OGLE-2019-BLG-0960 Lb: the Smallest Microlensing Planet

OGLE-2019-BLG-0960 was alerted by the OGLE-IV Early Warning System (Udalski 2003; Udalski et al. 2015) on 2019 June 23 at UT 21:04 (HJD' = HJD−2,450,000 = 8658.38). Its coordinates were R.A. = 18:16:03.43, decl.= −25:46:28.8 corresponding to (ℓ, b) = (6.10, −4.30). It is in OGLE-IV field BLG522, which was observed in the I band with a cadence of Γ ∼ 2 night⁻¹. This event was independently discovered as KMT-2019-BLG-1591 by KMTNet (Kim et al. 2018) on 2019 July 11 at UT 03:05 (HJD' = 8675.63) and as MOA-2019-BLG-324 by Microlensing Observations in Astrophysics (MOA; Bond et al. 2001) on 2019 July 17 at UT 16:07 (HJD' = 8682.17). This field is observed by KMTNet at a cadence of Γ ∼ 0.4 hr⁻¹. Most of the KMTNet observations were taken in the I band with about 10 in the V band for the purpose of measuring colors. MOA observed the field containing OGLE-2019-BLG-0960 using a custom R filter (which covers standard R and I bands) at a nominal cadence of Γ ∼ 0.6 hr⁻¹, but see below (Section 2.3).

OGLE-2019-BLG-0960 was selected for Spitzer observations and announced as a Spitzer target on HJD' = 8676.23 with the condition that it must reach I < 17.45 prior to HJD' = 8679.5 (i.e., the decision point for the next Spitzer target upload). Thus, it was initially selected as a "subjective, conditional" target, and observations began at HJD' = 8685.21. The following week HJD' ∼ 8687, the event met the "objective" criteria for selecting Spitzer targets. Thus, observations taken after HJD' = 8691.5 may be considered "objective." See Yee et al. (2015) for a detailed explanation of different types of selection, their implications, and criteria.

Once OGLE-2019-BLG-0960 was selected for Spitzer observations, we began to monitor it with the dual-channel imager, ANDICAM (DePoy et al. 2003) on the SMARTS 1.3 m telescope at CTIO in Chile (CT13). Observations were taken simultaneously in both the I and H bands for the purpose of tracking its evolution and obtaining an I–H color for the source. Starting at HJD' ∼ 8683.5, we increased the cadence to a few points per night to supplement the survey data in the hopes of better characterizing any planetary perturbations. We also began monitoring the event with the telescopes in the LCO 1 m network at SAAO (South Africa; LCOS) and SSO (Australia; LCOA01, LCOA02).

On 2019 July 20, the Spitzer team realized that the event was consistent with high magnification (A > 100) and sent an alert to the Microlensing Follow-Up Network (MicroFUN) at UT 13:06 (HJD' = 8685.05), encouraging dense follow-up observations to capture the peak of the event. Auckland, Farm Cove, and Kumeu Observatories in New Zealand (AO, FCO, and Kumeu, respectively) all responded to the alert and began intensive observations of the event. MOA also increased its cadence in response to the alert. At the same time, we scheduled dense observations with CT13 and the LCO 1 m telescopes at SAAO and SSO. Two days later at HJD' = 8687.05, we called off the alert because the event was declining and no significant perturbations had been observed over the peak. At this point, most follow-up observations ceased. However, we continued to observe this event at a cadence of a few per night with CT13, while Kumeu Observatory took an additional hour of normalizing data, which would prove crucial for recognizing and characterizing the planetary perturbation.

On 2019 July 23 (HJD' = 8688.06), a routine inspection of the CT13 data led to the discovery of a ∼0.5 mag outlier in the light curve. The outlier was confirmed by a data point in the KMTC light curve taken nearly simultaneously. This triggered additional dense observations with the LCO network. We did not realize the existence of Kumeu observations that characterized the falling side of the perturbation until HJD' ∼ 8689.05.

The data were reduced by each collaboration and the photometric error bars were renormalized according to the prescription in Yee et al. (2012). Table 1 gives the specific reduction method, the error renormalization factors, and other properties for each data set. In the case of KMTNet data from CTIO and SAAO, we restricted the data to points taken after HJD' = 8616 to control for possible effects of systematics on the parallax measurement. Likewise, data from KMTA were restricted to ±5 days from the peak of the event.

Table 1. Data with Corresponding Data Reduction Method and Rescaling Factors

Collaboration	Site	Filter	Coverage ($\mathrm{HJD}^{\prime}$)	Ndata	Reduction Method	k	${e}_{\min }$
OGLE		I	7799.9–8763.6	488	Woźniak (2000)	1.27	0.000
OGLE		V	7851.9–8723.6	26	Woźniak (2000)	⋯	⋯
MOA		Red	8537.2–8785.9	341	Bond et al. (2001)	1.14	0.005
KMTNet	SSO	I	8682.0–8692.0	19	pySIS a	0.80	0.000
	CTIO	I	8616.7–8777.6	312	pySIS	1.33	0.007
	SAAO	I	8616.4–8777.3	223	pySIS	1.46	0.000
LCO	SSO01	i	8684.0–8690.2	147	ISIS b	2.03	0.005
	SSO02	i	8688.1–8691.9	46	ISIS	1.04	0.000
μFUN	CTIO	i	8684.6–8692.7	23	ISIS	⋯	⋯
μFUN	SAAO	i	8685.4–9690.2	91	ISIS	0.86	0.004
μFUN	CT13	I	8678.5–8692.7	65	DoPHOT c	1.12	0.000
μFUN	CT13	H	8678.5–8692.7	343	DoPHOT	⋯	⋯
μFUN	AO	540–700 nm	8685.9–8686.9	51	DoPHOT	1.29	0.000
μFUN	FCO	unfiltered	8685.9–8686.9	75	DoPHOT	0.82	0.000
μFUN	Kumeu	540–700 nm	8685.9–8687.9	108	DoPHOT	1.29	0.000
Spitzer		L	8685.2–8712.0	25	Calchi Novati et al. (2015)	1.98	0.000

Notes. $\mathrm{HJD}^{\prime} =\mathrm{HJD} \mbox{-} 2,450,000$. The right-most two columns give the error renormalization factors k and ${e}_{\min }$ as described in Yee et al. (2012). CT13 H-band data are only used to determine the source color. LCO CTIO data are not used for the analysis due to systematics in the data.

