MOA-2020-BLG-208Lb: Cool Sub-Saturn-mass Planet within Predicted Desert

Gravitational microlensing (Mao & Paczynski 1991) provides a means for detecting planets that is sensitive to low-mass planets orbiting at moderate to large distances from their host star (Gould & Loeb 1992; Bennett & Rhie 1996), typically from 0.5 to 10 au. Such planets may be challenging to detect via other common exoplanet detection methods (e.g., photometric transits), hence gravitational microlensing helps provide a more complete understanding of planet statistics by providing access to another population of planets (Bennett 2008; Gaudi 2012).

The first planetary microlensing event was discovered by Bond et al. (2004). Expected to launch in 2026, the Nancy Grace Roman Space Telescope (Roman), a NASA flagship mission, will survey ∼10⁸ stars for microlensing events (Penny et al. 2019). With less than 200 planets discovered via microlensing thus far, each new planetary microlensing analysis facilitates the calibration of theory and influences the science goals and operational plan of large-scale missions such as Roman.

A recent statistical analysis of planetary signals discovered using gravitational microlensing implied that cold, Neptune-mass planets are likely to be the most common type of planets beyond the snow line (Suzuki et al. 2016; Jung et al. 2019). This was inferred from a break in the planet-to-host-star mass-ratio function for a mass ratio q ∼ 10⁻⁴, with the break resulting in a peak at Neptune-mass planets. Although the Suzuki et al. (2016) sample generally supports the planet distribution predictions from core accretion theory population synthesis models for planets beyond the snow line (Ida & Lin 2004; Mordasini et al. 2009), the existence of the peak in the planet-to-host-star mass-ratio distribution at Neptune-mass planets in the sample distribution presents issues. Specifically, these models of the planet distribution predict a dearth of sub-Saturn-mass planets, which conflicts with the microlensing observations (Suzuki et al. 2018). However, the Suzuki et al. (2016) sample consisted of only 30 exoplanets. This small sample size combined with the apparent contradiction emphasizes the importance of additional analyses, such as the one presented in this work. Formation models that propose a shortage of cold sub-Saturn-mass planets (Ida & Lin 2004; Mordasini et al. 2009; Ida et al. 2013) would be contested by such a population of planets (Suzuki et al. 2018; Zang et al. 2022; Terry et al. 2021), and revised formation models would be required (Ali-Dib et al. 2022).

The microlensing event MOA-2020-BLG-208 was discovered by the Microlensing Observations in Astrophysics (MOA) collaboration and first alerted on 2020 August 11. The event was located at the J2000 equatorial coordinates $\left({\rm{R}}.{\rm{A}}.,\ \mathrm{decl}.\right)\,=({17}^{{\rm{h}}}\,{53}^{{\rm{m}}}\,43\buildrel{\rm{s}}\over{.} 80,-32^\circ 35^{\prime} 21\buildrel{\prime\prime}\over{.} 52)$, and Galactic coordinates (l, b) = (3577569650, −33694423) in the MOA-II field "gb3." The MOA observations were performed using the purpose-built 1.8 m wide-field MOA telescope located at Mount John Observatory, New Zealand, and were taken at a 15 minutes cadence with the MOA-Red filter. The MOA-Red filter corresponds to a customized wide-band similar to a sum of the Kron-Cousins R and I bands (600–900 nm). Additional observations were made by MOA in the visual band using the MOA-V filter. The photometry in these filters was performed in real time by the MOA pipeline (Bond et al. 2001) based on the difference imaging method of Tomaney & Crotts (1996). A rereduction was carried out using the method detailed in Bond et al. (2017), resulting in photometry calibrated to phase-III of the Optical Gravitational Lensing Experiment (OGLE-III; Szymański et al. 2011).

Observations of the event, particularly of the primary lens event, were made by several other collaborations.

