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2003/10/15 1
(http://hubblesite.org/newscenter/archive/1995/14/image/a)
2003/10/15 2
1.
2.
3.
4.
2003/10/15 3
1.
Newton (1701) ....
Cavendish (1783), Laplace (1796), Soldner (1801)
N 2
20.85
GMc R
Newton
1”=1 1cm 2km
!
2003/10/15 4
(Einstein)
Schwarzschild
R 2
N
41.75
2
GMc R
Eddington (1919)
obs 1.60 0.31
2003/10/15 5
Walsh, Carswell, Weymann(1979)
Q0956+561 A,B
Image A
Image B
Image A
Image B
2003/10/15 6
Q0957+561 1/2
2003/10/15 7
Q0957+561 2/2
2003/10/15 8
Einstein cross
(Clover leaf)
H1413+1172237+0305
2003/10/15 9
Giant arc
A370 CL2244-02
2003/10/15 10
Einstein ring
1938+666 PKS1830-211
MG1131+0456
2003/10/15 11
Microlensing
2003/10/15 12
2.
•
•
•
•
•
2003/10/15 13
S OL
S
I
OSDLSD OLD
I S, , 1
= OS SD
= OS ID = -
= LSDOS S OS I LSD D D
I OL IDr
2003/10/15 14
LSS I 2
OL OS I
4 1GMDc D D
2 2
I OL I
4 4GM GMc r c D
LSE 2
OL OS
4GMDc D D
2
ES I
I
2 2
I S I E 0
2 2
I S S E12
4
( )
2003/10/15 15
I
1
2 2
I S S E12
4
2003/10/15 16
2
S
I+
I-
2 24S E
LSE 2
OS OL
42 2
G Dc DMD
S =0
2003/10/15 17
LS
2
OS OL
42
GMDc D D
17
OS 8kpc 2.4 10 kmD 0.002
32
(1045g)
22
OS 32 3.1 10 kmD 5.6
2003/10/15 18
I
2
ES I
I
2
LS LSEI I I S
OS I OS
, ,D DD D
LSI
OS
DD
LSI
OS
DD
I
S
I
I+
2003/10/15 19
2
4GM rc r
M r r
I
I
S
C
A
LSI
OS
DD
B
3
1
S
2003/10/15 20
S
0S
= ( I ) x ( S )
0 0IS
IS
I
0 S
SAS
2003/10/15 21
2 2
S E
2 2
S S E
2 1
22 4A
S 0
1, 0A A
A A
S
,A A
S 0E
Einstein ring
MG1131+0456
2003/10/15 22
LSS I I
OS
DD
S S
SLSS S I I I I
OS
DD
I
ILSS I
OS I
1
I I I I
DD
I U M
----- -----
IM (2 2 )
1 21
I
2 1
1
1M
2003/10/15 23
1 21
I
2 1
1 2
2 1
1
1
1 0
0 1
M
2003/10/15 24
critical line caustic
0
SAS 2 2 2
1 2
1det
1M
critical line
causticA
B
critical line
1/A = 0
caustic
critical line
S
2003/10/15 25
S=0 Einstein ring
caustic
2003/10/15 26
Einnstein
cross
Einnstein
cross
I
2003/10/15 27
BA
A
B
St t
S At t TS Bt t T
A BT T
1. A,B
( )
2. A,B
( )
A B AB
AB ABgeo pot
T T TT T
2003/10/15 28
(time delay)OL OS
L
LS
2 2
AB A S B Sgeo
1 12 2
1D DzD
Tc
L OL OS
LS
AB A Bpot
1 z D DTc D
L OL OS
LS
AB A S B S,1
,z
T D Dc D
LS LSOL OL
OS OS
,x yx y
D DD DD D
2
S S S12
, ,
2003/10/15 29
3.
•
•
•
2003/10/15 30
3.1.
http://vela.astro.ulg.ac.be/themes/extragal/gravlens/bibdat/engl/glc_homepage.html
1.
2. z3.
4.
5.
2003/10/15 31
LSS I OL I
OS
D DD
M,
( ? ?...)
