An anomaly detector with immediate feedback to hunt for planets of Earth mass and below by microlensing

Abstract

(abridged) The discovery of OGLE 2005-BLG-390Lb, the first cool rocky/icy exoplanet, impressively demonstrated the sensitivity of the microlensing technique to extra-solar planets below 10 M_earth. A planet of 1 M_earth in the same spot would have provided a detectable deviation with an amplitude of ~ 3 % and a duration of ~ 12 h. An early detection of a deviation could trigger higher-cadence sampling which would have allowed the discovery of an Earth-mass planet in this case. Here, we describe the implementation of an automated anomaly detector, embedded into the eSTAR system, that profits from immediate feedback provided by the robotic telescopes that form the RoboNet-1.0 network. It went into operation for the 2007 microlensing observing season. As part of our discussion about an optimal strategy for planet detection, we shed some new light on whether concentrating on highly-magnified events is promising and planets in the 'resonant' angular separation equal to the angular Einstein radius are revealed most easily. Given that sub-Neptune mass planets can be considered being common around the host stars probed by microlensing (preferentially M- and K-dwarfs), the higher number of events that can be monitored with a network of 2m telescopes and the increased detection efficiency for planets below 5 M_earth arising from an optimized strategy gives a common effort of current microlensing campaigns a fair chance to detect an Earth-mass planet (from the ground) ahead of the COROT or Kepler missions. The detection limit of gravitational microlensing extends even below 0.1 M_earth, but such planets are not very likely to be detected from current campaigns. However, these will be within the reach of high-cadence monitoring with a network of wide-field telescopes or a space-based telescope. Comment: 13 pages, 4 figures and 1 table. Accepted for publication in MNRAS

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arXiv:0706.2566v1 [astro-ph] 18 Jun 2007
Mon. Not. R. Astron. Soc. 000, 000–000 (0000)Printed 1 February 2008(MN LATEX style file v2.2)
An anomaly detector with immediate feedback to hunt for planets of
Earth mass and below by microlensing
M. Dominik,1⋆† N. J. Rattenbury,2A. Allan,3S. Mao,2D. M. Bramich,4
M. J. Burgdorf,5E. Kerins,2Y. Tsapras,5and Ł. Wyrzykowski6,7
1SUPA, University of St Andrews, School of Physics & Astronomy, North Haugh, St Andrews, KY16 9SS, United Kingdom
2Jodrell Bank Observatory, Macclesfield, Cheshire, SK11 9DL, United Kingdom
3School of Physics, University of Exeter, Stocker Road, Exeter EX4 4QL, United Kingdom
4Isaac Newton Group of Telescopes, Apartado de Correos 321, 38700 Santa Cruz de La Palma, Canary Islands, Spain
5Astrophysics Research Institute, Liverpool John Moores University, Twelve Quays House, Egerton Wharf, Birkenhead, CH41 1LD, United Kingdom
6Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA, United Kingdom
7Warsaw University Astronomical Observatory, Al. Ujazdowskie 4, 00-478 Warszawa, Poland
1 February 2008
ABSTRACT
The discovery of OGLE 2005-BLG-390Lb, the first cool rocky/icy exoplanet, impressively
demonstrated the sensitivity of the microlensing technique to extra-solar planets below
10 M⊕. A planet of 1 M⊕instead of the expected 5 M⊕for OGLE 2005-BLG-390Lb (with
an uncertainty factor of two) in the same spot would have provided a detectable deviation
with an amplitude of ∼ 3 per cent and a duration of ∼ 12 h. While a standard sampling
interval of 1.5 to 2.5 hours for microlensing follow-up observations appears to be insuffi-
cient for characterizing such light curve anomalies and thereby claiming the discovery of the
planets that caused these, an early detection of a deviation could trigger higher-cadence sam-
pling which would have allowed the discovery of an Earth-mass planet in this case. Here, we
describe the implementation of an automated anomaly detector, embedded into the eSTAR
system, that profits from immediate feedback provided by the robotic telescopes that form
the RoboNet-1.0 network. It went into operation for the 2007 microlensing observing season.