At UT 16:45 on 2020 September 6 (${{\rm{HJD}}}^{{\prime} }=9099.2$), the MAP Follow-Up Collaboration and the Microlensing Follow Up Network (μFUN) found that this event could become a high magnification within two days based on the real-time MOA data and thus conducted follow-up observations. Their follow-up observations were taken using the 1.0 m telescope of the Las Cumbres Observatory global network (Brown et al. 2013) located at the South African Astronomical Observatory, South Africa (LCOS), the 0.4 m telescopes at Auckland Observatory and Possum Observatory, the 0.36 m telescopes at Kumeu Observatory, Klein Karoo Observatory, Turitea Observatory and Farm Cove Observatory, and the 0.3 m Perth Exoplanet Survey Telescope in Australia. The LCOS data were reduced using a custom pipeline based on the ISIS package (Alard & Lupton 1998; Alard 2000; Zang et al. 2018), and the μFUN data were reduced using DoPHOT (Schechter et al. 1993).

In addition, this event was observed by the Korea Microlensing Telescope Network (KMTNet; Kim et al. 2016) with two 1.6 m telescopes located at the Siding Spring Observatory, Australia and the South African Astronomical Observatory, South Africa (the site at CTIO in Chile was closed due to Covid).

The Observing Microlensing Events of the Galaxy Automatically project (OMEGA) observed the event with 1.0 m telescopes located at the South African Astronomical Observatory, South Africa, using the Las Cumbres Observatory network of robotic telescopes (Brown et al. 2013). The OMEGA data includes SDSS-i' and SDSS-g' bands, with data reduced on a filter basis and uses the pyDanDIA photometry pipeline (pyDanDIA Contributors 2017).

The primary lens peak of the MOA-2020-BLG-208 event light curve (see Figure 1) generally resembles a Paczýnski curve (Paczynski 1986), which is the expected shape for a microlensing event with a single lens. The deviation from the Paczýnski curve occurs in the form of a secondary anomaly near $\mathrm{HJD}^{\prime} =9113.6$. ⁵²  The secondary anomaly in concert with the primary event is suggestive of a binary-lens system comprised of a star and a companion. To model the distribution of likely properties (e.g., mass ratio, orbital separation) that define this binary-lens system, we apply the method described by Bennett (2010). To perform this modeling and analysis process, we include photometric measurements from the 15 instrument data sets that are listed in Section 2.

Zoom In Zoom Out Reset image size

3.1. Model-fitting Parameters

The approach presented by Bennett (2010) uses an image-centered, ray-shooting method (Bennett & Rhie 1996) combined with the Metropolis–Hastings algorithm (Metropolis et al. 1953; Hastings 1970), which convergences to find χ² minima. The parameters of our modeling consist of: the Einstein crossing time (t_E); the time that the minimum separation of lens and source occurs (t₀); the minimum separation between source and lens as seen by the observer (u₀); the separation of the two masses of the binary-lens system during the event (s); the counter-clockwise angle between the lens-source relative motion projected onto the sky plane and the binary-lens axis (θ); the mass-ratio between the secondary lens and the primary lens (q); the source radius crossing time (t_*); two values to model parallax in polar coordinates (${r}_{{\pi }_{{\rm{E}}}}$ and ${\theta }_{{\pi }_{{\rm{E}}}}$); the source flux for each instrument i (f_s,i ); and the blend flux per instrument i (f_b,i ).

The parameters t_E, t₀ and u₀ are the common parameters for the single-lens model, while s, θ and q are the additional parameters for a binary-lens system model. Both length parameters, u₀ and s, are normalized by the angular Einstein radius θ_E, defined by

$$\begin{eqnarray}&&{\theta }_{{\rm{E}}}=\sqrt{\displaystyle \frac{4{{GM}}_{L}}{{c}^{2}{D}_{S}}\left(\displaystyle \frac{{D}_{S}}{{D}_{L}}-1\right)},\end{eqnarray} \tag{ 1 }$$

where G is the gravitational constant, M_L is the total mass of the lens system, c is the speed of light, D_S is the observer-source distance, and D_L is the observer-lens distance. The source radius crossing time, t_*, is a parameter used to take into account finite-source effects

$$\begin{eqnarray}&&{t}_{* }=\rho {t}_{{\rm{E}}}=\displaystyle \frac{{\theta }_{* }}{{\theta }_{{\rm{E}}}}{t}_{{\rm{E}}},\end{eqnarray} \tag{ 2 }$$

where ρ is the source angular radius in Einstein units, and θ_* is the source angular radius.

