EXTREME MAGNIFICATION MICROLENSING EVENT OGLE-2008-BLG-279: STRONG LIMITS ON PLANETARY COMPANIONS TO THE LENS STAR

A complete census of planets beyond the snow line will be crucial for testing the currently favored core-accretion theory of planet formation since that is the region where this model predicts that giant planets form. For example, Ida & Lin (2004) find that gas giant planets around solar-type stars preferentially form in the region between the snow line at 2.7 AU and ∼10 AU. While radial velocity and transit searches account for most of the more than 300 planets known to date, microlensing has the ability to probe a different region of parameter space that reaches far beyond the snow line and down to Earth-mass planets. Microlensing is most sensitive to planets near the Einstein ring radius, which Gould & Loeb (1992) showed lies just outside the snow line:

$$\begin{equation} r_{\rm {E}} \simeq 4\left(\frac{M_l}{M_{\odot }}\right)^{1/2} \;\mathrm{AU}, \end{equation} \tag{ 1 }$$

for reasonable assumptions. This sensitivity to planets beyond the snow line is demonstrated by the nine announced planets found by microlensing, which range in mass from super-Earths to Jupiters and more massive objects (Bond et al. 2004; Udalski et al. 2005; Beaulieu et al. 2006; Gould et al. 2006; Bennett et al. 2008; Dong et al. 2009; Gaudi et al. 2008; Janczak et al. 2009).

In high-magnification microlensing events (A ≳ 100), the images finely probe the full angular extent of the Einstein ring, making these events particularly sensitive to planets over a wide range of separations (Griest & Safizadeh 1998). Additionally, because the time of maximum sensitivity to planets (the peak of the event) can be determined in advance, intensive observations can be planned resulting in improved coverage of the event, particularly given limited resources. Even when a planet is not detected, the extreme sensitivity of such an event can be used to put broad constraints on planetary companions.

High magnification events are also useful because it is more likely that secondary effects such as the finite-source effect and terrestrial parallax can be measured (Gould 1997). These effects can be used to break several microlensing degeneracies and allow a measurement of the mass of the lens and its distance. This allows us to determine a true mass of a planet rather than the planet/star mass ratio and a true projected separation rather than a relative one. Thus, in addition to being more sensitive to planets, high-magnification events allow us to make more specific inferences about the nature of the system.

Previous work has empirically demonstrated the sensitivity of high-magnification events to giant planets by analyzing observed events without detected planets and explicitly computing the detection sensitivity of these events to planetary companions. The first high-magnification event to be analyzed in such a way was MACHO 1998-BLG-35 (Rhie et al. 2000). Rhie et al. (2000) found that planets with a Jupiter-mass ratio (q = 10⁻³) were excluded for projected separations in units of the Einstein ring radius of d=0.37–2.70. Since then, many other authors have analyzed the planet detection sensitivity of individual high-magnification events (Bond et al. 2002; Gaudi et al. 2002; Yoo et al. 2004; Abe et al. 2004; Dong et al. 2006; Batista et al. 2009). In particular, prior to the work presented here, the most sensitive event with the broadest constraints on planetary companions was MOA 2003-BLG-32, which reached a magnification of 520 (Abe et al. 2004). Dong et al. (2006) found that this event had sensitivity to giant planets out to d ≲ 4. ⁴⁰

This paper presents the analysis of OGLE-2008-BLG-279, which reached a magnification of A ∼ 1600 and was well covered over the peak, making it extremely sensitive to planetary companions. In fact, as we will show, this event has the greatest sensitivity to planetary companions of any event yet analyzed, and we can exclude planets over a wide range of separations and masses. Furthermore, this event exhibited finite-source effects and terrestrial parallax, allowing a measurement of the mass and distance to the lens. This allows us to place constraints on planets in terms of their mass and projected separation in physical units. We begin by describing the data collection and alert process in Section 2. In Section 3 we describe our fits to the light curve and the source parameters. We then go on to discuss the blended light and the shear contributed by a nearby star in Section 4. Finally, we place limits on planetary companions in Section 5. We conclude in Section 6.

