Download Gravitational Bending of Light and Its Measurement with

Практические приложения
фундаментальной теории
Д.ф.-м.н. Сергей М. Копейкин
Университет штата Миссури, США
1-я школа по астрометрии, Москва,
Октябрь 22-26, 2007
1
SIM
SIM PlanetQuest is designed as a space-based 9-m
baseline optical Michelson interferometer operating in the
visible waveband. This mission will open up many
areas of astrophysics, via astrometry with unprecedented
accuracy. Over a narrow field of view (1°), SIM aims to
achieve an accuracy of 1 µas in a single measurement.
1-я школа по астрометрии, Москва,
Октябрь 22-26, 2007
2
OBSS
OBSS is an astrometric satellite designed to determine with unprecedented accuracy the positions,
distances, and motions of a billion stars within our galaxy. It is a collaborative effort between the
U.S. Naval Observatory and several other institutions. OBSS will measure stellar positions to less
than 10 microarcseconds. (= the width of a typical strand of human hair from a distance of 650900 miles.)
1-я школа по астрометрии, Москва,
Октябрь 22-26, 2007
3
GAIA
Gaia will scan the sky continuously
according to a pre-defined pattern .
The satellite will rotate around its
spin axis at a rate of 60 arcsec/s,
equivalent to a spin period of 6
hours. The spin axis itself precesses
at a fixed angle of 45 degrees to the
Sun. The line of sight of the two
astrometric instruments are
separated by the 'basic angle', which
is 106.5 degrees.
During its operational lifetime, Gaia
will continuously scan the sky, roughly
along great circles, according to a
carefully selected pre-defined scanning
law. The characteristics of this law,
combined with the across-scan
dimension of the astrometric fields of
view, result in the above pattern for
the distribution of the predicted
number of transits on the sky in
ecliptic coordinates. The fixed solar
aspect angle (45 degrees) , i.e., the
angle between the Sun and Gaia's
spin axis, favours observations of stars
around ecliptic latitudes plus and
minus 90 - 45 = 45 degrees.
1-я школа по астрометрии, Москва,
Октябрь 22-26, 2007
4
JASMINE
JASMINE is abbreviation of the position astronomical satellite plan (Japan
Astrometry Satellite Mission for INfrared Exploration). It will surwey the
Milky Way and its buldge in the infrared band around 1 milli-micron, measure
positions, distances, and proper motion of several hundred million stars at high
accuracy approaching 10 microarcsecond.
1-я школа по астрометрии, Москва,
Октябрь 22-26, 2007
5
Square Kilometer Array (SKA)
The SKA will be an interferometric array of individual antenna stations, synthesizing an aperture
with diameter of up to several 1000 kilometers. The SKA is a new generation radio telescope that
will be 100 times as sensitive as the best present-day instruments. It will unlock information from
the very early Universe and, using novel capabilities, be able to undertake entirely new classes of
observation including VLBI with a microarcsecond resolution.
1-я школа по астрометрии, Москва,
Октябрь 22-26, 2007
6
VERA
VERA (VLBI
Exploration of
Radio
Astrometry) is
the first VLBI
array dedicated
to phasereferencing
astrometry.
S269 (Sharpless 269) is a massive star forming region toward
constellation Orion. VERA has successfully measured its
trigonometric parallax of 189 +/- 8 micro-arcsecond. This is the
smallest parallax ever measured, corresponding to a source distance
to 17,250 light year.
1-я школа по астрометрии, Москва,
Октябрь 22-26, 2007
7
Gravitational Light Deflection by
Multipoles in Cosmology
Object:
Abell 2218
Type:
Giant Arc
Image credits:
J.-P. Kneib, R.S. Ellis, I. Smail, W.J.
Couch, R.M. Sharples
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Октябрь 22-26, 2007
8
Gravitational Light-Ray Deflection by
Planetary Multipoles
•
•
•
•
Gravitational Field Model
Propagation of Light Model
Deflection Patterns
Interpretation of Observations and
Gravitational Physics
1-я школа по астрометрии, Москва,
Октябрь 22-26, 2007
9
Gravitational Field Model
1. Linearized general relativity
g    h
2. The harmonic gauge
 h

1 
   h  0
2
3. The gravity field equation (c = 1 units)
2
 2
  2   h  0
 t

1-я школа по астрометрии, Москва,
Октябрь 22-26, 2007
10
The metric, the waves, and the multipoles
2M
  2 I i (s) 
2
h00 
 i
 i j
r
x  r  x x
 I ij  ( s ) 

  ...
 r 

i
  ij  
4 I (s)   2 I (s) 
h0i  
 j
 ...

