last time · 2013. 12. 11. · (c)2013 van putten 1967 l s swift, hete ii: x-ray afterglows also to...
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![Page 1: Last time · 2013. 12. 11. · (c)2013 van Putten 1967 L S Swift, HETE II: X-ray afterglows also to short GRBs 050509B, 050709,… BeppoSax GRB 970228: Z-0.695 GRBs CC-SNe GRB-SNe](https://reader034.vdocuments.mx/reader034/viewer/2022052006/6019d189ad65f92e646f8b01/html5/thumbnails/1.jpg)
(c)2013 van Putten
Last time
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(c)2013 van Putten
1967
L
S
Swift, HETE II: X-
ray afterglows also
to short GRBs
050509B,
050709,…
BeppoSax GRB 970228: Z-0.695
GRBs
CC-SNe GRB-SNe are rare
making up < 1% of all
SN Ib/c and < 0.2% of
all CC-SNe
Swift Era LGRBs with no
association to massive
stars: GRB 059820A,
050911, 060418, 060505,
060614, 070125
Hyper energetic events
GRB 031203/SN2003lw
GRB030329/SN2003dh
defy max Erot of NS
CC-SNe are
diverse and
produce NS and
BH remnants
1/50 yr in MW
Thumbnail overview…
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(c)2013 van Putten
PN
SN
S
( )2
52
%1
1.5 spin, maximal erg103
Mc
M Sun×
BH
( )2
54
%30
10 spin, maximal erg106
Mc
M Sun×
NS
NS
BH
NS
low mass
rapidly
spinning BH
high mass
BHs with a
diversity in
spin
Orphan
LGRB
Orphan LGRB
Astronomical origin of GRBs
![Page 4: Last time · 2013. 12. 11. · (c)2013 van Putten 1967 L S Swift, HETE II: X-ray afterglows also to short GRBs 050509B, 050709,… BeppoSax GRB 970228: Z-0.695 GRBs CC-SNe GRB-SNe](https://reader034.vdocuments.mx/reader034/viewer/2022052006/6019d189ad65f92e646f8b01/html5/thumbnails/4.jpg)
(c)2013 van Putten
Detailed derivation of quadrupole emission formula
Lie derivative
Linearized Ricci tensor
Quadrupole GW emission
GW sources in the Local Transient Universe
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(c)2013 van Putten
Independent of Christoffel symbols
( ) ( ) ( ) ( ) fufufufufu
uw
b
b
c
c
b
b
c
c
b
b
c
c
b
b
c
c
b
b
bb
∂∂−∂∂=∂∂−∂∂=∂
=
ξξξξξ
ξ
],[
],[
b
c
cb
c
cb uuu ξξξ ∂−∂=],[
Vector fields from vector fields
fufu b
bb ∂→:
Vector field
Lie derivative (I)
Tp
ub
ξb
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(c)2013 van Putten
( )
c
a
aa
c
c
cbccb
b
b
bbb
b
bb
b
b
b
b
bb
b
b
bb
b
b
bb
b
bbb
uu
xuxux
x
uuuL
xx
x
x
xA
xx
xA
xx
ξξ
εξε
ε
ξεξ
ε
ξεξ
ε
εξ
ε
εξ
∂−∂=
−+
∂∂
=
−=
∂+=∂∂
+=
∂∂
=∂∂
=
∂−=∂∂
−=∂∂
=
−→
−
→
−
)()(
lim
11
11
'
'
1
*
0
''
1
''
'
'''
'
e.g. Stephani, H.
Taking along one’s own coordinate system along a vector field:
Simultaneous coordinate
transformation
Lagrangian displacement
Lie derivative (II)
![Page 7: Last time · 2013. 12. 11. · (c)2013 van Putten 1967 L S Swift, HETE II: X-ray afterglows also to short GRBs 050509B, 050709,… BeppoSax GRB 970228: Z-0.695 GRBs CC-SNe GRB-SNe](https://reader034.vdocuments.mx/reader034/viewer/2022052006/6019d189ad65f92e646f8b01/html5/thumbnails/7.jpg)
(c)2013 van Putten
c
e
e
cce
ec
c
c
c
cc
e
e
c
c
e
e
cc
cc
cc
c
c
c
e
ec
e
e
cce
ec
c
c
uvvuvLu
vLuuvuvvLuuLvvuL
vuuvvuvuL
ξξ
ξξ
ξξξ
ξ
ξξξξ
ξ
∂+∂=
+∂−∂=+≡
∂=∂+∂≡
c
acac
c
a vvvL ξξξ ∂+∂=
Lie derivative (III)
Independent of Christoffel symbols
A tensor vb is dual to a vector ub if ucvc is a scalar, e.g., vb=φ,b
![Page 8: Last time · 2013. 12. 11. · (c)2013 van Putten 1967 L S Swift, HETE II: X-ray afterglows also to short GRBs 050509B, 050709,… BeppoSax GRB 970228: Z-0.695 GRBs CC-SNe GRB-SNe](https://reader034.vdocuments.mx/reader034/viewer/2022052006/6019d189ad65f92e646f8b01/html5/thumbnails/8.jpg)
(c)2013 van Putten
c
abc
c
bacabc
c
c
abc
c
bacabc
c
ab
ggg
ggggL
ξξξ
ξξξξ
∇+∇+∇=
∂+∂+∂=
abbaabgL ξξξ ∇+∇=
0=∇+∇ abba ξξ
bξWe say is a Killing vector whenever
If so, it represents a symmetry of the metric tensor
Lie derivative and symmetries
