1. Error budget for the PRIMA conceptual design review Bob Tubbs, Richard Mathar, Murakawa Koji, Rudolf Le Poole, Jeff Meisner and Eric Bakker Leiden University and ASTRON

2. Executive summary Analysis work undertaken indicates that the design and operation of VLTI/PRIMA will have to be modified/improved in order to reach the goal of 10 μas accuracy. Key areas are: Main delay line and VCM performance Fringe tracking with wavefront corrugations Dependence of spectral response on wavefront corrugations across AT apertures Systematic gradients above Paranal

3. Executive summary Turbulence in the ducts and tunnels Accurate model for the refractive index of air Measurements of the difference in the colour of the correlated flux from each star Solutions to each of these problems must be found and tested to ensure that the 10 μas astrometric performance can be reached (in some cases solutions have already been suggested)

4. Executive summary The astrometric accuracy will be determined by the accuracy with which errors can be compensated – it will thus depend on how well the VLTI can be modelled, how accurately the refractive index of air can be calculated, how accurately the centroid wavelength of the observing band can be measured, etc.

5. Contents Interferometry introduction The need for an error budget Summary of principle contributions Error terms which must be reduced Potential problems Conclusions

6. Interferometry Interferometers measure the correlated flux (a complex number) from one or more sources on one or more baselines The phase of the measured correlated fluxes is corrupted by atmospheric fluctuations These correlated fluxes are usually normalised according to the source brightness, giving visibilities

7. Narrow angle interferometry If two stars lie within the isoplanatic separation angle then the correlated fluxes for the two stars are correlated with each other (the phases vary in harmony with each other) The phase of the cross-correlation is called the astrometric phase, and provides a measure of the separation between the stars Atmospheric errors can be eliminated by averaging this cross-correlation with time, if the stars are within one isoplanatic angle

8. Narrow angle interferometry

9. Coherent integration An additional benefit of narrow angle interferometry is that the phase of the correlated flux from a bright star can be tracked and used to correct fluctuations in the phase of a nearby fainter star This can allow long coherent integrations on the faint star, although it requires a bright primary star At K-band the improvement in limiting magnitude for the faint star is moderate (thermal background limitations)

10. Need for error budget The target of 10 μas accuracy is very challenging (requiring a total differential OPD accuracy of 10 nm, with individual contributions much smaller than this) For most of the path through the VLTI, the beams from the different stars are separated, passing through different air and reflecting off different mirrors

11. Principle contributions The principle contributions can be separated into random, zero-mean effects and systematic effects The zero-mean random terms produce requirements on the integration times and stability of the instrument between repeated measurements The systematic effects can only be eliminated with a good understanding of the instrument and again through good instrument stability

12. Zero-mean random terms 1 st order atmospheric, dome and tunnel seeing Photon shot noise, thermal background and readout noise Signal loss due to phase and group delay tracking errors Vibrations (zero mean to 1 st order) Polarisation effects (to 1 st order)

13. Zero-mean random terms Apart from VLTI internal seeing, the zero- mean random components of the OPD error will average out to the 10 nm level in 30-120 mins of observation, depending on the seeing and angular separation VLTI internal seeing is more problematic as it is applied separately to the two beams from each AT Measurements indicate that drifts of 1000s of nm occur on the timescales of beam switching

14. Eliminating systematic terms Many of the systematic terms will be reduced using careful calibration procedures such as: Regularly swapping the stellar beams using the AT derotator to eliminate systematic differences between the two beam paths after the derotator Monitoring the spectrum of the correlated flux from each star by using the FSUs as Fourier- transform spectrometers

15. Eliminating systematic terms Additional checks on residual systematic terms can be performed such as: Splitting the light from a single star in the image plane so that half of the light passes down the PS beam and half passes down the SeS beam (StS calibration mode) Repeated measurements of well known binary systems to check the system performance

16. Remaining systematic terms The remaining systematic terms generally come from 2 nd order effects or from combinations of multiple error terms, e.g.: The combined effects of stellar colour and differential dispersion in the main delay line The combined effects of the differential offset of the beam footprints on the mirrors before the derotator and figuring errors in these mirrors

