Imaging the heart of astrophysical objects with optical long-baseline interferometry

Abstract

The number of publications of aperture-synthesis images based on optical long-baseline interferometry measurements has recently increased due to easier access to visible and infrared interferometers. The interferometry technique has now reached a technical maturity level that opens new avenues for numerous astrophysical topics requiring milli-arcsecond model-independent imaging. In writing this paper our motivation was twofold: (1) review and publicize emblematic excerpts of the impressive corpus accumulated in the field of optical interferometry image reconstruction; (2) discuss future prospects for this technique by selecting four representative astrophysical science cases in order to review the potential benefits of using optical long-baseline interferometers.

For this second goal we have simulated interferometric data from those selected astrophysical environments and used state-of-the-art codes to provide the reconstructed images that are reachable with current or soon-to-be facilities. The image-reconstruction process was “blind” in the sense that reconstructors had no knowledge of the input brightness distributions. We discuss the impact of optical interferometry in those four astrophysical fields. We show that image-reconstruction software successfully provides accurate morphological information on a variety of astrophysical topics and review the current strengths and weaknesses of such reconstructions.

We investigate how to improve image reconstruction and the quality of the image possibly by upgrading the current facilities. We finally argue that optical interferometers and their corresponding instrumentation, existing or to come, with six to ten telescopes, should be well suited to provide images of complex sceneries.

Appendices

Appendix A: Noise model for ASPRO simulations

This appendix is aimed at describing the noise model used in ASPRO for creating the simulations used in this paper (see Sect. 5.2).

The noise model is based on a general scheme valid for spatially filtered recombiners where the detection of fringes is made on a detector with “pixels”. This scheme is valid for image-plane recombination, with fringes covering a surface of a pixel camera, and for pupil recombination where fringes are obtained on a few pixels detector by scanning in optical path difference.

The flux $\overline{N_{\mathrm{tot}}}$ from the object of magnitude m _b in a given bandwidth Δλ of a photometric band b is collected by N _tel telescopes of diameter D, transmitted with some instrumental transmission T _inst, and injected with some Strehl factor s due to incomplete correction of wavefront aberrations due to seeing into a spatial filter like an optical single mode fiber for example, during a time t _int. Thus:

$$ \overline{N_\mathrm{tot}} = \eta \, F_0 \, 10^{-0.4 m_b} \, T_\mathrm{inst} \, N_\mathrm{tel} \, s \, \frac{\pi D^2}{4} \, \Delta\lambda, $$

(5)

where F ₀ is the zero-magnitude flux in band b, expressed in ph s⁻¹ m⁻² μm⁻¹ transmitted through the atmosphere with an absorption η.

This flux is divided between the photometric flux and interferometric flux with a branching value b _i , where b _i equals 1 for recombiners which do not have simultaneous flux monitoring.

The N _tel photometric fluxes $\overline{N_{p}}=(1-b_{i}) \, \overline{N_{\mathrm{tot}}}/N_{\mathrm{tel}}$ are distributed on N _pix pixels. The interferometric flux $\overline{N_{i}}=b_{i} \, \overline{N_{\mathrm{tot}}}/N_{\mathrm{tel}}$ consists of N _f =N _tel (N _tel−1)/2 fringes that cover $N^{i}_{\mathrm{pix}}$ pixels. There are thus $N_{\mathrm{ppf}}=N^{i}_{\mathrm{pix}}/N_{f}$ pixels per fringe.

These fringes code the intrinsic visibility V(u,v,λ) degraded by the interferometer instrumental contrast and the atmosphere (through the jitter associated with the temporal coherence of the seeing). V(u,v,λ) and the derived interferometric observables are thus affected by the sum of the variance of the flux used to code the corresponding fringe in the interferometric flux and of the associated two photometric fluxes. For example, since the squared visibility estimator of a correlated flux $F_{c}^{ij}$ measured alongside with photometries F _i and F _j is $V^{2}=\frac{1}{4}\langle |F_{c}^{ij}|^{2}\rangle/\langle F_{i}F_{j}\rangle$, the associated variance is

$$ \sigma^2\bigl(V^2\bigr) = \overline{N_i} V(u,v,\lambda) + N_\mathrm{ppf} \sigma^2_\mathrm{det} + 2 \bigl( \overline{N_p}+N^p_\mathrm{pix} \sigma_\mathrm{det}^2 \bigr), $$

(6)

where σ _det is the readout noise of the detector.