![Page 9: Last time · 2013. 12. 11. · (c)2013 van Putten 1967 L S Swift, HETE II: X-ray afterglows also to short GRBs 050509B, 050709,… BeppoSax GRB 970228: Z-0.695 GRBs CC-SNe GRB-SNe](https://reader034.vdocuments.mx/reader034/viewer/2022052006/6019d189ad65f92e646f8b01/html5/thumbnails/9.jpg)
(c)2013 van Putten
If are the vector fields associated with two independent
coordinates (s,t):
( ) ( )0
],[
≡−=∂−=
∂∂−∂∂=∂
sttssa
a
t
a
c
c
a
a
c
c
a
a
b
b
fffvfu
fuvfvufvu
),( bb vu
0],[ =bvu
Lie derivative and coordinates
![Page 10: Last time · 2013. 12. 11. · (c)2013 van Putten 1967 L S Swift, HETE II: X-ray afterglows also to short GRBs 050509B, 050709,… BeppoSax GRB 970228: Z-0.695 GRBs CC-SNe GRB-SNe](https://reader034.vdocuments.mx/reader034/viewer/2022052006/6019d189ad65f92e646f8b01/html5/thumbnails/10.jpg)
(c)2013 van Putten
( )
[ ] ( )
hhhR
OhggggR
g
ggg
OR
hhhg
caecad
de
ace
e
ac
caeaccaeaec
de
b
d
adc
ccde
ded
dc
d
bca
d
acb
d
abc
ab
ab
ababab
∂∂−∂∂+∂∂−=
+∂∂−
−+∂=
∂==Γ
+Γ∂−Γ∂=
=+=
2
1
2
1
2
1
2
1
2
1
,
)(
2
,,,
,
2
η
ε
ε
ηη
Linearized Riemann and Ricci tensor
![Page 11: Last time · 2013. 12. 11. · (c)2013 van Putten 1967 L S Swift, HETE II: X-ray afterglows also to short GRBs 050509B, 050709,… BeppoSax GRB 970228: Z-0.695 GRBs CC-SNe GRB-SNe](https://reader034.vdocuments.mx/reader034/viewer/2022052006/6019d189ad65f92e646f8b01/html5/thumbnails/11.jpg)
(c)2013 van Putten
hhhh
hhhh
accaae
e
ace
e
a
accaaea
e
cea
e
∂∂−∂∂−∂∂+∂∂
∂∂−∂∂−∂∂+∂∂
4
1
4
1
2
1
2
1
4
1
4
1
2
1
2
1
hhh
hhR
abab
ea
e
cace
e
ac
η2
1
2
1)(
−=
∂∂+∂∂−=
:2
1)( hh caecad
de ∂∂−∂∂η
Linearized Ricci tensor
![Page 12: Last time · 2013. 12. 11. · (c)2013 van Putten 1967 L S Swift, HETE II: X-ray afterglows also to short GRBs 050509B, 050709,… BeppoSax GRB 970228: Z-0.695 GRBs CC-SNe GRB-SNe](https://reader034.vdocuments.mx/reader034/viewer/2022052006/6019d189ad65f92e646f8b01/html5/thumbnails/12.jpg)
(c)2013 van Putten
hhhhhR ababea
e
cace
e
ac η2
1,
2
1)( −=∂∂+∂∂−=
Gravitational radiation is in transverse waves. With
Choose
In transverse traceless gauge
abbaabgL ξξξ ∇+∇=
.2
1
0,under invariant ,
h
h
hhh
a
a
a
a
baaaab
a
b
abbaababbaabab
−=∂+∂
=∆+→−∂=∆
∂+∂+=∇+∇+→
ξµ
µµξξξ
ξξξξ
0,2
1=∂∂−= RhR ace
e
ac
linearized Ricci tensor - simplified
![Page 13: Last time · 2013. 12. 11. · (c)2013 van Putten 1967 L S Swift, HETE II: X-ray afterglows also to short GRBs 050509B, 050709,… BeppoSax GRB 970228: Z-0.695 GRBs CC-SNe GRB-SNe](https://reader034.vdocuments.mx/reader034/viewer/2022052006/6019d189ad65f92e646f8b01/html5/thumbnails/13.jpg)
(c)2013 van Putten
∫∫∫ −∂−=−==−= xdghhxdghRSxdgRS ab
abab
ab
4244
32
1
16
1
16
1
ππδ
π
About a solution of the Einstein equations, we have
[ ] ( )∫ −∂= xdghhS abcab
42
2
1
16
1
π
The metric fluctuation hab in Coulomb gauge :
[ ] ( )∫ −∂=↔ xdgS c
42
2
1ϕϕ
××
++
+×
×+ +=
−= ababab eheh
hh
hhh
0
0
00 =ah
[ ] ( ) ( )[ ] :16
1 422
∫ −∂+∂= ×+ xdghhhS ccab π ( ) ( )22000
16
1×+ ∂+∂=== hhttt tt
zzz
π
Time-average/orbital-average
Stress-energy tensor of GWs
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(c)2013 van Putten
0
02
2
2
2
==→==×=
====→=
∑∑
∑∑∑
Jdt
d
dt
dJjpx
Pdt
dp
dt
dmx
dt
dm
dt
ddxmd
AAA
A
A
A
A
A
A
µµ
No electric or magnetic dipole radiation in GWs
baabababab jAThTG πππ 482
1:8 22 −=∂↔=∂−=
Lienard-Wiechert potential for the radiation field produced by a source
ki
k
jkj
k
ij
k
iki
k
kjij TxTxxTxTxdT ∂+∂−=∂+∂= ∫∫∫ 2
1
2
13
( ) ijjiijjiij Idt
dxdxxTxdTxTxxdT
2
23002
0
300
0
3
2
1
2
1
2
1≡∂=+∂= ∫∫∫
|'|,'),'(4
),( 3 xxrxdrtxTr
trhV
TT
ij
TT
ij −=−= ∫
Making GWs
![Page 15: Last time · 2013. 12. 11. · (c)2013 van Putten 1967 L S Swift, HETE II: X-ray afterglows also to short GRBs 050509B, 050709,… BeppoSax GRB 970228: Z-0.695 GRBs CC-SNe GRB-SNe](https://reader034.vdocuments.mx/reader034/viewer/2022052006/6019d189ad65f92e646f8b01/html5/thumbnails/15.jpg)
(c)2013 van Putten
TT
ij
TT
ij Idt
d
rh
2
22= III ijij
T
ij δ3
1−=
TT
jk
TT
jk
TT
jk
TT
jk
z
Idt
dI
dt
d
