Astrophysical Applications of Gravitational Microlensing Dissertation Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Subo Dong, M.S. Graduate Program in Astronomy The Ohio State University 2009 Dissertation Committee: Professor Andrew Philip Gould, Advisor Professor Bernard Scott Gaudi Professor Krzysztof Zbigniew Stanek
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Astrophysical Applications of Gravitational Microlensing
Dissertation
Presented in Partial Fulfillment of the Requirements for the Degree Doctor ofPhilosophy in the Graduate School of The Ohio State University
By
Subo Dong, M.S.
Graduate Program in Astronomy
The Ohio State University
2009
Dissertation Committee:
Professor Andrew Philip Gould, Advisor
Professor Bernard Scott Gaudi
Professor Krzysztof Zbigniew Stanek
Copyright by
Subo Dong
2009
ABSTRACT
In this thesis, I present several astrophysical applications of Galactic and
cosmological microlensing.
The first few topics are on searching and characterizing extrasolar planets
by means of high-magnification microlensing events. The detection efficiency
analysis of the Amax ∼ 3000 event OGLE-2004-BLG-343 is presented. Due to
human error, intensive monitoring did not begin until 43 minutes after peak, at
which point the magnification had fallen to A ∼ 1200. It is shown that, had a
similar event been well sampled over the peak, it would have been sensitive to
almost all Neptune-mass planets over a factor of 5 in projected separation and even
would have had some sensitivity to Earth-mass planets. New algorithms optimized
for fast evaluation of binary-lens models with finite-sources effects have been
developed. These algorithms have enabled efficient and thorough parameter-space
searches in modeling planetary high-magnification events. The detection of the
cool, Jovian-mass planet MOA-2007-BLG-400Lb, discovered from an Amax = 628
event with severe finite-source effects, is reported. Detailed analysis yields a
fairly precise planet/star mass ratio of q = (2.5+0.5−0.3) × 10−3, while the planet/star
ii
projected separation is subject to a strong close/wide degeneracy. Photometric
and astrometric measurements from Hubble Space Telescope, as well as constraints
from higher order effects extracted from the ground-based light curve (microlens
parallax, planetary orbital motion and finite-source effects) are used to constrain the
nature of planetary event OGLE-2005-BLG-071Lb. Our primary analysis leads to
the conclusion that the host is an M = 0.46 ± 0.04 M⊙ M dwarf and that the planet
has mass Mp = 3.8 ± 0.4 MJupiter, which is likely to be the most massive planet yet
discovered that is hosted by an M dwarf.
Next a spaced-based microlens parallax is determined for the first time using
Spitzer and ground-based observations for binary-lens event OGLE-2005-SMC-001.
The parallax measurement yields a projected velocity v ∼ 230 km s−1, the typical
value expected for halo lenses, but an order of magnitude smaller than would be
expected for lenses lying in the Small Magellanic Cloud (SMC) itself.
Finally, I propose using quasar microlensing to probe Mg II and other
absorption “cloudlets” with sizes ∼ 10−4.0 − 10−2.0pc in the intergalactic medium.
I show that significant spectral variability on timescales of months to years can be
induced by such small-scale absorption “cloudlets” toward a microlensed quasar.
With numerical simulations, I demonstrate the feasibility of applying this method
to Q2237+0305, and I show that high-resolution spectra of this quasar in the near
future would provide a clear test of the existence of such metal-line absorbing
“cloudlets”.
iii
Dedicated to my mother and father
iv
ACKNOWLEDGMENTS
In modern times, most of the necessities of people’s daily lives rely upon others,
therefore, no matter how highly the lofty ideal of individual freedom is acclaimed,
most people have to choose their jobs according to the needs of the society rather
than following their inner voices (i.e., what they are interested in the most). I
consider myself very lucky to have five years’ opportunity to pursue my childhood
dream in becoming an astronomer with no financial burden. I am thankful to
everyone who has made this possible.
I thank Andy Gould for being an unparalleled advisor. He has taken immense
care in guiding me along my path to becoming a scientist. His influence and help
have permeated every aspect of my scientific activities as a graduate student.
Whenever I need discussions or advice (or whenever he perceives me as needing
them), he is always ready to offer them in most timely fashion. His excitement by
new findings, creativity in novel ideas, and self-driven “Gung-ho” working spirit
exhibit no trace of his age. It is difficult to describe how exhilarating an experience
it is to work with him.
v
I thank Andy for being a truthful person – for being truthful to himself, to
others, and to nature. I was initially intimidated by the way he spoke in morning
coffee and colloquia. But I have come to realize that, if one is offended by sincere
efforts to pursue the truth, or not offended by distortion or fabrication of facts and
careless or superficial analyses, he/she is not a real scientist. I thank Andy for
teaching me to be honest to data, to listen to what nature has to say rather than
projecting one’s prejudices onto nature.
I came to the United States certainly at an interesting time, when “the best and
brightest” in the White House and on the Wall Street have drastically transformed
the world’s social and economic orders. As a “Stranger in a Strange Land”, I
appreciate Andy for many illuminating discussions that shed light on the perplexing
and intriguing events in the U.S. and on the world stage. He observes the society at a
vantage point beyond his ethnical, national, and cultural background. I never have to
worry about offending him by expressing my honest views. I thank Andy for rallying
a group of people (mostly astronomers) to have discussions on real-world events in
a similar fashion as we discuss science. We attempt to distinguish between real and
bogus information from various sources and analyze the events by confronting ideas
against distilled evidence and data. Although I do not always agree with Andy, it
has always been such a great time talking to him on almost anything.
Next I thank my thesis committee members, Scott Gaudi and Kris Stanek.
vi
I first met Scott back when he was a postdoc at CfA. He has been providing
unreserved encouragement and support to me since then. I have benefited greatly
from his advice on science and professional development. I thank him for having
led me to see the bright side when I was in depression or frustration. I admire
his comprehensive knowledge in almost all branches of astronomy and his often
amazingly deep physical intuition.
I thank Kris for being friendly and humorous and willing to talk to me on any
topic. I thank him for inspiring discussions that led to my first original workable
science idea. I appreciate his insistence that scientific pictures should fundamentally
be simple. He always has great insights into complexity and sees through irrelevant
artificial obscuration. I hope I will be able to acquire even a fraction of his ability in
seizing the moments by recognizing fleeting opportunities in astronomy.
I am grateful to Rick Pogge, Chris Kochanek and many others in the department
for providing a lot of great technical help and having interesting science discussions
together. And I greatly appreciate many faculty in the department for creating an
intellectually stimulating environment which is also very student-friendly. Daily
coffee has become an indispensable part of my life. The department staff are
always helpful, and I especially thank David Will for solving many of my computer
problems.
vii
I have spent a great time with many graduate students and several postdocs at
OSU. I first thank Zheng for having provided me many great advice when I first got
to OSU, and for him being a role-model student and astronomer. He shows me how
high someone with a similar background as me can possibly achieve in astronomy. I
thank Jose, Vimal, Dale, Himel and others for having a lot of interesting discussions
and experience together. And I appreciate having many great office mates, Frank,
Molly, Ondrej, Kelly, Rob, Deokkeun, etc., during the last five years. Szymon and
Xinyu have been great friends and colleagues.
I could not possibly have finished my PhD work without many colleagues
outside Ohio State. In particular, I thank Andrzej Udalski for being an exemplary
astronomer. I never fail to be thrilled by the first-rate data he and the OGLE team
have produced. A lot of crucial data I have analyzed come from many “amateur”
astronomers: Jennie McCormick, Grant Christie and Berto Monard, to name a few.
They are amateurs only in the sense that they are not paid for their observations,
otherwise what they do are extraordinarily professional. Their fascination in the sky,
enthusiasm and dedication have been a constant source of inspiration to me. I thank
all my observer colleagues for offering me the privilege in being the first to see many
beautiful light curves and realize the exotic extrasolar worlds that they reveal. The
precious moments of discovery transcend anything else.
viii
I wish to express my deep gratitude to Profs. Dawei Yang, Tianyi Huang,
Qiusheng Gu and Dr. Jin Zhu for their selfless help and great guidance at various
important stages before I entered the graduate school.
At last, I thank my mother and father, who put my education, both in acquiring
knowledge and in being a decent member of the society, as the first priority in
their life. Without their unequivocal support and sacrifice, I would have never been
anywhere near the position I am at today to write this dissertation.
ix
VITA
March 18, 1982 . . . . . . . . . . . . . . . . Born – Chengde, Hebei, China
2008 – 2009 . . . . . . . . . . . . . . . . . . . . . Presidential Fellow, The Ohio State University
PUBLICATIONS
Research Publications
1. A. Udalski, et al. “A Jovian-Mass Planet in Microlensing Event OGLE-2005-BLG-071”, ApJL, 628, 109L, (2005).
2. Subo Dong, et al. “Planetary Detection Efficiency of the Magnification3000 Microlensing Event OGLE-2004-BLG-343”, ApJ, 642, 842, (2006).
3. A. Gould, A. Udalski, D. An, D. P. Bennett, A.-Y. Zhou, S. Dong, et al.“Microlens OGLE-2005-BLG-169 Implies That Cool Neptune-like Planets AreCommon”, ApJL, 644, L37, (2006).
x
4. Subo Dong, “Probing ∼100 AU Intergalactic Mg II Absorbing ‘Cloudlets’with Quasar Microlensing”, ApJ, 660, 206, (2007).
5. Subo Dong, et al. “First Space-Based Microlens Parallax Measurement:Spitzer Observations of OGLE-2005-SMC-001”, ApJ, 664, 862, (2007).
6. B.S. Gaudi, D. P. Bennett, A. Udalski, A. Gould, G. W. Christie, D.Maoz, S. Dong, et al. “Discovery of a Jupiter/Saturn Analog with GravitationalMicrolensing”, Science, 319, 927, (2008).
7. B. Scott Gaudi, Joseph Patterson, David S. Spiegel, Thomas Krajci, R.Koff, G. Pojmanski, Subo Dong, et al. “Discovery of a Very Bright, NearbyGravitational Microlensing Event”, ApJ, 677, 1268, (2008).
8. Subo Dong, et al. “OGLE-2005-BLG-071Lb, the Most Massive M DwarfPlanetary Companion?”, ApJ, 695, 970, (2009).
9. Subo Dong, et al. “Microlensing Event MOA-2007-BLG-400: Exhumingthe Buried Signature of a Cool, Jovian-Mass Planet”, ApJ, 698, 1826, (2009).
10. A.Gould, A.Udalski, B.Monard, K.Horne, Subo Dong, et al. “The ExtremeMicrolensing Event OGLE-2007-BLG-224: Terrestrial Parallax Observation of aThick-Disk Brown Dwarf”, ApJL, 698, L147, (2009).
If the light-curve parameters were well-constrained, the approaches of Gaudi
& Sackett (2000) and Rhie et al. (2000) would be very nearly equivalent, with the
former retaining a modest philosophical advantage, since it uses only the observed
light curve and does not require construction of light curves for hypothetical events.
43
However, because in our case these parameters are not well constrained, the Gaudi
& Sackett (2000) approach would require integration over all binary-lens parameters
(except Fs and Fb). Regardless of its possible philosophical advantages, this approach
is therefore computationally prohibitive in the present case. We therefore do not
follow Gaudi & Sackett (2000), but instead construct a binary light curve with the
same observational time sequence and photometric errors as the OGLE observations
of OGLE-2004-BLG-343, for each (b, q, α ; u0, ρ∗) combination and the associated
probability-weighted parameters alc: t0, tE, Fs and Fb in the m-th (u0, ρ∗) bin,
alc(weighted),m =
∑
j,kPm,j,kalc,j,k
∑
j,kPm,j,k
. (2.13)
Then each simulated binary light curve (b, q, α ; u0,m, ρ∗,m) is fitted to a
single-lens model with finite-source effects whose best fit yields χ2(b, q, α ; u0,m, ρ∗,m).
Another set of artificial binary light curves is generated under the assumption
that OGLE had triggered a dense series of observations following the internal
alert at HJD′ 3175.54508. These cover the peak of the event with the normal
OGLE frequency and are used to compare results with those obtained from the real
observations.
Magnification calculations for a binary lens with finite-source effects are very
time-consuming. Besides (b, q, α), our calculations are also performed on (u0, ρ∗)
grids, two more dimensions than in any previous search of a grid of models with
finite-source effects included. This makes our computations extremely expensive,
44
comparable to those of Gaudi et al. (2002), which equaled several years of processor
time. Therefore, we have developed two new binary-lens finite-source algorithms to
perform the calculations, as discussed in detail in Appendix A.
In principle, we should consider the full range of b, i.e., 0 < b < ∞; in practice,
it is not necessary to directly simulate b < 1 due to the famous b ↔ b−1 degeneracy
(Dominik 1999a; An 2005). Instead, we just map the b > 1 results onto b < 1
except for the isolated sensitive zones along the x−axis caused by planetary caustics
perturbations.
We define the planetary detection efficiency ǫ(b, q) as the probability that an
event with the same characteristics as OGLE-2004-BLG-343, except that the lens is
a planetary system with configuration of (b, q), is inconsistent with the single-lens
model (and hence would have been detected),
ǫ(b, q, α) = ∑
m
Θ[
χ2(b, q, α ; u0,m, ρ∗,m) − ∆χ2thres
]
× P [bin(u0,m, ρ∗,m)]
×∑
m
P [bin(u0,m, ρ∗,m)]−1 (2.14)
and
ǫ(b, q) =1
2π
∫ 2π
0ǫ(b, q, α)dα. (2.15)
45
2.4.2. Constraints on Planets
Figure 2.6 shows the planetary detection efficiency of OGLE-2004-BLG-343
for planets with mass ratios q = 10−3, 10−4, and 10−5, as a function of b, the
planet-star separation (normalized to θE), and α, the angle that the moving source
makes with the binary axis passing the primary lens star on its left. Different colors
indicate 10%, 25%, 50%, 75%, 90% and 100% efficiency. Note that the contours are
elongated along an axis that is roughly 60 from the vertical (i.e., the direction of
the impact parameter for α = 0). This reflects the fact that the point closest to
the peak occurs at t = 2453175.77626 when (t − t0)/tE = 2.16u0, and so when the
source-lens separation is at an angle tan−1 2.16 = 65. For q = 10−3, the region
of 100% efficiency extends through 360 within about one octave on either side of
the Einstein ring. However, at lower mass ratios there is 100% efficiency only in
restricted areas close to the Einstein ring and along the above-mentioned principal
axis.
Figure 2.7 summarizes an ensemble of all figures similar to Figure 2.6, but with
q ranging from 10−2.5 to 10−5.0 in 0.1 increments. To place this summary in a single
figure, we integrate over all angles α at fixed b. Comparison of this figure to Figure 8
from Gaudi et al. (2002) shows that the detection efficiency of OGLE-2004-BLG-343
is similar to that of MACHO-1998-BLG-35 and OGLE-1999-BUL-35 despite the fact
that their maximum magnifications are Amax ∼ 100, roughly 30 times lower than
46
OGLE-2004-BLG-343. Of course, part of the reason is that OGLE-2004-BLG-343
did not actually probe as close as u = u0 ∼ 1/3000 because no observations were
taken near the peak. However, observations were made at u ∼ 1/1200, about 12
times closer than in either of the two events analyzed by Gaudi et al. (2002). One
problem is that because the peak was not well covered, there are planet locations
that do not give rise to observed perturbations at all. But this fact only accounts for
the anisotropies seen in Figure 2.6. More fundamentally, even perturbations that do
occur in the regions that are sampled by the data can often be fitted to a point-lens
light curve by “adjusting” the portions of the light curve that are not sampled.
Note the central “spike” of reduced detection efficiency plots near b = 1. As
first pointed out by Bennett & Rhie (1996), this is due to the extreme weakness of
the caustic for nearly resonant (b ∼ 1) small mass-ratio (q ≪ 1) binary lenses.
2.4.3. No Planet Detected
Based on the detection efficiency levels we obtained in § 2.4.2, we fit the
observational data to binary-lens models to search for a planetary signal in the
regions with efficiency greater than zero from q = 10−5 to q = 10−2.5. We find
no binary-lens models satisfying our detection criteria. In fact, the total χ2
contributions to the best-fit single-lens model of the observational points over the
peak ( HJD = 2453175.5 − 2453176.0) are no more than 30, so even if all of these
47
deviations were due to a planetary perturbation, such a binary-lens solution would
not easily satisfy our ∆χ2 = 60 detection criteria. Therefore there are no planet
detections in OGLE-2004-BLG-343 data.
2.4.4. Fake Data
Partly to explore further the issue of imperfect coverage of the peak, and partly
to understand how well present microlensing experiments can probe for planets, we
now ask what would have been the detection efficiency of OGLE-2004-BLG-343 if
the internal alert issued on HJD′ 3175.54508 had been acted upon.
Of course, since the peak was not covered, we do not know exactly what u0
and ρ for this event are. However, for purposes of this exercise, we assume that they
are near the best fit as determined from a combination of the light-curve fitting and
the Galactic Monte Carlo, and for simplicity, we choose u0 = 0.00040, ρ∗ = 0.00040
which is very close to the best-fit combination. We then form a fake light curve
sampled at intervals of 4.3 minutes, starting from the alert and continuing to the
end of the actual observations that night. This sampling reflects the intense rate
of OGLE follow-up observations actually achieved during this event (see § 2.2). We
assume errors similar to those of the actual OGLE data at similar magnifications.
