NEARBY PLANETARY SYSTEMS AS LENSES DURING PREDICTED CLOSE PASSAGES TO BACKGROUND STARS

Equation (2) demonstrates that, for planets that are close to us, the Einstein radius, R_E = D_L θ_E, can be relatively small.⁵ The exact value of R_E depends on both the lens mass and on the value of D_L. Nevertheless, for a wide range of stellar masses and distances to the lens (within a few hundred parsecs), R_E can be smaller than an AU.

For a face-on circular orbit, the value of α is constant and is equal to the semimajor axis. In this case,

$$\begin{equation} P_{{\rm orb}}=(26.1\,{\rm days})\, \alpha ^\frac{3}{2} \left (\frac{M_\ast }{0.1\, M_\odot }\right)^\frac{1}{4} \left (\frac{D_L}{8\,{\rm pc}}\right)^\frac{3}{4} \left (1-\frac{D_L}{D_S}\right)^\frac{1}{4}. \end{equation} \tag{ 3 }$$

In general, the orbit will not be face-on, and may also be elliptical. Thus, α = α(t), and the semimajor axis is p(t) α(t), where the instantaneous value of p(t) can be larger than or smaller than unity, depending on the orbital inclination, eccentricity, and phase. For the purpose of considering specific well-defined cases, we will use circular face-on orbits as examples. We note, however, that when deriving models to fit observed light curves, the full range of possible orbits must be considered.

Short orbital periods mean that the orbital phase can change significantly during the interval when the lens is close to the source. For example, if R_E = 0.08 AU, then the time taken by a 10 km s⁻¹ lens to move through a distance of 2 R_E is 27.7 days. Thus, for a lens with M_* = 0.1 M_☉, located 8 pc away, a full orbital period can occur (if the orbit is circular and face-on, with α = 1) during the time required by the lens to move across an Einstein diameter. More revolutions can occur if α is smaller. In fact, because the ratio of P_orb to R_E is proportional to $D_L^{1/4}$, and to $M_\ast ^{-{1/4}}$, there are wide ranges of parameters for which orbital motion is important.

It is therefore more likely that regions in which the isomagnification contours are perturbed from the point-lens form will pass in front of the source, increasing the probability of planet detection. In some cases, the region in the lens plane with magnification deviations can pass over the source star more than once. The repetition of deviations makes them more likely to be detected and correctly identified.

Overall, the probability of planet detection can be significantly larger for HPMSs than it is for lens stars located several kiloparsecs away, the typical location for the lenses discovered by the lensing teams (e.g., the Optical Gravitational Lensing Experiment (Udalski 2003) and MOA (Bond et al. 2001) teams, both of which are presently active).

As a planet orbits a star, the star wobbles in its own smaller orbit. When the star serves as a lens, small changes in its position can make a measurable difference in the magnification it produces in a background star. The magnification enhances the effects of stellar wobble, making it more readily detectable.

In units of the Einstein angle, the wobble of the star has amplitude

$$\begin{equation} \delta = \frac{m_p}{M_J} \, {\alpha }\times f, \end{equation} \tag{ 4 }$$

where M_J is the mass of Jupiter, and the value of f depends on how much of the orbit is executed during the time when the magnification is detectable; if a full orbit is executed f could be as large as about 2, or much smaller, if the shift in orbital phase during the event is small.

Whether the stellar wobble is large enough to produce a detectable change in magnification depends on how close on the sky the source star is to the lens. Let β represent the projected distance between source and lens, expressed in units of the Einstein radius. The magnitude variation can be expressed as (A(β − δ) − A(β + δ)). The value of δ will generally be smaller than θ_E. For close approaches, where the reflexive motion is most magnified, the value of β is also small. Thus, the magnitude of the fractional change, ΔA/A in magnification can be written as follows:

$$\begin{equation} \frac{\Delta A}{A}= \left |\frac{\beta \, \delta }{\beta ^2-\delta ^2}\right |. \end{equation} \tag{ 5 }$$

For δ smaller than β, we may have ΔA/A = δ/β. Thus, ΔA/A may well correspond to a measurable change in the magnification. For close orbits (small values of α), this change will be small, but it will be periodic. For large α, the change may be large, but it will not repeat. Instead, it can be measured by a shift in the peak magnification relative to the baseline.

