deflection of light induced by the sun gravity field and measured with geodetic vlbi oleg titov...

Deflection of light induced by the Sun gravity field and measured with geodetic VLBI

Oleg Titov (Geoscience Australia)

Anastasiia Girdiuk (Institute of Applied Astronomy, RAS)


Outline

1. Introduction – General Relativity in geodetic VLBI

2. Gravitational delay vs light deflection angle

3. Observational results

4. Extra deflection with VLBI

5. Conclusion

Introduction

1. Geodetic Very Long Baseline Interferometry (VLBI) measures the group delay with accuracy up to 30 ps;

2. General relativity is presented by the gravitational delay and a coordinate term in the geometric delay;

3. Gravitational delay caused by the Sun gravitational field is about 40 ns near the Sun and about 400 ps in opposite direction;

,ln

23 srr

srr

22

11

c

GMgrav

))((1

1

2

)(1)(

1)(

2

21

)(

2

222

2

2

2

wVs

sVVb

wVVsb

c

ccccrc

GM

c

geom


Introduction

1. Geodetic Very Long Baseline Interferometry (VLBI) measures the group delay with accuracy up to 30 ps;

2. General relativity is presented by the gravitational delay and a coordinate term in the geometric delay;

3. Gravitational delay caused by the Sun gravitational field is about 40 ns near the Sun and about 400 ps in opposite direction;

,ln

)1(3 srr

srr

22

11

c

GMgrav

))((

11

2

)(1)(

1)(

2

)1(1

)(

2

222

2

2

2

wVs

sVVb

wVVsb

c

ccccrc

GM

c

geom


Introduction

1. Effect of general relativity is split into two separate terms;

2. Gravitational delay is expressed as a function of the barycentric distances and . Geometric delay is a function of baseline length

3. Light deflection angle is not presented – no link to the optical astrometry.

,ln

)1(3 srr

srr

22

11

c

GMgrav

))((

11

2

)(1)(

1)(

2

)1(1

)(

2

222

2

2

2

wVs

sVVb

wVVsb

c

ccccrc

GM

c

geom


Introduction

,ln

)1(3 srr

srr

22

11

c

GMgrav

))((

11

2

)(1)(

1)(

2

)1(1

)(

2

222

2

2

2

wVs

sVVb

wVVsb

c

ccccrc

GM

c

geom

))((1

1

2

)(1)(

1)(

2

)1(1

)(

2

222

2

2

2

wVs

sVVb

wVVsb

c

ccccrc

GM

cgrav

group

))((1

1

...

))((1

1

...)1()(

ln)1(

22

23

wVswVs

sb

srr

srr

22

11

cc

rc

GM

cc

GM

GRgroup



Light deflection angle vs gravitational delay

Q – quasar, B – deflecting body (Sun)

b

𝑟2

𝑟1

Q

B

s



Q – quasar, B – deflecting body (Sun)

.)cos1(

coscos

2

1

)cos1(

)cos1(2

coscos222

2

3

r

bb

rr

bbb

c

GMgrav

Acossinsincoscoscos

22

2222

32

222

33 )cos1(

cossinsin

)cos1(

)coscos1(

cos1

cossinsin2

r

Ab

c

GM

r

b

c

GMA

rc

GMbGR


rc

GM

cc

GMcoordgravGR 23

)(2ln

2 sb

srr

srr

22

11

Difference between two models


22

2222

32

222

3

3

)cos1(

cossinsin

2

)1(

)cos1(

)coscos1(

2

)1(

cos1

cossinsin)1(

r

Ab

c

GM

r

b

c

GM

A

rc

GMbGR

cos1

sin)1(2

rc

GM

...sincos Ac

bGR

Deflection angle for arbitrary …

…now explicitly presented in the equationfor the VLBI group delay


,ln

)1(3 srr

srr

22

11

c

GMgrav

22

2222

32

222

3

33

)cos1(

cossinsin

2

)1(

)cos1(

)coscos1(

2

)1(

cos1

cossinsin)1(cos

)1(

r

Ab

c

GM

r

b

c

GM

A

rc

GMb

rc

GMbgrav

))((1

1

...)1()(

ln)1(

2

23

wVs

sbsrr

srr

22

11

c

rcGM

ccGM

group




Observational results

1. Geodetic VLBI data from 1991-2001

2. Gravitational delay and coordinate term in geometric delay were switched off for a set of selected radio sources

3. Equatorial coordinates of the selected radio sources were estimated as daily parameters for each 24-hour VLBI sessions

4. The estimated coordinates are converted to the deflection angle and PPN parameter .


Light deflection angle 1803+784


Light deflection angle 1606+106


Light deflection angle 1334-127

Minimum elongation 2.6


Light deflection angle vs elongation 1606+106

cos1

sin22

rc

GM

1606+106

deg

20 40 60 80 100 120 140 160

arcsec

-0.005

0.000

0.005

0.010

0.015

0.020

0.025



1606+106 (1991-2001)

Elongation (deg)

30 40 50 60 70 80 90 100 110 120

-1.0

-0.5

0.0

0.5

1.0

cos1

sin)1(2

rc

GM

sin

cos12 GM

rc



1606+106 (1991-2001)