For those points that are brighter than the brightest OGLE point, the minimum
actual photometric errors are assigned. We also assume that the color information is
48
known exactly in this case to be V − I = 2.6. We then analyze these fake data in
exactly the same way that we analyze the real data. In contrast to the real data,
however, we do not find a finite range of z0 ≡ u0/ρ∗ that are consistent with the fake
data. Rather, we find that all consistent parameter combinations have z0 = 1 almost
identically. We therefore consider only a one-dimensional set of (u0, ρ∗) combinations
subject to this constraint.
Figure 2.8 is analogous to Figure 2.6 except that the panels show planet
sensitivities for q = 10−3, 10−4, 10−5, and 10−6, that is, an extra decade. In sharp
contrast to the real data, these sensitivities are basically symmetric in α, except for
the lowest value of q. Sensitivities at all mass ratios are dramatically improved.
For example, at q = 10−3, there is 100% detection efficiency over 1.7 dex in b
(1/7 ∼< b ∼< 7). Even at q = 10−5 (corresponding to an Earth-mass planet around an
M star), there is 100% efficiency over an octave about the Einstein radius.
Figure 2.9 is the fake-data analog of Figure 2.7. It shows that this event would
have been sensitive to extremely low mass-ratios, lower than those accessible to any
other technique other than pulsar timing.
2.4.5. Detection Efficiency in Physical Parameter Space
One of the advantages of the Monte Carlo approach of Yoo et al. (2004b) is
that it permits one to evaluate the planetary detection efficiency in the space of
49
the physical parameters, planet mass and projected physical separation (mp, r⊥),
rather than just the microlensing parameters (b, q). Figures 2.10 and 2.11 show
this detection efficiency for the real and fake data, respectively. The fraction
of Jupiter-mass planets that could have been detected from the actual data
stream is greater than 25% for 0.8AU ∼< r⊥ ∼< 10AU and is greater than 90% for
2AU ∼< r⊥ ∼< 6AU. There is also marginal sensitivity to Neptune-mass planets.
However, the detection efficiencies would have been significantly enhanced had the
FWHM around the peak been observed, as previously discussed by Rattenbury et
al. (2002). For the fake data, more than 90% of Jupiter-mass planets in the range
0.7AU ∼< r⊥ ∼< 20AU and more than 25% with 0.3AU ∼< r⊥ ∼< 30AU would have
been detected. Indeed, some sensitivity would have extended all the way down to
Earth-mass planets.
2.5. Luminous Lens?
Understanding the physical properties of their host stars is a major component
of the study of extra-solar planets. It is especially important to know the mass and
distance of the lens star for planets detected by microlensing because only then
can we accurately determine the planet’s mass and physical separation from the
star. Obtaining similar information for microlensing events that are unsuccessfully
searched for planets enables more precise estimates of the detection efficiency. There
50
are only two known ways to determine the mass and distance of the lens: either
measure both the microlensing parallax and the angular Einstein radius (which are
today possible for only a small subset of events) or directly image the lens. In most
cases the lens is either entirely invisible or is lost in the much brighter light of the
source.
A simple argument suggests, however, that in extremely high-magnification
events like OGLE-2004-BLG-343, the lens will often be easily visible and, indeed, it
is the lens that is unknowingly being monitored, with the source revealing itself only
in the course of the event. Events of magnification Amax require that the source be
much smaller than the Einstein radius, θ∗ ∼< 2θE/Amax. Since θE =√
κMπrel, large
θE requires a lens that is either massive or nearby, both of which suggest that it is
bright. On the other hand, a small θ∗ implies that the source is faint. Generally, if
a faint source and a bright potential lens are close on the sky, only the lens will be
seen, until it starts to strongly magnify the source. This has important implications
for the real time recognition of extreme magnification events, as we discuss in § 2.6.
Here we review the evidence as to whether the blended light in OGLE-2005-BLG-343
is in fact the lens.
As was true for OGLE-2003-BLG-175/MOA-2003-BLG-45 mentioned in § 2.1.3,
the blended light in OGLE-2004-BLG-343 lies in the “reddening sequence” of
foreground disk stars. It is certainly “dazzling” by any criterion, being about 50
51
times brighter than the source in I and 150 times brighter in V (see Fig. 2.2). Is the
blended light also due to the lens in this case?
There is one argument for this hypothesis and another against. We initiate
the first by estimating the mass and distance to the blend as follows. We
model the extinction due to dust at a distance x along the line of sight by
dAI/dx = 0.4 kpc−1e−qx and set q = 0.26 kpc−1 in order to reproduce the measured
extinction to the bulge AI(8 kpc) = 1.34. Using the Reid (1991) color-magnitude
relation, we then adjust the distance to the blend until it reproduces the observed
color and magnitude of the blend. We find a distance modulus of 12.6 (∼ 3.3 kpc),
and with the aid of the Cox (2000) mass-luminosity relation, we estimate a
corresponding mass Ml = 0.9 M⊙. Inspecting Figure 2.5, we see that this is almost
exactly the peak of the lens-distance distribution function predicted by combining
light-curve information and the Galactic model. This is quite striking because, in the
absence of light-curve information, the lens would be expected to be relatively close
to the source. From our Monte-Carlo simulation toward the line of sight of this
event, the total prior probability of the bulge-bulge events is about 1.5 times higher
than the prior probability of the bulge-disk events, and furthermore, only about 7%
of all events have lenses less than 3.3 kpc away (see green and purple histograms in
source and lens distance modulus panels of Fig. 2.5). It is only because the light
curve lacks obvious finite-source effects (despite its very high-magnification) that one
is forced to consider lenses with large θE, which generally drives one toward nearby
52
lenses in the foreground disk. Based on our experience analyzing many blended
microlensing events, the blended light is most often from a bulge star rather than
a disk star, which simply reflects the higher density of bulge stars. In brief, it is
quite unusual for lenses to be constrained to lie in the disk, and it is quite unusual
for events to be blended with foreground disk stars. This doubly unusual set of
circumstances would be more easily explained if the blend were the lens.
However, if the blend were the lens, then the source and lens would be aligned
to better than 1 mas during the event, and one would therefore expect that the
apparent position of the source would not change as the source first brightened
and then faded. In fact, we find that the apparent position does change by about
73 ± 9 mas. However, since the apparent source (i.e., combined source and blended
light at baseline) has a near neighbor at 830 mas, which is almost as bright as the
source/blend, it is quite possible that the lens actually is the blend, but that this
neighbor is corrupting the astrometry.
Thus, the issue cannot be definitively settled at present. However, it could be
resolved in principle by, for example, obtaining high-resolution images of the field a
decade after the event when the source and lens have separated sufficiently to both
be seen. If the blend is the lens, then they will be seen moving directly apart with
a proper motion given µ = θE/tE, where θE is derived from the estimated mass and
distance to the lens and tE is the event timescale.
53
Since the blend cannot be positively identified as the lens, we report our main
results using a purely probabilistic estimate of the lens parameters. However, for
completeness, we also report results here based on the assumption that the lens is
the blend. Compared to the previous simulation, in which we considered the full
mass function and full range of distances, we sample only the narrow intervals of
mass and distance that are consistent with the observed color and magnitude of the
lens/blend. To implement these restrictions, we repeat the Monte Carlo, but with
the additional constraint that the predicted apparent magnitudes agree with the
observed blend magnitude (with an error of 0.5 mag) and that the predicted colors
(using the above extinction law and the Reid 1991 color-magnitude relation) also
show good agreement with the observed color (with 0.2 mag error). These errors are,
of course, much larger than the observational errors. They are included to reflect the
fact that the theoretical predictions for color and magnitude at a given mass are not
absolutely accurate.
Figure 2.12 is the resulting version of Figure 2.10 when the Monte Carlo is
constrained to reproduce the blend color and magnitude. The sensitivity contours
are narrower and deeper, reflecting the fact that the diagram no longer averages over
a broad range of lens masses but rather is restricted effectively to a single mass (and
single distance).
54
2.6. Summary and Discussion
In this paper we present our analysis of microlensing event OGLE-2004-BLG-
343, with the highest peak magnification (Amax = 3000±1100) ever analyzed to date.
The light curve is consistent with the single-lens microlensing model, and no planet
has been detected in this event. We demonstrate that if the peak had been well
covered by the observations, the event would have had the best sensitivity to planets
to date, and it would even have had some sensitivity to Earth-mass planets (§ 2.4.4,
§ 2.4.5). However, this potential has not been fully realized due to human error
(§ 2.2), and OGLE-2004-BLG-343 turns out to be no more sensitive to planets than
a few other high-magnification events analyzed before (§ 2.4.2, § 2.4.5). Thus, while
ground-based microlensing surveys are technically sufficient to detect very low-mass
planets, the relatively short timescale of the sensitive regime of high-magnification
microlensing events demands a rapidity of response that is not consistently being
achieved. In the final paragraph below, we develop several suggestions to rectify this
situation.
In § 2.3 we show that finite-source effects are important in analyzing this
event, so we extend the method of Yoo et al. (2004b) to incorporate such effects in
planetary detection efficiency analysis. Moreover, since magnification calculations
of binary-lens models with finite-source effects are computationally remarkably
expensive, and applying previous finite-source algorithms, it would have taken
55
of order a year of CPU time to do the detection efficiency calculation required
by this event. We therefore develop two new binary-lens finite-source algorithms
(Appendix A) that are considerably more efficient than previous ones. The
“map-making” method (Appendix A.1) is an improvement on the conventional
inverse ray-shooting method, which proves to be especially efficient for use in
detection efficiency calculations, while the “loop-linking” method (Appendix A.2)
is more versatile and could be easily implemented in programs aimed at finding
best-fit finite-source binary-lens solutions. Using these algorithms, we were able to
complete the computations for this paper in about 4 processor-weeks, roughly an
order of magnitude faster than would have been required using previous algorithms.
Finally, we show in § 2.5 that the blend, which is a Galactic disk star, might
very possibly be the lens, and that this case also proves to be highly probable
from the Monte-Carlo simulation. However, it seems to contradict the astrometric
evidence, and we point out that this issue could in principle be solved by future
high-resolution images. Among the high-magnification events discovered by current
microlensing survey groups, it is very likely that the lens star, which is also the
apparent source, of those events is in the Galactic disk. Thus the blended light
is usually far brighter than the source, thereby increasing the difficulty in early
identification of such events. This fact motivates the first of several suggestions
aimed at improving the recognition of very high-magnification events:
56
1) When events are initially alerted they should be accompanied by instrumental
CMD of the surrounding field, with the location of the apparent “source”
highlighted. Events whose apparent sources lie on the “reddening sequence”
of foreground disk stars (see Fig. 2.2) have a high probability to actually be
lenses of more distant (and fainter) bulge sources. These events deserve special
attention even if their initial light curves appear prosaic.
2) For each such event it is possible to measure the color (but not immediately
the magnitude) of the source by the standard technique of obtaining two-band
photometry and measuring the slope of the relative fluxes in the two bands.
If the color is different from that of the apparent “source” at baseline, that
will prove that this baseline light is not primarily due to the source, and it will
increase the probability that this baseline object is the lens. Moreover, if the
source color is relatively red, it will show that the source is probably faint and
so is (1) most likely already fairly highly magnified (thereby making it possible
to detect above the foreground blended light) and (2) capable, potentially at
least, of being magnified to very high magnification (see § 2.5). This would
motivate obtaining more data while the event was still faint to help predict
its future behavior and would enable a guess as to how to “renormalize” the
event’s apparent magnification to its true magnification. This is important
because generally one cannot accurately determine this renormalization until
57
the event is within 0.4 mag (when the event is teff before its peak), at which
point it may well be too late to act on this knowledge.
3) Both survey groups and follow-up groups should issue alerts on suspected
high-magnification events guided by a relatively low threshold of confidence,
recognizing that this will lead to more “false alerts” than at present. If such
alerts are accompanied by a cautionary note, they will promote intergroup
discussions that could lead to more rapid identification of high-magnification
events without compromising the credibility of the group.
58
Fig. 2.1.— Light curve of OGLE-2004-BLG-343 near its peak on 2004 June 19(HJD 2,453,175.7467). Only OGLE I-band data (open circles) are u sed in mostof the analysis, except OGLE V -band data (open triangles) and µFUN H-band data(crosses) are used to constrain the color of the source star. All bands are linearlyrescaled so that Fs and Fb are the same as the OGLE I-band observations. The solidline shows the bes t-fit PSPL model. The upper right inset shows the peak of thelight curve, wit h the range of the simulated data points plotted by the thick line.
59
Fig. 2.2.— CMD of the OGLE-2004-BLG-343 field. Hipparcos main-sequence stars(blue dots), placed at 10−0.15/5R0 = 7.5 kpc and reddened by the reddening vectorderived from the clump, are displayed with the OGLE-II stars (black dots). The Reid(1991) relation is plotted by the red solid line over the Hipparcos stars. On the CMD,the magenta filled circle is the red clump and the green filled circle is the blendedstar. The large black filled circle is the OGLE V measurement of the source with of1σ error bars, which sets a lower limit for the source V − I color. The magenta filledcircle with error bars is the result of combining the (I −H)/(V − I) information (seeFig. 2.3) with the OGLE measurement.
60
Fig. 2.3.— (VOGLE − IOGLE)/(IOGLE − H2MASS) color-color diagram. All points arefrom matching 2MASS H-band data with OGLE-II V, I photometry in a field centeredon OGLE-2004-BLG-343. Stars on the giant branch are shown by open circles. Thesolid and dashed vertical lines represent the source IOGLE−H2MASS color transformedfrom its IOGLE−HµFUN value and its 1σ ranges. Their intersections with the diagonaltrack of stars give corresponding VOGLE − IOGLE colors, which are represented by thehorizontal lines.
61
Fig. 2.4.— Likelihood contours (1σ, 2σ, 3σ) for finite-source points-lens modelsrelative to the best-fit PSPL model. Contours with x-axis as log u−1
0 and log Amax
are displayed in solid and dashed lines, respectively.
62
b-b(posterior)b-d(posterior)
bigger finite-sourceeffects
b-b(prior)b-d(prior)
priorposterior
Fig. 2.5.— Probability distributions of u0, dereddened apparent I-band magnitudeof the source I0, proper motion µ, log(z0), source distance modulus, lens distancemodulus, absolute I-band magnitude of the source MI , and lens mass for Monte Carloevents toward the line of sight of OGLE-2004-BLG-343. Blue histograms representthe posterior probability distributions for bulge-disk microlensing events while redones represent the posterior probability distributions for bulge-bulge events. In thesource and lens distance-modulus panels, histograms in purple and green representthe prior probability distributions for bulge-disk and bulge-bulge events, respectively.The black Gaussian curves in the u0 and I0 panels show probability distributions fromPSPL light-curve fitting alone. In the lens mass panel, the dark green histogram showsthe prior probability distribution, while the orange histogram represents the posteriordistribution.
63
Fig. 2.6.— (For real data) Planetary detection efficiency for mass ratios q = 10−3,10−4, and 10−5 for OGLE-2004-BLG-343 as a function of the planet-star separationbx = b× cos α and by = b× sin α in the units of θE where α is the angle of planet-staraxis relative to the source-lens direction of motion. Different colors indicate 10% (red),25% (yellow), 50% (green), 75% (cyan), 90% (blue) and 100% (magenta) efficiency.The black circle is the Einstein ring, i.e., b = 1.
64
Fig. 2.7.— (For real data) Planet detection efficiency of OGLE-2004-BLG-343 as afunction of the planet-star separation b (in the units of rE) and planet-star mass ratioq. The contours indicate 25%, 50%, 75%, and 90% efficiency.
65
Fig. 2.8.— (For fake data) Planetary detection efficiency for mass ratios q = 10−3,10−4, 10−5 and 10−6 of OGLE-2004-BLG-343 augmented by simulated data pointscovering the peak as a function of the planet-star separation bx and by in the units ofθE. Different colors indicate 10% (red), 25% (yellow), 50% (green), 75% (cyan), 90%(blue) and 100% (magenta) efficiency. The black circle is the Einstein ring, i.e., b = 1.
66
Fig. 2.9.— (For fake data) Planetary detection efficiency of OGLE-2004-BLG-343augmented by simulated data points covering the peak. as a function of the planet-star separation b (in the units of θE) and planet-star mass ratio q. The contoursrepresent 25%, 50%, 75%, and 90% efficiency.
67
Fig. 2.10.— (For real data) Planetary detection efficiency as a function of r⊥, thephysical projected star-planet distance and mp, the planetary mass for OGLE-2004-BLG-343. The contours represent 25%, 50%, 75%, and 90% efficiency.
68
Fig. 2.11.— (For fake data) Planetary detection efficiency as a function of r⊥,the physical projected star-planet distance and mp, the planetary mass for OGLE-2004-BLG-343 augmented by simulated data points covering the peak. The contoursrepresent 25%, 50%, 75%, and 90% efficiency.
