Transcript Chap5-uLensing-cx - Groupe d`Astronomie et Astrophysique

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Chapter 5 – Microlensing
Contents
5.1 Introduction
5.2 Description
5.3 Caustic and critical curves
5.4 Other light curve effects
5.5 Microlens parallax and lens mass
5.6 Astrometric microlensing
5.7 Other configurations
5.8 Microlensing observations in practice
5.9 Exoplanet results
5.10 Summary of limitations and strengths
5.11 Future developments
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5.1 Introduction
 Principle
 Presence of matter distorts space-time. Light path is deflected as
a result. Gravity acts a lens.
 Light from background source can be amplified by a foreground
(lens) source.
 34 planets detected by this technique so far (as of February 2015)
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5.1 Introduction (2)
 Strong lensing
 Effects discernable at an individual object level.
 Macrolensing: multiple resolved images, arcs,
distorted/amplified sources.
 Microlensing: discrete multiple images are unresolved.
• Relevant for exoplanet detection.
• Requires extremely precise alignment of observer, source and lens to
within the angular Einstein radius, or ~1 mas.
• Primary lens mass is of order 1 M.
• Intensity variation of primary lens time scale: several weeks
• Secondary lens effect due to planet: several hours.
• Very low probability of event. Requires 100s millions sources to be
monitored simultaneously.
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5.1 Introduction (3)
 Weak lensing
 Effects discernable only in a statistical sense. Applies to a large
ensemble of source (galaxies).
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Contents
5.1 Introduction
5.2 Description
5.3 Caustic and critical curves
5.4 Other light curve effects
5.5 Microlens parallax and lens mass
5.6 Astrometric microlensing
5.7 Other configurations
5.8 Microlensing observations in practice
5.9 Exoplanet results
5.10 Summary of limitations and strengths
5.11 Future developments
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5.2 Description (1)
 Light bending
 Let a lensing mass ML with impact parameter b. The deflection
angle αGR is
(5.1)
on condition that
, where RS is the Schwarzchild radius
(5.2)
We have
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5.2 Description (2)
 Angle between source and lens
(5.3)
(5.4)
 Two solutions to this quadratic
equation.
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5.2 Description (3)
 Einstein radius (angular and linear)
(5.5)
(5.6)
Then equation 5.4 can be written
(5.7)
with two solutions
(5.8)
The angular separation between the two images is
The two sources are separated by
if
.
Rotationally symmetric configuration  Einstein ring.
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5.2 Description (4)
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5.2 Description (5)
 The Einstein radius in relevant numerical quantities
(5.10)
(5.11)
For a typical lens mass of ~ 1 M half way to the Galactic center (~8
kpc) where most of background sources are located, the Einstein
radius is ~ 1 mas. In linear scale, the Einstein is ~ 4 AU, coincidently
similar to the orbital radius of planets in the solar system. This is
particularly fortuitous for probing exoplanets.
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5.2 Description (6)
Magnification
Equation 5.7 can be written in unit of the Einstein radius
(5.7)
(5.13)
Equation 5.8 becomes
While brightening occurs through light deflection,
surface brightness is constant. The magnification
is the ratio of the image/source area. For a very
small source, we can consider a thin source
annulus with angle Δϕ. For a point lens, this thin
annulus will be mapped into two annuli, one
inside the Einstein radius and one outside.
Images of a thin annulus from
The dashed line is the Einstein ring.
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by a point lens on the plane of the sky.
is the angle subtended by the thin annulus.
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5.2 Description (7)
Magnification
The area of the source annulus is given by the product of the radial width
and the tangential length
. Similarly, each image area is
, and the magnification is given by
(5.12)
Then, we have
(5.14)
The total magnification is
(5.15)
for
and
for
. For a perfect alignment, A diverge
to infinity but it is limited in practice since the source has a finite size. The
highest magnification reported is ~ 3000 (Dong et al. 2006).
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5.2 Description (8)
Einstein crossing time
The total magnification varies as a function of time due to the relative
transverse motion between source, lens and observer. For a given relative
transverse velocity between source and lens,
, a typical scale for a lensing
event is given by the Einstein radius crossing time
(5.18)
One can also express the Einstein crossing time with the (unknown) lenssource relative proper motion
(5.19)
For a source in the Galactic bulge at ~ 8 kpc, a lend mass of ~ 1 M half way
to the source, the Einstein time scale for the microlensing event is ~35 days.
