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Can the masses of isolated planetary-mass gravitational lenses be measured by terrestrial parallax?

Recently Sumi et al. (2011) reported evidence for a large population of planetary-mass objects (PMOs) that are either unbound or orbit host stars in orbits>10 AU. Their result was deduced from the statistical distribution of durations of gravitational microlensing events observed by the MOA collaboration during 2006 and 2007. Here we study the feasibility of measuring the mass of an individual PMO through microlensing by examining a particular event, MOA-2011-BLG-274. This event was unusual as the duration was short, the magnification high, the source-size effect large and the angular Einstein radius small. Also, it was intensively monitored from widely separated locations under clear skies at low air masses. Choi et al. (2012) concluded that the lens of the event may have been a PMO but they did not attempt a measurement of its mass. We report here a re-analysis of the event using re-reduced data. We confirm the results of Choi et al. and attempt a measurement of the mass and distance of the lens using the terrestrial parallax effect. Evidence for terrestrial parallax is found at a 3 sigma level of confidence. The best fit to the data yields the mass and distance of the lens as 0.80 +/- 0.30 M_J and 0.80 +/- 0.25 kpc respectively. We exclude a host star to the lens out to a separation ~ 40 AU. Drawing on our analysis of MOA-2011-BLG-274 we propose observational strategies for future microlensing surveys to yield sharper results on PMOs including those down to super-Earth mass.


