol{X}}}_{{\rm{S}}},\end{eqnarray} \tag{ 15 }$$

where $\eta ={f}_{{\rm{S}}}/{f}_{\mathrm{base}}$ is the flux fraction of the source in the Gaia band and (f_S, f_base) are the flux of the source and the baseline object, respectively. We estimate η by noting that the peak of the Gaia passband is broadly consistent with that of the V band, and the typical photometric error of the Gaia observation is 2%. We thereby estimate η = 0.02 based on our result that the blend is 4.2 mag brighter than the source in the V band.

To find the lens parallax π_L, we adopt ${\pi }_{{\rm{S}}}=0.13\pm 0.01\,\mathrm{mas}$ from Nataf et al. (2013), and we renormalize the errors (by a factor 2.2) in Equation (11) to obtain π_base = 0.78 ± 0.40 mas. We then apply Equation (15) to π_base and find

$$\begin{eqnarray}&&{\pi }_{{\rm{L}}}=0.80\pm 0.40\,\mathrm{mas}.\end{eqnarray} \tag{ 16 }$$

The situation is substantially more complicated for the proper motion. First, the microlensing solution gives the amplitude of the lens–source relative proper motion in the geocentric frame, but the Gaia proper motion is in the heliocentric frame. We can relate these by

$$\begin{eqnarray}&&\begin{array}{l}{{\boldsymbol{\mu }}}_{\mathrm{rel},\mathrm{hel}}\equiv {{\boldsymbol{\mu }}}_{{\rm{L}},\mathrm{hel}}-{{\boldsymbol{\mu }}}_{{\rm{S}},\mathrm{hel}};\\ \ \times \ {{\boldsymbol{\mu }}}_{\mathrm{rel},\mathrm{hel}}={{\boldsymbol{\mu }}}_{\mathrm{rel}}+\displaystyle \frac{{\pi }_{\mathrm{rel}}}{\mathrm{au}}{{\boldsymbol{\nu }}}_{\oplus ,\perp },\end{array}\end{eqnarray} \tag{ 17 }$$

where ${{\boldsymbol{\nu }}}_{\oplus ,\perp }(N,E)=(-1.7,18.4)\,\mathrm{km}\,{{\rm{s}}}^{-1}$ is the projected velocity of Earth at t₀ and $({{\boldsymbol{\mu }}}_{{\rm{L}},\mathrm{hel}},{{\boldsymbol{\mu }}}_{{\rm{S}},\mathrm{hel}})$ are the heliocentric proper motions of the lens and source, respectively.

In principle, we could fully incorporate Equations (15) and (17) into the Bayesian analysis below. However, as we now show, for the case of OGLE-2018-BLG-1269, the lens proper motion is well approximated by ${{\boldsymbol{\mu }}}_{{\rm{L}},\mathrm{hel}}={{\boldsymbol{\mu }}}_{\mathrm{base}}$.

First, we combine Equations (15) and (17) to yield

$$\begin{eqnarray}&&\begin{array}{l}{{\boldsymbol{\mu }}}_{{\rm{L}},\mathrm{hel}}={{\boldsymbol{\mu }}}_{\mathrm{base}}+\eta [{{\boldsymbol{\mu }}}_{\mathrm{rel}}+\displaystyle \frac{{\pi }_{\mathrm{rel}}}{\mathrm{au}}{{\boldsymbol{\nu }}}_{\oplus ,\perp }],\\ {{\boldsymbol{\mu }}}_{{\rm{S}},\mathrm{hel}}={{\boldsymbol{\mu }}}_{\mathrm{base}}-(1-\eta )[{{\boldsymbol{\mu }}}_{\mathrm{rel}}+\displaystyle \frac{{\pi }_{\mathrm{rel}}}{\mathrm{au}}{{\boldsymbol{\nu }}}_{\oplus ,\perp }].\end{array}\end{eqnarray} \tag{ 18 }$$