A B C D, , , ,
2003/10/15 32
A
B
S A S AM
A
MAM
BM
1
B AM M
A B
A B1
B AM M
2003/10/15 33
Time delay
L OL OS
LS
AB A S B S,1
,z
T D Dc D
L OL OS
L S 0 0
LS 0
1+ 1, , ,
z D Df z z
cD H
AB obs
0 A SS B SL 0 0, , ,1
, ,T
zH f z
A,B
time delay, 0 0,0HHubble
2003/10/15 34
Q0957+561
0.36
1.41
A,B 6.5
A
B
B
A
Falco&Shapiro 1997
2003/10/15 35
Q0957+561
AB 417T
1 1
0 64 13kms MpcH
Q0957+561A,B
( )
2003/10/15 36
PG1115+080
2003/10/15 37
2237+0305
2237+0305
2003/10/15 38
B1608+656
0.63
1.39
0.9 2.1
BA 31 7
BC : 36 7
BD : 76 10
1 1
0 0 075kms Mpc 0.3, 0.7H
2003/10/15 39
3.2.
Lnd
E OL L Er D zS< E
S
2
2 0 LSE
0OL OS
3
2
zLL
H DN n r d dVc D D
:0 0.62
OL LD z
(Falco,Kochaneck,Munoz 1998)
2003/10/15 40
3.3. Microlensing
macho = massive compact halo object
0.002
LSE
OL OS
8[kpc]0.001
DMM D D
22 2
S E
2 2
S S E
2
4A A A
S
2003/10/15 41
2 2 2
S E 0
2 2 2
S S E 0 0
22
0 S E 0 0 OL E
2 2,
4 4p
uAu u
u V t t D
+
-Sr
I+r
I-r
microlens
Witt & Mao,1994
2 2
0 0
,
,
,,
,
2 cos
pd d I A uA
d d I
u u u2003/10/15 42
MicrolensMicrolens
transit
by-pass
2003/10/15 43
MicrolensMicrolensOGLE microlensing event candidate:
BUL_SC10 294229
http://sirius.astrouw.edu.pl/~ftp/ogle/ogle2/gb_lenses/gallery1.html
Field BUL_SC10
Star No 294229
RA (J2000.0) 18:20:20.60
Dec (J2000.0) -22:24:10.1
Remarks
t0 (HJD) 2450625.704 0.166
(1997-06-26.20 UT)
(day) 33.09 0.75
Amax 4.86 0.07
I0 18.950 0.010
Field BUL_SC10
Star No 294229
RA (J2000.0) 18:20:20.60
Dec (J2000.0) -22:24:10.1
Remarks
t0 (HJD) 2450625.704 0.166
(1997-06-26.20 UT)
(day) 33.09 0.75
Amax 4.86 0.07
I0 18.950 0.010
2003/10/15 44
MicrolensMicrolens
•
– R*, R= R* /rE
– type
– g
•
– R*, R= R* /rE
– type
– g
•Microlens
–Microlens M–Microlens = DOL/DOS
–Microlens V
•Microlens
–Microlens M–Microlens = DOL/DOS
–Microlens V
:
2tE= 2rE /V(event duration), 0(, R?)
Parallax rE /( 1- reduced Einstein ring raduis)
E= rE / DOL angular Einstein ring raduis)
:
2tE= 2rE /V(event duration), 0(, R?)
Parallax rE /( 1- reduced Einstein ring raduis)
E= rE / DOL angular Einstein ring raduis)
2003/10/15 45
MACHO (MAssive Compact Halo Object)
EROS (Experience de Recherche d'Objets Sombres)
OGLE (Optical Gravitational Lens Experiment)
AGAPE (Andromeda Galaxy and Amplified Pixels Experiment)
MOA (MACHO Observations in Astrophysics)
PLANET (Probing Lensing Anomalies NETwork)
DUO (Disk Unseen Objects)
GMAN (Global Microlensing Alert Network)
MIPS (Microlensing Planet Search)
2003/10/15 46
4.
