MICROLENSING EVENTS BY PROXIMA CENTAURI IN 2014 AND 2016: OPPORTUNITIES FOR MASS DETERMINATION AND POSSIBLE PLANET DETECTION^*

Proxima Centauri (α Centauri C, V645 Cen, GJ 551; Hipparcos parallax 771.64 ± 2.60 mas) is the nearest known star, lying 1.30 pc from the Sun. It is located at a projected angular separation of 22 and a linear distance of 0.07 pc (14,500 AU) from the bright and slightly more distant binary α Cen A+B. The distance, proper motion, and radial velocity (RV) of Proxima are closely similar to those of α Cen A+B, showing that the system forms a wide triple (Wertheimer & Laughlin 2006); however, Proxima's large separation from α Cen A+B means that it has evolved effectively as a single star. Proxima is a low-mass, M5 Ve dwarf and flare star, with an apparent magnitude of V = 11.05, a RV of −22.4 km s⁻¹ (Torres et al. 2006), and a proper motion of 38526 ± 00026 yr⁻¹ in position angle (P.A.) 281461 ± 0036, as measured by Hipparcos (van Leeuwen 2007; also see Benedict et al. 1999 ⁵).

1.1. Mass of Proxima Centauri

1.2. Planets Around Proxima Centauri

In this paper we discuss an alternative method for measuring masses of nearby single stars and for searching for planetary companions, and an upcoming opportunity to apply this method to Proxima Centauri. The special importance of microlensing events produced by very nearby stars was emphasized by Paczyński (1996). For such events the term "mesolensing" has been suggested (Di Stefano 2008), because the large angular Einstein radii of nearby stars, coupled with their typically large proper motions, yield relatively large probabilities of such events occurring. Searches for upcoming close stellar passages of high-proper-motion stars near background sources have been carried out by Salim & Gould (2000) and Proft et al. (2011), who have published lists of candidate events (see also Lépine & Di Stefano 2012, who discuss a specific event).

In the case of microlensing events due to nearby stars, the foreground star will almost always be much brighter than the background source, making it difficult to measure its astrometric shift of the fainter background star. The contamination from the foreground star reduces when the lens-source separation is larger, but then the astrometric signal also reduces, making the measurement again difficult. Measuring the astrometric signal will be easiest if we can find a situation where the centroid shift is large even when the lens–source angular separation is large. Such a situation arises for the very nearest stars, for which the angular Einstein radii are the largest.

To search for such very favorable events, we started with the Luyten Half Second (LHS) catalog (Luyten 1979), a compilation of all known stars with proper motions greater than 05 yr⁻¹. The coordinates in the LHS catalog are only approximate, so we used improved positions and proper motions determined by Lépine & Shara (2005; mostly for northern-hemisphere stars), and the revised positions and proper motions published by Bakos et al. (2002) for the remainder. Unlike the previous studies, we took the parallaxes of the LHS stars into account, if they are available in the SIMBAD database, since parallax significantly alters the circumstances of the events for very nearby stars. We then projected the positions of all ∼5000 LHS stars forward over the next 40 yr and searched for close passages (impact parameter <2'') near background stars contained in the GSC 2.3 catalog⁶ (Lasker et al. 2008).

Of the events that we have found will occur in the next few years, one of the most interesting is the close passage of Proxima Centauri in front of a pair of background source stars of V magnitudes 20 and 19.5, separated by 43. Closest approaches will occur in 2014 and 2016, respectively. Figure 1 shows images of the Proxima field from the Digitized Sky Survey⁷ infrared (IR) and Two Micron All Sky Survey (2MASS; Cutri et al. 2003) surveys, taken in 1997 and 2000, respectively. Proxima is the bright star to the left, marked with a blue circle, and moving rapidly to the right (west–northwest). A second green circle marks the pair of background stars.

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Although it might appear from a cursory inspection of the ground-based images in Figure 1 that close passages must occur frequently, passages as close as ∼15 are actually rare. Moreover, Proxima is exceptional in having an Einstein ring radius as large as 28 mas, because it is so close to the Earth. For more typical proper-motion stars at considerably larger distances than Proxima, only extremely close passages (<03) are of interest from a microlensing standpoint—and they would generally suffer considerably from large differences in brightnesses.

