FIRST SPACE-BASED MICROLENS PARALLAX MEASUREMENT OF AN ISOLATED STAR: SPITZER
OBSERVATIONS OF OGLE-2014-BLG-0939
J. C. Yee
1,10
, A. Udalski
2
, S. Calchi Novati
3,4,5,11
, A. Gould
6
, S. Carey
7
, R. Poleski
2,6
,B.S.Gaudi
6
, R. W. Pogge
6
,
J. Skowron
2
,S.Kozłowski
2
, P. Mróz
2
, P. Pietrukowicz
2
, G. PietrzyŃski
2,8
, M. K. SzymaŃski
2
, I. SoszyŃski
2
,
K. Ulaczyk
2
, and Ł. Wyrzykowski
2,9
1
Harvard-Smithsonian Center for Astrophysics, 60 Garden St., Cambridge, MA 02138, USA
2
Warsaw University Observatory, Al. Ujazdowskie 4, 00-478 Warszawa, Poland
3
NASA Exoplanet Science Institute, MS 100-22, California Institute of Technology, Pasadena, CA 91125, USA
4
Dipartimento di Fisica “E. R. Caianiello”, Università di Salerno, Via Giovanni Paolo II, I-84084 Fisciano (SA), Italy
5
Istituto Internazionale per gli Alti Studi Scientifici (IIASS), Via G. Pellegrino 19, I-84019 Vietri Sul Mare (SA), Italy
6
Department of Astronomy, Ohio State University, 140 W. 18th Ave., Columbus, OH 43210, USA
7
Spitzer Science Center, MS 220-6, California Institute of Technology, Pasadena, CA, USA
8
Universidad de Concepción, Departamento de Astronomia, Casilla 160–C, Concepción, Chile
9
Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK
Received 2014 October 21; accepted 2015 January 20; published 2015 March 24
ABSTRACT
We present the first space-based microlens parallax measurement of an isolated star. From the striking differences
in the lightcurve as seen from Earth and from Spitzer (
~1AU
to the west), we infer a projected velocity
~
-
v
˜
250 km s
hel
1
, which strongly favors a lens in the Galactic Disk with mass
=
MM0.23 0.07
and distance
=
D
3.1 0.4 kpc
L
. An ensemble of such measurements drawn from our ongoing program could be used to
measure the single-lens mass function including dark objects, and also is necessary for measuring the Galactic
distribution of planets since the ensemble reflects the underlying Galactic distribution of microlenses. We study the
application of the many ideas to break the four-fold degeneracy first predicted by Refsdal 50 years ago. We find
that this degeneracy is clearly broken, but by two unanticipated mechanisms: a weak constraint on the orbital
parallax from the ground-based data and a definitive measurement of the source proper motion.
Key words: gravitational lensing: micro
1. INTRODUCTION
When modern microlensing experiments were proposed
toward the Large Magellanic Cloud (Paczyn
ski 1986) and the
Galactic Bulge (Paczyn
ski 1991; Griest et al. 1991), it was
believed that the only information that could be extracted about
the lens mass M, distance D
L
, and transverse motion
μ
ge
o
would
come through their combination in a single measured
parameter, the Einstein timescale,
q
qk k=º º
t
μ
Mπ
G
c
M
;;
4
AU
8.14
mas
.(1)
E
E
geo
E
2
rel
2
Here
q
E
is the angular Einstein radius,
=-
--
πDDAU( )
LS
rel
11
is the lens–source relative parallax, and
μ
ge
o
is the lens–source
relative proper motion in the Earth frame at the peak of the
event. This would imply, in particular, that individual masses
could be estimated only to within an order of magnitude (e.g.,
Figure 1 of Gould 2000a).
It was quickly realized, however, that if two additional
potentially observable quantities could be measured,
q
E
and the
“microlens parallax vector”
π
E
, then these three quantities
could be disentangled (Gould 1992),
q
k
q
q
== =
μM
π
ππ
t
π
π
;; .(2)
E
E
rel E E
geo
E
E
E,geo
E
In modern notation, the microlens parallax vector is given by
(Gould 2000b),
q
º
μ
π
π
μ
.(3)
E
rel
E
Its amplitude quantifies the lens–source relative displacement
in the Einstein ring due to motion of the observer, while its
direction specifies the orientation of this displacement as the
event evolves. Hence,
π
E
is in principle measurable from
photometric deviations of the event relative to what is expected
from rectilinear motion. See Figure 1 of Gould & Horne (2013)
for a didactic explanation.