The planets many astronomers are most eager to discover are those that could harbor life. It is also desirable that lensing discover planets that could be habitable (Di Stefano 1999). Di Stefano & Night (2008) pointed out that, when the lens is a nearby dwarf star, planets that can produce detectable lensing signatures are more likely to lie in the zone of habitability. This should apply to the HPMSs for which we can predict events.

The zone for habitability is a concept that is based on the premise that a planet orbiting a star is more likely to harbor life when its surface temperature is such that it can support liquid water. This will be the case, when the flux incident on the planet from the star is comparable to the flux received by the Earth from the Sun. This condition on the flux is satisfied if

$$\begin{equation} D_L \, \left (1-\frac{D_L}{D_S}\right) = \frac{125\,{\rm pc}}{\alpha ^2}\, \left (\frac{M}{M_\odot }\right)^{2.5}. \end{equation} \tag{ 6 }$$

The habitable zone is an annulus that is large enough that some systems may have values of D_L and M_* roughly compatible with the habitable zone for a broad range of α values. In this case, one close passage might be able to discover multiple planets that are potentially habitable.

A point lens produces isomagnification contours that are circles centered at the position of the lens. The addition of a second mass changes the isomagnification contours in a way that can be highly nonlinear. In addition, caustic curves are introduced. When the track of the source passes behind a caustic curve, the number of images switches from three (five) to five (three); if the source is a point, the magnification becomes infinite at the time of crossing. The positions and sizes of the caustics are illustrated in Figure 2 (see also Wambsganss 1997); the planet used to compute the structures shown in Figure 1 had a mass equal to 0.001 that of the central star. The orbital separation in the bottom panel is α = 0.6. In this bottom panel we see that the caustic structures are small, so that it is unlikely that any given source track will encounter one or more of them. Furthermore, two of the structures are located more than an Einstein radius from the central star—in fact, much farther from the star than is the planet itself.

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Here we will consider close-orbit planets that represent the small-α continuation of Figure 2. We will call a planet a close-orbit planet if α < 0.5. This zone is shown in the left-bottom panel of Figure 1. Interestingly enough, the detection regime for close-orbit planets is much larger than the limited zone they occupy. This is illustrated in the right-bottom panel of Figure 1, and is explained below.

As the value of α decreases, the trend illustrated in Figure 2 continues. The caustics move out to larger distances from the star. They can be found in a small annulus centered on R_α = 1/α − α. The caustics themselves are very small, and it is unlikely that the source track will intersect them. There is, however, a more extended region around the caustics where the magnification is distorted from the point-lens form. The size of this region is comparable to but somewhat larger than the distance between the caustics, which increases as α decreases. Isomagnification contours from closer in are pulled out, and others from further out are pulled in. This means that, for α ∼ (0.2–0.3), for example, there can be deviations of a few percent that start when the source is somewhere between 4.8 R_E and 2.7 R_E. These are detectable for hours to days, and can repeat as the perturbed region is whipped around to follow the motion of the close-orbit planet.

Thus, the outer portion of the close-orbit detection regime starts at slightly larger values of u than R_α. It generally extends inward to the origin, since the reflexive motion of the lens star will repeat, making it possible to detect the periodicity and to infer the presence of the planet. The discussion above demonstrates that, for close-orbit planets, the detection regime is significantly larger than the zone within which the planets lie. Details can be found in Di Stefano (2012). Note again that the distinction between the zone and the regime is that the former contains the planets, while the latter is the region through which the source must pass if we are to detect evidence of the planets, as illustrated in Figure 1.

The panels shown in Figure 2 cover the ranges of separations from just over 0.5 R_E to just under 2.0 R_E. Planets located in this annulus are said to be in the zone for resonant lensing (Mao & Paczyński 1991; Gould & Loeb 1992). When the planets lie in the zone for resonant lensing, evidence for planets comes from light curves in which the source tracks pass near or behind the caustics, producing a significant deviation from the point-lens form. It is worth noting that the terms "resonant-orbit" or "zone for resonant lensing" do not refer to resonance in the dynamical sense. Instead, the term derives from the circumstance that discoveries of planets in this zone tend to be associated with caustic crossings which can produce large magnifications that rise and fall over very small time intervals.

The regime through which the source must pass in order to detect evidence of planets residing in the zone for resonant lensing extends from the origin to about 2 R_E. Thus, like the close-orbit regime, the resonant regime is larger than the zone within which the planets reside. The additional area is simply the relatively small area of the disk that extends from the origin to 0.5 R_E. Although small, this region contains a caustic structure that is associated with the central star. Thus, the track of any source that passes very close to the star crosses a caustic and can therefore exhibit evidence of the planet's presence (Griest & Safizadeh 1998).