On 2008 May 13 (HJD' ≡ HJD-2,450,000 = 4600.3604), the OGLE collaboration announced the discovery of a new microlensing event candidate OGLE-2008-BLG-279 at R.A. = 17^h58^m3617 decl. = −30°22'084 (J2000.0). This event was independently announced by the MOA collaboration on 2008 May 26 as MOA-2008-BLG-225. Based on the available OGLE and MOA data, μFUN began observations of this event on 2008 May 27 from the CTIO SMARTS 1.3 m in Chile, acquiring observations in both the V and I bands, and the next day identified it as likely to reach very high magnification 2 days hence. This event was monitored intensively over the peak by MOA, the PLANET collaboration, and many μFUN observatories. Specifically, the μFUN observatories Bronberg, Hunters Hill, Farm Cove, and Wise obtained data over the peak of this event (see Figure 1). OGLE-2008-BLG-279 peaked on 2008 May 30 at HJD' = 4617.3481 with a magnification A ∼ 1600.

Zoom In Zoom Out Reset image size

Because there were so many data sets, this analysis focuses on the μFUN data from observatories that covered the peak of the event (μFUN Bronberg (South Africa), Hunters Hill (Australia), Farm Cove (New Zealand), and Wise (Israel)) and PLANET Canopus (Australia) combined with the data from OGLE and MOA which cover both peak and baseline. We used the data from CTIO to measure the colors of the event but not in other analyses. Early fits of the data indicated that the Bronberg data from HJD'4617.0-4617.32 suffer from systematic residuals that are more severe than those seen in any of the other data, so these data were excluded from subsequent analysis.

The data were all reduced using difference imaging analysis (DIA; Wozniak 2000) with the exception of the CTIO data which were reduced using the DoPHOT package (Schechter et al. 1993). The uncertainties in all the data sets were normalized so that the χ²/degree of freedom ∼ 1, and we removed >3σ outliers whose deviations were not confirmed by near simultaneous data from other observatories. The normalization factors for each observatory are as follows: OGLE(1.8), MOA(1.0), μFUN Bronberg(1.4), μFUN Hunters Hill I(2.7) and U(1.5), μFUN Farm Cove(2.1), μFUN Wise(3.8), PLANET Canopus(4.6), and μFUN CTIO I(1.4) and V(2.0).

The data for OGLE-2008-BLG-279 appear to be consistent with a very high magnification, A = 1570 ± 120, single-lens microlensing event. We therefore begin by fitting the data with a point-lens model and then go on to place limits on planetary companions in Section 5. In this section, we describe our fits to the data and address the second-order, finite-source and parallax effects on the light curve.

3.1. Angular Einstein Ring Radius

From the V- and I-band images taken with CTIO both during the peak and after the event, we construct a color–magnitude diagram (CMD) of the event (Figure 2). We calibrate this CMD using stars that are also in the calibrated OGLE-III field. For the source, we measure [I, (V − I)] = [21.39 ± 0.09, 2.53 ± 0.01]. If we assume that the source is in the bulge and thus behind the same amount of dust as the clump, we can compute the dereddened color and magnitude. We measure the color and magnitude of the clump: [I, (V − I)]_cl = [16.48, 2.71]. The absolute color and magnitude of the clump are [M_I , (V − I)₀]_cl = [ − 0.20, 1.05], which at a distance of 8.0 kpc would appear to be [I, (V − I)]_0,cl = [14.32, 1.05]. We find A_I = I_cl − I_0,cl = 16.48 − 14.32 = 2.16 and E(V − I) = (V − I)_cl − (V − I)_0,cl = 1.66. We then calculate the dereddened color and magnitude of the source to be [I, (V − I)]₀ = [19.23, 0.87].

Zoom In Zoom Out Reset image size

The angular Einstein ring radius can be determined by combining information from the light curve and the CMD. Finite source effects in the light curve enable us to determine the ratio of the source size, θ_⋆, to the Einstein radius, θ_E:

$$\begin{equation} \rho _{\star } = \theta _{\star }/\theta _{{\rm E}}. \end{equation} \tag{ 2 }$$

We can then estimate θ_⋆ from the color and magnitude of the source measured from the CMD, and solve for θ_E.

