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Appendix A: Removal of detector artifacts from MIDI acquisition images

As Fig. A.1a shows, noticeable vertical stripes are present in the sky-subtracted frames of the MIDI acquisition data of L₂ Pup. These detector artifacts presumably result from the high brightness of L₂ Pup (the central region within 0.̋1 is saturated). No vertical stripes appear in the images of the PSF reference stars. The intensity of the vertical stripes is different in the upper

and lower regions with respect to the star, and it therefore is necessary to remove the stripes in the upper and lower regions separately. In each region, we removed the stripes as follows. First, for each column affected by the vertical stripes, we computed the median of the pixel values from the rows sufficiently far away from the star. Then we subtracted this median from all pixels in the column. We carried out this procedure for all columns that were affected by the stripes. Figure A.1 demonstrates that the image is nearly free from the stripes after this procedure.

	Fig. A.1 Removal of the detector artifacts in the MIDI acquisition images of L2 Pup. a) One of the sky-subtracted frame of L2 Pup, showing vertical stripes in the columns near the center. b) Same frame after the removal of the vertical stripes. The color scale in the diffraction core is saturated to clearly show the stripes.
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Appendix B: Image reconstruction

	Fig. B.1 Best-fit model consisting of a uniform-disk central star and an elliptical Gaussian shown in the same manner as Fig. 6.
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The observed visibilities plotted in Fig. 3a show a steep drop at low spatial frequencies ≲3 × 10⁶ rad^-1 (=baselines shorter than 6 m), which suggests a very extended component. The visibilities observed at spatial frequencies from ~6 × 10⁶ rad^-1 to 10⁷ rad^-1 (baselines from 13 to 22 m) may appear reminiscent of the first and the second visibility lobe expected from a uniform disk or limb-darkened disk. However, the visibility from a uniform disk without an extended component is 0.13 (or lower for a limb-darkened disk) in the extrema of the second visibility lobe, much lower than the observed values of ~0.3. With an extended component as revealed by the speckle data, the visibility would be even lower. We first attempted to explain these observed data using geometrical models. While simple geometrical models may not fit the data completely, they are useful for characterizing the approximate geometry of the object and can also be used as an initial model for the image reconstruction.

Appendix B.1: Geometrical model: uniform disk + elliptical Gaussian

We tried to fit the data with a geometrical model consisting of a uniform-disk-like central star and an elliptical Gaussian. The free parameters are the uniform-disk diameter of the central star (φ_⋆), the fractional flux contribution of the central star f_⋆, the widths of the elliptical Gaussian along the major and minor axes σ_major and σ_minor (the elliptical Gaussian is given by e^{− ((x/σ_major)2 + (y/σ_minor)2)}), and the position angle of its major axis PA (measured from North to East). We searched for the best-fit model by varying φ_⋆ = 8 ... 22 (mas) with Δφ_⋆ = 2 (mas), f_⋆ = 0.1 ... 0.7 with Δf_⋆ = 0.05, σ_major = 30 ... 100 (mas) with Δσ_major = 10 (mas), σ_minor = 10 ... 50 (mas) with Δσ_minor = 10 (mas), and PA = 70° ... 100° with ΔPA = 5° (the PA was limited to this range from the elongation of the image reconstructed from the speckle data alone). The best-fit model, which is plotted in Fig. B.1, is characterized by φ_⋆ = 12 mas, f_⋆ = 0.3, σ_major = 50 mas, σ_minor = 20 mas, and PA = 95° with a reduced χ² of 27.2. Figure B.1 reveals that the best-fit model cannot reproduce the observed visibilities at spatial frequencies of (0.6−2.0) × 10⁷ rad^-1 (baseline length = 13–44 m). Most of the observed visibilities at these spatial frequencies are noticeably higher than predicted by the model. This means that there is some sharp structure that is not seen in the smooth Gaussian model.