Next, we note that for typical final values of ${\pi }_{\mathrm{rel}}=0.4\,\mathrm{mas}$, we have $({\pi }_{\mathrm{rel}}/\mathrm{au}){\nu }_{\oplus ,\perp }\simeq 1.6\,\mathrm{mas}\,{\mathrm{yr}}^{-1}\ll {\mu }_{\mathrm{rel}}$; hence, regardless of the direction of ${{\boldsymbol{\mu }}}_{\mathrm{rel}}$, we have ${\mu }_{\mathrm{rel},\mathrm{hel}}\simeq {\mu }_{\mathrm{rel}}\simeq 8\,\mathrm{mas}\,{\mathrm{yr}}^{-1}$. Therefore, the second term in the first entry of Equation (18) is of order $\eta {\mu }_{\mathrm{rel}}\sim 0.16\,\mathrm{mas}\,{\mathrm{yr}}^{-1}$, which is a factor of about 4 smaller than the renormalized errors in ${{\boldsymbol{\mu }}}_{\mathrm{base}}$ ($\sim 0.7\,\mathrm{mas}\,{\mathrm{yr}}^{-1}$). Hence, we adopt ${{\boldsymbol{\mu }}}_{{\rm{L}},\mathrm{hel}}={{\boldsymbol{\mu }}}_{\mathrm{base}}$. Then, after renormalizing the errors (by a factor of 2.2) in Equation (10) and rotating to Galactic coordinates, we obtain

$$\begin{eqnarray}&&{{\boldsymbol{\mu }}}_{{\rm{L}}}(l,b)=(-1.86\pm 0.68,0.75\pm 0.57)\,\mathrm{mas}\,{\mathrm{yr}}^{-1}.\end{eqnarray} \tag{ 19 }$$

4.4.2. Bayesian Formalism

With the four measured constraints $({t}_{{\rm{E}}},{\theta }_{{\rm{E}}},{\pi }_{{\rm{L}}},{{\boldsymbol{\pi }}}_{{\rm{E}}})$, we now make a Bayesian analysis following the procedure of Jung et al. (2018). We first build a Galactic model with models of the mass function (MF), density profile (DP), and velocity distribution (VD) of astronomical objects. For the MF and DP, we adopt the models used in Jung et al. (2018). For the VD, we use the proper-motion distribution of stars measured by Gaia. For the source proper motion, we examine a Gaia CMD using red giant stars within 3' centered on the event direction and find their mean proper motion and standard deviation in Galactic coordinates:

$$\begin{eqnarray}&&{{\boldsymbol{\mu }}}_{{\rm{S}}}(l,b)=(-5.93\pm 3.10,0.03\pm 2.72)\,\mathrm{mas}\,{\mathrm{yr}}^{-1}.\end{eqnarray} \tag{ 20 }$$

For the lens proper motion, we employ Equation (19).

For each solution of u₀ > 0 and u₀ < 0, we draw 1 billion random events based on the adopted Galactic model. For each random event i, we then infer the four parameters ${({t}_{{\rm{E}}},{\theta }_{{\rm{E}}},{\pi }_{{\rm{L}}},{{\boldsymbol{\pi }}}_{{\rm{E}}})}_{i}$ and find the χ² difference between the inferred and measured values, i.e.,

$$\begin{eqnarray}&&\begin{array}{l}{\chi }_{\mathrm{gal},i}^{2}={\chi }_{i}^{2}({t}_{{\rm{E}}})+{\chi }_{i}^{2}({\theta }_{{\rm{E}}})+{\chi }_{i}^{2}({\pi }_{{\rm{L}}})+{\chi }_{p,i}^{2};\\ {\chi }_{p,i}^{2}=\displaystyle \sum {\left({a}_{i}-{a}_{0}\right)}_{j}{c}_{{jk}}^{-1}{\left({a}_{i}-{a}_{0}\right)}_{k},\end{array}\end{eqnarray} \tag{ 21 }$$

