DRAFT VERSION OCTOBER 31, 2018
Preprint typeset using LATEX style emulateapj v. 08/22/09
CRITERIA IN THE SELECTION OF TARGET EVENTS FOR PLANETARY MICROLENSING FOLLOW-UP
OBSERVATIONS
CHEONGHO HAN
Program of Brain Korea 21, Institute for Basic Science Research, Department of Physics,
Chungbuk National University, Chongju 361-763, Korea; cheongho@astroph.chungbuk.ac.kr
Draft version October 31, 2018
ABSTRACT
To provide criteria in the selection of target events preferable for planetary lensing follow-up observations,
we investigate the variation of the probability of detecting planetary signals depending on the observables
of the lensing magnification and source brightness. In estimating the probability, we consider variation of
the photometric precision by using a quantity defined as the ratio of the fractional deviation of the planetary
perturbation to the photometric precision. From this investigation, we find consistent result from previous
studies that the probability increases with the increase of the magnification. The increase rate is boosted at a
certain magnification at which perturbations caused by central caustic begin to occur. We find this boost occurs
at moderate magnifications of A . 20, implying that probability can be high even for events with moderate
magnifications. The probability increases as the source brightness increases. We find that the probability of
events associated with stars brighter than clump giants is not negligible even at magnifications as low as A ∼ 5.
In the absence of rare the prime target of very high-magnification events, we, therefore, recommend to observe
events with brightest source stars and highest magnifications among the alerted events. Due to the increase of
the source size with the increase of the brightness, however, the probability rapidly drops off beyond a certain
magnification, causing detections of low mass ratio planets (q . 10−4) difficult from the observations of events
involved with giant stars with magnifications A & 70.
Subject headings: gravitational lensing – planets and satellites: general
1. INTRODUCTION
With the advantages of being able to detect very low-mass
planets and those with separations from host stars that can-
not be covered by other methods, microlensing is one of
the most important methods that can detect and character-
ize extrasolar planets (Mao & Paczyński 1994; Gould & Loeb
1992). The microlensing planetary signal is a short duration
perturbation to the standard lensing light curve produced by
the primary star. To achieve high monitoring frequency re-
quired for the detection of the short-lived planetary signal,
current lensing experiments are employing early-warning sys-
tem to issue alerts of ongoing events in the early stage of
lensing magnification (Udalski et al. 1994; Bond et al. 2002)
and follow-up observations to intensively monitor the alerted
events (Dominik et al. 2002; Yoo et al. 2004). Under current
surveys, there exist in average & 50 alerted events at a certain
time (Dominik et al. 2002). Then, an important issue related
to the follow-up observation is which event should be moni-
tored for better chance of planet detections.
There have been several estimates of microlensing planet
detection efficiencies (Bolatto & Falco 1994; Bennett & Rhie
1996; Gaudi & Sackett 2000; Peale 2001). Most of these
works estimated the efficiency as a function of the instanta-
neous angular star-planet separation normalized by the angu-
lar Einstein radius, s, and planet/star mass ratio, q. However,
the efficiency determined in this way is of little use in the point
of view of observers who are actually carrying out follow-
up observations of lensing events. This is because the planet
parameters s and q are not known in the middle of lensing
magnification and thus they cannot be used as criteria in the
selection of target events for follow-up observations. Related
to the target selection, Griest & Safizadeh (1998) proposed a
useful criterion to observers. They pointed out that by focus-
ing on very high-magnification (A & 100) events, the proba-
bility of detecting planets in the lensing zone could be very
high. However, these events are rare and thus they cannot
be usually found in the list of alerted events. Therefore, it is
necessary to have criteria applicable to general lensing events
in the absence of very high-magnification events. To provide
such criteria, we investigate the dependency of the probability
of detecting planetary signals on the observables such as the
lensing magnification and source type.
The paper is organized as follows. In § 2, we briefly de-
scribe the basics of planetary microlensing. In § 3, we investi-
gate the variation of the probability of detecting planetary sig-
nals depending on the lensing magnification and source type
for events caused by planetary systems with different masses
and separations. We analyze the result and qualitatively ex-
plain the tendencies found from the investigation. Based on
the result of the investigation, we then present criteria for the
selection of target events preferable for follow-up observa-
tions. In § 4, we summarize the results and conclude.
2. BASICS OF PLANETARY LENSING
The lensing behavior of a planetary lens system is described
by the formalism of a binary lens with a very low-mass com-
panion. Because of the very small mass ratio, planetary a lens-
ing light curve is well described by that of a single lens of the
primary star for most of the event duration. However, a short-
duration perturbation can occur when the source star passes
the region around the caustics, that are the set of source po-
sitions at which the magnification of a point source becomes
infinite. The caustics of binary lensing form a single or multi-
ple sets of closed curves where each of which is composed of
concave curves (fold caustics) that meet at points (cusps).
For a planetary case, there exist two sets of disconnected
caustics: ‘central’ and ‘planetary’ caustics. The single central
http://arxiv.org/abs/0704.0968v1
2 MICROLENSING FOLLOW-UP CRITERIA
caustic is located close to the host star. It has a wedge shape
with four cusps and its size (width along the star-planet axis)
is related to the planet parameters by (Chung et al. 2005)
∆ξcc ∝
(s − 1/s)2
. (1)
For a given mass ratio, a pair of central caustics with sepa-
rations s and s−1 are identical to the first order of approxima-
tion (Dominik 1999; Griest & Safizadeh 1998; An 2005). The
planetary caustic is located away from the host star. The cen-
ter of the planetary caustic is located on the star-planet axis
and the position vector to the center of the planetary caustic
measured from the primary lens position is related to the lens-
source separation vector, s, by
rpc = s
. (2)
Then, the planetary caustic is located on the planet side, i.e.