3.1.1. Finite-source Effects

If the source passes very close to the lens star, finite-source effects will smooth out the peak of the light curve and allow a measurement of the source size ρ_⋆. Although finite-source effects are not obvious from a visual inspection of the light curve, including them yields a dramatic improvement in χ². In order to fit for finite-source effects, we first estimate the limb darkening of the source from its color and magnitude. We combine the color and magnitude of the source with the Yale–Yonsei isochrones (Demarque et al. 2004), assuming a distance of D_s = 8 kpc and solar metallicity, to estimate T_eff = 5250 K and log g = 4.5. We use these values to calculate the limb-darkening coefficients, u, from Claret (2000), assuming a microturblent velocity of 2 km s⁻¹. We calculate the linear limb-darkening parameters Γ_V and Γ_I using Γ = 2u/(3 − u) to find Γ_V = 0.65 and Γ_I = 0.47. We use these values in our finite-source fits to the data. We find that a point-lens fit including finite-source effects is preferred by Δχ² of 2647.85 over a fit assuming a point source. We search a grid of u₀ and ρ_⋆ near the minimum to confirm that this is a well-constrained result. We use z₀ = u₀/ρ_⋆ as a proxy for ρ_⋆ following Yoo et al. (2004). The resultant χ² map in the u₀–z₀ plane is shown in Figure 3. Our best-fit value for ρ_⋆ is (6.6 ± 0.6) × 10⁻⁴. For this value of ρ_⋆, z₀ is almost unity, indicating that the source just barely grazed the lens star. The other parameters for our best-fit including finite-source effects are given in Table 1.

Zoom In Zoom Out Reset image size

Table 1. Light Curve Fits

Effects				Fit Parameters
Finite-Source	Orbital Parallax	Terrestrial Parallax	−u0	−Δχ2	t0 − 4617.34 (days)	u0 (θE)	t E (days)	ρ⋆ (θE)	πE,E	πE,N
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)	(11)
✓				0.00	0.00783(7)	6.4(5) × 10−4	111.(9)	6.6(6) × 104	⋅⋅⋅	⋅⋅⋅
✓	✓	✓		164.50	0.00787(8)	6.6(5) × 10−4	106.(9)	6.8(6) × 104	−0.15(2)	0.02(2)
✓	✓	✓	✓	127.97	0.0081(1)	−6.4(6) × 10−4	109.(9)	6.7(6) × 104	0.11(2)	0.09(2)
✓	✓			15.52	0.00784(8)	8.(1) × 10−4	84.(12)	9.(1) × 104	1.5(4)	−0.3(2)
✓	✓		✓	15.51	0.00786(8)	−8.(1) × 10−4	84.(12)	9.(1) × 104	1.5(4)	−0.3(2)
✓		✓		166.40	0.00787(8)	6.9(6) × 10−4	101.(8)	7.2(6) × 104	−0.16(2)	0.03(2)
✓		✓	✓	129.59	0.0081(1)	−6.9(5) × 10−4	102.(8)	7.1(6) × 104	0.11(2)	0.11(3)

Notes. The first 4 columns indicate which effects were included in the point-lens fit. The Δχ² improvement for each fit (Column 5) is given relative to the best fit including finite-source effects but without parallax. There are 5731 data points in the fit light curve. The numbers in parentheses indicate the uncertainty in the final digit or digits of the fit parameters.

Download table as:  ASCIITypeset image

3.1.2. Source Size

We convert the dereddened color and magnitude of the source to (V − K) using Bessell & Brett (1988), and combine them with the surface brightness relations in Kervella et al. (2004) to derive a source size of θ_⋆ = 0.54 ± 0.4 μas. The uncertainty in θ_⋆ comes from two sources: the uncertainty in the flux and the uncertainty in the conversion from the observed (V − I) color to surface brightness. The uncertainty in the flux (i.e., the model fit parameter f_s,I ) is 8.5%, and we adopt 7% as the uncertainty due to the surface brightness conversion. From Equation (2), we find that $\theta _{\rm {E}} = \theta _{\star }/\rho _{\star } = 0.81\pm 0.07 \;\mathrm{mas}$. We also calculate the (geocentric) proper motion of the source: μ_geo = θ_E/t_E = 2.7 ± 0.2 mas yr⁻¹. Because the peak flux (∝f_s,I /ρ_⋆) and source crossing time ($\rho _{\star }t_{\rm {E}}$) are both essentially direct observables, and so are well constrained by the light curve, the fractional uncertainties in θ_E and μ_geo are comparable to the fractional uncertainty in θ_⋆. This result is generally applicable to point-lens/finite-source events and is discussed in detail in the Appendix.