where ${{\boldsymbol{a}}}_{i}={{\boldsymbol{\pi }}}_{{\rm{E}},i}={({\pi }_{{\rm{E}},N},{\pi }_{{\rm{E}},E})}_{i}$ is the inferred parallax, and a₀ and c_jk are the measured ${{\boldsymbol{\pi }}}_{{\rm{E}}}$ and its covariance matrix, respectively. We next find the likelihood of the event by ${\text{}}{P}_{i}=\exp (-{\chi }_{\mathrm{gal},i}^{2}/2)\times {{\rm{\Gamma }}}_{i}$, where ${{\rm{\Gamma }}}_{i}\propto {\theta }_{{\rm{E}},i}{\mu }_{\mathrm{rel},i}$ is the lensing event rate.

For each random event i, we also infer the lens position in the calibrated CMD, i.e., ${(V-I,I)}_{{\rm{L}}}$, in order to check whether the lens flux predicted from the Bayesian estimates is consistent with the blended flux. For this, we first construct a set of isochrones with different metallicities and ages (Spada et al. 2017), i.e., with $[\mathrm{Fe}/{\rm{H}}]=(-0.5,0.0,+0.3)$ and age = (2, 4, 6, 8, 10) Gyr. In each isochrone j, we next estimate the absolute I-band magnitude ${M}_{I,{\rm{L}},i,j}$ and intrinsic ${(V-I)}_{0,{\rm{L}},i,j}$ color of the lens from the inferred lens mass M_i. We then find the dereddened lens magnitude in the I and V bands by ${I}_{0,{\rm{L}},i,j}$ = ${M}_{I,{\rm{L}},i,j}\,+5\mathrm{log}({D}_{{\rm{L}},i}/\mathrm{pc})-5$ and ${V}_{0,{\rm{L}},i,j}$ = ${I}_{0,{\rm{L}},i,j}\,+{(V-I)}_{0,{\rm{L}},i,j}$. We next estimate the extinction to the lens ${A}_{\lambda ,{\rm{L}},i}$ using the partial extinction model (Bennett et al. 2015; Beaulieu et al. 2016),

$$\begin{eqnarray}&&{A}_{\lambda ,{\rm{L}},i}=\displaystyle \frac{1-{e}^{-\left|{D}_{{\rm{L}},i}/{\tau }_{\mathrm{dust}}\right|}}{1-{e}^{-\left|{D}_{{\rm{S}},i}/{\tau }_{\mathrm{dust}}\right|}}{A}_{\lambda ,{\rm{S}}},\end{eqnarray} \tag{ 22 }$$

where the index λ denotes the passband and ${\tau }_{\mathrm{dust}}\,=(0.12\,\mathrm{kpc})/\sin (b)$ is the dust scale height. Here ${A}_{\lambda ,{\rm{S}}}$ is the extinction to the source, for which we adopt ${A}_{I,{\rm{S}}}=1.98$ and $E{(V-I)}_{{\rm{S}}}=1.76$ from our CMD analysis (Equation (4)). We then derive ${(V-I,I)}_{{\rm{L}},i,j}$ using ${A}_{\lambda ,{\rm{L}},i}$ and bin the CMD by these lens positions with the likelihood P_i.

We emphasize that in this initial analysis, we completely ignore the constraints coming from the blended light. That is, we neither impose any constraint on the lens light (such as not to exceed the blended light) nor consider the possibility that the lens is responsible for the blended light. At this point, we simply "predict" the lens color and magnitude based on the $({t}_{{\rm{E}}},{\theta }_{{\rm{E}}},{\pi }_{{\rm{L}}},{{\boldsymbol{\pi }}}_{{\rm{E}}})$ (or $({t}_{{\rm{E}}},{\theta }_{{\rm{E}}},{\pi }_{{\rm{L}}})$) constraints, together with the Galactic model and model isochrones. We investigate the role of the blended light in constraining the solution only after comparing these predictions to the observed blended light.

