An Earth-mass planet in a time of COVID-19: KMT-2020-BLG-0414Lb

The 2L1S model requires three additional parameters e(s, q, α), which are respectively the separation of the two lens bodies scaled to θE, the mass ratio between these bodies and the angle of the source trajectory relative to the binary axis. For modeling, we use the advanced contour integration code (Bozza 2010; Bozza et al. 2018), VBBinaryLensing ⁵ . We initially carry out sparse grid searches for the parameters (log s, log q, α). The grid consists of 21 values equally spaced between –1.0 ≤ log s ≤ 1.0, 20 values equally spaced between 0° ≤ α < 360° and 61 values equally spaced between –6.0 ≤ log q ≤ 0.0. For each set of (log s, log q, α), we find the minimum χ² by Markov chain Monte Carlo (MCMC) χ² minimization utilizing the emcee ensemble sampler (Foreman-Mackey et al. 2013), with fixed log q, log s and free t₀, u₀, τE, ρ, α. We identify one local minimum at (log s,log q) ≃ (0.0, –5.1), similar to the case of Yee et al. (2021). We thus conduct a similar dense grid search as Yee et al. (2021) that consists of 51 values equally spaced between –0.02 ≤ log s ≤ 0.03 and 41 values equally spaced between –6.0 ≤ log q leq –4.0. Often, such a grid search yields two local minima at s > 1 and s < 1 (e.g., Jung et al. 2020; Yee et al. 2021), which must then be individually further explored and compared. However, in the present case, there is only one local minimum at s < 1, while the s > 1 model is disfavored by Δχ² > 900. We refine this minimum by allowing all parameters to vary. The parameters with their 68% uncertainty range from the MCMC are displayed in Table 2, and the fit and residuals are depicted in Figure 1. The very low mass ratio q ∼ 10⁻⁵ indicates that the companion is a very-low-mass planet.

Even without detailed analysis, the results for the static model, that are listed in Table 2, imply a large (and so potentially measurable) microlens parallax (Gould 1992, 2000),

$$\begin{eqnarray}&&{{\boldsymbol{\pi }}}_{{\rm{E}}}\equiv \displaystyle \frac{{\pi }_{{\rm{rel}}}}{{\theta }_{{\rm{E}}}}\displaystyle \frac{{{\boldsymbol{\mu }}}_{{\rm{rel}}}}{{\mu }_{{\rm{rel}}}}.\end{eqnarray} \tag{ 2 }$$

According to θ_* and the blended light in Section 5, θE = θ_*/ρ ≥ 1.68 mas and the lens light I_L ≥ 18.9 at 3σ level. The two limits correspond roughly to an M ∼ 0.5 M_⊙ at a distance of 1.2 kpc. Thus, we can expect ⁶

$$\begin{eqnarray}&&{\pi }_{{\rm{E}}}=\displaystyle \frac{{\theta }_{{\rm{E}}}}{\kappa {M}_{{\rm{L}}}}\ge \mathrm{0.41.}\end{eqnarray} \tag{ 3 }$$

Moreover, given the long Einstein timescale, θE ∼ 103 d, the projected velocity on the observer plane, $\tilde{v}\equiv {\rm{au}}/({\pi }_{E}{t}_{E})\,\le 42\,{\rm{km}}\,{{\rm{s}}}^{-1}$ is close to the changes of Earth's velocity over the course of the event, taking account of which could impact other parameters as well.

Therefore, it is essential to include microlens-parallax effects in the fit. We fit the annual microlens-parallax effect by introducing two additional parameters π_E,N and π_E,E, the north and east components of π _E in equatorial coordinates (Gould 2004). We also fit u₀ > 0 and u₀ < 0 models to consider the "ecliptic degeneracy" (Jiang et al. 2004; Poindexter et al. 2005). Table 2 lists the results of fitting the light curve with the microlens parallax effect. Because the annual parallax effect can be degenerate with the effects of lens orbital motion (Batista et al. 2011; Skowron et al. 2011), we also introduce two linearized parameters (ds/dt, dα/dt), the instantaneous changes in the separation and orientation of the two lens components defined at t₀. We restrict the MCMC trials to β < 0.8, where β is the ratio of projected kinetic to potential energy (Dong et al. 2009)

$$\begin{eqnarray}&&\beta \equiv \left|\displaystyle \frac{{{\rm{KE}}}_{\perp }}{{{\rm{PE}}}_{\perp }}\right|=\displaystyle \frac{\kappa {M}_{\odot }{{\rm{yr}}}^{2}}{8{\pi }^{2}}\displaystyle \frac{{\rho }_{{\rm{E}}}}{{\theta }_{{\rm{E}}}}{\gamma }^{2}{\left(\displaystyle \frac{s}{{\pi }_{{\rm{E}}}+{\pi }_{{\rm{S}}}/{\theta }_{{\rm{E}}}}\right)}^{3},\\ &&{\boldsymbol{\gamma }}\equiv \left(\displaystyle \frac{ds/dt}{s},\displaystyle \frac{d\alpha }{dt}\right),\end{eqnarray} \tag{ 4 }$$

where we adopt the source parallax πs = 0.128 mas based on the mean distance to clump giant stars in this direction (Nataf et al. 2013). See Table 2 for the results. We find that the addition of lens orbital motion effect provides improvements of Δχ² = 2.4 and 5.4 for the u₀ > 0 and u₀ < 0 solution, respectively, and π_E is basically the same compared to the "parallax" model.

