Download D - St Andrews Astronomy Group

Studying cool planets around distant low-mass
stars
Planet detection by gravitational microlensing
Martin Dominik
Royal Society University Research Fellow
SUPA, University of St Andrews, School of Physics & Astronomy
Deflection of light by gravity (1911)
bending angle
2GM
α=
c2ξ
measurable at Solar limb: α
0.″85
=
suggested: measure during Solar eclipse
Deflection of light by gravity (1915)
bending angle
4GM
α=
c2ξ
measurable at Solar limb: α
1.″7
=
confirmed by measurement
of stellar positions during Solar eclipse
(Eddington 1919)
Bending of starlight by stars
(Gravitational microlensing)
You are here
S
L
Deriving the gravitational lens equation
4GM
bending angle α =
c2ξ
side view
DS
DL
I−
L
O
η
S
ξ
α
I+
α
The gravitational lens equation
and its solutions
side view
4GM
bending angle α =
c2ξ
I−
η
angular Einstein
radius
4GM DS−DL
θE =
c2 DL DS
ξ
I+
ξ
x=
DL θE
η
y=
DS θE
two images
y = x −
1
1/2
(
)
1
x
(lens
equation)
x± = 2 [y ± (y2+4)1/2]
Image distortion and magnification
angular Einstein
radius
4GM DS−DL
θE =
c2 DL DS
observer’s view
(
1/2
)
with ‘typical’ DS ~ 8.5 kpc
and DL ~ 6.5 kpc
θE ~ 600 (M/M☉)1/2 μas
1
x± = 2 [y ± (y2+4)1/2]
lens equation relates radial coordinates, polar angle conserved
dx±
radial distortion
dy
total magnification
x±
tangential distortion
y
A(y) = ∑
±
x± dx±
y dy
=
y2+2
y (y2+4)1/2
Microlensing light curves
S
L
(t-t0)/tE
tE = θE/μ
y(t) = [u02 + (
t − t0
tE
)2]1/2
y2+2
A(y) = y (y2+4)1/2
A[y(t)] defined by u0, t0, tE
μ ~ 15 μas d-1
tE ~ 40 (M/M☉)1/2
days
Notes about gravitational lensing dated to 1912 on two pages of Einstein’s scratch notebook
Microlensing optical depth
quantify alignment by defining
optical depth τ = probability that given source star is inside Einstein circle
u < 1 A > 3/√5 ≈ 1.34
corresponds to brightening in excess of 34%
Solid angle of sky covered by NL Einstein
circles
(neglect
overlap)
NL πθE2
with mass volume density ρ(DL) and mass spectrum
f(M)
N→
∫ f(M)
DL2
ρ(DL)
M
∫f(M) dM = 1
dDL dΩ dM
for ρ = const., x = DL/DS
θE =
(
4GM DS−DL
D L DS
c2
1/2
)
I
τ = 4πG DS ρ x(1-x) dx = 2πG DS2
∫0
c2
3c2
ρ
2
DS ~ 8 kpc,Galactic disk ρ = 0.1
35% disk, 65% bulge
M☉
(pc)3
τ ~ 6 × 10-7 (Galactic disk)
τ ~ 2 × 10-6 (total)
Microlensing event rate
event time-scale tE ~ 20
daysstar moves on the sky by θE
during tE, source
average duration of ‘event’ with A >
1.34
π
<te> =
tE
2
(unit circle: area π, width 2, average
length )
π
2
event rate
Γ = NS
τ
= 2 × 10-5 NS (yr)-1
<te>
for 1000 events per year monitor NS ~ 5 × 107
stars
(Note: target observability shows seasonal variation)
~ 85 ongoing events at any time
Microlensing surveys are a major data processing
venture
1936: no computers
1965: computers not powerful enough
First reported microlensing event
MACHO LMC#1
Nature 365, 621 (October 1993)
Current microlensing surveys (2007)
Optical Gravitational Lensing Experiment
1.3m Warsaw Telescope, Las Campanas (Chile)
1.8m MOA Telescope, Mt John (New Zealand)
daily monitor
≳ 100 million stars,
-6
τ ~ 10 for microlensing event → ~1000 events alerted per
year
Lensing or eclipse ?