4.4.3. Bayesian Results

We finally investigate the posterior probabilities of the lens properties from all random events. We note that to check the contribution of the ${{\boldsymbol{\pi }}}_{{\rm{E}}}$ constraint on the Bayesian estimates, we also explore the posterior probabilities with $({t}_{{\rm{E}}},{\theta }_{{\rm{E}}},{\pi }_{{\rm{L}}})$ constraints.

The results from the constraints $({t}_{{\rm{E}}},{\theta }_{{\rm{E}}},{\pi }_{{\rm{L}}},{{\boldsymbol{\pi }}}_{{\rm{E}}})$ are shown in Figure 6 and listed in Table 2. Also listed are the total Galactic model probability ${P}_{\mathrm{tot}}=\sum {P}_{i}$ and net relative probability ${P}_{\mathrm{net}}={P}_{\mathrm{tot}}{P}_{\mathrm{lc}}$, where ${P}_{\mathrm{lc}}=\exp (-{\rm{\Delta }}{\chi }^{2}/2)$ is the relative fit probability and ${\rm{\Delta }}{\chi }^{2}$is the ${\chi }^{2}$ difference between the two solutions. Here ${\phi }_{\mathrm{hel}}={\tan }^{-1}\,[{\mu }_{\mathrm{rel},\mathrm{hel}}(E)/{\mu }_{\mathrm{rel},\mathrm{hel}}(N)]$ is the orientation angle of ${{\boldsymbol{\mu }}}_{\mathrm{rel},\mathrm{hel}}$.

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Table 2.  Physical Parameters

Parameters	${u}_{0}\gt 0$	${u}_{0}\lt 0$	Weighted
M1 $({M}_{\odot })$	${1.11}_{-0.34}^{+0.68}$	${1.18}_{-0.36}^{+0.78}$	${1.13}_{-0.35}^{+0.72}$
M2 $({M}_{{\rm{J}}})$	${0.67}_{-0.21}^{+0.41}$	${0.74}_{-0.23}^{+0.49}$	${0.69}_{-0.22}^{+0.44}$
a⊥ (au)	${4.51}_{-1.15}^{+1.66}$	${4.75}_{-1.21}^{+1.74}$	${4.61}_{-1.17}^{+1.70}$
${D}_{{\rm{L}}}$ (kpc)	${2.51}_{-0.61}^{+0.90}$	${2.64}_{-0.64}^{+0.94}$	${2.56}_{-0.62}^{+0.92}$
${\pi }_{{\rm{E}},N}$	${0.187}_{-0.071}^{+0.078}$	${0.174}_{-0.071}^{+0.074}$	${0.183}_{-0.071}^{+0.077}$
${\pi }_{{\rm{E}},E}$	${0.011}_{-0.017}^{+0.014}$	${0.029}_{-0.014}^{+0.011}$	${0.018}_{-0.019}^{+0.014}$
${\mu }_{\mathrm{rel},\mathrm{hel}}(N)$ $(\mathrm{mas}\,{\mathrm{yr}}^{-1})$	${8.01}_{-0.65}^{+0.59}$	${7.91}_{-0.68}^{+0.63}$	${7.97}_{-0.66}^{+0.62}$
${\mu }_{\mathrm{rel},\mathrm{hel}}(E)$ $(\mathrm{mas}\,{\mathrm{yr}}^{-1})$	${1.56}_{-0.56}^{+0.57}$	${2.37}_{-0.56}^{+0.63}$	${1.84}_{-0.68}^{+0.81}$
${\phi }_{\mathrm{hel}}$ (deg)	${11.02}_{-3.28}^{+2.89}$	${16.68}_{-2.63}^{+2.68}$	${13.00}_{-3.98}^{+4.14}$
${(V-I)}_{{\rm{L}}}$	${1.76}_{-0.20}^{+0.44}$	${1.74}_{-0.19}^{+0.41}$	${1.75}_{-0.20}^{+0.44}$
${I}_{{\rm{L}}}$	${17.40}_{-1.06}^{+1.15}$	${17.30}_{-1.03}^{+1.15}$	${17.36}_{-1.04}^{+1.16}$
${P}_{\mathrm{lc}}$	1.0	0.64	⋯
${P}_{\mathrm{tot}}$	416,913.4	382,315.5	⋯
${P}_{\mathrm{net}}$	416,913.4	244,681.9	⋯