3.2. Parallax

Given that we have a measurement for θ_E, if we can also measure microlens parallax, π_E, we can combine these quantities to derive the mass of the lens and its distance. The mass of the lens is given by

$$\begin{equation} M_l=\frac{\theta _{\rm {E}}}{\kappa \pi _{\rm {E}}}, \quad \kappa \equiv \frac{4G}{c^2 \;\mathrm{AU}} \simeq 8.14 \frac{\mathrm{mas}}{M_{\odot }}. \end{equation} \tag{ 3 }$$

Its distance D_l is

$$\begin{equation} \frac{1\;\mathrm{AU}}{D_l} = \pi _l = \pi _s+\pi _{\rm {rel}}, \end{equation} \tag{ 4 }$$

where π_l is the parallax of the lens, π_s = 0.125 mas is the parallax of the source (assuming a distance of D_s = 8 kpc), and π_rel = θ_Eπ_E.

Microlens parallax is the combination of two observable parallax effects in a microlensing event. Terrestrial parallax occurs because observatories located on different parts of the Earth have slightly different lines of sight toward the event and so observe slight differences in the peak magnification and in the timing of the peak, described by the parameters u₀ and t₀, respectively (Hardy & Walker 1995; Holz & Wald 1996). Orbital parallax occurs because the Earth moves in its orbit during the event, again, changing the apparent line of sight. Gould (1997) argued that one might expect to measure both finite-source effects and terrestrial parallax in extreme high-magnification events. We fit the light curve for both of the sources of parallax, including finite-source effects. Fitting for both kinds of parallax simultaneously yields a Δχ² improvement of 165 (see Table 1). We find ${\mbox{\boldmath $\pi $}}_{\rm E} = (\pi _{\rm {E,E}}, \pi _{\rm {E,N}}) = (-0.15\pm 0.02, 0.02\pm 0.02)$, where π_E,E and π_E,N are the projections of π_E in the east and north directions, respectively.

Smith et al. (2003) showed that for orbital parallax and a constant acceleration, u₀ has a sign degeneracy. This degeneracy may be broken if terrestrial parallax is observed (see also Gould 2004). In the fits described above, we assumed u₀>0. We repeat the parallax fit fixing u₀ < 0. We find that the +u₀ solution is preferred over the −u₀ case by Δχ² = 37 (see Table 1).

We perform a series of fits in order to isolate the source of the parallax signal, i.e., whether it is primarily due to orbital parallax or terrestrial parallax. We first fit the light curve for orbital parallax alone and then fit for terrestrial parallax alone. The results are given in Table 1. For +u₀, the orbital parallax fit gives (π_E,E, π_E,N) = (1.5 ± 0.4, − 0.3 ± 0.2) and a Δχ² improvement of ∼16 over the finite-source fit without parallax. In contrast, the +u₀ fit for terrestrial parallax alone yields Δχ² = 166 and (π_E,E, π_E,N) = (−0.16 ± 0.02, 0.03 ± 0.02). While the orbital and terrestrial parallaxes are nominally inconsistent at more than 3σ, from previous experience (Poindexter et al. 2005) we know that low-level orbital parallax can be caused by small systematic errors or xallarap (the orbital motion of the source due to a companion), so we ignore this discrepancy. From the Δχ² values, it is clear that terrestrial parallax dominates the microlens parallax signal in this event, so any spurious orbital parallax signal does not affect our final results.

We also confirm that the terrestrial parallax signal is seen in multiple observatories, and thus cannot be attributed to systematics in a single data set. To test this, we repeat the fits for parallax excluding the data from an individual observatory. If a data set is removed and the parallax becomes consistent with zero, then that observatory contributed significantly to the detection of the signal. Using this process of elimination, we find that the signal comes primarily from the MOA and Bronberg data sets.

Given the results of these various fits, we conclude that the best fit to the data is for the +u₀ solution, and we include both forms of parallax for internal consistency. Combining this parallax measurement with our measurement of θ_E from Section 3.1, we find M_l = 0.64 ±  0.1 M_☉ and D_l = 4.0 ±  0.6 kpc (π_rel = 0.13 ± 0.02 mas) using Equations (3) and (4).