Elongation (deg)

30 40 50 60

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

cos1

sin)1(2

rc

GM

sin

cos12 GM

rc



0229+131 (1991-2001)

Elongation (deg)

0 20 40 60 80 100 120 140

-0.4

-0.2

0.0

0.2

0.4

cos1

sin)1(2

rc

GM

sin

cos12 GM

rc



0229+131 (1991-2001)

Elongation (deg)

0 2 4 6 8 10 12 14

-0.10

-0.05

0.00

0.05

0.10

cos1

sin)1(2

rc

GM

sin

cos12 GM

rc

New formula: benefits

sincos Ac

bGR

1. Simple;

2. Consistent with the geometric part of group delay model;

3. Linked to the deflection angle directly;

4. Free off coordinate terms;

5. Partial derivate for is optimal;

6. Allows more flexibility;

7. Looking good as an alternative option for IERS Conventions


cos1

sin2

22

rc

GM


Outreach – “When GR works?”

Optical VLBI

During Solar eclipse Any time (365 days, 24/7)

Near the Sun Any elongation angle

Optical instrument Radio telescopes

Nice sunny day Any weather condition

~1”.75 0”.001 1”

Observations

If all the general relativity effects are switched off the VLBI group delay model,

the light deflection angle will be observed at any moment and at any

elongation of a radio sources to the Sun (the closer, the better).

The old good time (before 2002) when the radio sources were observed

close to the Sun (as close as 1.5). The PPN parameter was estimated

very accurately (0.001) from a standard 24-hour session, if the number of

delays is 100.

The accuracy would be about 0.0001, were a dedicated session for

observation of a strong radio source within 3 arranged.


The “minor” terms ( = 1)

The minor terms are not negligible in the small angle approximation ( 0)

AcR

b

Rc

GMA

c

b

Rc

GMGR 2cossin

2sincos

4 22

22

22

2222

32

222

3

3

)cos1(

cossinsin

2

)1(

)cos1(

)coscos1(

2

)1(

cos1

cossinsin)1(

r

Ab

c

GM

r

b

c

GM

A

rc

GMbGR

A

R

bA

c

bGR 2cossin

2cossin


The “minor” terms


Ac

bA

R

bA

c

bGR cossin2cossin

2cossin 2

''cos

2cossin

212

A

A

R

b

The “minor” terms for the Sun

For geodetic VLBI observation of the radio sources near the Solar limb The additional deflection angle is individual for all baselines

R (km) b=3000 km b=10000 km

700,000 (grazing light, 0.25) 1”.75 0”.0038 0.”0125

(1) 0”.45 0”.00024 0”.00078

(2) 0”.22 0”.00006 0”.00019

(5) 0”.09 0”.00001 0”.00003

''


''cos

2cossin

212

A

A

R

b

The “minor” terms for Jupiter

For geodetic VLBI observation of the radio sources near Jupiter

R (km) b=3000 km b=10000 km

70,000 (grazing light, 20”) 0”.016 0”.0004 0.”0011

210,000 (1’) 0”.0053 0”.00013 0”.00038

420,000 (2’) 0”.0026 0”.00006 0”.000182

=70.000 kmAngular size

016".042

Jup

Jup

Rc

GMFor grazing light

Mass g

''


The “minor” terms

The additional deflection angle is individual for all baselines and could

be detected with observations within 2 from the Sun and 2’ from Jupiter

22

2''

Rc

GMb

Total magnitude

22

4''

Rc

GMb


''cos

2cossin

212

A

A

R

b


Additional angle and its geometric interpretation

Q – apparent position;

Q’ – true position;

B – deflecting body,

b

𝑟2

𝑟1

Q

B

Q’

121

4

Rc

GM

222

4

Rc

GM

1R

2R

2

1

21 RR

21

2212

4

R

b

c

GM


Additional angle may be observed

Q – apparent position;

Q’ – true position;

B – deflecting body,

Q

Q’

B

B

Rc

GM2

4

b

R

R

Traditional deflection angle (unobservable)

Additional deflection angle (observable), if b/R ~ 0.1ARc

GMbcossin

2''

22


Close alignment of star and radio source 0758-737

Separation is less than 2”

0758-737


Conclusion

1. General relativity effects in VLBI could be expressed in term of the light deflection angle ;

2. The light deflection angle could be estimated for an arbitrary elongation from the Sun with high precision;

3. The equation for light deflection at arbitrary elongation was tested using a set of VLBI data;

4. The PPN parameter was estimated for several 24-hour sessions with accuracy 0.001.

5. Additional deflection angle could be observed for a close approach to the Sun or Jupiter;

6. A dedicated observational programs for detection of the additional angle are required.

Phone: +61 2 6249 9111

Web: www.ga.gov.au

Email: [email protected]

Address: Cnr Jerrabomberra Avenue and Hindmarsh Drive, Symonston ACT 2609

Postal Address: GPO Box 378, Canberra ACT 2601

Any Questions?

Thank you for your attention

deflection of light induced by the sun gravity field and measured with geodetic vlbi oleg titov...

Documents

geodetic vlbi slide

gravity field

gravitational field

geodetic vlbi difference

geodetic vlbi data

extra deflection

geometric delay

ras slide