If the planet lies in at larger values of α (α greater than about 2), we will refer to it as a wide-orbit planet. Wide-orbit planets can serve as almost independent lenses, and they can also be detected through deviations in the light curve associated with lensing by the central star (Di Stefano & Scalzo 1999a, 1999b).

Wide-orbit planets can be detected even for distant approaches, i.e., for large values of β. The reason is simply that dramatic events can be created when the source path comes close to the Einstein ring of the planet, which is, by definition, far from the central star. Such events can occur at times much earlier or later than the time of closest approach between the foreground and background star. Thus, the wide-orbit detection zone has an outer edge that is as wide as the widest orbit in the planetary system serving as lens. In addition, evidence of wide-orbit planets can be detected during close approaches between the foreground and background star, since the presence of a wide planet can affect the shape of the stellar-lens light curve. Thus, the wide-orbit detection regime extends inward to the origin. For any given wide-orbit planet, the detection regime may be viewed as a large-radius annulus, centered on the planet's position, combined with a disk, centered on the lens star. In fact, however, because a given planetary system may host a sequence of potentially detectable wide-orbit planets, it makes sense to view the detection regime as an extended region, as shown in the upper right panel of Figure 1.

5.1.1. Magnification Patterns for Wide-orbit Planets

A star and planet can each produce well-separated events for projected separations as small as 1.5 θ_E, although this result depends on the value of the mass ratio and on the photometric sensitivity of the observations. In this paper, we consider the inner boundary of the wide-planet zone to be approximately 2 θ_E.

The wider the planetary orbit, the less likely it is that the track of the source will pass behind both the planet and the central star (see Di Stefano & Scalzo 1999b for probability estimates). Nevertheless, even when only the planet-lens event occurs, the shape of the light curve associated with lensing by the planet can be influenced by the presence of the star until the value of α is fairly large. Figure 3, for example, shows that the 1% isomagnification contour is significantly distorted from a circular form, even for α ∼ 6. The significance of these distortions is twofold: they influence the light curve characteristics and, because they increase the linear dimensions of the region within which the magnification is significant, they increase the event probability.

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Similarly, the isomagnification contours associated with the star can be perturbed by the presence of a wide-orbit planet. The stellar-lens event may therefore show evidence of the planet, even in cases in which the planet does not produce an independent event. This is why the wide-orbit detection regime extends inward to the origin. In other words, the entire lens plane, from the origin out to the largest expected planetary orbit, comprises the wide-orbit detection regime.

5.1.2. Independent Events by Wide-orbit Planets

Consider a wide-orbit planet with projected orbital separation α > 2. Let the distance of closest approach, which will take place at time t₀, be equal to β. The path of the source will intersect a circular orbit of radius α θ_E at two times, given by the following equation:

$$\begin{equation} |t-t_0 | = 100\,{\rm days}\, \left (\frac{0.01\, \frac{\theta _E}{{\rm day}}}{\mu }\right)\, (\alpha ^2- \beta ^2)^{\frac{1}{2}}. \end{equation} \tag{ 7 }$$

Thus, at approximately these two times it is possible for a wide-orbit planet in a circular orbit of radius α to produce a lensing event. The event would last for a time roughly proportional the square root of the planet mass.⁷ For α ≫ 2 the light curve would have characteristics like those expected from an isolated lens. The closer the value of α to the inner edge of the wide-orbit zone, the more perturbed would be the light curve from the standard point-lens form.

Note that the independent planet-lens event can happen many days before or after the time of closes passage between the foreground and background stars. Equation (7) highlights an important feature of predicted events, or indeed any event in which the angle of closest approach is known. During any given day before or after the event, we can compute the specific value of α, the projected orbital separation, of the planet whose lensing signature we can detect on that day. This is so whenever we know the HPMS's proper motion μ, and can estimate the values of β and t₀. Actually, given μ, β, and t₀, we can compute a small range of possible values of α, since the size of the lensing region around the planet has a finite extent which can be computed for each set of values of q and α (Figure 3). Uncertainties in the path also produce uncertainties in the possible day-by-day values of α. Thus, if an event is detected on a specific day, say the 30th day after the main event, we can immediately estimate the value of α. The features of the associated light curve, including the value of τ_E, allow us to estimate the mass of the planet because the proper motion can be well measured.