sign(rpc) = sign(s), when s > 1, and on the opposite side, i.e.
sign(rpc) = −sign(s), when s < 1. When s > 1, there exists a
single planetary caustic and it has a diamond shape with four
cusps. When s< 1, there are two caustics and each caustic has
a triangular shape with three cusps. The size of the planetary
caustic is related to the planet parameters by
∆ξpc ∝
q1/2/(s
s2 − 1) for s > 1,
q1/2(κ0 − 1/κ0 +κ0/s2)cosθ0 for s < 1,
where κ(θ) =
[cos2θ± (s4 − sin2 2θ)1/2]/(s2 − 1/s2)
, θ0 =
[π±sin−1(31/2s2/2)]/2, and κ0 = κ(θ0) (Han 2006). The plan-
etary caustic is always bigger than the central caustic and the
size ratio between the two types of caustics, ∆ξcc/∆ξpc, be-
comes smaller as the mass of the planet becomes smaller and
the planet is located further away from the Einstein ring. The
planetary caustic is located within the Einstein ring of the pri-
mary when the planet is located in the range of separation
from the star of 0.6 . s . 1.6. The size of the caustic, which
is directly proportional to the planet detection efficiency, is
maximized when the planet is located in this range, and thus
this range is called as the ‘lensing zone’. As the position of
the planet approaches to the Einstein ring radius, s → 1, the
location of the planetary caustic approaches the position of
the central caustic. Then, the two types of caustic eventually
merge together, forming a single large one.
3. VARIATION OF DETECTABILITY
3.1. Quantification of Detectability
The quantity that has been often used in the previous es-
timation of the planet detection probability is the ‘fractional
deviation’ of the planetary lensing light curve from that of the
single lensing event of the primary, i.e.,
A − A0
. (4)
With this quantity, however, one cannot consider the variation
of the photometric precision depending on the lensing magni-
fication. In addition, it is difficult to consider the variation of
the detectability depending on the source type.
To consider the effect of source star brightness and its
lensing-induced variation on the planet detection probability,
we carry out our analysis based on a new quantity defined as
the ratio of the fractional deviation, ǫ, to the photometric pre-
cision, σ
, i.e,
D = |ǫ|
ν,S + Fν,B)1/2
(A − 1)F
, (5)
where F
ν,S and Fν,B represent the fluxes from the source star
and blended background stars, respectively. Here we as-
sume that photometry is carried out by using the difference
imaging method (Tomaney & Crotts 1996; Alard 1999). In
this technique, photometry of the lensed source star is con-
ducted on the subtracted image obtained by convolving two
images taken at different times after geometrically and pho-
tometrically aligning them. Then the signal from the lensed
star measured on the subtracted image is the flux variation
of the lensed source star, (A − 1)F
ν,S, while the noise orig-
inates from both the source and background blended stars,
ν,S + Fν,B. Under this definition of the planetary signal de-
tectability,D = 1 implies that the planetary signal is equivalent
to the photometric precision. Hereafter we refer the quantity
D as the ‘detectability’.
3.2. Contour Maps of Detectability
To see the variation of the detectability depending on the
separation parameter s, mass ratio q, and the types of involved
source star, we construct maps of detectability as a function
of the position in the source plane. Figure 1 shows example
maps. The individual sets of panels show the maps for events
associated with different types of source stars. All lengths
are normalized by the angular Einstein radius and ξ and η
represent the coordinates parallel with and normal to the star-
planet axis, respectively. A contours (yellow curve) is drawn
at the level of D = 3.0. The maps are centered at the position
of the primary lens star and the planet is located on the left.
The dotted arc in each panel represents the Einstein ring of the
primary star. The closed figures drawn by red curves represent
the caustics.
For the construction of the maps, we assume a mass of
the primary lens star of m = 0.3 M⊙ and distances to the
lens and source of DL = 6 kpc and DS = 8 kpc, respec-
tively. Then, the corresponding Einstein radius is rE =
(4Gm/c2)[(DL(DS − DL)/DS]
= 1.9 AU. For the source
stars, we test three different types of giant, clump giant, and
main-sequence stars. The assumed I-band absolute magni-
tudes of the individual types of stars are MI = 0.0, 1.5, and
3.6, respectively. With the assumed amount of extinction to-
ward the Galactic bulge field of AI = 1.0, these correspond
to the apparent magnitudes of I = 15.5, 17, and 19.1, respec-
tively. As the source type changes, not only the brightness
but also the size of the star changes. Source size affects the
planetary signal in lensing light curves (Bennett & Rhie 1996)
and thus we take account the finite source effect into consid-
eration. The assumed source radii of the individual types of
source stars are 10.0 R⊙, 3.0 R⊙, and 1.1 R⊙, respectively.
We assume that events are affected by blended flux equivalent
to that of a star with I = 20. We note that the adopted lens
and source parameters are the typical values of Galactic bulge
events that are being detected by the current lensing surveys
(Han & Gould 2003).
For the observational condition, we assume that images are
obtained by using 1 m telescopes, which are typical ones be-
ing used in current follow-up observations. We also assume
that the photon acquisition rate of each telescope is 10 pho-
tons per second for an I = 20 star and a combined image with
HAN 3
FIG. 1.— Contour maps of the detectability of the planetary signal, D, as a function of the position in the source plane for events caused by planetary systems
with various lens-source separations and mass ratios. The detectability represents the ratio of the fractional deviation of the planetary lensing light curve from
the single lensing light curve of the primary to the photometric precision. All lengths are normalized by the angular Einstein radius and ξ and η represent the
coordinates parallel with and normal to the star-planet axis, respectively. The individual sets of panels show the maps for events associated with different types
of source stars. Contours (yellow curve) are drawn at the level of D = 3.0. The maps are centered at the position of the primary lens star and the planet is located
on the left. The dotted arc in each panel represents the Einstein ring of the primary star. The closed figures drawn by red curves represent the caustics. For the
details about the assumed lens parameters and observational conditions, see § 3.2.