While both
q
E
and
π
E
are important, measurements of
π
E
are
more pressing for the following three reasons. First,
q
E
is very
frequently measured “automatically” in planetary and binary
events. Hence,
π
E
is the crucial missing link to obtain
individual masses for these high priority events, i.e., those
for which individual masses are the most important. Second,
q
E
is very rarely measurable in single-lens events, which means
that measuring
π
E
is the best way to obtain strong statistical
constraints on masses of the much larger population of (dark
and luminous) single lenses. Third, while
π
E
and
q
E
appear
symmetrically in Equation (2),
π
E
is actually much richer in
information than
q
E
. This is because the great majority of lenses
observed toward the Galactic Bulge have similar proper
motions within a factor ∼2of
~
-
μ 4masyr
1
. Thus, in the
limit that all microlens proper motions had exactly this value, a
measurement of
q
= μt
EE
would contain no additional
information, while
π
E
would completely determine the mass
k=Mμtπ
EE
. Although this limit does not strictly apply, an
ensemble of
π
E
measurements would constrain the mass
function very well (Han & Gould 1995).
The Astrophysical Journal, 802:76 (10pp), 2015 April 1 doi:10.1088/0004-637X/802/2/76
© 2015. The American Astronomical Society. All rights reserved.
10
Sagan Fellow.
11
Sagan Visiting Fellow.
1
There are two broad classes of methods by which parallax
might be measured. The first is to make a single time series
from an accelerated platform, either Earth (Gould 1992; Alcock
et al. 1995; Poindexter et al. 2005) or a satellite in low-Earth
(Honma 1999) or geosynchronous (Gould 2013) orbit. The
second is to make simultaneous observations from two (or
more) observatories, either on two platforms in solar orbit
(Refsdal 1966), or located at several places on Earth (Hardy &
Walker 1995; Holz & Wald 1996; Gould 1997). However, with
one exception, all of these methods are either subject to
extremely heavy selection bias or are impractical for the present
and near future. In particular, out of more than 10,000
microlensing events discovered to date, fewer than 100 have
π
E
measurements derived from Earth’s orbital motion, and
these are overwhelmingly events due to nearby lenses and with
abnormally long timescales (e.g., Table 1 of Gould et al. 2010).
Only two events have terrestrial parallax measurements (Gould
et al. 2009; Yee et al. 2009), and Gould & Yee (2013) showed
that these are subject to even more severe selection so that even
the two recorded measurements is unexpectedly high.
Hence, the only near-term prospect for obtaining a statistical
sample of microlens parallaxes from which to derive an
unbiased mass function, as originally outlined by Han & Gould
(1995), is by combining Earth-based observations with those of
a satellite in solar orbit. There are several major benefits to such
a study. First, it is the only way to obtain a mass-based census
of stellar, remnant, and planetary populations. Several
components of this population are dark or essentially dark
including free-floating planets, brown dwarfs, neutron stars,
and black holes and therefore are essentially undetectable by
any other method unless they are orbiting other objects. In
addition, even the luminous-star mass function of distant
populations (e.g., in the Galactic Bulge) is substantially more
difficult to study photometrically than is generally imagined.
For example, a large fraction of stars are fainter components in
binary systems, with separations that are too small to be
separately resolved, but whose periods are too long (or
primaries too faint) for study by the radial velocity technique.
In 2014, we were granted Director’s Discretionary Time for
a 100 hr pilot program to determine the feasibility of using
Spitzer as such a parallax satellite for microlenses observed
toward the Galactic Bulge. The main objective of this program
was to measure lens masses in planetary events. However,
especially in view of the fact that there is generally no way to
distinguish such planetary events from single-lens events in
advance, a secondary goal was to obtain parallaxes for an
ensemble of single-lens events. Prior to this program, there had
been only one space-based parallax measurement, which was
for a binary lens toward the Small Magellanic Cloud, OGLE-
2005-SMC-001 (Dong et al. 2007).
