1.1.

Nonredundant Masking

Nonredundant masking⁹ is well suited for detecting young, giant planets around more distant stars than those targeted by typical direct imaging surveys. NRM uses a pupil-plane mask to turn a conventional telescope into an interferometric array. The images are the interference fringes formed by the mask, which we Fourier transform to calculate complex visibilities. Since the mask is nonredundant, no two baselines have the same orientation and separation; information from each baseline is encoded in a unique location in Fourier space.

Nonredundancy means that the instrumental component of each Fourier phase can be written as a linear combination of two pupil phases. Calculating closure phases,¹⁰ sums of phases around baselines forming a triangle, eliminates these instrumental phases to first order, leaving behind sums of phases intrinsic to the target. (We note that if the instrumental pupil-plane phases for each subaperture are treated as static pistons, then closure phases cancel instrumental phase exactly. However, spatial and temporal pupil-plane phase variations, as well as amplitude variations such as scintillation, lead to higher order closure phase errors that require calibration.¹¹) Closure phases are particularly powerful for close-in companion detection because they are sensitive to asymmetries. Since closure phases are correlated, we project them into linearly independent combinations called kernel phases.^11–13 We also calculate squared visibilities, the powers on each mask baseline.

Despite the low throughput ($\sim 10\%$), NRM’s superior PSF characterization probes angular separations at and even within the diffraction limit. This angular resolution means that spatial scales of $\sim 10\mathrm{AU}$ can be resolved for stars $\sim 150\mathrm{pc}$ away. NRM has led to detections of both stellar¹⁴ and substellar companions,^15,16 as well as circumstellar disks^13,17,18 at these small angular separations. It led to the discovery of a promising system for planet formation studies, the protoplanetary candidates in LkCa 15.¹⁴ Dual-aperture LBTI masking observations recently resolved a solar-system-sized disk around the star MWC 349A,¹⁸ at a distance of $\gtrsim 1.2\mathrm{kpc}$ from Earth. With a maximum baseline of 23 m, this is a preview of NRM’s potential on 30-m class telescopes. NRM on current telescopes and on future facilities will expand the exoplanet detection parameter space.

1.2.

Filled-Aperture Kernel Phase

Extreme adaptive optics systems have made filled-aperture kernel phase¹² an interesting alternative to NRM. Kernel phase involves treating a conventional telescope as if it were a redundant array. Redundancy prevents Fourier phases from being written as a linear combination of pupil-plane phases in general. However, in the high Strehl regime (where instrumental pupil-plane phases are small), one can justify Taylor expanding the instrumental wavefront. This means that the redundant instrumental Fourier phases can be approximated as a linear operator on the pupil-plane phases. Finding the nullspace of this linear operator yields kernel phases, linearly independent combinations of Fourier phases that are robust to instrumental phase errors to first order. Like closure phase (and kernel phase) in NRM, filled-aperture kernel phase is sensitive to asymmetries and is thus powerful for close-in companion detection.

Kernel phase has been demonstrated on archival Hubble Space Telescope observations of ultracool dwarfs.¹⁹ In addition to confirming several known companions to these L dwarfs, it led to the detection of five binary systems at angular separations of $\sim 40$ to 80 mas at $\sim 1.1$ to $1.7\mu \mathrm{m}$. Kernel phase has also been applied to ground-based data from Keck²⁰ and Palomar,²¹ where it led to the detection of stellar and substellar companions at ($\sim 1$ to $2\lambda /D$). The Keck observations were taken at $M_s$ band, where the high sky background reduces the SNR of masking observations. These kernel phase datasets provided comparable resolution to that expected for NRM, with a shorter integration time.

1.3.

Outline of this Paper

Here we compare NRM and filled-aperture kernel phase in a controlled way, with a specific focus on exoplanet science. We generate simulated datasets for three imagers: NIRC2 on Keck, and NIRCam and NIRISS on James Webb Space Telescope (JWST). We also generate observations for two integral field spectrographs (IFSs): OSIRIS on Keck, and NIRSpec on JWST. Integral field spectroscopy is a particularly interesting kernel phase application, since dispersing the light and applying a pupil-plane mask would require long integration times for high signal-to-noise. Furthermore, when broadband kernel phase observations of bright stars may be unfeasible due to detector saturation, IFS kernel phase observations may not saturate.

We describe the simulated datasets in Sec. 2. We use the simulated observations to generate contrast curves for both filled-aperture kernel phase and NRM, and to place detection limits on planetary atmosphere and circumplanetary accretion disk models. We present these results in Sec. 3. In Secs. 4 and 5, we discuss the potential of each technique as a method for exoplanet detection and characterization.

2.1.