As part of our discussion about an optimal strategy for planet detection, we shed some new
light on whether concentrating on highly-magnified events is promising and planets in the
’resonant’ angular separation equal to the angular Einstein radius are revealed most easily.
Given that sub-Neptune mass planets can be considered being common around the host stars
probed by microlensing (preferentially M- and K-dwarfs), the higher number of events that
can be monitored with a network of 2m telescopes and the increased detection efficiency for
planets below 5 M⊕arising from an optimized strategy gives a common effort of current mi-
crolensing campaigns a fair chance to detect an Earth-mass planet (from the ground) ahead
of the COROT or Kepler missions. The detection limit of gravitational microlensing extends
even below 0.1 M⊕, but such planets are not very likely to be detected from current cam-
paigns. However, these will be within the reach of high-cadence monitoring with a network
of wide-field telescopes or a space-based telescope.
Key words: planetary systems – gravitational lensing – methods: observational.
1 INTRODUCTION
After Mao & Paczy´ nski (1991) first pointed out that microlens-
ing events can be used to infer the presence of extra-solar plan-
ets or place limits on their abundance, this technique has now
become established with several claimed detections (Bond et al.
2004; Udalski et al. 2005; Beaulieu et al. 2006; Gould et al. 2006).
⋆Royal Society University Research Fellow
† E-mail: md35@st-andrews.ac.uk
The discovery of OGLE 2005-BLG-390Lb (Beaulieu et al. 2006;
Dominik et al. 2006), estimated to be 5 times more massive than
Earth, with an uncertainty factor of two, under the lead of the
PLANET (Probing Lensing Anomalies NETwork)/RoboNet cam-
paign demonstrated that microlensing not only can detect massive
gas giants, but also planets that harbour a rocky/icy surface under a
thin atmosphere. Moreover, it provided the first observational hint
that cool rocky/icy planets are actually quite common, as previ-
ously predicted by simulations based on core-accretion models of
planet formation (Ida & Lin 2005).

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2
M. Dominik et al.
It was already estimated by Bennett & Rhie (1996) that there
is a non-negligible chance of 1–2 per cent for detecting an Earth-
mass planet located at about 2 AU from its host star by means of
observing a few-per-cent deviation in a microlensing light curve.
However, such a discovery requires photometric measurements on
a few hundred microlensing events, assuming that a fair fraction of
the host stars are orbited by such planets.
A sufficient number of events can only arise from monitoring
dense fields of stars. With a probability of ∼ 10−6for a star in the
Galactic bulge being magnified by more than 34 per cent at any
given time due to the bending of light caused by the gravitational
field of an intervening foreground star (Kiraga & Paczy´ nski 1994),
and such a microlensing event lasting of the order of a month, one
namely needs to monitor 107to108stars. Thiswasachieved by mi-
crolensing surveys like OGLE (Optical Gravitational Lensing Ex-
periment) (Udalski et al. 1992), MACHO (MAssive Compact Halo
Objects) (Alcock et al. 1993), EROS (Exp´ erience de la Recherche
d’Objets Sombres) (Aubourg et al. 1993) and MOA (Microlensing
Observations in Astrophysics) (Muraki et al. 1999) with a roughly
daily sampling. Moreover, all these surveys have been equipped
with real-time alert systems (Udalski et al. 1994; Udalski 2003;
Alcock et al. 1996; Glicenstein 2001; Bond et al. 2001) that notify
the scientific community about ongoing microlensing events. This
allowstoschedule follow-up observations that provide anincreased
photometric accuracy, a denser event sampling, and/or coverage
during epochs outside the target visibility from the telescope site
used by the respective survey campaign.