The microlens parallax is denoted π_E, with

$$\begin{eqnarray}&&{\pi }_{{\rm{E}}}=\displaystyle \frac{{\pi }_{\mathrm{rel}}}{{\theta }_{{\rm{E}}}},\end{eqnarray} \tag{ 3 }$$

where π_rel is the lens-source relative parallax (Gould 1992; Gould et al. 2004). Our model fits the parallax parameters in polar coordinates. The equivalent Cartesian coordinates are given by ${x}_{{\pi }_{E}}={r}_{{\pi }_{{\rm{E}}}}\ast \cos ({\theta }_{{\pi }_{{\rm{E}}}})$ and ${y}_{{\pi }_{E}}={r}_{{\pi }_{{\rm{E}}}}\ast \sin ({\theta }_{{\pi }_{{\rm{E}}}})$.

To account for blending with nearby unlensed stars and different photometric systems, we normalize the flux data from each data source independently to minimize the χ². As the observed brightness is linearly dependent on the blend and source fluxes, all fluxes from a single data source are normalized together to determine the normalization that produces the minimal χ². The total flux F_i (t) for time t for passband i is given by:

$$\begin{eqnarray}&&{F}_{i}(t)=A(t,{\boldsymbol{x}}){f}_{s,i}+{f}_{b,i},\end{eqnarray} \tag{ 4 }$$

where A(t, x ) is the magnification of the event at time t for a set of lens model parameters x = (t_E, t₀, u₀, s, q, t_*), f_s,i is the unlensed source flux for passband i, and f_b,i is the blended flux for passband i. Each instrument's passbands are independently normalized, as the relative scales of the source and blend fluxes are dependent on the instrument and the method used for difference imaging. For example, the method presented by Bond et al. (2017) is used to process the MOA data, which normalizes the target flux to match the flux of the nearest star-like object in the reference frame.

3.2. Fitting Procedure

In Figure 1, we show the best-fit model of our light-curve fitting. This model is a single-source binary-lens wide-orbit model with the lens parameters shown in the "Wide-model best-fit" column of Table 1. As a comparison, the best-fit single-source single-lens model is also shown in Figure 1. As expected, this single-source single-lens fit does not explain the anomalous data well.

Table 1. A Comparison of the Parameters for the Best-fit and Median MCMC Distribution Models for both Close and Wide Solutions

	Wide Model Distribution	Wide Model Best-fit	Close Model Distribution	Close Model Best Fit
χ2		2768.2		3247.1
tE (days)	${19.336}_{-0.071}^{+0.074}$	19.337	${20.226}_{-0.078}^{+0.079}$	20.218
q	${3.17}_{-0.27}^{+0.28}\times {10}^{-4}$	3.10 × 10−4	${5.72}_{-0.26}^{+0.26}\times {10}^{-4}$	5.86 × 10−4
${t}_{0}\,(\mathrm{HJD}^{\prime} )$	${9100.9076}_{-0.0010}^{+0.0010}$	9100.9076	${9100.8835}_{-0.0007}^{+0.0007}$	9100.8833
u0	${0.02772}_{-0.00018}^{+0.00018}$	0.02763	${0.02542}_{-0.00013}^{+0.00013}$	0.02551
t* (days)	${0.298}_{-0.014}^{+0.018}$	0.287	${0.110}_{-0.056}^{+0.020}$	0.130
s	${1.3807}_{-0.0018}^{+0.0018}$	1.3805	${0.73817}_{-0.00092}^{+0.00091}$	0.73809
θ (rad)	${3.112}_{-0.004}^{+0.004}$	3.113	${0.02837}_{-0.00676}^{+0.00793}$	0.02804
${r}_{{\pi }_{{\rm{E}}}}$	${0.0666}_{-0.0830}^{+0.1244}$	0.0433	${0.527}_{-0.039}^{+0.057}$	0.515
${\theta }_{{\pi }_{{\rm{E}}}}\,(\mathrm{rad})$	${2.92}_{-2.56}^{+0.38}$	3.18	$-{1.80}_{-0.32}^{+0.44}$	−1.88

Note. The MCMC distribution value also includes the upper and lower bounds of a 68.27% confidence interval.

Download table as:  ASCIITypeset image

In Table 1, we also show the best-fit single-source binary-lens close-orbit model for comparison. The light curve of the best-fit model for the close orbit is shown in Figure 2. Here we find the wide-orbit model is favored over the close-solution model with a difference in χ² of −478.8.

