Kepler Pixel Response Function Overview

• Kepler Pixel Response Function is the mapping of starlight onto CCD detectors, integrating the telescope’s optical PSF with the sub-pixel IPRF for flux conservation.
• It is computed through the convolution of the optical PSF and the intra-pixel response, calibrated via spot-scanning and data-driven regression techniques.
• This precise PRF characterization enhances photometric and astrometric accuracy, reducing flux and centroid errors and informing designs for future space missions.

The Kepler Pixel Response Function (PRF) is the end-to-end sensitivity kernel describing how starlight is mapped onto the Kepler Space Telescope's CCD detector array. Formally, it is the convolution of the optical point-spread function (PSF) of the telescope, incorporating all optical aberrations and spacecraft pointing jitter, with the sub-pixel–resolved intra-pixel response function (IPRF) of each 27 μm × 27 μm CCD pixel. The PRF quantitatively models the expected detector output as a function of a star’s sub-pixel centroid, enabling high-precision photometry and astrometry—especially in the undersampled, drift-prone Kepler and K2 regimes. Precise PRF characterization is foundational to the Kepler mission's SNR optimization, robust centroiding, and mitigation of inter- and intra-pixel sensitivity variations (Vorobiev et al., 2019, Bryson et al., 2010, Hogg et al., 2013).

1. Definition and Mathematical Formulation

The PRF is defined as the pixel-level mapping of incoming stellar flux, including all major instrumental and optical effects. For a star of monochromatic flux $F$ at continuous focal-plane position $(x_*, y_*)$, the PSF $\mathrm{PSF}(x, y; \lambda)$ represents the optical system response, normalized such that $\iint \mathrm{PSF}(x, y; \lambda)\,dx\,dy = 1$. The intra-pixel response function $\mathrm{IPRF}(x, y; \lambda)$ describes the relative sensitivity within a single pixel as a function of sub-pixel location and wavelength.

The total PRF is constructed as

$$\mathrm{PRF}(x, y; \lambda) = \iint \mathrm{PSF}(x', y'; \lambda)\; \mathrm{IPRF}(x - x',\, y - y'; \lambda)\; dx'\,dy'$$

This convolution predicts the response of the CCD to stellar images with arbitrary sub-pixel offsets. Discrete pixel-level responses are then given by integrating this function over the area corresponding to each pixel (Vorobiev et al., 2019, Hogg et al., 2013, Bryson et al., 2010).

A key property is normalization: for a properly normalized IPRF, the total PRF integrated over all pixels for a given source sums to unity, ensuring flux conservation even in the presence of sub-pixel sensitivity variations (Hogg et al., 2013).

2. Physical Origins: PSF, IPRF, and Spectral Dependence

Optical PSF

The telescope’s PSF is determined by the Schmidt optical design, chromatic aberrations, focus changes, and integrated pointing jitter over a typical long-cadence exposure (29.4 min). The PSF is empirically measured and super-sampled for precise modeling, incorporating channel-specific spatial variation (Bryson et al., 2010, Vorobiev et al., 2019).

Intra-Pixel Response Function (IPRF)

The IPRF quantifies sub-pixel sensitivity—critical for Kepler’s undersampled regime, where much stellar flux falls within a single pixel. Direct measurements on Kepler’s e2v CCD90 sensors using spot-scan techniques have revealed that pixels are substantially more sensitive at their centers, with responsivity dropping by ≈50% near edges and ≈70% at corners. This nonuniformity is strongly wavelength dependent:

• At short wavelengths (400–500 nm), photons are absorbed near the Si surface and generate significant lateral diffusion ($\sigma_\mathrm{diff} \approx 3$–5 μm), producing a gentle, nearly circular IPRF profile.
• At long wavelengths (≥700 nm), absorption occurs deeper in the depletion region, with minimal lateral diffusion ($\lesssim 1$ μm). The IPRF is dominated by the rectangular gate geometry, yielding a sharp, nearly square profile with a rapid fall-off at the pixel boundary.

Empirical IPRF maps are constructed on grids as fine as 27×27 (1 μm spacing), and the intrinsic IPRF is recovered by deconvolving the spot-illumination PSF (Lucy–Richardson deconvolution) (Vorobiev et al., 2019, Vorobiev et al., 2018).

Chromatic Effects

The PRF exhibits weak chromatic dependence due to both the optics and the CCD’s wavelength-dependent absorption profile. This translates to wavelength-dependent broadening or narrowing of the PRF core. Hedges et al. (Hedges et al., 2021) have demonstrated that effective wavelengths across different pixels of the same target can vary from 550 nm to 750 nm, with a mean of about 646 nm and $\sigma \approx 40$ nm, allowing coarse multi-wavelength photometry from Kepler data.

3. On-Orbit PRF Mapping and Representation

On-orbit, the PRF is measured by dithering bright, isolated targets over a grid of sub-pixel positions and fitting their pixel outputs to construct a piecewise-continuous polynomial model. Each $n\times n$ array of PRF pixels is subdivided into $(x_*, y_*)$0 sub-pixel patches, with 2D variable-order polynomials describing the flux distribution within each patch: $(x_*, y_*)$1 Optimal polynomial order in each patch is determined using a modified Akaike Information Criterion, and spatial variations across the CCD channel are modeled by storing five PRF “stamps” per output channel (center and corners) with barycentric and sub-pixel spline interpolations for arbitrary stellar positions (Bryson et al., 2010, Poleski et al., 2018).