Introduction
Three avenues of research have yielded evidence for isolated planetary-mass objects (PMOs) in recent years. Observations of star-forming regions by several groups indicate the presence of isolated objects with masses of a few Jupiter masses (Barrado-y-Navascues et al. 2001, Burgess et al. 2009, Haisch et al. 2010, Marsh et al. 2010, Delorme et al. 2012, Pena Ramirez et al. 2012, Scholz et al. 2012, Beichman et al. 2013, Liu et al. 2013).
Evidence has also been reported by gravitational microlensing for objects having approximately the mass of Jupiter distributed throughout the Galaxy (Sumi et al. 2011).
A surprisingly large number of these objects were inferred, nearly two for every star in the Galaxy. However, the evidence was based on the statistical properties of a large sample of microlensing events. Measurements of the masses of individual PMOs were not attempted.
Most recently, Luhman (2014) reported evidence for a nearby (∼2pc) Y type brown dwarf with temperature ∼250K, mass 3-10M J and age 1-10Gyr. The search by Luhman was aimed specifically at nearby brown dwarfs, and the result suggests a high spatial density of them, comparable to that reported by Sumi et al. (2011) for isolated PMOs with masses ∼ M J . It is possible that the object found by Luhman represents the high mass tail of the distribution found by Sumi et al. Further observations of nearby objects may therefore help to identify the objects found by Sumi et al.
The above results, in particular those by microlensing, raise a number of questions. Did these objects form in-situ, and could they therefore be better classified as sub-brown-dwarfs (Gahm et al. 2013)? Or could they be planets orbiting stars at larger radii than gravitational microlensing is sensitive to (Quanz et al. 2012)? Were they planets that formed around stars, and subsequently underwent ejection through planet-planet or planet-star interactions (Guillochon et al. 2011, Malmberg et al. 2011, Veras & Raymond 2012, Kaib et al. 2013?
Is the high number of putative objects of Jupiter mass accompanied by a larger population of terrestrial mass objects? Or does the high number of apparent detections imply some degree of bias in the statistical procedure used, or contamination by variable stars in the sample masquerading as gravitational microlensing events with low-mass lenses?
In this paper we report a re-analysis of MOA-2011-BLG-274, a microlensing event in which the lens was reported by Choi et al. (2012) as a possible PMO.
Our notation is as follows: The parameters r E , t E , and t 0 denote the Einstein radius, crossing time and time of closest approach between the lens and source stars, and θ E , θ min , and θ S denote the angular Einstein radius, the impact parameter between the lens and source stars in angular coordinates, and the angular radius of the source star respectively. D L and D S denote the distances to the lens and source stars. We define u min = θ min /θ E and ρ = θ S /θ E . Finally, for planetary events, we denote by q, d, and ψ the planet:star mass ratio, projected separation in units of r E and axis relative to the source star track respectively. The fundamental parameter of microlensing, the Einstein radius r E , is defined in Liebes (1964). Choi et al. reported analyses of several microlensing events where the lens geometrically transited the source, i.e. where θ min ≤ θ S or, equivalently, u min ≤ ρ. They referred to these well-aligned events as ones in which the lens 'passed over' the source star. In such events it is possible to measure the angular Einstein radius θ E of the event (Gould 1994, Alcock et al. 1997. Choi et al. found θ E ∼ 0.08 mas for MOA-2011-BLG-274, which is unusually small in comparison to typical microlensing events. They also reported that the Einstein radius crossing time, t E , for the event was unusually small, ∼ 2.7 days. As the Einstein radius for gravitational lensing is proportional to the square root of the mass of the lens, Choi et al. concluded that the lens of the event, i.e. MOA-2011-BLG-274L, could have been a PMO.
Here we extend the analysis of Choi et al. in a number of ways. The photometry for two of the datasets was improved by re-reductions. Secondly, instrument-specific limb-darkening coefficients of the source star were computed and used in the analysis.
Thirdly, an investigation into the possibility of recovering the lens mass and distance through measurement of terrestrial parallax was made following a prediction of Gould (1997). Fourthly, an independent search for satellites orbiting the putative PMO was carried out and the results compared to those of Choi et al. Fifthly, a search was made for a star in the vicinity of MOA-2011-BLG-274L that it might be orbiting. Finally, we discuss future observational strategies that could yield definitive measurements of the masses of individual PMOs and also uncover PMOs of lower mass than those reported by Sumi et al.
Our analysis utilised code based on magnification maps (Abe et al. 2013) that was written independently of the code used by Choi et al. The MicroFUN collaboration 2 responded promptly to the alert. They observed MOA-2011-BLG-274 intensively over a three hour period when the magnification rose from ∼ 40 to a peak value of ∼ 200 and then fell again to ∼ 40. The peak occurred over Australasia and MicroFUN monitored this period with a 0.4m telescope at the Auckland Observatory equipped with a custom red filter, a 0.36m telescope at Kumeu (in the Auckland region) equipped with a similar red filter, a 0.36m unfiltered telescope at the Farm Cove Observatory in Auckland, and a 0.3m unfiltered telescope at the PEST observatory in Perth.
Unfortunately, conditions were cloudy at Mt John during the peak of the event, although the sky was clear immediately prior to and immediately following the peak when the magnification was ∼ 40. Data were recorded with the 1.8m MOA telescope and the custom MOA red filter at these times, and also on previous and subsequent nights. In addition, data were obtained at magnifications ∼10 and lower in the I-band by the OGLE 3 and CTIO 4 1.3m telescopes in Chile. The larger datasets are shown in Figure 1.
To reduce computer runtime, the MOA data were binned into single night segments in the wings of the event. Also, to minimise the effects of possible drifts in the MOA data, all data taken prior to 2010 were excluded.
The data were generally reduced by the groups that supplied them. Thus the MOA and OGLE data were reduced using the codes of Bond et al. (2001) and Udalski (2003).
These codes utilise the difference imaging procedure of Alard andLupton (1998), andWozniak (2000). In the case of the MOA data the online reduction was replaced by an off-line reduction to optimise the precision through improved astrometry. The MicroFUN data were reduced by the DoPHOT code of Schechter et al. (1993) with the exception of the PEST data for which the latter reduction appeared relatively noisy. These data were therefore re-reduced using the more accurate difference imaging pySIS code of Albrow et al. (2009).
The uncertainties in the data from the Auckland Observatory, PEST and MOA telescopes were renormalised to force their contributions to χ 2 to be approximately equal to the numbers of data points. For the MOA data this involved adding uncertainties in quadrature and scaling them. The data from the other telescopes were already reasonably well normalised. A total of 1276 data points were used in the final analysis with 683 by MOA, 335 by OGLE, 113 by Auckland, 35 by Farm Cove, 77 by Kumeu, and 33 by PEST.