Keck NIRC2

We simulate kernel phases for filled-aperture and masked Keck NIRC2 observations at $K_s=2.15\mu \mathrm{m}$, $L^{\prime }=3.78\mu \mathrm{m}$, and $M_s=4.67\mu \mathrm{m}$. The filled-aperture and NRM modes yield 45 and 28 kernel phases, respectively. For each target observation, we simulate a cube of $n_{\text{frames}}$ frames, each of which is the sum of $n_{\mathrm{coadd}}$ coadds (see Tables 7Table 8Table 9Table 10Table 11–12). We add sky background, dark current, and readout noise to the images according to the number of coadds and exposure times; Table 1 lists the relevant noise parameters and other detector information for each bandpass. For all observations, we background subtract each frame and apply any window functions. We then calculate complex quantities for each kernel phase that are analogous to bispectra for closure phases (see Sec. 7). Like averaging bispectra for calculating mean closure phases, we average these complex quantities over the cube of images before taking the phase as the mean kernel phase for a single visit.

Table 1

NIRC2 noise and detector parameters.

We simulate OPD maps to account for quasistatic PSF aberrations. Keck AO is known to be affected by low-order residual wavefront errors; modeling these as segment piston errors has been shown to produce realistic PSFs.²⁶ To simulate PSFs, we create OPD maps that are a combination of low- and high-order residual wavefront errors. We generate low-order errors over the entire Keck pupil and on each individual mirror segment. For the entire pupil, we make combinations of the first 20 Zernike modes ($Z_n$; excluding tip and tilt), drawing each coefficient from a uniform distribution with a width [$2f(n)$] that depends on the Zernike order ($n$):

When we apply this prescription, we use arbitrary normalizations for $f(n)$ and $g$, and then rescale the final OPD maps to have an RMS tuned to match real observations. We also tune the dependence of the low-order Zernike coefficients on Zernike order ($\alpha$). The best OPD parameters are listed in Table 2. These lead to uncalibrated kernel phase scatters of $\sim 1.2\mathrm{deg}$ for $L^{\prime }$ NRM observations (compared to $\sim 1.0\mathrm{deg}$ to 1.5 deg for real observations), and Strehl values of 0.55, 0.85, and 0.9 at $K_s$, $L^{\prime }$, and $M_s$ (consistent with typical Strehls observed using NIRC2), respectively. They also match observed PSF radial profiles to within $\sim 4\%$ fractional error. Figures 1 and 2 show example images and power spectra, respectively, for NRM and kernel phase for each bandpass.

Fig. 1

Simulated individual (a) $K_s$, (b) $L^{\prime }$, and (c) $M_s$ images for the observing parameters listed in Table 1. The top row shows filled-aperture data, and the bottom row nonredundant masking data. The images are displayed on a square root scale to highlight structure in the wings of the PSF. The fields of view are different for the two techniques: the kernel phase images are 45 mas on a side, and the NRM images are 100 mas on a side.

Fig. 2

Simulated (a) $K_s$, (b) $L^{\prime }$, and (c) $M_s$ power spectra for a single frame using the observing parameters listed in Table 1. The top row shows filled-aperture kernel phase data, and the bottom row nonredundant masking data. Random noise from the large sky background is apparent in the $L^{\prime }$ and $M_s$ power spectra; building signal-to-noise in the long baselines requires combining more frames. The scattered points show the sampling locations for calculating kernel phases; they are shown in two colors since the Fourier transform is symmetric.

Keck’s low-order errors are quasistatic²⁷ and contribute to calibration errors in the final kernel phases. We thus generate calibrator observations after evolving the low-order and segment OPD Zernike coefficients by a factor drawn from a one-mean uniform distribution with width $2h$. Equations (3) and (4) describe this procedure:

Fig. 3

Example OPD evolution for (a) NIRC2 filled-aperture kernel phase and (b) nonredundant masking. The left shows an initial OPD made up of both low- and high-order wavefront errors, and the center panel shows the result of evolving the OPD map. The right side shows the difference between the two; it has an rms OPD of 60 nm.

We design a half night of observations for each target star magnitude and bandpass, accounting for overheads. We split the half night into a number of visits to target and PSF calibrator stars. The length of each visit depends on the number of photons collected from the star and sky; for a small number of source photons, kernel phase errors may be dominated by random noise. As the number of source photons increases, the random kernel phase scatter decreases, and the visit approaches the contrast limit—where quasi-static errors dominate over random errors. Longer integrations in this limit do not decrease the kernel phase scatter significantly. Thus, we design our visits to be either long enough to reach the contrast limit, or to take 3 h (half the observing time), since the observing is split evenly between target and PSF calibrator observations. Datasets on brighter stars will contain more visits than datasets on fainter stars. When multiple visits can fit into a half night, we treat them as snapshots at different parallactic angles that change in $5\text{-}\mathrm{deg}$ increments and have independent calibration errors.

