ASTR 210 · Lecture 07

Astrometry and Plate Solutions

Geometry plus calibration plus residual analysis

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.

Reading anchor: OpenStax Astronomy 2e, Chapter 19.2 (surveying the stars, parallax and astrometric fundamentals).

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.

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.

First question: what here is directly measured, and what is inferred from a model?

Terms for This Lecture

centroid

Definition: An estimated subpixel position of a source, usually computed from the intensity distribution around its image.

Why it matters here: Accurate centroids are the measured image coordinates used to fit a plate solution and track motion.

Relationships: Centroid uncertainty depends on point-spread function, SNR, sampling, and background, and propagates into astrometric residuals.

Boundary / common confusion: A centroid is not necessarily the brightest pixel and can be biased by blends, saturation, or asymmetric PSFs.

Evidence We Need to Explain

Model

Quantitative Tool

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

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.

Visual Reasoning

Compare the pixel-plane vector to the matched sky-plane vector.

  • Why must both vectors correspond to the same physical displacement?
  • What would a mismatch between the two vector lengths imply about the plate solution?
Lecture 07: Astrometry and Plate SolutionsPixel planeimage 1 (520.3, 411.2)image 2 (534.7, 415.8)Δpixel = 15.12 pxSky plane (RA, Dec)(132.1021°, 12.5510°)(132.0988°, 12.5521°)Δsky = 12.25 arcsecPlate scale = Δsky/Δpixel = 0.811 arcsec/pixel, derived from the same asteroid-track pair used in the worked example.
The measured pixel displacement and its matched sky-coordinate displacement.

Common Pitfall

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 Segment

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 or 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

Astrometry is geometry plus calibration plus residual analysis — a plate scale is only as good as the consistency check that follows it.

References