Spatio-spectral encoding of fringes in optical long ... - GSU Astronomy

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A&A 531, A110 (2011)Fig. 6. Bispectrum averaged over 250 short exposures of 20 ms (observationof the calibrator star HD 55185). Left: real part of the bispectrum,right: imaginary part of the bispectrum.The bispectrum is a complex valued function defined as:D (3) (u 1 , u 2 ,v 1 ,v 2 ) = Ĩ(u 1 ,v 1 )Ĩ(u 2 ,v 2 )Ĩ ∗ (u 1 + u 2 ,v 1 + v 2 ), (19)where Ĩ(u) is the Fourier transform of the fringe pattern and Ĩ ∗denotes the complex conjugate of Ĩ. The argument of the bispectrumis the closure phase Δφarg [ D (3) (u 1 , u 2 ,v 1 ,v 2 ) ] =Δφ = θ 12 + θ 23 − θ 13 , (20)where θ ij is visibility phase of baseline ij.The dimension of the bispectrum is always twice that of thepower spectrum. In the case of VEGA, the bispectrum will thenbe four dimensional as presented in Eq. (19). Because the smallnumber of telescopes (three or four) provides only a small numberof closure triangles, it is only necessary to calculate the bispectrumon a small number of points. In the case of three telescopes,the bispectrum is calculated around (U p1 , U p2 , V p1 , V p2 )as defined in Eq. (10).Figure 6 shows the real and imaginary parts of the bispectrumaveraged over 250 short exposures of 20 ms around thefrequencies (U p1 , U p2 , V p1 , V p2 ). The averaged bispectrum hasbeen projected on the plane defined by the axis (u 1 ,v 1 )inorderto ease the visualization of this four-dimensional function.We show a signal in the real part of the averaged bispectrumwhereas no signal is present in the imaginary part. This is ofcourse expected since the star is unresolved and therefore theclosure phase is zero.Figure 7 presents closure phase measurements for two calibratorstars (HD 46487 and HD 55185). The observations wereperformed using three telescopes (E1E2W2) at medium spectralresolution. Each estimate corresponds to 5 s of observation andto a spectral band of Δλ = 30 nm centered at λ = 735 nm. Themean and standard deviation of both series areΔφ HD 46487 = −0.091 ± 8.499 ◦ ,Δφ HD 55185 = 0.776 ± 5.625 ◦ . (21)No significant bias in the closure phase is detected from these estimates.The difference in the precision of the two sets is relatedto the photon noise, HD 55185 having a magnitude m V = 4.14compared to m V = 5.08 for HD 46487. These first estimates ofclosure phase also show that precision better than 1 ◦ ≈ 20 mradcan be achieved with VEGA by averaging phase closure estimatesover several minutes (10 mn for HD 46487 for example).3.5. Differential measurementsSpectro-interferometers such as VEGA permit us to determinethe differential visibility and phase. These instruments have beenClosure Phase (degree)Closure Phase (degree)50250-25HD46487-500 20 40 60 8050250-25HD55185-500 20 40 60 80Block NumberFig. 7. Estimate of the closure phase of calibrator stars HD 46487 andHD 55185 with VEGA with three telescopes (E1E2W2) at mediumspectral resolution (Δλ = 30 nm centered at λ = 735 nm) and fringetracking done with CLIMB. Each estimate corresponds to 5 s of observation(250 short exposures of 20 ms).intensively used, for example, to study the morphology and kinematicof circumstellar environments (Mourard et al. 1989; Berioet al. 1999; Meilland et al. 2007). A data reduction method wasdeveloped to derive these differential measurements for VEGA,which is described in detail in Mourard et al. (2009) fortwotelescope observations. In the case of three or four telescope observations,we use exactly the same method but for each highfrequency peak of the cross-spectrum.Here, we address the problem of the differential phases in thecase of three telescope observations. In Fig. 8, we present measurementsof differential phase for the observations of HD 55185(see the description of observation in Sect. 3.3). We used a referencechannel of Δλ 1 = 30 nm centered at λ 1 = 735 nm and asliding narrow channel of Δλ 2 = 0.6 nm. We averaged the crossspectrumof 20 000 short exposures of 20 ms (corresponding to400 s of observation) in order to estimate the differential phasewith a good signal to noise ratio (SNR). In theory, the differentialphase curves should be constant and equal to zero regardlessof the baseline (E1E2, E1W2, or E2W2), since the star is verypartially resolved. However, we detect linear and/or quadratictrends for each baseline. The differential phase is corrupted bythe OPD(χi (λ 1 )Δφ 12,i = θ λ1 ,i − θ λ2 ,i + 2π − χ )i(λ 2 ), (22)λ 1 λ 2where Δφ 12,i is the differential phase for the baseline i, θ λ j ,i is thephase of the object at the wavelength λ j for the baseline i, andχ i (λ j ) is the OPD of the baseline i. This OPD depends on thewavelength through the chromatism of the refractive index of air(Colavita et al. 2004;andCiddor 1996)χ i (λ) = n(λ)L i = (1 + N(λ)) L i = χ i + δχ i (λ), (23)where n(λ) is the refractive index of air, N(λ) represents the chromaticpart of n(λ), and L i is the path difference in air for thebaseline i. Although CHARA uses vacuum pipes for the beamtransportation, the delay lines are in open air, thus the differencein the positions of delay lines introduced a difference in the pathin air. In Eq. (23), χ i is the static part of the OPD as describedin Sect. 2 and δχ i (λ) is the chromatic OPD. Hence, insertingA110, page 6 of 9