The centroid of the light at baseline when the source is faint is different from the centroid at peak magnification, indicating that light from a third star is blended into the point-spread function (PSF). The measured color and magnitude of blended light are [I, (V − I)]_b = [17.21 ± 0.01, 2.32 ± 0.02]. Stars of this magnitude are relatively rare, and so the most plausible initial guess is that the third star is either a companion to the source or a companion to the lens. If the former, we can use the values of A_I and E(V − I) we found above to derive the intrinsic color of the blend: [I, (V − I)]_0,b = [15.05, 0.66]. This assumes that the blend is in the bulge at a distance of 8 kpc, giving an absolute magnitude of M_I,b = 0.53 and M_V,b = 1.19. Figure 4 shows this point (open square) compared to solar (Z=0.02) and sub-solar metallicity (Z=0.001) Yale–Yonsei isochrones at 1, 5, and 10 Gyr (Demarque et al. 2004). These isochrones show that the values of [M_V , (V − I)₀]_b may be consistent with a sub-giant that is a couple Gyr old, but a more precise determination of age is not possible since the age is degenerate with the unknown metallicity of the blend.

Zoom In Zoom Out Reset image size

If the blend is a companion to the lens, however, it lies in front of some fraction of the dust. In order to derive a dereddened color and absolute magnitude to this star, we need a model for the dust. We explore this scenario using a simple model for the extinction that is constant in the plane of the disk and decreases exponentially out of the plane with a scale height of H₀ = 100 pc:

$$\begin{equation} A_I(d) = K_1\left[1-\exp \left(\frac{-D\sin b}{H_0}\right)\right], \end{equation} \tag{ 5 }$$

where D is the distance to a given point along the line of sight, b is the Galactic latitude, and K₁ is a constant. We can solve for K₁ by substituting in the value of A_I that we find for the source at 8 kpc. We then model the selective extinction in a similar manner:

$$\begin{equation} E(V-I) = K_2\left[1-\exp \left(\frac{-D\sin b}{H_0}\right)\right], \end{equation} \tag{ 6 }$$

and solve for K₂ using the value of E(V − I) calculated for the source at 8 kpc. From Equations 5 and 6, we can recover the intrinsic color and magnitude of the blend assuming it is at various distances. In Figure 4, we plot a point assuming that the blend is at a distance of the lens, 4.0 kpc. By interpolating the isochrones and assuming a solar metallicity, we find that the blend is consistent with being a 1.4 M_☉ sub-giant companion to the lens with an age of 3.8 Gyr. For comparison, we also plot a line showing how the inferred color and magnitude of the blend vary with the assumed distance.

4.1. Astrometric Offset

4.2. Search for Shear

Because all stars have gravity, if the blend described above lies between the observer and the source, it will induce a shear γ in the light curve. We can estimate the size of the shear using the observed astrometric offset and assuming that the blend is a 1.4 M_☉ companion to the lens:

$$\begin{eqnarray} \hspace{-1pc}\gamma &=& \frac{\theta _{{\rm E},b}^2}{\Delta \theta ^2} = \frac{\kappa \pi _{{\rm rel},b} M_{b}}{\Delta \theta ^2}, \nonumber \\ \hspace{-1pc}&=&6.2\times 10^{-5} \left(\frac{\pi _{{\rm rel}}}{0.13\;\mathrm{mas}}\right)\left(\frac{M_{b}}{1.4\; M_{\odot }}\right)\left(\frac{\Delta \theta }{153\;\mathrm{mas}}\right)^{-2}. \end{eqnarray} \tag{ 7 }$$

Using the 1 σ upper limit on the separation (171 mas), we find a minimum shear of γ = 4.9 × 10⁻⁵ if the blend is a companion to the lens. To determine if this value is consistent with the light curve, we perform a series of fits to the data using binary-lens models that cover a wide range of potential shears. The effect of the shear is to introduce two small bumps into the light curve as the small binary caustic crosses the limb of the source, and this is indeed what we see in the binary-lens models we calculate.