FIG. 2.— Geometric representation of the probability of detecting planetary
signals, P. Under the definition of P as the average probability of detecting
planetary signals with a detectability greater than a threshold value Dth at
the time of observation with a magnification A, the probability corresponds
to the portion of the arclet(s) where the detectability is greater than a threshold
value out of a circle around the primary with a radius equal to the lens-source
separation corresponding to the magnification at the time of observation. The
individual circles in the upper panel correspond to the source positions at
which the lensing magnifications are A = 1.5 (pink), 3.0 (cyan), 5.0 (green),
and 10.0 (red), respectively. The curves in the bottom panels show the vari-
ation of the detectability as a function of the position angle (θ) of points on
the circles with corresponding colors in the upper panel. We set the thresh-
old detectability as Dth = 3.0, i.e. 3σ detection of the planetary signal. The
dashed circle represents the Einstein ring.
a total exposure time of 5 minutes is obtained from each set
of observations.
3.3. Probability of Detecting Planetary Signals
Based on the maps of detectability, we then investigate the
probability of detecting planetary signals as a function of the
lensing magnification. We define the probability P as the av-
erage probability of detecting planetary signals with a de-
tectability greater than a threshold value Dth at the time of
observation with a magnification A. Geometrically, this prob-
ability corresponds to the portion of the arclet(s) where the
detectability is greater than a threshold value out of a circle
FIG. 3.— Probability of detecting planetary signals as a function of lens-
ing magnification. The individual panels show the probabilities for events
involved with different types of source stars. The curves in each panel show
the variation of the probability for planets with different mass ratios and sep-
arations. We note that although not presented, the probabilities for planets
with separations s < 1 are similar to those of the corresponding planets with
s−1. The probability is defined the average probability of detecting plane-
tary signals with a detectability greater than a threshold value Dth at the time
of observation with a magnification A. We set the threshold detectability as
Dth = 3.0, i.e. 3σ detection of the planetary signal. We note that there is a
maximum magnification specific to the angular size of the source star and
thus the curves stop at certain magnifications.
around the primary with a radius equal to the lens-source sep-
aration corresponding to the magnification at the time of ob-
servation. This is illustrated in Figure 2. We note that the
magnification is a unique function of the absolute value of the
lens-source separation u1, and thus A = const corresponds to
a circle around the lens. The lens-source separation is related
to the magnification by
u(A) =
(1 − A−2)1/2
. (6)
We set the threshold detectability as Dth = 3.0, i.e. 3σ detec-
tion of the planetary signal.
1 Strictly speaking, the magnification depends additionally on the size of
the source star.
4 MICROLENSING FOLLOW-UP CRITERIA
TABLE 1
LIMITATION BY FINITE-SOURCE EFFECT
source type event type
giant A & 70 for planets with q . 10−3
clump giant A & 200 for planets with q . 5× 10−4
main-sequence A & 500 for planets with q . 10−4
NOTE. — Cases of planetary microlensing events
where detection of planetary signal is limited by finite
source effect. We note that “-” means the respective con-
figuration cannot be realized.
In Figure 3, we present the resulting probability as a func-
tion of magnification. The individual panels show the proba-
bilities for events involved with different types of source stars.
In each panel, we present the variations of the probability for
planets with different mass ratios and separations. We test six
different planetary separations of s = 1/1.6, 1/1.4, 1/1.2, 1.2,
1.4, and 1.6 as representative values for planets in the lensing
zone. For the mass ratio, we test five values of q = 5× 10−3,
10−3, 5× 10−4, 10−4, and 5× 10−5.
From the variation of the probability, we find the follow-
ing tendencies. First, we find that the probability increases
with the increase of the lensing magnification. This is con-
sistent with the result of K. Horne (private communication).
This tendency is due to three factors. First, the size of the
planetary caustic increases as it is located closer to the pri-
mary star. This can be seen in Figure 4, where we present the
relation between the location of the planetary caustic and its
size, which is obtained by using equations (2) and (3). Then,
higher chance of planetary perturbation is expected when the
source is located closer to the primary during which the lens-
ing magnification is high. Second, perturbation regions of
the same size cover a larger range of angle as the planetary
caustic moves closer to the lens. This also contributes to the
higher probability. Third, the photometric precision improves
with the increasing brightness of the source star due to lensing
magnification. As the photometric precision improves, it is
easier to detect small deviations induced by planets. The same
reason can explain the considerable size of the perturbation
region induced by central caustics. Perturbations induced by
the central caustics occur at high magnifications during which
the photometric precision is high. As a result, despite much
smaller size of the central caustic than that of the planetary
caustic, the central perturbation region is considerable and can
even be comparable to the perturbation region induced by the
planetary caustic. This can be seen in the detectability maps
presented in Figure 1.
However, the probability does not continue to increase with
the increase of the magnification. Instead, the probability
drops off rapidly beyond a certain magnification. This critical
value corresponds to the magnification at which finite-source
effect begins to wash out the planetary signal. In Table 1, we
present the cases where finite source effect limits planet detec-
tions. As a result, detections of planets with low mass ratios
would be difficult for events involved with giant source stars
with magnifications A & 70. We note that the finite source
effect also limits the maximum magnifications of events and
thus the curves in Figure 3 discontinue at a certain value.
