Volume 529, May 2011
|Number of page(s)||18|
|Published online||25 April 2011|
Our AMBER observations of Betelgeuse and the calibrator Sirius are summarized in Table A.1.
Log of AMBER observations of Betelgeuse and the calibrator Sirius with the E0-G0-H0 (16-32-48 m) baseline configuration. Seeing is in the visible.
Aperture synthesis imaging from optical/IR interferometric data depends on a number of parameters used in the image reconstruction process, such as the initial model, as well as the regularization scheme and prior, which represent a priori information about the object’s intensity distribution. Therefore, it is important to carry out the image reconstruction for simulated images to derive the appropriate reconstruction parameters before we attempt the image reconstruction from observed data. From simulated images, we generate simulated interferometric data by sampling the visibilities and CPs at the same uv points as interferometric observations. With the true images known for these simulated data, we can examine the appropriate reconstruction parameters that allow us to reconstruct the original images correctly.
Because MiRA is developed for two-dimensional image reconstruction, we took the following approach for the reconstruction of one-dimensional projection images: two-dimensional image reconstruction was carried out using an appropriate initial model and regularization parameters as described below. The reconstructed two-dimensional image was convolved with the clean beam, which is represented by a two-dimensional Gaussian with a FWHM of λ/Bmax = 9.8 mas, where Bmax is the maximum baseline length of the data. The one-dimensional projection image was obtained by integrating this convolved two-dimensional image in the direction perpendicular to the linear uv coverage.
We generated two simulated images that represent possible surface patterns of Betelgeuse: a simple limb-darkened disk and a uniform disk with a bright spot, a dark spot, and an extended halo, as shown in Figs. B.1a and B.2a, respectively. For both cases, the stellar angular diameter was set to be 42.5 mas, which is the limb-darkened disk diameter derived from all continuum
visibilities measured in 2009. The visibilities and CPs were computed from the simulated images at the same uv points as our AMBER observations, using the program of one of the authors (K.-H. Hofmann). Noise was also added to the simulated visibilities and CPs to achieve SNRs similar to the AMBER data. We tested different initial models, priors, and regularization schemes to find out the appropriate parameter range to reconstruct the one-dimensional projection image of the simulated data correctly. It turned out that uniform disks with angular diameters between 34 and 50 mas serve as good initial models. The prior used in the present work is a smoothed uniform disk described as
where r is the radial coordinate in mas, and rp and εp define the size and the smoothness of the edge (εp → 0 corresponds to a uniform disk), respectively. The appropriate values for rp and εp were found to be 10 ... 15 (mas) and 2 ... 3 (mas), respectively. Therefore, we used six different parameter sets for the image reconstruction of Betelgeuse by combining three diameters for the initial uniform-disk model (34, 42, and 50 mas) and two different parameter sets for the prior ((rp,εp = (10, 2) and (15, 3)). The final images and their uncertainties were obtained by taking the average and standard deviation, respectively, from the results reconstructed with these six parameter sets. The regularization using the maximum entropy method turned out to be appropriate for our reconstruction. We started the reconstruction with a high degree of regularization (μ = 105, see Thiébaut 2008, for the definition of μ) and reduced it gradually by a factor of 10 after every 500 iterations until the reduced χ2 reaches ~1 or MiRA stops the iteration. These tests with the simulated data also confirm the validity of our approach to reconstruct one-dimensional projection images using the MiRA software for two-dimensional image reconstruction.
Image reconstruction of the simulated data for a limb-darkened disk with the parameters derived in Sect. 3.1. a) Original two-dimensional image of the simulated data. The solid line represents the orientation of the linear uv coverage, while the dotted line represents the orientation perpendicular to it. b) Two-dimensional image of the simulated data convolved with the Gaussian beam with a FWHM of 9.8 mas. c) Comparison between the original and reconstructed one-dimensional projection images before convolving with the Gaussian beam. The one-dimensional projection images are obtained by integrating the two-dimensional images in the direction shown by the dotted lines in the panels a) and b). d) Comparison between the original and reconstructed one-dimensional projection images convolved with the Gaussian beam with a FWHM of 9.8 mas. e) The filled circles and triangles represent the visibilities from the original simulated data and the reconstructed image, respectively. f) The filled circles and triangles represent the CPs from the original simulated data and the reconstructed image, respectively. The abscissa is the spatial frequency of the longest baseline of each data set.
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Image reconstruction of the simulated data for a uniform disk with a bright spot, a dark spot, and a halo shown in the same manner as in Fig. B.1.
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Because the principle of the self-calibration technique using DP measurements is described in detail in Millour et al. (2011), we mention the actual procedure only briefly. Then we describe the modification we added to this technique to deal with an issue specific to the AMBER data of Betelgeuse.