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We find that the measured ${{\boldsymbol{\pi }}}_{{\rm{E}}}$ from modeling gives a strong constraint on the probability distributions. However, we also find that the host mass ranges of the two solutions (${u}_{0}\gt 0$ and ${u}_{0}\lt 0$) are somewhat different from each other. Because ${{\boldsymbol{\pi }}}_{{\rm{E}}}$ is the only prior constraint that differs significantly between the two solutions and is connected to the lens mass (Equation (2)), the difference would imply that the Galactic model priors disfavor one of the solutions. To check this, we also draw the two-dimensional (2D) likelihood ${ \mathcal L }$ for the lens parameters obtained from the $({t}_{{\rm{E}}},{\theta }_{{\rm{E}}},{\pi }_{{\rm{L}}})$ and $({t}_{{\rm{E}}},{\theta }_{{\rm{E}}},{\pi }_{{\rm{L}}},{{\boldsymbol{\pi }}}_{{\rm{E}}})$ constraints. See Figure 7. We note that the black and gray error bars in the three $({\pi }_{{\rm{E}},N},{\pi }_{{\rm{E}},E})$ planes are the errors of ${{\boldsymbol{\pi }}}_{{\rm{E}}}$ listed in Table 1 for the ${u}_{0}\gt 0$ and ${u}_{0}\lt 0$ solutions, respectively. From this figure, we find that the measured ${{\boldsymbol{\pi }}}_{{\rm{E}}}$ from both the ${u}_{0}\gt 0$ and ${u}_{0}\lt 0$ solutions are consistent at the $1\sigma$ level. This mutual consistency is reflected in the almost equal values of ${P}_{\mathrm{tot}}$ (ratio 0.92) in Table 2. Thus, our estimates should reflect the weighted average of the two solutions, although these hardly differ.

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Hence, these results, which do not yet incorporate constraints from the blended light, suggest that the host is a Sun-like star with ${M}_{1}={1.13}_{-0.35}^{+0.72}\,{M}_{\odot }$ located at a distance of ${D}_{{\rm{L}}}\,={2.56}_{-0.62}^{+0.92}\,\mathrm{kpc}$. Then, the microlensing companion is a planet with ${M}_{2}={0.69}_{-0.22}^{+0.44}\,{M}_{{\rm{J}}}$ separated (in projection) from the host by ${a}_{\perp }={4.61}_{-1.17}^{+1.70}\,\mathrm{au}$. That is, the planet is a cold giant lying beyond the snow line, i.e., ${a}_{{sl}}=2.7\,\mathrm{au}({M}_{1}/{M}_{\odot })\sim 3.1\,\mathrm{au}$.