Source star
It is customary to identify the source star in a gravitational microlensing event of high magnification from its de-reddened colour (V − I) s,0 and its de-reddened apparent magnitude I s,0 . These are normally determined by recording a few images of the event at high magnification in the V passband. These are combined with contemporaneous or near-contemporaneous images in the I passband to yield an instrumental and reddened (V − I) s,i colour and an instrumental and reddened apparent magnitude I s,i of the source star. These are then converted to non-instrumental and dereddened values (V − I) s,0 and I s,0 by using the position of the red clump on the colour magnitude diagram as a standard.
No V band images of MOA-2011-BLG-274 were taken at high magnification. A modified version by Gould et al. (2010) of the above procedure was therefore used.
Instrumental magnitudes of the source star were obtained from single lens fits to the light curves recorded by the unfiltered PEST telescope, R s,p,i , and the OGLE telescope, I s,o,i .
From these an instrumental colour R s,p,i -I s,o,i = 2.13 ± 0.04 was deduced.
Using field stars, two linear colour-colour relationships were found: one for R p,i -I c,i vs The angular radius θ S of the source star was determined from the above results and the surface brightness relationship of Kervella (2008) 5 . This yielded θ S = 0.87 ± 0.12 µas.
Equation (3) yields the source star radius r S and absolute magnitude M I as r S = 1.47 ± 0.24 r solar , The above results enable the source star to be classified. Recently  An effective temperature for the source star of 5700 ± 200K was estimated from its colour (Bessell et al, 1998). In comparison, Choi et al. (2012) reported a temperature of 6000K in their analysis. Limb-darkening coefficients were obtained from Kurucz's ATLAS9 stellar atmosphere models using the method described by Heyrovsky (2007). In Table 1 we present their values computed specifically for each light curve, taking into account the filter transmission, the CCD quantum efficiency, as well as the extinction towards the Galactic-bulge source. We point out that the values are in good agreement with those of

Choi et al. (2012). They quoted measurements and uncertainties for the Auckland and
Kumeu data only. These are within 1 and 2 standard deviations of ours.

Light curve without parallax
An attempt was initially made to model the data on MOA-2011-BLG-274 in a point source approximation but this proved to be entirely impossible. Very large departures were found that clearly showed the lens in the event had transited the source star. Figure 4 shows the best fit to the data assuming a finite, linearly limb-darkened source at 5700K, with limb-darkening coefficients given in Table 1, but with no allowance made for parallax. The parameters for this fit, and for others with source temperatures of 5500K and 5900K and square-root limb darkening, are given in Table 2. These are compared to earlier results obtained by Choi et al. (2012). The largest difference occurs in the value of t E which is to be expected following the re-reduction of the baseline data by MOA.
As seen from Table 2, the impact parameter u min is considerably smaller than the source size parameter ρ, implying the lens of MOA-2011-BLG-274 transited the source almost perfectly. For a perfect transit, i.e. u min = 0, we would have A max = 2θ E /θ S in the absence of limb darkening (Liebes, 1964). This would imply A max = 2θ E /θ S = 2/ρ = 190.
As seen from Figure 3, the effect of limb darkening is to effectively reduce the radius of the source slightly, and hence to increase A max slightly.
The angular Einstein radius is given by the relationship θ E = θ S /ρ and is included in 3.26d is also anomalously small. Typically t E ∼ 20d (Sumi et al. 2011, Paczynski 1996 and θ E ∼ 0.6mas for a one-third-solar mass lens half way to the bulge. and v t = 2.0 km/s. Table 2. Best fitting parameters without parallax. The above measurements independently suggest a small value for the mass of MOA-2011-BLG-274L, as the Einstein radius is proportional to the square root of the mass of the lens. But a small mass is not assured, as the above measurements do not fix the distance to the lens, D L , and the lens-mass depends on this as well. The magnitude of D L is the subject of the following sections.