Although the angular Einstein radius θE estimated from the parallax modeling is smaller than the value from the static model, the resulting parallax for 2L1S is still strongly inconsistent with the constraint of the blended light at about 3σ. We will further discuss the implication of the 2L1S results in Section 6.1.

In some cases, planetary (2L1S) light curves can be imitated by binary-source (1L2S) events (Gaudi 1998). We do not expect that this will be the case for KMT-2020-BLG-0414. because the planetary anomaly is mainly characterized by sharp changes in slope, rather than a smooth short-lived bump. Nevertheless, as a matter of due diligence, we search for such models including both microlens-parallax and microlens-xarallap effects (Griest & Hu 1992; Han & Gould 1997; Poindexter et al. 2005). We find that while the introduction of a second source yields a huge improvement with respect to the 1L1S model with Δχ² = χ²(1L1S) – χ²(1L2S) > 10 000, the 1L2S model still does not compete with the 2L1S model with Δχ² = χ²(1L2S) – χ²(2L1S) > 400.

We begin by conducting a grid search for static 3L1S solutions that is analogous to the one carried out previously for 2L1S solutions, but is substantially more computationally intensive. In a grid search (whether 2L1S or 3L1S), the lens geometry is held fixed at each grid point. However, for 2L1S, the geometry is specified by just two parameters, (s₂, q₂), whereas for 3L1S, five geometric parameters are required, (s₂, q₂, s₃, q₃, ψ). To reduce the grid of geometries from five to three dimensions, we consider a (70 × 70 × 180) grid in (s₃, q₃, ψ). We hold (s₂, q₂) fixed at the best-fit 2L1S model. We seed the remaining five parameters, (t₀, u₀, θE, ρ, α), at the best-fit 2L1S model, and we then allow these to vary. We initially consider only close models for the third body, i.e., s₃ < 1. For the grid-search phase, which relies on fixed geometries, we apply the map-making technique of Dong et al. (2006) to evaluate the magnifications. This grid search yields only one local region of candidate solutions.

We then seed an additional MCMC with the best grid point from this region and allow all 10 parameters to vary. Because the geometry now varies with each step in the MCMC, we apply the adaptive-image inverse-ray-shooting technique to evaluate the magnifications. The resulting parameters are shown in Table 3, and the model light curve is compared to the data in Figure 1. The corresponding caustic structure in the upper panel of Figure 2 demonstrates that KMT-2020-BLG-0414. is a classic case of "caustic factorization". The caustic is nearly the superposition of two well-known caustic types: a large resonant caustic associated with the planet, and a smaller, nearly Chang-Refsdal (Chang & Refsdal 1979), caustic associated with the third body. As is often the case, the two caustic structures "interact" and become intertwined where they overlap. Because of the caustic factors and q₃ ≪ 1, it is straightforward to guess the alternate wide (s₃ > 1) solution according to the prescription of Griest & Safizadeh (1998): ${s}_{3}\to {s}_{3}^{-1}$. We seed this guess into an MCMC to yield the alternate wide solution, whose parameters are given in Table 3 and whose geometry is displayed in the lower panel of Figure 2. The most striking difference between the 3L1S close and wide solutions is that s_{2, close} = 0.99914 ± 0.00011, while s_{2, wide} = 0.96882 ± 0.00018, which appears to be a "100σ" difference. We address this issue in Appendix A.

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As discussed in Section 3.2, the microlens-parallax parameters π _E (Eq. (2)) can be degenerate with the orbital-motion parameters γ (Eq. (4)). Hence, both should be considered together. However, as in that section, we proceed step-by-step, in part due to the increasing computational load as more parameters are introduced, and therefore the importance of understanding which are really necessary.

When only π _E is added to the 3L1S static model, there are 12 parameters. As was the case for 2L1S, adding parallax to the fit results in doubling the number of solutions, i.e., there is a ± u₀ pair of solutions for each of the close and wide solutions found in Section 4.1. Hence there are four solutions altogether. The resulting parameters and χ² values are given in Table 3. It is found that including parallax significantly improves the fit by Δχ² > 130. The wide solutions are disfavored by Δχ² > 17. Between the two close solutions, the u₀ > 0 solution is significantly favored by Δχ² ∼ 12. We note that the magnitude of π _E for 3L1S is substantially larger than the value estimated from the 2L1S modeling regardless of the lens-orbital effect. Figure 3 shows the cumulative distribution of Δχ² = χ²(static) – χ²(parallax) for the four solutions. Overall, Δχ² grows steadily over time, giving credence to the parallax measurement. An important feature of this diagram is that the contribution to Δχ² during the short time interval starting from one day before the peak and ending two days after the peak is about 40% of the total Δχ². By contrast, one normally expects the parallax signal to be dominated by the wings of the light curve. This time interval is essentially the duration of contact with the planetary caustic, in particular as the source "rides the caustic" for two days after the peak. This gives a plausible explanation for the sensitivity of π _E to the near-peak region of the light curve. Hence, it is essential to include orbital motion.

