An introduction to closure phases Michelson Summer Workshop Frontiers of Interferometry: Stars, disks, terrestrial planetsCrawl, walk, runOutlineOutlineStarting pointsThe atmosphereWhat this doesModelling this antenna-based corruptionWhat would pt. source visibility phases look like? Key ideas 1Quiz 1OutlineDefinition of the closure phase (i)Definition of closure phase (ii)Example of closure phasesA better representationWhat to do with multiple measurements?What’s the big deal?Key ideas 2Quiz 2OutlineWhat does a closure phase measure?A simple example (i): star and spotActual data for  Her.A simple example (ii): binary starA simple example (ii): binary starEqual binary – 1:1Almost equal binary – 1.05:1Unequal binary – 10:1What can we expect for more complex targets?Example 1 – YSO MWC 349Example 2 - YSO LkH101Key ideas 3Quiz 3Imaging with closure phasesSelf-calibrationImpact of closure phase dataKey ideas 4SummaryAn introduction to closure phasesMichelson Summer WorkshopFrontiers of Interferometry: Stars, disks, terrestrial planetsPasadena, USA,July 24th-28th2006C.A.HaniffAstrophysics Group, Department of Physics,University of Cambridge, UK25thJuly 2006Michelson Summer Workshop 225thJuly 2006C.A.Haniff – Closure phasesCrawl, walk, runCrawl, walk, runMichelson Summer Workshop 325thJuly 2006C.A.Haniff – Closure phasesOutlineOutline• Why closure phases:– The effect of the atmosphere.– Telescope-based errors.• Closure phases:– Definition.– Importance.• Uses:– Heuristic interpretation.– Toy examples.– Imaging.Michelson Summer Workshop 425thJuly 2006C.A.Haniff – Closure phasesOutlineOutline• Why closure phases:– The effect of the atmosphere.– Telescope-based errors.• Closure phases:– Definition.– Importance.• Uses:– Heuristic interpretation.– Toy examples.– Imaging.Michelson Summer Workshop 525thJuly 2006C.A.Haniff – Closure phasesStarting pointsStarting points• Interferometers measure the Fourier components of the sky brightness distribution. • These are complex quantities, and so we would like, in principle, to measure both the amplitude and phase of these Fourier components (i.e. the amplitude and phase of the visibility function).• The problem is that the atmosphere perturbs the electric fields sampled by the interferometric collectors and this impacts the measurements of the visbilityfunction. Note that, under all circumstances, the visibility phase can be measured – the problem is that its value no longer depends only on the source structure, but on lots else!Michelson Summer Workshop 625thJuly 2006C.A.Haniff – Closure phasesThe atmosphereThe atmosphere• A 12m × 12m patch of goodatmosphere. • Each contour represents oneradian of phase delay for lightat a wavelength of 2 microns.Michelson Summer Workshop 725thJuly 2006C.A.Haniff – Closure phasesWhat this doesWhat this does• So the effect of the atmosphere is to introduce an unknown phasedelay above each interferometriccollector.• In the cartoon on the right, we depict what happens when only oneaperture is affected.• This gives a shift in the output ofthe interferometer away from theexpected white-light position.Dyck et al. 1995 (IOTA).Michelson Summer Workshop 825thJuly 2006C.A.Haniff – Closure phasesModelling this antenna-based corruptionModelling this antenna-based corruption• Key idea is thatelectric field is alteredin amplitude and phase: Telescope Gain, e.g., coupling efficiency into single-mode fiber or reflectivity of mirror.Telescope phase shift, e.g., atmos-pheric piston, thermal drifts.Phase shift of detected fringe• Amplitude of complex visibility is scaled: |Vijtrue| → |Gi||Gj| |Vijtrue|• Phase of complex visibility is altered: Arg(Vijtrue) → Arg(Vijtrue) + ΦiG- ΦjGMichelson Summer Workshop 925thJuly 2006C.A.Haniff – Closure phasesWhat would pt. source visibility phases look like? What would pt. source visibility phases look like? • Different colors are different baselines.• x-axis direction represents time.Michelson Summer Workshop 1025thJuly 2006C.A.Haniff – Closure phasesKey ideas 1Key ideas 1• The atmosphere effects the phase of the radiation collected by the interferometer. • This in turn alters the phase of the measured complex visibility.• Measured optical/near-infrared complex visibilities have random phases that are unrelated to the source structure:– These cannot be easily made use of.Michelson Summer Workshop 1125thJuly 2006C.A.Haniff – Closure phasesQuiz 1Quiz 11. What does the model we have used to describe the effects of the atmosphere imply regarding the size of the collectors used in an interferometric array? (Look at slide 6 – “The atmosphere”)2. What would happen if we averaged the measured visibility phase on a given interferometer baseline for a long time, e.g. for 10 seconds?Michelson Summer Workshop 1225thJuly 2006C.A.Haniff – Closure phasesOutlineOutline• Why closure phases:– The effect of the atmosphere.– Telescope-based errors.• Closure phases:– Definition.– Importance.• Uses:– Heuristic interpretation.– Toy examples.– Imaging.Michelson Summer Workshop 1325thJuly 2006C.A.Haniff – Closure phasesDefinition of the closure phase (i)Definition of the closure phase (i)• Consider measurements of the threecomplex visibilities associated witha 3-element interferometer.• Let each complex visibility be corrupted by the atmosphere in the way we have described earlier:• Let’s examine the product of the three complex visibilities:Michelson Summer Workshop 1425thJuly 2006C.A.Haniff – Closure phasesDefinition of closure phase (ii)Definition of closure phase (ii)• So, the argument of this “triple product”:– Doesn’t contain any of the atmospheric error terms like ΦiG, ΦjGand ΦkG.– Has a value that is determined by the true values of the visibility function phases on the baselines that form this closed loop: Arg(Bijk) = Arg(Vijtrue) + Arg(Vjktrue) + Arg(Vkitrue).– Is hence a “good” observable.• This sum of visibility (Fourier) phases round a closed loop of baselines is called a “closure phase”.• It is perhaps more useful to refer to it as the argument of the triple product or of the “bispectrum”.Michelson Summer Workshop 1525thJuly 2006C.A.Haniff – Closure phasesExample of closure phasesExample of closure phases• The resistance of the closure phase to atmospheric perturbations can be