Second, as the magnification increases, the probability of
detecting planetary signal increases with two dramatically dif-
ferent rates of dP/d logA. We find that this abrupt change
of dP/d logA occurs due to the transition from the regime of
FIG. 4.— Variation of the size of the planetary caustic as a function of its
location. The value rpc represents the separation between the center of the
planetary caustic and the primary lens star. The sign of rpc is positive when
the caustic is on the planet side and vice versa. We note that the caustic size
at around rpc is not presented because the analytic expression in eq. (1) is not
valid in this region. In addition, there is no distinction between the planetary
and central caustics in this region.
perturbations induced by planetary caustics into the one of
perturbations induced by central caustics. The perturbation
region induced by the central caustic forms around the pri-
mary lens and thus the probability becomes very high once the
source star is in the central perturbation regime. The boost of
the increase rate occurs at different magnifications depending
on the planetary parameters and the types of involved source
stars. The critical magnification becomes lower as the mass
ratio of the planet increases and the separation of the planet
approaches the Einstein ring radius. In Table 2, we present
these critical magnifications. An important finding to be noted
is that the critical magnification occurs at moderate magnifi-
cations of . 20 for a significant fraction of events caused by
planetary systems with planets located in the lensing zone.
This implies that probability of detecting planetary signal can
be high even for events with moderate magnifications.
Third, the probability is higher for events involved with
brighter source stars. This is because of the improved pho-
tometric precision with the increase of the source brightness.
The difference in the probability depending on the source type
is especially important at low magnifications. For example,
the probabilities at a magnification of A = 5 for events caused
by a common planetary system with q = 10−3 and s = 1.2 but
associated with different source stars of giant, clump giant,
and main-sequence are P ∼ 20%, 10%, and 1%, respectively.
In the absence of high magnification events, therefore, the sec-
ond prime candidate event for follow-up observation is the
one involved with brightest source star. As the magnification
further increases and once the source star enters the central
perturbation region, the difference becomes less important.
4. SUMMARY AND CONCLUSION
For the purpose of providing useful criteria in the selec-
tion of target events preferable for planetary lensing follow-
up observations, we investigated the variation of the proba-
bility of detecting planetary lensing signals depending on the
observables of the lensing magnification and source bright-
ness. From this investigation, we found consistent result from
previous studies that the probability increases with the in-
crease of the lensing magnification due to the improvement
HAN 5
TABLE 2
CRITICAL MAGNIFICATIONS OF CENTRAL PERTURBATION
source planetary mass ratio
type separation q = 5× 10−3 q = 10−3 q = 5× 10−4 q = 10−4 q = 5× 10−5
s = 1.2, 1/1.2 A ∼ 2.2 A ∼ 7 A ∼ 8 A ∼ 22 A ∼ 22
giant s = 1.4, 1/1.4 A ∼ 2.5 A ∼ 8 A ∼ 12 – –
s = 1.6, 1/1.6 A ∼ 3.5 A ∼ 9 A ∼ 18 – –
clump s = 1.2, 1/1.2 A ∼ 7 A ∼ 8 A ∼ 11 A ∼ 30 A ∼ 60
giant s = 1.4, 1/1.4 A ∼ 8 A ∼ 12 A ∼ 17 A ∼ 60 A ∼ 80
s = 1.6, 1/1.6 A ∼ 9 A ∼ 16 A ∼ 20 A ∼ 745 –
main s = 1.2, 1/1.2 A ∼ 6 A ∼ 11 A ∼ 20 A ∼ 55 A ∼ 100
sequence s = 1.4, 1/1.4 A ∼ 8 A ∼ 20 A ∼ 30 A ∼ 100 A ∼ 150
s = 1.6, 1/1.6 A ∼ 11 A ∼ 30 A ∼ 40 A ∼ 150 A ∼ 200
NOTE. — Critical magnifications at which transition from the regime of perturbations induced
by planetary caustics into the one of perturbations induced by central caustics occur. We note that
the critical magnifications are . 20 in many cases.
of the photometric precision combined with the expansion of
the perturbation region. The increase rate of the probabil-
ity is boosted at a certain magnification at which perturba-
tion caused by the central caustic begins to occur. We found
that this boost occurs at moderate magnifications of A . 20
for a significant fraction of events caused by planetary sys-
tems with planets located in the lensing zone, implying that
probabilities can be high even for events with moderate mag-
nifications. The probability increases with the increase of the
source star brightness. We found that the probability of events
associated with source stars brighter than clump giants is not
negligible even at magnifications as low as A ∼ 5. In the ab-
sence of rare prime target of very high-magnification events
(A & 100), we, therefore, recommend to observe events with
brightest source stars and highest magnifications among the
alerted events. Due to the increase of the source size with the
increase of the brightness, however, the probability rapidly
drops off beyond a certain magnification. As a result, detec-
tions of planets with low mass ratios (q . 10−4) would be dif-
ficult for events involved with giant source stars with magni-
fications A & 70.
This work was supported by the Astrophysical Research
Center for the Structure and Evolution of the Cosmos (ARC-
SEC) of Korea Science and Engineering Foundation (KOSEF)
through Science Research Program (SRC) program.
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This figure "fig1.jpg" is available in "jpg"
 format from:
http://arxiv.org/ps/0704.0968v1
http://arxiv.org/ps/0704.0968v1

  To provide criteria in the selection of target events preferable for
planetary lensing follow-up observations, we investigate the variation of the
probability of detecting planetary signals depending on the observables of the
lensing magnification and source brightness. In estimating the probability, we
consider variation of the photometric precision by using a quantity defined as
the ratio of the fractional deviation of the planetary perturbation to the
photometric precision. From this investigation, we find consistent result from
previous studies that the probability increases with the increase of the
magnification. The increase rate is boosted at a certain magnification at which
perturbations caused by central caustic begin to occur. We find this boost
occurs at moderate magnifications of $A\lesssim 20$, implying that probability
can be high even for events with moderate magnifications. The probability
increases as the source brightness increases. We find that the probability of
events associated with stars brighter than clump giants is not negligible even
at magnifications as low as $A\sim 5$. In the absence of rare the prime target
of very high-magnification events, we, therefore, recommend to observe events
with brightest source stars and highest magnifications among the alerted
events. Due to the increase of the source size with the increase of the
brightness, however, the probability rapidly drops off beyond a certain
magnification, causing detections of low mass ratio planets ($q\lesssim
10^{-4}$) difficult from the observations of events involved with giant stars
with magnifications $A\gtrsim 70$.