^a Albrow et al. (2009). ^b Alard & Lupton (1998), Alard (2000), Zang et al. (2018). ^c Schechter et al. (1993).

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We search for the best-fit models following the method of Yang et al. (2020). In brief, we initially conduct a grid search over ($\mathrm{log}s,\mathrm{log}q$) to locate the solutions. For each set of ($\mathrm{log}s,\mathrm{log}q$), we fix $\mathrm{log}s,\mathrm{log}q$, and free t₀, u₀, t_E, α and ρ. For the initial values of t₀, u₀, and t_E, we use the values from 1L1S fitting to the light curve excluding the planetary anomaly. The initial values of α and ρ are estimated from the 1L1S parameters as follows:

$$\begin{eqnarray}\begin{array}{rcl}\alpha & = & {\tan }^{-1}\displaystyle \frac{{t}_{\mathrm{eff}}}{{t}_{0,\mathrm{anom}}-{t}_{0}}=0.266\,(15\buildrel{\circ}\over{.} 2);\\ {t}_{\mathrm{eff}} & = & {u}_{0}{t}_{{\rm{E}}}\end{array}\end{eqnarray} \tag{ 1 }$$

where t_0,anom = 8687.78. Using the method of Street et al. (2016), ρ can be estimated by

$$\begin{eqnarray}&&\rho \sim \displaystyle \frac{{t}_{0,\mathrm{anom}}-{t}_{\mathrm{cc}}}{{t}_{{\rm{E}}}}\times \sin \alpha =3.2\,\times \,{10}^{-4},\end{eqnarray} \tag{ 2 }$$

where t_cc ∼ 8687.70 is the time of caustic entry. Then, we search for the best-fit parameters using Markov chain Monte Carlo (MCMC) χ² minimization as implemented in the emcee ensemble sampler (Foreman-Mackey et al. 2013). We use the advanced contour integration code VBBinaryLensing (Bozza 2010) ⁴¹ to compute the binary-lens magnification. This code uses multiple rings to account for the limb-darkening effect and sets the number of rings automatically based on the required light-curve precision. We use linear limb-darkening coefficients for a T_eff = 5250 K star: Γ_I = 0.43, Γ_R = 0.52, Γ_L = 0.15 (Claret & Bloemen 2011).

This grid search yields only two local minima (left-hand panel of Figure 2). We then further refine these minima by allowing all seven parameters to vary in an MCMC. The results are shown in Table 2, and the caustics and source trajectories are shown in Figure 3. As with many events, the two solutions appear to be related by the famous s↔s⁻¹ (a.k.a. "close"/"wide") degeneracy (Griest & Safizadeh 1998; Dominik 1999) because one solution has s < 1 and the other s > 1. In fact, this is not the case (as we discuss in Section 6.3).

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Table 2. Parameters for 2L1S Models Using all Ground-based Data

Model	χ2/dof	t0	u0	tE	s	q	α	ρ	πE,N	πE,E	fS,I	fB,I
		($\mathrm{HJD}^{\prime}$)		(days)		(× 10−5)	(rad)	(×10−4)
(s > 1) :
Static	2099.2/1935	8686.4484	0.0060	61.5	1.028	1.41	0.270	3.20	⋯	⋯	0.1781	0.5391
		0.0005	0.0001	1.4	0.001	0.14	0.001	0.17	⋯	⋯	0.0042	0.0037
Parallax,	1934.1/1933	8686.4490	0.0061	61.8	1.028	1.43	0.273	3.23	0.395	−0.393	0.1773	0.5385
(u0 > 0)		0.0006	0.0001	1.4	0.001	0.14	0.001	0.18	0.155	0.039	0.0041	0.0036
Parallax,	1932.0/1933	8686.4496	−0.0061	61.4	1.029	1.48	−0.272	3.29	−0.351	−0.313	0.1784	0.5380
(u0 < 0)		0.0006	0.0001	1.4	0.001	0.14	0.001	0.16	0.173	0.027	0.0041	0.0036
(s < 1) :
Static	2101.2/1935	8686.4485	0.0060	61.9	0.997	1.23	0.269	2.97	⋯	⋯	0.1769	0.5401
		0.0005	0.0001	1.3	0.001	0.06	0.001	0.09	⋯	⋯	0.0040	0.0040
Parallax,	1933.9/1933	8686.4487	0.0059	63.0	0.996	1.27	0.272	2.97	0.464	−0.405	0.1736	0.5418
(u0 > 0)		0.0005	0.0001	1.4	0.001	0.07	0.001	0.10	0.146	0.039	0.0042	0.0037
Parallax,	1933.0/1933	8686.4494	−0.0060	62.0	0.997	1.27	−0.271	3.01	−0.440	−0.304	0.1766	0.5397
(u0 < 0)		0.0005	0.0001	1.3	0.001	0.07	0.001	0.10	0.161	0.027	0.0039	0.0034

Note. Uncertainties for each parameter are given in the second line for each solution.