The DP measured with AMBER at each uv point contains information on the phase of the complex visibility function and roughly represents the difference between the phase in a spectral feature and that in the continuum. However, two pieces of information are lost because of the atmospheric turbulence: the absolute phase offset and the linear phase gradient with respect to wavenumber. We can derive this lost phase offset and gradient by a linear fit to the phase (as a function of wavenumber) from the reconstructed continuum images, if the image reconstruction in the continuum is reliable and not sensitive to the reconstruction parameters. This is indeed the case for our image reconstruction of Betelgeuse in the continuum, as discussed in Sect. 3.4. Therefore, the phase in the CO lines can be restored from the continuum phase interpolated at the line spectral channels and the DPs measured in the lines. The image reconstruction is carried out with the measured visibilities and CPs as well as the restored phase. This process can be iterated, but our experiments show that the reconstructed images do not change after the first iteration.
We added the following modification to the technique presented in Millour et al. (2011). When the phase offset and gradient are derived by a linear fit to the phase of the reconstructed images, we only use the continuum spectral channels below 2.293 μm and those between the adjacent CO lines above 2.3 μm, instead of using the entire spectral channels, as Millour et al. (2011) did. The reason for this selection of the spectral channels is that the image reconstruction near the CO band head at 2.294 μm is so uncertain owing to the poor SNR in the data binned with a spectral resolution of 6000 that the inclusion of the spectral channels near the band head in the linear fit hampers the reliable derivation of the phase offset and gradient.
a) DP observed on the longest baseline in the data set #9 (Bp = 40.36 m) is plotted by the red solid line. The black line represents the scaled observed spectrum. The DP and spectrum are binned with a spectral resolution of 6000. The dashed line represents the linear fit at the selected continuum points as described in Appendix C. DP = 0 is shown by the dotted line. b) Enlarged view of panel a) for the CO lines. Note that the DP in the continuum points between the adjacent CO lines deviates from zero. c) DP after subtracting the linear fit to the continuum points as described in Appendix C is shown by the red solid line. The black line represents the scaled spectrum. The DP in the continuum points between the adjacent CO lines is now zero within the measurement errors.
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The inclusion of only the selected continuum channels has the following consequence. If we denote the continuum phase from the reconstructed continuum image at a given baseline and at the ith spectral channel as ϕc(i), the phase at the ith spectral channel, ϕ(i), is restored as
where DP(i) represents the differential phase at the ith spectral channel measured at the same baseline. At a continuum spectral channel denoted as ic, the restored phase ϕ(ic) should be equal to the phase from the reconstructed continuum image ϕc(ic). This is fulfilled if the measured DP in the continuum is zero. However, the measured DPs show noticeable non-zero values in the continuum, as exemplarily shown in Figs. C.1a and b. The reason for the non-zero DPs in the continuum is that amdlib derives differential phase by a linear fit to the instantaneous phase at all spectral channels. Owing to the strong deviation of the phase in many CO lines from that in the continuum, this linear fit does not go through all the continuum points. Therefore, the non-zero DPs in the continuum spectral channels lead to a systematic error in the phase restored in the continuum, which affects the subsequent image reconstruction. We found out
that the continuum one-dimensional projection image reconstructed using the restored phase shows a systematic wavelength dependence from the shortest to the longest wavelength of the observed spectral range, which is not seen in the continuum images reconstructed from the visibilities and CPs alone.
It is necessary to use the same spectral channels in the linear fit to the phase for the derivation of DP and for the derivation of the phase offset and gradient. Therefore, we refitted the DP from amdlib with a linear function (with respect to wavenumber) at the same continuum points as used for the derivation of the phase offset and gradient (dashed lines in Figs. C.1a and b) and subtracted the fitted linear function from the observed DP. This procedure enforces the DP in the continuum spectral channels to zero within the measurement errors, as shown in Fig. C.1c. The phase was restored using this “refitted” DP. The continuum one-dimensional projection images reconstructed using the refitted DPs do not show the aforementioned systematic wavelength dependence, which proves the validity of our procedure.
Figures D.1 and D.2 show the fit to the observed interferometric data for the one-dimensional projection image reconstruction in the CO line and in the CO (2, 0) band head shown in Figs. 5 and 6, respectively.
Comparison between the observed interferometric data and those from the one-dimensional projection image reconstruction for the CO line shown in Fig. 5. The first, second, third, and fourth columns show the comparison for the blue wing, line center, red wing, and continuum, respectively. The panels in the top row a)–d) show the observed CO line profile, and the filled circles denote the wavelength of the data shown in each column. In the remaining panels, the observed data are represented by the red circles, while the values from the image reconstruction are shown by the blue triangles. The reduced χ2 values for the fit including the complex visibilities, squared visibilities, and CPs are given in the panels m)–p).
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Comparison between the observed interferometric data and those from the one-dimensional projection image reconstruction near the CO (2, 0) band head shown in Fig. 6. The first, second, third, and fourth columns show the comparison for the continuum, blue side between the continuum and the band head, bottom of the band head, and red side of the band head, respectively. The panels in the top row a)–d) show the observed spectrum of the CO band head, and the filled circles denote the wavelength of the data shown in each column. In the remaining panels, the observed data are represented by the red circles, while the values from the image reconstruction are shown by the blue triangles. The reduced χ2 values for the fit including the complex visibilities, squared visibilities, and CPs are given in the panels m)–p).
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© ESO, 2011
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