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We now include both π _E and γ for the planet in the 3L1S fit, for a total of 14 chain parameters. We display the parameters of this fit in Table 3. It is found that including planet orbital motion significantly changes π _E in both magnitude and direction, and |πE| of the u₀ < 0 solution is ∼ 1.8 times greater than that of the u₀ > 0 solution. The close u₀ < 0 solution has the best fit to the observed data, while other solutions are only disfavored by Δχ² < 7. Thus, we cannot exclude any solution from the light-curve analysis. The ratio β of projected kinetic to potential energy is measured well, and all of the solutions have β < 0.1 at 3σ. This relatively low value of the ratio suggests that the planet and the host may be aligned along the line of sight.

The third body (with a brown-dwarf-like mass ratio q₃ ∼ 0.05) must also undergo orbital motion. In the wide solutions, the period would be of order 100 years, implying that orbital motion of the third body would not affect the lensing light curve. For the close solutions, the period would be of order 50 days, and thus its orbital motion could affect the light curve. Nevertheless, we do not attempt to model orbital motion of the third body for several reasons. First, the duration of its pronounced perturbation (∼ 0.3 days over the peak) is 10 times shorter than for the duration of the planetary signal, and its impact on the light curve is quadratic in the duration. Second, if there were clear prospects of a scientifically important result, such work would be warranted, but there are no such prospects (see Sect. 5.4). Finally, the results of 3L1S parallax + planet orbital motion required about two weeks of computations with 400 processors. The already prodigious use of computer time (which scales ∝ (n/2)! where n is the number of chain parameters) would increase by a factor eight. We therefore decline to pursue this aspect of the problem.

Figure 4 depicts the CMD of stars from the OGLE-III catalog (Szymański et al. 2011) located within a square region with one side length of 240'' centered at the location of KMT-2020-BLG-0414., together with the source position (blue) and the centroid of the red giant clump (red). We measure the centroid of the red giant clump as (V – I, I)_cl = (2.04 ± 0.01, 15.55 ± 0.02). For the intrinsic centroid of the red giant clump, we adopt (V – I, I)_cl,0 = (1.06, 14.35) (Bensby et al. 2013; Nataf et al. 2013). This implies that A_I = 1.20 and E(V – I) = 0.98 toward this direction. For the source color, which is independent of any model, we get (V – I)_S = 1.82 ± 0.01 by regression of MOA V versus R flux as the source magnification changes and a calibration to the OGLE-III scale using the field-star photometry from the same reductions. Because the source apparent brightness slightly depends on the model, for simplicity, we explicitly derive results for I_S = 19.12 and then present a scaling relation for different source magnitudes. From this procedure, we obtain the intrinsic color and brightness of the source as (V – I, I)_S,0 = (0.84 ± 0.03, 17.92 ± 0.03), suggesting that the source is a mid-G type dwarf (Bessell & Brett 1988). Considering the color/surface-brightness relation of Adams et al. (2018), we obtain

$$\begin{eqnarray}&&{\theta }_{* }=0.943\pm 0.047\,\mu {\rm{as}},\end{eqnarray} \tag{ 6 }$$

where the 5% error is given by table 3 of Adams et al. (2018). Then, for any particular model with source magnitude I_S, one can infer θ_* = 0.943 × 10^{−0.2(I_S – 19.12)}.

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For KMT-2020-BLG-0414., the baseline object was detected by the OGLE-III survey, (V, I)_base = (20.70 ± 0.08, 18.46 ± 0.03). We further consider the statistical errors due to the mottled background from unresolved stars (Park et al. 2004). We follow the approach of Ryu et al. (2020b) utilizing the GalSim package (Rowe et al. 2015) with the readout noise of Udalski et al. (2015). We find σ_I = 0.08 mag and σ_V = 0.012 mag, and thus the baseline object has (V, I)_base = (20.70 ± 0.14, 18.46 ± 0.09), yielding the blended light of ${(V-I,I)}_{{\rm{B}}}=({3.1}_{-0.6}^{+1.0},19.32\pm 0.20)$. This value is consistent with the lens properties that are predicted by the microlensing light-curve and CMD analyses. For example, using the median θE and πE values of the close u₀ < 0 solutions, the host mass M₁ = 0.25 M_⊙ and it would have rough intrinsic brightness and color of M_I ∼ 9.8, (V - I)₀ ∼ 3.0. Assuming an extinction curve with a scale height of 120 pc, it would have (A_I , E(V – I))_L = (0.38, 0.31) at the lens distance D_L = 0.74 kpc. These correspond to (V – I, I)_L ∼ (3.3, 19.5), which are displayed as the cyan point ("naive lens") in Figure 4 and is quite consistent with the blend.