• Microlens
•
2003/10/15 47
4.1. Microlens
sin , sinr r l lE a t E a t
rE lEStokes
2 2 2 2
0 0
0 0
,
2 cos , 2 sin
r l l r
r l r l
I a a Q a a
U a a V a a
R
2003/10/15 48
Chandrasekhar
•(Chandrasekhar 1960)
0US0QS
0, cos 2Q Q
0, sin 2U Q2
0 0 20 1 1Q I c R
SQ0=SU0=0 (;_;)
!! Algol (^_^)
2003/10/15 49
•Simmons,Newsam,Willis (1995) (SNW)
•Simmons,Willis,Newsam (1995)
•Agol (1996) [binary lens]
MicrolensMicrolens
Microlens VMicrolens V+
MicrolensMicrolens
2003/10/15 50
0US0QS
Microlens
2003/10/15 51
0US0QS
Microlens
QS US
2003/10/15 52
QS US
Microlens
2003/10/15 53
MicrolensMicrolens
QS US
Microlens
2003/10/15 54
MicrolensMicrolens
QS US
Microlens
2003/10/15 55
Microlens
QS US
2003/10/15 56
Microlens
QS US
2003/10/15 57
Microlens
QS US
2003/10/15 58
0
0 0
0 0
cos 2
sin
,
2
I
Q
U
p
S IS QS Q
Ad d
2 2
0 0
2 2 2 2
0 0 0 0
2 cos 2
2 cos 2 co 4,
sp
u uu u u
Au
22 2
0 0 0 E
2 2
0
u V t t r x
ysource
microlens
uo
ErEl
VR
00: Microlens Impact parameter
2003/10/15 59
2
0
2
0,
,Q U
I I
S Sp
S SD R u
R u22 2
0 0 0 E
2 2
0
u V t t r
SNW p R, u0
2003/10/15 60
Transit
R 0
E2V t r
2 2
E 0R V t r
R
(SNW)
2003/10/15 61
By-pass
(SNW)
R0
By-pass RBy-pass R
2003/10/15 62
Semi-Analytical Formulae
22
0 E 10
2
0
2
0 0
4 1 1 1
, 2
4
R
IS I r d c R
u k n F k
u u
Total Intensity; Witt, H.J., 1995
20
2 2
0 0
4 4,
4
u nn ku u
:
, :
F k
k n
2003/10/15 63
The Stokes Parameters SQ,SU
0
2
2 02
0 0 2 E20 2
0 0 0
6 6 2 2 2 2 4 4 2 2
0 0 0 0 0
22
0 0 0
22 2
0 0
0
0
2
0
0
,
,, 1 1
4
6 4
8 4
2
co
4
8
s2
sin2
R
Q
U
D R u
u k nD R u I c r d R
u u u
u u u u uF k
u u
SS
u
u u uE k
u:E k ( by Yoshida)2003/10/15 64
By-pass
SI , D R<u0, R<1 …
22
010 2
2 21 00 0
2 26 4 2
0 0 002
2 22 2 2
1 0 00 0 0
160 281 ,
15 5 2 4
304 3 6 5211
15 5 4 147 2 4
I p I
p I
uc RS A Sc uu u
u u uA Sc R RDc u uu u u
2
02
22
1 0
21
15 5 4
R ucpc u
2003/10/15 65
u0
22
0 02max 2 2
21 00
0.5210.0011
15 5 0.5 14
R Rcpc
1 20.64, 0.04c c
0 0 max ,p R max 0,R R p
SI 0
R2003/10/15 66
4.2.
• ( )
•
• Jacobian Matrix ( )
2003/10/15 67
(Friedmann-Lemaitre model)
(clumpy model)
clumpclump
FL 0;D z
0;D zDyer-Roeder distance
FL 0; 0;D z D z
clump
2003/10/15 68
OSS 1 1S 1 2S 2 S
01
OS1 S
101
N N
N
k k kk
D D D DDD DD
y y y y y
y y 01
101
ii
i k i k kk
D DD
y y y
0i i iDy
;ij i jD D z zDyer-Reoder distance
2003/10/15 69
i2
22
4
i
ii i S
ii
G dc
y yy y yy y
0i i iDy0iDy
0i iD
2003/10/15 70
Jacobian Matrix: A S 1 S
1 SS, S,
11
N
N N k k kk
M A I U A
11
ii
i k kk
A I U A0
ki kk k k
i k
DD
U = U
1 2
1 2 3
1 2 3
SS,
1 1 11
k kN
N k k kk k k
A I U I U I U
11 2
1 2 3
1 2 31 1 1 1 1
i
i
i
kk kN Ni
k k k ki k k k k
I U U U U
2003/10/15 71
1
Jacobian Matrix
11 2
1 2 3
1 2 3
S,
1 1 1 1 1
i
i
i
kk kN Ni
N k k k ki k k k k
A I U U U U
1
1
1
1
2003/10/15 72
Jacobian Matrix
1 2 3 1 2 3i ik k k k k k k ka a a aU U U U I
11 2
1 2 3
1 2 3
S,
1 1 1 1 1
S
1i
i
i
kk kN Ni
N k k k ki k k k k
N
a a a a
B z
A I
= I
2003/10/15 73
S S
NNB z B z
AS, N
N
1
1
0; ;41
0; 1
j j j
j j
j j j
k k kk k
k k k
D z D z zGa z zc D z z H z
S1, 0
zdVM dV z zN dz N
2003/10/15 74
S S, Slim N B zA A I
S S S0; 0;D z B z D z
Friedmann-Lemaitre
FL S S S0; 0;D z B z D z