It is of some interest to understand how the Bayesian analysis constrains the lens mass to a range of a factor of ∼2.4 at the 1σ level despite the fact that ${\pi }_{{\rm{E}}}$ varies by a factor of 10 at $1\sigma$ (see Figure 4), while $M={\theta }_{{\rm{E}}}/\kappa {\pi }_{{\rm{E}}}$. The main reason is that the direction of ${{\boldsymbol{\mu }}}_{\mathrm{rel}}$ (and so the direction of ${{\boldsymbol{\pi }}}_{{\rm{E}}}$) is reasonably well constrained by the Gaia measurement of ${{\boldsymbol{\mu }}}_{{\rm{L}}}$, together with the relatively large value of ${\mu }_{\mathrm{rel}}\simeq 8.3\,\mathrm{mas}\,{\mathrm{yr}}^{-1}$. This is illustrated in Figure 8, which shows ${{\boldsymbol{\mu }}}_{{\rm{L}}}$ and a blue circle to represent all possible ${{\boldsymbol{\mu }}}_{{\rm{S}}}$ values that are consistent with ${\mu }_{\mathrm{rel}}=| {{\boldsymbol{\mu }}}_{{\rm{L}}}-{{\boldsymbol{\mu }}}_{{\rm{S}}}|$. The magenta arc of this circle represents the $1\sigma$ range of ${\mu }_{{\rm{S}}}$ as given by Equation (20). The arc of allowed (at $1\sigma$) directions is shown by dashed lines. This same arc, rotated to equatorial coordinates (and displayed as lens–source rather than source–lens motion), is shown in Figure 4, with boundaries ϕ = 16°–43° (north through east). The first point to note is that, for both ${u}_{0}\gt 0$ and ${u}_{0}\lt 0$, this arc subtends a region that is almost entirely contained within the 1σ ${{\boldsymbol{\pi }}}_{{\rm{E}}}$ contours, implying mutual consistency between two constraints on the direction that are entirely independent. Second, for the ${u}_{0}\lt 0$ contours, which are almost perfectly vertical, we can evaluate the range of ${\pi }_{{\rm{E}}}$ as ${\pi }_{{\rm{E}}}(16^\circ )/{\pi }_{{\rm{E}}}(43^\circ )\,=\csc (16^\circ )/\csc (43^\circ )=2.47$. This confirms that the relatively tight constraints on M in the Bayesian analysis derive from the application of the directional constraint on lens–source relative motion (Figure 8) to the 1D parallax contours from the light-curve analysis (Figure 4). That is, the Bayesian mass estimate comes primarily from a combination of measured microlensing parameters and lens proper motion, while the Galactic model enters mainly by constraints on the source proper motion.

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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 ${M}_{\mathrm{iso}}$, intrinsic color ${(V-I)}_{0,\mathrm{iso}}$, and absolute magnitude ${M}_{I,\mathrm{iso}}$ to estimate the distance to the star ${D}_{\mathrm{iso}}$. We next find the dereddened I- and V-band magnitudes by ${I}_{0,\mathrm{iso}}$ = ${M}_{I,\mathrm{iso}}+5\mathrm{log}({D}_{\mathrm{iso}}/\mathrm{pc})-5$ and ${V}_{0,\mathrm{iso}}$ = ${I}_{0,\mathrm{iso}}+{(V-I)}_{0,\mathrm{iso}}$. We then find the position of the star ${(V-I,I)}_{\mathrm{iso}}$ in the calibrated CMD using the partial extinction model (Equation (22)). Finally, we build an observed isochrone with ${[M,D,(V-I),I]}_{\mathrm{iso}}$ from all stars listed in the model isochrone. For the three cases of $[\mathrm{Fe}/{\rm{H}}]=(-0.5,0.0,+0.3)$, 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 $([\mathrm{Fe}/{\rm{H}}],\mathrm{age})\,=(0.0,6\,\mathrm{Gyr})$ and $(+0.3,4.5\,\mathrm{Gyr})$ pass through the blend position with an offset of $(5.5,8.5)\times {10}^{-3}$, respectively. The estimated mass and distance to the blend are $({M}_{{\rm{b}}},{D}_{{\rm{b}}})=(1.16\,{M}_{\odot },2.49\,\mathrm{kpc})$ for $([\mathrm{Fe}/{\rm{H}}],\mathrm{age})=(0.0,6\,\mathrm{Gyr})$ and $({M}_{{\rm{b}}},{D}_{{\rm{b}}})\,=(1.38\,{M}_{\odot },2.80\,\mathrm{kpc})$ for $([\mathrm{Fe}/{\rm{H}}],\mathrm{age})=(+0.3,4.5\,\mathrm{Gyr})$. These estimates imply that for typical disk populations with $0\leqslant [\mathrm{Fe}/{\rm{H}}]\leqslant 0.3$, ${M}_{{\rm{b}}}$ and ${D}_{{\rm{b}}}$ are in the range of $1.16\,\leqslant {M}_{{\rm{b}}}/{M}_{\odot }\leqslant 1.38$ and $2.49\leqslant {D}_{{\rm{b}}}/\mathrm{kpc}\leqslant 2.80$, 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.