DRAFT VERSION OCTOBER 31, 2018
Preprint typeset using LATEX style emulateapj v. 08/22/09
CRITERIA IN THE SELECTION OF TARGET EVENTS FOR PLANETARY MICROLENSING FOLLOW-UP
OBSERVATIONS
CHEONGHO HAN
Program of Brain Korea 21, Institute for Basic Science Research, Department of Physics,
Chungbuk National University, Chongju 361-763, Korea; cheongho@astroph.chungbuk.ac.kr
Draft version October 31, 2018
ABSTRACT
To provide criteria in the selection of target events preferable for planetary lensing follow-up observations,
we investigate the variation of the probability of detecting planetary signals depending on the observables
of the lensing magnification and source brightness. In estimating the probability, we consider variation of
the photometric precision by using a quantity defined as the ratio of the fractional deviation of the planetary
perturbation to the photometric precision. From this investigation, we find consistent result from previous
studies that the probability increases with the increase of the magnification. The increase rate is boosted at a
certain magnification at which perturbations caused by central caustic begin to occur. We find this boost occurs
at moderate magnifications of A . 20, implying that probability can be high even for events with moderate
magnifications. The probability increases as the source brightness increases. We find that the probability of
events associated with stars brighter than clump giants is not negligible even at magnifications as low as A ∼ 5.
In the absence of rare the prime target of very high-magnification events, we, therefore, recommend to observe
events with brightest source stars and highest magnifications among the alerted events. Due to the increase of
the source size with the increase of the brightness, however, the probability rapidly drops off beyond a certain
magnification, causing detections of low mass ratio planets (q . 10−4) difficult from the observations of events
involved with giant stars with magnifications A & 70.
Subject headings: gravitational lensing – planets and satellites: general
1. INTRODUCTION
With the advantages of being able to detect very low-mass
planets and those with separations from host stars that can-
not be covered by other methods, microlensing is one of
the most important methods that can detect and character-
ize extrasolar planets (Mao & Paczyński 1994; Gould & Loeb
1992). The microlensing planetary signal is a short duration
perturbation to the standard lensing light curve produced by
the primary star. To achieve high monitoring frequency re-
quired for the detection of the short-lived planetary signal,
current lensing experiments are employing early-warning sys-
tem to issue alerts of ongoing events in the early stage of
lensing magnification (Udalski et al. 1994; Bond et al. 2002)
and follow-up observations to intensively monitor the alerted
events (Dominik et al. 2002; Yoo et al. 2004). Under current
surveys, there exist in average & 50 alerted events at a certain
time (Dominik et al. 2002). Then, an important issue related
to the follow-up observation is which event should be moni-
tored for better chance of planet detections.
There have been several estimates of microlensing planet
detection efficiencies (Bolatto & Falco 1994; Bennett & Rhie
1996; Gaudi & Sackett 2000; Peale 2001). Most of these
works estimated the efficiency as a function of the instanta-
neous angular star-planet separation normalized by the angu-
lar Einstein radius, s, and planet/star mass ratio, q. However,
the efficiency determined in this way is of little use in the point
of view of observers who are actually carrying out follow-
up observations of lensing events. This is because the planet
parameters s and q are not known in the middle of lensing
magnification and thus they cannot be used as criteria in the
selection of target events for follow-up observations. Related
to the target selection, Griest & Safizadeh (1998) proposed a
useful criterion to observers. They pointed out that by focus-
ing on very high-magnification (A & 100) events, the proba-
bility of detecting planets in the lensing zone could be very
high. However, these events are rare and thus they cannot
be usually found in the list of alerted events. Therefore, it is
necessary to have criteria applicable to general lensing events
in the absence of very high-magnification events. To provide
such criteria, we investigate the dependency of the probability
of detecting planetary signals on the observables such as the
lensing magnification and source type.
The paper is organized as follows. In § 2, we briefly de-
scribe the basics of planetary microlensing. In § 3, we investi-
gate the variation of the probability of detecting planetary sig-
nals depending on the lensing magnification and source type
for events caused by planetary systems with different masses
and separations. We analyze the result and qualitatively ex-
plain the tendencies found from the investigation. Based on
the result of the investigation, we then present criteria for the
selection of target events preferable for follow-up observa-
tions. In § 4, we summarize the results and conclude.
2. BASICS OF PLANETARY LENSING
The lensing behavior of a planetary lens system is described
by the formalism of a binary lens with a very low-mass com-
panion. Because of the very small mass ratio, planetary a lens-
ing light curve is well described by that of a single lens of the
primary star for most of the event duration. However, a short-
duration perturbation can occur when the source star passes
the region around the caustics, that are the set of source po-
sitions at which the magnification of a point source becomes
infinite. The caustics of binary lensing form a single or multi-
ple sets of closed curves where each of which is composed of
concave curves (fold caustics) that meet at points (cusps).
