2003/10/15 1

yoshidah@fmu.ac.jp

(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