Lecture 07: Astrometry and Plate Solutions

ASTR 210 Observational Astronomy and Data Reduction

Context and Why This Matters

Pixels become right ascension and declination only after an image is matched to a reference catalog through a plate solution (World Coordinate System). Astrometry underlies proper motions, orbit determination for asteroids and comets, and the reference frames used for every other kind of positional measurement.

Learning Goals

  1. Explain how a plate solution links pixel coordinates to sky coordinates.
  2. Derive a plate scale from two matched image/catalog coordinate pairs.
  3. State what residual analysis after a plate solution tells you about systematic error.

Opening Phenomenon

An asteroid imaged twice, minutes apart, shifts noticeably in pixel coordinates. Converting that pixel shift into an angular sky motion requires knowing the plate scale (arcseconds per pixel), which is itself derived from the same data.

Terms Developed in Context

These terms form the measurement chain for Astrometry and Plate Solutions: each connects an observable to a model choice, and each has a boundary that must be respected when making a scientific claim.

centroid

An estimated subpixel position of a source, usually computed from the intensity distribution around its image. Accurate centroids are the measured image coordinates used to fit a plate solution and track motion.

Centroid uncertainty depends on point-spread function, SNR, sampling, and background, and propagates into astrometric residuals. A centroid is not necessarily the brightest pixel and can be biased by blends, saturation, or asymmetric PSFs.

Developed in this lecture: The term helps interpret this evidence: Image 1: (x, y) = (520.3, 411.2) px corresponds to (RA, Dec) = (132.1021°, 12.5510°). It also constrains the working model: A plate solution is a geometric transformation, calibrated using catalog stars, from pixel coordinates to sky coordinates.

plate scale

The angular sky distance represented by one detector pixel, typically in arcseconds per pixel. It converts measured pixel separations into angular separations and sets the sampling of the seeing disk.

Locally, \(s=\Delta\theta/\Delta p\); a full WCS also includes rotation and distortion. A single constant scale is only an approximation when optics introduce significant field distortion.

Developed in this lecture: The term helps interpret this evidence: Image 1: (x, y) = (520.3, 411.2) px corresponds to (RA, Dec) = (132.1021°, 12.5510°). It also constrains the working model: A plate solution is a geometric transformation, calibrated using catalog stars, from pixel coordinates to sky coordinates.

World Coordinate System (WCS)

Metadata and a transformation that map image pixel coordinates to celestial coordinates. WCS lets an image be compared with catalogs and allows positions to be reported on the sky rather than in detector pixels.

A plate solution fits transformation parameters using matched reference stars and records them in image headers. A WCS can be present but inaccurate; residuals and reference coverage must be inspected.

Developed in this lecture: The term helps interpret this evidence: Image 1: (x, y) = (520.3, 411.2) px corresponds to (RA, Dec) = (132.1021°, 12.5510°). It also constrains the working model: A plate solution is a geometric transformation, calibrated using catalog stars, from pixel coordinates to sky coordinates.

astrometric residual

The positional difference between a reference catalog coordinate and the coordinate predicted by the fitted plate solution. Residuals diagnose whether the transformation is accurate enough for the intended position or motion measurement.

Residual vectors reveal translation, rotation, scale, and distortion errors; their RMS summarizes typical fit scatter. A small global RMS can hide local patterns or a few misidentified reference stars.

Developed in this lecture: The term helps interpret this evidence: Image 1: (x, y) = (520.3, 411.2) px corresponds to (RA, Dec) = (132.1021°, 12.5510°). It also constrains the working model: A plate solution is a geometric transformation, calibrated using catalog stars, from pixel coordinates to sky coordinates.

proper motion

The angular change in a star’s sky position per unit time after accounting for coordinate conventions. It is an astrophysical signal measured by comparing calibrated positions across epochs.

Proper motion is \(\mu=\Delta\theta/\Delta t\) and must be distinguished from parallax and plate-solution error. It is only the transverse angular motion; converting to tangential speed requires distance.

Developed in this lecture: The term helps interpret this evidence: Image 1: (x, y) = (520.3, 411.2) px corresponds to (RA, Dec) = (132.1021°, 12.5510°). It also constrains the working model: A plate solution is a geometric transformation, calibrated using catalog stars, from pixel coordinates to sky coordinates.

reference catalog

A tabulated set of sources with externally calibrated celestial positions and associated quality information. It anchors the pixel-to-sky transformation and supplies the standards against which residuals are computed.

Matched catalog stars constrain translation, scale, rotation, and distortion in the plate solution. Catalog positions have uncertainties, epochs, flags, and selection effects; they are not error-free truth.

Developed in this lecture: The term helps interpret this evidence: Image 1: (x, y) = (520.3, 411.2) px corresponds to (RA, Dec) = (132.1021°, 12.5510°). It also constrains the working model: A plate solution is a geometric transformation, calibrated using catalog stars, from pixel coordinates to sky coordinates.

pixel-to-sky transformation

A mathematical mapping from detector coordinates \((x,y)\) to celestial coordinates such as right ascension and declination. It is the model that turns measured centroids into scientifically comparable sky positions.

The transformation may range from a linear scale/rotation model to a polynomial distortion model encoded in WCS. A transformation fitted over one region or instrument configuration should not be extrapolated without checking residuals.

Developed in this lecture: The term helps interpret this evidence: Image 1: (x, y) = (520.3, 411.2) px corresponds to (RA, Dec) = (132.1021°, 12.5510°). It also constrains the working model: A plate solution is a geometric transformation, calibrated using catalog stars, from pixel coordinates to sky coordinates.

Evidence

Model

Working Equation

\[ \text{plate scale} = \dfrac{\Delta(\text{sky separation, arcsec})}{\Delta(\text{pixel separation, px})} \]

Worked Example

  1. Pixel displacement: Δx = 14.4 px, Δy = 4.6 px, so the total pixel separation is √(14.4² + 4.6²) = 15.12 px.
  2. Sky displacement: ΔRA·cos(Dec) = 11.596 arcsec, ΔDec = 3.960 arcsec, so the total angular separation is 12.25 arcsec.
  3. Plate scale = 12.25 arcsec / 15.12 px = 0.811 arcsec/pixel.
  4. This plate scale, once derived, can be applied to any other pixel measurement on the same image without re-deriving it from scratch — but only if the optical setup has not changed.

Common Misconception

Treating a measured centroid as the final position without asking about the plate-solution residual. A centroid can be very precise (small statistical uncertainty) yet still biased by an uncorrected optical distortion.

Active-Learning Guidance

Given a third image/catalog coordinate pair, check whether it predicts the same plate scale derived above to within about 5%, and discuss what a large discrepancy would imply.

Lab / Observing Connection

Lab 05 (Astrometry and Motion) extends this two-point plate-scale derivation to a full three-image asteroid track and asks for a proper-motion estimate.

Synthesis Questions

References