D. Mourard et al.: Spatio-spectral encoding of fringes in optical long-baseline interferometry60Differential Phase (degree)Differential Closure Phase (degree)40200-20-40-60720 725 730 735 740 745 7506040200-20-40-60720 725 730 735 740 745 750Wavelength (nm)Fig. 8. Top: plots represent the differential phases in a narrow spectralband of 0.6 nm with respect to the wavelength (baseline E1E2 in black,E2W2 in red, E1W2 in green). Bottom: plot represents the differentialclosure phase deduced from the three individual differential phases.Each estimate corresponds to 400 s of observation of calibrator starHD 55185 using VEGA with three telescopes (E1E2W2) at mediumspectral resolution and fringe tracking done with CLIMB. Individualerror bars are not plotted for clarity. On baselines E1E2 and E2W2, theerrors are at the level of 0.5 ◦ , whereas on the baseline E1W2 they arecloser to 5 ◦ .Eqs. (23)into(22)givesΔφ 12,i = θ λ1 ,i − θ λ2 ,i + 2πχ i( 1λ 1− 1 λ 2)+ φ disp,12 , (24)where the dispersive differential phase is( δχi (λ 1 )φ disp,12 = 2π − δχ )i(λ 2 )· (25)λ 1 λ 2Hence, if the interferometer operates in vacuum and if the OPDis servoing to zero, the third and fourth terms in Eq. (24) disappearand the differential phase depends only on the phase ofthe object. In the case of VEGA, the OPDs of each baselineare servoing to a non-zero value and the delay lines operate inair. Therefore, the differential phases are corrupted by both thestatic OPD and the dispersive differential phase φ disp,12 ,whichexplains the linear and/or quadratic trends of the curves presentedin Fig. 8.Extraction of the astrophysical signal requires the removalof these linear and/or quadratic trends. This is usually doneby fitting and subtracting a low-order polynomial function ora model of the dispersive differential phase to the differentialphase (Matter et al. 2010). The main drawback of this methodis the loss of both the linear and/or quadratic terms in the astrophysicalsignal. For studies of circumstellar discs for example,it is not a major problem because the astrophysical signal ismore complex and confined only to spectral lines. So the neighboringcontinuum could be used to remove the linear and/orquadratic trends. However, other astrophysical studies, such asbinary stars, require us to measure signatures in the differentialphase that are very close to linear or quadratic trends. In thatcase, the differential phase could not be used directly. We defineda new interferometric observable, which is the differentialclosure phaseΔφ (3)12 =Δφ 12,1 +Δφ 12,2 − Δφ 12,3 . (26)Fig. 9. Sky coverage of potential calibrator stars well adapted toVEGA/CHARA in 3T/4T mode with CLIMB as external IR group delaytracking. All these stars have a spectral type ranging from O (cross),B (star), A (diamond), to F (triangle).It can easily be shown that this quantity does not depend on theOPD but is the difference in the classical closure phase of eachspectral bandΔφ (3)12 =Δφ 1 − Δφ 2 , (27)where Δφ 1 and Δφ 2 are respectively the classical closure phaseof the reference channel and of the narrow channel. Figure 8(bottom plot) illustrates that the linear and/or quadratic trendsdue to the OPD have disappeared in the differential closurephase. The mean of the differential closure phase in one narrowchannel is Δφ (3)12 = 0.2 ± 11 ◦ all over the spectral band of 30 nm.4. Discussion of possible VEGA/3T-4T sciencecases4.1. 3T-4T operation issuesEven if progress in fringe tracking allows fainter objects to beobserved and/or lower visibilities to be recorded, this requiresfinding calibrators that are bright enough for both the scientificinstrument and the fringe tracker. Moreover, using very longbaselines makes the choice of calibrators much more difficultsince lots of them are partially resolved. The calibrators mustalso be close to the scientific targets in terms of both distanceand spectral type to avoid any bias in the calibrated interferometricobservables. As an example, we selected calibrators amongthe 9500 stars (declination larger than −30 ◦ and magnitude Vbrighter than 7.0) in the VEGA internal catalog by applying filtersto parameters such as magnitude and diameter so as to fit theVEGA capabilities. Angular diameters were calculated using aspectrophotometric polynomial law based on the V magnitudeand the (V − K) color index (Bonneau et al. 2006b). Accordingto the current limiting magnitude, we considered only the starsbrighter than 6.5intheV band and brighter than 7 in the K band(for CLIMB that performs the group delay tracking). We identified1866 potential calibrators in all parts of the sky with anangular diameter smaller than 0.4 mas providing a high visibilitymeasurement even on the largest baselines. All of them areOBAF stars (Fig. 9). Figure 10 shows the histogram of the numberof calibrators found per regions of sky of 1h in right ascensionand 5 ◦ in declination.A110, page 7 of 9
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