For a planetary case, there exist two sets of disconnected
caustics: ‘central’ and ‘planetary’ caustics. The single central
http://arxiv.org/abs/0704.0968v1
2 MICROLENSING FOLLOW-UP CRITERIA
caustic is located close to the host star. It has a wedge shape
with four cusps and its size (width along the star-planet axis)
is related to the planet parameters by (Chung et al. 2005)
∆ξcc ∝
(s − 1/s)2
. (1)
For a given mass ratio, a pair of central caustics with sepa-
rations s and s−1 are identical to the first order of approxima-
tion (Dominik 1999; Griest & Safizadeh 1998; An 2005). The
planetary caustic is located away from the host star. The cen-
ter of the planetary caustic is located on the star-planet axis
and the position vector to the center of the planetary caustic
measured from the primary lens position is related to the lens-
source separation vector, s, by
rpc = s
. (2)
Then, the planetary caustic is located on the planet side, i.e.
sign(rpc) = sign(s), when s > 1, and on the opposite side, i.e.
sign(rpc) = −sign(s), when s < 1. When s > 1, there exists a
single planetary caustic and it has a diamond shape with four
cusps. When s< 1, there are two caustics and each caustic has
a triangular shape with three cusps. The size of the planetary
caustic is related to the planet parameters by
∆ξpc ∝
q1/2/(s
s2 − 1) for s > 1,
q1/2(κ0 − 1/κ0 +κ0/s2)cosθ0 for s < 1,
where κ(θ) =
[cos2θ± (s4 − sin2 2θ)1/2]/(s2 − 1/s2)
, θ0 =
[π±sin−1(31/2s2/2)]/2, and κ0 = κ(θ0) (Han 2006). The plan-
etary caustic is always bigger than the central caustic and the
size ratio between the two types of caustics, ∆ξcc/∆ξpc, be-
comes smaller as the mass of the planet becomes smaller and
the planet is located further away from the Einstein ring. The
planetary caustic is located within the Einstein ring of the pri-
mary when the planet is located in the range of separation
from the star of 0.6 . s . 1.6. The size of the caustic, which
is directly proportional to the planet detection efficiency, is
maximized when the planet is located in this range, and thus
this range is called as the ‘lensing zone’. As the position of
the planet approaches to the Einstein ring radius, s → 1, the
location of the planetary caustic approaches the position of
the central caustic. Then, the two types of caustic eventually
merge together, forming a single large one.
3. VARIATION OF DETECTABILITY
3.1. Quantification of Detectability
The quantity that has been often used in the previous es-
timation of the planet detection probability is the ‘fractional
deviation’ of the planetary lensing light curve from that of the
single lensing event of the primary, i.e.,
A − A0
. (4)
With this quantity, however, one cannot consider the variation
of the photometric precision depending on the lensing magni-
fication. In addition, it is difficult to consider the variation of
the detectability depending on the source type.
To consider the effect of source star brightness and its
lensing-induced variation on the planet detection probability,
we carry out our analysis based on a new quantity defined as
the ratio of the fractional deviation, ǫ, to the photometric pre-
cision, σ
, i.e,
D = |ǫ|
ν,S + Fν,B)1/2
(A − 1)F
, (5)
where F
ν,S and Fν,B represent the fluxes from the source star
and blended background stars, respectively. Here we as-
sume that photometry is carried out by using the difference
imaging method (Tomaney & Crotts 1996; Alard 1999). In
this technique, photometry of the lensed source star is con-
ducted on the subtracted image obtained by convolving two
images taken at different times after geometrically and pho-
tometrically aligning them. Then the signal from the lensed
star measured on the subtracted image is the flux variation
of the lensed source star, (A − 1)F
ν,S, while the noise orig-
inates from both the source and background blended stars,
ν,S + Fν,B. Under this definition of the planetary signal de-
tectability,D = 1 implies that the planetary signal is equivalent
to the photometric precision. Hereafter we refer the quantity
D as the ‘detectability’.
3.2. Contour Maps of Detectability
To see the variation of the detectability depending on the
separation parameter s, mass ratio q, and the types of involved
source star, we construct maps of detectability as a function
of the position in the source plane. Figure 1 shows example
maps. The individual sets of panels show the maps for events
associated with different types of source stars. All lengths
are normalized by the angular Einstein radius and ξ and η
represent the coordinates parallel with and normal to the star-
planet axis, respectively. A contours (yellow curve) is drawn
at the level of D = 3.0. The maps are centered at the position
of the primary lens star and the planet is located on the left.
The dotted arc in each panel represents the Einstein ring of the
primary star. The closed figures drawn by red curves represent
the caustics.
For the construction of the maps, we assume a mass of
the primary lens star of m = 0.3 M⊙ and distances to the
lens and source of DL = 6 kpc and DS = 8 kpc, respec-
tively. Then, the corresponding Einstein radius is rE =
(4Gm/c2)[(DL(DS − DL)/DS]
= 1.9 AU. For the source
stars, we test three different types of giant, clump giant, and
main-sequence stars. The assumed I-band absolute magni-
tudes of the individual types of stars are MI = 0.0, 1.5, and
3.6, respectively. With the assumed amount of extinction to-
ward the Galactic bulge field of AI = 1.0, these correspond
to the apparent magnitudes of I = 15.5, 17, and 19.1, respec-
tively. As the source type changes, not only the brightness
but also the size of the star changes. Source size affects the
planetary signal in lensing light curves (Bennett & Rhie 1996)
and thus we take account the finite source effect into consid-
eration. The assumed source radii of the individual types of
source stars are 10.0 R⊙, 3.0 R⊙, and 1.1 R⊙, respectively.
We assume that events are affected by blended flux equivalent
to that of a star with I = 20. We note that the adopted lens
and source parameters are the typical values of Galactic bulge
events that are being detected by the current lensing surveys
(Han & Gould 2003).