Discontinuities between sub-pixel polynomial patches are handled via smoothing functions at patch boundaries.

4. Calibration, Fitting, and Data-Driven Approaches

Laboratory Calibration

Spot-scanning with sub-3 μm spots across each pixel at multiple wavelengths provides direct laboratory measurements of the IPRF. Errors in IPRF measurements are dominated by source drift, CCD read noise, and stage repeatability. Repeated scans show that $(x_*, y_*)$2 near pixel centers, rising to ≈1% at edges (Vorobiev et al., 2019, Vorobiev et al., 2018).

Data-Driven and Physics-Driven Fitting

Image modeling in Kepler/K2 analysis alternates between physics-driven (PSF+IPRF convolution, per-pixel flat-field) and fully data-driven (flexible regularized regressions) approaches:

• The iterative physics-driven procedure alternates fitting for fluxes, PSF parameters, per-pixel intra-pixel sensitivities, and backgrounds using alternating-least-squares or MCMC. Sub-pixel sensitivity maps $(x_*, y_*)$3 are regularized (e.g., Tikhonov) to avoid overfitting.
• The Causal Pixel Model and variants for crowded K2 fields integrate the channel-specific, spatially interpolated PRF into their data-driven decorrelation pipelines, yielding order-of-magnitude improvements in photometric precision relative to difference-imaging methods (Poleski et al., 2018).
• Data-driven, pixel-by-pixel linear models employing a basis of cotrending vectors, low-order polynomials in centroid position, and B-splines have enabled disentangling instrumental systematics from astrophysical variability and even extracting color information from the weak chromatic dependence of the PRF (Hedges et al., 2021).

5. Impact on Photometric and Astrometric Precision

Comprehensive PRF modeling yields substantial gains for both photometry and astrometry:

• Photometry: Centroid drift in under-sampled regimes (e.g., ≈0.5 px in K2) would induce flux variations ≲500 ppm across pixel boundaries if IPRF is ignored. Applying the measured PRF reduces these errors to ≲50 ppm, restoring nearly original Kepler precision (≈30 ppm over 6 hr for 10th mag stars) (Vorobiev et al., 2019, Hogg et al., 2013, Bryson et al., 2010).
• Astrometry: Centroid offsets due to unmodeled intra-pixel sensitivity can reach 0.05 px (≈200 mas). Employing the empirically determined PRF reduces centroid biases to <0.005 px (≲1 mas), enabling milliarcsecond localization and improved deblending (Vorobiev et al., 2019, Bryson et al., 2010).
• Multi-wavelength analysis: PRF chromaticity enables “color aperture photometry” and astrophysical validation of variable and exoplanet candidates, allowing cross-validation of achromatic transits (planets) versus chromatic features (eclipsing binaries, flares) (Hedges et al., 2021).

6. Applications and Extensions to Other Missions

Precise PRF characterization enables critical operations and science tasks for Kepler and other missions:

Application Domain	Role of PRF Modeling	Quantitative Impact
Optimal Pixel Selection	Selection of photometric apertures	Photon-limited photometry at 20–50 ppm
PRF-based Centroiding	High-precision astrometry, plate scale	Sub-millipixel centroiding
Crowded-field Photometry	Decontamination in dense K2 or bulge fields	Factor of 6–14 improvement in precision
Data-driven Systematics Removal	Regression of instrumental trends	Residuals ≈200 ppm (bright)
Multi-band Photometry	Extraction of per-pixel effective λ$(x_*, y_*)$4	σ(λ$(x_*, y_*)$5) ≈ 40 nm

Direct IPRF measurement, spot-scan-convolved PRF construction, and data-driven PRF regression pipelines are readily transferrable to missions such as TESS. For TESS, whose 21 μm pixels and broader passband (600–1000 nm) further exacerbate sub-pixel sensitivity variations, analogous spot-scan and analytic methods are being adopted to support millimagnitude photometric precision and milliarcsecond centroiding (Vorobiev et al., 2019, Vorobiev et al., 2018, Hedges et al., 2021).

7. Concluding Synopsis

The Kepler PRF, built as the convolution of empirically measured, wavelength-dependent IPRF maps with a precisely reconstructed telescope+spacecraft PSF, is fundamental to extracting high-fidelity light curves and accurate source positions from highly undersampled, crowded, and dynamic focal-plane data. PRF-based modeling corrects sub-pixel systematics at the tens-of-ppm level, restoring the mission’s design photometric and astrometric performance. As new missions adopt similarly undersampled architectures, the PRF methodology—melding direct detector characterization, analytic convolution, and flexible data-driven regression—remains central to precision time-domain astrophysics (Vorobiev et al., 2019, Bryson et al., 2010, Hogg et al., 2013, Poleski et al., 2018, Hedges et al., 2021, Vorobiev et al., 2018).