The PLANET (Probing Lensing Anomalies NETwork) col-
laboration1established the first telescope network capable of
round-the-clock nearly-continuous high-precision monitoring of
microlensing events (Albrow et al. 1998) with the goal to detect
gas giant planets and to determine their abundance. For being able
to detect deviations of 5 per cent, PLANET aims at a 1-2 per cent
photometric accuracy. Witha typical sampling interval of 1.5 to 2.5
hrs allowing a characterization of planetary anomalies on the basis
of at least 10-15 data points taken while these last, the required ex-
posure time then limitsthe number of events that can be monitored.
For bright (giant) stars, exposure times of a few minutes are suffi-
cient, so that PLANET can monitor about 20 events each night or
75 events per observing season, but this reduces to about 6 events
each night or 20 events per season for fainter stars, for which expo-
sure times reach 20 min (Dominik et al. 2002). In 1999, MACHO
and OGLE-II together provided about 100 microlensing alerts, out
of which only 7 were on giant source stars. This severely limited
PLANET in its planet detection capabilities: rather than 75 events,
only about 25 could be monitored per season. The OGLE-III up-
grade, in effect from 2002, had a major impact on the potential
of microlensing planet searches, paving the way towards the now
nearly 1000 microlensing events per year provided by the alert sys-
tems of the OGLE2and MOA3surveys. The much larger number
of events arising from this upgrade allowed OGLE itself to obtain
meaningful constraints on planets of Jupiter mass (Tsapras et al.
2003; Snodgrass et al. 2004), while OGLE and MOA have even
demonstrated that such planets can in fact be detected by their sur-
veys (Bond et al. 2004). However, for studying less massive plan-
ets, their sampling is insufficient. At the same time, the OGLE-III
upgrade enabled PLANET to exploit its full theoretical capability,
1http://planet.iap.fr
2http://ogle.astrouw.edu.pl/ogle3/ews/ews.html
3http://www.massey.ac.nz/˜iabond/alert/alert.html
and moreover, it gave PLANET a reliable chance to detect planets
of a few Earth masses provided that these are not rare around the
stars that cause the microlensing events. The discovery of OGLE
2005-BLG-390Lb (Beaulieu et al. 2006; Dominik et al. 2006) ex-
plicitly proved the sensitivity of the PLANET observations to plan-
ets in that mass range.
Microlensing events are also regularly monitored by the
MicroFUN (Microlensing Follow-Up Network) team4. However,
rather than exploiting a permanent network, MicroFUN concen-
trates on particularly promising events and activates target-of-
opportunity observations should such an event be in progress. Be-
sides 1m-class telescopes, their stand-by network includes a larger
number of small (down to 0.3m diameter) telescopes operated by
amateur astronomers, which are well suited to observe the peaks of
events over which the source star makes a bright target.
Since the PLANET network is restricted in its capabilities
of monitoring ∼25 per cent of the currently alerted events with
the observational requirements, the planet detection rate could be
boosted by using larger (2m) telescopes or clusters of 1m-class
telescopes. In fact, such an upgrade is required in order to ob-
tain a sample that allows a reliable test of models of the forma-
tion and evolution of planets around K- and M-dwarfs. RoboNet-
1.05(Burgdorf et al. 2007) marks the prototype of a network of 2m
robotic telescopes, not only allowing a fast response time, but also
a flexible scheduling by means of the multi-agent contract model
provided by the eSTAR project6(Allan, Naylor & Saunders 2006;
Allan et al., 2006). eSTAR is a key player in the Heterogeneous
Telescope Networks (HTN) consortium and involved in the IVOA
(International Virtual Observatory Alliance) standards process.