Zoom In Zoom Out Reset image size

Figure 3 shows the source trajectory and magnification pattern of the wide-orbit best-fit solution. Figure 4 shows the equivalent for the close-orbit best-fit solution.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

The distribution of MCMC states for the run with covariance can is shown in Figure 5 for the wide-orbit solution, and Figure 6 for the close-orbit solution.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Next, we obtain the dereddened color and corrected magnitude of the source star from the instrumental MOA-II magnitudes, which are calibrated via the procedure described by Bond et al. (2017). The color–magnitude diagram of the stars near MOA-2020-BLG-208 is shown in Figure 7. Using the unextincted red clump centroid as predicted by Nataf et al. (2013), we estimate the color and magnitudes shown in Table 2. Using the Boyajian et al. (2014) restricted analysis of stars with 3900 < T_eff < 7000, we use the color and magnitude to estimate the angular size of the source star. Using this, we estimate the Einstein radius of the system. These and other estimated source and source-lens properties are shown in Table 2.

Zoom In Zoom Out Reset image size

Table 2. Source and Lens-source System Properties

Property	MCMC Median
Source magnitude IS,0	${15.008}_{-0.077}^{+0.075}$
Source magnitude KS,0	${13.79}_{-0.17}^{+0.12}$
Source color (V − I)S,0	${0.997}_{-0.100}^{+0.092}$
Source angular radius θ⋆ (μas)	${4.13}_{-0.42}^{+0.45}$
Einstein radius θE (mas)	${0.267}_{-0.029}^{+0.033}$
Lens-source proper motion μrel,G (mas yr−1)	${5.04}_{-0.55}^{+0.62}$

Download table as:  ASCIITypeset image

Table 3. This Table Shows the Median Values of our Galactic-model Distribution as well as the Upper and Lower Bounds of a 68.27% Confidence Interval (1σ)

	Prior Uniform in M	Prior Proportional to M
Source distance (kpc)	${9.15}_{-1.16}^{+1.04}$	${9.02}_{-1.09}^{+1.04}$
I-band lens magnitude	${24.0}_{-2.8}^{+1.7}$	${22.3}_{-2.1}^{+2.3}$
Planet mass (M⊕)	${46}_{-24}^{+42}$	${69}_{-34}^{+37}$
Host mass (M⊙)	${0.43}_{-0.23}^{+0.39}$	${0.66}_{-0.32}^{+0.35}$
Projected separation (au)	${2.74}_{-0.47}^{+0.43}$	${2.88}_{-0.41}^{+0.40}$
Lens distance (kpc)	${7.49}_{-1.13}^{+0.99}$	${7.81}_{-0.93}^{+0.93}$
K-band lens magnitude	${20.7}_{-2.1}^{+1.3}$	${19.5}_{-1.7}^{+1.7}$

Note. These results are for the wide-orbit model.

Download table as:  ASCIITypeset image

Various properties of the lens system, such as lens mass and lens distance, cannot be directly inferred without microlensing parallax and lens brightness measurements. These measurements do not exist for the MOA-2020-BLG-208 event, as parallax is not reliably detected and the angular source size cannot be directly measured. Indeed, the wide-solution microlensing parallax is not well constrained by our light curve fitting procedure, as is seen by the high uncertainty in the parallax parameters ${r}_{{\pi }_{{\rm{E}}}}$ and ${\theta }_{{\pi }_{{\rm{E}}}}$ in Table 1. Hence, we estimate additional lens system properties using the Bayesian analysis galactic model provided by Bennett et al. (2014). This model allows its posterior distributions to be influenced by a prior based on the host mass. Specifically, it allows the posterior distributions to rely on either a mass function that assumes that all stars have an equal planet-hosting probability or one that assumes planets are more likely to orbit around more massive stars.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

We infer the lens properties posterior distributions using the MCMC states described in Section 3 as the input, excluding the parallax parameters. Along with the MCMC states as input, we provide the model with the estimated angular source radius and I-band extinction that we calculate as described in Section 2. To facilitate high-angular resolution follow-up observations, we additionally run the galactic-model inference for the K-band, using an extinction that we calculate via the method provided by Gonzalez et al. (2012) and Nishiyama et al. (2009). For MOA-2020-BLG-208, we calculate this extinction to be A_K = 0.2062. For both bands, we infer the lens parameters using two different input mass priors. The difference between these two priors comes in the form of varying α in the power-law stellar-mass function defined by P ∝ M^α .

