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On the systematics in apparent proper motions of radio sources observed by VLBI
V. Raposo-Pulido, S. Lambert, N. Capitaine, T. Nilsson, R. Heinkelmann, H. Schuh


Motivation
With the improvement of the VLBI system and the increase of number of VLBI observations, it is possible to identify systematic effects on radio source positions. The current celestial reference frame only takes into account the positions of the radio sources. However, some radio sources show apparent proper motions, thus it is not possible to consider their positions stable in time. The estimation and analysis of the systematic effects can provide us the physical reasons of these variations and an alternative way to correct the positions. Such a study can be used to improve the next realization of the celestial reference frame: ICRF3.


The study
Estimate the systematics in apparent proper motions with Vievs. Compare results obtained with VieVS against results of Calc/Solve comparison using same approach used by Titov and Lambert (2013)

Approach applied by Titov and Lambert (2013) based on a the three-step procedure. The steps are: 1. Estimation of source coordinate time series from analysis of VLBI sessions. 2. Least square method to fit proper motions to the source coordinate time series. 3. Fit of spherical harmonics to the proper motions of the sources
T&L (2013) ­ Calc/Solve

Present study ­ VieVS


1. Radio source time series estimation

VieVS Time span: 1979-Dec/2013 Number of sessions: 4677

Calc/Solve Time span: 1979-Feb/2013 Number of sessions: 5632

With VieVS, sessions wich are not suitable for reliable EOP determination were excluded: http://lupus.gsfc.nasa.gov/files_IVS-AC/eop_exclusion.txt Sources with less than three observations were excluded from the analysis. Models following the IERS 2010 Conventions Cut-off elevation angle: 5є Terrestrial datum: NNR/NNT constraints to VTRF2008 stations without breaks (in total)


For the sources:
T&L(2011)
NNR Yes No s [mas] 2000 2000 Dip. ampl. [mas/yr] 6.4 ±1.5 6.5 ±1.5 a (є) 263 ±11 263 ±11 d (є) -20 ±12 -23 ±12

Constraining each source to the ICRF2 position using very loose constraints (s=2as) is equivalent to apply loose NNR constraint with the same weight (1/s2)
4 solutions are computed with VieVS: each source is tied to ICRF2 positions using: s=10-5 rad (~ 2000 mas) s=10-7 rad (~20 mas) s=10-6 rad (~ 200 mas) s=10-8 rad (~ 2 mas) and compared with the T&L (2013) solution (loose NNR constraint with s=10-5 rad)


2. Proper motions fitted to the time series
STEPS a. Sessions with RMS 100ps b. Sources (spec. handling excl.) c. Sources for veloc. (after first iterative process*) with >10 sessions 10-5 rad 4556 3288 545 10-6 rad 4557 3288 545 10-7 rad 4562 3187 545 10-8 rad 4555 3137 543 T&L(2013) 5583 3596 707

*1st Iterative process to remove outliers source i if |dacosdij - meandacosd| > T1*sd |ddij - meandd| > T1*sddij

acosdij

session j excluded for the source i

T1 = 90 ci2 = [(xij ­ mean_x)2 / sij2 ] 1 (T&L, 2013)


Comparison of proper motions depending on the constraints
Velocities 3000 constraint to the sources (s =10-5 rad)
(mas / y r)

Velocities 100 50
(mas / y r)
c os d

2000 1000

loose NNR constraint (s=10-5 rad)

0 -50 -100 -150 500 constraint to the sources (s=10-6 rad) loose NNR constraint (s=10-5 rad)

ma

c os d

0

-1000 500

1000

1500

2000

2500

3000

3500

4000

4500

ma

1000

1500

2000

2500

3000

3500

4000

4500

500

100 50

md (mas / y r)

0

md (mas / y r)

0 -50 -100 500

-500 500

1000

1500

2000

2500 3000 Number of sessions

3500

4000

4500

1000

1500

2000

2500 3000 Number of sessions

3500

4000

4500


Comparison of proper motions depending on the constraints
Velocities 60 40
(mas / y r)

Velocities 60 constraint to the sources (s=10-7 rad)
(mas / y r)

loose NNR constraint (s=10-5 rad)

40 20 0 -20 -40 500

constraint to the sources (s =10-8 rad) loose NNR constraint (s=10-5 rad)