dtd
Ed
ht
ht
T
3
3
3
32
0
8
1
counting) double avoid to1/2 (extra 32
1
π
π
=Ω
∂∂
∂∂
=
In a given direction, there are 2 polarization modes. They represent
2/5 times the contributions of all 5 (=6-1) components in traceless
part of T
ijI
’t Hooft 2002
T
jk
T
jk
T
jk
T
jkGW Idt
dI
dt
dI
dt
dI
dt
dL
3
3
3
3
3
3
3
3
5
1
8
14
5
2=××=→
ππ
Counting degrees of freedom
![Page 16: Last time · 2013. 12. 11. · (c)2013 van Putten 1967 L S Swift, HETE II: X-ray afterglows also to short GRBs 050509B, 050709,… BeppoSax GRB 970228: Z-0.695 GRBs CC-SNe GRB-SNe](https://reader034.vdocuments.mx/reader034/viewer/2022052006/6019d189ad65f92e646f8b01/html5/thumbnails/16.jpg)
(c)2013 van Putten
( ) ( ) ( ) 24622
2
26
3
3
3
3
3
3
3
3
322cos22cos22
12 µϕϕµ aaI
dt
dI
dt
d
Idt
dI
dt
d
T
jk
T
jk
ij
T
ij
Ω=+
Ω=
=
GW emission in binary motion
( ) ( )
( ) const.const.const. const.2cos2
1
const.2cos2
1cos
,,,,5
1
2
222
22
2
11
21
21213
212
3
3
3
3
===+−=
+=+=
+=+=
+=ΩΩ==
yxxyzzyy
xx
T
jk
T
jkGW
IIIaI
aamamI
mm
mmaaa
a
mmtI
dt
dI
dt
dL
ϕµ
ϕµϕ
µϕ
trace of moment of inertia reduces to a constant
![Page 17: Last time · 2013. 12. 11. · (c)2013 van Putten 1967 L S Swift, HETE II: X-ray afterglows also to short GRBs 050509B, 050709,… BeppoSax GRB 970228: Z-0.695 GRBs CC-SNe GRB-SNe](https://reader034.vdocuments.mx/reader034/viewer/2022052006/6019d189ad65f92e646f8b01/html5/thumbnails/17.jpg)
(c)2013 van Putten
( )5
23
21246
3
3
3
3
5
32
5
32
5
1
a
mmaI
dt
dI
dt
dL T
jk
T
jkGW
µµ
+=Ω==
1-595
s erg106.3 ×=G
c
( )25
3/10
5
32~
5
32
≅Ω=M
m
a
MMLGW
δ
( )2/32/1
5/1
21
5/3
2
5/3
1 ,~ −≅Ω
+= aM
mm
mmM
Quadrupole emission from an inhomogeneity
Appreciable luminosities arise ONLY for
high-density matter formed in extreme
transient events, like CC-SNe and binary
mergers. The disk-mass (and its
inhomogeneities) in generic accretion
systems (e.g. GRS1915+105 are too small.
Making a wave in a high density disk or torus
Chirp mass
![Page 18: Last time · 2013. 12. 11. · (c)2013 van Putten 1967 L S Swift, HETE II: X-ray afterglows also to short GRBs 050509B, 050709,… BeppoSax GRB 970228: Z-0.695 GRBs CC-SNe GRB-SNe](https://reader034.vdocuments.mx/reader034/viewer/2022052006/6019d189ad65f92e646f8b01/html5/thumbnails/18.jpg)
(c)2013 van Putten
LF:
Inspiral of NS/NS
binaries, stellar mass
BHs
HF:
Emissions
from extreme
transient
emissions?
VHF:
Quasi-normal
mode ringing
of stellar
mass BH
burst-sources in
the local universe
HF Shot-noise
2/11 −−∝ lasern PLh
Present and Advanced sensitivities
![Page 19: Last time · 2013. 12. 11. · (c)2013 van Putten 1967 L S Swift, HETE II: X-ray afterglows also to short GRBs 050509B, 050709,… BeppoSax GRB 970228: Z-0.695 GRBs CC-SNe GRB-SNe](https://reader034.vdocuments.mx/reader034/viewer/2022052006/6019d189ad65f92e646f8b01/html5/thumbnails/19.jpg)
(c)2013 van Puttenvan Putten 2002
At a critical slenderness: m=1,2,... instability
Ma
mδb
mδ
m=2
Formation of mass-inhomogeneitiesVan Putten, 2002, ApJ, 575, L71
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(c)2013 van Putten
Negative chirp during black hole spindownVan Putten, 2008, ApJ, 684, L91
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(c)2013 van Putten
+
+
GW-modes
LIGO detector
Unraveling the
mysterious explosion
mechanism of CC-SNe
Observational strategy: CC-SN triggers from optical-radio surveys
![Page 22: Last time · 2013. 12. 11. · (c)2013 van Putten 1967 L S Swift, HETE II: X-ray afterglows also to short GRBs 050509B, 050709,… BeppoSax GRB 970228: Z-0.695 GRBs CC-SNe GRB-SNe](https://reader034.vdocuments.mx/reader034/viewer/2022052006/6019d189ad65f92e646f8b01/html5/thumbnails/22.jpg)
(c)2013 van Putten
+= −
M
Mmetf SunTtGW
chirp
10
2]2971.27029.0[)( 90/5.7
Chirp diagramVan Putten, 2009, MNRAS,
396, L81
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(c)2013 van Putten
Nearby CC-SNe –어디있어, 얼마나자주?