To account for NIRC2 overheads, we treat each visit as a collection of frames and each frame as the sum of a number of coadds. We design each coadd to be long enough to reach 5000 to 10,000 maximum counts—this is to minimize overheads but remain in a linear regime on the detector. We assume a $512\times 512\text{pixel}$ subframe, which results in a single coadd overhead of 0.05 s. We enforce a minimum frame time of 20 s, which adds an additional overhead of 2.18 s every 20 s. We assume that the observations are dithered every 200 s to enable sky subtraction, which adds an overhead of 6 s for each dither. Lastly, we assume target acquisition overheads of $\sim 90$ s per pointing.²⁸ Tables 7Table 8Table 9Table 10Table 11–12 in Sec. 7 provide frame, coadd, and dither details for each target star brightness. Figures 24 and 25 show histograms of the raw (uncalibrated) and calibrated kernel phases for each technique and bandpass for two different target star brightnesses.

2.2.

JWST NIRCam and NIRISS

We use the Pandeia engine²⁹ and WebbPSF software³⁰ to simulate NIRCam kernel phase and NIRISS aperture masking (AMI) observations to compare the two techniques. NIRCam imaging and NIRISS AMI yield 21 and 15 kernel phases, respectively. Since exoplanets are relatively bright in the near- to midinfrared, we simulate data for filters centered on wavelengths $\gtrsim 4\mu \mathrm{m}$ that can be used with NIRISS AMI and NIRCam imaging (F430M and F480M). Since slew and telescope roll overheads are large for JWST, we do not allow the number of visits for the different target stars to vary like we do for NIRC2. We rather assume that the observations are made up of two pairs of target—PSF calibrator visits taken at different telescope roll angles 45 deg apart. We assume that the length of each visit is 1.5 h, for a total observation time of 6 h, excluding slew and roll angle overheads. When designing the observations, we account for detector overheads within each 1.5-h visit.

JWST visits are split into sets of integrations, each of which is composed of a number of groups. To calculate overheads, for each target magnitude, we first find the maximum number of groups ($n_g$) that can be used in a single integration before saturation. For NIRCam imaging, this means choosing the readout mode that provides the maximum integration time, but for NIRISS AMI only one readout mode is available. We then assume that each visit is composed of the maximum number of $n_g$-group integrations that can be acquired in 1.5 h ($n_i$), given that each integration comes with a readout overhead (0.0494 s for NIRCam in a sub64P subarray, and 0.0745 s for NIRISS in a sub80 subarray). When the remaining time after $n_i$ integrations allows for more than a single group, we add an additional integration containing ($n_{g,\mathrm{rem}}$) groups. Tables 13Table 14Table 15–16 in Sec. 6 list $n_g$, $n_i$, and $n_{g,\mathrm{rem}}$ for each instrument.

We use Pandeia’s “detector” output for the observational parameters in Tables 13Table 14Table 15–16. This generates a long-exposure image for the entire visit that contains the Poisson noise from the target and JWST’s “medium” thermal background, and read noise. We note that the data product for each real JWST exposure will be a cube of integrations, allowing for coadding of smaller pieces of each exposure, or for averaging kernel phases over all short integrations. We do not explore different options for coadding or averaging of short-exposure kernel phases using Pandeia. However, we do test the effects of using a single summed image versus averaging many short-exposure images using our Keck simulations. Contrast curves generated from these two extreme options are nearly identical in the bright limit, where OPD errors dominate. In the limit where random noise dominates, averaging short exposures performed better by $\sim 0.5\mathrm{mag}$; averaging kernel phases generated from short exposures would thus improve the achievable contrast for the faintest simulated targets ($\gtrsim 11\mathrm{th}$ mag). For real JWST observations, averaging the kernel phases for many short exposures will be more practical, since effects such as pointing jitter and noise sources such as cosmic rays will degrade long-exposure datasets. Tables 3 and 4 list the relevant noise and detector parameters for these observations.

Table 2

NIRC2 OPD parameters.

Table 3

NIRCam noise and detector parameters.

For each pointing, we generate science target and calibrator frames using different OPD maps to simulate PSF evolution. We begin by randomly choosing one of the 10 OPD maps included in WebbPSF. We then fit a hexike basis³¹ to each mirror segment, including up to 100 coefficients. We evolve each of the hexike coefficients ($C_n$) by a factor drawn from a one-mean uniform distribution with a width ($2h$) tuned to result in an rms residual WFEs of $\sim 10\mathrm{nm}$ [see Eq. (5)]. This is consistent with the thermal evolution expected for JWST over approximate hour long timescales.³²

Fig. 4

Example JWST OPD evolution. The left panels show one of the 10 predicted (a) NIRCam and (b) NIRISS OPD files from WebbPSF. The center panels show the resulting OPDs after fitting hexikes to each JWST mirror segment and evolving their coefficients by a small fraction in random directions. The right panels show the difference between the first two OPD maps and have RMS residual OPDs of $\sim 10\mathrm{nm}$.