Here we report on the first space-based parallax measure-
ment of an isolated lens, OGLE-2014-BLG-0939 L. This
measurement serves as a pathfinder and as a benchmark to
test ideas that have been discussed in the literature for almost
50 years about how to resolve degeneracies in such events.
1.1. Degeneracies in Space-based Microlens Parallaxes
As already pointed out by Refsdal (1966), space-based
microlensing parallaxes are subject to a four-fold discrete
degeneracy. This is because, to zeroth order, the satellite has a
fixed separation from Earth projected on the plane of the sky
^
D
, and hence they measure identical Einstein timescales
==
Å
tt t
EE,satE,
. Since the flux evolution F(t) of a single-lens
Figure 1. Lightcurve of OGLE-2014-BLG-0939 as seen by OGLE from Earth
(black) and Spitzer (red)
~1AU
to the west. While both are well-represented
by Paczyn
ski (1986) curves (blue), they have substantially different maximum
magnifications and times of maximum, whose differences yield a measurement
of the “microlens parallax” vector
π
E
. The dashed portion of the Spitzer curve
extends the model to what Spitzer could have observed if it were not prevented
from doing so by its Sun-angle constraints. Light curves are aligned to the
OGLE I-band scale (as is customary), even though Spitzer observations are at
3.6 μm. Lower panel shows residuals.
Table 1
μLens Parameters (Free F
B
)
Parameter Unit
-+
u
0, , --
u
0, , ++
u
0, , +-
u
0, ,
c
2
dof L 273.1/ 273.7/ 281.5/ 290.2/
265 265 265 265
-
t
6800
0
day 36.22 36.20 36.06 35.95
L 0.11 0.11 0.11 0.11
u
0
L 0.922 −0.913 0.897 −0.843
L 0.132 0.129 0.125 0.110
t
E
day 22.87 22.99 22.91 23.87
L 2.14 2.12 2.10 2.04
π
E,N
L −0.248 0.220 −1.370 1.325
L 0.072 0.067 0.172 0.158
π
E,E
L 0.234 0.238 −0.060 0.024
L 0.028 0.030 0.025 0.018
v
˜
hel,N
km s
−1
−162.3 156.9 −55.5 54.2
L 7.2 5.5 2.2 2.1
v
˜
hel,E
km s
−1
181.6 199.7 26.6 29.9
L 37.2 39.5 0.7 0.8
F
S,OGLE
L 13.20 12.95 12.51 11.09
L 3.77 3.63 3.42 2.75
F
B,OGLE
L −2.19 −1.93 −1.49 −0.08
L 3.77 3.62 3.42 2.75
F
Spitze
r
S,
L 4.31 4.37 3.32 3.30
L 1.10 1.12 0.72 0.69
F
SpitzerB,
L −0.08 −0.15 0.96 1.02
1.21 1.22 0.81 0.79
2
The Astrophysical Journal, 802:76 (10pp), 2015 April 1 Yee et al.
microlensing event is given by (Paczyn
ski 1986),
=+ =
=+
+
+
-
Ft FAt F Aut
ut u
() () ;
[
()
]
;
[
()
]
,(4)
SB
u
uu
tt
t
2
4
2
0
2
()
2
42
0
2
E
2
they are therefore distinguished only by different times of peak
t
0
and different impact parameters u
0
(in addition to the
nuisance parameters F
S
and F
B
, the source and blended-light
fluxes, respectively). The microlens parallax
π
E
can nominally
be derived from these differences,
=
æ
è
ç
ç
ç
ç
D
D
ö
ø
÷
÷
÷
÷
^
π
D
t
t
u
AU
,, (5)
E
0
E
0
where
D
=-
Å
tt t
00,sat0,
,
D
=-
Å
uu u
00,sat0,
, and where the
x-axis of the coordinate system is set by the Earth–satellite
vector
^
D
. The problem is that while
D
t
0
is unambiguously
determined from this procedure, u
0
is actually a signed quantity
whose amplitude is recovered from simple point-lens events
but whose sign is not (since it appears only quadratically in
Equation (4)). Hence, there are two solutions
D
= -
- Å
∣∣∣∣uuu())
0, , 0,sat 0,
for which the satellite and
Earth observe the source trajectory on the same side of the lens
as each other (with the “ ± ” designating which side this is),
and two others
D
= +
+ Å
∣∣∣∣uuu())
0, , 0,sat 0,
for which the
source trajectories are seen on opposite sides of the lens
(Gould 1994, Figure 1).