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Regardless, the two solutions are very similar in their physical implications. First, the mass ratio is q = 1.27 ± 0.07 or q = 1.45 ± 0.15 × 10⁻⁵, making this the smallest mass-ratio planet discovered by microlensing to date. Second, although the two solutions are distinct, s is very close to 1, so the separation of the planet from the host star is similar in the two cases. Third, the relatively long Einstein timescale, t_E ∼ 62 days, suggests that it may be possible to measure the annual parallax effect for this event.

Thus, we are immediately led to introduce two additional parameters π _E = (π_E,N, π_E,E); i.e., the north and east components of the parallax vector (Gould 2004) in equatorial coordinates. Because the effect of Earth's motion can be partially mimicked by lens orbital motion (Batista et al. 2011; Skowron et al. 2011), we initially introduce two further additional parameters γ = ((ds/dt)/s, d α/dt); i.e., the instantaneous angular velocity at t₀. We find that γ is poorly constrained, so we impose a limit on the ratio of projected kinetic to potential energy (Dong et al. 2009)

$$\begin{eqnarray}&&\beta =\left|\displaystyle \frac{{{KE}}_{\perp }}{{{PE}}_{\perp }}\right|=\displaystyle \frac{\kappa {M}_{\odot }{\mathrm{yr}}^{2}}{8\,{\pi }^{2}}\displaystyle \frac{{\pi }_{{\rm{E}}}}{{\theta }_{{\rm{E}}}}{\gamma }^{2}{\left(\displaystyle \frac{s}{{\pi }_{{\rm{E}}}+{\pi }_{s}/{\theta }_{{\rm{E}}}}\right)}^{3},\end{eqnarray} \tag{ 3 }$$

where π_s is the source parallax. We set a requirement that β < 0.8 and do not accept trials with larger values. ⁴² We find that γ is neither significantly constrained in the fit nor significantly correlated with π _E, so we suppress these two degrees of freedom.

The final parallax solutions are summarized in Table 2. Figure 4 shows the cumulative Δχ² diagram between the best-fit parallax and static solutions. All of the survey data sets that span the event (OGLE, MOA, KMTC, KMTS) contribute positively and consistently to the parallax signal. This meets our expectation that a real parallax signal should affect data from all observatories and makes it extremely unlikely that the signal has a non-astrophysical origin.

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In Figure 5, we show the χ² contours projected on the π _E plane, which are in the form of an ellipse with a large axis ratio a/b ∼ 8. Such "1D parallax" measurements were predicted by Gould et al. (1994) and explored in greater detail by Smith et al. (2003) and Gould (2004) for moderately long microlensing events. They arise because π_E,∥, the component of π _E that is parallel to the projected position the Sun at t₀, is directly constrained by the asymmetry in the light curve that is induced by Earth's instantaneous acceleration near the peak of the event. The first clear example of this in a planetary event was OGLE-2005-BLG-071 (Udalski et al. 2005; Dong et al. 2009). Indeed, the orientations (north through east) of the short axes of the ellipses for OGLE-2019-BLG-0960 are close to the projected position of the Sun. Hence, π_E,∥ is both relatively large and well constrained from the ground-based photometry, even though there are substantial uncertainties in the total value of π_E.

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In principle, a short duration bump on an otherwise point-lens-like light curve can also be caused by a binary source with a large flux ratio (Gaudi 1998). Hence, we also search for 1L2S solutions. For this search, we introduce the additional parameters t_0,2 and u_0,2 to describe the trajectory of the second source and ρ₂ for its radius. We also introduce q_F,I, the flux ratio between the two sources, as a fit parameter. The results of these fits are given in Table 3 and show that such solutions are disfavored by Δχ² > 1000. Figure 6 shows that the best-fit 1L2S model fails to reproduce the decrease in magnification (relative to a point lens) seen before and after the bump.

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Table 3. Parameters for 1L2S Models Using all Ground-based Data and Ground-based Survey Data

Model	χ2/dof	t0,1	t0,2	u0,1	u0,2	tE	ρ1	ρ2	πE,N	πE,E	qF,I	fS,I	fB,I
		($\mathrm{HJD}^{\prime}$)	($\mathrm{HJD}^{\prime}$)			(days)	( × 10−4)	( × 10−4)
All,	2952.0/1931	8686.4425	8687.8064	0.0056	0.0003	66.1	9.5	3.1	0.341	−0.308	0.0054	0.1645	0.5501
(u0 > 0)		0.0005	0.0012	0.0004	0.0002	1.4	33.2	92.0	0.174	0.059	0.0028	0.0024	0.0028
All,	2951.4/1931	8686.4431	8687.8076	−0.0056	−0.0003	65.6	6.4	3.0	−0.234	−0.233	0.0053	0.1656	0.5499
(u0 < 0)		0.0005	0.0011	0.0004	0.0002	1.3	30.2	82.1	0.370	0.079	0.0025	0.0022	0.0026
Survey,	1616.0/1323	8686.4406	8687.7951	0.0065	0.0000	66.3	61.4	0.1	−0.051	−0.224	0.0001	0.1641	0.5510
(u0 > 0)		0.0017	0.0116	0.0003	0.3024	0.5	13.4	273.3	0.299	0.053	0.0009	0.0166	0.0172
Survey,	1614.9/1323	8686.4386	8687.7953	−0.0057	−0.0000	67.2	33.4	0.1	−0.622	−0.244	0.0001	0.1614	0.5545
(u0 < 0)		0.0018	0.0126	0.0003	0.3401	0.5	17.1	265.0	0.387	0.070	0.0008	0.0171	0.0180