Conversely, if no detection is made on a given day, we know what type of orbit is not producing an event. Lack of evidence for an event on any given day does not translate to evidence for lack of a planet. It does, however, allow us to place limits on the possible presence of a planet as follows. We consider the times the observations that have been made and, for each, the minimum value of the magnification to which the observation would have been sensitive. Then, by generating a large number of orbits for each value of α and each value of q, including a range of orbital eccentricities and inclinations, we can compute what fraction of the time a planet of each type would be detected. If the observations would have detected the planet 100% of the time, then the failure to observe the planet means that it is not there. If, however, the probability of detecting the planet is ${\cal P}$, then we can say that there is now a probability of only ${1-{\cal P}}$ that such a planet orbits the lens star. This numerical approach can be very accurate, but general planning can be facilitated even by less accurate analytic estimates.

A simple analytic approach to computing the probability that a wide-orbit planet will produce a separate short-duration event is to compare the size of the region within which a planet produces significant effects with the size of the orbit. This ignores velocity effects, but can nevertheless provide a reasonable estimate of the fraction, ${\cal P}$, of initial phases likely to produce an isolated planet-lens event.

To compute this fraction, we simply need to compute the ratio between (1) the size of the region around the planet that can produce detectable lensing effects, and, (2) the size of the orbit. We will express each in units of the Einstein radius. The denominator, the size of the orbit, is simply 2πα. Developing a good approximation for the value of the numerator requires some care. First, the size of the region within which lensing can be detected depends on the sensitivity of the observations. If only magnifications of 34% (6%, 1%) or more are detectable, then the width of the lensing region is 1 × (2 R_E) [2 × (2 R_E), 3.5 × (2 R_E)]. The factor to the left in each case corresponds to the distance, in Einstein radii, at which the magnification becomes detectable. We have taken this into account by introducing a factor F/2, which is unity when deviations of 6% are detectable, 0.5 if deviations must be larger than 34% are detectable, 3.5/2 = 1.75 if deviations must be larger than only 1% to be detectable.

A second factor influences the size of the lensing region associated with the planet. This is the size of the orbit. For values of α in the range 2–6, the lensing region is significantly stretched, with the amount of stretching also dependent on the mass ratio, q. For larger values of α the stretching is minimal. To take this into account, we have introduced the factor E(q_i, α)/1.5, which has the value roughly equal to 1 for α = 3 and q = 0.001.

With these definitions, the probability, ${\cal P}$, that a wide-orbit planet will produce an isolated event is

$$\begin{equation} {\cal P}=0.36\, \frac{F}{2} \sum \limits _{i=1}^n \left(\frac{q_i}{0.01}\right)^{\frac{1}{2}{}} \, \left (\frac{1}{\alpha _i}\right)\, \left (\frac{E(q_i,\alpha _i)}{1.5}\right), \end{equation} \tag{ 8 }$$

where the sum is over wide-orbit planets. For α = 3, the contribution is 0.12. A more accurate estimate of the probability would include the effects of orbital motion and parallax.

5.1.3. Detecting Evidence of the Planet during the Stellar-lens Event

Fits to light curves produced by the star have the potential to identify deviations from the point-lens form and to discover wide-orbit planets. This is the case even when the planet itself does not produce a separate detectable event.

The influence of the planet could be manifest through a slight distortion of the isomagnification contours surrounding the star. Such effects are likely to be most prominent in the low-magnification portion of the light curve.

The planet could also be detected through the reflexive motion of the star. As shown in Equation (4), the magnitude of the positional shift is small for small values of α. Fortunately, for small α, direct detection of the planet is more likely (Equation (8)). Also for small α, if the relative speed is not too high, the phase could change significantly enough during a close approach to affect the light curve shape. The wider the orbit, the larger the reflexive motion, and the more likely it is to produce deviations in the stellar-lens light curve from the pure point-lens form. This is illustrated in Figure 4.

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If the foreground star happens to have planets with orbits much smaller than the Einstein radius, the lensing signatures can be significantly perturbed from the point-lens form. This is because there is a small region within which the isomagnification contours are distorted from the circular shape they would have had in the absence of the planet. These perturbations are maximized at a distance R_α = (1/α − α) > 1.5 from the center of mass. In the absence of orbital motion, there would be only a small probability that the distorted region would pass in front of the source. The orbital period is short for small values of α, however, especially when the planetary system serving as a lens is nearby. Thus, even though the deviations tend to be small, repetition confers an advantage.

