Speckle imaging relies on short exposures obtained with digital detectors, each of which will contribute both stellar signal and noise from the instrument, star, and sky to the pending Fourier analysis and thus to the final contrast curves and reconstructed images. Our data reduction practices and reduced pipeline products are described in detail in Howell et al. (2011) and Scott et al. (2021).

In order to observe fainter stars and their companions and reach larger Δ magnitude contrast levels, target stars are observed using longer total integration times (as in normal CCD imaging), but with ’Alopeke and Zorro we gain this increased integration time by taking additional sets of fast exposures. Each set consists of 1,000, 0.06 s exposures stored together as one multi-extension FITS file. For bright stars, three sets are the minimum number we use, with one additional set used for the bright PSF standard. As the target becomes fainter, we balance total integration time per target with useful contrast limits using a developed guideline for the number of sets to take during our usual observing practices. Table 1 lists a range of target R magnitude values along with our usual number of sets taken for each magnitude bin. The values in Table 1 are based on observational experience and assume the following rules of thumb: seeing is ≤1.0″, airmass is better than 1.4, and R is brighter than ∼14^th magnitude. Our typical observational parameters allow us to observe about 50–100 stars per night, given good conditions and a 10 h night. We have observed, under very good conditions (seeing ∼0.4″) and not near full moon, targets as faint as R = 19 (e.g., mid-to late M stars, Quasi-Stellar Objects) and obtained favorable results with good contrast levels by using about 1 h of total on-source integration time.

Previous papers (Horch et al., 2009, 2010; Howell et al., 2011) have discussed the relationships related to speckle decorrelation stating its linear dependence on the separation of two close point sources (stars) and its inverse dependence on the size of the isoplanatic patch. Taking ρ as the separation of two point sources (arssec), s as the seeing value (arcsec) and δs and the size of the isoplanatic patch (arcsec), we can define q′ (arcsec²) as follows: q′ = ρ/δs = ρs where s is inversely related to δs. Horch et al. (2010) and Scott et al. (2021) show that speckle decorrelation sets in when the seeing increases above ∼1.2” and/or q′ approaches ∼0.6 arcsec², being certainly of concern when q′ is > 0.8 arcsec². We mention q′ here as it can provide limits on the contrast level obtained. For example, observations taken during poor seeing can make the Fourier analysis less than ideal as the fringes become smeared together (Horch et al., 2010). High levels of the sky background, which is brighter in the blue, can dilute the ability to detect fringes during the Fourier analysis and thus close binaries (e.g., Scott et al., 2018). Therefore, besides the integration time, the seeing and sky background are important factors when estimating the expected contrast curve of a speckle observation.

As a proxy for the resulting S/N in a specific speckle final reconstructed image, we use the standard CCD equation modified to represent the particular image parameters and data collection procedures specific to speckle observations. In addition, the companion detection depends on the ability to robustly determine the background level and detect interferometric fringes in the processed short-exposure images. We will use known theoretical relationships (Dainty and Greenaway, 1979) for interferometric S/N determinations to provide modifications to the usual S/N equation, allowing us to calculate S/N estimates for speckle observations at Gemini. We will examine a few typical cases, that of bright (near full moon) and grey (full to quarter moon) sky and seeing values of 1.0 and 0.5″. The ranges considered herein provide a useful guide to allow an observer to estimate the desired result, such as the contrast limit, for a given observation.

We start by using Gemini’s listed values for their optical CCD imager, GMOS, as described in the GMOS integration time calculator ³ (ITC). We use the Gemini Observatory definitions for sky brightness and seeing ⁴ —that is image quality (IQ) yielding 0.5″ seeing is called 20%–Best and 1.0″ seeing is called 85–Poor. For the sky background brightness, we adopt 80%–Grey for times of quarter to near full moon or times of night before/after moon rise/set and Any/Bright for times of near full moon. We use the GMOS ITC values of the point spread function (PSF) spatial size at the focal plane for each seeing value considered and the sky background photons received per unit area for our modeled sky conditions. We assume an airmass of < 1.4 for all observations, a G5V star spectral energy distribution, and observations made in the SDSS r filter. We divide the ITC reported GMOS flux by the ratio of the respective filter FWHM values (SDSS r vs. our narrow bandpass, 40 nm) and adjust for the ratio of pixel size (GMOS has 0.08″ pixels and our speckle cameras have 0.01″ pixels–a factor of 64 in area).

The CCD S/N equation for a point source is given in §4.4 of Howell (2006) as S N = N ∗ N ∗ + n pix × N S + N D + N R 2 where N _* is the total number of photons gathered from the source, n _pix is the number of pixels used for this S/N calculation, N _S is the number of photons per pixel from the background or sky, N _D is the number of dark current electrons per pixel, and N _R is the number of electrons per pixel resulting from read noise. Zorro and ‘Alopeke use EMCCD imagers as described in Scott et al. (2021). For speckle imaging with an EMCCD, we can set the read noise ( N R 2 ) and the dark current (N _D ) values to zero, as a 60 ms exposure has no dark current buildup, and using the Electron Multiplication (EM) gain stage in the output essentially reduces the few electron read noise inherent in the CCD to zero at the output.

Thus, our S/N equation can be reduced to S N = N ∗ N ∗ + n pix × N S which illustrates why the sky background level is a critical element here given the large number of pixels which the observed stellar PSF illuminates. For example, a PSF of 0.5″ FWHM covers 4,352 pixels in our images (which have plate scale of 0.01”/pixel), and at 1.0″ FWHM, 19,200 pixels are in the PSF. Thus, n _pix × N _S can become substantial.