![Page 24: Last time · 2013. 12. 11. · (c)2013 van Putten 1967 L S Swift, HETE II: X-ray afterglows also to short GRBs 050509B, 050709,… BeppoSax GRB 970228: Z-0.695 GRBs CC-SNe GRB-SNe](https://reader034.vdocuments.mx/reader034/viewer/2022052006/6019d189ad65f92e646f8b01/html5/thumbnails/24.jpg)
(c)2013 van Putten
Challenge: initial sensitivity range ~ tens of Mpc
D = 200 Mpc, N ~ 10 million
D = 10 Mpc, N ~ few hundred
CfA redshift survey(SAO 1998)
GW detectors are “near sighted”
![Page 25: Last time · 2013. 12. 11. · (c)2013 van Putten 1967 L S Swift, HETE II: X-ray afterglows also to short GRBs 050509B, 050709,… BeppoSax GRB 970228: Z-0.695 GRBs CC-SNe GRB-SNe](https://reader034.vdocuments.mx/reader034/viewer/2022052006/6019d189ad65f92e646f8b01/html5/thumbnails/25.jpg)
(c)2013 van Putten
Van Putten, 2004, ApJ, 611, L81
Guetta & Della Valle 2007, ApJ, 657,
L73
In rare instances, CC-SNe produce GRBs
Branching ratio of SN Ib/c:
~ 0.2-4 %
Relative supernova rates:
SN Ia :SN II : SN Ib/c ~ 50:50:10
(depends on survey, e.g., 68:22:7 in PTF)
Remnants of CC-SNe: bubbles and shells
of the LMC (HI-Halpha)
Kim, S., et al. (1999,2003)
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(c)2013 van Putten
Local rates: Milky WayDiehl, R., et al., 2006,
Nature, 439, 45
CC-SNe rate: 1.9(+/- 1.1) events / century
1.809 MeV line
![Page 27: Last time · 2013. 12. 11. · (c)2013 van Putten 1967 L S Swift, HETE II: X-ray afterglows also to short GRBs 050509B, 050709,… BeppoSax GRB 970228: Z-0.695 GRBs CC-SNe GRB-SNe](https://reader034.vdocuments.mx/reader034/viewer/2022052006/6019d189ad65f92e646f8b01/html5/thumbnails/27.jpg)
(c)2013 van Putten
M51: farmland of CC-SNe at 8 Mpc
![Page 28: Last time · 2013. 12. 11. · (c)2013 van Putten 1967 L S Swift, HETE II: X-ray afterglows also to short GRBs 050509B, 050709,… BeppoSax GRB 970228: Z-0.695 GRBs CC-SNe GRB-SNe](https://reader034.vdocuments.mx/reader034/viewer/2022052006/6019d189ad65f92e646f8b01/html5/thumbnails/28.jpg)
(c)2013 van Putten
CC-SNe are potentially powerful broad band sources of GWs,
especially those associated with long GRBs. Nearby events are
relevant to LIGO-Virgo and KAGRA
Believed to be aspherical and radio-loud. Program for combined
optical-radio monitoring of nearby (e.g. KKT) galaxies.
Efficient surveys (maximal use of telescope time) would exploit correlations of CC-SN rates and galaxy properties, i.e., identify galaxies
with potentially high yields like M51 or M82
EM counterparts to GW searches Van Putten, 2001, Phys.
Rev. Lett., 87, 091101;
2004, JKAS, 45, S77; Van
Putten, et al., 2004, Phys.
Rev. D, 69, 044007; van
Putten, et al., 2011, Phys.