69
Fig. 2.12.— Planetary detection efficiency as a function of r⊥, the physical projectedstar-planet distance, and mp, the planetary mass for OGLE-2004-BLG-343, byassuming that the blended light is due to the lens. The contours represent 25%,50%, 75%, and 90% efficiency.
We therefore adopt results from the Wozniak-based reductions, noting that they
may subject to systematic errors ∼< 1σ.
Figure 3.3 shows the source trajectory and the central caustics as well as
the differences in magnification between the best-fit planetary model and its
corresponding single-lens model. This geometry nicely accounts for the main features
of the point-lens residuals seen in Figure 3.1. The regions beyond the “back walls”
(long segments) of the caustic are somewhat de-magnified, which accounts for the
initial depression of the light curve. As the source crosses the “back wall” of the
caustic, it spikes. After the source has exited the caustic, it continues to suffer
additional magnification due to the “ridge” of magnification that extends from the
trailing cusp.
We also conducted a similar blind search as above, but concentrating on the
regime d < 1. As expected, we recover the well-known d ↔ d−1 degeneracy, and
89
find a solution with essentially the same mass ratio q = (2.6 ± 0.4) × 10−3, but with
d = 2.9−1 = 0.34+0.03−0.02, and the wide solution is slightly preferred by ∆χ2 = 0.2.
Thus, although each solution is well-localized to its respective minimum, this
discrete degeneracy implies that the projected separation can take on two values
that differ by a factor of ∼ 8.5. The severity of the degeneracy can be traced to
the planetary parameters. Although the planet/star mass ratio is quite large, which
tends to reduce the severity of the degeneracy, the planet lies quite far from the
Einstein ring, which tends to make it more severe. Actually, a better measure of the
overall expected asymmetry between the d and d−1 solutions is the short diameter
w, which in this case is small, implying a severe degeneracy. Indeed, the caustic
structure and magnification pattern of the two solutions are nearly identical. In this
case, the large size of the source has competing influences on the ability to resolve
the degeneracy. On one hand, the large size of the source serves to suppress the
planetary deviations, thus making subtle differences more difficult to distinguish.
On the other hand, the large source implies that a large fraction of the planetary
perturbation region is probed. In this case, the source probes essentially the entire
region of significant planetary perturbation, as can be seen in Figure 3.3. This is
important for distinguishing between the solutions, as the largest difference between
the magnification patterns of the two degenerate solutions occurs in the region near
the tip of the arrow-shaped caustic (Griest & Safizadeh 1998). From Figure 3.3
90
it is clear that this region would have been entirely missed if the source had been
substantially smaller than the caustic.
3.4. Finite-Source Effects
In addition to (d, q), the model also yields the source radius relative to the
Einstein radius,
ρ = θ∗/θE = (3.29 ± 0.08) × 10−3, (3.4)
We then follow the standard (Yoo et al. 2004a) technique to determine the angular
source radius,
θ∗ = 1.05 ± 0.05 µas. (3.5)
That is, we first adopt [(V − I)0, I0]clump = (1.00, 14.32) for the dereddened position
of the clump. We then measure the offset of the source relative to the clump centroid
∆[(V − I), I] = (−0.19, 3.25), to obtain [(V − I)0, I0]s = (0.81, 17.57). See Figure
3.5. The instrumental source color is derived from model-independent regression of
the V and I flux, while the instrumental magnitude is obtained from the light-curve
model. We convert (V − I) to (V − K) using the color-color relations of Bessell &
Brett (1988), yielding (V − K)0 = 1.75, and then obtain equation (3.5) using the
color/surface-brightness relations of Kervella et al. (2004). Combining equations
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(3.4) and (3.5) gives θE = θ∗/ρ = 0.32 mas. And combining this with the definition
θ2E = κMπrel, where M is the lens mass, πrel is the source-lens relative parallax,
and κ = 4G/c2AU ∼ 8.1 mas M−1⊙ , together with the measured Einstein timescale,
tE = 14.3 ± 0.3days, we obtain
M = 0.10 M⊙
(
πrel
125 µas
)−1
(3.6)
and
µrel =θE
tE= 8 mas yr−1. (3.7)
The relatively high lens-source relative proper motion µrel is mildly suggestive of a
foreground disk lens, but still quite consistent with a bulge lens. Since πrel = 125µas
corresponds to a lens distance DL = 4 kpc (assuming source distance DS = 8 kpc),
equation (3.6) implies that if the lens did lie in the foreground, then it would be a
very low-mass star or a brown dwarf.
Assuming that the source lies at a Galactocentric distance modulus 14.52, its
dereddened color and magnitude imply that [(V − I)0,MI ] = (0.81, 3.07), making it
a subgiant.
92
3.5. Limb Darkening
As illustrated in Figure 3.1, the principal deviations from a point-lens light
curve occur at the limb of the star. This prompts us to investigate the degree to
which the planetary solution is influenced by our treatment of limb darkening. The
results that we report are based on a fit to the H-band surface brightness profile of
the form
S(ϑ)
Sλ
= 1 − Γ(
1 − 3
2cos ϑ
)
− Λ(
1 − 5
4cos1/2 ϑ
)
, (3.8)
where Γ and Λ are the linear and square-root parameters, respectively, and where
ϑ is the angle between the normal to the stellar surface and the line of sight, i.e.,
sin ϑ = θ/θ∗. See An & Han (2002) for the relation between (Γ, Λ) and the usual
(c, d) formalism.
In deriving the reported results, we fix the H-band limb-darkening parameters
(Γ, Λ) = (−0.15, 0.69), corresponding to (c, d) = (−0.21, 0.79) from Claret (2000)
for a star with effective temperature Teff = 5325 K and log g = 4.0. These stellar
parameters are suggested by comparison to Yale-Yonsei isochrones (Demarque et
al. 2004) for the dereddened color and absolute magnitude reported in § 3.4. We
also perform fits in which Γ and Λ are allowed to be completely free. From these
fits, we find that our best-fit model has (Γ, Λ) = (−0.64, 1.47). Γ and Λ are highly
correlated, so their individual values are not of interest, and the surface-brightness
93
profiles generated by these two sets of (Γ, Λ) are qualitatively similar. In the present
context, however, the key point is that when we fix the limb-darkening parameters
at the Claret (2000) values, the contours in Figure 3.4 remain essentially identical
and the best fit values change by much less than 1σ.
Because of lower point-density and the aforementioned problems with the
I data over the peak, we only attempt a linear limb-darkening fit, i.e., we use
equation (3.8) with Λ ≡ 0, and we adopt Γ = 0.47 from Claret (2000).
3.6. Blended Light
In the crowded fields of the Galactic bulge, the photometered light of a
microlensing event rarely comes solely from the lensed source. Rather there is
typically additional light that is blended with the source but is not being lensed.
This light can arise from unrelated stars that happen to be projected close enough
to the line of sight to be blended with the source, or it can come from companions
to the source, companions to the lens, or the lens itself. This last possibility is most
interesting because, if the lens flux can be isolated and measured, it provides strong
constraints on the lens properties, and in this case would enable a complete solution
of the lens mass and distance, when combined with the measurement of θE (e.g.,
Bennett et al. 2007).
94
To investigate the blended light, we begin by using the method of Gould &
An (2002) to construct an image of the field with the source (but not the blended
light) removed, and compare this to a baseline image, which of course contains both
the source and the blended light (see Fig. 3.7). In these images, the source/blend is
immersed in the wings of a bright star (roughly 3.7 mag brighter than the source),
which lies about 2′′ away. On the baseline image, the source/blend is noticeable
against this background, but hardly distinct. On the source-subtracted image, the
blend is not directly discernible.
To make a quantitative estimate of the blend flux, we fit the region in the
immediate vicinity of the bright star to the form F = a1 + a2 × PSF, where “PSF”
is the point-spread-function determined from the DIA analysis. We then subtract
the best fit flux profile from the image. This leaves a clear residual at the position
of the source/blend in the baseline image, but just noise in the source-subtracted
image. We add all the flux in a 1.8′′ square centered on the lens, finding 564 ADU
and −28 ADU, respectively. We conclude that the I-band flux blend/source ratio is
fb/fs < 0.05. Of course, even if we had detected blended light, it would be impossible
to tell whether it was directly coincident with the source. If it were, this would imply
that this light would be directly associated with the event, i.e., being either the lens
itself or a companion to the lens or the source. Hence, this measurement is an upper
limit on the light from the lens in two senses: no light is definitively measured, and
if it were we do not know that it came from the lens. Combining this limit with
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equation (3.6), and assuming the lens is a main-sequence star, it must then be less
massive than M < 0.75 M⊙, and so must have relative parallax πrel > 15 mas. This
implies a lens-source separation DS − DL > 1 kpc, which certainly does not exclude
bulge lenses. Indeed, if the lens were a K dwarf in the Galactic bulge, it would
saturate this limit.
3.7. Discussion
MOA-2007-BLG-400 is the first high-magnification microlensing event for which
the central caustic generated by a planetary companion to the lens is completely
enveloped by the source. As a comparison, the planetary caustic of OGLE-2005-
BLG-390 (Beaulieu et al. 2006) is smaller than its clump-giant source star in angular
size. When the planetary caustics is covered by the source, the finite-source effects
broaden the “classic” Gould & Loeb (1992) planetary perturbation features (Gaudi
& Gould 1997). By contrast, planet-induced deviations in MOA-2007-BLG-400 are
mostly obliterated, rather than being broadened, because the source crosses the
central caustic rather than the planetary caustic. We showed, nevertheless, that the
planetary character of the event can be inferred directly from the light-curve features
and that the standard microlensing planetary parameters (d, q) = (2.9, 2.5 × 10−3)
can be measured with good precision, up to the standard close/wide d ↔ d−1
degeneracy. We demonstrated that, in this case, the close/wide degeneracy is quite
96
severe, and the wide solution is only preferred by ∆χ2 = 0.2. This is unfortunate,
since the separations of the two solutions differ by a factor of ∼ 8.5. We argued that
the severity of this degeneracy was primarily related to the intrinsic parameters of
the planet, rather than being primarily a result of the large source size.
Although the mass ratio alone is of considerable interest for planet formation
theories, one would also like to be able to translate the standard microlensing
parameters to physical parameters, i.e., the planet mass mp = qM , and planet-star
projected separation r⊥ = dθEDL. Clearly this requires measuring the lens mass
M and distance DL. In this case, the pronounced finite source effects have already
permitted a measurement of the Einstein radius θE = 0.32 mas, which gives a relation
between the mass and lens-source relative parallax (eq. [3.6]). This essentially
yields a relation between the lens mass and distance, since the source distance is
close enough to the Galactic center that knowing DL is equivalent to knowing πrel.
Therefore, a complete solution could be determined by measuring either M or DL,
or some combination of the two.
One way to obtain an independent relation between the lens mass and distance
is to measure the microlens parallax, πE. There are two potential ways of measuring
πE. First, one can measure distortions in the light curve arising from the acceleration
of the Earth as it moves along its orbit. Unfortunately, this is out of the question
in this case because the event is so short that these distortions are immeasurably
small. Second, one can measure the effects of terrestrial parallax, which gives
97
rise to differences between the light curves simultaneously observed from two or
more observatories separated by a significant fraction of the diameter of the Earth.
Practically, measuring these differences requires a high-magnification event, which
would appear to make this event quite promising. Unfortunately, although we
obtained simultaneous observations from two observatories separated by several
hundred kilometers during the peak of the event, one of these datasets suffers from
large systematic errors and an unknown time zero point, rendering it unusable for
this purpose.
The only available alternative for breaking the degeneracy between the lens
mass and distance would be to measure the lens flux, either under the glare of the
source or, at a later date, to separately resolve it after it has moved away from the
line of sight to the source (Alcock et al. 2001a; Kozlowski et al. 2007). Panels (e)
and (f) of Figure 3.6 show the Bayesian estimates of the lens brightness in I-band
and H-band, respectively. If the lens flux is at least 2% of the source flux, then the
former kind of measurement could be obtained from a single epoch Hubble Space
Telescope observation, provided it were carried out in the reasonably near future. At
roughly 99% probability, the blended light would be either perfectly aligned with
the source (and so associated with the event) or well separated from it. HST images
can be photometrically aligned to the ground-based images using comparison stars
with an accuracy of better than 1%. Hence, photometry of the source+blend would
detect the blend, unless it were at least 4 mag fainter than the source. In principle,
98
the blend could be a companion to either the source or lens. Various arguments can
be used to constrain either of those scenarios. We do not explore those here, but see
Dong et al. (2009). If the lens is not detectable by current epoch HST observations
(or no HST observations are taken), then it will be detectable by ground-based AO
H-band observations in about 5 years. This is because the lens-source relative proper
motion is measured to be µrel = 8 mas yr−1, and the diffraction limit at H band on
a 10m telescope is roughly 35 mas. If the lens proves to be extremely faint, then a
wider separation (and hence a few years more time baseline) would be required.
In the absence of additional observational constraints, we must rely on a
Bayesian analysis to estimate the properties of the host star and planet, which
incorporates priors on the distribution of lens masses, distances, and velocities
(Dominik 2006; Dong et al. 2006). This is a standard procedure, which we only
briefly summarize here. We adopt a Han & Gould (1995) model for the Galactic
bar, a double-exponential disk with a scale height of 325 pc, and a scale length
of 3.5 kpc, as well as other Galactic model parameters as described in Bennett &
Rhie (2002). We incorporate constraints from our measurement of the lens angular
Einstein radius θE and the event timescale, as well as limits on the microlens parallax
and I-band magnitude of the lens. In practice, only the measurements of θE and tE
provide interesting constraints on these distributions. In addition, we include the
small penalty on the close solution, exp(−∆χ2/2), where the wide solution is favored
99
by ∆χ2 = 0.2. For the estimates of the planet semimajor axis, we assume circular
orbits and that the orbital phases and cos(inclinations) are randomly distributed.
The resulting probability densities for the physical properties of the host
star, as well as selected properties of the planet, are shown in Figure 3.6. The
Bayesian analysis suggests a host star of mass M = 0.30+0.19−0.12M⊙ at distance of
DL = 5.8+0.6−0.8 kpc. In other words, given the available constraints, the host is most
likely an M-dwarf, probably in the foreground Galactic bulge. Given that the
planet/star mass ratio is measured quite precisely, the probability distribution for
the planet mass is essentially just a rescaled version of the probability distribution
for the host star mass. We find mp = 0.83+0.49−0.31 MJup. The close/wide degeneracy
is apparent in the probability distribution for the semimajor axis a. We estimate
aclose = 0.72+0.38−0.16 AU for the close solution, and awide = 6.5+3.2
−1.2 AU for the wide
solution. The equilibrium temperatures for these orbits are Teq.,close = 103+28−26 K and
Teq.,wide = 34 ± 9 K for the close and wide solutions, respectively.
Thus our Bayesian analysis suggests that this system is mostly likely a
bulge mid-M-dwarf, with a Jovian-mass planetary companion. The semimajor
axis of the planetary companion is poorly constrained primarily because of the
close/wide degeneracy, but the implied equilibrium temperatures are cooler than the
condensation temperature of water. Specifically we find that Teq ∼< 173 K at 2σ
level. Alternatively, if we assume the snow line is given by asnow = 2.7 AU(M/M⊙),
we find for this system a snow line distance of ∼ 0.84 AU, essentially the same as
100
the inferred semimajor axis of the close solution. Thus this planet is quite likely to
be located close to or beyond the snow line of the system.
Although we cannot distinguish between the close and wide solutions for the
planet separation, theoretical prejudice in the context of the core-accretion scenario
would suggest that a gas-giant planet would be more likely to form just outside
the snow line, thus preferring the close solution. However, we have essentially no
observational constraints on the frequency and distribution of Jupiter-mass planets
at the separations implied by the wide solution (∼ 5.3 − 9.7 AU), for such low-mass
primaries. Unfortunately, the prospects for empirically resolving the close/wide
degeneracy in the future are poor. The only possible method of doing this would
be to measure the radial velocity signature of the planet. Given the faintness of
the host star (see §3.6 and Fig. 3.6), this will likely be impossible with current or
near-future technology.
The mere existence of a gas-giant planet orbiting a mid-M-dwarf is largely
unexpected in the core-accretion scenario, as formation of such planets is thought
to be inhibited in such low-mass primaries (Laughlin et al. 2004). Observationally,
however, although the frequency of Jovian companions to M-dwarfs with a ∼< 3 AU
does appear to be smaller than the corresponding frequency of such companions
to FGK dwarfs (Endl et al. 2006; Johnson et al. 2007b; Cumming et al. 2008),
several Jovian-mass companions to M dwarfs are known (see Dong et al. 2009 for
a discussion), so this system would not be unprecedented. Furthermore, it must
101
be kept in mind that the estimates of stellar (and so planet) mass depend on the
validity of the priors, and even in this context have considerable uncertainties.
Most of the ambiguities in the interpretation of this event would be removed
with a measurement of the host star mass and distance, which could be obtained by
combining our measurement of θE with a measurement of the lens light as outlined
above. The Bayesian analysis informs the likelihood of success of such an endeavor.