r
x 
r



2 I ij  ( s )
hij   ij h00 
 ...
r
the retarded time:
I i (s)  MxPi (s)
s t r
I ij (s)  MxPi (s) xPj  (s)  J ij (s)
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11
The Light-ray Perturbations
The light-ray geodesic
dK 

 
K  K  0
d
The wave vector decomposition
dx
K 
 k   
d



1   h h h
       
2
x
x
 x



The Christoffel symbols

The unperturbed equation
of light ray
dk 
0
d
The perturbed equation
of light ray
d 
1

  h k      h k  k 
d
2
x


1-я школа по астрометрии, Москва,
Октябрь 22-26, 2007


12
The Unperturbed Light-ray Trajectory
x Ni ( )  k i   i
r   2 d2
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13
The light-ray deflection angle
d 2 x i 1   h d  
1 i
1 i j p 
 k k

 k hi  k h00  k k k h jp 
2
i
d
2

d 
2
2



dx j
    k kj
  i
d
i
i
j
i
i
 i   Sun
  Mi   Di   Qi
 Mi 
4M
1   ni
d
j



4 I (s) i j
4k I j ( s ) i
i
j
 
n n m m 
n
d2
d
i
D
j
4 I  jp ( s ) i j p
i
j
p
i
j p
i
p j
 
n
n
n

n
m
m

m
m
n

m
m
n
d3
i
Q

Time argument is the retarded time: s = t - r

The slowly evolving "Coulomb component"
of the gravitational field can not transfer
information about position of the source of the
gravitational field with the speed (of gravity)
faster than the speed of light.
1-я школа по астрометрии, Москва,
Октябрь 22-26, 2007
14
The Minkowski diagram of the interaction of gravity and light
Observer
Star’s world line
Future gravity null cone
Observer
Future gravity null cone
Future gravity null cone
Future gravity null cone
Future gravity null cone
Planet’s world line
1-я школа по астрометрии, Москва,
Октябрь 22-26, 2007
Observer’s
world line
15
The bi-characteristic interaction of gravity and
light in general relativity
Any type of gravitational field obeys the principle of causality, so that even the slowly
evolving "Coulomb component" of the gravitational field can not transfer information
about position of the source of the gravitational field with the speed faster than the
speed of light.
1-я школа по астрометрии, Москва,
Октябрь 22-26, 2007
16
Retardation of gravitational field in a light-ray deflection
experiment
Observer and planet are at rest
Planet moves uniformly relative to observer
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17
The deflection equations and the central inverse mapping
 M   1  cos  n
D 

L
( z  n) n  ( z  m) m
d







R2
 Q  J 2 2 ( s  n) 2  ( s  m) 2 n  2( s  n)( s  m)m
d
L2
  2 ( z  n) 2  ( z  m) 2 n  2( z  n)( z  m)m
d
   limb

R
d
 limb  4 1  k  v P
 MR
1-я школа по астрометрии, Москва,
Октябрь 22-26, 2007
18
Snapshot deflection patterns
Monopole
Dipole
Quadrupole
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19
Dynamic deflection patterns
Circle
  2r cos 
r
2M
X0
March 21, 1988
Treuhaft & Lowe
DSN JPL NASA
Cardioid
Caley’s sextic
  p1  cos 2 
  qL cos 3  3 cos  
L
pr
X0
r L 