In one case, we set α = 0, which provides a mass prior that assumes all stars have an equal probability of hosting planets. This is a common prior assumption in many existing planetary microlensing analyses and related statistical population studies (e.g., Cassan et al. 2012). Figure 8 shows the Galactic model posterior distributions with α = 0.

However, several works have suggested that the probability of hosting a planet scales in proportion to the host star mass, including cases in radial velocity samples (Johnson et al. 2007, 2010), in direct imaging samples (Nielsen et al. 2019), and in revised analyses of microlensing samples with additional follow-up observations (Bhattacharya et al. 2021), suggesting mass prior with α = 0 may be incorrect. Thus, following the analysis of Silva et al. (2022), we also present an analysis using the power-law stellar-mass function where α = 1, which provides a mass prior that assumes more massive stars have a greater probability of hosting planets. Figure 9 shows the Galactic model posterior distributions with α = 0. The median values of the distributions as well as the upper and lower bounds of a 68.27% confidence interval are shown in Table 3.

In this Bayesian analysis, we assume that the planet-hosting probability does not depend on Galactocentric distance (Koshimoto et al. 2021b).

The galactic model by Bennett et al. (2014) does not consider the possibility of a nearby source. Therefore, we performed a brief analysis using the galactic model by Koshimoto et al. (2021a, 2021b) and Koshimoto & Ranc (2022), which does include the possibility of nearby sources. From this, we found that probability of a source distance D_S < 4 kpc is less than 1.31 × 10⁻⁶, suggesting a nearby source is unlikely.

6.1. Evidence Against a Binary-source Model

We also attempted to fit the observed data to a binary-source model. The results of this modeling suggest the observed data does not fit well to a binary-source model.

For the binary-source modeling, we restricted the data sources to observatories that obtained data near the anomalous event, namely, MOA and KMT data. Each of these instruments have relatively narrow passbands. With these data, we fit both a binary-lens and binary-source model. The resulting binary-lens model fit was similar to the wide-solution model shown in Section 4. A comparison of these two models is shown in Table 4.

Table 4. This Table Shows a Comparison of Binary-source-single-lens and Single-source-binary-lens Models

	Binary-source-single-lens Model Best Fit	Single-source-binary-lens Model Best Fit
χ2	1579.5	1425.3
tE (days)	19.039	19.377
${t}_{0}\,(\mathrm{HJD}^{\prime} )$	9100.8982	9100.9058
u0	0.02708	0.02785
t* (days)	0.0623	0.324
${u}_{0,{s}_{2}}$	0.00481
${t}_{0,{s}_{2}}$	9110
${f}_{I,{s}_{2}}$	0.00990
${f}_{R,{s}_{2}}$	0.0358
q		3.91 × 10−4
s		1.3794
θ (rad)		3.118

Download table as:  ASCIITypeset image

The χ² value of the binary-source model is 154.1 more than that of the binary-lens model, suggesting a strong preference toward the binary-lens model. Furthermore, this best-fit binary-source model attributes an improbable blue color to the second source (see Figure 10). Attempts to restrict the second source to a more likely color result in significant increases to the χ² value of the model fit.

Zoom In Zoom Out Reset image size

With both the significant χ² preference toward the binary-lens model and the best-fit binary-source model having unlikely physical parameters, we infer this event is unlikely to have been caused by a binary source.