This analysis suggests that, if the host is a main-sequence star, its magnitude will
be IL = 23.9+0.8−1.0 and HL = 21.4+0.7
−1.0, which corresponds to 0.6% and 1.7% of the
source flux, respectively. If initial efforts to detect the lens fail, more aggressive
observations would certainly be warranted: microlensing is the most sensitive
method for detecting planets around very low-mass stars simply because it is the
only method that does not rely on light from the host (or the planet itself) to detect
the planet. And given equation (3.6), even an M dwarf at the very bottom of the
main sequence M = 0.08 M⊙, would lie at DL = 3.5 kpc and so would be H ∼ 24.
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HJD - 2454354.0
Fig. 3.1.— Top: Lightcurve of MOA-2007-BLG-400 with data from µFUN CTIO(Chile) simultaneously taken in H (cyan), I (DIA, black). Models are shown for apoint lens (blue) and planet-star system (red). There are 5 50-second H exposures foreach 300-second I (or V – not shown) exposure in 6 minutes cycles. Some I-band dataat the peak suffer from saturation, and those points are therefore removed from theanalysis (see text). Middle: Residuals for best-fit point-lens model and its differencewith the planetary model. Note that in the top panel, the H data are shown asobserved, while the I data are aligned. Normally, such alignment is straightforwardbecause microlensing of point sources is achromatic. However, here there is significantchromaticity due to different limb-darkening. The I-band points in the top panel areactually the residuals to the I-band limb-darkened model (middle panel), added tothe H-band model curve (top panel). Bottom: Residuals from a point-lens modelwith the same parameters as the planetary model, which can be directly compared tothe “magnification map” in Fig. 3.3. These “didactic residuals” are naturally morepronounced than those from the best-fit point lens.
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Fig. 3.2.— Comparisons of residuals to the best-fit point-lens models between twophotometric reductions of µFUN H-band data, using the DIA packages developedby Wozniak (2000) (top) and Bond et al. (2001) (bottom). The red curve in eachpanel represents the difference of the best-fit planetary and point-lens models for thatpanel’s reduction. Both reductions agree on the main planetary features, but eachpackage introduces its own systematics. For example, the systematic deviations fromthe planetary model at HJD ∼ 2454354.57 shown in the top panel are not supportedby the reductions of the Bond’s package. During 2454354.46 < HJD < 2454354.51,most images have low transparency (∼< 50%), which causes relatively large scatter inBond’s DIA reductions. In comparison, the reduction by Wozniak’s DIA has smallerscatter during this period. However, the data exhibit some low-level systematics,which are not supported by the other reduction.
104
Fig. 3.3.— Magnification differences between of best-fit planetary model [(q, d) =(0.0025, 2.9) and (q, d) = (0.0026, 0.34) being nearly identical] and single-lens models,in units of the measured source size, ρ = 0.0033 Einstein radii. Contours show 1%, 2%,3%, and 4%, deviations in the positive (brown) and negative (blue) directions. Toppanel: Single-lens geometry (t0, u0, tE) is taken to be the same as in the planetarymodel, with no finite source effects. Caustic (contour of infinite magnification) isshown in white. The deviations are very pronounced. Bottom Panel: Same as toppanel, but including finite-source effects, which now explain the main features of thelight curve. The trajectory begins with a negative deviation, then hits a narrow“brown ridge” causing the spike seen in the bottom panel of Fig. 3.1, as the edgeof the source first hits the caustic. Then there are essentially no deviations (white)while the source covers the caustic. The caustic exit induces a narrow “blue ridge”corresponding to the negative-deviation spike seen in Fig 3.1. Finally, the source runsalong the long “brown ridge” corresponding to the prolonged post-peak mild excessseen in Fig 3.1.
105
Fig. 3.4.— Contours of ∆χ2 = 1, 4, 9 relative to the minimum as a function of planet-star mass ratio q and projected planet-star separation d (top), as well as “short causticdiameter” (see Fig. 3.3) w (bottom). w (in units of θE) is a function of q and d (seetext). The solution shown here corresponds to q = (2.5+0.5
−0.3)×10−3 and d = 2.9±0.2 (ord = 0.34+0.03
−0.02). These values of d correspond to physical separations and equilibriumtemperatures of ∼ 5.3 − 9.7 AU, ∼ 34 K and ∼ 0.6 − 1.1 AU, ∼ 103 K for the closeand wide solutions, respectively.
106
(V−I) [instrumental]
I [in
stru
men
tal]
−.6 −.4 −.2 0 .2
22
20
18
16
Fig. 3.5.— Instrumental color-magnitude diagram of field containing MOA-2007-BLG-400. The color and magnitude of the source (black filled circle) are derived fromthe fit to the light curve, which also yields an upper limit for the I-band blendedflux (green triangle). The large error bar on the latter point indicates a completelack of information about its V -band flux. The clump centroid is indicated in redsquare. From the source-clump offset, we estimate [I, (V − I)]0,s = (17.57, 0.81),implying it has angular radius θ∗ = 1.05 µas. Assuming the source lies at 8 kpc, ithas [MI , (V − I)]0,s = (3.07, 0.81), making it a subgiant. The lack of blended lightallows us to place an upper limit on the lens flux, which implies that it has massM < 0.75 M⊙. See text.
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Fig. 3.6.— Bayesian relative probability densities for the physical properties of theplanet MOA-2007-BLG-400Lb and its host star. (a) Mass of the host star. (b) Planetsemimajor axis. (c) Distance to the planet/star system. (d) Equilibrium temperatureof the planet. (e) I-band magnitude of the host star. (f) H-band magnitude of thehost star. In panel (a), we also show the probability density for the planet mass, whichis essentially a rescaling of that of the host star, because the mass ratio is measuredso precisely q = (2.5+0.5
−0.3) × 10−3. In all panels, the solid vertical lines show themedians, and the 68.3% and 95.4% confidence intervals are enclosed in the dark andlight shaded regions, respectively. In panel (b) and (d), the probability distributionsfor wide and close degenerate solutions are computed separately, and then the closesolution is weighted by exp(−∆χ2/2), where ∆χ2 = 0.2 is the difference betweenthem. These distributions are derived assuming priors obtained from standard modelsof the mass, velocity, and density distributions of stars in the Galactic bulge and disk,and include constraints from the measurements of lens angular Einstein radius θE andthe timescale of the event tE, as well as limits on the I-band magnitude of the lens.In practice, only the measurements of θE and tE provide interesting constraints onthese distributions.
108
B) BASELINE A) PEAKPEAK
D) BASELINE - SOURCE C) PEAK - BASELINE
Fig. 3.7.— Constraining the blended flux from CTIO I-band images. A good-seeingbaseline image B (upper right) is subtracted from an image taken at magnification245 (A, upper left), and the resulting image C is shown in the bottom-left panel.The white circles on images A and C indicate the source positions. Then C is scaledby 1/244 in flux and subtracted from image B to generate image D, which has thesource contribution removed from the baseline image. As described in the text, we fitPSF to the stars close to the source, and subtract them from images B and D. Aftersubtraction, the flux sums in a 1.8” square (shown as black boxes on right two panels)are 564 ADU and -28 ADU, respectively. Non-detection of the blend constrains theblend-source flux ratio to be less than 5%.
109
Model1 t0 − tref2 u0 tE d q α3 ρ
day day deg
Close 0.08107 0.00025 14.41 0.34 0.0026 227.06 0.00326
We obtain similar results with F555W, but with understandably larger errorbars
since the astrometry is more precise for the microlens in F814W.
Combining equations (4.3) and (4.4), the lens proper motion in the heliocentric
frame is therefore
µl = µgeo + µs +v⊕πrel
AU. (4.25)
For each MCMC realization, πrel is known, so we can convert the lens proper motion
to the velocity of the lens in the heliocentric frame vl,hel and also in the frame of
local standard of rest vl,LSR (we ignore the rotation of the galactic bulge). The
lens velocity in the LSR is estimated to be vl,LSR = 103 ± 15 km s−1. This raises
the possibility of the lens being in the thick disk, in which the stars are typically
metal-poor. As shown in Figure 4.7, the constraints we have cannot resolve the
metallicity of the lens star.
143
Planetary Orbital Motion
Wide/Close Degeneracy Binary-lens light curves in general exhibit a
well-known “close-wide” symmetry (Dominik 1999b; An 2005). Even for some
well-covered caustics-crossing events (e.g., Albrow et al. 1999a), there are quite
degenerate sets of solutions between wide and close binaries. In Paper I, we found
that the best-fit point-source wide-binary solution was preferred over close-binary
solutions by ∆χ2 ∼ 22. But this did not necessarily mean that the wide-close binary
degeneracy was broken, since the two classes of binaries may be influenced differently
by higher order effects. We find that the χ2 difference between best-fit wide and
close solutions is within 1 from “MCMC A” and 2.1 (positive u0) or 2.2 (negative
u0) from “MCMC B”.
However, orbital motion of the planet is subject to additional dynamical
constraints: the projected velocity of the planet should be no greater than the escape
velocity of the system: v⊥ ≤ vesc, where,
v⊥ =√
d2 + (ωd)2AU
πl
θE, (4.26)
vesc =
√
2GM
r≤ vesc,⊥ ≡
√
2GM
dθEDl
=
√
πl
2dπE
c, (4.27)
and where r is the instantaneous 3-dimensional planet-star physical separation. Note
that in the last step, we have used equation (4.1).
144
We then calculate the probability distribution of the ratio
v2⊥
v2esc,⊥
= 2AU2
c2
d3[(d/d)2 + ω2]
[πE + (πs/θE)]3πE
θE
(4.28)
for an ensemble of MCMC realizations for both wide and close solutions. Figure 4.9
shows probability distributions of the projected velocity r⊥γ in the units of critical
velocity vc,⊥, where r⊥γ is the instantaneous velocity of the planet on the sky, which
is further discussed in Appendix C and vc,⊥ = vesc,⊥/√
2. The dotted circle encloses
the solutions that are allowed by the escape velocity criteria, and the solutions that
are inside the solid line are consistent with circular orbital motion. We find that the
best-fit close-binary solutions are physically allowed while the best-fit wide-binary
solutions are excluded by these physical constraints at 1.6 σ. The physically excluded
best-fit wide solutions are favored by ∆χ2 = 2.1 (or 2.2) over the close solutions,
so by putting physical constraints, the degenerate solutions are statistically not
distinguishable at 1σ.
Circular Planetary Orbits and Planetary Parameters Planetary deviations
in microlensing light curves are intrinsically short, so in most cases, only the
instantaneous projected distance between the planet and the host star can be
extracted. As shown in § 4.3.6, for this event, we tentatively measure the
instantaneous projected velocity of the planet thanks to the relatively long
(∼ 4 days) duration of the planetary signal. One cannot solve for the full set of
145
orbital parameters just from the instantaneous projected position and velocity.
However, as we show in Appendix C, we can tentatively derive orbital parameters by
assuming that the planet follows a circular orbit around the host star. In Figure 4.10,
we show the probability distributions of the semimajor axis, inclination, amplitude
of radial velocity, and equilibrium temperature of the planet derived from “MCMC
B” for both wide and close solutions. The equilibrium temperature is defined to
be Teq ≡ (Lbol/Lbol,⊙)1/4(2a/R⊙)−1/2T⊙, where Lbol is the bolometric luminosity of
the host, a is the planet semimajor axis, and Lbol,⊙, R⊙, and T⊙ are the luminosity,
radius, and effective temperature of the Sun, respectively. This would give the Earth
an equilibrium temperature of Teq = 285 K. In calculating these probabilities, we
assign a flat (Opik’s Law) prior for the semimajor axis and assume that the orbits
are randomly oriented, that is, with a uniform prior on cos i.
4.3.7. Constraints on a Non-Luminous Lens
In § 4.3.5, we noted that the blended light must lie within 15 mas of the source:
otherwise the HST images would appear extended. We argued that the blended
light must be associated with the event (either the lens itself or a companion to
either the source or lens), since the chance of such an alignment by a random
field star is < 0.07%. In fact, even stronger constraints can be placed on the
blend-source separation using the arguments of § 4.3.4. These are somewhat more
complicated and depend on the blend-source relative parallax, so we do not consider
146
the general case (which would only be of interest to further reduce the already very
low probability of a random interloper) but restrict attention to companions of the
source and lens. We begin with the simpler source-companion case.
Blend As Source Companion
As we reported in § 4.3.4, there were two HST measurements of the astrometric
offset between the V and I light centroids, dating from 0.09 and 0.84 years after peak,
respectively. In that section, we examined the implications of these measurements
under the hypothesis that the blend is the lens. We therefore ignored the first
measurement because the lens source separation at that epoch is much better
constrained by the microlensing event itself than by the astrometric measurement.
However, as we now examine the hypothesis that the blend is a companion to the
source, both epochs must be considered equally. Most of the weight (86%) comes
from the second observation, partly because the astrometric errors are slightly
smaller, but mainly because the blend contributes about twice the fractional light,
which itself reduces the error on the inferred separation by a factor of 2. Under
this hypothesis, we find a best-fit source-companion separation of 5 mas, with a
companion position angle (north through east) of 280. The (isotropic) error is
3 mas. Approximating the companion-source relative motion as rectilinear, this
measurement strictly applies to an epoch 0.73 years after the event, but of course
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the intrinsic source-companion relative motion must be very small compared to the
errors in this measurement.
There would be nothing unusual about such a source-companion projected
separation, roughly 40 ± 25 AU in physical units. Indeed, the local G-star binary
distribution function peaks close to this value (Duquennoy & Mayor 1991).
The derived separation is also marginally consistent with the companion
generating a xallarap signal that mimics the parallax signal in our dominant
interpretation. The semi-major axis of the orbit would have to be about 0.8 AU to
mimic the 1-year period of the Earth, which corresponds to a maximum angular
separation of about 100 µas, which is compatible with the astrometric measurements
at the 1.6 σ level.
Another potential constraint comes from comparing the color difference with
the magnitude difference of the source and blend. We find that the source is about
0.5 ± 0.5 mag too bright to be on the same main sequence. However, first, this is
only a 1σ difference, which is not significant. Second, both the sign and magnitude
of the difference are compatible with the source being a slightly evolved turnoff star,
which is consistent with its color.
The only present evidence against the source-companion hypothesis is that the
astrometric offset between V and I HST images changes between the two epochs,
and that the direction and amplitude of this change is consistent with other evidence
148
of the proper motion of the lens. Since this is only a P = 1.7% effect, it cannot be
regarded as conclusive. However, additional HST observations at a later epoch could
definitively confirm or rule out this hypothesis.
Blend As A Lens Companion
A similar, but somewhat more complicated line of reasoning essentially rules
out the hypothesis that the blend is a companion to the lens, at least if the lens is
luminous. The primary difference is that the event itself places very strong lower
limits on how close a companion can be to the lens.
A companion with separation (in units of θE) d ≫ 1 induces a Chang-Refsdal
(1979) caustic, which is fully characterized by the gravitational shear γ = q/d2. We
find that the light-curve distortions induced by this shear would be easily noticed
unless γ < 0.0035, that is,
γ =qc
d2c
=qcθ
2E
θ2c
< 0.0035, (4.29)
where qc = Mc/M is the ratio of the companion mass to the lens mass and dc = θc/θE
is the ratio of the lens-companion separation to the Einstein radius. Equivalently,
θc > 19(
qc
1.3
)1/2
θE. (4.30)
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Here, we have normalized qc to the minimum mass ratio required for the companion
to dominate the light assuming that both are main-sequence stars. (We will also
consider completely dark lenses below).
We now show that equation (4.30) is inconsistent with the astrometric data. If a
lens companion is assumed to generate the blend light, then essentially the same line
of reasoning given in § 4.3.7 implies that 0.73 years after the event, this companion
lies 5 mas from the source, at position angle 280 and with an isotropic error of 3
mas. The one wrinkle is that we should now take account of the relative-parallax
term in equation (4.15), whereas this was identically zero (and so was ignored) for
the source-companion case. However, this term is only about 1.8πrel and hence is
quite small compared to the measurement errors for typical πrel ∼< 0.2mas. We will
therefore ignore this term in the interest of simplicity, except when we explicitly
consider the case of large πrel further below.
Of course, the lens itself moves during this interval. From the parallax
measurement alone (i.e. without attributing the V/I astrometric displacement to
lens motion), it is known that the lens is moving in the same general direction,
i.e., with position angle roughly 210. In assessing the amplitude of this motion we
consider only the constraints from finite-source effects (and ignore the astrometric
displacement). These constraints yield a hard lower limit on θE (from lack of
pronounced finite-source effects) of θE > 0.6 mas, which corresponds to a proper
motion µ = 3.1 mas yr−1. At this extreme value (and allowing for 2σ uncertainty
150
in the direction of lens motion as well in the measurement of the companion
position), the maximum lens-companion separation is 11.4 mas (i.e., 19 θE),
which is just ruled out by equation (4.30). At larger θE, the lens-companion
scenario is excluded more robustly. For example, in the limit of large θE, we have
θc = µ × 0.73 yr = θE(0.73 yr/tE) = 3.9θE, which is clearly ruled out by equation
(4.30).
Then we note that any scenario involving values of πrel that are large enough
that they cannot be ignored in this analysis (πrel ∼> 0.5 mas), must also have very
large θE = πrel/πE ∼> 1 mas, a regime in which the lens-companion is easily excluded.