For most applications, only the second of these two
degeneracies is important. That is, the two solutions
D
-
u
0, ,
have the same amplitude of parallax
π
E
(as do the two solutions
D
+
u
0, ,
) and so yield the same lens mass and distance. In each
case, the solutions differ only in the direction of lens–source
motion, which is usually not of major interest. However, the
two sets of solutions can yield radically different
π
E
. Hence, if
these sets of solutions really cannot be distinguished, the value
of the parallax measurement is seriously undermined. As a
result, considerable work has been applied over two decades to
figuring out how to break these degeneracies.
Before reviewing this work, however, one should note an
important exacerbation of the underlying problem. If the four
solutions are placed in the
DDtt u
(
,)
0E 0
plane, they of course
all lie along a vertical line of constant
D
t
0
. As pointed out by
Gould (1995), the error ellipses are also elongated in the
vertical direction. This is because u
0
is strongly correlated with
the nuisance parameters F
S
and F
B
(since all three enter
Equation (4) symmetrically in
-tt
(
)
0
) while t
0
, which enters
anti-symmetrically, is not strongly correlated with other
parameters. This continuous degeneracy enhances the prob-
ability that the discretely degenerate solutions will overlap and
become a continuous degeneracy.
Four ideas have been proposed to break the
D
u
0
four-fold
degeneracy.
1.1.1. Measurement of
Dt
E
Gould (1995) proposed to break the degeneracy by using the
fact that the Earth–satellite separation changes with time, and
therefore
¹
Å
tt
E,sat E,
. For near-circular, near-ecliptic orbits
(characteristic of both Spitzer and Kepler), this works quite
well for targets near the ecliptic poles (Boutreux & Gould 1996)
because the difference in timescales
D
t
E
is directly proportional
to
D
u
0
. However, it becomes increasing problematic for targets
close to the ecliptic, like the Galactic Bulge (Gaudi &
Gould 1997), because for targets directly on the ecliptic,
D
t
E
does not depend at all on
D
u
0
to linear order. That is,
D
u
0
completely disappears from Equation (2.3) of Gould (1995).
1.1.2. Photometric Alignment of Space and Ground Observations
Gould (1995) also proposed to equip the satellite with a
camera having identical photometric response to one on the
ground, which would guarantee that
=
Å
FF
SS,sat ,
and so
effectively insulate
D
-
u
0, ,
from uncertainties in F
S
by forcing
the two
D
-
u
0, ,
solutions to move together in a highly-
correlated way as F
S
is varied over its allowed range. While
this idea would be quite difficult to implement, Yee et al.
(2012) demonstrated that observations in different bands could
be aligned quite tightly with each other based on color–color
diagrams of reference stars. As a practical matter, it is not
obvious that this technique can be applied to Spitzer
observations because Yee et al. ( 2012) predicted F
S
for a
certain band by interpolating between two other measured
bands, whereas predicting Spitzerʼs
μ
3
.6
m
F
S
requires
considerable extrapolation from ground-based bands.
1.1.3. Combining 1D Parallaxes from Space and Ground
Gould (1999) suggested that the robust one-dimensional
(1D) parallax information along the
^
D
(i.e.,
D
t
0
) direction
from Earth–satellite observations could be combined with
robust 1D information along the direction of Earth’s projected
acceleration from ground-based observations (Gould
et al. 1994) to break the
D
u
0
degeneracy. This idea was
specifically motivated by the possibility of Spitzer parallax
observations toward the Magellanic Clouds, which are at high
ecliptic latitude where these two directions are nearly
orthogonal. As he noted, it is substantially more difficult to
apply this approach toward the Bulge where the two directions
are close to parallel.
1.1.4. High-magnification Events (As Seen From Earth)
Gould & Yee (2012) pointed out that for sufficiently high-
magnification events as observed from Earth
Å
∣∣u
(
1
)
0,
,we
have
Å
∣
∣∣
∣
uu
0, 0,sat
and therefore
DD
- +
∣
∣∣
∣
uu
0, , 0, ,
,so
that there is no degeneracy in the amplitude of
π
E
, although the
direction degeneracy persists. Moreover, if one of the satellite
observations were actually made near
Å
t
0,
, then only 1–3
satellite observations would be required. They therefore
advocated targeting such events. However, since OGLE-
2014-BLG-0939 was not a high-magnification event, this idea
is not directly relevant here and is included only for
completeness.