We also check the astrometric alignment between the source and the baseline object from KMTA imaging. We find that the baseline object lies (0.18'', 0.01'') west and south of the source. Because the source position is derived from difference image analysis on highly magnified images, the uncertainty in the source position (≲ 0.01'') is negligible relative to the error in the baseline position. We estimate the error of baseline position by the fractional astrometric error being equal to the fractional photometric error (Jung et al. 2020), σ_ast = 0.39 σ_I FWHM = 0.09''. Hence, the baseline object is astrometrically consistent with the source (and thus lens) at the 2σ level. Thus, it is plausible that most or all of the blended light is due to the lens.

The alignment between the source and the baseline object can be immediately checked (i.e., 2021 bulge season) by the Hubble Space Telescope (HST) or by ground-based adaptive optics (AO) mounted on large ground-based telescopes (e.g., Keck, Subaru). Even if the alignment were demonstrated, the blended light could in principle come from a stellar companion to either the source or the lens. However, if the alignment is ≲ 50 mas, an additional stellar companion to the lens would have generated significant deviations on the peak, so the confirmation that the blend is well aligned to the lens could probably rule out the lens-companion scenario. Because the source and the blended light have significantly different colors, the possibility of the source companion can be checked by a measurement of the astrometric offset in different bands (Bennett et al. 2006). It is a priori unlikely that the blended light is primarily due to ambient stars that are unassociated with the event because of the low surface density of stars relative to the 180 mas offset. If high-resolution imaging nevertheless showed that this were the case, it would imply that the parallax was even larger and the lens was less massive and closer than the best-estimated values ⁷ .

In the Bayesian analysis, we choose the log-normal initial mass function of Chabrier (2003) as the mass distribution of the lens. For the bulge and disk stellar number density, we choose the model used by Zhu et al. (2017) and Bennett et al. (2014), respectively. For the dynamical distribution of the disk lens, we assume the disk lenses follow a rotation of 240 km s⁻¹ (Reid et al. 2014) with the velocity dispersion used by Han et al. (2020a). For the source and bulge lens dynamical distributions, we examine a Gaia CMD (Gaia Collaboration et al. 2016, 2018) utilizing the stars within 5' and derive the proper motion of "clump" stars (16.4 < G < 17, 1.9 < B_p – R_p < 2.35). We obtain (in the heliocentric frame)

$$\begin{eqnarray}&&\langle {{\boldsymbol{\mu }}}_{{\rm{bulge}}}(\ell,b)\rangle =(-6.19,-0.36)\pm (0.15,0.13)\,{{\rm{mas}}{\rm{yr}}}^{-1},\end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray}&&\sigma ({{\boldsymbol{\mu }}}_{{\rm{bulge}}})=(2.78,2.39)\pm (0.11,0.09)\,{{\rm{mas}}{\rm{yr}}}^{-1}.\end{eqnarray} \tag{ 8 }$$

We create a sample of 5 × 10⁹ simulated events from the Galactic model. We weight each simulated event, i, by

$$\begin{eqnarray}&&{\omega }_{{\rm{Gal}},i}={\Gamma }_{i}{ {\mathcal L} }_{i}({t}_{{\rm{E}}}){ {\mathcal L} }_{i}({\theta }_{{\rm{E}}}){ {\mathcal L} }_{i}({\pi }_{{\rm{E}},{\rm{N}}},{\pi }_{{\rm{E}},{\rm{E}}}),\end{eqnarray} \tag{ 9 }$$

where Γ_i ∝ θ_E,i × μ_rel,i is the microlensing event rate, and ${ {\mathcal L} }_{i}({t}_{{\rm{E}}})$, ${ {\mathcal L} }_{i}({\theta }_{{\rm{E}}})$ and ${ {\mathcal L} }_{i}({\pi }_{{\rm{E}},{\rm{N}}},{\pi }_{{\rm{E}},{\rm{E}}})$ are the likelihoods of its inferred parameters given the distributions of these quantities. In addition, we adopt the 3σ upper limit of the blended light, I_{B, limit} = 18.9, as the upper limit of the lens flux. We then adopt the mass-luminosity relation of Wang et al. (2018),

$$\begin{eqnarray}&&{M}_{I}=4.4-8.5\mathrm{log}(\displaystyle \frac{{M}_{{\rm{L}}}}{{M}_{\odot }}),\end{eqnarray} \tag{ 10 }$$

where M_I is the absolute magnitude in the I band, and reject trial events for which the lens properties obey

$$\begin{eqnarray}&&{M}_{I}+5\mathrm{log}\displaystyle \frac{{D}_{{\rm{L}}}}{10{\rm{pc}}}+{A}_{I,{D}_{{\rm{L}}}}\lt {I}_{{\rm{B}},{\rm{limit}}},\end{eqnarray} \tag{ 11 }$$

where A_{I, DL} is the extinction at D_L, which is derived by an extinction curve with a scale height of 120 pc.