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5.2 Description (9)
Einstein crossing time
(5.20)
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Contents
5.1 Introduction
5.2 Description
5.3 Caustic and critical curves
5.4 Other light curve effects
5.5 Microlens parallax and lens mass
5.6 Astrometric microlensing
5.7 Other configurations
5.8 Microlensing observations in practice
5.9 Exoplanet results
5.10 Summary of limitations and strengths
5.11 Future developments
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5.3 Caustics and critical curves (1)
 Critical curves are regions in the lens plane where the
magnification is infinite. Corresponding regions in the source
plane as mapped by the lens equation are termed caustics.
 For a single point lens, the caustic is the single point behind
and the critical curves (positions of the images of these
caustics) is the Einstein ring.
 High-magnification microlensing events occur when the source
comes near to a caustic.
 For a binary lens, a star and an orbiting planet, the caustics
and shapes can be formulated in terms of the planet/star
ratio q=Mp/M★, the angular star-planet separation d in units of
θE and α, the angle of the source trajectory relative to te binary
axis.
 The planet induces a secondary peak in the smooth light curve. For
Earth-mass planets, the time scale of the secondary microlensing
event is 3-5 hours.
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5.3 Caustics and critical curves (2)
Position of the planetary caustics
 Close to primary mass lens since planet mass is much smaller.
 Planet has large effect when its position is near one of the images.
 Let xc be the position of the caustics and d the star-planet separation
in the lens plane, then the lens equation is written
 Inverting this equation yields
(5.21)
 This equation is invariant in 1/d, i.e. two star-planet separation
(d, -1/d) yields the same caustic position.
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5.3 Caustics and critical curves (3)
Light curve classification
RE refers to the primary.
Source trajectory through magnification
regions of both primary and planet.
Source trajectory through magnification
region of the planet.
Source trajectory very close to the primary.
Source trajectory very close to the primary.
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Contents
5.1 Introduction
5.2 Description
5.3 Caustic and critical curves
5.4 Other light curve effects
5.5 Microlens parallax and lens mass
5.6 Astrometric microlensing
5.7 Other configurations
5.8 Microlensing observations in practice
5.9 Exoplanet results
5.10 Summary of limitations and strengths
5.11 Future developments
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5.4 Other light curve effects (1)
 Six microlensing parameters to fit:
Other effects
 Finite source size θ★. Add a 7th parameter:
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5.4 Other light curve effects (2)
 Limb darkening and star spots
 Orbital motion of a binary lens
 Three cases: rotating binary lens, rotating binary source and a
rotating oberver (Earth orbiting the Sun)
 Most dramatic effects for a rotating binary lens.
 First claimed planetary microlensing event MACHO-97-BLG-41
was revised with an improved fit involving a binary with a period
of 1.5 yr.
 Orbital motion of a star-planet lens
 Effect seen in the data for the outer planet in the first multiple
planetary microlensing event, OGLE-2006-BLG-109L (Gaudi et
al. 2008)
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5.4 Other light curve effects (3)
 Blending
 Due to a physical companion of the source star, lens itself or
superposition of another (non-lensing) object along line-of-sight.
Separation between light from source and lens is possible once
angular separation has increased by a few mas, several years
after the lens event. Difference in color between the source and
the lens will result in a small displacement of the centroid with
wavelength.
• Yields an estimate of the host star spectral type and, with
assumption on underlying stellar population, DL, hence a complete
solution to lens equation.
 Relative source-lense transverse motion
 Cannot be determined from light curve but possible by measuring small
change in the elongation of the image, .e.g. with stable PSF from HST.
• Yield the relative angular proper motion μLS, hence the Einstein radius from
equation 5.19.
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Contents
5.1 Introduction
5.2 Description
5.3 Caustic and critical curves
5.4 Other light curve effects
5.5 Microlens parallax and lens mass
5.6 Astrometric microlensing
5.7 Other configurations
5.8 Microlensing observations in practice
5.9 Exoplanet results
5.10 Summary of limitations and strengths
5.11 Future developments
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5.5 Microlens parallax and lens mass (1)
 Fitting of the microlensing light curve yields 6 orn7 parameters
including q, the planet/host star mass ratio. Determination of the
planet mass thus requires an estimate of the host star mass. This
can done through the microlens parallax framework.