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We measure the intrinsic color and magnitude of the source by finding its location in a color–magnitude diagram (CMD) relative to the red clump (Yoo et al. 2004). We do this in two ways. We first use the OGLE-IV photometry to find (V − I, I)_S = (1.53, 19.81) ± (0.04, 0.03), and we then find the centroid of the red clump is (V − I, I)_cl = (1.73, 15.24) ± (0.02, 0.04) (see Figure 8). The intrinsic color and magnitude of the red clump along this line of sight is (V − I, I)_0,cl = (1.06, 14.27) (Bensby et al. 2011; Nataf et al. 2013). This implies that A_I = 0.97 and E(V − I) = 0.67 for this field. Hence, assuming the source experiences the same reddening and extinction as the clump, we find (V − I, I)_0,S = (0.86, 18.84) ± (0.05, 0.05). The OGLE magnitudes reported here and elsewhere have all been calibrated from the OGLE-IV system to the standard OGLE-III Johnson-Cousins system. Using the color/surface-brightness relation of Adams et al. (2018), we obtain θ_* = 0.628 ±0.040 μas.

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As a check, we also derive the (V − I) color from the KMTC data. We obtain (V − I)_0,S = 0.81, which is in good agreement with the OGLE-IV color.

We also construct an I–H versus I CMD by cross-matching the CT13 I-band and the VVV H-band stars within a 4' × 4' square centered on the event (see Figure 8). We measure the centroid of the red giant clump as (I − H, I)_cl = (3.81 ± 0.01, 17.04 ± 0.03), and find the intrinsic centroid of the red giant clump to be (I − H, I)_0,cl = (1.30, 14.27) (Nataf et al. 2013, 2016). For the source color, which is independent of any model, we first get (I − H) _CT13,S = − 1.766 ± 0.012 by regression of CT13 H versus I flux as the source magnification changes. We then measure the offset between CT13 H-band stars and VVV stars to be H_CT13–H_VVV = 5.199 ± 0.004, yielding (I − H)_S = 3.433 ± 0.013. Therefore, we obtain the intrinsic color and brightness of the source (I − H, I)_0,S = (0.92 ± 0.03, 18.85 ± 0.05). Then, from Adams et al. (2018), we obtain θ_* = 0.623 ± 0.037 μas.

As our final value, we adopt the weighted mean of the OGLE and CT13 measurements, θ_*,0 = 0.625 ± 0.028. For the source apparent brightness, we note that the source magnitude depends on the model. For simplicity, we have explicitly derived results for I _OGLE,S = 19.81. Then, for different source magnitudes, one can derive θ_* of a solution with a particular I_S by ${\theta }_{* }={\theta }_{* ,0}\times {10}^{-0.2({I}_{{\rm{S}}}-19.81)}$. We use this formula to account for slight variations in ${f}_{{\rm{S}},{I}_{\mathrm{OGLE}}}$ for different degenerate solutions. Finally, the measured value of ρ implies θ_E = 1.9 − 2.1 mas, depending on the solution (see Table 4).

Independent of any other constraints, the observed flux constrains the mass of the lens (unless the lens is a stellar remnant). First, there is a constraint from the blend flux. Because the blend consists of light from stars other than the lensed source (i.e., the lens and any other stars blended into the PSF), the lens cannot be brighter than the blend. In principle, because the background in the bulge consists of unresolved stars rather than blank sky, OGLE-2019-BLG-0960 could be sitting on a "hole" in the background (Park et al. 2004) and, therefore, the blend could be slightly brighter than measured. To place a 3σ upper limit on the flux from the lens, we take account of this effect in addition to the ordinary photon-based error in the measurement of the flux from the baseline object. These must be handled separately because the photon noise is Gaussian but the effect of the mottled background, including a possible "hole," is highly non-Gaussian.

For the photon noise, we estimate the photometric error in the baseline object to be 0.020 mag. Subtracting the measured source flux (taking account of its very small error), and propagating the errors, we find that this term contributes 0.0285 mag at 1σ and so 0.085 mag at 3σ.

To account for the mottled background, we follow the method of Gould et al. (2020). We model the bulge stellar distribution using the Holtzman et al. (1998) luminosity function, scaled down by a factor 0.50 according to the bulge projected clump-star density map of Nataf et al. (2013) and D. Nataf (2019, private communication). We estimate that OGLE-IV imaging captures field stars I < 20 but leaves fainter stars unresolved, and that this map is constructed from FWHM ∼ 09 images. We then evaluate the impact of random field stars on the baseline-object (hence, blend) photometry by Monte Carlo. We find that at (1, 2, 3)σ [i.e., (84.1, 97.7, 99.9) percentiles], the blend could be brighter than its naive value by (0.08, 0.12, 0.13) mag. Hence, the combined three sigma limit due to both photon noise and mottled background is 0.16 mag. Hence, the 3σ limit on the brightness of the lens is I_L > 18.38.

Figure 9 shows the resulting limits on the lens mass and distance. For this figure, we have converted the limit on I_L to a limit on M_L using Pecaut & Mamajek (2013). To show the effect of dust, we have calculated these limits for both I = 18.38 and I₀ = I − A_I = 17.41.

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Second, the color of the baseline object provides an additional constraint on the lens. The baseline object (measured when the source is unmagnified) consists of light from the source, the lens, companions to those stars, and ambient stars blended into the PSF. We have measured the astrometric offset between the baseline object in higher-resolution CFHT imaging and the KMTNet microlensing event. We find Δθ_CFHT = (23.1, 7.6) ± (12.2, 18.4) mas, which is consistent with zero offset. Hence, all or most of the light from the baseline object (and blend) is likely to be associated with the event as either the source, the lens, or a companion to one or both of them.