If one aims at the discovery of Earth-mass planets, the stan-
dard follow-up sampling of 1.5 hrs usually does not produce the
amount of data required to characterize the corresponding signals,
and with less frequent sampling one even faces a significant risk
of missing any hint for a deviation from an ordinary microlensing
light curve. However, planets of Earth mass and even below can be
discovered by shortening the sampling interval to ∼10 min once a
regularly sampled point is suspected to depart from a model light
curve that represents a system without planet. In order to properly
trigger such anomaly alerts, all incoming data need to be checked
immediately, and prompt action needs to be taken within less than
∼15 min. The amount of data and the required response time for
achieving a good detection efficiency for Earth-mass planets are
however prohibitive for relying on human inspection. Therefore,
we here describe the implementation of an automated anomaly de-
tector that exploits the opportunities of immediate response and
flexible scheduling of a network of robotic telescopes. A first sim-
ilar warning system, dubbed EEWS, had been installed by OGLE
in 2003 (Udalski 2003), which however involves further human in-
spection and operates with a single telescope. In contrast, our de-
sign needs to succeed without any human intervention and take
care of a heterogeneous telescope network. The underlying algo-
rithm follows previous experience on the assessment of anomalies.
We explicitly aim at reaching a significant detection efficiency to
Earth-mass planets with the current survey/follow-up strategy of
microlensing planet searches.
This paper is organized as follows. In Sect. 2 we describe the
modelling of ordinary microlensing events with particular empha-
4http://www.astronomy.ohio-state.edu/˜microfun/
5http://www.astro.livjm.ac.uk/RoboNet/
6http://www.estar.org.uk

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A microlensing hunt for Earth-mass planets
3
sis on the importance of robust parameter estimates, not confused
by outliers, in order to properly identify real deviations. While
Sect. 3 deals with the general strategy for detecting low-mass plan-
ets by microlensing, we derive a suitable concept for an anomaly
detector in Sect. 4. The embedding of the SIGNALMEN anomaly
detector, that went into operation for the 2007 microlensing cam-
paign, into the eSTAR project is discussed in Sect. 5, before its al-
gorithm is described in Sect. 6. Sect. 7 then discusses the prospects
of the SIGNALMEN anomaly detector for discovering planets of
Earth mass and below. In Sect. 8, we provide a short summary and
final conclusions.
The Appendix makes a point on the inability to detect planets
at the resonant separation in some of the observed events that was
not discussed earlier.
2 ORDINARY LIGHT CURVES AND ANOMALIES
The bending of light due to the gravitational field of a foreground
’lens’ star with mass M at distance DL causes an observed back-
ground ’source’ star at distance DS to be magnified by (Einstein
1936)
A(u) =
u2+ 2
u√u2+ 4,
(1)
if both objects are separated on the sky by the angle uθE with θE
denoting the angular Einstein radius
θE =
?
With the assumption that lens and source star move uniformly,
where µ is the absolute value of their relative proper motion, the
separation angle can be parametrized as
4GM
c2
(D−1
L
− D−1
S).
(2)
u(t) =
?
u2
0+
?t − t0
tE
?2
,
(3)
with u0 denotes the closest approach at epoch t0, and tE = θE/µ
is a characteristic event time-scale.
Each set of observations with a specific telescope and filter
comprises a data archive s of observed fluxes F[s]
bars σFi[s] at epochs t[s]
flux F[s]
light curves
i
and their error
Sand background
i. Withthe source flux F[s]
Bdepending on the data archive s, one observes symmetric
F[s](t) = F[s]
SA[u(t)] + F[s]
B
(4)
peaking at t0.
Estimates for (t0,tE,u0,F[s]
minimizing
S,F[s]
B) can then be obtained by
χ2=
m
?
k=1
n[k]
?
i=1
?
F[k](t) − F[k]
σF[k]
i
i
?2
.
(5)
While we use the CERN library routine MINUIT for determining
(t0,tE,u0),the source and background fluxes F[s]
choice of (t0,tE,u0) simply follow from linear regression as
Sand F[s]
Bfor any
FS
=
?A(ti)Fi
?[A(ti)]2
σ2
i
?
?
1
σ2
i−?A(ti)
1
σ2
σ2
i
?Fi
σ2
i
σ2
i
σ2
ii−
??A(ti)
?2,
FB
=
?[A(ti)]2
?[A(ti)]2
σ2
i
?Fi
σ2
i
σ2
i−?A(ti)
?