 Le us consider the geometry of the light bending for an impact
parameter b=RE
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5.5 Microlens parallax and lens mass (2)
 We have the following relations:
,
, with
(5.24)
(5.25)
where ωrel is the relative parallax
(5.26)
by analogy with the usual trigonometric
parallax ω=AU/d (ω in radian and d in AU)
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5.5 Microlens parallax and lens mass (3)
 The microlens parallax ωE expresses the size of the Earth orbit
relative to the Einstein radius of the microlensing event projected
onto the observer plane
(5.27)
 Equations 5.24-5.27 can be rearranged to give
(5.28)
(5.28)
(5.29)
(5.30)
 If both θE and ωE can be determined, then the lens (host star) can be
established from equation 5.28. The planet mass follows from the
mass ratio parameter q.
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5.5 Microlens parallax and lens mass (4)
 How does one constrain θE ?
 From finite source size effects. Light curve fitting yields:
If the source star is visible and unblended, its angular radius θ★
can be estimated from the angular size-color relation based on its
magnitude and color (Yoo et al. 2004).
 Measurements of the relative lens-source proper motion through
multi-color imaging secured several months/years after the
microlens event.
 How does one constrain ωE ?
 From observations of the microlensing light curve over an
extented observer baseline either from
• Combined observations from the ground and space
• Effects of the non-linear motion of the Earth’s orbit; can be
measured only for (relatively rare) high-magnification events.
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5.5 Microlens parallax and lens mass (5)
Example of a microlens parallax measurement
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Contents
5.1 Introduction
5.2 Description
5.3 Caustic and critical curves
5.4 Other light curve effects
5.5 Microlens parallax and lens mass
5.6 Astrometric microlensing
5.7 Other configurations
5.8 Microlensing observations in practice
5.9 Exoplanet results
5.10 Summary of limitations and strengths
5.11 Future developments
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5.6 Astrometric microlensing
 In addition to the photometric magnification, microlensed images
also leads to a small motion of their photocenter, typically by a
fraction of a mas. The astrometric of the centroid is, from Eqn 5.15,
(5.32)
 The maximum deflection angle is with u=√2
(5.33)
For a typical bulge lens, with θE~300 μas, the shift is ~0.1 mas.
 Astrometric detection through high-proper motion stars
 The idea is to follow a nearby high-proper motion star (e.g. Bernard’s star;
~10’’/yr) over several years and identify, through μas astrometry (e.g. with GAIA),
potential microlensing events.
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Contents
5.1 Introduction
5.2 Description
5.3 Caustic and critical curves
5.4 Other light curve effects
5.5 Microlens parallax and lens mass
5.6 Astrometric microlensing
5.7 Other configurations
5.8 Microlensing observations in practice
5.9 Exoplanet results
5.10 Summary of limitations and strengths
5.11 Future developments
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5.7 Other configurations
Other source/lens configurations that might be observed in
the future.
 Planet orbiting the source star.
 Provides information on atmospheric composition, satellites and
ring. Planets could be detected as they cross the caustics of the
foreground lens. Theoretically possible to detect H2O and CH4 of
a close-in Jupiter (Spiegel et al. 2005).
 Satellite orbiting a planet
 Feasible under favourable conditions.
 Planet orbiting a binary system
 Free-floating planets
 Microlensed transiting planets
 Planetasimal disks
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Contents
5.1 Introduction
5.2 Description
5.3 Caustic and critical curves
5.4 Other light curve effects
5.5 Microlens parallax and lens mass
5.6 Astrometric microlensing
5.7 Other configurations
5.8 Microlensing observations in practice
5.9 Exoplanet results
5.10 Summary of limitations and strengths
5.11 Future developments
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5.8 Microlensing in practice
 Large-scale observing programmes focus on Baades’s Window
in the central Galactic bulge.
 Pros: high stellar density to maximize number of microlensing
events. Region of relatively low extinction.
 Cons: crowding and blending.
 Observing strategy: Two-step mode
1.
2.
Wide-angle survey detects the early stages of a microlensing event
using relatively coarse temporal sampling.
Once deviation is significant, an alert is issued and an array of
smaller, follow-up narrow-angle telescopes distributed in Earth
longitudes follow the events with high-precision photometry and a
much denser time coverage.
 Two major teams

MOA (Microlensing Observations in Astrophysics); NZ/Japan
•

OGLE (Optical Gravitational Lens Experiment)
•

Phase I: 0.6m telescope; Phase II: 2m telescope. 20 sq degrees.
