Deflection of light by gravity (1911)

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Deflection of light by gravity (1911)
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Deflection of light by gravity (1915)
4GM
a
c2?
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Bending of starlight by stars (Gravitational
microlensing)
You are here
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Deriving the gravitational lens equation
side view
I-
L
O
S
a
a
I
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The gravitational lens equation
and its solutions
side view
angular Einstein radius
)
(
1/2
DS-DL
4GM
?E
DL DS
c2
?
?
y
x
(lens equation)
DL ?E
DS ?E
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Image distortion and magnification
observers view
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x y (y24)1/2
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lens equation relates radial coordinates, polar
angle conserved
y22
x
dx
A(y) ?

total magnification
y (y24)1/2
y
dy

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Microlensing light curves
Ay(t) defined by u0, t0, tE
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Microlensing optical depth
quantify alignment by defining
optical depth t probability that given source
star is inside Einstein circle
u lt 1 ? A gt 3/v5 1.34
corresponds to brightening in excess of 34
Solid angle of sky covered by NL Einstein circles
NL p?E2
(neglect overlap)
with mass volume density ?(DL) and mass spectrum
f(M)
for ? const., x DL/DS
t 6 10-7
(Galactic disk)
35 disk, 65 bulge
t 2 10-6
(total)
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Microlensing event rate
event time-scale tE 20 days
during tE, source star moves on the sky by ?E
average duration of event with A gt 1.34
event rate
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First reported microlensing event
Nature 365, 621 (October 1993)
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Current microlensing surveys (2007)
1.8m MOA Telescope, Mt John (New Zealand)
daily monitor ? 100 million stars,
t 10-6 for microlensing event ? 1000 events
alerted per year
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Lensing or eclipse ?
foreground object occults the background object
foreground object occults the lensing images of
the background object
Condition for eclipse
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 eclipsing
stellar binaries
Consequences
Text
Prediction from data prior to first caustic peak
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Which host stars?
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Multiple point-mass lens
in weak-field limit, superposition of deflection
terms, but not of light curves
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The binary point-mass lens
completely characterized by two dimensionless
parameters (d,q)
time-scale of planetary deviations orbital
period
solving for (x1,x2) leads to 5th-order complex
polynomial
either 3 or 5 images
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Magnification and caustics
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Planetary-regime binary-lens caustics
and excess magnification
q 10-2
red caustics
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Caustics and excess magnification (II)
q 10-2
q 10-3
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Caustics and excess magnification (III)
Detection efficiency
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Planetary deviations
tE 20 d
tE 20 d
wide
close
d
d
q 10-2
q 10-3
q 10-4
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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 ??
main-sequence star R? 1 R? vs giant R? 15 R?
tE 20 d
tE 20 d
planetary caustic
central caustic
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