Parallax I
The importance of measuring the lens distance D L may be seen if we examine the range of values that M L takes if, as a worst case scenario, we eschew any knowledge of D L .
In that case we may use Einstein's equation for the mass of a lens as a function of θ E and D L as follows It therefore appears ideal for investigation of the measurability of terrestrial parallax. In a previous detection of terrestrial parallax (Gould et al. 2009), A max was 2,500 and t E was 7 days.
The related effect of orbital parallax, in which the Earth's non-rectilinear motion about the Sun is taken into account, can also be used to determine the distance to the lens in microlensing events (Gould 1992, Alcock et al. 1995. Orbital parallax is normally detectable in events with relatively long Einstein times, but t E was anomalously short in MOA-2011-BLG-274. We may therefore anticipate that, if parallax is detectable in MOA-2011-BLG-274, it will have been caused predominantly by terrestrial parallax. The effects of parallax (both terrestrial and orbital) may be quantified by the two-dimensional vector π E with east and north components π E,E and π E,N respectively (Gould 2004). The magnitude of the parallax vector |π E | is defined to be AU/r E wherer E is the radius of the Einstein ring projected back to the observer's plane, and its direction is defined to be the direction of motion of the lens projected onto the observer's plane.
Best fits to the data for MOA-2011-BLG-274 were initially found over a coarse grid of values of the parallax plane with step sizes of 1 in both the easterly and northerly directions. Two minima were found for (π E,E ,π E,N ) at approximately (-3,+13) and (-2,+8) for positive and negative values of u min respectively. The results for linear limb darkening and a source star temperature of 5700K are shown in Figure 5. Fine grids were plotted over smaller regions of the parallax plane with step sizes of 0.1 for π E,E and π E,N for the best fitting model. These yielded the results in Table 3. The best fit light curves with parallax included are shown in Figure 6.  Table 3.
The values of parallax for all models in Table 3 are an order of magnitude larger than found in most previous events. This implies an unusually small value for the Einstein radius projected back to the observer planer E . MOA-2011-BLG-274 thus exhibited unusual behaviour on three counts, an unusually small angular Einstein radius θ E , an unusually short Einstein radius crossing time t E , and an unusually large value of parallax. We note that the larger uncertainty of π E,N compared to that π E,E reflects the longer baseline between the telescopes in the east-west direction compared to the north-south direction.
Uncertainties for u min , ρ, t 0 , and t E were found by marginalising each parameter as shown in Figure 7.
It is clear from Figure 5 that a detection of parallax of moderate significance only has been achieved, with a δχ 2 improvement of 10 in comparison to the solution without parallax. This corresponds to 3σ.
In view of this, additional checks were made. Effects of differential refraction were searched for in the data obtained with the filterless PEST and FCO telescopes by examining the light curves of field stars of similar colour to the source star. The zenith angle for the PEST telescope decreased monotonically from approximately 55 • to approximately 33 • during the main night, but for the FCO telescope it was less than 20 • throughout the night.
Comparison stars for the PEST telescope were found to drift by less than approximately ±5 milli-magnitudes during the night, and for the FCO telescope the drift was less than ±10 milli-magnitudes. No attempt was made to correct for these effects.  Table 3.
The uncertainty of each measurement was taken to be a quarter of the width of the fitted curve 2 sigma above the minium.