σ2
i
?A(ti)Fi
σ2
i
σ2
i
1
σ2
i−
??A(ti)
?2
,
(6)
where the summations run from 1 to n[k], σi ≡ σFi, and the in-
dex [k] has been dropped. Any archive s can only be included if it
contains at least 3 data points.
The characteristic form of the light curve described by Eq. (4)
is based on the assumption that both source and lens star are sin-
gle point-like objects that are moving uniformly with respect to
each other as seen from Earth. Apart from planets orbiting the lens
star, significant deviations, so-called anomalies, can however also
be caused by binarity or multiplicity of lens or source, the finite
angular size of the stars, or the revolution of the Earth (parallax
effect).
Since it is our primary goal to detect light curve anomalies,
it is essential to ensure that our adopted model is reasonably cor-
rect. However, frequently our data do not allow strong constraints
to be placed on the model, in particular during early phases of the
event. It is a well-known fact that OGLE announce a fair fraction
of their events with the prediction of quite high peak magnification,
whereas it turns out later that most of these peak at much lower
magnifications. As studied in some detail by Albrow (2004), this is
related to the fact that χ2-minimization is equivalent to obtaining a
maximum-likelihood estimate of the model parameters if the data
are assumed to follow a Gaussian distribution, which is biased, i.e.
its expectation value does not coincide with the true expectation
value of the considered quantity. Using the statistics of previously
observed OGLE events, a Bayesian estimate that can be obtained
by adding an effective penalty function to χ2comes closer to the
expectation value (Albrow 2004). While the estimated value can be
tuned by this, one does not fully get around the problem of large
indeterminacy of the model parameters.
A further problem arises from the necessity to avoid that our
model is driven towards data outliers. Otherwise, real anomalies
would be missed while points matching an ordinary light curve
would seem deviant. As a consequence, we would face the problem
ofnot beingabletodistinguishbetweenongoing anomalies andfur-
ther data requiring an adjustment of model parameters. Therefore,
we apply a more sophisticated algorithm for estimating the model
parameters that is rather invulnerable to outliers.
The model can be made to follow the bulk of the data by
downweighting points according to their respective residual (e.g.
Hoaglin, Mosteller & Tukey 1983) as follows. With the residuals
r[k]
i
=F[k](t) − F[k]
σF[k]
i
i
(7)
and the median of their absolute values ˜ r[k]for each data archive,
we give further (bi-square) weight
w[k]
i
=





?
1 −
?
r[k]
i
K ˜ r[k]
?2?2
for
|r[k]
|r[k]
i| < K ˜ r[k]
0
for
i| ? K ˜ r[k]
(8)
to each data point, where we adopt K = 6 for the tuning constant.
The choice of the weights, Eq. (8), means that data points whose
absolute residuals exceeds K times their median are ignored. This
procedure is repeated until the formal χ2converges. However, we
need todeal withnon-linear models which are prone to several pos-
sible χ2minima. In contrast to linear models, it can therefore hap-
pen that this procedure leads to periodic switching between differ-

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4
M. Dominik et al.
ent minima, where nevertheless a subsequence converges to each
of these. In this case, we have to live with the absence of a unique
minimum and choose that one with the lowest χ2. With the formal
χ2not being dominated by outliers, we can also reliably adjust the
relative weight between different data archives k after each itera-
tion step, so that all (χ2)[k]/n[k]coincide, preventing the estima-
tion of model parameters being influenced by the collective over-
or underestimation of error bars.