Appendix B.2: Geometrical model: uniform disk + elliptical ring

We then tried a model consisting of the uniform-disk-like central star and an elliptical ring, because a ring gives rise to visibilities much higher than a uniform disk or limb-darkened disk at long baselines. The free parameters are the uniform-disk diameter of the central star φ_⋆, the fractional flux contribution of the central star f_⋆, the semi-major and semi-minor axes of the elliptical ring (R_major and R_minor, respectively), and its position angle PA. The width of the ring was set to be 10% of its radius at each position angle. We searched for the best-fit model by varying φ_⋆ = 8 ... 22 (mas) with Δφ_⋆ = 2 (mas), f_⋆ = 0.1 ... 0.6 with Δf_⋆ = 0.05, R_major = 30 ... 100 (mas) with ΔR_major = 10 (mas), R_minor = 10 ... 50 (mas) with ΔR_major = 10 (mas), and PA = 70° ... 100° with

ΔPA = 5°. Figure B.2 shows a comparison of the best-fit ring model with the observed data. This model is characterized by φ_⋆ = 16 mas, f_⋆ = 0.6, R_major = 60 mas, R_minor = 40 mas, and PA = 95° with the reduced χ² = 54.8. While the visibilities at (0.6−1) × 10⁷ rad^-1 from this model is as high as or even higher than the observed data, the fit to the speckle data and the data at longer baselines is worse than the above star + Gaussian model as shown by the worse reduced χ².

	Fig. B.2 Best-fit model consisting of a uniform-disk central star and an elliptical ring shown in the same manner as Fig. 6.
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Appendix B.3: Geometrical model: obscured uniform disk + elliptical Gaussian

We found out that the observed data are much better reproduced if the southern half of the aforementioned uniform-disk-like central star + elliptical Gaussian model is obscured. In this model, the intensity of the uniform-disk star + elliptical Gaussian model I₀(x,y) (x and y are the coordinates on the sky with the origin at the central star) is modified as follows:

where ε is a parameter to smoothen the obscuration edge. We set ε to be 0.2, which decreases the intensity to zero over ~2 mas in the y direction. The free parameters are the uniform-disk diameter of the central star, the fractional flux contribution of the central star f_⋆, the widths of the elliptical Gaussian along the major and minor axes σ_major and σ_minor, and the position angle of its major axis PA. We searched for the best-fit model by varying φ_⋆ = 8 ... 22 (mas) with Δφ_⋆ = 2 (mas), f_⋆ = 0.2 ... 0.6 with Δf_⋆ = 0.05, σ_major = 30 ... 100 (mas) with Δσ_major = 10 (mas), σ_minor = 10 ... 50 (mas) with Δσ_minor = 10 (mas), and PA = 70° ... 100° with ΔPA = 5°.

The best-fit model is characterized by φ_⋆ = 20 mas, f_⋆ = 0.45, σ_major = 70 mas, σ_minor = 30 mas, and PA = 85°. As Fig. B.3 shows, this model can reproduce the observed visibilities much better than the above two models. The uncertainties in f_⋆, σ_major, σ_minor, and PA are ±0.05, ±10 mas, ±10 mas, and ±5°. The uniform-disk diameter of the central star is in the range between 18 and 20 mas. This agrees with the 17.5 mas derived by K14, given the difference in the data and the model used by them and us. The reduced χ² of this model is 16.2, which is much better than the star + Gaussian or star + ring models, but still much larger than 1. This is because the fit to the visibilities observed at the longest baselines (spatial frequencies higher than ~2 × 10⁷ rad^-1) and to the observed closure phases are not satisfactory. However, this disagreement can be due to small-scale structures that are not included in our geometrical model, but can be modeled by the image reconstruction.

	Fig. B.3 Best-fit model consisting of a half-obscured uniform-disk central star and an elliptical Gaussian shown in the same manner as Fig. 6.
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Appendix B.4: MiRA parameters

We used the best-fit half-obscured star + elliptical Gaussian model as the initial model. The regularization scheme of the maximum entropy method was adopted, with the prior being the Gaussian with the same widths as the initial model. The degree of regularization was set to μ = 10³ (see Thiébaut 2008 for details of the regularization scheme and the definition of μ). We also reconstructed the image with the total variation regularization scheme, but the image shows no noticeable differences.

© ESO, 2015