Because this is the first space-based parallax measurement
for a single-lens event, we systematically study the role of all
these ideas (except the last) for both characterizing and
breaking the degeneracies in practice. We note at the outset that
two of these methods are adversely affected by the Bulge being
close to the ecliptic, and that this problem is more pronounced
for OGLE-2014-BLG-0939 than for typical events because it
lies just
+
◦
2.0
from the ecliptic, i.e., about 3 times closer to it
than Baade’s Window.
3
The Astrophysical Journal, 802:76 (10pp), 2015 April 1 Yee et al.
2. OBSERVATIONS
2.1. OGLE Observations
On 2014 May 28, the Optical Gravitational Lens Experiment
(OGLE) alerted the community to a new microlensing event
OGLE-2014-BLG-0939 based on observations with the 1.4
deg
2
camera on its 1.3 m Warsaw Telescope at the Las
Campanas Observatory in Chile using its Early Warning
System real-time event detection software (Udalski et al. 1994;
Udalski 2003). Most observations were in I band, but with
three V band observations during the magnified portion of the
event to determine the source color. At equatorial coordinates
(17:47:12.25, −21:22:58.7), this event lies in OGLE field
BLG630, which implies that it is observed at relatively low
cadence, roughly once per two nights.
2.2. Spitzer Observations
The structure of our Spitzer observing protocol is described
in detail in Section 3.1 of Udalski et al. (2014). In brief,
observations were made during 38 2.63 hr windows between
HJD
¢
º
HJD
-
=2450000 6814.0
and 6850.0. Each observa-
tion consisted of six dithered 30 s exposures in a fixed pattern
using the 3.6 μm channel on the IRAC camera. Observation
sequences were uploaded to Spitzer operations on Mondays at
UT 15:00, for observations to be carried out Thursday to
Wednesday (with slight variations). As described in Udalski
et al. (2014), JCY and AG balanced various criteria to
determine which targets to observe and how often. In general,
there were too many targets to be able to observe all viable
targets during each epoch.
At the decision time (June 2 UT 15:00, HJD
¢
6811.1) for the
first week of Spitzer observations, OGLE-2014-BLG-0939 was
poorly understood, with acceptable fits having Earth-based
peaks over the range
Å
t6807 6845
0,
, i.e., from well
before to (effectively, see below) the end of the Spitzer
observing interval. Nevertheless, it was put in the “daily”
category and observed during all eight epochs, in part because
the source was bright, implying good precision Spitzer
photometry. The following week, it was degraded to “low”
priority because it was unclear that it would have low enough
u
0
for an effective parallax measurement, and if u
0
were low
enough, the peak would be well in the future. However, due to
a transcription error, it was left in the “daily” file and observed
during all six epochs. By upload time for the third week it was
clear first that
Å
t
0,
would occur during or near these
observations and second that the amplitude would be low
(i.e., relatively high impact parameter
~
Å
u 1
0,
). These
considerations pulled in opposite directions, resulting in
“moderate” priority and so observations during six out of
eight epochs. The fact that the predicted peak (from Earth) was
expected to occur at the beginning of the fourth week led to
classifying the event as “daily ” , and so it was observed in all
seven epochs. Because OGLE-2014-BLG-0939 lies relatively
far to the west, it moved out of the Spitzer observing window
(set by the Sun angle) during the final week. Hence it was
observed during all four of the available epochs (out of eight
total). Hence, OGLE-2014-BLG-0939 was observed relatively
uniformly, close to once per day, during the entire interval that
it was observable, from 6814.1 to 6845.7.
3. LIGHTCURVE ANALYSIS
The analysis of the lightcurve is straightforward because the
magnification for a single-lens can be written in closed form
(Equation (4 )), i.e.,
=+ +
A
uuu(2)(4)
24212
. While the
argument u in this equation is not as simple as in the case of
rectilinear motion illustrated in Equation (4), the deviations
from that formula due to Earth’s motion are easily incorporated
(Gould 2004). Spitzerʼs offset from the center of Earth is
treated just as any other observatory, except that it is much
larger, i.e., of order AU rather than
Å
R
. We adopt the inertial
frame that is coincident with the position and velocity of Earth
at the peak of the event, i.e.,
¢
=
H
JD 6836.06
. Any frame will
yield equivalent results (after a suitable transformation of
parameters). However, this (quite standard) geocentric frame
permits direct comparison with the results from Earth-only
observations, which turns out to be crucial to understanding the
degeneracies.