Rev. D, 83, 044046
![Page 29: Last time · 2013. 12. 11. · (c)2013 van Putten 1967 L S Swift, HETE II: X-ray afterglows also to short GRBs 050509B, 050709,… BeppoSax GRB 970228: Z-0.695 GRBs CC-SNe GRB-SNe](https://reader034.vdocuments.mx/reader034/viewer/2022052006/6019d189ad65f92e646f8b01/html5/thumbnails/29.jpg)
(c)2013 van Putten
Famaey, B., &
McGaugh, S.S., 2012,
Living ReviewsMissing mass problem
![Page 30: Last time · 2013. 12. 11. · (c)2013 van Putten 1967 L S Swift, HETE II: X-ray afterglows also to short GRBs 050509B, 050709,… BeppoSax GRB 970228: Z-0.695 GRBs CC-SNe GRB-SNe](https://reader034.vdocuments.mx/reader034/viewer/2022052006/6019d189ad65f92e646f8b01/html5/thumbnails/30.jpg)
(c)2013 van Putten
Galaxy rotation curves Famaey, B., &
McGaugh, S.S., 2012,
Living Reviews
![Page 31: Last time · 2013. 12. 11. · (c)2013 van Putten 1967 L S Swift, HETE II: X-ray afterglows also to short GRBs 050509B, 050709,… BeppoSax GRB 970228: Z-0.695 GRBs CC-SNe GRB-SNe](https://reader034.vdocuments.mx/reader034/viewer/2022052006/6019d189ad65f92e646f8b01/html5/thumbnails/31.jpg)
(c)2013 van Putten
Galaxy rotation curves Famaey, B., &
McGaugh, S.S., 2012,
Living Reviews
![Page 32: Last time · 2013. 12. 11. · (c)2013 van Putten 1967 L S Swift, HETE II: X-ray afterglows also to short GRBs 050509B, 050709,… BeppoSax GRB 970228: Z-0.695 GRBs CC-SNe GRB-SNe](https://reader034.vdocuments.mx/reader034/viewer/2022052006/6019d189ad65f92e646f8b01/html5/thumbnails/32.jpg)
(c)2013 van Putten
Characteristic acceleration in weak gravity Famaey, B., &
McGaugh, S.S., 2012,
Living Reviews
![Page 33: Last time · 2013. 12. 11. · (c)2013 van Putten 1967 L S Swift, HETE II: X-ray afterglows also to short GRBs 050509B, 050709,… BeppoSax GRB 970228: Z-0.695 GRBs CC-SNe GRB-SNe](https://reader034.vdocuments.mx/reader034/viewer/2022052006/6019d189ad65f92e646f8b01/html5/thumbnails/33.jpg)
(c)2013 van Putten
Scales in gravity Famaey, B., &
McGaugh, S.S., 2012,
Living Reviews
![Page 34: Last time · 2013. 12. 11. · (c)2013 van Putten 1967 L S Swift, HETE II: X-ray afterglows also to short GRBs 050509B, 050709,… BeppoSax GRB 970228: Z-0.695 GRBs CC-SNe GRB-SNe](https://reader034.vdocuments.mx/reader034/viewer/2022052006/6019d189ad65f92e646f8b01/html5/thumbnails/34.jpg)
(c)2013 van Putten
Famaey, B., &
McGaugh, S.S., 2012,
Living ReviewsCosmological baryon distribution
![Page 35: Last time · 2013. 12. 11. · (c)2013 van Putten 1967 L S Swift, HETE II: X-ray afterglows also to short GRBs 050509B, 050709,… BeppoSax GRB 970228: Z-0.695 GRBs CC-SNe GRB-SNe](https://reader034.vdocuments.mx/reader034/viewer/2022052006/6019d189ad65f92e646f8b01/html5/thumbnails/35.jpg)
(c)2013 van Putten
Lensing in curved spacetime
![Page 36: Last time · 2013. 12. 11. · (c)2013 van Putten 1967 L S Swift, HETE II: X-ray afterglows also to short GRBs 050509B, 050709,… BeppoSax GRB 970228: Z-0.695 GRBs CC-SNe GRB-SNe](https://reader034.vdocuments.mx/reader034/viewer/2022052006/6019d189ad65f92e646f8b01/html5/thumbnails/36.jpg)
(c)2013 van Putten
)sin(2
12
1 22222
1
22 ϕθθ ddrdrr
Mdt
r
Mds ++
−+
−−=−
gH RR 2=
ba
ab
ba
ab
a
a
a
a dxdxgdsuuguud
dx
d
dxL ===
= 22 ,ττ
Lensing in Schwarzschild line element
![Page 37: Last time · 2013. 12. 11. · (c)2013 van Putten 1967 L S Swift, HETE II: X-ray afterglows also to short GRBs 050509B, 050709,… BeppoSax GRB 970228: Z-0.695 GRBs CC-SNe GRB-SNe](https://reader034.vdocuments.mx/reader034/viewer/2022052006/6019d189ad65f92e646f8b01/html5/thumbnails/37.jpg)
(c)2013 van Putten
Aging is slower nearby the surface of the earth
than far out in space: gravitational redshift
Orbital precession (Mercury, PSR1913+16, 4U1820-30)
Bending of light by gravitation: gravitational lensing
Inner most stable circular orbit: critical inner radius of accretion disks
Circular photon-orbits
Event horizon
periastron
aphelion
Some phenomenology
![Page 38: Last time · 2013. 12. 11. · (c)2013 van Putten 1967 L S Swift, HETE II: X-ray afterglows also to short GRBs 050509B, 050709,… BeppoSax GRB 970228: Z-0.695 GRBs CC-SNe GRB-SNe](https://reader034.vdocuments.mx/reader034/viewer/2022052006/6019d189ad65f92e646f8b01/html5/thumbnails/38.jpg)
(c)2013 van Putten
Chandra X-ray image (false colors) with
radio (contour) (Chartas et al. 2001)
Discovery: Dennis
Walsh, Bob Carswell,and
Ray Weymann (1979)
Rhee, G., et al. (NASA)
Lensed AGN
![Page 39: Last time · 2013. 12. 11. · (c)2013 van Putten 1967 L S Swift, HETE II: X-ray afterglows also to short GRBs 050509B, 050709,… BeppoSax GRB 970228: Z-0.695 GRBs CC-SNe GRB-SNe](https://reader034.vdocuments.mx/reader034/viewer/2022052006/6019d189ad65f92e646f8b01/html5/thumbnails/39.jpg)
(c)2013 van Putten
4 images