1.3m telescope in Chile
Plus several teams for follow-up observations
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Contents
5.1 Introduction
5.2 Description
5.3 Caustic and critical curves
5.4 Other light curve effects
5.5 Microlens parallax and lens mass
5.6 Astrometric microlensing
5.7 Other configurations
5.8 Microlensing observations in practice
5.9 Exoplanet results
5.10 Summary of limitations and strengths
5.11 Future developments
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5.9 Exoplanet results (1)
 Naming convention
 After the team who first reported the event (ex: OGLE-2003-BLG235)
 The lens and source can be specified with the suffixes ‘L’ and ‘S’.
Additioncal capital or lower case letters designate companions of
stellar or planetary mass respectively
• OGLE-2006-BLG-109LA, …Lb, …Lc designate specifically the star,
and the two known planets of this multiple system.
 First detection: OGLE-2003-BLG-235 (Bond et al. 2004)
 Original etsimates: Mp~1.5 MJ, DL=5.2 kpc refined to Mp~2.6 MJ,
DL=5.8 kpc, after follow-up HST observations to constrain
spectral type of the host (K) star.
 First low-mass microlensing event (Beaulieu et al. 2006)
 Mp~5.5 ME, DL=6.6 kpc, M host.
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5.9 Exoplanet results (2)
First multiple system: OGLE-2006-BLG-109 (Gaudi etal. 2008)
MP=0.7 and 0.27 MJ
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5.9 Exoplanet results (3)
 Statistical results (Gould et al. 2010)
 First constraints on giant/ice giant frequency beyond the snow
line.
 Based on only 6 detections with mean mass ratio q=5x10-4
 Consistent with a flat distribution in log a (semi-major axis)
 Planet frequency 8x that of Doppler surveys at 25x the orbital
separation.
 Constraint on super-Earth population from Cassan et al. (2012).
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Contents
5.1 Introduction
5.2 Description
5.3 Caustic and critical curves
5.4 Other light curve effects
5.5 Microlens parallax and lens mass
5.6 Astrometric microlensing
5.7 Other configurations
5.8 Microlensing observations in practice
5.9 Exoplanet results
5.10 Summary of limitations and strengths
5.11 Future developments
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5.10 Summary of limitations and strengths
 Limitations
 Low-probability (10-8), non-repeatable events.
 Planetary events are short-lived (Jupiter ~1 day; Earth: a few hours)
• Dense temporal sampling required.
 Planet typically at several kpc. Distance difficult to estimate without
additional constraints (microlens parallax, knowledge of host star spectral
type)
 Planet parameters (mass, orbital radius) scale with (possibly uncertain)
properties of the host star.
 Strengths
 Given high frequency monitoring and high photometric accuracy (e.g. in
space), the method is sensitive of Earth-mass planets over wide separations.
 To first oder, magnification amplitude is independant of the planet mass
 Detection of multiple systems possible through well-sampled high
magnification events.
 Technique largely unbiased in terms of host star properties. Planets and
their host star should therefore be found in proportion of their actual
frequency in the Galactic disk
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Contents
5.1 Introduction
5.2 Description
5.3 Caustic and critical curves
5.4 Other light curve effects
5.5 Microlens parallax and lens mass
5.6 Astrometric microlensing
5.7 Other configurations
5.8 Microlensing observations in practice
5.9 Exoplanet results
5.10 Summary of limitations and strengths
5.11 Future developments
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5.11 Future developments (1)
 Ground-based
 A network of 2m telescopes across the Earth to monitor several square degrees
every 10 mins.
 6000 events per yr. 100-fold increase in the number of events proved.
 Better sensitivity to wide-separaion and free-floating planets
 Sspace-based
 Advantages of space environmeny: less crowding/blending, exquisite
photometric sensitivity.
 Sensitivity to Earth mass planets.
 Euclid (ESA)
• Dark energy mission, 1.2m telescope
• Launch date: 2020
• Will include a 3-12 month microlensing program (Beaulieu et al. 2010).
 WFIRST (NASA)
• Dark energy mission, 2.4m telescope
• Launch date: 2024
• Microlensing survey planned. Should detect Mars-like planets.
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5.11 Future developments- Euclid (2)
(Credit: Jean-Philippe Beaulieu
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5.11 Future developments- Euclid (3)
Euclid will detect Earth mass planets
(Credit: Jean-Philippe Beaulieu
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5.11 Future developments- Euclid (4)
Euclid will detect free-floating planets
(Credit: Jean-Philippe Beaulieu
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