20 0 -20 -40 500

c os d

ma

1000

1500

2000

2500

3000

3500

4000

4500

ma

c os d

1000

1500

2000

2500

3000

3500

4000

4500

40 20
md (mas / y r)

40 20
md (mas / y r)

0

0

-20 -40 500

-20 -40 500

1000

1500

2000

2500 3000 Number of sessions

3500

4000

4500

1000

1500

2000

2500 3000 Number of sessions

3500

4000

4500


3. Spherical harmonics fitted to the proper motion field
Dma cosd = - d1sina + d2cosa + r1cosasind + r2sinasind - r3cosd Dmd = - d1cosasind - d2sinasind + d3cosd - r1sina + r2cosa
(Dma cosd , Dmd) systematic part of the proper motion field (d1, d2, d3) electric harmonics of degree 1 acceleration of the SSB (r1, r2, r3) magnetic harmonics of degree 1 global rotation STEPS c. Sources for VSH* (after second iterative process**) 10-5 rad 320 10-6 rad 407 10-7 rad 388 10-8 rad 425 T&L(2013) 427

*VSH= vector of spherical harmonics

**2nd Iterative process to exclude unstable sources proper motion i if residual_veloci > T2*RMS(residuals) source i excluded T2 = 7 (T&L, 2013) from the VSH estimation


Vector of spherical harmonics for the approaches developed and the reference solution T&L 2013
VSH d1 [mas/yr] d2 [mas/yr] d3 [mas/yr] r1 [mas/yr] r2 [mas/yr] r3 [mas/yr] 10-5 rad 58.6 ±18.8 -63.4 ±18.6 -20.8 ±10.0 77.3 ±11.9 214.1 ±13.3 -1151.5 ±37.1 10-6 rad -0.2 ±1.9 -5.8 ±1.6 0.8 ±1.3 0.31 ±1.5 -2.4 ±1.8 -20.9 ±1.5 10-7 rad -0.6 ±0.6 -5.7 ±0.7 1.1 ±0.7 2.8 ±0.7 0.6 ±0.7 -2.0 ±0.5 10-8 rad 0.0 ±0.4 -4.5 ±0.4 0.8 ±0.4 2.5 ±0.4 0.4 ±0.4 0.6 ±0.3 T&L (2013) -0.4 ±0.7 -5.7 ±0.8 -2.8 ±0.9 -1.1 ±0.9

1.4 ±0.8
0.7 ±0.6

loose tight


Source distribution and ampitude of the dipole depending on the constraints
120 100
c ons traint to the s ourc es (s=10-5 rad) c ons traint to the s ourc es (s=10-6 rad) c ons traint to the s ourc es (s=10-7 rad) c ons traint to the s ourc es (s=10-8 rad)

7 6 5 4

Number of sources

80 60 40 20 0

loos e NNR c ons traint (s=10-5 rad)

Dipole Amplitude (mas/yr)

constraint to the sources
3

loose NNR constraint
2 1 0

-8

-90

-70

-50

-30

-10 10 d (є)

30

50

70

90

Log10 of the constraint (rad)

-7

-6

-5

Distribution of radio sources w.r.t. the declination

Dipole amplitude w.r.t. the constraint


Summary and conclusions
The number of suitable sources to estimate velocities was ~ 550 (out of ~3200 sources) in most of the cases the sources were observed in less than 11 sessions, most of them, VCS sources. Systematics in apparent proper motions were estimated with VieVS using different weights. This preliminary study aims to be the first step to create the final configuration for systematics estimation. The best agreement w.r.t T&L (2013) is when VieVS applies a constraint of 10-7 rad ~ 20mas. However, there are significant differences for the third component of the dipole (d3). These should be study in depth.
A comparison between different software and approaches for the estimation of very small effects, such as the galactic aberration effect, from VLBI observations are essential.


Future goals
Determine the most reliable results for galactic aberration and for gravitational wave systematics on the proper motions. Develop other approaches to estimate the systematic effects and compare them. Estimating the radio source velocities directly in a global solution with VieVS, and the estimating the vector spherical harmonics by a fit to the velocities. Estimating the vector spherical harmonics directly in the global solution.


Thank you for your attention
Acknowledgements:
· Travel funding: Local Organizing Committee ECORAS Project · This work were performed during a stay by Virginia RaposoPulido at the Paris Observatory, financed by Paris Observatory.