![Page 40: Last time · 2013. 12. 11. · (c)2013 van Putten 1967 L S Swift, HETE II: X-ray afterglows also to short GRBs 050509B, 050709,… BeppoSax GRB 970228: Z-0.695 GRBs CC-SNe GRB-SNe](https://reader034.vdocuments.mx/reader034/viewer/2022052006/6019d189ad65f92e646f8b01/html5/thumbnails/40.jpg)
(c)2013 van Putten
( )
b
Mr
rOrrb
v
v
44][
)()(2~)2cos(2
1
2
3sin
)2cos(2
1
2
3
sin
serieson perturbatiRegular
2
1
0
==±∞=∆⇒
∞→++
++=⇒
+=
=
εϕ
ϕεϕϕεϕ
ϕ
ϕ
ϕb
Photon trajectories
10
2
2
)(3''
3''
1,
vvv
buvvvv
M
rb
M
εε
κκκ
κε
+=
==+
=+
==
ϕ∆
![Page 41: Last time · 2013. 12. 11. · (c)2013 van Putten 1967 L S Swift, HETE II: X-ray afterglows also to short GRBs 050509B, 050709,… BeppoSax GRB 970228: Z-0.695 GRBs CC-SNe GRB-SNe](https://reader034.vdocuments.mx/reader034/viewer/2022052006/6019d189ad65f92e646f8b01/html5/thumbnails/41.jpg)
(c)2013 van Putten
Young et al. 1980
Turner, Ostriker & Gott (1984)
Blandford & Narayan (1986,1992)
ξ
∧
αθ
αβ
ξα
M=
∧
Source-image plane Observer’s planeLensing object
ξ
dsDdD
sDLens equation: βαθ +=
αα sds DD =∧
Image (“mirage”)
source
observer
Lensed trajectory
Unlensed trajectory
Photon trajectories
![Page 42: Last time · 2013. 12. 11. · (c)2013 van Putten 1967 L S Swift, HETE II: X-ray afterglows also to short GRBs 050509B, 050709,… BeppoSax GRB 970228: Z-0.695 GRBs CC-SNe GRB-SNe](https://reader034.vdocuments.mx/reader034/viewer/2022052006/6019d189ad65f92e646f8b01/html5/thumbnails/42.jpg)
(c)2013 van Putten
arcsec-milliGpc110
32
)(
)(,~
411
2
1
,4
2/12/1
6
2
2
2
−
==
<<±
>>−
+±=
=
==−
D
M
M
D
R
D
DDD
DD
DM
sun
g
E
EE
EE
E
ds
sd
d
s
ds
θ
θβθ
θββθ
ββθ
βθ
θξβξ
θ (all vectors are are aligned)
one image outside,
one image insideEθ
Einstein ring
(source near axis)
Point mass
![Page 43: Last time · 2013. 12. 11. · (c)2013 van Putten 1967 L S Swift, HETE II: X-ray afterglows also to short GRBs 050509B, 050709,… BeppoSax GRB 970228: Z-0.695 GRBs CC-SNe GRB-SNe](https://reader034.vdocuments.mx/reader034/viewer/2022052006/6019d189ad65f92e646f8b01/html5/thumbnails/43.jpg)
(c)2013 van Putten
==
−−=
=−
ds
sd
ds
ds
D
DDD
D
M
etravel tim
DD
DM
θθψ
θψβθθτ
βθ
θ
ln4
)(
)()(2
1)( theis
4 of integral The
2
Find images in terms of local extrema of photon travel times
Fermat’s principle
![Page 44: Last time · 2013. 12. 11. · (c)2013 van Putten 1967 L S Swift, HETE II: X-ray afterglows also to short GRBs 050509B, 050709,… BeppoSax GRB 970228: Z-0.695 GRBs CC-SNe GRB-SNe](https://reader034.vdocuments.mx/reader034/viewer/2022052006/6019d189ad65f92e646f8b01/html5/thumbnails/44.jpg)
(c)2013 van Putten
Numerical example
![Page 45: Last time · 2013. 12. 11. · (c)2013 van Putten 1967 L S Swift, HETE II: X-ray afterglows also to short GRBs 050509B, 050709,… BeppoSax GRB 970228: Z-0.695 GRBs CC-SNe GRB-SNe](https://reader034.vdocuments.mx/reader034/viewer/2022052006/6019d189ad65f92e646f8b01/html5/thumbnails/45.jpg)
(c)2013 van Putten
αβ ire=Source positions
quadrupole mass-moment
(a binary of galaxies or a
model for an elliptical galaxy)
observer
Quadrupole lens
![Page 46: Last time · 2013. 12. 11. · (c)2013 van Putten 1967 L S Swift, HETE II: X-ray afterglows also to short GRBs 050509B, 050709,… BeppoSax GRB 970228: Z-0.695 GRBs CC-SNe GRB-SNe](https://reader034.vdocuments.mx/reader034/viewer/2022052006/6019d189ad65f92e646f8b01/html5/thumbnails/46.jpg)
(c)2013 van Putten
Multiple images
(not to scale)
![Page 47: Last time · 2013. 12. 11. · (c)2013 van Putten 1967 L S Swift, HETE II: X-ray afterglows also to short GRBs 050509B, 050709,… BeppoSax GRB 970228: Z-0.695 GRBs CC-SNe GRB-SNe](https://reader034.vdocuments.mx/reader034/viewer/2022052006/6019d189ad65f92e646f8b01/html5/thumbnails/47.jpg)
(c)2013 van Putten
0 1 2 3 4 5 6-16
-15.5
-15
-14.5
-14
-13.5
-13
-12.5
-12Merger and annihilation of a minimum and saddle point
τ(θ)
S1
L1
S2
L2
S1
angle along Einstein ring
β=0
β=0.1*eiπ/4
β=0.2*eiπ/4
β=0.3*eiπ/4
Merger of S and L in quadrupole lensing
![Page 48: Last time · 2013. 12. 11. · (c)2013 van Putten 1967 L S Swift, HETE II: X-ray afterglows also to short GRBs 050509B, 050709,… BeppoSax GRB 970228: Z-0.695 GRBs CC-SNe GRB-SNe](https://reader034.vdocuments.mx/reader034/viewer/2022052006/6019d189ad65f92e646f8b01/html5/thumbnails/48.jpg)
(c)2013 van Putten
Cre
ate
extrem
um
+ s
addle
poin
t
Split
an e
xtrem
um
into
tw
oS
X
X
X XS
Always create two new extra extrema (leaving a total of 2n+1)
On the boundary
Boundary reconnection
Burke 1981, Blandford &
Narayan 1986
ϕcos:Limacon bar += 2||||:Lemniscate ccpcp =+−
X (high,low), S out of change in topology
![Page 49: Last time · 2013. 12. 11. · (c)2013 van Putten 1967 L S Swift, HETE II: X-ray afterglows also to short GRBs 050509B, 050709,… BeppoSax GRB 970228: Z-0.695 GRBs CC-SNe GRB-SNe](https://reader034.vdocuments.mx/reader034/viewer/2022052006/6019d189ad65f92e646f8b01/html5/thumbnails/49.jpg)
(c)2013 van Putten
( )
inside) (const.