3 DETECTION OF LOW-MASS PLANETS
It was pointed out by Mao & Paczy´ nski (1991) that planets orbit-
ing the lens star can reveal their existence by causing significant
deviations to microlensing light curves. They also found that the
probability to detect a planet becomes resonant if the angular sep-
aration from its host star is comparable to the angular Einstein
radius θE, which reflects the fact that the detection of planets is
aided by the tidal field of their host star. However, as pointed out
in the Appendix, for a given event, in particular for larger im-
pact parameters, the detection probability of smaller planets can
actually drop to zero for angular separations close to θE rather
than reaching a maximum. In such case, only slightly wider or
closer separations can be probed. It is a lucky coincidence that the
gravitational radius of stars and distances within the Milky Way
combine in such a way that the angular Einstein radius converts
to a projected separation DLθE ∼ 2 AU for M = 0.3 M⊙,
the typical mass of the lens stars, assuming DS ∼ 8.5 kpc and
DL ∼ 6.5 kpc. Gould & Loeb (1992) quantified the prospects
for detecting planets from microlensing signatures by finding that
Jupiter-mass planets distributed uniformly within angular separa-
tions 0.6 θE ? dθE ? 1.6 θE, comprising the so-called lensing
zone, have a probability of 15 per cent of being detected among
microlensing events with peak magnifications A0 ? 1.34, corre-
sponding to the source entering the Einstein ring (of angular radius
θE) of the lens star, i.e. u0 ? 1. As shown by Griest & Safizadeh
(1998), this probability increases significantly if one restricts the
attention to events with larger peak magnifications, where about
80 per cent is reached for A0 ? 10. Since the area subtended on
the sky by angular source positions that correspond to a significant
deviation decreases towards smaller planet masses, both a shorter
duration of the planetary signal and a smaller probability toobserve
it result. In contrast, the signal amplitude is only limited by the fi-
nite angular size of the source, where significant signal reductions
start arising once it becomes comparable or larger than the size of
the region for which a point source provides a significant deviation.
However, Bennett & Rhie (1996) estimatedthat Earth-mass planets
still have a 1–2 per cent chance of providing a signal in excess of a
few per cent.
Planets around the lens star affect the light curve only by
means of two dimensionless parameters, namely the planet-to-star
mass ratio q and the separation parameter d, where dθE is the in-
stantaneous angular separation of the planets from its host star (i.e.
the lens star). With typical relative proper motions between lens
and source stars of µ ∼ 15 µasd−1, microlensing events on Galac-
tic bulge stars are usually observable for about a month or two,
whereas planetary deviations last between a few hours and a few
days, depending on the mass of the planet. In contrast to other in-
direct techniques, microlensing therefore obtains a snapshot mea-
surement of the planet rather than having to wait for it to complete
its orbit. This gives microlensing the unique capability of probing
planets in wide orbits whose periods otherwise easily exceed the
life-time of a project or its investigator.
With many events on offer from the OGLE and MOA sur-
veys and only limited resources available for follow-up observa-
tions, one needs to make a choice which of these to monitor and
how frequently to sample each event. With the goal to maximize
the number of detections of planetary deviations, a prioritizational-
gorithm that spreads the available observing time over the potential
targetshasbeen devised byHorne(2007),which formsacentral en-
gine of the RoboNet observing strategy. Any such strategy must be
based on observables, model parameters arising from the collected
data, or any other data statistics. As Horne (2007) pointed out, each
data point carries a detection zone with it, composed of the angular
positions for which a planet would have caused a detectable devia-
tion. Unless finite-source effects begin diminishing the detectabil-
ity of planets (Han 2007), detection zones grow with the current
magnification. Moreover, the same photometric accuracy can be
achieved with smaller exposure times for brighter targets. An ef-
ficient prioritization algorithm therefore needs to be based on both
the current magnification and brightness along with the time when
the last observation was carried out, where taking into account the
latter avoids obtaining redundant information. Such a prioritization
of events however does not consider how well an observed devia-
tion allows to constrain its nature of origin and it also assumes that
the model parameters of the ordinary light curve are known exactly.