qL  
2  X 0 
September 8, 2002
Fomalont & Kopeikin
VLBA+MPfRA
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2
Not measured as yet
SIM? SKA? Gaia?
JASMINE? VERA?
20
Measuring the dipolar deflection of light.
The «speed of gravity» experiment.
Position of Jupiter taken from
the JPL ephemerides
Position of Jupiter
determined from the
gravitational deflection
of light by Jupiter
10 microarcseconds = the width of a
typical strand of a human hair from a
distance of 650 miles.
Measured with 20% of accuracy, thus, proving a
direct evidence that the null cone is a bi-characteristic
hypersurface (speed of gravity = speed of light)
1-я школа по астрометрии, Москва,
Октябрь 22-26, 2007
21
Gravitational deflection maps gravity field
Optical position of Jupiter
Gravitational position of Jupiter
measured from the light deflection of stars
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Октябрь 22-26, 2007
22
The «Speed of Gravity» Experiment
The Team
Edward Fomalont (NRAO)
(observation + data processing)
Sergei Kopeikin (UMC)
1-я школа по астрометрии, Москва,
(theory + interpretation)Октябрь 22-26, 2007
23
The Very Long Baseline Array
Mauna Kea
Hawaii
Owens Valley
California
Brewster
Washington
North Liberty
Iowa
Hancock
New Hampshire
Effelsberg, Germany
Green Bank,
Virginia
24
Kitt Peak
Arizona
Pie Town
New Mexico
Fort Davis
Texas
Los Alamos
New Mexico
St. Croix
Virgin Islands
Basic Interferometry
in about one minute for
‘sufficiently strong’
radio sources (S/N>1)
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Октябрь 22-26, 2007
25
VLBI Limitations to Positional Accuracy
• Location of Radio Telescope
Position on earth (~ 1 cm)
Earth Rotation and orientation (~ 5 cm)
• Time synchronization (~ 50 psec)
• Array stability (~ 5 cm)
• Propagation in troposphere and ionosphere
Very variable in time and space (~ 5 cm in 10 min)
CONVERSION FACTORS:
1 cm = 30 psec = 300 microarcsec
0.03cm = 1 psec = 10 microarcsec
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26
The Experiment
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27
Motion of Jupiter
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28
Source Structure Stability Over Experiment
Source Stability
2 mas ticks
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29
Effect of Troposphere
Two calibrators – phase-referencing technique.
Factor
of 3 increase
in wave
accuracy
using
2accuracy
calibrators
Reconstruction
of
the
front
with
1-я школа по астрометрии, Москва,
Октябрь
2007 hair at 500 miles !
10 microarcseconds
– a22-26,
human
30
Measured Delays for Each Source
r
e
s
i
d
u
a
l
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Октябрь 22-26, 2007
31
Residual Delays for J0842 Compared on Several
Days for a Few Baselines
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32
Jupiter’s retarded position from the gravitational time delay (green points)
versus Jupiter’s retarded position from JPL ephemeris (magenta)
magnetosphere
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33
Results of Experiment
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34
Detection of Ultra-Long Gravitational Waves via
Astrometry
Sergei M. Kopeikin and Valeri V. Makarov
Is the large-scale structure of spacetime
shaped by relic gravitational waves?
&
How to detect them ?
1-я школа по астрометрии, Москва,
Октябрь 22-26, 2007
35
Short and Ultra-Long Gravitational Waves
Orbiting binary stars, supernova explosions, coalescence of binary neutron stars
and encounters of stars in dense clusters are thought to generate ripples in
spacetime propagating far from the source (~ 1/r law). They cause tensor-type 2
transverse deflection of light rays from background sources (quasars, masers,
etc.). As a rule, General Relativity predicts negligibly small astrometric effects
from these sources (Kopeikin et al, Phys. Rev. D, 1999, 2001, 2002).
Plane long-period cosmological waves, on the other hand, may be detectable at
the (sub) μas level of accuracy via specific patterns of proper motions of light
sources, expressed mostly by vector spherical harmonics of a second order.
m
   S1 

1
m  1
secular aberration
m
T1 
1
m  1
residual rotation
m
 S2 
2
m  2
m
 T2  ...
2
m  2
gravitational waves
(quadrupole modes)
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Октябрь 22-26, 2007
36
Patterns of apparent proper motions of quasars
Simulated proper motion field of
distant quasars induced by a set
of plane cosmological waves.
Gravitational waves of different
phase and orientation superpose
to make up a complicated vector
field pattern mostly represented
by several vector spherical
harmonics of second order. The
amplitude of the effect depends
on the dimensionless strain of the
GW packet. Such a pattern, if it
exists, could be determined by
precision global astrometry at a
high signal-to-noise ratio if the
astrometric grid includes > 100
quasars.
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37
Differential Wide-Angle Astrometry of Quasars
Detection of the apparent motion of a very distant light source
induced by the propagating long wave can be done with an
astrometric instrument capable of (sub) microarcsecond
relative measurements over wide angles (~ 90°)
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38
Tentative design
Two Michelson interferometers
with articulating siderostats and
mutually orthogonal baseline
vectors, plus two guide
interferometers to monitor attitude
changes by locking on guide stars,
tied into a rigid external metrology
system
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39
Triangulation grid of quasars
• 6 quasars separated by ~ 90 degrees
make a simple celestial triangulation grid
• >100 quasars needed to investigate the
global pattern of gravitational waves
• Since only relative proper motions are
required, a set of independent triangulation
sextuplets is sufficient
• Proper motion amplitude proportional to
frequency ω and dimensionless stress h
• Estimate: primordial gravitational
waves of h = 3·10-9 at ω = 10 -2 yr -1 or of
h = 3·10 -11 at ω = 1 yr -1 can be detected
by the global astrometric triangulation
• Conclusions:
- alternative detection of GW;
- the method is feasible;
- deeper study is required.
1-я школа по астрометрии, Москва,
Октябрь 22-26, 2007
40