outside) (||ln2)(
inside) (0
outside) (2
''
1
'
1
'
'
'
1)('
)(Re'lnRe'ln)(
,2'
const.''ln4
''ln)'(4
)(
2
0
2
0
0
''
0
∴
=⇒
=
−−
=−
=
≡−=−=
====
+−=−=
∫∫
∫∫
∫∫
∫∫
==
z
zzzI
z
zzdz
zzzz
i
iz
dz
zzzf
zfdzzdzzzI
D
DDDDdDdsM
dzzD
Dd
D
D
ds
sdd
RD
d
Rz
s
ds
s
ds
d
ππ
ϕϕ
µπθµµ
ϕµ
θθθθµθψ
ππ
θ
)(
''
Rd
d
DD
dDRd
≡
= θϕ observer
=ring) inside(0
ring) outside(ln4
)(θθψ D
M
Lensing by a ring
![Page 50: Last time · 2013. 12. 11. · (c)2013 van Putten 1967 L S Swift, HETE II: X-ray afterglows also to short GRBs 050509B, 050709,… BeppoSax GRB 970228: Z-0.695 GRBs CC-SNe GRB-SNe](https://reader034.vdocuments.mx/reader034/viewer/2022052006/6019d189ad65f92e646f8b01/html5/thumbnails/50.jpg)
(c)2013 van Putten
>
<=
=
)()ln(4
)()ln(4
)(
22
22
d
dd
dd
D
MD
D
DM
θθθ
θθθθπµ
θψ
θπµ
'' θϕ dDRd d=
observer
Lensing by a uniform disk
![Page 51: Last time · 2013. 12. 11. · (c)2013 van Putten 1967 L S Swift, HETE II: X-ray afterglows also to short GRBs 050509B, 050709,… BeppoSax GRB 970228: Z-0.695 GRBs CC-SNe GRB-SNe](https://reader034.vdocuments.mx/reader034/viewer/2022052006/6019d189ad65f92e646f8b01/html5/thumbnails/51.jpg)
(c)2013 van Putten
Gpc10kpc5
25.1
disk of radius nalgravitatio4
1
4
1)4(
observer)-emitter(symmetry 1414
later) on this more-distancesangular (,4
,44
1
11
2
,
1**
222
−
−
=
=
===
−=⇔
−=
+==
===
sun
dd
dd
d
gd
ddsd
ds
dsd
d
dds
ddssd
ds
s
ds
sdEdenclE
M
MR
RR
RR
RDD
D
DD
D
DD
DDDDD
D
D
DDD
D
D
D
M
µπ
µπµπ
πµ
θπµθ
*
dD*
dsD
Minimal focusing distances
![Page 52: Last time · 2013. 12. 11. · (c)2013 van Putten 1967 L S Swift, HETE II: X-ray afterglows also to short GRBs 050509B, 050709,… BeppoSax GRB 970228: Z-0.695 GRBs CC-SNe GRB-SNe](https://reader034.vdocuments.mx/reader034/viewer/2022052006/6019d189ad65f92e646f8b01/html5/thumbnails/52.jpg)
(c)2013 van Putten
Elliptical galaxy modeled by two disks of uniform density
Oops! – too close cq. insuffient lensing mass
Einstein radius
M=0.1
Lensing by two elliptical galaxies
![Page 53: Last time · 2013. 12. 11. · (c)2013 van Putten 1967 L S Swift, HETE II: X-ray afterglows also to short GRBs 050509B, 050709,… BeppoSax GRB 970228: Z-0.695 GRBs CC-SNe GRB-SNe](https://reader034.vdocuments.mx/reader034/viewer/2022052006/6019d189ad65f92e646f8b01/html5/thumbnails/53.jpg)
(c)2013 van Putten
M=0.5 M=1
Lensing by two elliptical galaxies
![Page 54: Last time · 2013. 12. 11. · (c)2013 van Putten 1967 L S Swift, HETE II: X-ray afterglows also to short GRBs 050509B, 050709,… BeppoSax GRB 970228: Z-0.695 GRBs CC-SNe GRB-SNe](https://reader034.vdocuments.mx/reader034/viewer/2022052006/6019d189ad65f92e646f8b01/html5/thumbnails/54.jpg)
(c)2013 van Putten
H
M=4
L,S
LS are no more
Lensing by two elliptical galaxies
![Page 55: Last time · 2013. 12. 11. · (c)2013 van Putten 1967 L S Swift, HETE II: X-ray afterglows also to short GRBs 050509B, 050709,… BeppoSax GRB 970228: Z-0.695 GRBs CC-SNe GRB-SNe](https://reader034.vdocuments.mx/reader034/viewer/2022052006/6019d189ad65f92e646f8b01/html5/thumbnails/55.jpg)
(c)2013 van Putten
νθ ∝d
[ ]
[ ] .0det
:i.e.