6.2. Close-orbit Model Orbital Motion

6.3. Degenerate Close-orbit Model Trajectory

Recent exoplanet surveys have provided a plethora of planetary systems, many of which challenge established formation models (Gaudi et al. 2021). Among these surveys, gravitational microlensing surveys are apt to probe planets on significantly wider orbits (Kane 2011). Of particular note is the statistical inferences of the population of cold sub-Saturn-mass planets. Based on a limited number of detections, survey sensitivity analyses (Suzuki et al. 2016; Jung et al. 2019) suggest planets with a host-star mass ratio, q, larger than 10⁻⁴ are common. Formation models that propose a shortage of cold sub-Saturn-mass planets (Ida & Lin 2004; Mordasini et al. 2009; Ida et al. 2013) would be contested by such a population of planets (Suzuki et al. 2018; Zang et al. 2022; Terry et al. 2021), and revised formation models would be required (Ali-Dib et al. 2022). Notably, Suzuki et al. (2018) presents a mass-ratio gap in models from Ida & Lin (2004) and Mordasini et al. (2009), which is not identifiable in microlensing observations from Suzuki et al. (2016). Our analysis suggests a planet mass that contributes to the planet population in this gap with $q={3.17}_{-0.26}^{+0.28}\times {10}^{-4}$ (${m}_{\mathrm{planet}}={69}_{-34}^{+37}\,{M}_{\oplus }$ when the mass prior uses α = 1 and ${m}_{\mathrm{planet}}={46}_{-24}^{+42}\,{M}_{\oplus }$ when α = 0). Note that the planet mass is sub-Saturn even though the mass-ratio $q={3.17}_{-0.26}^{+0.28}\times {10}^{-4}$ is slightly above that of Saturn. However, this still falls within of the mass-ratio gap described by Suzuki et al. (2018).

As the event was discovered via the MOA alert system, as well as the planet model having a Δχ² < 100 compared to the single-lens fit and data from other observatories supporting the planet model—the event qualifies for inclusion in the extended MOA-II exoplanet microlensing sample. The extended MOA-II sample is an upcoming statistical analysis of cold exoplanets detected by the MOA-II survey and is the expansion of the Suzuki et al. (2016) sample analysis. Future high-resolution angular follow up of the lens and source may help contribute to the population of events that can be used to clarify which of the two mass priors used in this work are more appropriate.

In this work, we presented the analysis of the MOA-2020-BLG-208 gravitational microlensing event including the discovery and characterization of a new planet with an estimated mass similar to Saturn. As a cold Saturn-mass planet, this planet contributes to the evidence supporting the need for revised planetary formation models. The planet also qualifies for inclusion in the extended MOA-II exoplanet microlensing sample. We have provided evidence that the anomaly in the event is best described by a planet in a wide-solution orbit as opposed to other potential models, such as a binary-source model. Our characterization was derived both from light-curve-modeling analysis and galactic-model analysis. Notably, our galactic modeling included results from two potential mass-law priors.

We note that observations used in this analysis were obtained during early phases of the COVID-19 pandemic, which introduced logistical challenges for many observatories. Several collaborating observatories were unable to take measurements during this period, and those that did endured additional obstacles to do so. The MOA project is supported by JSPS KAKENHI grant No. JSPS24253004, JSPS26247023, JSPS23340064, JSPS15H00781, JP16H06287, and JP17H02871. Work by C.R. was supported by a Research fellowship of the Alexander von Humboldt Foundation.

This research uses data obtained through the Telescope Access Program (TAP), which has been funded by the TAP member institutes. W.Zang, S.M., and W.Zhu acknowledge support by the National Science Foundation of China (grant No. 12133005). W.Zhu acknowledges the science research grants from the China Manned Space Project with No. CMS-CSST-2021-A11. This research has made use of the KMTNet system operated by the Korea Astronomy and Space Science Institute (KASI) and the data were obtained at the host site of SSO in Australia. J.C.Y. acknowledges support from N.S.F grant No. AST-2108414. Y.S. acknowledges support from BSF grant No. 2020740. Work by C.H. was supported by the grants of National Research Foundation of Korea (2020R1A4A2002885 and 2019R1A2C2085965). E.B. gratefully acknowledges support from NASA grant 80NSSC19K0291. E.B.'s work was carried out within the framework of the ANR project COLD-WORLDS supported by the French National Agency for Research with the reference ANR-18-CE31-0002. Ł.W., K.A.R., K.K., and P.Z. acknowledge the support from the Polish National Science Centre (NCN) grants Harmonia No. 2018/30/M/ST9/00311 and Daina No. 2017/27/L/ST9/03221 as well as the European Union's Horizon 2020 research and innovation program under grant agreement No. 101004719 (OPTICON-RadioNet Pilot, ORP) and MNiSW grant DIR/WK/2018/12. Y.T. acknowledges the support of DFG priority program SPP 1992 "Exploring the Diversity of Extrasolar Planets" (TS 356/3-1).