The one major loophole to this argument is that the lens may be a stellar
remnant (white dwarf, neutron star, or black hole), in which case it could be more
massive than the companion despite the latter’s greater luminosity.
4.3.8. Xallarap Effects and Binary Source
Binary source motion can give rise to distortions of the light curve, called
“xallarap” effects. One can always find a set of xallarap parameters to perfectly
mimic parallax distortions caused by the Earth’s motion (Smith et al. 2003).
However, it is a priori unlikely for the binary source to have such parameters, so
if the parallax signal is real, one would expect the xallarap fits to converge to the
Earth parameters. For simplicity, we assume that the binary source is in circular
151
orbit. We extensively search the parameter space on a grid of 5 xallarap parameters,
namely, the period of binary motion P , the phase λ and complement of inclination
β of the binary orbit, which corresponds to the ecliptic longitude and latitude in
the parallax interpretation of the light curve, as well as (ξE,E, ξE,N), which are the
counterparts of (πE,E, πE,N) of the microlens parallax. We take advantage of the
two exact degeneracies found by Poindexter et al. (2005) to reduce the range of the
parameter search. One exact degeneracy takes λ′ = λ + π and χE′ = −χE, while
all other parameters remain the same. The other takes β′ = −β, u0′ = −u0 and
ξ′E,N = −ξE,N (the sign of α should be changed accordingly as well). Therefore we
restrict our search to solutions with positive u0 and with π ≤ λ ≤ 2π. In modeling
xallarap, planetary orbital motion is neglected. In Figure 4.11, the χ2 distribution
for best-fit xallarap solutions as a function of period is displayed in a dotted line,
and the xallarap solution with a period of 1 year has a ∆χ2 = 0.5 larger than the
best fit at 0.9 year. Figure 4.12 shows that, for the xallarap solutions with period of
1 year, the best fit has a ∆χ2 = 3.2 less than the best-fit parallax solution (displayed
as a black circle point) and its orbital parameters are close to the ecliptic coordinates
of event (λ = 268, β = −11). Therefore, the overall best-fit xallarap solution has
∆χ2 = 3.7 smaller than that of the parallax solution (whose χ2 value is displayed as
a filled dot in Fig. 4.11) for 3 extra degrees of freedom, which gives a probability
of 30%. The close proximity between the best-fit xallarap parameters and those
of the Earth can be regarded as good evidence of the parallax interpretation. The
152
slight preference of xallarap could simply be statistical fluctuation or reflect low-level
systematics in the light curve (commonly found in the analysis by Poindexter et al.
2005).
We also devise another test on the plausibility of xallarap. In § 4.3.5, we argued
that the blend is unlikely to be a random interloper unrelated to either the source or
the lens. If the source were in a binary, then the blend would naturally be explained
as the companion of the source star. Then from the blend’s position on the CMD,
its mass would be mc ∼ 0.9 M⊙. By definition, ξE is the size of the source’s orbit as
in the units of rE (the Einstein radius projected on the source plane),
ξE =as
rE
=amc
(mc + ms)rE
, (4.31)
where a is the semimajor axis of the binary orbit, and ms and mc are the masses of
the source and its companion, respectively. Then we apply Kepler’s Third Law:
(
P
yr
)2 m3c
M⊙(mc + ms)2=(
ξErE
AU
)3
. (4.32)
Once the masses of the source and companion are known, the product of ξE and
rE are determined for a given binary orbital period P . And in the present case,
rE/AU = θEDs = θ∗/ρDs = 4.5 × 10−3/ρ. By adopting ms = 1M⊙, mc = 0.9M⊙, for
each set of P and ρ, there is a uniquely determined ξE from equation (4.32). We then
apply this constraint in the xallarap fitting for a series of periods. The minimum χ2s
for each period from the fittings are shown in solid line in Figure 4.11. The best-fit
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solution has ∆χ2 ∼ 1.0 less than the best-fit parallax solution for two extra degrees
of freedom. Although as compared to the test described in the previous paragraph,
the current test implies a higher probability that the data are explained by parallax
(rather than xallarap) effects, it still does not rule out xallarap.
4.4. Summary and Future Prospects
Our primary interpretation of the OGLE-2005-BLG-071 data assumes that the
light-curve distortions are due to parallax rather than xallarap and that the blended
light is due to the lens itself rather than a companion to the source. Under these
assumptions, the lens is fairly tightly constrained to be a foreground M dwarf, with
mass M = 0.46 ± 0.04 M⊙ and distance Dl = 3.2 ± 0.4 kpc, which has thick-disk
kinematics (vLSR ∼ 103 km s−1). As we discuss below, future observations might
help to constrain its metallicity. The microlens modeling suffers from a well-known
wide-close binary degeneracy. The best-fit wide-binary solutions are slightly favored
over the close-binary solutions, however, from dynamical constraints on planetary
orbital motion, the physically allowed solutions are not distinguishable within 1 σ.
For the wide-binary model, we obtain a planet of mass Mp = 3.8 ± 0.4 MJupiter
at projected separation r⊥ = 3.6 ± 0.2 AU. The planet then has an equilibrium
temperature of about T = 55 K, i.e. similar to Neptune. In the degenerate
154
close-binary solutions, the planet is closer to the star and so hotter, and the
estimates are: Mp = 3.4 ± 0.4 MJupiter, r⊥ = 2.1 ± 0.1 AU and T ∼ 71 K.
As we have explored in considerable detail, it is possible that one or both of
these assumptions is incorrect. However, future astrometric measurements that are
made after the lens and source have had a chance to separate, will largely resolve
both ambiguities. Moreover, such measurements will put much tighter constraints
on the metallicity of the lens (assuming that it proves to be the blended light).
First, the astrometric measurements made 0.84 yr after the event detected
motion suggests that there was still 1.7% chance that the blend did not move relative
to the source. A later measurement that detected this motion at higher confidence
would rule out the hypothesis that the blend is a companion to the source. We
argued in § 4.3.7 that the blend could not be a companion to a main-sequence
lens. Therefore, the only possibilities that would remain are that the lens is the
blend, that the lens is a remnant (e.g., white dwarf), or that the blend is a random
interloper (probability < 10−3). As we briefly summarize below, a future astrometric
measurement could strongly constrain the remnant-lens hypothesis as well.
Of course, it is also possible that future astrometry will reveal that the blend
does not moving with respect to the source, in which case the blend would be a
companion to the source. Thus, either way, these measurements would largely
resolve the nature of the blended light.
155
Second, by identifying the nature of blend, these measurements will largely,
but not entirely, resolve the issue of parallax vs. xallarap. If the blend proves not to
be associated with the source, then any xallarap-inducing companion would have to
be considerably less luminous, and so (unless it were a neutron star) less massive
than the mc = 0.9 M⊙ that we assumed in evaluating equation (4.32). Moreover,
stronger constraints on rE (rhs of eq. [4.32]) would be available from the astrometric
measurements. Hence, the xallarap option would be either excluded or very strongly
constrained by this test.
On the other hand, if the blend were confirmed to be a source companion, then
essentially all higher order constraints on the nature of the lens would disappear.
The parallax “measurement” would then very plausibly be explained by xallarap,
while the “extra information” about θE that is presently assumed to come from the
blend proper-motion measurement would likewise evaporate.
These considerations strongly argue for making a future high-precision
astrometric measurement. Recall that in the HST measurements reported in § 4.3.5,
the source and blend were not separately resolved: the relative motion was inferred
from the offset between the V and I centroids, which are displaced because the source
and blend have different colors. Due to its well-controlled PSF, HST is capable of
detecting the broadening of the PSF even if the separation of the lens and source is
a fraction of the FWHM. Assuming that the proper motion is µgeo ∼ 4.4 mas yr−1,
and based on our simulations in § 4.3.5, such broadening would be confidently
156
detectable about 5 years after the event (see also Bennett et al. 2007 for analytic
PSF broadening estimates). Ten years after the event, the net displacement would
be ∼ 40 mas. This compares to a diffraction limited FWHM of 40 mas for H band on
a ground-based 10m telescope and would therefore enable full resolution. The I −H
color of the source is extremely well determined (0.01 mag) from simultaneous I and
H data taken during the event from the CTIO/SMARTS 1.3m in Chile. Hence,
the flux allocation of the partially or fully resolved blend and source stars would
be known. The direct detection of a partially or fully resolved lens will provide
precise photometric and astrometry measurements (see Kozlowski et al. 2007 for one
such example), which will enable much tighter constraints on the mass, distance
and projected velocity of the lens. It also opens up the possibility of determining
the metallicity of the host star by taking into account both non-photometric and
photometric constraints. If, as indicated by the projected velocity measurement, it
is a thick-disk star, then it will be one of the few such stars found to harbor a planet
(Haywood 2008).
As remarked above, a definitive detection of the blend’s proper motion would
still leave open the possibility that it was a companion to the lens, and not the
lens itself. In this case, the lens would have to be a remnant. Without going into
detail, the astrometric measurement would simultaneously improve the blend color
measurement as well as giving a proper-motion estimate (albeit with large errors
because the blend-source offset at the peak of the event would then not be known).
157
It could then be asked whether the parallax, proper-motion, and photometric
data could be consistently explained by any combination of remnant lens and
main-sequence companion. This analysis would depend critically on the values of
the measurements, so we do not explore it further here. We simply note that this
scenario could also be strongly constrained by future astrometry.
4.5. Discussion
With the measurements presented here, and the precision with which these
measurements allow us to determine the properties of the planet OGLE-2005-BLG-
071Lb and its host, it is now possible to place this system in the context of similar
planetary systems discovered by radial velocity (RV) surveys. Of course, the kind
of information that can be inferred about the planetary systems discovered via RV
differs somewhat from that presented here. For example, for planets discovered via
RV, it is generally only possible to infer a lower limit to the planet mass, unless
the planets happen to transit or produce a detectable astrometric signal. Mutatis
mutandis, for planets discovered via microlensing, it is generally only possible to
measure the projected separation at the time of the event, even in the case for which
the microlensing mass degeneracy is broken as it is here (although see Gaudi et al.
2008).
158
With these caveats in mind, we can compare the properties of OGLE-2005-
BLG-071Lb and its host star with similar RV systems. It is interesting to note that
the fractional uncertainties in the host mass and distance of OGLE-2005-BLG-071Lb
are comparable to those of some of the systems listed in Table 4.6.
OGLE-2005-BLG-071Lb is one of only eight Jovian-mass (0.2MJupiter <
Mp < 13MJupiter) planets that have been detected orbiting M dwarf hosts (i.e.,
M∗ < 0.55 M⊙) (Marcy et al. 1998, 2001; Delfosse et al. 1998; Butler et al. 2006;
Johnson et al. 2007b; Bailey et al. 2008). Table 4.6 summarizes the planetary
and host-star properties of the known M dwarf/Jovian-mass planetary systems.
OGLE-2005-BLG-071Lb is likely the most massive known planet orbiting an M
dwarf.
As suggested by the small number of systems listed in Table 4.6, and shown
quantitatively by several recent studies, the frequency of relatively short-period
P ∼< 2000 days, Jupiter-mass companions to M dwarfs appears to be ∼ 3 − 5 times
lower than such companions to FGK dwarfs (Butler et al. 2006; Endl et al. 2006;
Johnson et al. 2007b; Cumming et al. 2008). This paucity, which has been shown
to be statistically significant, is expected in the core-accretion model of planet
formation, which generally predicts that Jovian companions to M dwarfs should be
rare, since for lower mass stars, the dynamical time at the sites of planet formation is
longer, whereas the amount of raw material available for planet formation is smaller
(Laughlin et al. 2004; Ida & Lin 2005; Kennedy & Kenyon 2008, but see Kornet
159
et al. 2006). Thus, these planets typically do not reach sufficient mass to accrete a
massive gaseous envelope over the lifetime of the disk. Consequently, such models
also predict that in the outer regions of their planetary systems, lower mass stars
should host a much larger population of ‘failed Jupiters,’ cores of mass ∼< 10 M⊕
(Laughlin et al. 2004; Ida & Lin 2005). Such a population was indeed identified
based on two microlensing planet discoveries (Beaulieu et al. 2006; Gould et al.
2006).
Our detection of a ∼ 4 MJupiter companion to an M dwarf may therefore present
a difficulty for the core-accretion scenario. While we do not have a constraint on the
metallicity of the host, the fact that it is likely a member of the thick disk suggests
that its metallicity may be subsolar. If so, this would pose an additional difficulty for
the core-accretion scenario, which also predicts that massive planets should be rarer
around metal-poor stars (Ida & Lin 2004), as has been demonstrated observationally
(Santos et al. 2004; Fischer & Valenti 2005). This might imply that a different
mechanism is responsible for planet formation in the OGLE-2005-BLG-071L system,
such as the gravitational instability mechanism (Boss 2002, 2006).
One way to escape these potential difficulties is if the host lens is actually a
stellar remnant, such as white dwarf. The progenitors of remnants are generally
more massive stars, which are both predicted (Ida & Lin 2005; Kennedy & Kenyon
2008) and observed (Johnson et al. 2007a,b) to have a higher incidence of massive
planets. As we discussed above, future astrometric measurements could constrain
160
both the low-metallicity and remnant-lens hypotheses. These measurements are
therefore critical.
Although it is difficult to draw robust conclusions from a single system,
there are now four published detections of Jovian-mass planetary companions
with microlensing (Bond et al. 2004; Gaudi et al. 2008), and several additional
such planets have been detected that are currently being analyzed. It is therefore
reasonable to expect several detections per year (Gould 2009), and thus that it will
soon be possible to use microlensing to constrain the frequency of massive planetary
companions. These constraints are complementary to those from RV, since the
microlensing detection method is less biased with respect to host star mass (Gould
2000a), and furthermore probes a different region of parameter space, namely cool
planets beyond the snow line with equilibrium temperatures similar to the giant
planets in our solar system (see, e.g. Gould et al. 2007 and Gould 2009).
161
RoboNet FTN R
PLANET Canopus I
MOA I
I
I
Clear
V
I
ClearOGLE V
OGLE I
3479 3480 3481 3482 348316
15.8
15.6
15.4
15.2
15
Fig. 4.1.— Main panel: all available ground-based data of the microlensing eventOGLE-2005-BLG-071. HST ACS HRC observations in F814W and F555W weretaken at two epochs, once when the source was magnified by A ∼ 2 (arrow), andagain at HJD = 2453788.2 (at baseline). Planetary models that include (solid) andexcludes (dotted) microlens parallax are shown. Zoom at bottom: triple-peak featurethat reveals the presence of the planet. Each of the three peaks corresponds to thesource passing by a cusp of the central caustic induced by the planet. Upper inset:trajectory of the source relative to the lens system in the units of angular Einsteinradius θE. The lens star is at (0, 0), and the star-planet axis is parallel to the x-axis.The best-fit angular size of the source star in units of θE is ρ ∼ 0.0006, too small tobe resolved in this figure.
162
Fig. 4.2.— Probability contours (∆χ2 = 1, 4) of microlens parallax parametersderived from MCMC simulations for wide-binary (in solid line) and close-binary (in adashed line) solutions. Fig. 2 and eq. (12) in Gould (2004) imply that πE,⊥ is definedso that πE,‖ and πE,⊥ form a right-handed coordinate system.
163
Fig. 4.3.— CMD for the OGLE-2005-BLG-071 field. Black dots are the stars withthe OGLE I-band and V -band observations. The red point and green points show thecenter of red clump and the source, respectively. The errors in their fluxes and colorsare too small to be visible on the graph. Cyan points are the stars in the ACS field,which are photometrically aligned with OGLE stars using 10 common stars. Themagenta point with error bars show the color and magnitude of the blended light.
164
Fig. 4.4.— HST ACS astrometric measurements of the target star in F814W (red)and F555W (blue) filters in 2005 (filled dots) and 2006 (open dots). The centerpositions of the big circles show mean values of the 4 dithered observations in eachfilter at each epoch while radii of the circle represent the 1σ errors.
165
finite-sourceastrometryjoint
wide
wideparallaxjoint
finite-sourceastrometryjoint
close
closeparallaxjoint
Fig. 4.5.— Upper two panels show posterior probability contours at ∆χ2 = 1 (solidline) and 4 (dotted line) for relative lens-source proper motion µgeo. The left panelis for wide-binary solutions and the right one is for close-binary. The green contoursshow the probability distributions constrained by the finite-source effects. The blackcontours are derived from HST astrometry measurements assuming πrel = 0.2 mas.The red contours show the joint probability distributions from both constraints. Thelower two panels show the posterior probability distribution of the position angle φµ
geo
of the relative lens-source proper motion for wide-binary and close-binary solutions,respectively. The histogram in red is derived from the red contours of joint probabilityfor finite source and astrometry constraints in the upper panel. The blue histogramrepresents that of the microlens parallax. They mildly disagree at 2.5σ.
166
Fig. 4.6.— Differences between OGLE V and HST F555W magnitudes for thematched stars are plotted against their V magnitudes measured by OGLE. Tocalculate the offset, we add a 0.017 mag “cosmic error” in quadrature to each point inorder to reduce χ2/dof to unity. The open circles represent the stars used to establishthe final transformation, and the filled point shows an “outlier”.