The light curves associated with close-orbit planets have the same overall shape as light curves produced by the central star. To detect evidence of the planets, we must be sensitive to small deviations of the type studied in Di Stefano (2012). The deviations have a characteristic up-down–up-down shape, and for nearby lenses, will generally repeat. Figures 3–6 of Di Stefano (2012) illustrate the deviations in the isomagnification contours and light curves for a "hot Jupiter," and for a Neptune-mass and an Earth-mass planet located in the habitable zone of a dwarf star. In Figures 5 and 6 of this paper, we show the light curves for a variety of planets and planetary orbits, plotting the deviations, A_pl − A_pt, from the point-lens light curve produced by the central star alone. The specific foreground star for which the calculations were done is VB 10; the central star is, therefore, a dwarf star of mass 0.075 M_☉. The value of R_E is roughly 0.08 AU. Here we explain the patterns.

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The most distinctive features are repeating deviations that start in the wings (i.e., the low-magnification portion) of the light curve. For each value of α, the deviations start as the source star approaches a distance R_α from the central star. The deviations are larger for larger values of α. In fact, if the observations are able to reliably detect deviations of size δ, then close-orbit planets with semimajor axes in the range between α_small and 0.5 can be detected, where

$$\begin{equation} \alpha _{{\rm small}}=0.84\, \delta ^{\frac{1}{4}}. \end{equation} \tag{ 9 }$$

This effect is illustrated in the early-time and late-time deviations seen in the light curves of Figures 5 and 6, where the size and location of the planet-induced perturbations clearly depend on the value of α.

Detectability is determined not just by the size of the perturbations, but also by their duration. The duration depends on the value of the orbital period and on the mass ratio. While shorter orbital periods increase detectability by increasing the number of repetitions, the faster motion associated with them decreases the duration of perturbations. On the other hand, increasing the mass ratio q between the planet and star always increases the duration. Let T_dev represent the duration of perturbations. Di Stefano (2012) shows that

$$\begin{equation} T_{{\rm dev}}= 2.5\, \xi \, P_{{\rm orb}} 10^{[0.5\, {\rm Log}_{10}(q) - 0.2]}, \end{equation} \tag{ 10 }$$

where the value of ξ depends on the observing setup, and can be taken to be of order unity in this discussion. (See Di Stefano 2012 for more details.) Figure 5 displays the early-time and late-time perturbations associated with close-orbit planets. It shows that like wide-orbit planets, close-orbit planets can produce deviations both in advance of and after the closest approach. The time at which the deviations start is the time at which the source star enters the region with magnification perturbations at R_α. Let t_max be the time at which deviations related to the planet begin (before the time of closest passage) or end (after the time of closest passage), representing the maximum time difference between the time of closest approach and the time at which deviations occur. If we measure time in days we have the analog to Equation (3) for close-orbit planets:

$$\begin{equation} |t_{{\rm max}}-t_0 | \sim 100\,{\rm days}\, \left (\frac{0.01\, \frac{\theta _E}{{\rm day}}}{\mu }\right)\, \left (\left (\frac{1}{\alpha } - \alpha \right)^2 - \beta ^2 \right)^{\frac{1}{2}}. \end{equation} \tag{ 11 }$$

The smaller the value of α, the larger the value of R_α, and the longer the difference in time between the closest approach and the start of close-planet-lens deviations. This is shown in Figure 5, which plots the difference between the magnification which includes the effect of the planet, A_pl, and the point-lens magnification, A_pt, expected if there is no planet. Figure 5 illustrates that the effects diminish in size as the value of α becomes smaller. Thus the effects of a 3-Jupiter-mass planet at α = 0.15 would be challenging for ground-based observations, while a Saturn-mass planet could easily be discerned for α larger than about 0.3. The early-time and late-time perturbations are also shown in the top two panels of Figure 6.

The bottom panel of Figure 6 illustrates that the deviations produced by close-orbit planets can be detected even when the distance of closest approach is larger. The example shown in this panel has β = 5. It is interesting to note that A_pt reaches a maximum value of only 1.00275. Thus, if we had plotted only the raw magnification, A_pl, the light curve would be essentially the same as the one shown. Thus, the signature of the planet would be an oscillatory deviation from baseline, of finite duration.

Another similarity with the wide-orbit case is that we can compute a specific value of α for a planet whose lensing signature we are capable of detecting (given the sensitivity and cadence of sampling) on a given day. Thus, when we do not find evidence of planets, we can place quantifiable limits on their existence.

