Imaging submilliarcsecond stellar features with intensity interferometry using air Cherenkov telescope arrays

Mon. Not. R. Astron. Soc. 424, 1006–1011 (2012) doi:10.1111/j.1365-2966.2012.21263.x Imaging submilliarcsecond stellar features with intensity interferometry using air Cherenkov telescope arrays Paul D. Nuñez,1 Richard Holmes,2 David Kieda,1 Janvida Rou1 and Stephan LeBohec1 1 Department of Physics & Astronomy, University of Utah, 115 South 1400 East, Salt Lake City, UT 84112-0830, USA 2 Nutronics Inc., 3357 Chasen Drive, Cameron Park, CA 95682, USA Accepted 2012 May 8. Received 2012 April 11; in original form 2012 January 27 ABSTRACT Key words: instrumentation: interferometers – techniques: high angular resolution – techniques: image processing – techniques: interferometric – stars: imaging. 1 I N T RO D U C T I O N Stellar intensity interferometry (SII) has seen a revival due to the extraordinary (u, v) coverage that future air Cherenkov telescope arrays (CTA) will provide (Holder & LeBohec 2006). The angular resolution that can be achieved is as fine as 0.06 mas at the longest baselines (1.4 km) and the shortest optical wavelengths (∼400 nm) (Nuñez et al. 2012). The possible improvement in angular resolution by an order of magnitude and increased sensitivity for hot (> 6000 K) stellar objects (Dravins et al. 2010) has motivated the exploration of SII capabilities to investigate several science topics. These include diameter measurements, stellar rotation, gravity darkening, mass-loss and mass transfer (see Dravins et al. 2010 for more details). Intensity interferometry consists in measuring the squared modulus of the complex mutual degree of coherence between detector pairs from the cross-correlation between the light intensity fluctuations. Therefore, only the magnitude of the Fourier transform of the stellar radiance distribution is accessible. This poses a challenge for performing model-independent imaging. However, recent developments in phase recovery techniques from simulated SII data will make model-independent imaging possible (Holmes, Nuñez & LeBohec 2010; Nuñez et al. 2012). In previous work, we have shown that overall shapes and dimensions can be reconstructed with sub-per cent accuracies by using a Cauchy–Riemann (CR) phase reconstruction algorithm (Nuñez et al. 2012). The use of this phase  E-mail: reconstruction algorithm along with iterative post-processing routines also allows for further details to be imaged. This paper is organized as follows. Section 2 outlines the generation of simplified pristine stellar images and the simulation of intensity interferometry data. Then, phase recovery is briefly outlined and image post-processing is presented in Section 3. Results are then presented in Section 4 for images with increasing degrees of pristine image complexity. 2 P R I S T I N E S T E L L A R I M AG E S A N D DATA S I M U L AT I O N Pristine images of disc-like stars are first generated.1 These images correspond to blackbodies of a specified temperature containing an arbitrary number of ‘star spots’ of variable size, temperature and location at the surface of the spherical star in this three-dimensional model. The simulated stellar surface is then projected on to a plane, so that spots located near the edge of the visible half-sphere appear more elongated than those located near the centre. Additionally, limb darkening is included by assuming that the stellar atmosphere has a constant opacity. More details of the simulated images are presented in Section 4. From these pristine images, simulated SII array data are obtained by computing the Fourier transform of the ‘pristine’ image so that the squared modulus of the degree of coherence can be found between telescope pairs in a large imaging air Cherenkov telescope 1 The original ‘pristine’ image consists of 2048 × 2048 pixels corresponding to ∼10 mas × 10 mas of angular extension and a wavelength of λ = 400 nm.  C 2012 The Authors C 2012 RAS Monthly Notices of the Royal Astronomical Society  Recent proposals have been advanced to apply imaging air Cherenkov telescope arrays to stellar intensity interferometry (SII). Of particular interest is the possibility of model-independent image recovery afforded by the good (u, v)-plane coverage of these arrays, as well as recent developments in phase retrieval techniques. The capabilities of these instruments used as SII receivers have already been explored for simple stellar objects, and here the focus is on reconstructing stellar images with non-uniform radiance distributions. We find that hot stars (T > 6000 K) containing hot and/or cool localized regions (T ∼ 500 K) as small as ∼ 0.1 mas can be imaged at short wavelengths (λ = 400 nm). Imaging submilliarcsecond stellar features Figure 1. Schematic of the GS error-reduction algorithm. 3 C R P H A S E R E C OV E RY A N D P O S T- P RO C E S S I N G RO U T I N E S The CR phase recovery algorithm consists in using the theory of analytic functions to relate the magnitude and the phase of the Fourier transform (Holmes & Belen’kii 2004). That is, in one dimension, the CR equations relate the magnitude and the phase differentials along the real and imaginary directions in the complex plane. For details on the two-dimensional CR phase recovery algorithm, see Nuñez et al. (2012, section 5). The resulting reconstructed image with this estimated phase is sometimes not ideal, and so is taken as a first guess for the iterative algorithms that are now described. The Gerchberg–Saxton algorithm (GS; Gerchberg & Saxton 1972), also known as the error-reduction algorithm, is an iterative procedure. Starting from a reasonable guess of the image, the algorithm consists in going back and forth between image and Fourier space, each time imposing general restrictions in each domain. Fig. 1 describes the GS algorithm. Starting from an image Ok , the first step consists in taking the Fourier transform to obtain something of the form Mk eiφk . Next, Fourier constraints can then be applied, i.e. the magnitude is replaced by that given by the data, and the phase of the Fourier transform is maintained. Next, the inverse Fourier transform is calculated and constraints can be imposed in image space. 2 A preliminary design of the CTA was used (Bernlöhr 2008; CTA Con- sortium 2010). See figure 2 in Nuñez et al. (2012) for the exact array configuration used in these simulations.  C 2012 The Authors, MNRAS 424, 1006–1011 C 2012 RAS Monthly Notices of the Royal Astronomical Society  The constraints in image space can be very general. The image constraint that we impose comprises applying a mask, so that only pixels within a certain region are allowed to have positive non-zero values. For the images presented below, the mask is a circular region whose radius is typically found by measuring the radius of the first guess obtained from the CR approach. In all reconstructions where the GS is used, we perform 50 iterations, and found that more iterations typically do not produce significant changes in the reconstru (...truncated)