167
wide
close
V-I
1.8 (best-estimate)
2.0
2.1
2.3
2.6
Fig. 4.7.— Posterior probability distribution of lens mass M and relative lens-source parallax πrel from MCMC simulations discussed in § 4.3.6. The constraintsinclude those from parallax effects, finite-source effects and relative proper motionmeasurements from HST astrometry. The ∆χ2 = 1, 4, 9 contours are displayed insolid, dotted and dashed lines, respectively. Both wide-binary (magenta) and close-binary (blue) solutions are shown. The lines in black, red and green represent thepredicted M and πrel from the isochrones for different metal abundances: [M/H] = 0(black), −0.5 (red), −1.0 (green). The points on these lines correspond to the observedI-band magnitude I = 21.3 and various V −I values V −I = 1.8 (best estimate, filleddots), 2.0 (0.5 σ, filled triangle), 2.1 (1.0 σ, filled squares), 2.3 (1.5 σ, filled pentagons),and 2.6 (2.0 σ, filled hexagons)
168
Fig. 4.8.— Posterior probability distribution of lens mass M and relative lens-sourceparallax πrel from MCMC simulations assuming that the blended light comes fromthe lens star. The ∆χ2 = 1, 4 contours are displayed in a solid line for wide solutions,and in a dotted line for close solutions.
169
Fig. 4.9.— Probability contours of projected velocity r⊥γ (defined in Appendix C)in the units of vc,⊥ for both close-binary (upper panel) and wide-binary (lower panel)solutions. All the solutions that are outside the dotted circle are physically rejected asthe velocities exceed the escape velocity of the system. The boundary in a solid lineinside the dotted circle encloses the solutions for which circular orbits are allowed.
170
Fig. 4.10.— Probability distributions of planetary parameters (semimajor axis a,equilibrium temperature, cosine of the inclination, and amplitude of radial velocity ofthe lens star) from MCMC realizations assuming circular orbital motion. Histogramsin black and red represent the close-binary and wide-binary solutions, respectively.Dotted and dashed histograms represent the two degenerate solutions for each MCMCrealization discussed in Appendix C.
171
Fig. 4.11.— χ2 distributions for best-fit xallarap solutions at fixed binary-sourceorbital periods P . The solid and dotted lines represent xallarap fits with and withoutdynamical constraints described in § 4.3.8. The best-fit parallax solution is shown asa filled dot at period of 1 year. All of the fits shown in this figure assume no planetaryorbital motion.
172
Fig. 4.12.— Results of xallarap fits by fixing binary orbital phase λ and complementof inclination β at period P = 1 yr and u0 > 0. The plot is color-coded for solutionswith ∆χ2 within 1 (black), 4 (red), 9 (green), 16 (blue), 25 (magenta), 49 (yellow)of the best fit. The Earth parameters are indicated by black circles. Because of aperfect symmetry (u0 → −u0 and α → −α), the upper black circle represents Earthparameter (λ = 268, β = −11) for the case u0 < 0. Comparison of parallax withxallarap must be made with the better of the two, that is, the lower one.
Fig. 5.1.— Standard (Paczynski 1986) microlensing fit to the light curve of OGLE-2005-SMC-001, with data from OGLE I and V in Chile, µFUN I and V in Chile,Auckland clear-filter in New Zealand, and the Spitzer satellite 3.6 µm at ∼ 0.2 AUfrom Earth. The data are binned by the day. All data are photometrically alignedwith the (approximately calibrated) OGLE data. The residuals are severe indicatingthat substantial physical effects are not being modeled. The models do not includeparallax, but when parallax is included, the resulting figure is essentially identical.
239
−1 −.5 0 .5 1−1
−.5
0
.5
1(B)
Time (days)2.
5 lo
g(A
)
−40 −20 0 20 400
.5
1
1.5
2
(A)Earth
Time (days)
2.5
log(
Fs*
A+
Fb)
−40 −20 0 20 400
.5
1
1.5
2
(C)Spitzeru0 < 0
Time (days)−40 −20 0 20 400
.5
1
1.5
2
(D)Spitzeru0 > 0
Fig. 5.2.— Why 4 (not 3) Spitzer observations are needed to measure πE =(πE,‖, πE,⊥). Panel A shows Earth-based light curve of hypothetical event (black curve)with πE = (0.4,−0.2), u0 = −0.2, and tE = 40 days, together with the corresponding(red) lightcurve with zero parallax. From the asymmetry of the lightcurve, one canmeasure πE,‖ = 0.4 and |u0| = 0.2, but no information can be extracted aboutπE,⊥ or the sign of u0. Indeed, 9 other curves are shown with various values ofthese parameters, and all are degenerate with the black curve. Panel (B) shows thetrajectories of all ten models in the geocentric frame (Gould 2004) that generatethese degenerate curves. Solid and dashed curves indicate positive and negative u0,respectively, with πE,⊥ = −0.4,−0.2, 0, +0.2, +0.4 (green, black, magenta, cyan, blue).Motion is toward positive x, while the Sun lies directly toward negative x. Dotsindicate 5 day intervals. Panel C shows full light curves as would be seen by Spitzer,located 0.2 AU from the Earth at a projected angle 60 from the Sun, for the 5 u0 < 0trajectories in Panel B. The source flux Fs and blended flux Fb are fit from the twofilled circles and a third point at baseline (not shown) as advocated by Gould (1999).Note that these two points (plus baseline) pick out the “true” (black) trajectory, fromamong other solutions that are consistent with the ground-based data with u0 < 0, butPanel D shows that these points alone would pick out the magenta trajectory amongu0 > 0 solutions, which has a different πE,⊥ from the “true” solution. However, afourth measurement open circle would rule out this magenta u0 > 0 curve and soconfirm the black u0 < 0 curve.
240
Fig. 5.3.— Parallax πE = (πE,N , πE,E) 1 σ error ellipses for all discrete solutions forOGLE-2005-SMC-001. The left-hand panels show fits excluding the Spitzer data,while the right-hand panels include these data. The upper panels show fits withblending as a free parameter whereas the lower panels fix the OGLE blending at zero.The ellipses are coded by ∆χ2 (relative to each global minimum), with ∆χ2 < 1 (red),1 < ∆χ2 < 4 (green), 4 < ∆χ2 < 9 (cyan), 9 < ∆χ2 < 16 (magenta), ∆χ2 > 16(blue). Close- and wide-binary solutions are represented by bold and dashed curves,respectively. Most of the “free-blend, no-Spitzer” solutions are highly degeneratealong the πE,⊥ direction (33 north through east), as predicted from theory, becauseonly the orthogonal (πE,‖) direction is well constrained from ground-based data. Atseen from the two upper panels, the Spitzer observations reduce the errors in the πE,⊥
direction by a factor ∼ 3 when the blending is a free parameter. However, fixing theblending (lower panels) already removes this freedom, so Spitzer observations thenhave only a modest additional effect.
241
OGLE-2005-SMC-001
OGLE I
OGLE V
CTIO I
AUCKLAND
CTIO V
SPITZER
1.5
2
2.5
3
3560 3580 3600 3620 364017
16.5
16
15.5
15
3560 3580 3600 3620 36400.06
0.04
0.02
0
-0.02
-0.04
-0.06
Fig. 5.4.— Best-fit binary microlensing model for OGLE-2005-SMC-001 togetherwith the same data shown in Fig. 5.1. The model includes microlens parallax (twoparameters) and binary rotation (two parameters). The models for ground-basedand Spitzer observations are plotted in blue and red, respectively. All data arein the units of 2.5 log(A), where A is the magnification. Ground-based data arealso photometrically aligned with the (approximately calibrated) OGLE data. Theresiduals show no major systematic trends.
242
6 4 2 0 -2 -4 -6-6
-4
-2
0
2
4
6
Fig. 5.5.— Likelihood contours of the inverse projected velocity Λ ≡ v/v2 for SMClenses together with Λ values for light-curve solutions found by MCMC. The latterare color-coded for solutions with ∆χ2 within 1, 4, and 9 of the global minimum. Thelikelihood contours are spaced by factors of 5.
243
6 4 2 0 -2 -4 -6-6
-4
-2
0
2
4
6
Fig. 5.6.— Likelihood contours of the inverse projected velocity Λ ≡ v/v2 for halolenses together with Λ values for light-curve solutions found by MCMC. Similar toFigure 5.5 except in this case the contours are color coded, with black, red, yellow,green, cyan, blue, magenta, going from highest to lowest.
244
Chapter 6
Probing ∼ 100AU Intergalactic Mg II Absorbing
“Cloudlets” with Quasar Microlensing
6.1. Introduction
Mg II absorbers toward quasar sight lines have been systematically studied since
Lanzetta et al. (1987) (for more recent studies, see Zibetti et al. 2005 and references
therein). Similar Mg II absorbers were subsequently seen in gamma-ray burst (GRB)
spectra. Prochter et al. (2006) compared strong Mg II absorbers toward GRB and
quasar sight lines, and found a significantly higher incidence toward the former.
They proposed three possible effects to explain this discrepancy: (1) obscuration
of faint quasars by dust associated with the absorbers; (2) Mg II absorbers being
intrinsic to GRBs; (3) gravitational lensing of the GRB by the absorbers. However,
they concluded that none of these effects provide a satisfactory explanation.
Frank et al. (2006) proposed a simple geometric solution to this puzzle. They
argued that if Mg II absorption systems are fragmented on scales ∼< 1016cm, similar
to the beam sizes of GRBs, then the observed difference in the frequency of Mg II
245
absorbers would simply reflect the difference in the average beam sizes between
GRBs and quasars, with quasars being several times larger. This explanation
predicts that absorption features due to intervening Mg II cloud fragments should
evolve as the size of GRB afterglow changes, an effect that has now been observed
by Hao et al. (2006). However, the structures of Mg II absorbers down to scales of
∼ 1016cm cannot be directly inferred from their spectral features. Rauch et al. (2002)
put the strongest upper limits on Mg II absorber sizes to date. They observed the
spectra of three images of Q2237+0305 (Huchra et al. 1985) and found that each line
of sight contained individual Mg II absorbers at approximately the same redshift,
but with distinct spectral features. Thus these absorbers are part of a complex that
extends at least ∼ 500pc, but the sizes of the individual “cloudlets” must be smaller
than 200 − 300pc, corresponding to the separations of the macro-images.
Some, if not all, strongly lensed quasars are also gravitationally microlensed
by compact stellar-mass objects in the lensing galaxy (Wambsganss 2006), and
Q2237+0305 was the first lensed quasar found to exhibit significant microlensing
variability (Corrigan et al. 1991; Wozniak et al. 2000). The macro-image of a
microlensed quasar is split into many micro-images, and when the source moves over
the caustic networks induced by the microlenses, those micro-images expand, shrink,
appear, disappear and experience drastic astrometric shifts over timescales of months
or years (Treyer & Wambsganss 2004). The angular sizes of major micro-images are
usually of the same order as those of the quasars, and during the shape and position
246
changes of these images, absorption structures of similar scale along their sight lines
will likely imprint significant variations on the spectrum.
Brewer & Lewis (2005) pioneered the theoretical investigation of quasar
microlensing as a probe of the sub-parsec structure of intergalactic absorption
systems. They concluded that variation in the strength of the absorption lines over
timescales of years or decades caused by microlensing can be used to probe the
structures of Lyman α clouds and associated metal-line absorption systems on scales
∼< 0.1pc. However, as I will show, they significantly underestimated the relevant
timescales for spectral variability given the sizes of the systems they considered.
Thus, they substantially overestimated the scales of absorption structures that
microlensing can effectively probe.
In § 6.2, I lay out the basic theoretical framework of the method. Then in
§ 6.3, I present a numerical simulation of the microlensing of Q2237+0305. I show
that micro-images of this quasar can be used to probe structures of Mg II and other
metal-line absorbing clouds on scales of ∼ 1014 − 1016cm by monitoring the spectral
variations of absorption lines over months or years. Finally in § 6.4, I summarize the
results and discuss their implications.
247
6.2. Varying Microlensed Quasar Image as A “Ruler”
I begin with a brief summary of notation. Subscripts “l”, “s”, “o” and “c”
refer to the lens, source, observer and absorption cloud planes, respectively. The
superscript “ray” is used to refer to the light ray on the cloud plane to distinguish it
from the cloud. The angular diameter distance between object x and y is denoted
Dxy and is always positive regardless of which is closer; in particular, Dx refers to the
angular diameter distance between the observer and object x. The vector angular
position of object x is denoted θx, while its redshift is denoted zx.
Consider an absorbing cloud that is confronted with a “bundle of light rays”
making their way from the source to the lens to the observer. Let rrayc be the position
vector of a point on the plane of the cloud. The line depth 〈Aλ〉 of an absorption
line centered at wavelength λ is given by:
〈Aλ〉 =∫
σλ(rrayc )Aλ(r
rayc )d2rray
c
/∫
σλ(rrayc )d2rray
c (6.1)
where σλ(rrayc ) is the surface density of the “ray bundles” on the plane of the
absorption cloud with the rays weighted by the surface brightness profile of the
quasar and Aλ(rrayc ) is the absorption fraction at rray
c for light of wavelength λ
(Brewer & Lewis 2005).
248
6.2.1. Basic Geometric Configurations and Motions
The absorption cloud can be located either between the lens and observer or
between the lens and source. In the former case, the angular position of the light ray
at the cloud plane is θrayc = θi, so rray
c = θiDc. The projected light rays on the cloud
plane maintain the exact shapes of the quasar images, and their physical extents are
proportional to the distance to the observer.
If the cloud is between the lens and the source, it can be easily shown that θrayc
is given by (Brewer & Lewis 2005):
θrayc =
(
1 − DlcDs
DlsDc
)
θi +DlcDs
DlsDc
θs, (6.2)
so
rrayc =
(
Dc −Dlc
Dls
Ds
)
θi +Dlc
Dls
rs. (6.3)
The lens-source relative proper motion, with time as measured by the observer, is
given (in other notation) by Kayser et al. (1986),
µls =1
1 + zs
vs
Ds
− 1
1 + zl
vl
Dl
+1
1 + zl
voDls
DlDs
(6.4)
where vs, vl, vo are the transverse velocities of the source, lens and observer, relative
to the cosmic microwave background (CMB). In particular,
vo = vCMB − (vCMB · z)z, (6.5)
249
where z is the unit vector in the direction of the lens and vCMB is the heliocentric
CMB dipole velocity (Kochanek 2004).
The formulae in this subject can be greatly simplified by proper choice of
notation. To this end, I defined the “absolute” proper motion of an object x moving
at transverse velocity vx to be:
µabs,x =1
1 + zx
vx
Dx
, (6.6)
I also define the “reflex proper motion” of an object x relative to the observer-lens
axis to be:
µo,l,x = sgn(zx − zl)1
1 + zl
voDlx
DlDx
, (6.7)
Then equation (6.4) can then be simplified,
µls = µabs,s − µabs,l + µo,l,s. (6.8)
Similarly, the lens-cloud relative proper motion is given by:
µlc = µabs,c − µabs,l + µo,l,c (6.9)
Note that the last term has a different sign depending on whether the cloud is farther
or closer than the lens (see eq. [6.7]).
250
6.2.2. Bulk Motion of the Un-microlensed “Ray
Bundles”
Equations (6.2), (6.8) and (6.9) are the key formulae needed to carry out the
simulation in § 6.3. In this section, I discuss the bulk motion and relevant timescales
of the macro-image and its associated “ray bundles” on the cloud plane for the
underlying case that the image is not perturbed by microlenses.
When the source moves at µls, the angular positions of the rays that compose
the images are also changing with respect to the observer-lens axis. If the
macro-image is unperturbed by the microlenses, the relative proper motion µli is
simply given by (Kochanek et al. 1996),
µli = M · µls, (6.10)
where M is the magnification tensor. At the same time, the intersection of “ray
bundles” with the cloud plane also change its angular position relative to the lens.
If the cloud plane is between the lens and the source, then from equation (6.2), the
“ray bundle”-lens relative proper motion µraylc is a linear combination of µls and µli
weighted by distances,
µraylc =
(
1 − DlcDs
DlsDc
)
µli +DlcDs
DlsDc
µls. (6.11)
251
Substituting equation (6.10) into equation (6.11) yields the bulk proper motion of
the un-microlensed “ray bundle” relative to the lens,
µraylc,bulk =
[(
1 − DlcDs
DlsDc
)
M +DlcDs
DlsDc
I]
· µls. (6.12)
where I is the unit tensor.
When the “ray bundles” are between the source and the lens, their relative bulk
proper motion is simply,
µraylc,bulk = µli = M · µls. (6.13)
There are two major effects that may induce variability of absorption lines
toward a lensed quasar. One is the creation, destruction, distortion and astrometric
shifts of micro-images, which probe structures similar to the size of the micro-images.