17. Systematic terms The dependence of the spectral sensitivity of the FSUs on the seeing and STRAP performance (due to spatial filtering effects) The difference in the atmospheric refraction along the paths to the two different stars 2 nd order effects on the phase measured at the FSU from seeing, photon shot noise and polarisation effects

18. Stellar colour and dispersion If the correlated fluxes from the stars have different colours, then the differential OPD depends on the position of the main delay line (MDL) (Richard Mathar will discuss this) The colour of the correlated flux must be well known (the centroid wavelength for the observations must be known to ~0.2nm – either using the FSUs as Fourier transform spectrometers, or by measuring the stellar SEDs accurately and knowing the PRIMA instrument response very well)

19. Beam walk before de-rotator This will only be significant on M4, which is very close to an image plane and before the de- rotator The beams from the two stars will reflect off different parts of this mirror, so that figuring errors at the 5 nm level will cause significant errors in the astrometry It may be necessary to map the figuring errors in the AT M4 mirrors at the nm level

20. Spectral sensitivity of FSUs Image plane obstructions in the VLTI light beams make spectral throughput of the VLTI depend on the seeing, the atmospheric refraction and the STRAP performance Image plane obstructions include the FSU spatial filters and the star separator roof mirror when operating in StS calibration mode The spectral throughput must be accurately known for atmospheric dispersion corrections

21. Spectral sensitivity of FSUs The StS roof mirror is the easiest component to describe here: It acts as a Schlieren detector (knife edge) – (Schlieren is a trick used by optics manufacturers to convert wavefront phase perturbations across the aperture into wavefront amplitude perturbations across the aperture, making them visible to the eye) The effect on the astrometric phase is also quite complicated and needs modelling

22. Spectral sensitivity of FSUs In order to investigate the effect of the StS roof mirror, a simple simulation was undertaken

23. Spectral sensitivity of FSUs Adding atmospheric refraction

24. Spectral sensitivity of FSUs Spatial filtering

25. Spectral sensitivity of FSUs The FSU spatial filter causes the spectral sensitivity to vary with the seeing and STRAP performance, and causing the centroid wavelength to fluctuate by ~5nm RMS on short timescales The roof mirror causes the difference in spectral sensitivity to vary with the amount of atmospheric refraction when in StS calibration mode, producing a shift of up to 10 nm in the centroid wavelength of the observations (strongly dependent on the seeing)

26. Other terms The difference in atmospheric refraction along the different beams will be discussed by Richard Mathar 2 nd order effects on the phases measured will require further information about the FSU performance Additional terms are discussed in the error budget document

27. Potential problems for PRIMA Potential problems which may prevent PRIMA from operating have been attached to the error budget workpackage. These include: Difficulty in operating the VCMs due to problems with the delay line supports (also discussed by Rudolf Le Poole) Refractive index fluctuations in the delay line tunnel due to airflow (discussed by Rudolf Le Poole)

28. The MDL Tunnel is built from 20m sections Deformation of MDL tunnel

31. Currently this deformation prevents use of the VCMs The VCMs are essential to PRIMA astrometry (otherwise the beam walk on mirrors becomes large, leading to large OPD variations, and the pupil is not re-imaged in front of the DDL) This will be discussed in detail by Rudolf

32. Refractive index fluctuations Airflow in tunnels will be discussed by Rudolf

33. Fringe tracking problems Numerical simulations indicate that fringe tracking may be unreliable with large apertures using the FSU design envisaged The principle problem relates to the break-up of the stellar images into speckles Each speckle has a different optical phase, and the speckle patterns are different in the different group delay tracking spectral channels

34. Phase in the image plane

35. Fringe tracking problems Even for an AT-size aperture, it may be necessary to average the group delay over many atmospheric coherence times in order to get a “sensible” number

36. Fringe tracking problems The high frequency fringe motion is dominated by the effects of the image breaking up into speckles

37. Fringe tracking The fringe tracking performance will be dominated by effects which can only be studied using numerical simulations It will be essential to incorporate the wavefront corrugations across the AT apertures in any modelling of PRIMA performance

38. Conclusions A summary of some of the key terms in the error budget has been presented There are several potential problems which could halt the PRIMA astrometry project A number of areas require more detailed analysis, to determine whether error terms can be adequately compensated or not Numerical simulations will be required in order to estimate the fringe-tracking performance of PRIMA