If the effect on the microlensing light curve is dominated by
a single planet, the lens system can be fairly approximated as a
binary system consisting of the star and this planet. Gravitational
lensing by a binary point-mass lens has been studied in great de-
tail for equal masses by Schneider & Weiß (1986) and later gen-
eralized for arbitrary mass ratios by Erdl & Schneider (1993). On
the other hand, Chang & Refsdal (1979) have discussed lensing by
bodies of different mass scales. While their target of interest was
the brightness variation of individual images of QSOsthat are grav-
itationally lensed by an intervening galaxy, a very similar situation
arises for planets orbiting a lens star. Similarly to individual stars
in the galaxy splitting an image due to lensing by the galaxy as
a whole into ’micro-lensing’, a planet can further split one of the
two images due to lensing by its host star if it roughly coincides
in angular position with that image. Dominik (1999) has further
investigated the transition towards extreme mass ratios and shown
how the case described by Chang & Refsdal (1979), the so-called
Chang-Refsdal lens, isapproached. The derived expansions intose-
ries have later been used by Bozza (1999) for discussing the case of
multiple planets. Binary lenses in general and planetary systems in
particular createasystem of extended caustics, consistingof thean-
gular positions forwhich apoint-likesource starwould beinfinitely
magnified. Whilesufficientlysmallsources passing thecaustics can
provide quite spectacular signals, planets are more likely to already
reveal their existence on entering a much larger region surrounding
these.
Forlessmassiveplanets,thereareusually twoseparateregions
for positions of the source star that lead to detectable planetary sig-
nals, which are related to two types of caustics. Only if the angular
separation of the planet from its host star is in a close vicinity to
the angular Einstein radius θE, where the corresponding range is
broader for more massive planets, a single caustic results and these
regions merge. Otherwise, there are one or two planetary caustics
which are located around positions for which bending of its light
due to the gravitational field of the lens star causes the source to
have image at the position of the planet, and a central caustic which
can be found near the lens star (Griest & Safizadeh 1998; Dominik

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A microlensing hunt for Earth-mass planets
5
Figure 1. Model light curve of microlensing event OGLE 2005-BLG-390
along with data taken with the Danish 1.54m at ESO LaSilla (Chile), red,
the Perth 0.6m (Western Australia), blue, and the Canopus 1.0m (Tas-
mania), cyan, by PLANET, the Faulkes North 2.0m (Hawaii), green, by
RoboNet-1.0, the OGLE 1.3m (Chile), black, and the MOA 0.6m (New
Zealand), brown, where ∆m = 2.5 lgA(t) has been plotted along with
mi = 2.5lgAi. The ∼ 15 per cent deviation lasting about a day re-
vealed the existence of a planet with m ∼ 5.5 M⊕(uncertain to a factor
two), while an Earth-mass planet in the same spot would have caused a 3
per cent deviation lasting about 12 hours (thin line). The time-scale of this
event is tE= 11.0 d, while d = 1.610 and q = 7.6 × 10−5. Moreover,
u0= 0.359, t0= 31.231 July 2005 UT, and the angle between the vector
from the planet to its host star and the source trajectory is α = 157.9◦,
where the less centre of mass is to the right hand side. Finally, the source
star moves by its own radius relative to the lens within t⋆ = 0.282 d. The
dotted line refers to a model light curve in the absence of a planet.
1999). As Bozza (1999) demonstrated, the planetary caustics asso-
ciated with different planets are almost always separated and any
kind of interference between these is quite unlikely. In contrast,
Gaudi et al. (1998) pointed out that the central caustic is always af-
fected by the combined action of all planets. However, it is likely,
although not guaranteed, that there is a hierarchical order among
the effects of different planets, so that a linear superposition is a
fair approximation (Rattenbury et al. 2002; Han 2005).