The noise model used in ASPRO takes also into account the possibility of increasing the integration time to keep observations in a photon-dominated regime, when a fringe tracker is present.

Finally, no detailed calibration error was computed. We took instead an additional visibility and closure-phase threshold error set to 0.002 in visibility. and 0.1 degree in closure phase.

Appendix B: Computing fidelities

Testing image-reconstruction software is out of scope of this paper. In this appendix, we follow the approach described in ALMA Memo 398 (F. Gueth, private communication) to evaluate the quality of the reconstructed images presented in this paper. One of the possible methods to compare original (convolved to the interferometer resolution) and reconstructed image is to compute the fidelity of the image. This can be done either in the direct image plane or in the spatial frequency (u,v) plane. Such a pixel to pixel comparison requires subpixel alignment to limit the effect of sharp transitions.

In the image plane this fidelity can be expressed as

$$ \mathcal{F}(x,y)= \frac{\mathrm{abs}(\mathit{Model} (x, y))}{\max(\mathit{Diff}(x, y), \mathit{Threshold})}, $$

(7)

where Model(x,y) describes the object “true” brightness distribution and

$$ \mathit{Diff}(x,y)=\mathit{Model}(x,y)- \mathit{Reconstructed}(x-\Delta x,y-\Delta y) $$

(8)

describes the difference between the model and the reconstructed image shifted by the offset (Δx,Δy) to have the images centered. For a proper comparison images are normalized to the total intensity contain in the image. Threshold is defined here as 0.7 rms[Diff(x,y)] which provides an estimation of the threshold noise of this difference. For example a pixel fidelity of 100 corresponds to a difference of 1 % between the model and reconstructed image. We have computed such a fidelity for three of our objects: the supergiant, the evolved star and the active galactic nuclei. Figures 10 and 11 offer different ways to visualize fidelity. The first row shows the cumulated average fidelity over the image for the three different objects. The second row displays the original model image filtered by fidelity level, i.e. only the pixels with fidelity above a certain level are displayed. Finally Table 6 shows the average fidelity of the pixels whose intensity is above a certain fraction of the total image intensity.

Fig. 10

Top row: cumulated average fidelity for the supergiant and evolved star cases. Blue dashed, green dashed-dotted, red dotted vertical lines: number of pixels with intensity greater than a given fraction of the total image intensity, respectively: 4×10⁻⁴, 1×10⁻⁴, 4×10⁻⁵ for the supergiant (left), 5×10⁻³, 1×10⁻³, 5×10⁻⁴ for the evolved star (right). Horizontal lines share the same color code: median fidelity for pixels having intensity greater than a given fraction of image total intensity. Central row: original model image filtered with a fidelity threshold (supergiant: 10, evolved star: 10). Bottom row: fidelity profile along the solid cut line in central row. Orange solid line: fidelity; green dashed-dotted line: model image; dashed blue: reconstructed image. Units are given in fidelity, image profiles have been scaled to fit the figure

Fig. 11

Top: cumulated average fidelity for the AGN case. Blue dashed, green dashed-dotted, red dotted vertical lines: number of pixels with intensity greater than a given fraction of the total image intensity i.e. 1×10⁻³, 5×10⁻⁴, 1×10⁻⁴. Horizontal lines share the same color code: median fidelity for pixels having intensity greater than a given fraction of image total intensity. Central: original model image filtered with a fidelity threshold (AGN: 3). Bottom: fidelity profile along the solid cut line in central row. Orange solid line: fidelity; green dashed-dotted line: model image; dashed blue: reconstructed image. Units are given in fidelity, image profiles have been scaled to fit the figure

Table 6 For a given fraction of the brightest pixels (col 1) the corresponding number of pixels, median and average fidelity is given for each object