Fig. 5

Simulated (a) NIRCam and NIRISS integrations and (b) power spectra. This dataset uses the F430M filters and residual OPD files comparable to those in Fig. 4. The scattered points show the sampling locations for calculating kernel phase; they are shown in two colors since the Fourier transform is symmetric.

2.3.

Keck OSIRIS

OSIRIS is a $K$ band integral field spectrograph on Keck. We simulate OSIRIS kernel phase observations in the Kn3 bandpass, since it covers the accretion-tracing emission line Br-$\gamma$, which would be of interest for observing protoplanet candidates. This mode contains 433 0.25 nm wide wavelength bins between 2.121 and $2.229\mu \mathrm{m}$. With one subaperture per mirror segment, it yields 45 kernel phases. We assume 6 h of available observing time and account for readout, dither, and slew overheads. We optimize our observing strategy in an identical manner to NIRC2, designing visits that either reach the contrast limit or a length of 3 h including overheads. We keep the peak counts per frame under 30,000 (the OSIRIS saturation limit) and use a readout time of 0.829 s per frame, a dither overhead of 6 s, and a target acquisition overhead of 90 s. We use an OPD evolution prescription identical to NIRC2, and account for detector, sky, and Poisson noise in the same way as for NIRC2. Table 5 lists the relevant detector and noise parameters for OSIRIS, and Table 17 lists the number of frames and integration times for each target brightness. Figure 6 shows an example image and power spectrum, and Fig. 28 shows kernel phase histograms for targets with $K_s$ band brightnesses of 6 and 11 magnitudes.

Table 4

NIRISS noise and detector parameters.

Fig. 6

Example-simulated (a) OSIRIS image and (b) power spectrum.

2.4.

JWST NIRSpec

We use the Pandeia engine and WebbPSF software to generate simulated data for NIRSpec IFU observations. We simulate observations for stars with target magnitudes between $\sim 6$ and 13. We use the G395H grating and the F290LP filter, which yield $\gtrsim 4000$ wavelength bins from 2.87 to $5.27\mu \mathrm{m}$. We follow the same methods for OPD evolution and calibration as for NIRCam / NIRISS, evolving the hexike coefficients to achieve 10 nm rms residual OPD [$h=0.11$ in Eq. (5)]. We use the NRSRAPID readout mode. Table 18 lists the observation parameters for each target brightness.

Since the NIRSpec spaxels are 0.1 arc sec across, they do not Nyquist sample the PSF. We thus generate a nine-point small grid dither for each target observation, with the goal of reconstructing a Nyquist sampled image. To ensure adequate signal to noise for each dither position, we allocate 3 h for each target and calibrator observation and assume a single observing roll angle. As we did for NIRCam and NIRISS, we find the maximum number of groups ($n_g$) that can be included in an integration before saturation. We then split the observations up into nine sets (one for each dither) of $n_{\mathrm{int}}$ integrations, each of which has an associated frame-time overhead. Table 6 lists the relevant detector and noise parameters, and 18 lists the observing parameters for each target star magnitude.

Table 5

OSIRIS noise and detector parameters.

Table 6

NIRSpec noise and detector parameters.

We use the first nine available pointings in the NIRSpec SMALL CYCLING pattern to create a simulated small grid dither. We assign the flux at each pixel in each undersampled image to the angular coordinates located at the center of that pixel. We then use a linear interpolation to assign fluxes to the angular coordinates at the centers of the pixels in the Nyquist sampled image (which have sizes of $\sim {0.03}^{″}$). Figure 7 shows example undersampled and interpolated images, compared to a simulated Nyquist sampled PSF. This interpolation does not account for the spatial extent of each pixel; using a more sophisticated interpolation scheme may improve the results. We also note that up to 60 pointings can be used in the CYCLING pattern, and having more pointings may improve the reconstructed PSF quality. Since the PSF reconstruction is imperfect and some signal is lost on the highest spatial frequencies, we do not include the longest baselines in the kernel phase projection model. This makes the number of observable kernel phases 18. This modified projection degrades the kernel phase effective resolution, but decreases the noise significantly (see Fig. 8). Figure 29 shows kernel phase histograms for targets with $L^{\prime }$ band brightnesses of 6 and 11 magnitudes.

Fig. 7

(a) Undersampled and interpolated simulated NIRSpec images, with (b) a simulated Nyquist sampled PSF for comparison. The bottom right panel shows the radial profiles for the Nyquist sampled (black, thin line), and the interpolated (purple, thick line) PSF. While similar large-scale changes are seen in both, the interpolated image does not have the same small-scale variations as the Nyquist sampled PSF.