While the absence of any deviations near the peak of extreme
highly-magnified ordinary events that are related to the source po-
tentially approaching the central caustic poses strict limits on the
abundance of low-mass planets (Abe et al. 2004; Dong et al. 2006),
their actual discovery from this kind of deviations suffers from
several complications. While the linear size of the detection re-
gion around planetary caustics scales with the square root of the
planet mass, it is proportional to the planet mass itself for the
central caustic (Chang & Refsdal 1979; Griest & Safizadeh 1998;
Dominik 1999; Chung et al. 2005; Han 2006). Therefore, the finite
angular size of the source star is more likely to cause a significant
reduction of the signal amplitude. Moreover, the characterization
of the nature of origin for such deviations is significantly more dif-
ficult than for deviations related to planetary caustics. The latter
provide further information by means of the time elapsed between
the peak of the background ordinary light curve and the deviation,
whereascentral-caustic deviations involveahigher degree of model
degeneracies with more prominent finite-source and parallax ef-
fects. In any case, a promising sensitivity to Earth-mass planets is
only reached for lens-source impact parameters u0 ? 5 × 10−4,
which occur at a rate of less than one per year.
On the other hand, the non-negligible probability to detect
planetary signals if the source passes in the vicinity of planetary
caustics offers a fair chance of detecting a planet of Earth-mass by
also making use of the large number of events that exhibit lower
magnifications at a given time. Given these facts, it is not a surprise
that the first sub-Neptune mass planet whose existence could be
reported on the basis of microlensing observations, OGLE 2005-
BLG-390Lb (Beaulieu et al. 2006), produced a 15 to 20 per cent
signal at a magnification A ∼ 1.3 about 10 days after an observed
peak at magnification A0 ∼ 3 (see Fig. 1) rather than a deviation
within a highly-magnified peak.
While the mass of OGLE 2005-BLG-390Lb is about 5 M⊕,
uncertain to about a factor of two (Dominik 2006), a planet of
1 M⊕ in the same spot would still have produced a signal with
an amplitude of ∼ 3 per cent, lasting ∼ 12 h rather than about
twice that long. The actual sampling would have been insufficient
for discovering such a planet in this configuration, but the situation
would have been different had we decreased our sampling inter-
val to 10-15 min on the suspicion of a first deviation. This case
explicitly shows how an anomaly detector can help us in not miss-
ing short-lasting small deviations (related to low-mass planets). By
requiring an initial sampling that is just dense enough for an on-
going anomaly being alerted before most of it has passed, it more-
over allows to monitor a sufficient number of events for provid-
ing a reasonable number of planet discoveries. The main gain of
the anomaly detector will indeed be achieved for detecting planets
from perturbations related to planetary caustics at lower and mod-
erate magnification, whereas a high-cadence sampling can already
be scheduled a-priori for (predictable) high magnifications without
the need for any further alert.
The ability of detecting an anomaly depends on how well ear-
lier data constrain the model describing an ordinary light curve. For
large model parameter uncertainties, it becomes hard to distinguish
a real deviation from a necessary model revision due to a previous
misestimate, for which χ2adjustments are not a reliable indica-
tor due to the intricate parameter space and poor knowledge about
the measurement uncertainties. Therefore, the anomaly detection is
more efficient after the peak of a microlensing has passed rather
than prior to it (c.f. Udalski 2003), where the ability is particularly
vulnerable to data gaps. Thus, if the increased detection efficiency
for low-mass planets that is achieved by means of the anomaly de-
tector is a relevant goal for a monitoring strategy, it is sensible to
give preference to events past peak over those pre peak for compa-
rablemagnifications. Although itismoredifficulttodecide whether
a deviation from a previous model is real or due to a model mises-
timate if constraints on its parameters are weaker, it is more likely
that a suspected deviation occurs and is reported. This has the by-
effect that more data will be collected in this case, which in turn
strengthens the model parameter constraints. Despite the fact that
the higher magnification around the peak allows for accurate data
being taken with shorter exposure times, the weak constraints on
the position of the peak make it rather difficult to detect an ongoing
anomaly there, unless the peak region is monitored quite densely
and no data gaps occur.
4 CONCEPT FOR AN ANOMALY DETECTOR
If reported data deviate from the expected light curve, this could
either mean that there is a real effect, the deviation could be of sta-
tistical nature, or the data could simply be erratic by any means. It
is therefore impossible to arrive at an appropriate judgement about
the presence of anomalies on the basis of a single deviating data