The other is the bulk motion of the “ray bundles” relative to the cloud. On the
cloud plane, this motion has an angular speed of ∆µlc,bulk = µraylc,bulk − µlc, and the
time tcc required for the “ray bundles” to cross a cloud of transverse size Rc is,
tcc =Rc
Dc|∆µbulk,lc|=
Rc
Dc|µraybulk,lc − µlc|
. (6.14)
6.3. Application to Q2237+0305
In October 1998, Rauch et al. (2002) obtained high-resolution Keck spectra of
images A, B and C of Q2237+0305. They found Mg II absorption lines at redshifts
252
of z = 0.5656 and z = 0.827 in the spectra of all three images, but absorption profiles
of the individual sight lines differed (e.g., Figs. 6 and 10 of their paper). Therefore,
they concluded that the Mg II complexes giving rise to these absorption features
must be larger than ∼ 0.5kpc, while the individual Mg II components must be
smaller than ∼ 200 − 300h−150 pc. Q2237+0305 is also one of the most observed and
best studied lensed quasars with obvious microlensing features, and the properties
of the system are well known. These factors make it an ideal object to investigate.
The comprehensive statistical study of this lens by Kochanek (2004) showed
that the size of the quasar is ∼ 1015h−1cm − 1016h−1cm. Mortonson et al. (2005)
demonstrated that the source size has a much more significant effect on microlensing
models than the source brightness profile. For simplicity, in the simulation, I model
the source as uniform disks with four different sizes: 1015h−1cm, 3 × 1015h−1cm,
5 × 1015h−1cm and 1016h−1cm. Uniform grids of rays are traced from the image
plane to the source plane (Wambsganss 1990). Because structures of interest have
similar sizes as the source, finite-source effects must be taken into account. The
grid size used has an angular scale 1/10 of the smallest source. Kochanek (2004)
demonstrated that a Salpeter mass function cannot be distinguished from a uniform
mass distribution and found the mean stellar mass to be 〈M 〉 ∼ 0.037h−1M⊙. For
simplicity, I assign all stars in the simulation the same mass of 0.04h−1M⊙. Rays
are shot from a region extending 47〈θE〉 on each side, where 〈θE〉 is the Einstein
radius of a 0.04h−1M⊙ star. I adopt a convergence and shear for image A of
253
(κ, γ) = (0.394, 0.395) from Kochanek (2004), and set the stellar surface density
κ∗ = κ. Four different trajectories, oriented at 0, 30, 60 and 90 degrees with respect
to the direction of the shear are studied. Positions on both the image plane and the
source plane are recorded once a ray falls within a distance of 2 times the largest
source size from any trajectory on the source plane. A total length of 5〈θE〉 along
each trajectory is considered.
Throughout the paper, I adopt a ΛCDM cosmology with Ωm = 0.3, ΩΛ = 0.7
and H0 = 100h km s−1 Mpc−1. The lens and source are at redshifts
zl = 0.0394 and zs = 1.695 (Huchra et al. 1985). These imply (Ds, Dl, Dls) =
(1223, 113, 1180)h−1 Mpc. Based on Kochanek (2004), I adopt transverse velocities
of the lens, source and observer of (vl, vs, vo) = (300, 140, 62)km s−1. The lens
and source absolute proper motions and the source reflex proper motion are
(µabs,l, µabs,s, µo,l,s) = (0.54, 0.009, 0.11) h µas yr−1. So the lens absolute proper
motion dominates the lens-source relative proper motion. It can easily be shown
that the absolute and reflex proper motions of the cloud are much smaller than the
absolute proper motion of the lens unless the cloud redshift is close to or smaller
than the lens redshift. For Q2237+0305, the lens redshift is very small compared
to the source, so most likely zc ≫ zl. Thus, in following analysis, I focus on clouds
with redshift zc > zl, and ignore the absolute and reflex proper motions of the source
254
and the cloud. Under these assumptions, the source and the cloud share the same
relative proper motion,
µlc = µls = −µabs,l (6.15)
In the simulation, as a practical matter, I hold the positions of the observer,
lens galaxy (as well as its microlensing star field) fixed, and allow the source to move
through the source plane at µls. By equation (6.15), the cloud then has the same
relative proper motion as the source. At any given time, the angular positions of
the “ray bundles” are calculated using equation (6.2). Then by simply subtracting
the angular position of the source at that time, the “ray bundles” positions are
transformed to the reference frame of the cloud.
During short timescales, microlensing causes centroid shifts of the macro-image
(Lewis & Ibata 1998; Treyer & Wambsganss 2004) with respect to the steady bulk
motion of “ray bundles” relative to the cloud, which is described in § 6.2.2. If the
direction of relative lens-source proper motion is the same as the lens shear, then by
subtracting µls from equation (6.12), one finds that the “ray bundles” have their
maximum bulk proper motion relative to the cloud,
∆µbulk,lc,max =(
1 − DlcDs
DlsDc
)
(
1
1 − κ − γ− 1
)
µls
∼ 2(
1 − 1.037Dlc
Dc
)
h µas yr−1. (6.16)
255
If the source moves perpendicular relative to the lens shear, ∆µlc =
[1 − DLCDS/(DLSDC)]|1/(1 − κ + γ) − 1|µls, which is approximately 0 for
the (κ, γ) of image A.
Figures 6.1 and 6.2 show the results for a source trajectory that is parallel
to the shear direction. The bottom panel of Figure 6.1 shows the magnification
pattern on the source plane, with a series of 4 concentric circles centered at 5 source
positions; and the top panels show the images at these positions relative to the
source (which has the same proper motion as the cloud). Different colors represent
different source sizes. The middle panel of Figure 6.1 shows the light curves for the
4 source sizes with the blue dashed lines used to mark the times for the five source
positions. Figure 6.2 shows the “ray bundle” positions in the cloud frame at redshifts
1.69, 0.83, 0.57, 0.1. The 5 different columns show the 5 positions corresponding to
those in Figure 6.1.
In the top row of Figure 6.2, one can see that the “ray bundles” at zc = 1.69,
which is very close to the quasar, have almost exactly the same size and shape
as the source and that the bundles show almost no bulk motion. The density of
“ray bundles” clearly have the imprints from the magnification pattern shown in
the bottom panel of Figure 6.1. So if an absorption cloudlet has a size similar to
or somewhat smaller than a source that is sitting directly behind it, the depth of
its corresponding absorption line will change dramatically as the source crosses the
caustics. The magnification close to a fold caustic is proportional to the inverse
256
square root of the distance from it, so for a cloud with angular size θc < θs, the
fractional change in absorption line depth caused by the caustics scales as (θc/θs)3/4.
Hence, for a cloud close to the quasar redshift, structures on scale ∼ 1014−1016h−1cm
(depending on the source size) will be probed over few-month to few-year timescales
(i.e., the timescale of typical caustics crossings).
The fourth row of panels of Figure 6.2 shows “ray bundles” for zc = 0.1, which
is close to the lens redshift. A distinct difference between these “ray bundles”
and the ones at zc = 1.69 is that they are split into many groups of bundles,
which correspond to the micro-images in the top panel of Figure 6.1. I dub these
groups of “ray bundles” as micro-images in the following discussions. Most of these
micro-images are stretched one-dimensionally, and most rays are concentrated in
a few major micro-images. The micro-images in different columns have drastically
different morphologies and positions. If there are cloudlets of similar size as these
micro-images distributed on the cloud plane, then the absorption spectra will show
multi-component absorption features at any given time. These components will
experience drastic changes in line depth, with some components disappearing and
other new components appearing during the course of months or years, as the source
crosses the microlensing caustics. Another important characteristic is that the bulk
of these micro-images are moving in the same direction as the source. This motion
is described by equation (6.16), which yields ∼ 0.8 h µas yr. So from equation (6.14),
structures as large as ∼ 3.0n × 1016 h−1 cm will be crossed in n × 10 h−1 yr by the
257
bulk motion of the “ray bundles”. Hence, the effects caused by the micro-images
and the bulk motion of the bundles together probe scales of ∼ 1014 − 1016h−1cm on
timescales of months to years.
The second and third row of panels in Figure 6.2 refer to clouds at intermediate
redshifts between the lens and the source. Their redshifts, zc = 0.83 and zc = 0.57,
are close to the Mg II absorption systems observed by Rauch et al. (2002). As
expected, the characteristics of these “ray bundles” are intermediate between those
shown in rows 1 and 4. The overall shapes of the micro-images are close to that
of the source, with magnification patterns imprinted on them. And they also
clearly show multiple components, which appear and disappear as the source crosses
caustics. The angular sizes of the images are close to that of the (unmagnified)
quasar. These micro-images could probe clouds with angular size from a factor of
few smaller to a factor of few larger than the source size, which corresponds to scales
of ∼ 1014 − 1016h−1cm. From equation (6.14) and (6.16), the bulk motion of the
bundles will probe cloud structures ∼ 0.8n × 1016 h−1 cm and ∼ 1.3n × 1016 h−1 cm
during n × 10 h−1 yr for zc = 0.83 and zc = 0.57, respectively.
For a source trajectory that is perpendicular to the shear, the “ray bundles”
will exhibit almost no bulk motion relative to the cloud, while the magnitude
of bulk motion for intermediate trajectories is a fraction that is of the parallel
case, depending on the angles of motion relative to the shear direction. And
the micro-images of these trajectories share similar properties with those of the
258
trajectory that is parallel to the shear. Figure 6.2 shows that micro-image motions
are on the same order of magnitude as the bulk motion of the macro-image over the
timescales considered. Therefore, trajectory direction only has a modest impact on
the scales at which cloud can be probed.
These results are contrast significantly with those of Brewer & Lewis (2005),
who claimed that quasar microlensing for image A of Q2237+0305 can induce
considerable variability of absorption lines associated with structures as large as
0.1pc during the course of years to decades. According to their analysis, the effect is
largest when the cloud is very close to the source, and for example, the timescale of
line strength variation for a 0.1pc cloud very close to the source is given as ∼ 16.2yr.
However, in their analysis, they effectively assumed the relative lens-cloud proper
motion µlc = 0. This would lead to a timescale tcc ∼ RcDl/(vlDs) ∼ 16(Rc/0.1pc)yr
for an absorption cloud near the source redshift (they adopted vl = 600km s−1),
which is in agreement with column 4 of their Table 1. In fact, I showed that, when
peculiar velocities of the source and cloud are ignored, µlc = µls (eq. [6.15]), which
is not negligible. Even considering realistic peculiar motions of the source and the
cloud, it still leads to time scales that are more than one order of magnitude slower
than those predicted for µlc = 0. In addition, when the cloud is close to the source,
the angular sizes of the clouds they considered are orders of magnitudes larger than
their source size, so effects of changes in magnification pattern on the source plane
259
alone have very little impact as well. Therefore, Brewer & Lewis (2005) significantly
overestimated the cloud size to which microlensing is effectively sensitive.
6.4. Discussion and Conclusion
I have shown that there are two effects that might induce variation of
absorption lines along the sight lines to lensed quasars. One effect is caused by
the drastic morphological and positional changes of micro-images when the source
crosses the caustic network. The other effect is due to the bulk motion of the
“ray bundles” relative to the absorption clouds. I have laid out a basic framework
in studying these effects for microlensed quasars in general. And in particular, I
performed numerical simulations to apply the method to image A of Q2237+0305.
I demonstrated that the combinations of these two effects probe 1014 − 1016 h−1 cm
absorption cloudlets between the lens and the source over timescales of months to
years. The existence of these cloudlets will be revealed by either changes in line
depths or appearances/disappearances of multi-absorption components. Spectra
should preferably be taken during the course of caustic crossings, which can be
inferred from photometric monitoring programs of lensed quasars. In fact, the Mg II
lines observed by Rauch et al. (2002) about 8 years ago already show different
multi-components along sight lines to three different macro-images, implying they
might be caused by fragmented cloudlets with similar sizes as the micro-images. A
260
similar high-resolution spectrum taken in the near future would provide a definitive
test of the existence for structures of Mg II or other metal-line absorbers at the scales
of 1014 − 1016 h−1 cm. If the spectral variations are indeed observed, a statistical
study similar to Kochanek (2004) will be required to infer the properties of the
cloudlets. Moreover, a time series of spectra may provide additional constraints to
quasar models that are currently based only on photometric-monitoring data.
261
Fig. 6.1.— Caustics network (bottom panel), light curve (middle panel) and imagesrelative to the source position (top panel) for a source trajectory that is parallel to thelens shear (the direction of x-axis). Different colors represent the 4 different sourcesizes.
262
Fig. 6.2.— Physical positions of “ray bundles” in the frame of cloud at redshifts 0.1,0.57, 0.83 and 1.69. The 5 different columns correspond to the source positions shownin Figure 6.1. x-axis is the direction of the lens shear.
263
Appendix A
Two New Finite-Source Algorithms
To model planetary light curves, we develop two new binary-lens finite-source
algorithms. The first algorithm, called “map-making”, is the main work horse. For
a fixed (b, q) geometry, it can successfully evaluate the finite-source magnification of
almost all data points on the light curve and can robustly identify those points for
which it fails. The second algorithm, called “loop-linking”, is much less efficient than
map-making but is entirely robust. We use loop-linking whenever the map-making
routine decides it cannot robustly evaluate the magnification of a point. In addition,
at the present time, map-making does not work for resonant lensing geometries, i.e.,
geometries for which the caustic has six cusps. For planetary mass ratios, resonant
lensing occurs when the planet is very close to the Einstein radius, b ∼ 1. We use
loop-linking in these cases also.
A.1. Map-Making
Map-making has two components: a core function that evaluates the
magnification and a set of auxiliary functions that test whether the measurement
264
is being made accurately. If a light-curve point fails these tests, it is sent to
loop-linking.
Finite-source effects are important when the source passes over or close to
a caustic. Otherwise, the magnification can be evaluated using the point-source
approximation, which is many orders of magnitude faster than finite-source
evaluations. Hence, the main control issue is to ensure that any point that falls
close to a caustic is evaluated using a finite-source algorithm or at least is tested to
determine whether this is necessary. For very high magnification events, the peak
points will always pass close to the central caustic. Hence, the core function of
map-making is to “map” an Einstein-ring annulus in the image plane that covers
essentially all of the possible images of sources that come close to the central caustic.
The method must also take account of the planetary caustic(s), but we address that
problem further below.
We begin by inverse ray-shooting an annulus defined by APSPL > Amin, where
APSPL is the Paczynski (1986) magnification due to a point source by a point lens
and Amin is a suitably chosen threshold. For OGLE-2000-BLG-343, we find that
Amin = 75 covers the caustic-approaching points in essentially all cases. The choice
of the density of the ray-shooting map is described below. Each such “shot” results
in a four-element vector (xi, yi, xs, ys). We divide the portion of the source plane
covered by this map into a rectangular grid with k = 1, . . . , Ng elements. We choose
the size of each element to be equal to the smallest source radius being evaluated by
265
the map. Hence, each “shot” is assigned to some definite grid element k(xs, ys). We
then sort the “shots” by k. For each light-curve point to be evaluated, we first find
the grid elements that overlap the source. We then read sequentially through the
sorted file1 from the beginning of the element’s “shots” to the end. For each “shot”,
we ask whether its (xs, ys) lies within the source. If so, we weight that point by the
limb-darkened profile of the source at that radius. Note that a source with arbitrary
shape and surface brightness profile would be done just as easily.
For each light-curve point, we first determine whether at least one of the images
of the center of the source lies in the annulus. In practice, for our case, the points
satisfying this condition are just those on the night of the peak, but for other events
this would have to be determined on a point-by-point basis. We divide those points
with images outside the annulus into two classes, depending on whether they lie
inside or outside two or three rectangles that we “draw”, one around each caustic.
Each rectangle is larger than the maximum extent of the caustics by a factor of 1.5 in
each direction. If the source center lies outside all of these rectangles, we assume that
the point-source approximation applies and evaluate the magnification accordingly.
If it lies inside one of the rectangles (and so either near or inside one of the caustics),
we perform the following test to see whether the point-source approximation holds.
1Whether this “file” should actually be an external disk file or an array in internal memory
depends on both the size of the available internal memory and the total number of points evaluated
in each lens geometry. In our case, we used internal arrays.
266
We evaluate the point-source magnifications at five positions, namely, the source
center A(0, 0), two positions along the source x-axis A(±λρ∗, 0), and two positions
along the source y-axis A(0,±λρ∗), where λ ≤ 1 is a parameter. We demand that
∣
∣
∣
∣
A(λρ∗, 0) + A(−λρ∗, 0)
2A(0, 0)− 1
∣
∣
∣
∣
+∣
∣
∣
∣
A(0, λρ∗) + A(0,−λρ∗)
2A(0, 0)− 1
∣
∣
∣
∣
< 4σ, (A.1)
where σ is the maximum permitted error (defined in § A.3). We use a minimum of
five values in order to ensure that the magnification pattern interior to the source is
reasonably well sampled; the precise value is chosen empirically and is a compromise
between computing speed and accuracy. We require the number of values to be at
least 2[ρ∗/√
q] in order to ensure that small, well-localized perturbations interior to
the source caused by low mass ratio companions are not missed.
If a point passes this test, the magnification pattern in the neighborhood of the
point is adequately represented by a gradient and so the point-source approximation
holds. Points failing this test are sent to loop linking.
The remaining points, those with at least one image center lying in the annulus,
are almost all evaluated using the sorted grid as described above. However, we must
ensure that the annulus really covers all of the images. We conduct several tests to
this end.