Fig. 8

(a) Simulated NIRSpec power spectrum of an interpolated Nyquist sampled image, displayed on a log scale to highlight the features at high spatial frequencies. The hollow circles show the $(u,v)$ sampling points we use to calculate kernel phases. Their histogram is shown in purple in (b). The red $x$’s show the $(u,v)$ points that would have been used for a perfect Nyquist sampled observation, but that we omit from the kernel phase model to reduce the observed scatter. The black histogram shows the kernel phases calculated for a point source when these $(u,v)$ points are included in the kernel phase model. The standard deviations of the purple and black histograms are 1.1 deg and 5.2 deg, respectively.

3.1.

NIRC2

As a check, we compare the contrast curves for real Keck NRM observations of PSF calibrators to those for simulated observations with the same target star brightness and exposure parameters. The $L^{\prime }$ target star magnitude is 5.8, and the observation consists of a single cube of twenty 20-s images. The results are shown in Fig. 9; the contrast curves are nearly identical.

Fig. 9

Contrast curves for a single target observation made from (a) real $L^{\prime }$ NIRC2 NRM observations and (b) simulated observations. The contrast curves are nearly identical, with $5\sigma$ contrast of $>5.5$ magnitudes.

Figure 10 compares the simulated NIRC2 contrast curves for NRM and kernel phase at two representative target brightnesses: apparent magnitudes of 6 and 11. The sixth magnitude case represents the contrast-limited regime for $K_s$ and $L^{\prime }$; comparing the NRM and kernel phase contrast curves here shows that kernel phase performance degrades at low Strehl. Kernel phase can achieve comparable contrast to NRM in the bright limit at high ($\gtrsim 0.85$) Strehl. This is evident in the observed kernel phase scatter (see Fig. 24)—the calibrated scatter is much lower for NRM observations at $K_s$ band, but they are comparable at $L^{\prime }$. When NRM and kernel phase are both in the contrast limit, NRM provides slightly higher contrast very close to the core of the PSF (i.e., within $\sim 80$ mas or $\lambda /D$ for the $L^{\prime }$ $M_{*}=6.0$ contrast curves).

Fig. 10

Kernel phase (thick lines) and NRM (thin lines) for (a) $K_s$, (b) $L^{\prime }$, and (c) $M_s$ bands. Solid purple lines show observations of 6th apparent magnitude stars, while black dashed lines show 11th apparent magnitude stars.

The $L^{\prime }$ contrast curves for the $M_{*}=11$ star, and all of the $M_s$ contrast curves demonstrate kernel phase and NRM’s different behaviors with the same sky brightness. The NRM PSF is spread over more pixels than the filled-aperture PSF. Thus, contrast degrades more quickly with an increasing sky brightness for NRM observations. This is also apparent in the kernel phase histograms (see Fig. 25); the raw scatter in the $L^{\prime }$ and $M_s$ observations is larger for NRM than for kernel phase. Furthermore, calibration decreases the $L^{\prime }$ kernel phase scatter more significantly for filled aperture observations than for NRM observations. This shows that random noise dominates the NRM observations at lower sky background levels compared to filled-aperture observations.

The contrast as a function of stellar apparent magnitude supports these points. Figure 11 shows this: at $K_s$, the achievable contrast is constant with stellar brightness, and higher for NRM than for kernel phase. At $L^{\prime }$, the kernel phase observations stay close to the contrast limit until an apparent magnitude of $L^{\prime }\sim 9$, while NRM degrades more quickly. Both techniques degrade quickly at $M_s$, although kernel phase provides higher achievable contrast.

Fig. 11

The colorscale shows the $5\sigma$ single companion contrast for NRM (top) and filled-aperture kernel phase observations (bottom), as a function of target star apparent magnitude and angular separation. $K_s$, $L^{\prime }$, and $M_s$ bands, respectively. Gray-shaded regions are regions of the parameter space where no companion model was significant at the $5\sigma$ level, compared to the null (single point source) model. The gray-dotted lines show 1 magnitude increments in companion contrast.

We convert our $5\sigma$ contrast curves to hot-start planet mass limits using DUSTY models,³³ and to planet mass times accretion rate limits using circumplanetary accretion disk models.^3,4 For the DUSTY models, we assume an age of $\sim 1\mathrm{Myr}$, since the apparent magnitude range we explore corresponds to the absolute magnitudes of $\sim 2\mathrm{Myr}$ old stars with masses ≲$5{\mathrm{M}}_{\odot }$²⁵ at the distance of Taurus ($\sim 140\mathrm{pc}$⁸). For the circumplanetary accretion disk models, we assume full disks with an inner disk radius of 2 Jupiter radii.³ Figures 12 and 13 show the planet and accretion disk limits, respectively, for a range of separations and stellar absolute magnitudes assuming a distance of 140 pc. Depending on the stellar magnitude, both NRM and kernel phase at $K_s$ and $L^{\prime }$ can reach planet masses of a few to a few tens of Jupiter masses, or planet masses times accretion rates of a few times $\sim {10}^{-6}$ to $\sim {10}^{-5}{\mathrm{M}}_{\mathrm{J}}^2{\mathrm{yr}}^{-1}$.

