4. From interference fringes to stellar diameters
Observations were carried out at the Whipple Observatory on Mount Hopkins in October 1995, February and April 1996. In this section, we describe the observing procedure that we have adopted and the calibration process.
4.1. Observing procedure
Observing sequences unfold on a routine basis. A short-stroke delay line sweeps through the zero optical path difference (OPD thereafter) position at a speed which is computed in order to keep the OPD rate constant (the fringe frequency is typically of the order of 300 Hz). The OPD length of the sequences is about 100 µm. At the end of the scan, shutters block the beams in the two arms and a sequence of detector dark current is recorded in each output. It will be used for signal and noise calibration in the data reduction process (Coude du Foresto et al. 1997). In regular turbulence conditions, fringes are acquired every four seconds. The standard observation of a source is a batch of 100 scans that spreads over a few minutes.
The key point of astronomical interferometry is the calibration of all losses of coherence. In order to build an estimate of the losses, observations of sources are interleaved with reference stars.
4.2. Fringe calibration
4.2.1. Diameter of reference stars
Since few stellar diameters have been measured up to now (less than one hundred), it was necessary to use indirect estimates of the diameter of reference sources in most cases. Our estimates are based on a scale of stellar diameters at for giants by Dyck et al. (1996a). The diameter at a given K magnitude is derived by assuming that the surface of the star is proportional to the brightness of the star. Although the accuracy of the scale is estimated by the authors to be of the order of 10%, we suggest and adopt an accuracy of 5%. We have checked this a priori accuracy by comparing the photometric scale with our own results and we have found an average dispersion less than 5% between the two.
The diameters and associated accuracies of reference stars used for calibration in this paper are listed in Table 1. Stars whose diameter estimate was derived from the spectral scale are labelled with 'a'. Others, labelled with a 'b', have been directly measured, at the same wavelength, with the I2T interferometer (Di Benedetto & Rabbia 1987).
Table 1. Reference stars.
4.2.2. Computation of the transfer function
The transfer function is directly computed from the contrast of the fringes acquired on calibrators. It is the ratio between the contrast of the fringe packet and the expected visibility. Since the fringe visibility of calibrators is larger than 50%, the expected visibility is computed from a uniform disk model of the source. This will not generate an error larger than 1%. We thus measure a discrete estimate of the temporal evolution of the transfer function during the night. As the observations of scientific sources are bracketed by the observations of the reference sources it is necessary to build an estimate of the losses of coherence at the time when scientific objects are observed from non simultaneous data. To do so, the procedure is to interpolate two successive measures of the transfer function to get its evolution in the interval. We have made the assumption that this evolution is linear, that is to say that the time between two observations of reference stars is short enough so that the transfer function can be described by its first derivative.
We have represented the evolution of the (instrumental) transfer function in the two interferometric outputs in Fig. 1. error bars include the dispersion of the distribution of data points and the uncertainty on the diameter of reference stars. Because the loss of coherence generated by turbulence has been removed, the average values of the instrumental contrast in the two outputs are quite high (90% and 60%). They are not equal to 100% because of residual polarization mismatches between the beams, and a degradation of the photometer time response at 300 Hz. As long as the instrumental set-up does not change, the transfer function is very stable and the fluctuations are well within the error bars. The fluctuations are smooth enough so that we can interpolate the transfer function as explained. Uncertainties on the diameter of calibrators may cause the apparent transfer function to vary but detecting this error would require a better accuracy on the tranfer function than we can achieve yet.
4.3. Visibilities and associated errors
The visibility estimates in the two outputs are the ratios of the estimates of the fringe packets contrasts and the estimates of the transfer functions. The final estimate of the visibility is the weighted mean of the visibilities in the two outputs. The final error bar is obtained by summing the reciprocals of the individual variances.