To refine the impact parameters and times of closest approach, we obtained Hubble Space Telescope (HST) images of the field on 2012 October 1. Observations were taken with the UVIS channel of the Wide Field Camera 3 (WFC3) in the F475W, F555W, F606W, and F814W filters. Since the background stars are >8 mag fainter than Proxima, we obtained two sets of observations, a short one with an exposure time of 0.5 s (the minimum allowed exposure time with WFC3) in each filter and a long one with an exposure time of 100 to 200 s. An extra set of short and long exposures were obtained in the F555W filter, which were dithered by about 4'' with respect to the first set. The two long exposures obtained in the F555W filter were used to produce a cosmic-ray cleaned image of the field. Since Proxima is heavily saturated in the longer exposures, care was taken to choose a telescope orientation such that neither the diffraction spikes nor the charge bleeding from Proxima would affect the background sources. Proxima was not saturated in the short exposures in the F475W and F555W filters, but was slightly (≲30%) saturated in the F606W and F814W filters. The short exposures were used for photometry of Proxima itself, and the long exposures were used for photometry of the background stars. Even though we have used a large aperture to take the charge bleeding into account in our photometry in the saturated images (Gilliland 1994) in the F814W and F606W filters, the photometric uncertainties in these filters are expected to be higher. The photometric magnitudes for Proxima (on the Vega-mag system) and the source stars along with the estimated errors are listed in Table 1. We also examined the 2MASS source catalog for the IR magnitudes of the sources. The 2MASS catalog shows a single source at this position, with magnitudes of J = 16.089, H = 15.436, and K_s = 15.764, and uncertainties of about ±0.07 mag. These magnitudes most likely represent the combined light of both background sources.

Table 1. Details of the Source Stars

Parameter	Source 1	Source 2	Proximaa
R.A. (J2000)	14:29:34.693	14:29:34.268
Decl. (J2000)	−62:40:33.46	−62:40:34.91
F475W ("B")	21.26	20.55	12.03
F555W ("V")	20.36	19.89	11.33
F606W ("wide-V")	19.61	19.29	(10.44a)
F814W ("I")	17.78	17.93	(7.25a)
Date of closest approach	2014.80 ± 0.03	2016.16 ± 0.03
	(2014 Oct 20 ± 10 days)	(2016 Feb 26 ± 10 days)
Impact parameter	16 ± 01	05 ± 01

Note. ^aThe magnitudes of Proxima in the F606W and F814W filters are determined from saturated images which are uncertain by ±0.5 mag. All other magnitudes have uncertainty of ±0.05 mag.

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Figure 2 shows our long-exposure HST image taken in the F555W (V) filter. Also shown is the future path of Proxima (green line), calculated using the Hipparcos proper motion and parallax (ESA 1997), and taking into account the stellar positions as observed by HST on 2012 October 1. The uncertainty in Proxima's path is less than the width of the line during the time interval shown here. As the figure shows, Proxima will actually pass between the two background stars labeled as "Source 1" and "Source 2," affording two independent opportunities to measure the relativistic light deflection and search for effects of planetary companions. The closest passages will occur in 2014 October, with an impact parameter of 16 ± 01, and in 2016 February, with an impact parameter of 05 ± 01. This prediction is confirmed through a second set of HST observations that we obtained in 2013 March. Details of the predicted close encounters are given in Table 1. (We have an approved program to obtain further HST observations at 10 different epochs during 2014–2017. The predictions given in Table 1 will improve as we obtain further HST observations, and the improved predictions will be made available at this Web site: http://www.stsci.edu/~ksahu. The exact time of the future HST observations will be adjusted based on results obtained from previous observations, in order to maximize the scientific return.)

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We note from Table 1 that the colors of both sources are similar to or bluer than Proxima, but they are at least 8.5 mag fainter than Proxima. That would imply that the distances to these source stars are >60 pc, much larger than the distance to Proxima. In our current analysis of predicting the closest approaches, we have ignored the parallax and proper motion of the source stars, which are expected to be small. This can be rectified if our future HST observations show any parallax or proper motion.

When a background star (the source) passes within or close to the Einstein ring of a foreground object (the lens), it splits into two images and the combined brightness exceeds that of the source star as depicted in Figure 3. As explained in more detail later, the sources will pass several Einstein ring radii from Proxima itself, but could pass very close to or within the Einstein ring of a planet. We provide a general formalism here, which is valid for small as well as large impact parameters. (For more details, see Schneider et al. 1992; Paczyński 1996; Dominik & Sahu 2000.)

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The angular radius of the Einstein ring of a point lens can be written as

$$\begin{equation} \theta _\mathrm{E} = \left ({ 4\,GM \over {{c^2 d_\pi }} }\right) ^{1/2}, \end{equation} \tag{ 1 }$$

where M is the lens mass and d_π is the parallax distance of the lens as measured with respect to the more distant source, given by

$$\begin{equation} {1\over {d_\pi }} = {1\over D_L} - {1\over D_S}, \end{equation} \tag{ 2 }$$

D_L and D_S being the distances from the Earth to the lens and the source, respectively.