In this case, the baseline object in OGLE is very red: (V − I)_base = 2.18 ± 0.13. In the KMTC pyDIA CMD, the baseline object is the same, very red, color within errors. Because the source is blue relative to the clump, this necessarily implies that the blended light is at least as red or even redder than the baseline object. Hence, either the lens is also very red or it is much fainter than the blend. Even if the lens is behind all of the dust, this implies (V − I)_0,L > 1.12 at 3σ, assuming E(V − I) = 0.67. Then, using Pecaut & Mamajek (2013) to convert this color to a mass limit gives a limit of M_L ≲ 0.73M_⊙. This limit is shown in Figure 9 as is the limit for (V − I)_L > 1.79, which would imply the lens is in front of all of the dust.

If we combine the flux constraints on the mass of the lens with the measurement of θ_E, we can then place a constraint on the microlens parallax and the distance to the lens:

$$\begin{eqnarray}&&{M}_{{\rm{L}}}=\displaystyle \frac{{\theta }_{{\rm{E}}}}{\kappa {\pi }_{{\rm{E}}}};\quad {\pi }_{\mathrm{rel}}=\displaystyle \frac{{\theta }_{{\rm{E}}}^{2}}{\kappa M}=\left(\displaystyle \frac{\mathrm{au}}{{D}_{{\rm{L}}}}-\displaystyle \frac{\mathrm{au}}{{D}_{{\rm{S}}}}\right)\end{eqnarray} \tag{ 6 }$$

where we assume that the source is at the mean distance to the red clump; i.e., D_S = 7.56 kpc. Given that θ_*,0 ≃ 0.625 μas, the Einstein radius is quite large θ_E = θ_*/ρ ≃ 1.9 mas. Hence, a mass limit of M_L < 0.73 M_⊙ (from (V − I)_base) implies π_E > 0.32 and D_L < 1.35 kpc. However, this estimate is not fully self-consistent because, even if the lens were behind all of the dust, a 0.73 M_⊙ star at 1.35 kpc would be brighter than the observed blend (i.e., in Figure 9 the intersection of the cyan and magenta lines is above the black lines). Combining the blended light constraint with the θ_E measurement results in an upper limit on the lens mass of ∼0.55M_⊙, so π_E > 0.42 and D_L ≲ 1 kpc (intersection of the black and magenta lines in Figure 9).

From the ground-based parallax alone, we have a robust measurement of π_E,∥. This gives a lower limit on the microlens parallax of π_E ≳ 0.3. Combined with θ_E, π_E,∥ alone implies M_L ≲ 0.78M_⊙. These limits are consistent with our expectations based on θ_E and the blended light. The mass–distance relation calculated from the full ground-based π_E for the (s > 1, u₀ < 0) solution is shown in Figure 9 with 1σ limits. This shows that there are regions of overlap between this relation and the θ_E relation that are consistent with the constraints from the blended light.

However, the constraints from the blend and baseline object support the conclusion that the low-amplitude signal in the Spitzer light curve should be treated with caution. The preferred value of the Spitzer-"only" parallax gives π_E = 0.25 (red line in Figure 9). This value is in tension with the ground-based parallax values at ∼3σ but, when combined with θ_E, it also implies M_L ≃ 0.93 M_⊙, which is too blue and too bright for the blended light. ⁴³ At the same time, the full 7σ arc from the Spitzer-"only" parallaxes includes regions that overlap with the ground-based parallax contours at <2σ (see Figure 5) and which is also more compatible with the blended light. Thus, this tension may be resolved in the future once the constraints from the Spitzer light curve are better understood.

For now, we focus on constraints on the lens properties that may be derived from the ground-based analysis. Table 4 gives the physical properties of the lens derived from each solution to the microlensing light curve.

These values are both consistent with the constraints from the blended light and baseline object and suggest that the lens is responsible for most of the blend. For example, for the (s > 1), (u₀ < 0) solution, we have M_L = 0.50 ± 0.12M_⊙ and D_L = 0.98 ± 0.2 kpc. For a 0.5 M_⊙ star, Pecaut & Mamajek (2013) give M_I = 7.6 and (V − I)₀ = 2.0. Hence, for a lens at 0.98 kpc,

$$\begin{eqnarray}&&{I}_{{\rm{L}}}=17.6+{A}_{I}{f}_{\mathrm{dust}},\end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray}&&{(V-I)}_{{\rm{L}}}=2.0+E(V-I){f}_{\mathrm{dust}},\end{eqnarray} \tag{ 8 }$$

where f_dust is the fraction of the dust in front of the lens. We make the naive assumption that the dust follows an exponential profile such that ${f}_{\mathrm{dust}}=1-\exp (-{D}_{{\rm{L}}}/(1.334\ \mathrm{kpc}))$. Then, this yields I_L = 18.07 and (V − I)_L = 2.35, which is shown as the cyan point ("naive lens") in Figure 8. The solid cyan line shows the values of I_L and (V − I)_L calculated for θ_E = 1.90 mas for the 1σ ranges for π_E. The dotted lines show the results evaluated at θ_E ± 1σ. Because of the simplified assumption about the dust, these calculations are simply meant to be illustrative. Nevertheless, they indicate that, within the uncertainties, the lens mass and distance are consistent with the blended light constraints within 1σ and suggest that the lens could be responsible for most or all of the blended light.