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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.

In order to evaluate the three possibilities, i.e., that the blend light

• 1.  
  is dominated by host,
• 2.  
  is dominated by a stellar companion to the host (and the host contributes relatively little light), and
• 3.  
  receives comparable contributions from the host and a stellar companion,

we first divide all single and binary systems containing a subgiant (or possibly late-turnoff) star into six classes:

• (A)  
  single stars,
• (B)  
  binaries with orbital periods $P\lt {10}^{4}\,$ days,
• (C)  
  binaries with orbital periods $P\gt {10}^{5.3}\,$ days,
• (D)  
  binaries with ${10}^{4}\lt P/\mathrm{day}\lt {10}^{5.3}$ and mass ratios $Q\lt 0.5$,
• (E)  
  binaries with ${10}^{4}\lt P/\mathrm{day}\lt {10}^{5.3}$ and $0.5\lt Q\,\lt 0.9$, and
• (F)  
  binaries with ${10}^{4}\lt P/\mathrm{day}\lt {10}^{5.3}$ and $0.9\lt Q\lt 1$.

Using the statistics of Duquennoy & Mayor (1991) for solar-type stars, we estimate relative fractions $(0.36,0.20,0.28,0.10,0.05,0.01)$ for classes $({\rm{A}},\ {\rm{B}},\ {\rm{C}},\ {\rm{D}},\ {\rm{E}},\ {\rm{F}})$, respectively.

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

Class (C) is excluded for cases (2) and (3) because the centroid of light would be displaced from the lens 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 the events of case (1) but not cases (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 1 mag, 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 ∼0.1–0.5 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 the turnoff/subgiant star is dominated by the host, i.e., classes (A), (C), and (D) and part of class (E)) and case (2) (blended light from the 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 the 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 the observed 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 [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 weight each event by the same $({t}_{{\rm{E}}},{\theta }_{{\rm{E}}},{\pi }_{{\rm{L}}},{{\boldsymbol{\pi }}}_{{\rm{E}}})$ 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 where the blend is dominated by the host light. We consider the same (3 × 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 1σ, 2σ, and 3σ contours in the lens mass–distance plane for all accepted+rejected (black, dark gray, light gray) and accepted-only (red, yellow, green) simulated events. We first focus on the five $[\mathrm{Fe}/{\rm{H}}]=-0.5$ 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.

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Table 3.  Rate of Acceptance

Age (Gyr)	$[\mathrm{Fe}/{\rm{H}}]=-0.5$	$[\mathrm{Fe}/{\rm{H}}]=0.0$	$[\mathrm{Fe}/{\rm{H}}]=+0.3$
2	$3.56\times {10}^{-6}$	$1.15\times {10}^{-4}$	$2.17\times {10}^{-3}$
4	$4.32\times {10}^{-4}$	$1.86\times {10}^{-3}$	$5.97\times {10}^{-3}$
6	$1.67\times {10}^{-3}$	$1.43\times {10}^{-2}$	$2.71\times {10}^{-2}$
8	$2.24\times {10}^{-3}$	$2.97\times {10}^{-2}$	$5.69\times {10}^{-2}$
10	$2.37\times {10}^{-3}$	$3.72\times {10}^{-2}$	$8.19\times {10}^{-2}$

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We next focus on the [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%, and 3.7% for the 6, 8, and 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 on the subgiant branch for case (2), we would need to take account of the fact that main-sequence stars are more common than subgiants.

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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 of ∼5 due to slower evolution on the turnoff, its contribution would still be small.

Thus, considering that both $[\mathrm{Fe}/{\rm{H}}]=0.0$ and +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) and (3), we estimate that from this quantitative analysis alone, the probability for cases (2) and (3) relative to case (1) is ${p}_{a}\lt 5 \%$.

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) and (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 blended light is $(1-{p}_{a}\times {p}_{b})\gt 99.6 \%$.

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.