For the observational condition, we assume that images are
obtained by using 1 m telescopes, which are typical ones be-
ing used in current follow-up observations. We also assume
that the photon acquisition rate of each telescope is 10 pho-
tons per second for an I = 20 star and a combined image with
HAN 3
FIG. 1.— Contour maps of the detectability of the planetary signal, D, as a function of the position in the source plane for events caused by planetary systems
with various lens-source separations and mass ratios. The detectability represents the ratio of the fractional deviation of the planetary lensing light curve from
the single lensing light curve of the primary to the photometric precision. All lengths are normalized by the angular Einstein radius and ξ and η represent the
coordinates parallel with and normal to the star-planet axis, respectively. The individual sets of panels show the maps for events associated with different types
of source stars. Contours (yellow curve) are drawn at the level of D = 3.0. The maps are centered at the position of the primary lens star and the planet is located
on the left. The dotted arc in each panel represents the Einstein ring of the primary star. The closed figures drawn by red curves represent the caustics. For the
details about the assumed lens parameters and observational conditions, see § 3.2.
FIG. 2.— Geometric representation of the probability of detecting planetary
signals, P. Under the definition of P as the average probability of detecting
planetary signals with a detectability greater than a threshold value Dth at
the time of observation with a magnification A, the probability corresponds
to the portion of the arclet(s) where the detectability is greater than a threshold
value out of a circle around the primary with a radius equal to the lens-source
separation corresponding to the magnification at the time of observation. The
individual circles in the upper panel correspond to the source positions at
which the lensing magnifications are A = 1.5 (pink), 3.0 (cyan), 5.0 (green),
and 10.0 (red), respectively. The curves in the bottom panels show the vari-
ation of the detectability as a function of the position angle (θ) of points on
the circles with corresponding colors in the upper panel. We set the thresh-
old detectability as Dth = 3.0, i.e. 3σ detection of the planetary signal. The
dashed circle represents the Einstein ring.
a total exposure time of 5 minutes is obtained from each set
of observations.
3.3. Probability of Detecting Planetary Signals
Based on the maps of detectability, we then investigate the
probability of detecting planetary signals as a function of the
lensing magnification. We define the probability P as the av-
erage probability of detecting planetary signals with a de-
tectability greater than a threshold value Dth at the time of
observation with a magnification A. Geometrically, this prob-
ability corresponds to the portion of the arclet(s) where the
detectability is greater than a threshold value out of a circle
FIG. 3.— Probability of detecting planetary signals as a function of lens-
ing magnification. The individual panels show the probabilities for events
involved with different types of source stars. The curves in each panel show
the variation of the probability for planets with different mass ratios and sep-
arations. We note that although not presented, the probabilities for planets
with separations s < 1 are similar to those of the corresponding planets with
s−1. The probability is defined the average probability of detecting plane-
tary signals with a detectability greater than a threshold value Dth at the time
of observation with a magnification A. We set the threshold detectability as
Dth = 3.0, i.e. 3σ detection of the planetary signal. We note that there is a
maximum magnification specific to the angular size of the source star and
thus the curves stop at certain magnifications.
around the primary with a radius equal to the lens-source sep-
aration corresponding to the magnification at the time of ob-
servation. This is illustrated in Figure 2. We note that the
magnification is a unique function of the absolute value of the
lens-source separation u1, and thus A = const corresponds to
a circle around the lens. The lens-source separation is related
to the magnification by
u(A) =
(1 − A−2)1/2
. (6)
We set the threshold detectability as Dth = 3.0, i.e. 3σ detec-
tion of the planetary signal.
1 Strictly speaking, the magnification depends additionally on the size of
the source star.
4 MICROLENSING FOLLOW-UP CRITERIA
TABLE 1
LIMITATION BY FINITE-SOURCE EFFECT
source type event type
giant A & 70 for planets with q . 10−3
clump giant A & 200 for planets with q . 5× 10−4
main-sequence A & 500 for planets with q . 10−4
NOTE. — Cases of planetary microlensing events
where detection of planetary signal is limited by finite
source effect. We note that “-” means the respective con-
figuration cannot be realized.
In Figure 3, we present the resulting probability as a func-
tion of magnification. The individual panels show the proba-
bilities for events involved with different types of source stars.
In each panel, we present the variations of the probability for
planets with different mass ratios and separations. We test six
different planetary separations of s = 1/1.6, 1/1.4, 1/1.2, 1.2,
1.4, and 1.6 as representative values for planets in the lensing
zone. For the mass ratio, we test five values of q = 5× 10−3,
10−3, 5× 10−4, 10−4, and 5× 10−5.
From the variation of the probability, we find the follow-
ing tendencies. First, we find that the probability increases
with the increase of the lensing magnification. This is con-
sistent with the result of K. Horne (private communication).
This tendency is due to three factors. First, the size of the
planetary caustic increases as it is located closer to the pri-
mary star. This can be seen in Figure 4, where we present the
relation between the location of the planetary caustic and its
size, which is obtained by using equations (2) and (3). Then,
higher chance of planetary perturbation is expected when the
source is located closer to the primary during which the lens-
ing magnification is high. Second, perturbation regions of
the same size cover a larger range of angle as the planetary
caustic moves closer to the lens. This also contributes to the
higher probability. Third, the photometric precision improves
with the increasing brightness of the source star due to lensing
magnification. As the photometric precision improves, it is
easier to detect small deviations induced by planets. The same
reason can explain the considerable size of the perturbation
region induced by central caustics. Perturbations induced by
the central caustics occur at high magnifications during which
the photometric precision is high. As a result, despite much
smaller size of the central caustic than that of the planetary
caustic, the central perturbation region is considerable and can
even be comparable to the perturbation region induced by the
planetary caustic. This can be seen in the detectability maps
presented in Figure 1.