Because the separation between the lens and a companion is large (B = Δθ/θ_E ≫ 1), the shear can be approximated as γ ≃ Q/B², where Q = M_b /M_l is the mass ratio of the companion and the lens. This reduces the number of parameters that need to be considered from three to two: γ and α, the angular position of the blend with respect to the motion of the source. We use a grid search of γ and α to place limits on the shear. For each combination of γ and α, we generate a binary light curve in the limit B ≫ 1 that satisfies Q = γB² and fit it to the data using a Markov Chain Monte Carlo with 1000 links. We bin the data over the peak to reduce the computing time. We compute the difference in χ² between the binary model and the best-fit finite-source point-lens model. Figure 5 shows the results of the grid search overplotted with the upper and lower limits on the shear assuming the blend is a companion to the lens. From this figure, we infer that a shear of 6.2 × 10⁻⁵ is inconsistent with our data since it is in a region where the fit is worse by Δχ²>36.

Zoom In Zoom Out Reset image size

The two minima in the χ² map at γ ∼ 10⁻⁴, α = π/2, π are well defined but appear to be due to a single, deviant data point. Fits to the data with these binary models show improvement in the fit to this data point, but the residuals from these fits for the other data points are large and show increased structure. Thus, we believe these minima to be spurious and conclude that the maximum shear that is consistent with our data (Δχ² ⩽ 9) is γ_max = 1.6 × 10⁻⁵.

Since we have ruled out the scenario where the blend is a companion to the lens, we need to ask what possible explanations for the blend are consistent both with γ_max and with the observed color and magnitude. Given γ_max, we can place constraints on the distance to the blend, D_b , for a given mass. The distance is given by

$$\begin{equation} D_{b} = \frac{1}{\pi _{b}}, \end{equation} \tag{ 8 }$$

$$\begin{equation} \mathrm{where}\quad \pi _{b} = \pi _s +\pi _{{\rm rel},b} = \pi _s + \frac{\gamma (\Delta \theta)^2}{\kappa M_{b}}. \end{equation} \tag{ 9 }$$

If we assume M_b = 1 M_☉, γ = γ_max, and use previously stated values for the other parameters, we find D_b >5.8 kpc. A metal-poor sub-giant with this mass located at or beyond this distance would be consistent with the observed color and magnitude of the blend given the simple extinction model described above. However, other explanations are also possible. For example, if the mass of the blend were decreased, π_b would increase, and a slightly closer distance would be permitted. Thus, we cannot definitively identify the source of the blended light. However, given that γ_max is very small, we can ignore any potential shear contribution in later analysis.

We use the method described by Rhie et al. (2000) to quantify the sensitivity of this event to planets. This approach is used for events such as this one for which the residuals are consistent with a point lens. Rather than fitting binary models for planetary companions to our data as advocated by Gaudi & Sackett (2000), we generate a binary model from the data and fit it with a point-lens model. When the single-lens parameters are well constrained (as is the case with OGLE-2008-BLG-279), these two approaches are essentially equivalent (see the discussion in Gaudi et al. 2002 and Dong et al. 2006). We create a magnification map assuming an impact parameter, d, and star/planet mass ratio, q, using a lens with the characteristics from our finite-source fit. The method for creating the magnification map is described in detail in Dong et al. (2006, 2009). For each epoch of our data, we generate a magnification due to the binary lens assuming some position angle, α, of the source's trajectory relative to the axis of the binary and assign it the uncertainty of the datum at that epoch. As in Section 4.2, we use binned data for this analysis.

For q = 10⁻³, 10⁻⁴, 10⁻⁵,  and 10⁻⁶ we search a grid of d, α and compute the Δχ². Based on the systematics in our data, we choose a threshold Δχ² _min = 160 (Gaudi & Sackett 2000). For Δχ²>Δχ² _min, the fit is excluded by our data, and we are sensitive to a planet of mass ratio q at that location. We repeat the analysis using unbinned data for a small subset of points and confirm that the Δχ² for fits with the unbinned data is comparable to fits with binned data. Figure 6 shows the sensitivity maps for four values of q. These maps show good sensitivity to planets with mass ratios q = 10⁻³, 10⁻⁴,  and 10⁻⁵ and some sensitivity to planets with q = 10⁻⁶. For our measured value of M_l = 0.64 M_☉, a mass ratio of q = 10⁻³ corresponds to a planet mass m_p = 0.67 M_Jup and a mass ratio of q = 10⁻⁶ corresponds to m_p ≃ 2 M_Mars. The results bear a striking resemblance to the hypothetical planet sensitivity of the A_max ∼ 3000 event OGLE-2004-BLG-343 if it had been observed over the peak (Dong et al. 2006). In particular, this event shows nearly uniform sensitivity to planets at all angles α for large mass ratios. The hexagonal shape of the sensitivity map is the imprint of the difference between the magnification maps of planetary-lens models and their corresponding single-lens models (see upper panel of Figure 3 in Dong et al. 2009).