Note that, by selecting VB 10 as an example, we have chosen a star with particularly high proper motion. Such stars are the most likely to produce events that can be predicted. The high proper motion does decrease the time duration of deviations, which makes detection more challenging. Other cases are shown in Di Stefano (2012) and in a future paper, based on Di Stefano et al. (2012).

In addition to the repeating small deviations in the wings of the light curves, Figure 5 and the top two panels of Figure 6 show another interesting effect. This is that, for small values of β, there is a second region displaying significant perturbations. This region is near the light curve's peak, when the background star makes its closest approach to the foreground star. It corresponds to the reflexive motion of the central star and is larger for larger orbits. The size of the reflexive effects also increases with the mass of the close-orbit planet.

Just as there is a largest orbital separation for a wide-orbit planet, there is a smallest orbital separation for a close-orbit planet. To avoid catastrophic tidal disruption a planet must satisfy the condition r_h ≳ 2r_p, where r_h is the Hill radius given by r_h ≈ a(m_p/3M_*)^1/3. Thus the minimum orbital separation of a planet around VB 10 is roughly given by

$$\begin{equation} a_{{\rm min}}\sim 0.3\,R_\odot \frac{r_p}{r_J}\left(\frac{M_*}{0.075\,M_{\odot }}\right)^{1/3}\left(\frac{0.001\,M_{\odot }}{m_p}\right)^{1/3}. \end{equation} \tag{ 12 }$$

This corresponds to α_min ∼ 0.03 for the VB 10 case. The deviations produced by a planet in such a close orbit would occur when the position of the source star on the sky was approximately 30 R_E from the position of the foreground star. Their magnitude would be so small that they would be difficult to detect with today's technology.

Event probabilities. In Figures 5 and 6, each panel contains nine independent light curves, slightly displaced from each other so that the variations in each can be resolved. The nine light curves in each panel differ from each other only in that a different initial phase was chosen for each. The effects were similar in all cases. Had the perturbations been potentially detectable for only a small fraction of initial phases, then the probability of planet detection would be comparable to the value of that fraction. The ubiquity of detectable results indicates that, if the observational strategy is capable of detecting deviations caused by planets with a given value of α, orbital period, and q, then it will detect such planets with certainty. Thus, either close-orbit planets will be discovered, or limits on their existence can always be placed, if the observing conditions are favorable and the monitoring program is well designed. The values of α for which we can derive limits or else guarantee a detection are determined by the photometric sensitivity, while those of q are determined by the cadence of sampling.

When the possibility for discovering planets via microlensing was first discussed (Mao & Paczyński 1991; Gould & Loeb 1992), attention focused on passages of the source star near or behind caustic structures like those shown in Figure 2. The associated light curves can show dramatic effects, although these can be moderated by the smoothing effect associated with the finite size of the source. Until now, microlensing searches for planets have focused on finding planets in the resonant zone. Lensing events are found by observing teams monitoring wide fields. Events judged to have a high probability of exhibiting planet-lens signatures are targeted for almost continuous monitoring during an interval (which can be as short as hours) when the magnification is deemed likely to be significantly influenced by the presence of planets. The 16 planets discovered via lensing have been found through such methods and are squarely in or near the borders of the zone for resonant lensing.

Light curves. The associated light curves are therefore familiar from both calculations and observations. The new element introduced when the lens is a nearby foreground star is that the orbital size for values of α in the resonant zone can be small enough to make the orbital periods relatively short. Thus, rotation can play a significant role in determining the form of the light curves. This is illustrated in Figure 7. Because of the effect of orbital motion, the individual light curves plotted there exhibit more structure than typical for more distant lenses with planets in the zone for resonant lensing.

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Event probabilities. The closer the approach, the higher the probability of detecting events in which the source has passed near a caustic structure. This is illustrated in Figure 7, where light curves for a 5 mas approach of VB 10 to a background star are shown. For α = 1.2 (top panel) and α = 0.8 (bottom panel), 10 randomly selected initial phases were each used to generate a light curve. For α = 1.2, eight of the light curves showed significant deviations, three involving caustic crossings (those exhibiting wall-like rises and falls). For α = 0.8, all 10 light curves displayed significant variations, with an even larger number of caustic crossings than for α = 1.2. This illustrates that, for close passages between a background star and a nearby planetary system, the probability of detecting any planets inhabiting the zone for resonant lensing is close to unity.