The hypothesis that the blend is the lens could be tested immediately with adaptive optics observations. These observations could give a stronger constraint associating the blended light with the event, as well as a more precise measurement of that light by resolving out unrelated stars. Given the high lens-source relative proper motion (μ_rel = 11.4 ± 0.8 mas yr⁻¹ or μ_rel = 12.2 ± 0.7 mas yr⁻¹), it will be possible by ∼2025 to separately resolve the source and lens (e.g., Terry et al. 2021), provided that the lens is responsible for a substantial fraction of the blended light. Such observations would confirm that the blended light is moving with the lens; i.e., that the separation matches the expectations from the lens-source relative proper motion given in Table 4. As discussed in Section 4, this would also give an independent measurement of the direction of that motion, which could be compared to the constraints from the parallax.

Naively, we would not expect such a small mass-ratio planet to be found in such a low-cadence survey field unless, as in this case, there are additional follow-up observations. Such planetary perturbations would typically last

$$\begin{eqnarray}&&{t}_{{\rm{p}}}\sim \sqrt{q}{t}_{{\rm{E}}}=4.8\ \mathrm{hr}{\left(\displaystyle \frac{q}{{10}^{-4}}\right)}^{1/2}\left(\displaystyle \frac{{t}_{{\rm{E}}}}{20\ \mathrm{days}}\right).\end{eqnarray} \tag{ 9 }$$

By contrast, all of the surveys observe this field at a cadence of less than one observation per hour (see Section 2). The highest survey cadence is for the MOA survey at 0.6 hr⁻¹, which was not able to observe the night of the anomaly. Hence, the mass ratio of this planet is an order of magnitude smaller than the expected minimum mass ratio for this field if we require three observations during the anomaly. Indeed, there is only a single survey observation over the main bump caused by the planet. However, because we know that the planet is real, we can ask whether this planet can be recovered and characterized with the sparse survey data (Yee et al. 2012). Then, we can consider whether or not this is evidence that short duration signals with only one to a few observations during the anomaly (which is the most likely case for the smallest planets) in general can be reliably and meaningfully recovered from sparse survey fields.

Therefore, we repeat the search for solutions using only the survey data (i.e., the data from OGLE, MOA, and KMTNet). The grid search over s, q, and α finds the previous two solutions but also a second pair of degenerate solutions (see Table 5 and Figure 2). The new set of solutions corresponds to the case in which the source is much smaller than the caustic and the single KMTC data point at 8687.796 falls in the trough between two caustic crossings (see Figure 10). Hence, this is an observational degeneracy that could be resolved with the addition of more data. Nevertheless, all four solutions lead to the same conclusions regarding the planetary parameters: the planetary perturbation is created by an extremely small mass-ratio planet at s ∼ 1. We also find that the 1L2S solution is still strongly ruled out (by Δχ² ∼ 300, see Table 3).

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Table 5. Best-fit Parameters for the Four Solutions Using Only Ground-based Survey Data

Model	χ2/dof	t0	u0	tE	s	q	α	ρ	πE,N	πE,E	fS,I	fB,I
		($\mathrm{HJD}^{\prime}$)		(days)		( × 10−5)	(rad)	( × 10−4)
(s > 1) :
Single	1325.3/1324	8686.4504	0.0059	62.6	1.027	1.41	0.271	3.12	0.093	−0.352	0.1747	0.5408
		0.0022	0.0002	1.5	0.007	0.16	0.002	0.28	0.288	0.046	0.0043	0.0037
Double	1323.8/1324	8686.4513	0.0060	62.0	1.028	1.69	0.269	1.75	0.032	−0.334	0.1767	0.5390
		0.0021	0.0002	1.1	0.004	0.32	0.003	0.63	0.293	0.047	0.0033	0.0030
(s < 1) :
Single	1325.2/1324	8686.4506	0.0059	62.6	0.994	1.43	0.269	3.26	0.081	−0.351	0.1748	0.5407
		0.0016	0.0002	1.4	0.007	0.20	0.002	0.25	0.351	0.042	0.0042	0.0037
Double	1323.3/1324	8686.4522	0.0060	61.3	0.994	1.89	0.264	0.78	0.077	−0.332	0.1787	0.5373
		0.0027	0.0001	1.1	0.007	0.37	0.004	0.30	0.301	0.046	0.0031	0.0028

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In some ways, it is not surprising that the planet parameters are so well constrained. Because observations are taken every ∼2.5 hr, the maximum duration of the bump is ∼5 hr. Then, given Equation (9) and t_E ∼ 62 days, we would expect q ≲ 1.2 × 10⁻⁵. Furthermore, the fact that the event does not produce a perturbation over the peak, despite the extremely high magnification, suggests a planetary caustic-like perturbation. At the same time, the perturbation occurs only 1.4 days after the peak, which places the "planetary" caustic very close to the central caustic (i.e., in a resonant or close-to-resonant configuration).

However, if the planetary perturbation were found and characterized based on a single outlier, then this would raise substantial questions as to the believability (or publishability) of the planet. In this case, we are fortunate to have extensive follow-up data, so the existence of the perturbation is well-supported. However, the detection of this planet in the survey data is strengthened by several other factors. First, this event has a longer timescale than average (t_E = 62 days), which enhances the duration of the perturbation via Equation (9), and hence the likelihood of obtaining observations during the anomaly. Second, similar to KMT-2019-BLG-0842 (Jung et al.2020), the source trajectory passes at an oblique angle with respect to the binary axis, which further lengthens the duration of the perturbation. As a result, we see from Figure 10 that there are five additional points that are significantly demagnified relative to the point-lens light curve. Hence, the perturbation lasts for almost a day and the detection does not, in fact, rely solely on a single outlier.