Lens distance and mass
Figure 8, which is adapted from Gould (2000), shows the Einstein radius r E projected simultaneously to both the observer planer E and the source planer E . Both of the projected radii were measured. We havẽ where the values of π E,E , π E,N , and ρ are given in the first row of Table 3, and r S is given in Equation (5). The confidence levels are 1 sigma.
It is apparent from Figure 8 that the values ofr E ,r E , and D S jointly determine the Einstein radius r E and the distance to the lens D L . The figure leads to a 'resistor' equation which implies r E = 0.067 ± 0.020 AU. Also, Finally, the mass of the lens M L is given by Einstein's equation written in terms ofr E and The lens of MOA-2011-BLG-274 thus has a mass similar to that of Jupiter with uncertainty arising mainly from uncertainty in the distance to the lens, D L , which itself arose from uncertainty in the measured value of parallax.  Table 3 correspond to a hypothetical point at the centre of the Earth (as in Figure 9). The times of maximum magnification recorded by each telescope according to the best fit to the data (i.e. the first model in Table 3) also assist to visualize MOA-2011-BLG-274.
They were determined from the light curves for each telescope shown in Figure 6 and they are listed in Table 4. They are clearly consistent with an event travelling northwards.
The projected distance between Mt John and the Auckland observatories along the above trajectory at 12 • west of north is 690 km, so at a speed of 39 km/s we anticipate an 18 second difference between the observed peak magnifications, in agreement with the times in Table 4. The peak magnification observed at Farm Cove was higher than that observed at  Figure 6) over that without (Figure 4).
In the present event we expect orbital parallax to be undetectably small as the event occurred at the end of June when the Earth's orbit is nearly perpendicular to the line of sight to the Galactic bulge, and because the Einstein crossing time was short. We therefore expect terrestrial parallax to dominate. The clearest manifestation of the latter is the different times of peak magnification it causes for telescopes at different locations on the Earth's surface. An analysis that focuses on these times may therefore be helpful.
With this in mind, the data for each telescope were individually fitted to single-lens, finite-source, linearly limb-darkened, non-parallax light curves for a sequence of values of t 0 .
In this procedure u min , ρ, and t E were allowed to vary. The results are shown in Figure 10.
These plots enable the effect of parallax to be seen by the naked eye. They were used to extract the peak times recorded by each telescope together with their uncertainties, as given in Table 5.
Comparison of the entries in Tables 4 and 5 shows that the times recorded by the individual observatories agree closely with those determined by the combined analysis in  fit to the data for all telescopes shown in Figure 6.
The peak times were taken to be the mid-times when the magnification was 185 for the given site on the ascending and descending branches of the event. No information is available from the OGLE telescope as it was in daylight at the peak of the event.  Cove observatory where there is a 2σ discrepancy. We assume this occurred as a statistical fluctuation.
The above highlights the accuracy of the timing needed to carry out a successful measurement of terrestrial parallax in events like MOA-2011-BLG-274. The required accuracy is of order a few seconds for the peak of the light curve for each telescope. This is undoubtedly demanding, but we note that the analysis is expected to be relatively immune to small uncertainties in limb darkening, sky transmittance and CCD spectral response of each telescope, and also to small uncertainties in the best values of u min , ρ, and t E used in the fitting procedure. This follows because only the axis of symmetry of each light curve was extracted from the data, and small errors in the above quantities should not affect this appreciably, especially as each telescope observed the ascending and descending branches of the light curve almost symmetrically.
Suppose, as discussed in §5, the lens for the event was actually a brown dwarf at a distance ≥ 6 kpc. Figure 8 then predicts a velocity of the lens in the observer plane ≥ 1050 kms −1 . Whilst such a velocity may be possible with high velocity stars in the bulge, it would imply timing differences between the various telescopes in Table 5 ≤ 1s which appears unlikely.
Checks were made of the sampling procedure used to determine the light curve. The