As expected (Refsdal 1966), the fit yields four distinct
minima, which are listed in Table 1. The best fit is shown in
Figure 1. The remaining three fits look almost identical and so
are not shown to avoid clutter.
We note that the degeneracy between the
D
-
u
0, ,
and
D
+
u
0, ,
is marginally broken, with the latter two disfavored by
c
D
= 8
2
and 17, respectively. However, the two
D
-
u
0, ,
solutions are consistent with each other at
s<
1
.
In Table 1, we have fit with blending as a free parameter for
both observatories. The results show that for the preferred
solutions, the best-fit blending for OGLE is negative but
consistent with zero at the
s
1
level. A low level of negative
blending is permitted because the baseline photometry is
carried out against a mottled background of unresolved turnoff
stars, and the source can in principle land on a “hole” in this
background. However, plausible levels of negative blending
due to this effect are
~-F 0.
2
S
(on a flux scale of I = 18
corresponding to one flux unit), which is an order of magnitude
Figure 2. Four-fold degeneracy in the heliocentric projected velocity
=+
Å^
v
vv
˜˜
hel geo ,
where
=
ππt
v
˜
AU
geo E,geo
E
2
E
and
Å^
v
,
is the velocity of
Earth projected on the sky at the peak of the event. Solutions are labeled
(, )
by their
Du
0
degeneracy. Two smaller
v
˜
hel
+(, )
are disfavored by
cD=8
2
and 17. Note that the error ellipses for these are quite small and partly
obscured by the “arrow heads”. The dashed curves show the
s
1
error for the
expected direction
v
˜
hel
(same as
μ
hel
) based on the measured proper motion of
the source and the assumption that the lens is in the Galactic Disk. This proper
motion measurement decisively breaks the degeneracy.
4
The Astrophysical Journal, 802:76 (10pp), 2015 April 1 Yee et al.
smaller than what is observed. The most likely explanation is
that the blending is very small or zero and has fluctuated below
zero in the fit because of the relatively large errors in this
quantity, which are typical for low-amplitude microlensing
events.
In addition to the parameter values, in Table 1 we also list
the heliocentric projected velocity
v
˜
hel
,
=+ =
Å^
π
π
t
vvv v˜˜ ;˜
AU
,(6)
hel geo , geo
E,geo
E
2
E,geo
where
-
Å^
-
v
(N, E) ( 0.5, 28.9) km s
,
1
is the projected
velocity of Earth at the peak of the event and where we have
explicitly noted that
π
E
and
t
E
are evaluated in the geocentric
frame (as in Table 1). Figure 2 shows the projected velocities
and
s
1
error ellipses for each of the four solutions.
We also show in Table 2 the parameter values and errors
under the assumption that F
B
= 0. As expected from the fact
that F
B
was consistent with zero, the central values hardly
change after application of this restriction. Note also that while
the errors in u
0
,
t
E
, and
π
EE,
(all of which are strongly
correlated with F
B
) shrink dramatically under this assumption,
the errors in
v
˜
hel
hardly change. This is because the east
component of
v
˜
hel
(the one that is heavily correlated with
t
E
) is
directly determined from
D
t
0
together with the physical
separation between Spitzer and Earth at the times of the
respective peaks, both of which are direct empirical quantities,
which do not depend on the fitted Einstein timescale
t
E
.
4. INTERPRETATION
Here we illustrate the power of measuring
π
E
for estimating
the mass and distance, even when
q
E
is not measured or
constrained by considering the specific example of OGLE-
2014-BLG-0939.
The
D
-
u
0, ,
solutions are significantly favored by
c
2
so we
consider these first. The solutions are nearly identical except
that u
0
and
π
NE,
reverse sign. This is expected under the
“ecliptic degeneracy” (Skowron et al. 2011), which is
particularly strong in the present case because the source lies
only 2° from the ecliptic.