In Table 5, we list the angular Einstein radius, θE, the microlens parallax, πE, the relative proper motion, μ_rel, the estimated masses of the individual lens components, M₁, M₂ and M₃, the distance to the lens, D_L, and the projected separations to the host, a_⊥,2 and a_⊥,3, for all of the four 3L1S solutions from the Bayesian analysis. For both solutions, it is found that the host is probably an M₁ ∼ 0.3 M_⊙ M-type dwarf, the second body is an Earth-mass terrestrial planet at a projected separation of a_⊥,2 ∼ 1.5 au, and the third body is an object at the planet/brown-dwarf boundary at a projected separation of a_⊥,3 ∼ 0.15 au for the close solutions and a_⊥,3 ∼ 15 au for the wide solutions. The lens system is located in the Galactic disk with a distance of D_L ∼ 1.0 kpc.

Table 5. Physical Parameters

Solutions	Close u0 > 0	Close u0 < 0	Wide u0 > 0	Wide u0 < 0
θE (mas)	${1.475}_{-0.091}^{+0.087}$	${1.586}_{-0.074}^{+0.074}$	${1.489}_{-0.083}^{+0.077}$	${1.574}_{-0.060}^{+0.057}$
πE	${0.470}_{-0.096}^{+0.106}$	${0.715}_{-0.105}^{+0.096}$	${0.513}_{-0.097}^{+0.114}$	${0.840}_{-0.142}^{+0.158}$
M1 (M⊙)	${0.364}_{-0.057}^{+0.072}$	${0.259}_{-0.034}^{+0.041}$	${0.340}_{-0.053}^{+0.063}$	${0.217}_{-0.037}^{+0.045}$
M2 (M⊕)	${1.21}_{-0.22}^{+0.28}$	${0.96}_{-0.13}^{+0.15}$	${1.28}_{-0.21}^{+0.25}$	${0.86}_{-0.13}^{+0.19}$
M3 (MJ )	${23.3}_{-3.9}^{+4.7}$	${15.4}_{-2.5}^{+3.3}$	${16.9}_{-3.0}^{+3.4}$	${12.5}_{-2.1}^{+2.6}$
DL (kpc)	${1.22}_{-0.23}^{+0.32}$	${0.80}_{-0.10}^{+0.12}$	${1.12}_{-0.20}^{+0.27}$	${0.69}_{-0.11}^{+0.13}$
a⊥, 2 (au)	${1.79}_{-0.28}^{+0.36}$	${1.26}_{-0.14}^{+0.19}$	${1.62}_{-0.26}^{+0.30}$	${1.04}_{-0.16}^{+0.19}$
a⊥, 3 (au)	${0.16}_{-0.03}^{+0.03}$	${0.12}_{-0.02}^{+0.02}$	${16.6}_{-2.7}^{+3.2}$	${11.4}_{-1.8}^{+2.1}$
μhel, N (mas yr−1)	${4.92}_{-0.81}^{+0.51}$	$-{2.82}_{-0.82}^{+0.94}$	${5.22}_{-0.76}^{+0.49}$	$-{3.58}_{-0.94}^{+1.41}$
μhel, E (mas yr−1)	${6.12}_{-1.92}^{+1.76}$	${12.07}_{-1.21}^{+1.38}$	${6.91}_{-1.72}^{+1.58}$	${13.13}_{-1.30}^{+1.67}$

The highly degenerate 3L1S solutions and their very different π _E, with large uncertainties, indicate that the characteristics of the lens system cannot be completely determined by our work. The ambiguous elements mainly include two aspects: the mass and distance of the lens system and the projected separation of the third body. They can be clarified by future high-resolution photometric and spectroscopic observations.

For the lens mass and distance, they can be unambiguously determined by conducting high-resolution AO imaging observations when the source and lens are resolved. Bhattacharya et al. (2018) resolved the source and lens of OGLE-2012-BLG-0950 employing Keck AO and the HST when they were separated by ∼ 34 mas, in a case for which the source and lens had approximately equal brightness. In the present case, the blended light and source also have approximately equal brightness, so the lens and source can be resolved in 2024 provided that the lens contributes a significant part of the blended light. The lens-source resolution can yield the measurement of the lens flux and the lens-source relative proper motion vector, μ_rel (e.g., Alcock et al. 2001; Kozłowski et al. 2007; Batista et al. 2015). The lens flux would provide an independent mass-distance relationship, and μ _rel would further constrain θE (Eq. (1)) and πE (Eq. (2)) by its magnitude and direction, respectively. Together, these can significantly reduce the uncertainties of the physical lens parameters.

For the orbit of the third body, future spectroscopy observations to measure the radial velocity (RV) can determine whether the third body lies inside or outside the planetary orbit. Following the procedure of Han et al. (2019b), we estimate that the close solution has an RV amplitude vsin(i) of order 1 km s⁻¹ with a period of order 0.15 yr, and the wide solution has a vsin(i) of order 100 m s⁻¹ with a period of order 100 yr. These RV amplitudes and differences are big enough to be distinguished by high-resolution spectrometers on VLT/Espresso or future 30 m telescopes. Note that if the lens indeed has comparable brightness to the source, then these observations can be made immediately, i.e., long before the lens and source separate on the sky, because the lens and source RVs likely differ by tens or hundreds of km s⁻¹.