Host star
The preceding discussion indicates that the lens in MOA-2011-BLG-274L may be an isolated PMO or a sub brown dwarf. In principle, it may also be a planet orbiting a host star at such a large distance that the host did not affect the microlensing light curve appreciably.
To test this possibility simulations were conducted with a star of mass 300 times the mass of the planet, i.e. approximately one-third solar mass or the most likely value, located at various distances out to 1000 Einstein radii from the planet, i.e. out to approximately 70 AU from the planet.
Our simulations, and those in the following section, were conducted using the magnification map procedure described in Abe et al (2013) and the on-line reduction of the images from the MOA telescope. The planet was placed at the origin in the lens plane, and its possible host star with a star:planet mass ratio of q 2 was placed on the y-axis at y 2 . Both q 2 and y 2 were assumed to be large in the sense that q 2 ≫ 1 and y 2 ≫ 1. Thus the host star was assumed to be much heavier than the planet, and its separation from the  planet was assumed to be much larger than the Einstein radius of the planet.
Under these conditions, the point of maximum magnification in the source plane is no longer directly behind the lens. It is shifted by a factor of q 2 /y 2 Einstein radii towards the larger second body (Dominik 1999). Also, it becomes an astroid shaped four-fold caustic, which is transited by the source in the best fit. Figure 12 shows a typical magnification map, a four-fold caustic, when the host star is detectable (q 2 = 300 and y 2 = 400). The horizontal and vertical diameters of the caustic are similar to the diameter of the source star, and the effect of the caustic on the light curve is detectable. For host stars at greater distances with y 2 ≥ 500 the size of the caustic diminishes and it fits completely inside the source star. Its effect on the light curve is then undetectable. Similar behaviour is well-known in the case of lensing by an individual star on the outskirts of a galaxy (Chang and Refsdal, 1979).
The results are shown in

Satellites
Conceivably, isolated PMOs may have satellites orbiting them, sometimes referred to as 'exomoons'. Searches by microlensing have been carried out for such objects, e.g. (Bennett et al., 2013). Choi et al. (2012) already reported a search for satellites to MOA-2011-BLG-274L. Using our magnification map procedure we were able to confirm the results reported by Choi et al. We did not find any evidence for the presence of satellites, and we were able to exclude them in the regions found by Choi et al.
In fact, we derived slightly larger exclusion regions than those reported by Choi et al.
We found a small but finite exclusion zone for satellites with a satellite:host mass ratio of

Discussion and future observations
This is the first time that a direct measurement of the mass of an isolated PMO has been attempted. The masses of PMOs found previously in star forming regions were derived from theoretical cooling curves, and the masses of PMOs found previously by microlensing were deduced using statistical arguments. However, this could necessitate a change of current observing strategies. The critical observations by backyard telescopes of MOA-2011-BLG-274 were made continuously. If future events were observed with larger telescopes, these observations would also need to be made continuously, or nearly continuously, in order to realise a significant improvement in accuracy over the backyard observations reported here.
At present this is not a strategy that is generally employed, but it could be implemented by groups such as the Las Cumbres Observatory Global Telescope Network (LCOGT) 6 .
Certainly the geographical location of the LCOGT telescopes is ideal, with three sites in the southern hemisphere and two in the north.
A second problem is the difficulty of triggering efficiently on scarce, rapid events.  Gould and Yee (2013). With this low rate, a network such as LCOGT would need to trigger efficiently on suitable events.
The fact that this was done successfully in the case of MOA-2011-BLG-274 demonstrates it is possible.
Another possibility would be for a network such as the Korean Microlensing Telescope Network (KMTNet) 7 to monitor two (say) fields alternately rather than the greater number they presently plan to observe (Henderson et al, 2014). This would yield exquisite data on fewer events, but it should include a sample of high-quality transiting PMO events. This option would also enjoy sensitivity to PMOs of lower masses than those reported by Sumi et al. It is now known that bound terrestrial planets outnumber bound giant planets by a factor of a few (Cassan et al. 2012). If a similar distinction applies to isolated PMOs, then the KMTNet may be able to detect PMOs down to super-Earth mass, assuming that their strategy could be modified.

Conclusion
We re-analysed the short time-scale, high-magnification microlensing event MOA-2011-BLG-274 that was monitored by 0.3m-class and larger telescopes. We used re-reduced photometry and confirmed the possible interpretation of the event by Choi et al. (2012) in terms of an isolated PMO. We attempted to carry out a measurement of terrestrial parallax in the event and obtained a result at the 3σ level of confidence. This corresponded to a mass of 0.80± 0.30 M J for the lens of the event.
We proposed observational strategies employing high-cadence observations with 1m-class telescopes to enable higher quality measurements to be made in future events.
These observations would require a concerted effort by the microlensing community but