Condition for
eclipse:
foreground object occults the lensing images of the background
foreground object occults the background object
object
angular radius
θ = RL/DL
vs. angular Einstein radius
θE =
(
4GM DS−DL
c2
DL DS
1/2
)
of intervening object
θ ≪ θE : lensing (brightening)
θ ≫ θE : eclipse (dimming)
Lensing regime: DL/DS ≈ 1/2, region broadens with increasing DS
Eclipse regime: DS − DL ≫ DL or DS ≫ DL
eclipsing planets around observed (source) stars
microlensing planets around lens stars
Consequences:
eclipsing stellar binaries
Which host stars?
Microlensing detects planets around lens stars
Host stars are selected by means of chance alignment
Their mass distribution is related to stellar mass function
Stellar mass probed by microlensing
Microlensing prefers detecting planets around red-dwarf stars
Multiple point-mass lens
lens equation (2D) for single point-mass
lens
x
y = x −
|x|2
θ
x=
θE
two-dimensional position angles β (source) and θ
(image)
angular Einstein radius
total mass M,
θE =
(
4GM DS−DL
c2
DL DS
i-th object with mass fraction mi at
x(i),
β
y=
θE
1/2
)
∑ mi
=1
in weak-field limit, superposition of deflection terms, but not of light
curves
x − x(i)
y = x − ∑ mi
|x − x(i)|2
The binary point-mass lens
angular separation
δ = d θE
mass ratio
q = m2/m1
m2 = 1 − m1
completely characterized by two dimensionless parameters
(d,q)
time-scale of planetary deviations ≪ orbital period
m1 =
in centre-of-mass
system:
y1 = x1 −
1
1+q
y2 = x2 −
1
1+q
x(1)
q
1+q
x1 −
( x1 −
1
1+q
q
1+q
=
m2 =
(
d
2
d ) + x2
2
x2
( x1 −
q
1+q
2
d ) + x2
2
q
1+q
q
1+q
d, 0)
−
q
1+q
−
q
1+q
x(2)
= −(
1
1+q
1
1+q
x1 +
( x1 +
1
1+q
d, 0)
d
2
d ) + x22
x2
( x1 +
1
1+q
2
d ) + x22
solving for (x1,x2) leads to 5th-order complex polynomial
either 3 or 5 images
Magnification and caustics
magnificatio
A(y) =
n
A(y) → ∞
for det (
caustics
∂y
∂x
∑j | det (
) (xj) = 0
{yc = y(xc) |det (
∂y
∂x
∂y
∂x
) (xj)
|
-1
for at least one j
) (xc) = 0 }
point lens: critical
points
x = 1caustic
(Einstein
circle), of
map to
y = 0at(point
at position
lens)
3 topologies for a binary lens
“resonance” at d ~ 1
Planetary-regime binary-lens caustics
and excess magnification
for q ≪ 1: two-scale problem
lens star produces 2 images
if planet close to one of these, it creates a further split
maximal effect if lens star creates image near the
planet
tidal field of the star creates extended ‘planetary
caustic’
there is also a ‘central caustic’ close to the lens star
for d ~ 1, both merge into a single caustic
d[1] planet-star separation in units of
stellar angular Einstein radius
position of planetary caustic(fulfills lens equation)
D = d[1] −
1
d[1]
inside Einstein circle (|D| < 1) for (√5−1)/2 < d[1] < (√5+1)/2
(“lensing zone”)
red: caustics
green shades: brighter
shading levels: more than 1%, 2%, 5%, 10%
blue shades: dimmer
q = 10-2
Caustics and excess magnification (II)
q = 10-2
q = 10-3
Caustics and excess magnification (III)
Detection efficiency
q = 10-2
q = 10-3
q = 10-4
light curve determined by 1D cut through 2D magnification map
for event with (u0,t0,tE), only a fraction of all trajectory angles lead to detectable
signal
detection efficiency ε (d,q; u0,t0,tE) = P ( detectable signal in event (u0,t0,tE) | planet
(d,q) )
Planetary deviations
ρ = θ/θE = 0.025
tE = 20 d
(giant R ~ 15
R☉)
tE = 20 d
wide
close
[d]
q = 10-2
[d]
q = 10-3
q = 10-4
Planetary signals
Linear size of deviation regions scale with q1/2 (planetary caustic) or q (central caustic)
For point-like sources, both signal duration and probability scale with this factor
Signal amplitude limited by finite angular source size θ
main-sequence star R ~ 1 R☉ vs giant R ~ 15
R☉
tE = 20 d
planetary caustic
tE = 20 d
central caustic