If the angular separation between the lens and the (undeflected) source is Δθ, then the two images of the source will be located at separations of

$$\begin{equation} {\theta }_{\pm } = 0.5 \, \left [u \pm \sqrt{u^2 + 4}\right ] \theta _\mathrm{E} \end{equation} \tag{ 3 }$$

relative to the position of the lens, and

$$\begin{equation} u \equiv \Delta \theta / \theta _\mathrm{E}. \end{equation} \tag{ 4 }$$

Note that the source, the lens, and the two images of the source all lie on a straight line; the positive sign implies the same side as the source from the lens, and the negative sign implies the opposite side. The angular distances of the two images from the original position of the source are given by (θ_± − Δθ).

Figure 3 schematically shows images as the source (shown by the red dots) passes though the Einstein ring (blue circle) of the lens. The two images are shown in black, one outside the Einstein ring (major image), and the other inside (minor image).

The amplifications of the two images of the source are given by

$$\begin{equation} A_{\pm } = 0.5 \, \left [ {{u^2 +2} \over {u \sqrt{4+u^2}}} \pm 1 \right ]. \end{equation} \tag{ 5 }$$

The combined amplification can be written as

$$\begin{equation} A = A_+ + A_- = {{u^2 +2} \over {u \sqrt{4+u^2}}}. \end{equation} \tag{ 6 }$$

For u ≪ 1, the combined amplification can be approximated as

$$\begin{equation} A \simeq {1\over u}, \end{equation} \tag{ 7 }$$

and for u ≫ 1

$$\begin{equation} A \simeq {1 + {2\over u^4}}. \end{equation} \tag{ 8 }$$

The intensity-weighted centroid of the two images is given by

$$\begin{equation} {\theta } = {{A_+ {\theta }_+ + A_- {\theta }_-} \over {A}}. \end{equation} \tag{ 9 }$$

If the two images are unresolved, then the centroid shift with respect to the original position of the source can now be written as

$$\begin{equation} \delta \theta = {\theta } - \Delta \theta = {u \over {u^2 + 2}} \ \theta _\mathrm{E}. \end{equation} \tag{ 10 }$$

The locations of the two images, and of their centroid, as functions of u are shown in the top panel of Figure 4, and the corresponding magnifications are shown in the bottom panel. Note that δθ increases with u when $u < \sqrt{2}$, after which it decreases with u. As the figure shows, when u ≫ 1 (as in the case of Proxima), A₋ ≃ 0 and A ≃ A₊, so that the position of the major image can be taken as the centroid of the combined image, and its amplification as the amplification of the combined image.

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For u ≫ 1, the centroid shift can now be written as

$$\begin{equation} \delta \theta \simeq {\theta _\mathrm{E} \over {u}} = {{\theta _\mathrm{E}^2} \over {\Delta \theta }}. \end{equation} \tag{ 11 }$$

Adopting a distance of 1.3 pc and a mass of 0.12 M_☉ for Proxima, and assuming D_S ≫ D_L in Equation (1), we find that its Einstein ring radius is θ_E ≃ 28 mas. Thus a background source lying at an angular distance Δθ from Proxima will have its apparent position shifted by

$$\begin{equation} {\rm \delta \theta \approx { {(28\,mas)^2 \over {\Delta \theta }} } }. \end{equation} \tag{ 12 }$$

Using Equations (1) and (11), we can express the mass of the lens as

$$\begin{equation} M = {{\theta _\mathrm{E}^2 c^2 d_\pi } \over {4 G}} \simeq {{ \delta \theta \, c^2 d_\pi \Delta \theta } \over 4G}. \end{equation} \tag{ 13 }$$

The angular separation Δθ and the centroid shift δθ can be determined from the observations taken before and during the event. To measure the mass of the lens using Equation (13), the remaining required parameter is the parallax distance d_π, depending only on the difference between the parallaxes of the lens and the source. The parallax of the lens is already known, and our planned HST observations will directly constrain the source parallaxes (expected to be very small as noted above).

For the upcoming close encounters of Proxima Centauri, the sources lie on either side, with impact parameters of 0.5 and 1.6 arcsec, which correspond to u ≃ 18 and 57, respectively. For such large u, the major image will lie within ∼1.5 mas of the undeflected source location. The brightness of the minor image, which will lie within ∼1 mas of Proxima itself, will be <2 × 10⁻⁵ that of the primary image, and thus can be ignored.