As a start, we take our CCD S/N calculation as above and check it against the GMOS ITC. Using GMOS values (e.g., pixel scale, filter) our calculations obtain answers that match those of the Gemini GMOS ITC to within 2%. Thus, we conclude that our implementation of Gemini telescope and GMOS observational parameters is accurate.

A stellar PSF at our speckle imaging plate scale covers a large area, and with a 0.06 s integration, the “PSF” is not a uniform Gaussian 2-D shape, but a set of many hundred randomly placed speckles, each containing few photons (e.g., Labeyrie, 1970; Scott et al., 2018, show illustrative speckle images). As the angular resolution achievable by an optical system becomes better, more photons are concentrated into each speckle, providing an increase in the S/N per frame. For our observations reported herein, this increase occurs with the better seeing and the larger collecting area of the Gemini 8-m telescope.

This complex speckle pattern S/N situation has been examined for some limiting cases by Welsh (1995), Petrov et al. (1986), and Dainty & Greenaway (1979). Following the discussions presented in these works, especially the latter, we modify the standard S/N equation such that N _* is now the average number of detected photons per speckle (n _s ), where the number of speckles per frame is approximately given by ( n s / f r a m e ) ∼ ( D / r 0 ) 2 * ( 1 / 0.435 ) , where D is the mirror diameter and r ₀ is the Fried parameter which varies as λ ^6/5 (Fried, 1966). Since r ₀ indicates the length over which an incoming wavefront can be considered planar, the ratio ( D / r 0 ) 2 gives the ratio of the areas collecting the total photon flux (the telescope mirror) compared with the approximate area of each speckle. The 1/0.435 term is needed as we approximate each speckle PSF as a Gaussian distribution rather than its statistically proper log-normal distribution. Additionally, we normalize the image intensity to unity at maximum, we use the bright star approximation of Dainty and Greenaway (that is, we divide the calculated S/N by 2 to approximate the realized S/N in the interference fringes which cover about one-half of the image area), and we approximate each speckle as a Gaussian distribution. We then solve the S/N equation for stars with R-band magnitudes of 10, 12, 14, and 16, two sky conditions (bright and grey), and two values of seeing (0.5 and 1.0″). Using the GMOS ITC, a 10th magnitude star in our 832 (40 nm wide) filter gives N_* = 805,815 photons/sec and sky counts/sec of 600/3481 (grey/bright) at 0.5″ seeing and of 2653/15376 (grey/bright) at 1.0″ seeing. The other N_* values are simply scaled by magnitude. Figure 1 shows our results for the S/N of a 10th through 16th magnitude star observed under the typical good sky conditions listed above for two sky brightness levels and two seeing values.

We can see that for bright stars (10^th to 12^th magnitude), the S/N is acceptable to very good for all cases under consideration. At 14^th magnitude, poor seeing (∼ 1″) severely limits the resulting S/N in all cases considered herein. The bright sky during full moon can be compensated for by observing 10 or more 1000 frame sets especially under conditions of good seeing. At 16th magnitude, we see that only for very good seeing and grey skies is it possible to achieve decent S/N results, and then only by using 15–20 or more sets. Fainter targets have been successfully observed under good seeing conditions (0.4–0.6″) and dark to grey skies (Howell et al., 2016; 2021b). The S/N values calculated here will become less accurate as the target magnitude becomes fainter than 14th, since we used the “bright star” approximation in our S/N calculations as mentioned above and described in Dainty & Greenaway (1979).

As mentioned above, each short exposure represents an instantaneous view of the atmospheric distortions of the star’s wavefront frozen in time. From these images, the Fourier summation of the images into a power spectrum yields a diffraction limited view of the scene near the star, revealing other light sources (companions) within that field of view. Given the large areal coverage of the instantaneous PSF, we can use the S/N result above, noting that the S/N is proportional to 1/seeing², to predict that a change in seeing of ±0.55″ will yield a change in the final achieved contrast level of about ±1.2 magnitudes at an angular separation of 0.5″ and ±2.5 magnitudes at 1.0″ (see Figure 1 and §3). Since any fringes produced by a multiple source will be of higher contrast as the flux ratio of the two (or more) sources approaches unity, nearly equal brightness binaries have an advantage over a pair of stars that have a delta magnitude value of, say, 6. Additionally, the S/N for a target, all else being equal, increases as the square root of the total integration time (i.e., number of sets) (e.g., Howell, 2006).

While we have used SDSS r filter parameters (which is centered at 630 nm), scaled to our 40 nm wide speckle filters, we would predict (and will see below) that better results will always be obtained at 832 nm, as the wavelength dependence of the Fried parameter r ₀ makes it about 1.8 times larger in the 832 nm sky. This larger “seeing cell” (isoplanatic patch) in turn directly correlates to the number of speckles in each image at each wavelength, being again about 1.8 times less at 832 nm than 562 nm, thus giving a higher S/N per speckle in the redder images. As is well known observationally (e.g., Kellerer and Tokovinin, 2007), better sky behavior in the red translates into the fact that the S/N is about 1.5–2 times better at 832 nm than at 562 nm, all else being equal. While we did not solve the equations for each specific wavelength, we will see below that the observational results at 832 nm are better in all aspects than those at 562 nm, and that our calculated change of ∼2 magnitudes of contrast for each 0.5″ seeing change matches the observations.