However, the probability does not continue to increase with
the increase of the magnification. Instead, the probability
drops off rapidly beyond a certain magnification. This critical
value corresponds to the magnification at which finite-source
effect begins to wash out the planetary signal. In Table 1, we
present the cases where finite source effect limits planet detec-
tions. As a result, detections of planets with low mass ratios
would be difficult for events involved with giant source stars
with magnifications A & 70. We note that the finite source
effect also limits the maximum magnifications of events and
thus the curves in Figure 3 discontinue at a certain value.
Second, as the magnification increases, the probability of
detecting planetary signal increases with two dramatically dif-
ferent rates of dP/d logA. We find that this abrupt change
of dP/d logA occurs due to the transition from the regime of
FIG. 4.— Variation of the size of the planetary caustic as a function of its
location. The value rpc represents the separation between the center of the
planetary caustic and the primary lens star. The sign of rpc is positive when
the caustic is on the planet side and vice versa. We note that the caustic size
at around rpc is not presented because the analytic expression in eq. (1) is not
valid in this region. In addition, there is no distinction between the planetary
and central caustics in this region.
perturbations induced by planetary caustics into the one of
perturbations induced by central caustics. The perturbation
region induced by the central caustic forms around the pri-
mary lens and thus the probability becomes very high once the
source star is in the central perturbation regime. The boost of
the increase rate occurs at different magnifications depending
on the planetary parameters and the types of involved source
stars. The critical magnification becomes lower as the mass
ratio of the planet increases and the separation of the planet
approaches the Einstein ring radius. In Table 2, we present
these critical magnifications. An important finding to be noted
is that the critical magnification occurs at moderate magnifi-
cations of . 20 for a significant fraction of events caused by
planetary systems with planets located in the lensing zone.
This implies that probability of detecting planetary signal can
be high even for events with moderate magnifications.
Third, the probability is higher for events involved with
brighter source stars. This is because of the improved pho-
tometric precision with the increase of the source brightness.
The difference in the probability depending on the source type
is especially important at low magnifications. For example,
the probabilities at a magnification of A = 5 for events caused
by a common planetary system with q = 10−3 and s = 1.2 but
associated with different source stars of giant, clump giant,
and main-sequence are P ∼ 20%, 10%, and 1%, respectively.
In the absence of high magnification events, therefore, the sec-
ond prime candidate event for follow-up observation is the
one involved with brightest source star. As the magnification
further increases and once the source star enters the central
perturbation region, the difference becomes less important.
4. SUMMARY AND CONCLUSION
For the purpose of providing useful criteria in the selec-
tion of target events preferable for planetary lensing follow-
up observations, we investigated the variation of the proba-
bility of detecting planetary lensing signals depending on the
observables of the lensing magnification and source bright-
ness. From this investigation, we found consistent result from
previous studies that the probability increases with the in-
crease of the lensing magnification due to the improvement
HAN 5
TABLE 2
CRITICAL MAGNIFICATIONS OF CENTRAL PERTURBATION
source planetary mass ratio
type separation q = 5× 10−3 q = 10−3 q = 5× 10−4 q = 10−4 q = 5× 10−5
s = 1.2, 1/1.2 A ∼ 2.2 A ∼ 7 A ∼ 8 A ∼ 22 A ∼ 22
giant s = 1.4, 1/1.4 A ∼ 2.5 A ∼ 8 A ∼ 12 – –
s = 1.6, 1/1.6 A ∼ 3.5 A ∼ 9 A ∼ 18 – –
clump s = 1.2, 1/1.2 A ∼ 7 A ∼ 8 A ∼ 11 A ∼ 30 A ∼ 60
giant s = 1.4, 1/1.4 A ∼ 8 A ∼ 12 A ∼ 17 A ∼ 60 A ∼ 80
s = 1.6, 1/1.6 A ∼ 9 A ∼ 16 A ∼ 20 A ∼ 745 –
main s = 1.2, 1/1.2 A ∼ 6 A ∼ 11 A ∼ 20 A ∼ 55 A ∼ 100
sequence s = 1.4, 1/1.4 A ∼ 8 A ∼ 20 A ∼ 30 A ∼ 100 A ∼ 150
s = 1.6, 1/1.6 A ∼ 11 A ∼ 30 A ∼ 40 A ∼ 150 A ∼ 200
NOTE. — Critical magnifications at which transition from the regime of perturbations induced
by planetary caustics into the one of perturbations induced by central caustics occur. We note that
the critical magnifications are . 20 in many cases.
of the photometric precision combined with the expansion of
the perturbation region. The increase rate of the probabil-
ity is boosted at a certain magnification at which perturba-
tion caused by the central caustic begins to occur. We found
that this boost occurs at moderate magnifications of A . 20
for a significant fraction of events caused by planetary sys-
tems with planets located in the lensing zone, implying that
probabilities can be high even for events with moderate mag-
nifications. The probability increases with the increase of the
source star brightness. We found that the probability of events
associated with source stars brighter than clump giants is not
negligible even at magnifications as low as A ∼ 5. In the ab-
sence of rare prime target of very high-magnification events
(A & 100), we, therefore, recommend to observe events with
brightest source stars and highest magnifications among the
alerted events. Due to the increase of the source size with the
increase of the brightness, however, the probability rapidly
drops off beyond a certain magnification. As a result, detec-
tions of planets with low mass ratios (q . 10−4) would be dif-
ficult for events involved with giant source stars with magni-
fications A & 70.
This work was supported by the Astrophysical Research
Center for the Structure and Evolution of the Cosmos (ARC-
SEC) of Korea Science and Engineering Foundation (KOSEF)
through Science Research Program (SRC) program.
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