In Figures 5 and 6 we plot the centroid shifts of Sources 1 and 2 as functions of time, under the assumption that Proxima has a mass of 0.12 M_☉. The maximum centroid shift for Source 1 is ∼0.5 mas at closest approach, and for Source 2 it is ∼1.5 mas. As seen from Equation (12), the centroid shift scales inversely with the angular separation between Proxima and the source star, so the deflections are already underway as we write! Assuming that the centroid shifts of the two sources can be measured with an accuracy of 200 mas per epoch, Proxima's mass can be measured with an accuracy of ∼5%.

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If Proxima has planetary companions, it may be possible to detect them—and measure their masses—through the extra astrometric shifts of one or both of the background stars (Safizadeh et al. 1999; Han & Lee 2002). Using Equation (1), we can write the size of the angular Einstein ring due to a planet around Proxima as

$$\begin{equation} (\theta _\mathrm{E})_{\rm planet} \simeq 8 \ \left ({M_{\rm planet} \over M_{\rm Jup}}\right)^{1/2} \, {\rm mas}. \end{equation} \tag{ 14 }$$

The maximum possible extra deflection due to a Jupiter-mass planet is $(\sqrt{2}/4) \, \theta _\mathrm{E} \simeq 2.8$ mas, which would occur when the angular separation between the background source's deflected image and the planet is $\sqrt{2} \ \theta _\mathrm{E} \simeq 11$ mas.

Figure 6 shows an example of how the gravitational deflection of Source 2 might be modified by the presence of a planetary companion of Proxima under favorable circumstances. We have assumed a Jupiter-mass companion in a face-on circular orbit with a separation of 0.8 AU, which has a closest approach of 14 mas from the background star. The dashed lines in both panels show the deflections when this hypothetical planet is included in the calculations. The effect of the motion of the planet in its 751 day orbit is included. As the figure illustrates, such a planet would produce a relatively large distortion of the deflection curve by about 0.5 mas, but lasting only about three days.

Let us now calculate the sizes of the regions around Proxima in which planets can be detected through their astrometric signals. If the minimum astrometric deflection that can be measured is δ_min, then the corresponding maximum angular distance between the source and the planet can be written, using Equation (9), as

$$\begin{equation} \phi _{\rm max} = {{(\theta _\mathrm{E})_{\rm planet}^2} \over {\delta _{\rm min}}} \simeq {{(8\,{\rm mas})^2} \over {\delta _{\rm min}}} \, \left ({{M_{\rm planet}} \over {M_{\rm Jup}}} \right). \end{equation} \tag{ 15 }$$

This equation implies, for example, that if δ_min = 0.03 mas (detectable with HST spatial scanning as discussed above), the deflection due to a Jupiter-mass planet would be detectable if the planet passes within 2'' of either background source.

Figure 7 illustrates the spatial regions around Proxima in which planets of various masses will be detectable during the upcoming approaches of the two background stars. Here we show the trajectories of the two stars in a Proxima-centered reference frame. Dashed curves refer to Source 1, and solid curves to Source 2. The sets of blue curves surrounding both trajectories, with separations of ∼2'' from the stars, enclose the regions in which a Jupiter-mass planet is detectable with a deflection exceeding 0.03 mas. The red curves surrounding the two trajectories enclose the regions in which the deflection due to a 10 M_Earth planet is detectable.

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Even though astrometric microlensing provides an exciting possibility of detecting planets around Proxima, the probability of such a detection is small for two reasons. First, the microlensing signal due to a planet lasts only about three days. Second, there is a zone close to Proxima where it becomes difficult to distinguish the signal due to a planet from that due to Proxima itself. Detection of the planetary signal is easier when the planet passes closer to the source star, and is most favorable when the planet passes through a point where its separation from the source is much less than the planet's separation from Proxima.

If Proxima has a planet which passes within a few mas of either source, photometric signatures of microlensing due to the planet may be detectable. As noted earlier, the Einstein ring of a Jupiter-mass planet has a radius of about 8 mas. For a photometric accuracy of 1%, the planetary signature will be detected for an amplification of A ⩾ 1.01, which is achieved if the impact parameter u ⩽ 3.5 (Equation (4)). This requires the planet to lie within an angular distance ζ of the source, where

$$\begin{equation} \zeta = 3.5 \, (\theta _\mathrm{E})_{\rm planet} \simeq 28 \, {\left ({M_{\rm planet}\over {M_{\rm Jup}}} \right)^{0.5}} \, \rm \, mas. \end{equation} \tag{ 16 }$$

Thus any Jovian-mass planet that passes within 28 mas of either background star (corresponding to a linear separation of ⩽0.036 AU) can be detected through precision photometry.