Fig. 12

Five $\sigma$ planet mass limits using 1 Myr DUSTY models (solid lines) for all NIRC2 bandpasses and both techniques. The color scale shows the $5\sigma$ absolute companion magnitude, calculated using the contrasts shown in Fig. 11 and by translating the stellar apparent magnitudes to absolute magnitudes at 140 pc. Gray-shaded regions are regions of the parameter space where no companion model was significant at the $5\sigma$ level, compared to the null (single point source) model. The gray-dotted lines show 1 magnitude increments in absolute companion magnitude. The changing noise regimes (OPD error dominated to random error dominated) in the $L^{\prime }$ and $M_s$ NRM simulations cause the achievable contrast to fall off nonuniformly as the target brightness changes (see Fig. 11, upper center and upper right panels). This causes local regions in the $L^{\prime }$ and $M_s$ stellar magnitude—separation parameter space to be better suited for detecting lower planet masses (see upper center and upper right panels here, colorscale and contours).

Fig. 13

Five $\sigma$ circumplanetary accretion disk (planet mass times accretion rate) limits (solid lines) for all NIRC2 bandpasses and both techniques. The color scale shows the $5\sigma$ absolute companion magnitude, calculated using the contrasts shown in Fig. 11 and by translating the stellar apparent magnitudes to absolute magnitudes at 140 pc. Gray-shaded regions are regions of the parameter space where no companion model was significant at the $5\sigma$ level, compared to the null (single point source) model. The gray-dotted lines show 1 magnitude increments in absolute companion magnitude. The changing noise regimes (OPD error dominated to random error dominated) in the $L^{\prime }$ and $M_s$ NRM simulations cause the achievable contrast to fall off nonuniformly as the target brightness changes (see Fig. 11, upper center and upper right panels). This causes local regions in the $L^{\prime }$ and $M_s$ stellar magnitude—separation parameter space to be better suited for detecting lower planet masses times accretion rates (see upper center and upper right panels here, colorscale and contours).

In the bright limit, $L^{\prime }$ NRM and $L^{\prime }$ kernel phase are both sensitive to lower planet masses and accretion rates than $K_s$ NRM. For fainter stars ($\gtrsim \mathrm{t}\mathrm{e}\mathrm{n}\text{th}$ apparent magnitude or $\gtrsim \text{fifth}$ absolute magnitude), $K_s$ NRM outperforms $L^{\prime }$ NRM due to the higher sky background at $L^{\prime }$. However, for these stellar brightnesses, $L^{\prime }$ kernel phase performs best. Despite brighter expected fluxes for hot-start planets at $M_s$, the high sky background prevents the detection of planets less massive than $\sim 10{\mathrm{M}}_{\mathrm{J}}$. We note that these mass limits look worse for older planets, and for models other than hot-start models. For example, the “warm-start” absolute magnitudes predicted by Spiegel and Burrows³⁴ are not detectable even for 1 Myr old $10{\mathrm{M}}_{\mathrm{J}}$ planets with either technique and at any bandpass.

3.2.

JWST: NIRCam Kernel Phase and NIRISS Aperture Masking Interferometry

Figure 14 shows representative contrast curves for $M_{*}=6.3$ and $M_{*}=11.3$ apparent magnitude stars for NIRCam and NIRISS. Again, in the bright limit ($M_{*}=6.3$), NRM outperforms kernel phase within $\lambda /D$ ($\sim 150$ mas), by 0.5 to 1.0 magnitudes. At larger separations, kernel phase provides comparable contrast. The faint $M_{*}=11.3$ mag contrast curves show that at lower signal-to-noise, NIRCam kernel phase can provide higher contrast than NRM on NIRISS. Figures 26 and 27 support this. The raw and calibrated kernel phase scatters for NIRCam and NIRISS are comparable in the bright case, but in the faint case, the NIRCam kernel phases calibrate to a lower noise level than the NIRISS kernel phases. This is apparent in Fig. 15, which shows that NIRISS’s contrast falls off as the stellar apparent magnitude becomes $>\sim 9$, but NIRCam’s contrast does not. We also note that for apparent magnitudes $<\sim 6$ to 6.5, NIRCam is at its saturation limit and thus kernel phase cannot be used.

Fig. 14

NIRCam kernel phase (thick lines) and NIRISS NRM (thin lines) for (a) F430M and (b) F480M bands. Solid purple lines show observations of 6.3 apparent magnitude stars, while black dashed lines show 11.3 apparent magnitude stars.