In Section 3, we adopted the perspective that there are no data over peak, and we, in particular, asked how the event would have been analyzed and reported in the absence of such data. Of course, one obvious difference is that the data fit well with the 2L1S model and so there would have been no report of a "third body", but here we see that there would also have been a report of a "major puzzle" about this parallax inconsistency. The derived parallax values in Table 3 (πE = 0.229 ± 0.042 for the u₀ > 0 solution or πE = 0.173 ± 0.040 for the u₀ < 0 solution) are strongly inconsistent with Equation (3), which was derived from a combination of the well-measured value of θE and the photometric constraints on blended light. Most likely, this would have been "explained" as due to "probable low-level systematics in the KMTA and/or MOA data". Perhaps there would have been some additional commentary about the possibility that the host was a white dwarf, which would enable evasion of the photometric constraints. Then, it is very likely that the final reported parameters would have been given as the static solution, while the physical parameters would have been derived from a Bayesian analysis that simply ignored the parallax measurement. Finally, it would have been pointed out that the apparent discrepancies could ultimately be resolved by future high-resolution follow-up observations.

While the scenarios laid out in the previous paragraph are necessarily somewhat speculative, we believe that the great majority of people familiar with the microlensing-planet literature would broadly agree with this assessment of what would have happened. We recall them here in detail because all of the "explanations", "caveats", etc., in the previous paragraph are completely wrong. In fact, once the peak data are added back into the light curve and a third body is included in the modeling, the supposed "tension" surrounding Equation (3) entirely disappears. We note that the problem of the impact of undetected "third bodies" on microlensing solutions was analyzed by Zhu et al. (2014a). While that study focused on the impact on the measured parameters of the detected planet, the present work shows that higher-order parameters can be affected as well.

Over the past two years, the previously empty "tip" of the (log s,log q) diagram has been gradually populated by new discoveries. While KMT-2019-BLG-0842Lb lies just below the "pile up" associated with the Jung et al. (2019) "break" (see Fig. 5), three other recent discoveries (KMT-2018-BLG-0029Lb, OGLE-2019-BLG-0960Lb, KMT-2020-BLG-0414Lb), with q ∼ (6.0, 4.5, 3.6) q_⊕, all lie well below it. All four of these recent discoveries were detected via resonant caustics. Moreover, for three of these (all except KMT-2018-BLG-0029Lb), the source trajectory angle with respect to the binary axis was oblique. For two of the four (OGLE-2019-BLG-0960Lb and KMT-2020-BLG-0414Lb), followup observations played a major or dominant role in the detection and characterization.

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Due to inhomogeneous (and difficult to model) selection, it is probably not possible to draw precise statistical conclusions from this particular sample. However, the broad features listed above tend to support the idea that there is not a strong break below Neptune mass ratios, as had been previously conjectured. Rather, the conditions and lens geometries required for such detections are rare, including resonant caustic geometries and dense coverage over/near the peak of high-magnification events, often aided by oblique trajectories. In two cases, the dense coverage was provided by KMTNet high (Γ = 1 hr⁻¹), or very high (Γ = 4 hr⁻¹) 3-continent monitoring, and in the other two cases by dense 3-continent follow-up observations.

These characteristics suggest two paths forward for probing the very low-q cold-planet population. First, as discussed by Yee et al. (2021), it should be possible to construct a rigorous statistical sample by systematic investigation of all KMTNet high-magnification events, where the threshold for "high-magnification" would be set at some definite value, like A_max > 20. Each event with pipeline parameters near or above this boundary would require tender-loving care (TLC) reductions to determine if it were truly in the sample, and, if so, it would be subjected to a dense grid search for planets. Both KMT-2018-BLG-0029Lb and KMT-2019-BLG-0842Lb would definitely be recovered by such a search, and it is possible that OGLE-2019-BLG-0960Lb would be recovered as well. These reductions are costly in human effort, but if restricted to high-magnification events, the project would be feasible.

Second, KMT-2020-BLG-0414Lb indicates that follow-up observations of high-magnification events in low-cadence fields can detect the most extreme systems. It is challenging, but not impossible, to derive rigorous statistical conclusions from such a sample. For example, Gould et al. (2010) demonstrated that their sample of 13 very high-magnification (A_max > 200) events was statistically well-grounded because they showed that the μFUN follow-up campaign randomly sampled half of the underlying OGLE-III A_max > 200 events during 2004–2008. This achievement required prodigious human effort because, in the majority of cases, it was not possible to determine which events would be very high magnification based on (often sparse) OGLE-III data alone. Hence, it was necessary to "patrol" a much larger set of OGLE events with 1.3 m SMARTS telescope observations to even determine which events should be densely monitored. Even choosing the events to be patrolled required several hours of human effort per day. This level of effort, for 1–2 planets per year, was the main reason for abandoning this approach once massive surveys were underway.

