MOA-2010-BLG-073L: AN M-DWARF WITH A SUBSTELLAR COMPANION AT THE PLANET/BROWN DWARF BOUNDARY

The mass function of individual compact objects (brown dwarfs and planets) in the Galaxy remains poorly understood, particularly at the low-mass end. Brown dwarfs are commonly defined as objects with masses between the deuterium- and hydrogen-burning limits (DBL and HBL, respectively), but these can be hard to detect, being intrinsically faint and fading further as they cool over time. Below the DBL, the mass function for individual objects is even more poorly measured. Unbound, free-floating objects of planetary mass have been discovered via direct imaging of clusters (for example, in σ Orionis; Béjar et al. 2012) and in the field (e.g., the 6–25 M_J object reported by Kirkpatrick et al. 2006). Sumi et al. (2011) reported a population of planets that are either unbound or at very wide separations, discovered when their gravity caused short-timescale microlensing events.

At least partially as a result of these poor constraints, the origin of low-mass compact objects remains unclear. Although traditionally thought of as separate classes of objects, planets and brown dwarfs form a continuous scale of mass and are best distinguished by the circumstances of their formation (Burrows et al. 2001; Chabrier et al. 2005, 2011; Sahlmann et al. 2010). Planets form in disks of material orbiting a protostellar object and may subsequently migrate to different period orbits. Brown dwarfs, on the other hand, are considered to be the extreme low-mass end of the star formation process by fragmentation of locally overdense cores caused by turbulence in a cloud (Chabrier et al. 2011), which can themselves form protoplanetary disks (Klein et al. 2003; Scholz et al. 2006). This mechanism may also produce objects of a few Jupiter masses. For a recent review, see Luhman (2012).

The mass function of free-floating low-mass objects is likely to be different from those bound to stars. Marcy & Butler (2000) identified a paucity of brown dwarfs orbiting close (<3 AU) to their host stars, a region where Jovian-mass planets are commonly found. This "brown dwarf desert" may represent the gap between the largest objects that can form in protoplanetary disks and the smallest objects that can concurrently collapse/condense next to a star.

Two different theories have been proposed to explain the formation of Jovian planets in disks (for a review, see Zhou et al. 2012). The core accretion model predicts that planets form from protoplanetary cores, growing up to tens of Jupiter masses (e.g., Mordasini et al. 2009; Baraffe et al. 2010), but predicts few giant planets around M-dwarfs. Higher mass stars are thought to have disks with enhanced surface densities that allow the cores to grow more rapidly (Laughlin et al. 2004), as do disks with a high fraction of dust, leading to enhanced planet formation around high-metallicity stars (Ida & Lin 2004). Alternatively, the model of planet formation via gravitational instabilities in the disk (e.g., Boss 2006) tends to favor the formation of more massive planets, in generally wider orbits.

A number of lines of evidence support the core accretion theory. There is a well-established correlation of increasing planet frequency with stellar metallicity (Santos et al. 2001; Fischer & Valenti 2005; Maldonado et al. 2012). The results of radial velocity surveys imply that there is a dearth of M-dwarf stars with massive, close-in planets. In part, this reflects an observational bias against these faint objects, but the sample is sufficiently large that a real statistical trend is emerging (Cumming et al. 2008; Johnson et al. 2010; Bonfils et al. 2011), for companions with P < 2000 days. Recent spectroscopic and Kepler results have confirmed the prediction of a rapid increase in frequency for planets with small radii (down to 2 R_⊕) and P < 50 days for all spectral types and found that these small planets are several times more common around stars of late spectral type (Bonfils et al. 2011; Howard et al. 2012).

However, the core accretion model has difficulty forming massive planets at large orbital radii, and such systems have been discovered, for example, HR 8799 (Marois et al. 2008). Furthermore, a number of planets have been found orbiting M-dwarf hosts at larger orbital separations, for example, Dong et al. (2006), Forveille et al. (2011), and Batista et al. (2011). These systems may instead form through gravitational instability in the disk, which can account for companions up to several Jupiter masses around M-dwarfs (Boss 2006).

To better understand the formation mechanisms of heavy substellar companions in bound systems, we need to trace the distributions of the physical and orbital properties (such as mass ratio, orbital separation, occurrence frequency) of a significant number of systems. Yet relatively few bound brown dwarf companions have been reported, despite their being easy to detect at close orbital separations (the "brown dwarf desert").

Microlensing offers a complementary window onto brown dwarf and planet formation by probing for cooler companions of all masses in orbital radii between ∼0.2 and 10 AU, separations that are difficult or time-consuming to explore by other methods (Shin et al. 2012b). It can probe the companion mass function down to M- and brown dwarf hosts and is sensitive to companions from nearly equal mass down to terrestrial masses.

Sixteen systems have been published to date,⁶⁷ and thanks to large-scale galactic lensing surveys and efficient follow-up, each season's bulge observing campaign is now producing a regular yield of new discoveries (e.g., Bachelet et al. 2012; Yee et al. 2012; Miyake et al. 2012). Of these 16 companions, 3 are giant planets orbiting M-dwarf stars: OGLE-2005-BLG-071Lb, a 3.8 M_J planet (Dong et al. 2009); MOA-2009-BLG-0387Lb, with an M_P = 2.5 M_J planet (Batista et al. 2011); and MOA-2011-BLG-293Lb, which hosts a 2.4 M_J companion (Yee et al. 2012).

Here we present the newly discovered system, MOA-2010-BLG-073L, an M-dwarf star with a companion whose mass is close to the DBL of ∼12.6 M_J. The discovery and follow-up observations are described in Section 2, and we discuss the variability of the source star in Section 3. We describe our analysis in Section 4, from which we derive the physical properties of the lens in Section 5. Finally, we discuss our findings in Section 6.

The microlensing event, MOA-2010-BLG-073, was first announced by the Microlensing Observations in Astrophysics⁶⁸ (MOA; Bond et al. 2001 and Sumi et al. 2003 on the 1.8 m telescope at Mt. John Observatory, New Zealand) survey on 2010 March 18. A background source star in the Galactic bulge, α = 18:10:11.342, δ = −26:31:22.544 (J2000.0), previously having a mean baseline magnitude of I ∼ 16.5 mag, was discovered to be rising smoothly in brightness consistent with a point-source, point-lens (PSPL) microlensing event. However, on 2010 May 3 the event was found to show an anomalous brightening of ∼0.5 mag and an alert was issued (K. Furusawa 2010, private communication).

Microlensing follow-up teams worldwide—RoboNet-II⁶⁹ (Tsapras et al. 2009), μFUN⁷⁰ (Gould et al. 2006), PLANET⁷¹ (Beaulieu et al. 2006), and MiNDSTEp⁷² (Dominik et al. 2010)—responded to provide intensive coverage of the event for the duration of the anomaly (∼2 days) and monitored the event as it returned to baseline, over the course of the next ∼2 months.

In addition to the MOA data, taken with the wide-band "MOA-Red" filter (corresponding to R+I bandpasses), the event was observed from several other sites in New Zealand. The 0.41 m telescope at Auckland Observatory, the 0.35 m at Kumeu Observatory, and the 0.41 m Possum Observatory all used R-band filters while the 0.304 m Molehill Astronomical Observatory (MAO), the 0.35 m telescope at Farm Cove Observatory (FCO), and the 0.4 m telescope at Vintage Lane Observatory (VLO) all observed it unfiltered. The event was then picked up from three sites in Australia, first in I band by the 1 m Canopus telescope in Tasmania followed by the 2 m Faulkes Telescope South (FTS), where an SDSS-i filter was used. The 0.6 m telescope in Perth also observed in I band. Of the observing sites around longitude zero, the event was imaged from the 1 m telescope at the South African Astronomical Observatory (SAAO) using an I-band filter and in J, H, and K_S by the 1.4 m Infra Red Survey Facility (IRSF), also at SAAO. The 2 m Liverpool Telescope (LT) observed the event in SDSS-i from the Canary Islands.

As darkness fell in the Americas, a number of Chilean telescopes picked up the observing baton: the SMARTS 1.3 m at the Cerro Tololo Interamerican Observatory (CTIO) obtained data in V, I, and H bands with the andicam camera, and the Danish 1.54 m used an I filter. Though in the midst of commissioning the new OGLE-IV camera at the time, the 1.3 m Warsaw telescope also covered the event in I band.⁷³ The 1 m Mt. Lemmon Telescope in Arizona imaged the event in the I band, and in the extreme west, the 2 m Faulkes Telescope North (FTN) in Hawaii used an SDSS-i filter to complete the 24 hr coverage of the event from the Pacific. Table 1 summarizes the data obtained, which are plotted in Figure 1.

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The high density of Galactic bulge star fields and the consequent degree of overlap (or blending) in stellar point-spread functions (PSF) have long since made difference image analysis (DIA) the photometry method of choice among microlensing teams. Both MOA and OGLE make their photometry available to the community, automatically reducing their data with their custom pipelines described, respectively, in Bond et al. (2001) and Udalski et al. (2003). The RoboNet data (from FTN, FTS, and the LT) were reduced with the project's automated data reduction pipeline, which is based around the DanDIA package (Bramich 2008). This software was also later used to reduce data from Canopus, the Perth 0.6 m, the SAAO 1 m, and the H-band data from CTIO, while the DIAPL package was used to process the images from the Danish telescope. The PLANET team released their photometry (produced by the WISIS pipeline) in real time via their Web site, and the Pysis DIA pipeline (Albrow et al. 2009) was used for later re-reduction of these data sets.

MOA-2010-BLG-073 was present in the fields of the OGLE-II and OGLE-III surveys, so the source star's I-band photometric record extends from 1998 to 2006 (Figure 2). OGLE-IV was in the commissioning phase when this event took place. From this excellent baseline it was immediately clear that the source is variable over many-month timescales. This raised the possibility that shorter-term variability might obfuscate the microlensing signal, making it difficult to determine its properties.

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To investigate this possibility, we performed a search for periodicities in the baseline OGLE-II and OGLE-III data, excluding the lensing event, using the ANOVA algorithm (Schwarzenberg-Czerny 1996). Due to the seasonal gaps in the baseline, we analyzed the OGLE-II data in yearly subsets as it is the best sampled, searching for periods between P = 0.5 and 200 days. As Figure 3 demonstrates, there are no significant or persistent periodicities, other than the expected integer multiples of the 1 day sampling alias.

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We then combined the OGLE-II and III data sets in order to search for periods up to P = 4000 days. Figure 2 indicates a slight (∼0.04 mag) magnitude offset between the OGLE-II and III data. This can occur as a residual of OGLE's photometric calibration between the two surveys, but it might also be the result of the intrinsic stellar variation. Therefore, we performed a search for periods between 0.5 and 4000 days based on the combined data both with and without this offset (estimated visually). In both cases, the periodogram was dominated by the window function; the only significant power was found in the peaks marking multiples of the 1 day alias, plus one peak at extreme low frequency corresponding to the finite length of the data set. We conclude that this star is an irregular, long-term variable, most likely as a result of pulsations.

However, there remained the possibility that the star could be irregularly variable on timescales comparable to that of the lensing event. To test this possibility, we binned the OGLE-II light curve on a range of timescales between 2 and 1200 days. To each bin we fitted two functions, one of constant brightness and one with a linear slope, and calculated the weighted rms photometric scatter around each function per bin. We plot the average and maximum rms (calculated over all bins in the light curve) against the width of the bins in time in Figure 4. On timescales shorter than 200 days the rms scatter in the binned light curve is reasonably constant, implying no significant short-term variability. The longer-term trend becomes clearly evident in the constant brightness curves for timescales longer than 400 days. We note that for bin widths between ∼800 and 1200 days, this curve has an rms actually exceeding that of the whole light curve; this is because these bins were sufficiently wide that the first bin included the majority of the data and the most variable sections of the light curve. As the bin widths became longer, they included more data points from the relatively stable section toward the end of the OGLE-II data set, and the rms drops. The deviation of the OGLE light curve from a constant brightness exceeds 3σ for timescales longer than ∼750 days. However, it deviates from a linear slope by ⩽1.6σ for timescales less than 1000 days, so we represent this variation as a gradient in the light curve over the duration of the event.

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The most intuitive way to account for the variation of the source was to measure the gradient of the light curve taken at baseline immediately before and after the event. Unfortunately, only one of the available data sets covered these periods. Fitting a straight-line model (via a nonlinear least-squares Marquardt–Levenberg algorithm) to the MOA 2009 and 2011 season data, we measured a slope of 0.018 mag yr⁻¹. However, the rms scatter in the residuals of this fit was 0.016 mag, making it difficult to properly determine the slope. Additionally, in order to remove this slope from the other data sets, it would be necessary to transform the fluxes measured by each telescope onto the same scale as the MOA data. We attempted this via a linear regression approach but found that significant residuals remained. These contributed to overall higher χ² values when the corrected data were fit with binary lensing models. We therefore adopted the alternative method of incorporating the slope as an additional parameter in our lensing model that we fit to the original, uncorrected data, and we describe this approach in the following sections.

In the analysis of this event, we used the established modeling software developed by S. Dong and C. Han (Dong et al. 2006; Shin et al. 2012a).

4.1. Initial Parameters

As a starting point for our analysis, we needed approximate values for the three parameters of the standard model for a PSPL event (not yet including the slope; this is discussed in Section 4.8): t₀, the time of peak magnification occurring at the closest projected separation between the lens and source, u₀, and the Einstein radius crossing time, t_E. Following standard convention, all distances are quoted in units of the angular Einstein radius, θ_E, of the lens.

To estimate these parameters, we combined all available data sets into a single light curve using the following scaling to take account of the varying degrees of PSF blending from different instruments:

$$\begin{equation} \protect f(t,k) = A(t)f_{s}(k) + f_{b}(k), \end{equation} \tag{ 1 }$$

where f(t, k) is the measured flux of the target at time t from data set k, A(t) is the lensing magnification at that time, f_s(k) is the flux of the source star, and f_b(k) represents the flux of all stars blended with the source in the data set. A regression fit was used to measure f_s and f_b for each data set, producing an aligned light curve. Although the resulting parameter estimates are somewhat different from their "true" values due to the existence of the anomalous deviation on the light curve, they provided a starting point in parameter space.

We note that two additional parameters can contribute to a PSPL model. They are the lens parallax parameters π_{E, N} and π_{E, E} that account for the light-curve deviation caused by the motion of Earth in its orbit over the course of the event. The vector microlens parallax ${{\bm \pi} _{{\rm E}}} = {\rm AU}/\widetilde{r_{{\rm E}}}$, where $\widetilde{r_{{\rm E}}}$ is the Einstein radius projected onto the observer plane and

$$\begin{equation} \protect {{\bm \pi} }_{\rm E} \equiv (\pi _{{\rm E},N}, \pi _{{\rm E},E}) \equiv (\cos {\phi _{\pi }}, \sin {\phi _{\pi }})\pi _{{\rm E}}, \end{equation} \tag{ 2 }$$

where ϕ_π represents the direction of lens motion relative to the source as a counterclockwise angle, north through east. However, this is generally significant only for long-timescale (∼months) events and was included at a later stage (see Section 4.6).

4.2. Finite Source

The sharp spike feature in the light curve is indicative of the source closely approaching or crossing a caustic. In these circumstances, it cannot be approximated by a point light source and must be treated as a disk of finite angular radius, ρ, with wavelength-dependent limb darkening. This is addressed within our software using the ray-shooting approach (Kayser et al. 1986): the path of light rays is traced from the image plane back to the source, taking into account the bending of the trajectory according to the lens equation. If a ray is found to "land" within the radius of the source, its intensity is computed taking limb darkening into account. We derived this from the linear limb-darkening law:

$$\begin{equation} \protect I_{\lambda }(\cos {\phi }) = I_{\lambda }(1)\left[1 - u_{\lambda }(1-\cos {\phi })\right], \end{equation} \tag{ 3 }$$

where I_λ is the intensity of the source at radius ϕ from the center, relative to the central intensity I_λ(1) in the same wavelength, λ, scaled by the coefficient u_λ. While more accurate limb-darkening models are available, they are not commonly used in microlensing analyses due to the complexity introduced by combining data from many sources (Bachelet et al. 2012 discussed this in more detail). The values of u_λ for each passband were calculated from the Kurucz ATLAS9 stellar atmosphere models presented by Kurucz (1979) using the method of Heyrovský (2007). However, within the microlensing community and software, Equation (3) more commonly follows the formalism derived by Albrow et al. (1999):

$$\begin{equation} \protect I_{\lambda } = \frac{F_{\lambda }}{\pi \theta _{*}^{2}} \left[1-\Gamma _{\lambda }\left(1-\frac{3}{2}\cos {\phi }\right)\right], \end{equation} \tag{ 4 }$$

where F_λ is the total flux from the source in a given passband and ϕ is the angle between the line of sight to the observer and the normal to the stellar surface. The limb-darkening coefficient, Γ_λ, is related to u_λ by

$$\begin{equation} \protect \Gamma _{\lambda } = \frac{2u_{\lambda }}{3-u_{\lambda }}. \end{equation} \tag{ 5 }$$

The values of u_λ and Γ_λ applied for each data set are presented in Table 1. The lensing magnification is then computed as the ratio of the number of rays reaching the source plane relative to the number in the image plane. This approach is only required while the source is close to the caustic. At larger separations, the software employs a semi-analytic hexadecapole approximation to the finite source calculation to improve computation speeds (Pejcha & Heyrovský 2009; Gould 2008).

4.3. Standard Binary Model Grid Search

4.4. Optimized Standard Binary Model

4.5. Normalization of Photometric Errors

When fitting microlensing events, the reduced χ² of the fit on a per data set basis, χ²_red, typically produces a range of values both less than and exceeding the expected unity value. This can occur as different groups have slightly different ways of estimating photometric errors, but it can lead to over- or underemphasis being placed on particular data sets during the modeling process.

A common technique to address this issue is to arrive at a complete model for the event and then use this model to renormalize the original photometric errors of each data set, e_orig, according to the expression

$$\begin{equation} \protect e_{{\rm new}} = a_{0}\sqrt{e_{{\rm orig}}^{2} + a_{1}^{2}}. \end{equation} \tag{ 6 }$$

We first conducted the sequence of models described in the following sections in order to find the best model for the event. We then set the coefficients a₀, a₁ such that the χ²_red relative to that model equaled unity; the adopted values are given in Table 1. We then repeated our MCMC fitting process, starting with the standard binary model and systematically adding parameters in to determine the extent of improvement in the model χ² in each case. We compare models in Tables 2 and 3, and we plot the residuals (data − model) in Figure 1. In the following sections, the χ² values given are those post-renormalization.

Table 2. The χ² for the Best-fitting Model in Each Class, Comparing u₀ > 0 and u₀ < 0 Solutions

Model	χ2
	u0 > 0	u0 < 0
Standard	6331.950	6331.950
Standard+Parallax	6064.773	6099.487
Standard+Parallax+Orbital Motion	5544.071	5690.003
Standard+Parallax+Orbital Motion with Slope	4782.367	4802.606
Standard+Slope	5334.378	5334.378

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Table 3. The Best-fitting Parameters

Parameter	Standard	Standard	Standard	Standard+Parallax	Standard+Parallax
(Units)		+Slope	+Parallax	+Orbital Motion	+Orbital Motion+Slope
χ2	6331.950	5334.378	6064.773	5544.071	4782.367
Δχ2a	1549.583	552.011	1282.406	761.704	0.0
t0 (HJD')b	5344.32	5344.38	5344.47	5344.83	5344.69
	±0.01	±0.01	±0.02	±0.03	±0.02
u0	0.4089	0.403	0.403	0.381	0.386
	±0.0009	±0.001	±0.001	±0.001	±0.001
tE (days)	43.82	44.84	43.49	43.4	44.3
	±0.08	±0.09	±0.09	±0.1	±0.1
s0	0.7692	0.7725	0.7717	0.7792	0.7750
	±0.0005	±0.0005	±0.0006	±0.0007	±0.0007
q	0.0705	0.0677	0.0695	0.0683	0.0654
	±0.0005	±0.0005	±0.0006	±0.0006	±0.0006
α0	0.180	0.171	0.198	0.297	0.221
	±0.003	±0.003	±0.003	±0.006	± 0.007
ρ	0.01963	0.01912	0.01931	0.0163	0.0165
	±0.00006	±0.00007	±0.00008	±0.0001	±0.0001
πE, N			0.18	0.96	0.37
			±0.02	±0.04	±0.05
πE, E			−0.124	0.09	0.01
			±0.007	±0.01	±0.01
ds/dt (yr−1)				0.53	0.49
				±0.02	±0.02
dα/dt (yr−1)				−1.21	−0.37
				±0.06	±0.08
Slope (mag yr−1)		−0.0160			−0.0153
		±0.0004			±0.0004

Notes. ^aImprovement in χ² relative to that of the best-fitting model. ^bAll timestamps are abbreviated to HJD' = HJD −2,450,000.0.

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4.6. Parallax

4.7. Lens Orbital Motion

4.8. Sloping Baseline

4.9. Second-order Effects

4.10. Final Model

All our models are compared in Table 2, and the parameters of the best-fitting models are presented in Table 3 and Figure 1. We plot the χ² for each link in the MCMC chain for all parameters against one another in the best-fitting model in Figure 5. This plot was used as a diagnostic throughout the fitting process, as any correlations between parameters display distinct trends as the chain moves toward the minimum. The caustic structure changed during the course of the event, so in Figure 6 we show the structure at two distinct times: the first at the time of the first caustic crossing during the anomaly, and the second at the time of closest approach.

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The purpose of this model is to ultimately arrive at the physical parameters of the lens and source, which can be achieved using the known relations between these and the lensing parameters obtained from the modeling.

Chiefly of interest is the mass of the lensing system, M_{L, tot}, which can be determined explicitly for events where parallax is measurable provided that the angular extent of the Einstein radius, θ_E, is known from

$$\begin{equation} M_{L,{\rm tot}} = \frac{c^{2}}{4G}\widetilde{r}_{{\rm E}}\theta _{{\rm E}} = \frac{c^{2}{\rm AU}}{4G}\frac{\theta _{{\rm E}}}{\pi _{{\rm E}}}, \end{equation} \tag{ 7 }$$

where $\widetilde{r}_{{\rm E}}$ is the Einstein radius projected from the source onto the observer's plane. The model parameter ρ represents the angular size of the source star θ_S in units of the angular Einstein radius θ_E. We derive this from the crossing time taken for the source to travel behind the lens, t_* = μθ_S, where μ is the relative source–lens proper motion. ρ can then be written as

$$\begin{equation} \rho = \frac{t_{*}}{t_{{\rm E}}} = \frac{\theta _{S}}{\theta _{{\rm E}}}. \end{equation} \tag{ 8 }$$

These parameters also yield the distance to the lens,

$$\begin{equation} \frac{{\rm AU}}{D_{L}} \equiv \pi _{L} = \theta _{{\rm E}}\pi _{{\rm E}} + \frac{{\rm AU}}{D_{S}}, \end{equation} \tag{ 9 }$$

which in turn yields the projected separation between the lens components,

$$\begin{equation} a_{\perp } = s_{0}D_{L}\theta _{{\rm E}}, \end{equation} \tag{ 10 }$$

and the relative proper motion between lens and source, when combined with t_E,

$$\begin{equation} \mu =\frac{\theta _{{\rm E}}}{t_{{\rm E}}}. \end{equation} \tag{ 11 }$$

The appreciable lens orbital motion during this event also allows us to test whether the companion object is bound to the primary lensing mass, via the ratio of its kinetic to potential energy:

$$\begin{equation} {\rm KE}/{\rm PE} = \frac{(s_{0}R_{{\rm E}})^{3}\gamma ^{2}}{8\pi ^{2}M_{L}}, \end{equation} \tag{ 12 }$$

where γ relates the two lens orbital parameters, γ² = (ds/dt/s₀)² + (dα/dt)², and where the masses are in units of M_☉, the distances in AU, and time measured in years.

However, these expressions include two key terms that are as yet unknown: θ_S, the angular source radius, and D_S, the distance to the source. In order to extract the physical characteristics of the lens, we therefore turned our attention to the characteristics of the source.

5.1. Source Star

Long-exposure V, I images were acquired by the CTIO 1.3 m at several epochs, which enabled us to plot the color–magnitude diagram for the field including the source star (Figure 7). By observing the event at different levels of lensing magnification, these data can be incorporated into the model that yields the source and blended light fluxes, f_s, f_b for those data, and hence the instrumental magnitudes and colors of the source and blend. But we note that these uncalibrated fluxes also suffer from the high degree of extinction along the line of sight to the Galactic bulge. To calculate the dereddened color, (V − I)_{S, 0}, and magnitude, I_{S, 0}, of the source, we needed to calibrate the instrumental fluxes f_s and f_b relative to a standard candle.

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Figure 7 clearly shows a locus of stars centered at I_{RC, inst} = 15.821 ± 0.05 mag, (V − I)_{RC, inst} = −0.350 ± 0.05 mag. This consists of a clump of red giant stars, for which stellar theory predicts a stable absolute luminosity, varying only slightly with age and chemical composition. Their frequent occurrence makes these objects useful as standard candles. Stanek et al. (1998) established photometric calibrations for red clump magnitudes, which were later refined by Alves et al. (2002) using Hipparcos data. Most recently, Nataf et al. (2012) were able to measure the dereddened apparent magnitude of the red clump stars at the Galactocentric distance, I_{RC, 0} = 14.443. By mapping the distances, D_RC, to red clump stars in the Galactic bar as a function of Galactic longitude, l, they found an apparent viewing angle on the bar of ϕ_Bar = 40°,

$$\begin{equation} \frac{R_{0}}{D_{{\rm RC}}} = \frac{\sin {\phi + l}}{\phi } = \cos {l} + \sin {l}\cot {\phi }, \end{equation} \tag{ 13 }$$

where Nataf et al. (2012) measured R₀ to be 8.20 kpc. For the field of MOA-2010-BLG-073 (Galactic coordinates: l = 481030, b = −350131), we derive D_RC = 7.48 kpc, and we assumed that the source star lies behind the same amount of dust as the red clump stars and at the same distance. Scaling the dereddened apparent magnitude of the red clump stars, I_{RC, 0}, appropriately for the slightly closer distance of the stars in this field, I_{RC, app}= I_{RC, 0}+ ΔI, where

$$\begin{equation} \Delta I = 5\log _{10}{R_{0}/D_{{\rm RC}}}. \end{equation} \tag{ 14 }$$

We found ΔI = 0.20 mag, and so the distance modulus to the red clump and the source in this field is I_{RC, app}= 14.24 mag. Bensby et al. (2011) determined the intrinsic (V − I)_{RC, 0} = 1.09 mag for red clump stars, so we were able to derive their absolute V magnitude of M_{V, RC, O} = 0.97 mag. Combining these results with the measured Δ(V − I)_inst and ΔV_inst between the source and red clump in the CTIO data, we then derived the dereddened color (V − I)_{S, 0} and magnitude I_{S, 0} of the source, summarized in Table 4.

Table 4. The Physical Parameters of the Lensing System and Source Star, Derived from the Model Including Slope, Parallax, and Orbital Motion, plus Color Information

Parameter	Units	Value
θS	μas	9.143 ± 0.792
θE	mas	0.557 ± 0.09
RS	R☉	14.7 ± 1.3
ML, 1	M☉	0.16 ± 0.03
ML, 2	MJ	11.0 ± 2.0
ML, tot	M☉	0.17 ± 0.03
DL	kpc	2.8 ± 0.4
a⊥	AU	1.21 ± 0.16
KE/PE		0.079
Proper motion	mas yr−1	4.60 ± 0.4
IS, inst	mag	15.554 ± 0.007
VS, inst	mag	15.335 ± 0.007
(V − I)S, inst	mag	−0.22 ± 0.01
IRC, inst	mag	15.821 ± 0.05
(V − I)RC, inst	mag	−0.350 ± 0.05
IRC, 0	mag	14.443
(V − I)RC, 0	mag	1.09
IS, 0	mag	13.976
VS, 0	mag	15.197
(V − I)S, 0	mag	1.221 ± 0.051
(V − K)S, 0	mag	2.852
KS, 0	mag	12.345
J (2MASS)	mag	13.686 ± 0.053
H (2MASS)	mag	12.926 ± 0.057
KS (2MASS)	mag	12.642 ± 0.054

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Bessell & Brett (1988) provided a relationship between (V − I) and (V − K) color indices, and Kervella et al. (2004) related (V − K) to angular radius for giant and dwarf stars. Having thus determined D_S and θ_S, we derived the physical parameters of the lensing system, which are also summarized in Table 4. The color and large source radius of 14.7 ± 1.3 R_☉ imply that this star is a K-type giant, which is consistent with the observed photometric variability. Jorissen et al. (1997) found that red giant stars with spectral types later than early-K are all variable, with amplitudes increasing from microvariability to several magnitudes toward cooler temperatures and timescales from days to years. Kiss et al. (2006) note that irregular photometric variability may be caused by large convection cells, or may actually be the result of a number of simultaneous periodic pulsation modes, and many examples have been identified from time-domain surveys (e.g., Wray et al. 2004; Woźniak et al. 2004; Eyer & Blake 2005; Ciechanowska et al. 2006). We note that the star was detected by the Two Micron All Sky Survey (2MASS; Skrutskie et al. 2006) as source 2MASS J18101138−2631226 with colors (J − H) = 0.76 ± 0.078 mag and (H − K_S) = 0.284 ± 0.079 mag. Although the MASS field is crowded, the star's PSF is distinct and their photometry for it has the best-quality AAA flag. These colors are consistent with a giant star and with our derived value for K_{S, 0} = 12.345 mag, when we take into account AK_S = 0.24 mag from the vvv survey (Gonzalez et al. 2012).

Nataf et al. (2012) explain that their value of our viewing angle of the Galactic bar is a "soft upper bound" because the distance along the plane to the greatest density of stars along the line of sight to a triaxial bar structure is less than the distance to the structure's major axis on the far side and greater on the near side. As the physical parameters derived for MOA-2012-BLG-073 are somewhat dependent on the value of our viewing angle of the Galactic bar and Nataf et al. (2012) quote consistent results with values as low as 25°, we explored the potential impact of this on our results. A reduced viewing angle would produce a smaller distance to the source, changing its dereddened magnitude and color. The resulting increase in source radius produces a corresponding reduction in the value of D_L and increases in the lens masses. However, we found that the physical parameter values do not change by more than the errors quoted in Table 4, implying that this is not the dominant source of uncertainty. Finally, we computed the physical parameters derived from the best-fitting u₀ < 0 model for comparison and found that the masses derived changed by <1σ.

Earth's movement during this relatively long timescale (t_E = 44.3 days) microlensing event resulted in a gradual shift in our perspective on the lensing system, breaking the symmetry of the caustic. Meanwhile, the change in projected separation of the lensing objects modified the shape of the caustic just as the source's trajectory happened to pass close by. If not for these subtle variations, it can be seen from Figure 6 that a source trajectory u₀ > 0 would produce exactly the same light curve as a u₀ < 0 trajectory. As it is, for this event, the u₀ > 0 solution best explains our observations, though the difference in χ² relative to the corresponding u₀ < 0 solution is very small (Δχ² = 20.2) compared with the χ² of both fits.

The measurable parallax signature enables us to determine the masses of the lensing bodies. The primary lensing object has M_{L, 1} = 0.16 ± 0.03 M_☉, making it an M-dwarf star. The companion's mass is M_{L, 2} = 11.0 ± 2.0 M_J. This places it in the brown dwarf desert (see Figure 8), though we note that this traditionally refers to close-in companions, and since microlensing and direct imaging measure only the projected separation, we know only their minimum orbital semimajor radii. Regardless, it is clear that MOA-2010-BLG-073L b is close to the mass threshold for deuterium burning (∼0.012 M_☉ = 12.6 M_J) quoted as the nominal boundary between planets and brown dwarfs established by the IAU (Chabrier et al. 2005). So what kind of object is it?

No further orbital or metallicity information is available for this event, which might have shed light on its evolutionary history. Theoretical isochrones predict that a star of this low mass will not have lost a significant amount of material over its lifetime, so we can say that MOA-2010-BLG-073L b formed as a high mass ratio binary. Models of protoplanetary disks have the expectation that disk mass, M_D, will scale linearly with star mass M_* (Williams & Cieza 2011), M_D/M_* ∼ 1% at young ages, but this would limit M_D—and hence M_{P, max}—to ∼1–10 M_J in the case of MOA-2010-BLG-073L. So it seems questionable whether such a massive companion could have formed in a protostellar disk, via either core accretion or gravitational instability.

Bonnell et al. (2008) and Kroupa & Bouvier (2003) discuss a number of mechanisms that can produce an M-dwarf/brown dwarf binary following gravitational fragmentation in a molecular cloud.

• 1.  
  Embryo rejection model. The nascent binary was ejected from a dynamically unstable multiple protostellar system, leading to the loss of its accretion envelope before the secondary component could acquire enough mass to become a star.
• 2.  
  Collision model. The binary was prematurely ejected from a larger protostellar system by the close passage of another star.
• 3.  
  Photoevaporation model. The accretion envelope around the binary was photoevaporated by the nearby presence of a massive star in the birth cluster before the secondary could accrete enough mass to become a star.
• 4.  
  "Star-like" model. The object formed as a normal stellar binary with low-mass components.

The embryo rejection model predicts (Bate et al. 2002) that the maximum separation of binaries surviving this process is a_max ∼ 10–20 AU. We cannot rule out this scenario as we only measure the projected separation of the lens, which is nevertheless below a_max. Since MOA-2010-BLG-073L is a field object, we have no information regarding the proximity of other stars during its birth, so the collisional and photoevaporation models are equally plausible. However, we note that Whitworth & Stamatellos (2006) derived a minimum mass for primary fragmentation components of 0.004 M_☉ ∼ 4.2 M_J. As this threshold is below the mass of MOA-2010-BLG-073L b, the simplest explanation is that the companion is an extremely low mass product of star formation. However, we note that the mass ratio of this system would match that of an object in the brown dwarf desert if the host were a more massive star. Recent surveys (e.g., Dieterich et al. 2012; Evans et al. 2012, though restricted to massive companions with a > 10 AU) hint that the brown dwarf desert may extend to M-dwarfs and beyond 3 AU, which would make MOA-2010-BLG-073L a rare example of its type.

R.A.S. is grateful to the Chungbuk University group and especially to C. Han and J.-Y. Choi for their advice and hospitality in Korea, where much of this work was completed. R.A.S. also expresses appreciation for Y. Tsapras, K. Horne, M. Hundertmark, S. Dong, and P. Fouqué for many useful discussions. K.A., D.M.B., M.D., K.H., M.H., C.L., C.S., R.A.S., and Y.T. thank the Qatar Foundation for support from QNRF grant NPRP-09-476-1-078. Work by C. Han was supported by the Creative Research Initiative Program (2009-0081561) of the National Research Foundation of Korea. C.S. received funding from the European Union Seventh Framework Programme (FP7/2007-2013) under grant agreement no. 268421. The OGLE project has received funding from the European Research Council under the European Community's Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement no. 246678 to AU. This publication makes use of data products from the Exoplanet Encyclopeida and the Two Micron All Sky Survey, which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center/California Institute of Technology, funded by the National Aeronautics and Space Administration and the National Science Foundation. M.O.A. acknowledges funding JSPS20340052, JSPS22403003, and JSPS19340058. T.C.H. gratefully acknowledges financial support from the Korea Research Council for Fundamental Science and Technology (KRCF) through the Young Research Scientist Fellowship Program. C.U.L. and T.C.H. acknowledge financial support from KASI (Korea Astronomy and Space Science Institute) grant number 2012-1-410-02. D.R. (boursier FRIA), F.F. (boursier ARC), and J. Surdej acknowledge support from the Communauté francaise de Belgique-Actions de recherche concertées-Académie universitaire Wallonie-Europe. A. Gould acknowledges support from NSF AST-1103471. B. S. Gaudi, A. Gould, and R. W. Pogge acknowledge support from NASA grant NNX12AB99G. Work by J. C. Yee is supported by a National Science Foundation Graduate Research Fellowship under grant No. 2009068160. S.D. is supported through a Ralph E. and Doris M. Hansmann Membership at the IAS and NSF grant AST-0807444.

Facilities: FTN - Faulkes Telescope North, Liverpool:2m - Liverpool 2 meter telesope at Observatorio del Roque de los Muchachos, CTIO:1.3m - Cerro Tololo Inter-American Observatory's 1.3 meter Telescope, MtCanopus:1m - Mt. Canopus Observatory 1m Telescope, Elizabeth - South Africa Astronomical Observatory's 1m Elizabeth Telescope, Danish 1.54m Telescope -