Fig. 15

The colorscale shows the $5\sigma$ single companion contrast for (a) NIRISS NRM and (b) NIRCam filled-aperture kernel phase observations, as a function of target star apparent magnitude and angular separation. The gray-dotted lines show 1 magnitude increments in companion contrast. The top and bottom panels show F430M and F480M bands, respectively. Gray-shaded regions are regions of the parameter space where either no companion model was significant at the $5\sigma$ level, compared to the null (single point source) model, or regions where the target magnitude was below the bright limit.

Figures 16 and 17 show the $5\sigma$ planet and circumplanetary accretion disk limits for a range of stellar absolute magnitudes and angular separations. The absolute magnitudes were calculated assuming a distance of 140 pc. The greater stability of JWST leads to higher contrast than that achieved with NIRC2, making lower mass planets and lower accretion rates more accessible for brighter stars. Planet masses of $1{\mathrm{M}}_{\mathrm{J}}$ are detectable for more than half the stellar absolute magnitudes in the hot-start case, and for the high stellar absolute magnitudes in the warm-start case for NIRCam. The range of detectable planet masses times accretion rates reaches below ${10}^{-7}{\mathrm{M}}_{\mathrm{J}}^2{\mathrm{yr}}^{-1}$. Comparing the top and bottom panels of Figs. 16 and 17 shows that, with NIRISS NRM in the bright limit ($M_{\mathrm{abs}}$≲ 2 to 3), slightly lower mass (accretion rate) planets can be detected at smaller angular separations. The maximum detectable absolute planet magnitude is higher for NIRCam, as a result of the improved contrast for fainter stars; this is particularly useful for detecting planets that have formed via nonhot-start scenarios.

Fig. 16

Five $\sigma$ planet mass limits using 1 Myr DUSTY models (solid lines) and 1 Myr warm-start Spiegel and Burrows (2012) models (dashed lines) for the F430M and F480M bandpasses and both techniques/instruments. The color scale shows the $5\sigma$ absolute companion magnitude, calculated using the contrasts shown in Fig. 15 and by translating the stellar apparent magnitudes to absolute magnitudes at 140 pc. The gray-dotted lines show 1 magnitude increments in absolute companion magnitude. For clarity, since the 1 Myr warm-start models are tightly clustered in absolute magnitude, we only contour the $1{\mathrm{M}}_{\mathrm{J}}$ model. Warm-start planets with masses $<10{\mathrm{M}}_{\mathrm{J}}$ are only detectable in the NIRCam observations for high stellar absolute magnitudes. Gray-shaded regions are regions of the parameter space where either no companion model was significant at the $5\sigma$ level, compared to the null (single point source) model, or regions where the target magnitude was below the bright limit.

Fig. 17

Five $\sigma$ circumplanetary accretion disk (planet mass times accretion rate) limits (solid lines) for all NIRC2 bandpasses and both techniques. The color scale shows the $5\sigma$ absolute companion magnitude, calculated using the contrasts shown in Fig. 15 and by translating the stellar apparent magnitudes to absolute magnitudes at 140 pc. The gray-dotted lines show 1 magnitude increments in absolute companion magnitude. Gray-shaded regions are regions of the parameter space where either no companion model was significant at the $5\sigma$ level, compared to the null (single point source) model, or regions where the target magnitude was below the bright limit.

3.3.

Keck OSIRIS

Figure 18 shows the achievable kernel phase contrast for OSIRIS as a function of ${\mathrm{K}}_{\mathrm{s}}$ stellar apparent magnitude for the central wavelength bin in the Kn3 bandpass ($\lambda =2.175\mu \mathrm{m}$). Contrasts of $\sim 5$ magnitudes are detectable at the $5\sigma$ level. This is lower than the achievable contrast for NIRC2 at broadband $K_s$, and the calibrated kernel phase scatter is also higher for OSIRIS (see Fig. 28) for target stars with the same brightness. This may be due to the fact that a single OSIRIS wavelength bin has 1700 times lower throughput than NIRC2: random noise sources from the sky and detector may be a more significant noise source for OSIRIS, degrading contrast.

Fig. 18

The colorscale shows the $5\sigma$ single companion contrast for OSIRIS kernel phase observations as a function of target star apparent magnitude and angular separation. The gray-dotted lines show 1 magnitude increments in companion contrast.

Figures 19 and 20 translate these contrast limits to planet mass limits as a function of stellar absolute magnitude. We again assume a distance of 140 pc. With OSIRIS, planet masses of 5 to $20{\mathrm{M}}_{\mathrm{J}}$ are detectable for stellar absolute magnitudes $>\sim 2$. Warm start models with masses $<10{\mathrm{M}}_{\mathrm{J}}$ are not detectable. Planet masses times accretion rates of $\sim {10}^{-5}{\mathrm{M}}_{\mathrm{J}}^2{\mathrm{yr}}^{-1}$ are detectable for the fainter stars ($K_s\sim 10$; $M_{K_s}\sim 5$). We note that these detection limits would be worse for the shortest wavelength bin ($\lambda =2.121\mu \mathrm{m}$) and better for the longest wavelength bin ($\lambda =2.229\mu \mathrm{m}$) due to the wavelength dependence of the Strehl ratio.