However, with the advent of KMTNet's 3-continent survey, together with (beginning in full in 2019) its real-time Alert-Finder system (Kim et al. 2018), it has become possible to identify rising high-magnification events with little or no "patrolling". The LCOGT has eight 1 m robotic telescopes at the same sites as KMTNet and a reaction time of about 15 minutes, which allows the rapid necessary coverage for high-magnification events. μFUN is mainly composed of amateur volunteers and has different sites ⁸ from the current surveys, which can provide targeted coverage for high-magnification events when the survey sites have poor weather. Indeed, our 2020 follow-up program was meant to be a pilot for such a patrol-free (or patrol-lite) high-magnification (A ≳ 20) planet search. This original idea was undermined by COVID-19, which resulted in the closure of two out of three KMTNet observatories for the bulk of the 2020 season. Substantial patrolling was therefore required in this season. For KMT-2020-BLG-0414Lb, the beginning of follow-up observations at "A_now > 10" was actually the result of such patrolling. Nevertheless, the detection of KMT-2020-BLG-0414Lb affirms that this approach is viable.

For KMT-2020-BLG-0414., thanks to its oblique source trajectory and resonant caustic, the signal of the q ∼ 10⁻⁵ planet lasted for about five days (9038 < HJD' < 9043) and ${\chi }_{1{\rm{L}}1{\rm{S}}}^{2}-{\chi }_{2{\rm{L}}1{\rm{S}}}^{2}\gt {10}^{4}$ (without the MOA data on the peak). Two other very low-q planets, KMT-2019-BLG-0842Lb and OGLE-2019-BLG-0960Lb, also have oblique source trajectories and resonant caustics. Thus, it is of considerable interest to compute the planet sensitivities of KMT-2020-BLG-0414. with different source trajectories (α) and caustic structures (s) and to quantitatively determine the contributions of the oblique source trajectory and resonant caustic in the detections of q ∼ 10⁻⁵ planets. In addition, Yee et al. (2009) studied the planet sensitivities of the extreme high-magnification microlensing event OGLE-2008-BLG-279, which has similar 1L1S parameters (u₀ ∼ 0.0007, τE ∼ 100, ρ ∼ 0.0007) to KMT-2020-BLG-0414.. Because the two events have different followup data coverages (see below), comparing the planet sensitivity of the two events can help provide guidance for the followup strategy of very low-q planets.

We apply the method of Rhie et al. (2000) to evaluate the planet sensitivities, which was also followed by Yee et al. (2009). In brief, we first simulate 2L1S curves from a grid of (α, s), using (t₀, u₀, τE, ρ, q, I_S, I_B) of the best-fit 3L1S model with parallax and planet orbital motion. The epochs and the corresponding residuals of the mock 2L1S curves are the same as the real observed data. The grid consists of 360 values equally spaced between 0° ≤ α ≤ 359° and 151 values equally spaced between 0.5 ≤ s ≤ 2.0. Yee et al. (2009) generated 2L1S curves for q = 10⁻³, 10⁻⁴, 10⁻⁵, 10⁻⁶. Here we only study q = 1.08 × 10⁻⁵, because in the present context, we are only interested in the sensitivity to q ∼ 10⁻⁵ planets. We then fit the mock 2L1S curves using the 1L1S model allowing (t₀, u₀, τE, ρ) to vary and compute the Δχ² relative to the curve without any planet.

The resulting Δχ² as a function of (s cosα, s sinα), depicted in Figure 6, has three important features compared to the lower-left panel of figure 6 of Yee et al. (2009). First, for both cases, the resonant caustics have the highest sensitivities to a q ∼ 10⁻⁵ planet, while the observed data are sensitive to a q ∼ 10⁻⁵ planet over a wide range, with |log s| ≲ 0.14 applying a detection threshold of $\Delta {\chi }_{{\rm{\min }}}^{2}\sim 100$. Second, KMT-2020-BLG-0414. is only sensitive to q ∼ 10⁻⁵ planets from oblique source trajectories (|sinα| ≲ 0.5), while OGLE-2008-BLG-279 is quite sensitive for all source trajectories. The planetary signals of perpendicular and nearly-perpendicular source trajectories mainly occur on the peak, so the different α distributions are due to the different data coverage on the very high-magnification region. OGLE-2008-BLG-279 has nearly continual coverage during A ≥ 100, while KMT-2020-BLG-0414. has no coverage during ∼ 40% of the A ≥ 100 light curve, and its peak was only observed by MOA with a cadence of Γ ∼ 4 hr⁻¹. Third, for oblique source trajectories, KMT-2020-BLG-0414. has a much higher sensitivity, with several hundred times higher Δχ² for some resonant caustics. The planetary signals of resonant caustics together with oblique source trajectories are mainly detectable at intermediate magnification. OGLE-2008-BLG-279 has only very sparse coverage on A < 100, while KMT-2020-BLG-0414. was observed several hundred times more during 20 < A < 100. These features are qualitatively similar to those obtained in Dong et al. (2006), where they compared the sensitivity of q = 10⁻⁵ planets from actual data for the extremely high-magnification microlensing event OGLE-2004-BLG-343 with incomplete coverage over peak (see their fig. 6) with those from simulated data with complete coverage (see their fig. 8) using the method of Rhie et al. (2000).