Zoom In Zoom Out Reset image size

Figure 7 shows a map of the planet detection efficiency for this event. The efficiency is the percentage of trajectories, α, at a given mass ratio and separation that have Δχ²>Δχ² _min (Gaudi & Sackett 2000). The efficiency contours are all quite close together because of the angular symmetry described above for the planet sensitivity maps. Because we measure the distance to the lens, we know the projected separation, r_⊥, in physical units:

Zoom In Zoom Out Reset image size

$$\begin{equation} r_{\perp }=d\theta _{\rm {E}} D_l. \end{equation} \tag{ 10 }$$

Since we know M_l , we also know the planet mass, m_p = qM_l . We can rule out Neptune-mass planets with projected separations of 1.5–7.2 AU (d = 0.5–2.2) and Jupiter-mass planets with separations of 0.54–19.5 AU (d = 0.2–6.0). We are also able to detect Earth-mass planets near the Einstein ring, although the efficiency is low. The region where this event is sensitive to giant planets probes well beyond the snow line of this star, which we estimate to be at 1.1 AU assuming a_snow = 2.7 AU(M_⋆/M_☉)² (Ida & Lin 2004). The observed absence of planets, especially Neptunes, immediately beyond the snow line of this star is interesting given that core-accretion theory predicts that Neptune-mass planets should preferentially form around low-mass stars (Laughlin et al. 2004; Ida & Lin 2005).

It is also interesting to consider how the sensitivity of this event to planets compares to the sensitivity of other planet-search techniques. Obviously, because of the long timescales involved, most transit searches barely probe the region of sensitivity for this event. As a space-based mission, the Kepler satellite has the best opportunity to probe some of the microlensing parameter space using transits. Using Equation (21) from Gaudi & Winn (2007), we can estimate Kepler's sensitivity to transits around this star:

$$\begin{equation} m_p = 0.22\left(\frac{{{\rm S}/{\rm N}}}{10}\right)^{3/2}\left(\frac{a}{1\;\mathrm{AU}}\right)^{3/4}10^{0.3(m_V-12)} M_{\mathrm{Earth}}, \end{equation} \tag{ 11 }$$

where (S/N) is the signal-to-noise ratio, a is the semimajor axis of the planet, and m_V is the apparent magnitude of the star. We have assumed that the density of the planet is the same as the density of the Earth and the stellar mass–radius relation R_⋆ = kM^0.8 _⋆ (Cox 2000, p. 389). Kepler is also limited by its mission lifetime of 3.5 yr. For periods longer than this, it becomes increasingly unlikely that Kepler will observe a transit (Yee & Gaudi 2008). This limits the sensitivity to planets within ∼2 AU where the period is less than the mission lifetime. These boundaries are plotted in Figure 7.

For comparison, we can also estimate the sensitivity of the radial velocity technique to planets around a star of this mass assuming circular orbits and an edge-on system. Radial velocity is sensitive to planets of mass

$$\begin{equation} m_p = 8.9 \left(\frac{\sigma _{\rm {RV}}}{1\;\mathrm{m \;s^{-1}}}\right)\left(\frac{{{\rm S}/{\rm N}}}{10}\right)\left(\frac{N}{100}\right)^{-1/2}\left(\frac{a}{1\;\mathrm{AU}}\right)^{1/2} M_{\mathrm{Earth}}, \end{equation} \tag{ 12 }$$

where σ_RV is the precision, and N is the number of observations. The limit of radial velocity sensitivity is plotted in Figure 7 as a function of separation assuming a precision of 1 m s⁻¹. Additionally, we can consider how this microlensing event compares to the sensitivity of a space-based astrometry mission with microarcsecond precision (σ_a = 3 μas):

$$\begin{eqnarray} m_p &=& 6.4 \left(\frac{\sigma _a}{3\;\mu \mathrm{as}}\right)\left(\frac{{{\rm S}/{\rm N}}}{10}\right)\left(\frac{N}{100}\right)^{-1/2}\nonumber\\ &&\times\,\left(\frac{a}{1\;\mathrm{AU}}\right)^{-1}\left(\frac{d}{10\,}\right) M_{\mathrm{Earth}}. \end{eqnarray} \tag{ 13 }$$