Fig. 19

Five $\sigma$ planet mass limits using 1 Myr DUSTY models (solid lines) for OSIRIS kernel phase. The color scale shows the $5\sigma$ absolute companion magnitude, calculated using the contrasts shown in Fig. 18 and by translating the stellar apparent magnitudes to absolute magnitudes at 140 pc. The gray-dotted lines show 1 magnitude increments in absolute companion magnitude.

Fig. 20

Five $\sigma$ circumplanetary accretion disk (planet mass times accretion rate) limits (solid lines) for OSIRIS. The color scale shows the $5\sigma$ absolute companion magnitude, calculated using the contrasts shown in Fig. 18 and by translating the stellar apparent magnitudes to absolute magnitudes at 140 pc. The gray-dotted lines show 1 magnitude increments in absolute companion magnitude.

3.4.

JWST NIRSpec

Figure 21 shows the achievable kernel phase contrast for JWST NIRSpec as a function of $L^{\prime }$ stellar apparent magnitude for the central wavelength bin ($\lambda =4.07\mu \mathrm{m}$), and Fig. 29 shows example histograms of the raw and calibrated kernel phases. Contrasts of $\sim 5$ magnitudes are detectable at the $5\sigma$ level for all target star magnitudes. Figures 22 and 23 translate these contrast limits to planet mass limits as a function of stellar absolute magnitude, assuming a distance of 140 pc. With NIRSpec, planet masses of a few $M_J$ are detectable for stellar absolute magnitudes $>\sim 4$ to 5. Warm start models with masses $<10{\mathrm{M}}_{\mathrm{J}}$ are not detectable. Planet masses times accretion rates of $\sim {10}^{-5}{\mathrm{M}}_{\mathrm{J}}^2{\mathrm{yr}}^{-1}$ are detectable for all target star absolute magnitudes, and lower accretion rates $\sim$ a few times ${10}^{-7}{\mathrm{M}}_{\mathrm{J}}^2{\mathrm{yr}}^{-1}$ to $\sim {10}^{-6}{\mathrm{M}}_{\mathrm{J}}^2{\mathrm{yr}}^{-1}$ are detectable for stars with absolute magnitudes $>\sim 5$. We note that these detection limits would be worse for the shortest wavelength bin ($\lambda =2.87\mu \mathrm{m}$), partly due to lower Strehl, and partly due to the fact that the PSF is more poorly sampled in this wavelength bin. They would be better for the longest wavelength bin ($\lambda =5.27\mu \mathrm{m}$) where the Strehl is higher and the PSF is larger.

Fig. 21

The colorscale shows the $5\sigma$ single companion contrast for NIRSpec kernel phase observations as a function of target star apparent magnitude and angular separation. The gray-dotted lines show 1 magnitude increments in companion contrast.

Fig. 22

Five $\sigma$ planet mass limits using 1 Myr DUSTY models (solid lines) for NIRSpec kernel phase. The color scale shows the $5\sigma$ absolute companion magnitude, calculated using the contrasts shown in Fig. 21 and by translating the stellar apparent magnitudes to absolute magnitudes at 140 pc. The gray-dotted lines show 1 magnitude increments in absolute companion magnitude.

Fig. 23

Five $\sigma$ circumplanetary accretion disk (planet mass times accretion rate) limits (solid lines) for NIRSpec kernel phase. The color scale shows the $5\sigma$ absolute companion magnitude, calculated using the contrasts shown in Fig. 21 and by translating the stellar apparent magnitudes to absolute magnitudes at 140 pc. The gray-dotted lines show 1 magnitude increments in absolute companion magnitude.

6.1.

Projection

We use the “Martinache” projection¹² to calculate kernel phases; this forms orthonormal combinations of Fourier phases that eliminate instrumental phase. The projection is based on representing Fourier phases as linear combinations of pupil-plane phases:

6.2.

Weighted Averaging

In interferometric observations, closure phases are calculated from bispectra, the product of three complex visibilities on baselines forming a triangle.¹⁰ When closure phases are calculated for a cube of images, the bispectra are averaged over the cube, and the phase of the average bispectrum is taken as the average closure phase.³⁷ Averaging bispectra rather than phases has been demonstrated to perform better in noisy conditions.³⁸ We thus generalize this form of vector averaging, which upweights higher signal-to-noise observations by including amplitude information in the average, from closure phase to kernel phase.

For each frame, for the $i$’th kernel phase, we calculate a complex quantity by taking a weighted product of complex visibilities: