Kepler P-PRF: Instrument Response Model

• P-PRF is a composite response model that maps stellar light to pixel signals by integrating the optical PSF, pointing jitter, and systematic instrument effects.
• It employs a piecewise-continuous, two-dimensional polynomial mesh to achieve sub-pixel resolution, ensuring smooth transitions between calibration patches.
• The model enables optimal pixel selection and precise centroiding, underpinning Kepler’s ability to detect Earth-size exoplanet transits with ~100 ppm precision.

The Kepler Pixel Response Function (PRF) is a composite instrument response model essential to achieving the photometric precision required by the Kepler mission to detect transits of Earth-size exoplanets. Unlike a standard optical point spread function (PSF), the PRF formally incorporates not only the PSF but also spacecraft pointing jitter and additional instrumental and systematic effects during the nominal 29.4-minute cadence. The representation, construction, and applications of the PRF are central to maximally exploiting the Kepler photometer's capability for detecting sub-100 ppm transit signals. The structure and detail of the PRF modeling pipeline further serve as a methodological reference point for photometric analysis in large-scale exoplanet surveys.

1. Definition and Physical Basis

The Kepler PRF is defined as the system-level response mapping incident stellar flux to the observed signal in each detector pixel, formally encompassing:

• The optical PSF, capturing spatial distribution arising from telescope diffraction and aberrations.
• Pointing jitter, encoding temporal smearing within an integration capsule due to attitude instabilities.
• Systematic effects, e.g., charge transfer inefficiencies or shutterless smear, occurring during readout or integration.

This composite approach ensures that the PRF accurately models how a point source illuminates pixels during a nominal observation, yielding predictions of pixel signal as a continuous function of a star’s intra-pixel (sub-pixel) position. Accurate modeling of these combined effects underpins precision photometric extraction and centroid determination, which are crucial for planet transit searches and spacecraft calibration.

2. Mathematical Representation: Piecewise-Polynomial Sub-Pixel Mesh

To achieve sub-pixel resolution, the PRF is represented not as a global analytic or empirical function but as an array of two-dimensional, piecewise-continuous polynomials. For an n×n pixel region centered on a star (with n typically 11 to 15 depending on the channel), each pixel is subdivided into an m×m sub-pixel mesh (with m commonly 6), leading to a total of n²×m² unique polynomial patches per output channel.

Formally, within the (i,j)th pixel and (s,t)th sub-pixel, the local response is:

$\mathrm{PRF}_{i,j,s,t}(x, y)$

where $x, y$ are real-valued sub-pixel coordinates. Polynomial orders are selected via a modified Akaike Information Criterion minimizing:

$2c + N\log(\mu(o)) + \frac{2c(c-1)}{N-c-1}$

with $c$ the number of coefficients at order $o$, $N$ sample count, and $\mu(o)$ mean-square error, thus formally balancing model complexity and fit quality.

Boundary discontinuities between polynomial patches are addressed in two stages:

• Overlapping fit phase: Fitting uses overlapping data from neighboring sub-pixel regions.
• Evaluation phase smoothing: At evaluation, responses near boundaries are blended via an infinitely differentiable weight function $w(z)$, with:

$$w(z) = \frac{f(z)}{f(z) + f(1-z)},\quad f(z) = \begin{cases} \exp(-1/z^{a}) & z > 0 \ 0 & z \leq 0 \end{cases}$$

ensuring smooth transitions and continuity in response derivatives.

3. PRF Construction and On-Orbit Super-Resolution

Empirical determination of the PRF employs on-orbit calibration, using observations of bright, isolated stars across the focal plane. Super-resolution is achieved by aggregating many nominally undersampled star images with different sub-pixel positions, a technique akin to dithered imaging. The model is constructed separately at five anchor locations per output channel (center and four corners) to account for intra-channel variation. At runtime, linear spatial interpolation of these five PRFs yields a high-fidelity, local response model for arbitrary positions within a channel.

Because polynomial fitting is carried out per patch and only limited by available calibration data, the far PRF wings require careful background subtraction. To counteract contamination by background sources in outer regions—especially in small calibration sets—distant-wing fits are sometimes replaced with solutions from larger calibration data sets in a controlled "hybrid" approach.

4. Applications: Optimal Pixel Selection and Astrometric Centroiding

4.1. Photometric Aperture Optimization

For downlink efficiency and SNR maximization, Kepler must select an optimal subset of pixels per star. Using the locally appropriate PRF, two synthetic images are generated:

• A pure target image,
• A scene image including all catalogued sources and modeled background.

Pixels are ranked by per-pixel SNR, accounting for shot noise, read noise, etc. Pixels are iteratively accumulated in order of SNR contribution until the SNR stops increasing, defining the aperture. This procedure ensures that photometric extraction for a given star is signal-maximal within strict telemetry constraints.

4.2. PRF-Fitted Centroids for Attitude and Plate Scale Tracking

The PRF is used to perform non-linear least squares fits to pixel-level star images, thereby extracting robust and precise centroids that reflect the star's sub-pixel position on the detector ($\sim0.1$ millipixel stability). This centroid information is fundamental for:

• Continuous calibration of the focal plane geometry model,
• Correction of instrumental effects such as attitude jitter,
• Refinement of the spacecraft's plate scale and pointing knowledge.

5. Spatial Variation and Iterative Model Registration

The PRF is not invariant over the field of view due to optical and systematic effects. Thus, per-channel variation is modeled by interpolating between five anchor PRFs as described above. Moreover, there is a tight coupling between the PRF determination and the focal plane geometry (FPG): precise PRF centroiding is only possible with accurate FPG, but FPG cannot be constructed without high-quality centroids. An iterative process "pins" the PRF centroid to a reference point during convergence, aligning geometric and photometric self-consistency.

6. Addressing Model Discontinuities and Data Limitations

Primary challenges in the PRF approach are:

• Patch boundary smoothness: Smoothed blending at sub-pixel patch boundaries via $w(z)$ weighting avoids artificial steps in the predicted response.
• Data sparsity and background contamination: For regions with weak calibration data or high background, the hybrid approach ensures statistical stability.
• System evolution and flexibility: Given long-term on-orbit drift or instrument evolution, the methodology's modularity allows for efficient periodic recalibration or update.

7. Impact on Photometric Precision and Kepler Science

The rigorous construction and maintenance of the Kepler PRF is a foundational component enabling the mission’s reported photometric precision of $\sim$100 ppm. PRF-based synthetic aperture definition and centroid tracking minimize systematic noise, maximize signal extraction fidelity, and support accurate attitude knowledge. This level of modeling directly enables the detection and characterization of transits from Earth-size planets, supporting both the mission's primary scientific objective and long-term photometric stability.

The methodological approach—particularly the use of piecewise-continuous polynomial meshes and super-resolution calibration—sets a benchmark for future photometric surveys requiring sub-pixel calibration and precision-limited by instrumental, not astrophysical, factors.