4.4. Selection of data
Since the contribution of atmospheric random phase errors to contrast error has been removed, the main source of noise on fringe contrast is detector noise. As a consequence, the accuracy on the average contrast of fringes in a batch of one hundred interferograms (the statistical accuracy) is usually better than 1% for most sources. As the final estimate of the error on the visibilities is derived from statistical accuracies, it is mandatory to check that these are meaningful. The only way to do so is to compare the simultaneous estimates of the visibility in the two outputs. If the difference between the two is consistent with the error bars then the data and the error bars are declared good. Otherwise the quality of the calibration is poor and the data are rejected. Specifically, a test is built with the residual of the fit of the two visibilities by the weighted mean. if and are the error bars of the two visibilities and then the residual at the optimum is:
The data are rejected if:
When observing conditions are not optimum, the instrumental transfer function evolves rapidly. The assumption of Sect. 4.2.2on the smoothness of its variations then becomes wrong and visibilities may be badly calibrated. When conditions of observation are satisfactory, however, the rejection rate is smaller than 10%.
4.5. Stellar diameters
4.5.1. Uniform disk diameters
The data are fitted with a uniform disk model. The results of the fit in terms of uniform disk diameter are listed in the fifth column of Table 2. The error on the diameter is computed by varying the of the fit until it increases to . We have chosen to present two interesting examples of model fit in Fig. 2. As Boo is one of the brightest stars that we have observed, the error bars are very small. They are nevertheless fully consistent with the dispersion of data points about the best fit curve. BK Vir, a more typical case, can also be very well fitted by a uniform disk model even though the source is fainter and the error bars are larger.
Table 2. Details of observations and results.
A quality factor of the adjustment is obtained by computing the ratio of the residual of the fit () and of the number of data points minus the number of degree of freedom in the fit, that is to say one (last column of Table 2). If this latter quantity is larger than 1 then error bars are statistically too small, by a factor equal to the square root of that quantity, and the model does not represent well the data. This occurs twice in the data and is not, therefore, representative of the general quality of the fits. It is legitimate to evaluate the quality of the calibration with the quality of the fits with a uniform disk model as most giants can be described by such a simple model at spatial frequencies smaller than the first null of the visibility function. We thus conclude that the data are well calibrated.
4.5.2. Limb-darkened disk diameters
The uniform disk diameter is a biased estimate of the real stellar diameter. The stellar limb is darker than the center of the stellar disk at wavelengths of interest here, and the averaging effect of a uniform model leads to underestimate the diameter of the star. This effect is smaller in the K band than it is in the visible. But it must be taken into account otherwise effective temperature estimates will also be biased. The bias depends upon the spatial resolution of the interferometer. We have calibrated this bias on a few stars for which we have measured high quality visibility points. To do so, we have fitted our data with limb-darkened disk models published in the literature (Manduca 1979, Scholz & Takeda 1987). In all cases the fit is of slightly better quality when the model disk is limb-darkened than when it is uniform. The average result is a ratio between the uniform and the limb-darkened disk diameters of 1.035 with a dispersion of 0.01. From the grids of published limb darkening predictions for stars in the 3000-4500 K range, we have made a best effort to estimate the ratio of limb darkening corrected diameter to uniform disk diameter. To do so, we have fitted limb darkened disk visibilities before the first null by a uniform disk model. This yields an average ratio of 1.03 with a dispersion of 0.01. Our results are thus coherent with predictions and, in the following, the real diameter of stars will be computed from the uniform disk diameter by applying the 1.035 scaling factor.
4.5.3. Check against other work
In order to intercompare angular diameters recorded at different wavelengths, it is necessary to invoke limb darkening corrections. The expected variation in apparent diameter from visible to infrared is of the order of 5% or more. Up to now, only two measurements of limb darkening have been performed: on Sirius A (Hanbury Brown et al. 1974) and on Arcturus (Quirrenbach et al. 1996). With our calibration in K we find a mas limb darkened diameter for Arcturus. This is compatible with the multi-wavelength interferometric result of Quirrenbach et al. (1996) at , 0.50, 0.55, 0.70 and 0.80 µm since they find limb darkened diameters in the range to mas.
While limb darkening for Tau has not been directly measured, the variation of apparent diameter with color has been found to follow the expected limb darkening variation (Ridgway et al. 1982). The compilation of several multi-wavelength studies by lunar occultation yields a limb darkened diameter of mas (White & Kreidl 1984). More recently, a limb darkened diameter of was determined by Michelson interferometry observations of Tau (Di Benedetto & Rabbia 1987). Once again, these results are compatible with our current result of mas.
© European Southern Observatory (ESO) 1998
Online publication: February 16, 1998