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The planet sensitivity studies of KMT-2020-BLG-0414. and OGLE-2008-BLG-279 provide two important implications for followup observations. First, for the very high-magnification region (A ≳ 100), dense or even continual observations are of critical importance. Second, the followup observations should begin at intermediate magnification (e.g., the current threshold of the LCOGT & μFUN followup program is A = 20), in order to capture the signals of oblique source trajectories. Assuming at least four data points are required to characterize the planetary caustic crossing, the minimum required cadence at intermediate magnification is

$$\begin{eqnarray}&&{\Gamma }_{{\rm{\min }}}=\displaystyle \frac{2}{{t}_{* }\sin \alpha }\simeq \displaystyle \frac{2{A}_{{\rm{now}}}}{{t}_{* }{A}_{{\rm{\max }}}},\end{eqnarray} \tag{ 12 }$$

where t_* = ρτE = θ_*/μ_rel is "source-crossing timescale", and one can adopt a relatively conservative lower limit on t_* by adopting μ_rel = 10 mas yr⁻¹ and estimate θ_* by the rough source position on the CMD. On the declining part of the light curve, Γ_min can be most accurately estimated because A_max is well known unless the rising side has almost no coverage due to weather. On the rising side, Γ_min is still more uncertain because A_max is generally not known well before the peak. Nevertheless, for events that eventually reach very high magnification, it is often possible to reliably predict that

$$\begin{eqnarray}&&{A}_{{\rm{\max }}}\gtrsim n\times {A}_{{\rm{now}}},\end{eqnarray} \tag{ 13 }$$

where n ∼ 3 is a typical value. In addition, this cadence should be reached by followup alone, without a combination of survey cadence. First, independent observations of followup and survey can doubly confirm the planetary signal. Second, part of the observing window on the planetary perturbation could be lost due to weather. Third, if an anomaly has occurred or the quality of the real-time survey photometry is not good enough, Γ_min could be underestimated.

Including KMT-2020-BLG-0414., 15 triple-lens events have been published. Among these, six lens systems contain two planets orbiting a star and seven consist of a planet in a binary-star system (for details, see table 1 of Han et al. 2021). OGLE-2016-BLG-0613L (Han et al. 2017) and KMT-2020-BLG-0414L are composed of a star, a low-mass brown dwarf and a planet ⁹ . Among the eight systems containing a host and two low mass-ratio (q < 0.1) companions, five were detected by a high-magnification event: OGLE-2006-BLG-109 (u₀ = 0.0035, Gaudi et al. 2008; Bennett et al. 2010), OGLE-2012-BLG-0026 (u₀ = 0.0088, Han et al. 2013; Beaulieu et al. 2016; Madsen & Zhu 2019), OGLE-2018-BLG-0532 (u₀ = 0.0079, Ryu et al. 2020b), KMT-2019-BLG-1953 (u₀ = 0.0007, Han et al. 2020b) and KMT-2020-BLG-0414. (u₀ = 0.0007) ¹⁰ . High-magnification events provide an efficient channel for multiple low mass-ratio companions because the source trajectory goes very close to the host, where each planet/BD induces distortions in the magnification profile via their central or resonant caustics (Gaudi et al. 1998).

There have been only two microlensing studies about the occurrence rate of systems with multiple low mass-ratio companions. Using the one two-planet system OGLE-2006-BLG-109L among the 13 μFUN A > 200 sample, Gould et al. (2010) found that the frequency of solar-like systems is 1/6 for all planetary systems. Utilizing the two multiplanetary systems OGLE-2006-BLG-109L and OGLE-2014-BLG-1722L, and the detection efficiency of the six-year MOA survey combined with the four-year μFUN follow-up observations, Suzuki et al. (2018) estimated that 6% ± 2% of stars host two cold giant planets. The detection of KMT-2020-BLG-0414L suggests that the new follow-up program for high-magnification events can also give an estimate of the occurrence rate of systems with multiple low mass-ratio companions.

In addition, although the new follow-up program aims to observe events located in KMTNet Γ ≤ 1 hr⁻¹ fields, it is still important to follow up very high-magnification events located in KMTNet Γ = 4 hr⁻¹ fields. First, a simulation of Zhu et al. (2014b) indicates that even KMTNet Γ = 6 hr⁻¹ cadence is not intensive enough to capture the subtle anomalies for very high-magnification events. This has been demonstrated by the two less secure multiplanetary events OGLE-2018-BLG-0532 and KMT-2019-BLG-1953, for which a cadence of Γ = 4 hr⁻¹ was only barely adequate to detect the second planet. Second, the survey observations could suffer from nonlinearity or saturation on the very bright peak of very high-magnification events. For KMT-2020-BLG-0414., its I = 11.1 peak is too bright for the current surveys with their normal exposure time (60 s for KMTNet with three 1.6 m telescopes, 100-120 s for OGLE with a 1.3 m telescope and 60 s for MOA with a 1.8 m telescope). The LCOGT and most of the μFUN telescopes have smaller apertures and more flexible exposure time, so the new follow-up program can provide supplementary coverage for the very bright peak of very high-magnification events in KMTNet Γ = 4 hr⁻¹ fields.