First, we demand that no more than one of the three or five images of the
source center lie outside the annulus. In a binary lens, there is usually one image
267
that is associated with the companion and that is highly demagnified. Hence, it can
generally be ignored, so the fact that it falls outside the annulus does not present a
problem. If more than one image center lies outside the annulus, the point is sent to
loop-linking. Second, it is possible that an image of the center of the source could
lie inside the annulus, but the corresponding image of another point on the source
lies outside. In this case, there would be some intermediate point that lay directly
on the boundary. To guard against this possibility, we mark the “shots” lying within
one grid step of the boundaries of the annulus, and if any of these boundary “shots”
fall in the source, we send the point to loop-linking. Finally, it is possible that
even though the center of the source lies outside the caustic (and so has only three
images), there are other parts of the source that lie inside the caustic and so have
two additional images. If these images lay entirely outside the annulus, the previous
checks would fail. However, of necessity, some of these source points lie directly on
the caustic, and so their images lie directly on the critical curve. Hence, as long as
the critical curve is entirely covered by the annulus, at least some of each of these
two new images will lie inside the annulus and the “boundary test” just mentioned
can robustly determine whether any of these images extend outside the annulus. For
each (b, q) geometry, we directly check whether the annulus covers the critical curve
associated with the central caustic by evaluating the critical curve locus using the
algorithm of Witt (1990).
268
A.2. Loop-Linking
Loop-linking is a hybrid of two methods: inverse ray-shooting and Stokes’s
theorem. In the first method (which was also used above in “map-making”), one
finds the source location corresponding to each point in the image plane. Those
that fall inside the source are counted (and weighted according to the local surface
brightness), while those that land outside the source are not. The main shortcoming
of inverse ray-shooting is that one must ensure that the ensemble of “shots” actually
covers the entire image of the source without covering so much additional “blank
space” that the method becomes computationally unwieldy.
In the Stokes’s theorem approach, one maps the boundary of (a polygon-
approximation of) the source into the image plane, which for a binary lens yields
either three or five closed polygons. These image polygons form the (interior or
exterior) boundaries of one to five images. If one assumes uniform surface brightness,
the ratio of the combined areas of these images (which can be evaluated using
Stokes’s theorem) to the area of the source polygon is the magnification.
There are two principal problems with the Stokes’s theorem approach. First,
sources generally cannot be approximated as having uniform surface brightness.
This problem can be resolved simply by breaking the source into a set of annuli, each
of which is reasonably approximated as having uniform surface brightness. However,
this multiplies the computation time by the number of annuli. Second, there can
269
be numerical problems of several types if the source boundary passes over or close
to a cusp. First, the lens solver, which returns the image positions given the source
position, can simply fail in these regions. This at least has the advantage that one
can recognize that there is a problem and perhaps try some neighboring points. The
second problem is that even though the boundary of the source passes directly over
a cusp, it is possible that none of the vertices of its polygon approximation lie within
the caustic. The polygonal image boundary will then fail to surround the two new
images of the source that arise inside the caustic, so these will not be included in
the area of the image. Various steps can be taken to mitigate this problem, but the
problem is most severe for very low-mass planets (which are of the greatest interest
in the present context), so complete elimination of this problem is really an uphill
battle.
The basic idea of loop-linking is to map a polygon that is slightly larger than
the source onto the image plane, and then to inverse ray shoot the interior regions
of the resulting image-plane polygons. This minimizes the image-plane region
to be shot compared to other inverse-ray shooting techniques. It is, of course,
more time-consuming than the standard Stokes’s theorem technique, but it can
accommodate arbitrary surface-brightness profiles and is more robust. As we detail
below, loop-linking can fail at any of several steps. However, these failures are
always recognizable, and recovery from them is always possible simply by repeating
the procedure with a slightly larger source polygon.
270
Following Gould & Gaucherel (1997), the vertices of the source polygon are
each mapped to an array of three or five image positions, each with an associated
parity. If the lens solver fails to return three or five image positions, the evaluation is
repeated beginning with a larger source polygon. For each successive pair of arrays,
we “link” the closest pair of images that has the same parity and repeat this process
until all images in these two arrays are linked. An exception occurs when one array
has three images and the other has five images, in which case two images are left
unmatched. Repeating this procedure for all successive pairs of arrays produces a set
of linked “strands”, each with either positive or negative parity. The first element of
positive-parity strands and the last element of negative-element strands are labeled
“beginnings” and the others are labeled “ends”. Then the closest “beginning” and
“end” images are linked and this process is repeated until all “beginnings” and
“ends” are exhausted. The result is a set of two to five linked loops. As with the
standard Stokes’s theorem approach, it is possible that a source-polygon edge crosses
a cusp without either vertex being inside the caustic. Then the corresponding
image-polygon edge would pass inside the image of the source, which would cause
us to underestimate the magnification. We check for this possibility by inverse ray
shooting the image-polygon boundary (sampled with the same linear density as we
later sample the images) back into the source plane. If any of these points land in
the source, we restart the calculation with a larger source polygon.
271
We then use these looped links to efficiently locate the region in the image
plane to do inverse ray shooting. We first examine all of the links to find the largest
difference, ∆ymax, between the y-coordinates of the two vertices of any link. Next, we
sort the m = 1, . . . , n links by the lower y-coordinate of their two vertices, y−m. One
then knows that the upper vertex obeys y+m ≤ y−
m + ∆ymax. Hence, for each y-value
of the inverse ray shooting grid, we know that only links with y−m ≤ y ≤ y−
m + ∆ymax
can intersect this value. These links can quickly be identified by reading through
the sorted list from y−m = y − ∆ymax to y−
m = y. The x value of each of these
crossing links is easily evaluated. Successive pairs of x’s then bracket the regions (at
this value of y) where inverse rays must be shot. As a check, we demand that the
first of each of these bracketing links is an upward-going link and the second is a
downward-going link.
A.3. Algorithm Parameters
Before implementing the two algorithms described above, one must first specify
values for certain parameters. Both algorithms involve inverse ray shooting and
hence require that a sampling density by specified. Let g be the grid size in
units of the Einstein radius. For magnification A ≫ 1, the image can be crudely
approximated as two long strands whose total length is ℓ = 4Aρ∗ and hence of mean
width (πρ2∗A)/ℓ = (π/4)ρ∗. If, for simplicity, we assume that the strand is aligned
272
with the grid, then there will be a total of ℓ/g grid tracks running across the strand.
Each will have two edges, and on each edge there will be an “error” of 12−1/2 in the
“proper” number of grid points due to the fact that this number must be an integer,
whereas the actual distance across the strand is a real number. Hence, the total
number of grid points will be in error by [(2ℓ/g)/12]1/2, while the total number itself
is π(ρ∗/g)2A. This implies a fractional error σ,
σ−2 =[π(ρ∗/g)2A]2
(2ℓ/g)/12=
3π2
2A(ρ∗/g)3. (A.2)
In fact, the error will be slightly smaller than given by equation (A.2) in part
because the strand is not aligned with the grid, so the total number of tracks across
the strands will be lower than 2ℓ/g, and in part because the “discretization errors”
at the boundary take place on limb-darkened parts of the star, which have lower
surface brightness, so fluctuations here have lower impact. Hence, an upper limit to
the grid size required to achieve a fractional error σ is
g
ρ∗
=(
3π2
2
)1/3
σ2/3A1/3. (A.3)
For each loop-linking point, we know the approximate magnification A because
we know the weighted parameters of the single-lens model. We generally set σ at
1/3 of the measurement error, so that the (squared) numerical noise is an order of
magnitude smaller than that due to observational error. Using equation (A.3) we
can then determine the grid size.
273
For the map-making method, the situation is slightly more complicated. Instead
of evaluating one particular point as in the loop-linking method, all of the points on
the source plane with the same (b, q, u0, ρ) are evaluated within one map. Therefore,
the grid size for this map is the minimum value from equation (A.3) to achieve the
required accuracy for all of these points. We derive the following equation from
equation (A.3) to determine the grid size in the map-making method:
g =[
3π2
2
Q(u0)
u0
]1/3
ρ∗ (A.4)
where
Q(u0) = minAi>75
F (ti) − Fb(u0)
Fmax(ti) − Fb(u0)σ2
i
(A.5)
We find that Q(u0) = 2.65×10−7 is independent of u0 for both the observational
and simulated data of OGLE 2004-BLG-343. In principle, one could determine a
minimum g for all (u0, ρ) combinations and generate only one map for a given (b, q)
geometry, but this would render the calculation unnecessarily long for most (u0, ρ)
combinations. Instead we evaluate g for each (u0, ρ) pair and create several maps,
one for each ensemble of (u0, ρ) pairs with similar g’s. The sizes of the ensembles
should be set to minimize the total time spent generating, loading and employing
maps. Hence, they will vary depending on the application.
274
Appendix B
MOA-2003-BLG-32/OGLE-2003-BLG-219
MOA-2003-BLG-32/OGLE-2003-BLG-219, with a peak magnification
Amax = 525 ± 75 is most sensitive to low-mass planets to date (Abe et al. 2004).
However, instead of fitting the simulated binary-lens light curves to single-lens
models, Abe et al. (2004) obtain their ∆χ2 by directly subtracting the χ2 of a
simulated binary-lens light curve from that of the light curve that is the best fit
to the data. Since the source star of this event could reside in the Sagittarius
dwarf galaxy, which makes the Galactic modeling rather complicated, we do not
attempt to apply our entire method to this event. We calculate planet exclusion
regions with the same ∆χ2 thresholds (60 for q = 10−3 and 40 for the q < 10−3) as
Abe et al. (2004) but using our method of obtaining ∆χ2 by fitting the simulated
binary-lens light curves to single-lens models. Figure B.1 shows our results for
the exclusion regions at planet-star mass ratios q = 10−5, 10−4, and 10−3 for
MOA-2003-BLG-32/OGLE-2003-BLG-219. The exclusion region we have obtained
at q = 10−3 is about 1/4 in vertical direction and 1/9 in horizontal direction relative
to the corresponding region in Abe et al. (2004), and the size of our exclusion
region at 10−4 is about 60% in each dimension relative to that in Abe et al. (2004).
275
Although according to our analysis, Abe et al. (2004) overestimate the sensitivity of
MOA-2003-BLG-32/OGLE-2003-BLG-219 to both Jupiter-mass and Neptune-mass
planets, their estimates of sensitivity to Earth-mass planets are basically consistent
with our results and MOA-2003-BLG-32/OGLE-2003-BLG-219 nevertheless retains
the best sensitivity to planets to date.
276
Fig. B.1.— Planetary exclusion regions for microlensing event MOA-2003-32/OGLE-2003-BLG-219 as a function of projected coordinate bx and by at planet-star massratio q = 10−5 (green), 10−4 (red) and 10−3 (blue). The source size (normalized to θE)ρ∗ is equal to 0.0007. The black circle is the Einstein ring, i.e., b = 1.
277
Appendix C
Extracting Orbital Parameters for Circular
Planetary Orbit
OGLE-2005-BLG-071 is the first planetary microlensing event for which the
effects of planetary orbital motion in the light curve have been fully analyzed. The
distortions of the light curve due to the orbital motion are modeled by ω and b as
discussed in § 4.3.2. In addition, the lens mass M and distance Dl are determined,
so we can directly convert the microlens light-curve parameters that are normalized
to the Einstein radius to physical parameters. In this section, we show that under
the assumption of a circular planetary orbit, the planetary orbital parameters can
be deduced from the light-curve parameters. Let r⊥ = DlθEd be the projected
star-planet separation and let r⊥γ be the instantaneous planet velocity in the
plane of the sky, i.e. r⊥γ⊥ = r⊥ω is the velocity perpendicular to this axis and
r⊥γ‖ = r⊥d/d is the velocity parallel to this axis. Let a be the semi-major axis and
define the ı, , k directions as the instantaneous star-planet-axis on the sky plane,
278
the direction into the sky, and k = ı × . Then the instantaneous velocity of the
planet is
v =
√
GM
a[cos θk + sin θ(cos φı − sin φ)], (C.1)
where φ is the angle between star-planet-observer (i.e., r⊥ = a sin φ) and θ is the
angle of the velocity relative to the k direction on the plane that is perpendicular to
the planet-star-axis. We thus obtain
γ⊥ =
√
GM
a3
cos θ
sin φ, γ‖ =
√
GM
a3sin θ cot φ. (C.2)
To facilitate the derivation, we define
A ≡ γ‖
γ⊥
= − tan θ cos φ, B ≡ r3⊥γ2
⊥
GM= cos2 θ sin φ, (C.3)
which yield as an equation for sin φ:
B = F (sin φ); F (x) =x(1 − x2)
A2 + 1 − x2. (C.4)
Note that F ′(sin φ) = 0 when sin2 φ∗ = (3/2)A2 + 1 − |A|√
(9/4)A2 + 2 . So
equation (C.4) has two degenerate solutions when B < F (sin φ∗) and has no
solutions when B > F (sin φ∗). Subsequently, one obtains,
a =r⊥
sin φ, cos i = − sin φ cos θ, K =
√
GM
aq sin i, (C.5)
279
where i is the inclination and K is the amplitude of radial velocity.
The Jacobian matrix used to transform from P (r⊥, γ⊥, γ‖) to P (a, φ, θ) is given
below,
∂(r⊥, γ⊥, γ‖)
∂(a, φ, θ)=
GM
a3
∣
∣
∣
∣
∣
sin φ a cos φ 0
− 32a
cos θsin φ
− cos θ cos φsin2 φ
− sin θsin φ
− 32a
sin θ cot φ − sin θsin2 φ
cos θ cot φ
∣
∣
∣
∣
∣
=GM
a3cot2 φ
(
1
2− sin2 θ tan2 φ
)
. (C.6)
Then for an arbitrary function H(a),
∂(r⊥, γ⊥, γ‖)
∂(H(a), cos φ, θ)=
∂(r⊥, γ⊥, γ‖)
∂(a, φ, θ)× 1
sin φH ′(a), (C.7)
which, for the special case of a flat distribution, H(a) = ln a, yields,
∂(r⊥, γ⊥, γ‖)
∂(ln(a), cos φ, θ)=
GM
r2⊥
cos2 φ
sin φ.(
1
2− sin2 θ tan2 φ
)
(C.8)
280
Appendix D
Failure of Elliptical-Source Models for
MOA-2007-BLG-400
Because MOA-2007-BLG-400 is the first microlensing event with a completely
buried caustic, it is important to rule out other potential causes of the deviations
seen in the light curve (apart from a planetary companion to the lens). The
principal features of these deviations are the twin “spikes” in the residuals, which are
approximately centered on the times when the lens enters and exits the source. In
the model, these crossings occur at about HJD’ = 4354.53 and HJD’ = 4354.63, i.e.,
very close to the spikes in Figure 2.1. In principle, one might be able to induce such
spikes by displacing the model source crossing times from the true times. The only
real way to achieve this (while still optimizing the overall fit parameters) would be if
the source were actually elliptical, but were modeled as a circle (which, of course, is
the norm).
One argument against this hypothesis is the similarity of the I and H residuals
(§ 2.3). If the source were an ellipsoidal variable, then one would expect color
gradients due to “gravity darkening”.
281
Nevertheless, we carried out two types of investigation of this possibility. First,
we modeled the light curve as an elliptical source magnified by a point (non-binary)
lens. In addition to the linear flux parameters (source flux plus blended flux for each
observatory) there are 6 model parameters, the three standard point-lens parameters
(t0, u0, tE), plus the source semimajor and semiminor axes (ρa, ρb) and the angle of
the source trajectory relative to the source major axis, α. We find that the elliptical
source reduces χ2 by about 200, but it does not remove the “spikes” from the
residuals, which was the primary motivation for introducing it. Instead, essentially
all of the χ2 improvement comes from eliminating the asymmetries from the rest
of the light curve. Recall, however, that the planetary model removes both these
asymmetries and the “spikes”. Moreover, the best-fit axis ratio is quite extreme,
ρb/ρa = 0.7, which would produce very noticeable ellipsoidal variations unless the
binary were being viewed pole on.
Next we looked for sinusoidal variations in the baseline light curve. The
individual OGLE errorbars at baseline are smaller than for MOA, and since
ellipsoidal variations are strictly periodic, the longer OGLE baseline (about
T = 2000 days versus T = 800 days for MOA) does a better job of isolating this
signal from various possible systematics. Therefore, for this purpose, the OGLE
data are more suitable than MOA. The OGLE data are essentially all baseline (only
two magnified points out of 452). Their periodogram shows several spikes at the
0.01 mag level, and a maximum ∆χ2 = 20, which are consistent with noise. The
282
width of the spikes is extremely narrow, consistent with the theoretical expectation
for uniformly sampled data of σ(P )/P 2 ∼√
24/∆χ2/(2πT ) ∼ 10−4 day−1, indicating
that the data set is behaving normally.
In brief, our investigation finds no convincing evidence for ellipticity of the
source, certainly not for the several tens of percent deviation from circular that
would be needed to significantly ameliorate the deviations seen near peak in the
light curve. Moreover, even arbitrary source ellipticities cannot reproduce the light
curve’s most striking features: the two “spikes” in the residuals that occur when the
lens crosses the source boundary.
283
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