thatprovided on,continuati l)l/numerica(analyticaby uniquely )( ofbranch aTrack
0)),((
Then
.0)),((
distance) source:..(parameter control aConsider
locations. source toresponsein extremum isolatedan ofy trajectorThe
0::),( of points Extremal
1
≠
−=
=+=
=
=∂=
−
θθ
θλθθ
θλθθθ
θ
θ
τ
ττλθ
λθ
τλθ
τλλθτλ
λλθτλ
τθτ
d
d
d
d
d
d
ge
GΛ
Confluence of images (isolated extrema)
![Page 56: Last time · 2013. 12. 11. · (c)2013 van Putten 1967 L S Swift, HETE II: X-ray afterglows also to short GRBs 050509B, 050709,… BeppoSax GRB 970228: Z-0.695 GRBs CC-SNe GRB-SNe](https://reader034.vdocuments.mx/reader034/viewer/2022052006/6019d189ad65f92e646f8b01/html5/thumbnails/56.jpg)
(c)2013 van Putten
[ ]
same) theremains imagescomplex ofnumber total(The
2."by jumps images real ofnumber thesurfaces, caustic a across"
:Precisely 2.by jumps produced images ofnumber thesurface, caustic a Across
sky. in the positions image possible for ),( positions source Solve
0det,0
equations 3 thesolving space source-3Din preimages are caustics The
θβ
ττ θθθ
allDds
==
One real root
Two complex
rootsThree real rootsTwo real roots
(one double)
Caustics: changing multiplicity of images
![Page 57: Last time · 2013. 12. 11. · (c)2013 van Putten 1967 L S Swift, HETE II: X-ray afterglows also to short GRBs 050509B, 050709,… BeppoSax GRB 970228: Z-0.695 GRBs CC-SNe GRB-SNe](https://reader034.vdocuments.mx/reader034/viewer/2022052006/6019d189ad65f92e646f8b01/html5/thumbnails/57.jpg)
(c)2013 van Putten
[ ]
.spanKer and spanIm definesIt
leavesreduction one-rank a andconfluent are images twoHere, .point a is
0det :confluent are images When two
⊥=∈=
=
=
ξτθξτ
ξξτ
τ
θθθθ
θθ
θθ
d
nbifurcatio
T
Trajectory of confluent images
Trajectory of source:
generator of a caustic surface
)(),( λλβ sdD
)(λθ
Trajectory of confluent images
![Page 58: Last time · 2013. 12. 11. · (c)2013 van Putten 1967 L S Swift, HETE II: X-ray afterglows also to short GRBs 050509B, 050709,… BeppoSax GRB 970228: Z-0.695 GRBs CC-SNe GRB-SNe](https://reader034.vdocuments.mx/reader034/viewer/2022052006/6019d189ad65f92e646f8b01/html5/thumbnails/58.jpg)
(c)2013 van Putten
( )
0:0
:0
)(2
1
1
2
=−=
−==
−−=
−λψτ
λψθβτ
θλψβθτ
θθθθ
θθ
HHH det)(tr4
1tr
2
1 21 −±=−λ
Travel time
Fermat’s principle
Eigenvalue problem
Hessian of potential
Away from lens
Apex
µµ
ψ θθ
2tr, == HH
HH
D
M
d
det4)(tr
0
2
2
=
=
=
µ
πµ
D
M
M
dM
4
'ln)(
=
−= ∫
λ
θθθψ
Hdet1 −±=−±λ
Generators of a caustic
![Page 59: Last time · 2013. 12. 11. · (c)2013 van Putten 1967 L S Swift, HETE II: X-ray afterglows also to short GRBs 050509B, 050709,… BeppoSax GRB 970228: Z-0.695 GRBs CC-SNe GRB-SNe](https://reader034.vdocuments.mx/reader034/viewer/2022052006/6019d189ad65f92e646f8b01/html5/thumbnails/59.jpg)
(c)2013 van Putten
.)det of isocurves are theseregion, source (Outside
.)( seigenvalue theof isocurves
: fixedat caustic a ofsection -Cross
0
H
Dds
λθλ =+
d
ddsd
DM
DDD
−=+=
+
−
λψλθβ θ
/4,1
:),( sDβθ →
Image of a constant distance cross section of a caustic
![Page 60: Last time · 2013. 12. 11. · (c)2013 van Putten 1967 L S Swift, HETE II: X-ray afterglows also to short GRBs 050509B, 050709,… BeppoSax GRB 970228: Z-0.695 GRBs CC-SNe GRB-SNe](https://reader034.vdocuments.mx/reader034/viewer/2022052006/6019d189ad65f92e646f8b01/html5/thumbnails/60.jpg)
(c)2013 van Putten
Caustic surfaceImages of constant Dds-caustics
Caustic surface of a quadrupole lens at (+/-p,0,0)
![Page 61: Last time · 2013. 12. 11. · (c)2013 van Putten 1967 L S Swift, HETE II: X-ray afterglows also to short GRBs 050509B, 050709,… BeppoSax GRB 970228: Z-0.695 GRBs CC-SNe GRB-SNe](https://reader034.vdocuments.mx/reader034/viewer/2022052006/6019d189ad65f92e646f8b01/html5/thumbnails/61.jpg)
(c)2013 van Putten
[ ] [ ]
[ ][ ]θθ
θθ
θβθθθβθθ
θβθθ
θ
ττ
βθ
τδβτδβττδθ
δβτδθτ
τ
det
1det
)1(
:0
:0
1
11
==∂∂
≡−=−=
=+
=
−
−−
Fermat’s principle
introduces the linearized relation
and hence an expression for the amplification (of brightness, expressed per unit
of opening angle) in terms of the determinant of the associated Jacobian
Amplification