The Einstein-ring crossing time is given by T_E = θ_E/A, where A is the proper motion of the planet relative to the background star. Assuming the proper motion of Proxima for the planet, the crossing time is

$$\begin{equation} (T_E)_{\rm planet} \simeq 2.7 \, {\left ({M_{\rm planet}\over {M_{\rm Jup}}} \right)^{0.5}} \, \rm days. \end{equation} \tag{ 17 }$$

The duration of the photometric event can be written as

$$\begin{equation} T_\mathrm{D} = 2 \, T_\mathrm{E} \, (b^2 - u^2)^{0.5}, \end{equation} \tag{ 18 }$$

where b is the source–lens separation, in units of θ_E, at which the amplification is detectable. An amplification of A = 1.01 corresponds to b = 3.5. Thus T_D can be up to seven times T_E, or, for a Jupiter-mass planet, ∼19 days.

In Figure 7 the green lines surrounding each background source trajectory enclose the regions within which a 1% photometric amplification by a Jovian planet would be detectable. These zones are very small, so the probability of detecting photometric microlensing by a planet is extremely small.

Successful observations of these events will be challenging, because the background stars are ⩾8.5 mag fainter than Proxima and will require accurate astrometry and photometry at separations from Proxima ranging from 05 to ∼2''. Here we discuss the feasibility of the observations from space and from the ground.

7.1. Space-based Astrometry and Photometry

For the upcoming events, the minimum values of Δθ will be 16 and 05, corresponding to maximum deflections of 0.5 and 1.5 mas. Such displacements are routinely measured using cameras on board HST; in fact, astrometric accuracies of 0.2 mas have been achieved in single measurements through direct imaging with the Advanced Camera for Surveys and WFC3 (e.g., Anderson & King 2006; Bellini et al. 2011), and spatial-scanning modes have been developed recently with WFC3 that can achieve astrometric accuracies of about 0.03 mas (Riess et al. 2014).

The fact that these observations can be obtained easily with HST even in the presence of Proxima is illustrated in Figure 8. In order to achieve astrometric accuracy of 0.2 mas per observation, we need a signal-to-noise ratio (S/N) of about 300 for each background star, which can be achieved in an exposure time (WFC3, F555W filter) of 70 s. The left panel shows a 200 s exposure on Proxima taken with WFC3 in F555W filter, which is thrice the required exposure time. The white circle has a radius of 12.5 pixels (05). The right-hand panel shows the radial profile of this star, greatly magnified to show the signal in the wings of the stellar image. The counts drop below 5000 electrons at a radial distance of 12.5 pixels, at which point the background becomes negligible compared to a star with S/N = 300. Since the source star will actually be at a separation of 05 even at closest approach, saturation effects are thus unimportant provided the diffraction spikes and charge bleeding are avoided.

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These events will conveniently occur during the operational lifetime of Gaia, ESA's astrometry mission, recently successfully launched. Gaia can observe down to V ≃ 20, and its astrometric accuracy is expected to be 0.1–0.3 mas depending on the source magnitude (Gare et al. 2010; Prusti 2012; Liu et al. 2012). Assuming Gaia can observe the fainter background star in the presence of the much brighter Proxima, the mass of Proxima should be measured very well.

7.2. Ground-based Astrometry and Photometry

We have shown that the nearest star, Proxima Centauri, will pass close to two background stars in 2014 and 2016, with impact parameters of about 16 and 05. Because Proxima is so near, its angular Einstein ring radius is large (∼28 mas) and will lead to detectable relativistic light deflections of the images of the background stars even at those angular separations. Measurement of the astrometric shifts offers a unique opportunity for the determination of the mass of Proxima with an accuracy of ∼5%.

Although the background stars are ⩾8.5 mag fainter than Proxima, the large contrast is mitigated by the relatively large separations at which the gravitational deflection is still detectable, and well within the capabilities of the HST.

The upcoming events also offer the opportunity to detect and determine the masses of planetary companions, either through additional astrometric shifts, or in rare circumstances through a photometric microlensing event, leading to a brightening of the source star. These events would have durations of a few hours to several days.

We thank Rémi Soummer, Will Clarkson, and Rosanne Di Stefano for useful discussions. Partial support for this research was provided by NASA through grant GO-12985 from the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555. K.S. acknowledges support from the European Southern Observatory, the Institute for Theory and Computation at the Harvard-Smithsonian Center for Astrophysics, and the Institute for Advanced Study at Princeton for sabbatical visits, during which parts of this work were carried out. M.D. is thankful to the Qatar National Research Fund (QNRF), member of the Qatar Foundation, for support through grant NPRP 09-476-1-078.

Facility: HST (WFC3) - Hubble Space Telescope satellite