The magnitude of
~
-
v
˜
250 km s
hel
1
strongly favors a
Galactic disk lens at intermediate distances, an inference that
follows from the relation between
v
˜
and
μ
=μ π
v˜
AU
.(7)
rel
If the lens were in the Bulge (
π 0.0
2
rel
), then this would
imply relative proper motion
=
-
μπ1.05 mas yr ( 0.02)
hel
1
rel
.
This compares to typical Bulge lens–source proper motions
~
-
μ 4masyr
1
. Since the probability of an event scales as
µμ
2
, Bulge lenses are strongly disfavored but not ruled out by this
argument. On the other hand, for nearby lenses
ππ
(
)
Lrel
, the
projected velocity
v
˜
is nearly equal to the space velocity of the
lens in the frame of the Sun,
^
v
. Since there are very few stars
moving at these speeds, this hypothesis is also disfavored.
At intermediate distances, we would expect that the lens–
source motion would be dominated by the fact that both the
observer and lens partake in the same flat rotation curve. Thus,
apart from the peculiar motion of the Sun and the lens (and
random “noise” introduced by the proper motion of the source),
we expect the lens to be moving in the direction of Galactic
rotation (
~
30
east of north ) at the proper motion of SgrA*,
=
-
μ 6.4 mas yr
SgrA
*
1
. In fact, one of these two solutions
D
--
u
()
0, ,
does show motion similar to this direction (52° east
of north), making it the preferred solution.
To make a more precise comparison between the expected
and observed heliocentric motions, we measure the proper
motion of the “source” (actually the “baseline object” that is
coincident with the source) using four years of OGLE-IV data.
We find
=- -
-
μ (N, E) ( 0.64 0.45, 5.31 0.45)mas yr . (8)
S,hel
1
In principle it is possible that this “baseline object” is a blend of
two or more stars. However, because the blending F
B
from the
microlensing fit is consistent with zero and because the surface
density of stars that are bright enough to substantially affect the
proper motion measurement is small, we tentatively assume
that the proper motion of the microlensed source and this
“baseline object” are the same.
Thus, ignoring the peculiar motion of the lens, we then
expect
f=-
=
-
μμμ
ˆ
(6.2 0.5, 8.5 0.5)mas yr (9)
Sexp,hel SgrA
*
,hel
1
where
f
ˆ
is the direction of Galactic rotation. The direction of
this proper motion is
◦◦
5
3.9 2.7
east of north. We show
immediately below that when account is taken of the dispersion
in lens peculiar motions, the error bar widens to
=
- ◦◦
μ
μ
tan 53 . 9 8. 5. (10)
1
exp,hel,E
exp,hel,N
This
s1
range of proper motions is shown on Figure 2, which
Table 2
μLens Parameters (
=F 0
B,OGLE
)
Parameter Unit
-+
u
0, , --
u
0, , ++
u
0, , +-
u
0, ,
c
2
dof L 273.6/ 274.1/ 281.8/ 290.2/
266 266 266 266
-
t
6800
0
day 36.22 36.20 36.07 35.95
L 0.11 0.11 0.10 0.11
u
0
L 0.840 −0.840 0.840 −0.840
L 0.002 0.002 0.002 0.002
t
E
day 24.29 24.27 23.92 23.93
L 0.16 0.16 0.15 0.15
π
E,N
L −0.214 0.192 −1.292 1.321
L 0.044 0.043 0.029 0.029
π
E,E
L 0.217 0.222 −0.052 0.024
L 0.006 0.008 0.018 0.033
v
˜
hel,N
km s
−1
−164.9 158.3 −56.4 54.3
L 4.8 4.7 1.3 1.3
v
˜
hel,E
km s
−1
195.5 212.4 26.7 29.9
L 34.2 36.3 0.7 0.8
F
S,OGLE
L 11.01 11.01 11.01 11.01
L 0.00 0.00 0.00 0.02
F
B,OGLE
L 0.00 0.00 0.00 0.00
L 0.00 0.00 0.00 0.00
F
Spitze
r
S,
L 3.85 3.93 3.10 3.29
L 0.68 0.69 0.47 0.50
F
SpitzerB,
L 0.34 0.25 1.15 1.04
L 0.87 0.88 0.64 0.66
5
The Astrophysical Journal, 802:76 (10pp), 2015 April 1 Yee et al.