We assume circular face-on orbits. We show the limiting mass as a function of the semimajor axis in Figure 7 for 3 μas precision. While these contours encompass a large region of the parameter space, they do not take into account the time it takes to make the observations, which increases with increasing semimajor axis. Furthermore, we only expect this kind of astrometric precision from a future space mission, whereas this event shows that microlensing is currently capable of finding these planets from the ground. This discussion shows that microlensing is sensitive to planets in regions not probed by transits and radial velocity and will be particularly important for finding planets at wide separations where the periods are long. For example, for the semimajor axis a = 4 AU (near the maximum sensitivity shown in Figure 7), the period is P ≃ 10 yr.

The extreme magnification microlensing event OGLE-2008-BLG-279 allowed us to place broad constraints on planets around the lens star. Even with a more conservative detection threshold (Δχ²>160), this event is more sensitive than any previously analyzed event (the prior record holder was MOA-2003-BLG-32; Abe et al. 2004). Furthermore, because we observe both parallax and finite-source effects in this event, we are able to measure the mass and distance of an isolated star (M_l = 0.64 ± 0.10 M_☉, D_l = 4.0 ± 0.6 kpc). Using these properties of the lens star, we convert the mass ratio and projected separation to physical units. We can exclude giant planets around the lens star in the entire region where they are expected to form, out beyond the snow line. For example, Jupiter-mass planets are excluded from 0.54–19.5 AU. Events like this that can detect or exclude a broad range of planetary systems out beyond the snow line will be important for determining the planet frequency at large separations and constraining models of planet formation and migration.

We acknowledge the following support: NSF AST-0757888 (A.G., S.D., J.C.Y.), NASA NNG04GL51G (D.D., A.G., and R.P.), Polish MNiSW N20303032/4275 (A.U.), Korea Astronomy and Space Science Institute (B.-G.P. and C.-U.L.), Creative Research Initiative Program (2009-008561) of Korea Science and Engineering Foundation (C.H.), and David Warren for his support of Mt. Canopus Observatory.

In the present case, the fractional errors in θ_⋆, μ, and θ_E are all very nearly the same, although for somewhat different reasons. Since the same convergence of errors is likely to occur in many point-lens/finite-source events, we briefly summarize why this is the case. We first write (generally),

$$\begin{eqnarray*} &&\theta _{\star } = \sqrt{f_s}/Z, \end{eqnarray*}$$

where f_s is the source flux as determined from the model, and Z is the remaining set of factors, which generally include the surface brightness of the source, uncertainties due to the calibration of the source flux, and numerical constants. Next, we write

$$\begin{eqnarray*} &&\mu = {\theta _{\rm E}\over t_{\rm E}} = {\theta _{\star }\over t_{\star }} = {\sqrt{f_s}\over Z}\,{1\over t_{\star }} \qquad \theta _{\rm E} = {\theta _{\star }\over \rho } = {1\over Z\sqrt{f_s}}\,f_{\rm grand}, \end{eqnarray*}$$

where f_grand ≡ f_s /ρ and t_⋆ ≡ ρt_E. We note that for point-lens events with strongly detected finite source effects, t_⋆ and f_grand are quasi-observables, and so have extremely small errors. For example, if u₀ = 0, then 2t_⋆ is just the observed source crossing time while 2f_grand[1 + (3π/8 − 1)Γ] is the observed peak flux. Even for u₀ ≠ 0, these quantities are very strongly constrained, with errors $\sigma _{f_{\rm grand}} = 0.4\%$ and $\sigma _{t_{\star }}=0.3\%$ in the present case. Since the errors in f_s and Z are independent, the fractional errors in θ_⋆, μ, and θ_E are each equal to $[(1/4)(\sigma _{f_s}/f_s)^2 + (\sigma _Z/Z)^2]^{1/2}$. In the present case, $\sigma _{f_s}/f_s$ is given by the fitting code to be 8.5%, while we estimate σ_Z /Z to be 7%, and therefore find a net error in all three quantities (θ_*, θ_E, and μ) of 8%.