In terms of the broader implications for characterizing planetary perturbations based on sparse survey data, it is interesting to compare this event to other cases. The first such case was described in Dominik & Hirshfeld (1996), who showed that a number of different 2L1S models could explain two outliers in MACHO-LMC-001. Another case was described in Gaudi & Han (2004), who found multiple 2L1S models that fit a single outlier in OGLE-2002-BLG-055. One major contrast between these two events and OGLE-2019-BLG-0960 is the cadence of observations relative to the Einstein timescale. In both MACHO-LMC-001 and OGLE-2002-BLG-055, observations were only obtained once per few days during the perturbation, which allows for a much broader range of potential models. For example, the perturbation could always be much shorter, resulting in a relatively weak constraint on q using Equation (9). Hence, even though the survey observations of OGLE-2019-BLG-0960 were relatively sparse compared to the duration of the perturbation, they are dense compared to t_E, leading to good constraints on the planet. However, further investigation of short and sparsely observed perturbations (but which might have well-constrained models) is needed to determine whether or not they are publishable in general.

Given that there appear to be a number of factors that make this planetary perturbation "special," we now consider how detections of small planets in general can be enhanced.

Previously, Abe et al. (2013) had suggested that the sensitivity of microlensing events to small planets was enhanced for moderate magnification events (A > 50). Indeed, examining the published microlensing planets ⁴⁴ with q < q_br = 1.7 × 10⁻⁴ supports this conclusion. Here, we include events with multiple degenerate solutions if all solutions are planetary and at least one solution has q < q_br. Seven out of 19 planets with q < q_br have been in events with u₀ ≤ 0.02 (i.e., A ≥ 50). Another seven are found in events with 0.02 < u₀ ≤ 0.1 and two more in events with 0.1 < u₀ ≤ 0.2. The remaining three planets were all found in "Hollywood" events, in which the source was a giant.

The second thing to notice about the known small planets (see Figure 11) is that about half of them are found in events with resonant caustics and many of the others have separations that are close to resonance. On the one hand, resonant caustics are larger than planetary caustics (the size of the resonant caustic scales as q^1/3 whereas the size of the planetary caustic scales as q^1/2; Dominik 1998; Han 2006), which enhances the probability that the images of the source are perturbed. On the other hand, resonant caustics are generally expected to produce weaker perturbations (Gaudi 2012), which might not be significant enough to be detected. Empirically, Figure 11 shows that the majority of the detected planets are in resonant or near-resonant configurations, which suggests that the enhanced cross-section is the dominant effect and extends to configurations outside of resonance. It also suggests that the "enhancement" for detectability extends to planets with separations s that are well outside the formal range for resonant caustics.

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The enhanced cross-section for a perturbation for caustics close to, but outside of resonance, comes about because the perturbed magnification pattern extends out from and stretches all the way between the planetary and central caustics. Figure 12 compares the width of the caustics and the width of the region for which the magnification of a star+planet lens deviates by at least 10% compared to the star alone. For this exercise, we choose a 10% deviation as a reasonable threshold for producing a detectable perturbation. The "width of the caustics" is measured as the horizontal extent of the caustics (i.e., measured along the binary axis) and, if the caustic is not resonant, is the sum of the widths of the planetary and central caustics. The width of the deviation from a point lens is measured horizontally along a slice through the binary axis.

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We usually think of the cross-section for detecting planetary perturbations as being equal to (or proportional to) the size of the caustics. However, Figure 12 shows that when measured in terms of a 10% deviation, the horizontal cross-section is largest for separations just outside of resonance and is significantly larger at that boundary than one would expect from strict proportionality. The decreased width of the 10% perturbation region for $\mathrm{log}s=0$ reflects the relative "weakness" of true resonant caustics. ⁴⁵ The maximum horizontal extent of a 10% deviation for fixed q peaks at ∼$3\mathrm{log}{s}_{\mathrm{resonant}}$ for s < 1 and ∼$1.8\mathrm{log}{s}_{\mathrm{resonant}}$ for s > 1, where s_resonant is the boundary between resonant and non-resonant caustics (i.e., Equations (57) and (58) from Dominik 1998). Even though caustics in this region are not formally resonant, they are clearly influencing each other. Thus, we refer to caustics in this range as "semi-resonant."

Abe et al. (2013) noted this effect for caustic structures with s < 1 and referred to it as a "cooperative effect." For such caustics, there is an extended "trough" (negative magnification deviation) along the binary axis connecting the planetary and central caustics (bottom panel of Figure 13). However, we also see this effect for s > 1 caustics but in the form of a narrow "ridge" (positive magnification deviation) connecting the planetary and central caustics (top panel of Figure 13). These effects can also be seen in Figure 10 of Gaudi (2010). In terms of detectability, it may be that s < 1 caustic perturbations are easier to detect because the vertical extent of the 10% deviation region is larger, ⁴⁶ and thus a typical perturbation will last longer. Empirically, we do see that there are more s < 1 planets than s > 1 planets for q < 1.7 × 10⁻⁴, but the total numbers are still too small for this to be a statistically meaningful result. By contrast, Jung et al. (2020) noted that published perturbations crossing "ridges" found in s > 1 events tend to have source trajectories with oblique angles. Such trajectories will result in proportionally longer perturbations, but represent only a small fraction of possible trajectories. This suggests that many such perturbations are missed due to insufficient cadence.

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In terms of detection strategy, Figure 14 shows the maximum value of u for which the magnification pattern due to a planet is perturbed by at least 10% relative to a point lens. Because the orientation of the caustic structure relative to the source trajectory is random, this suggests that events should be monitored continuously for τ ≡ ( t − t ₀)/ t _E ∼ ± u to probe this full structure. In the case of OGLE-2019-BLG-0960, we called off the alert at τ ∼ +0.01, and thus the planet was nearly missed. This investigation suggests that to search for planets with $\mathrm{log}q\lt -4.5$, events should be densely monitored for τ = ±0.2, i.e., A > 5. Indeed, Han et al. (2021) report the detection of a planet in KMT-2018-BLG-1025 with q ∼ 10⁻⁴ in which the perturbation occurred at τ ∼ 0.05.

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Figure 11 also shows that many planets have multiple, degenerate solutions. These degenerate solutions usually have one solution with s < 1 and another with s > 1, and are often referred to as arising from the "close"/"wide" degeneracy. However, we will see that this degeneracy is more complex than the degeneracy that was first described by Griest & Safizadeh (1998), and expanded upon by Dominik (1998) and An (2005).

As an example, the two solutions for OGLE-2019-BLG-0960 nominally have the hallmarks of the "close"/"wide" degeneracy for central caustics. First, the event is in the "high-magnification" regime (u₀ < 0.01). Second, one of the solutions has s < 1 while the other has s > 1. However, the two solutions are not centered around $\mathrm{log}s=0$, i.e., ${s}_{\mathrm{close}}\ne {s}_{\mathrm{wide}}^{-1}$. Furthermore, the "close"/"wide" degeneracy was derived by Griest & Safizadeh (1998) and Dominik (1998) from the lensing equation in the regime where $| \mathrm{log}s|$ is large; i.e., the regime in which the central and planetary caustics are well-separated from each other. Griest & Safizadeh (1998) even state, "We expect the formula to break down when q → 1 or x_p → 1" (where they use x_p in place of s). OGLE-2019-BLG-0960 is clearly not in this regime: in both solutions, s ∼ 1 and the caustic structure is resonant. Thus, it is interesting to consider whether this is in fact a case of the "close"/"wide" degeneracy.

Because the two solutions are not centered around $\mathrm{log}s=0$, they may in fact be more analogous to the "inner"/"outer" degeneracy described by Calchi Novati et al. (2019) in which the source trajectory may pass either inside or outside the planetary caustic relative to the central caustic. An examination of the caustic structures for OGLE-2019-BLG-0960 (see Figure 3), shows that the "close" solution passes to the outside of the caustic, whereas the "wide" solution passes over the bridge in the resonant caustic created in between what would be the central and planetary caustics. Two other small planets, OGLE-2016-BLG-1195Lb and KMT-2019-BLG-0842Lb, also show this degeneracy (Bond et al. 2017; Shvartzvald et al. 2017; Jung et al. 2020), and it also arises in the cases of MOA-2016-BLG-319 (Han et al. 2018) and OGLE-2016-BLG-1227 (Han et al. 2020).

The "inner"/"outer" degeneracy in these six events appears to result from the planetary caustic degeneracy that was first derived in the Chang–Refsdal limit (Chang & Refsdal 1979) by Gaudi & Gould (1997). This degeneracy is intrinsic, i.e., rooted in mathematical symmetries in the lens equation rather than "accidental," which refers to degeneracies arising due to insufficient observational coverage of the perturbation. However, it was again derived in the limit that $| \mathrm{log}s|$ was large, and it was expected to break down as $| \mathrm{log}s| \to 0$. None of these events is in the limit that $| \mathrm{log}s|$ is large, yet a degeneracy persists. This suggests that while the "close"/"wide" degeneracy of Griest & Safizadeh (1998) and the "inner"/"outer" degeneracy of Gaudi & Gould (1997) were derived in the limit that $| \mathrm{log}s|$ is large, as $| \mathrm{log}s| \to 0$, instead of breaking down, the two degeneracies merge.

To investigate this possibility in more detail, we review the known microlensing planets with two degenerate "geometric" solutions. These may refer to either two distinct caustic topologies or two distinct source trajectories relative to the caustic. As shown in Figure 15, very few of the planets with one s < 1 and one s > 1 solution can be described as having well-separated caustics; i.e., are in the "close"/"wide" regime as described by Griest & Safizadeh (1998). The vast majority of planets with degenerate solutions are either in the resonant or semi-resonant regime, similar to OGLE-2019-BLG-0960. This suggests that some events referred to in the literature as exhibiting the "close"/"wide" degeneracy may be better classified as having the "inner"/"outer" degeneracy. At the same time, the continuum of solutions suggests that the two types of solutions may be part of a continuous degeneracy. We note that subsequent to the initial submission of this paper, and based partly upon it, Hwang et al. (2021) introduced "s_± = s^† ± Δs" notation that both quantifies this continuum and would allow one to specify distinct inner/outer versus close/wide regimes within this continuum. Also subsequent to our original submission, An (2021) carried out an analytic investigation into this question.

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Events in the resonant or semi-resonant regime share the characteristics that the degenerate solutions tend not to be perfectly symmetric about $| \mathrm{log}s| =0$. Also, in contrast to the Griest & Safizadeh (1998) limit, the degenerate solutions tend to have slightly different values of q. This is especially true for events in the moderate magnification regime (0.01 < u₀ < 0.1), which is precisely the region where we expect the smallest planets to be found. Fortunately, because $| \mathrm{log}s|$ is very close to 0 in both cases, this degeneracy does not meaningfully affect the interpretation of the projected separation between the planet and its host star. However, the small differences in q will have to be taken into account when interpreting statistical samples of such planets.