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Sandbox Physics
Measuring the depth of the sky

L3 · Stellar astrometry

Separate parallax and proper motion in noisy positions

Sandbox Physics ·

One position cannot separate a star’s own drift from the effect of Earth’s changing viewpoint. Positions at different dates reveal the difference between a straight drift and a repeating annual curve.

Each date supplies two equations

At observation ii, the date tit_i and known ecliptic latitude determine parallax factors PxiP_{xi} and PyiP_{yi}. Five constants remain unknown: two starting positions, two proper motions and one parallax.

Put them in a parameter column θ=(x0,y0,μx,μy,ϖ)T\boldsymbol\theta=(x_0,y_0,\mu_x,\mu_y,\varpi)^{\mathsf T}. Each eastward equation uses the eastward starting position, eastward motion and parallax. Each northward equation uses its corresponding starting position and motion, sharing the same parallax.

Twenty-five observations give fifty coordinate measurements. Their coefficients form a fifty-by-five design matrix AA. This matrix describes how every parameter affects every measurement. The observation column z\boldsymbol z holds the measured positions.

(xiyi)=(10ti0Pxi010tiPyi)θ+ϵi\begin{pmatrix}x_i\\y_i\end{pmatrix}=\begin{pmatrix}1&0&t_i&0&P_{xi}\\0&1&0&t_i&P_{yi}\end{pmatrix}\boldsymbol\theta+\boldsymbol\epsilon_i
The error column has eastward and northward components. Positions and parallax use milliarcseconds; proper motions use milliarcseconds per year.

A tilted error ellipse changes the weights

Variance is the mean squared measurement error across repeated observations. Its square root is the standard deviation σ\sigma. If an eastward error tends to accompany a southward error, the two coordinates are correlated. Covariance measures their joint variation; dividing by the two standard deviations gives a dimensionless correlation coefficient.

The synthetic record uses equal standard deviations in both coordinates and within-epoch correlation minus 0.35. Its error ellipse is tilted. Displacements along the long axis are more plausible than equal displacements along the short axis. Different dates are independent, so the full covariance CC consists of two-by-two blocks.

An ordinary squared distance treats those two residual directions alike. Generalized least squares uses the inverse covariance to distinguish them. It asks whether the residual is large compared with the uncertainty in that particular direction.

Ci=σ2(1ρρ1),ρ=−0.35C_i=\sigma^2\begin{pmatrix}1&\rho\\\rho&1\end{pmatrix},\qquad\rho=-0.35
This is correlation between two coordinates in one synthetic observation. Gaia’s catalogue correlations instead describe errors among its fitted astrometric parameters.

Minimize the uncertainty-weighted residuals

For proposed parameters, the predicted observations are AθA\boldsymbol\theta. Subtract them from the observed positions to obtain residuals. Their covariance-weighted squared sum is chi-square. Smaller values indicate a closer fit relative to the stated measurement errors.

Chi-square is quadratic in the five parameters. Set its derivative with respect to each parameter to zero and obtain five linear normal equations. Form their coefficient matrix and right-hand side, then solve. This small system is inexpensive enough to update continuously in a browser.

When the Gaussian noise variance is known, the model is correct and the design matrix has full rank, the estimate is unbiased. Its covariance has the analytic form below. The square root of the parallax diagonal entry is the reported parallax uncertainty.

χ2=(z−Aθ)TC−1(z−Aθ)\chi^2=(\boldsymbol z-A\boldsymbol\theta)^{\mathsf T}C^{-1}(\boldsymbol z-A\boldsymbol\theta)
Chi-square is dimensionless and measures residual size relative to the measurement uncertainty.
θ^=(ATC−1A)−1ATC−1z,K=(ATC−1A)−1\widehat{\boldsymbol\theta}=(A^{\mathsf T}C^{-1}A)^{-1}A^{\mathsf T}C^{-1}\boldsymbol z,\qquad K=(A^{\mathsf T}C^{-1}A)^{-1}
The noise variance is known, so no residual-based variance scale is multiplied into this covariance. Production numerical work commonly solves using QR or Cholesky factorizations; this small teaching system uses pivoted elimination.

Dates determine which motions can be separated

Imagine observing only on the same calendar date each year. The parallax factors repeat, and a constant annual displacement can be absorbed into the starting position. More years do not supply the missing orbital phases. More measurements cannot replace geometry that was never observed.

A very short observation span presents a related problem. A small part of a sinusoid resembles a straight line, so parallax and proper motion can compensate for each other. Observations across seasons and years make the distinction visible.

The synthetic sequence uses twenty-five equally spaced dates spanning 2.4 years. A reference epoch near the middle reduces entanglement between starting position and proper motion. Increasing the noise reduces precision. The dates still determine which geometric patterns can be distinguished.

Test intervals by repeating the measurement

The default true parallax is fifty milliarcseconds. A new noise draw produces new measured positions and another parallax estimate. Under the stated model, subtract the true parallax from the estimate and divide by the reported standard error. The result should follow a standard normal distribution.

About ninety-five percent of repeated records should place the true value inside the estimate plus or minus 1.96 standard errors. This is coverage. It describes the performance of intervals across repeated measurements.

The frozen default uses distance twenty parsecs, latitude thirty-eight degrees, eastward motion eighteen and northward motion 7.2 milliarcseconds per year, with four-milliarcsecond coordinate errors and seed 3817. The fitted parallax is 49.562 milliarcseconds with standard error 0.907; fitted eastward motion is 18.499 milliarcseconds per year. Residual chi-square is 44.64 for forty-five degrees of freedom. The model script and exported default reproduce these numbers. The accompanying test generates twelve hundred independent records. It checks parallax-interval coverage, the mean standardized error and the mean residual chi-square. Fifty coordinate measurements minus five fitted parameters leave forty-five residual degrees of freedom. Correctly specified noise gives mean chi-square near forty-five.

Real data can contain binary motion, calibration error or correlations between dates. The coverage test validates the stated synthetic model; real Gaia errors require their own checks.

Reading: Lindegren et al. · Lindegren et al.

Try the geometry

Keep the noise draws and change the signal

  1. Compare the nearby and distant presets. Record true distance, fitted parallax and standard error, then explain which changes most strongly.
  2. Choose The wobble in noise and inspect how much wider the parallax uncertainty becomes.
  3. Export the record and identify fifty coordinate values, five parameters, twenty-five dates and the within-epoch correlation.

Exercises and explained answers

A fit gives parallax fifty-two milliarcseconds with standard error 1.5. Does its approximate ninety-five-percent interval include the true value fifty?

The interval is about 49.1 to 54.9 milliarcseconds, so it includes fifty. One successful interval cannot establish coverage; that requires repeated records.

Why does ignoring a correlation not always underestimate parameter uncertainty?

Correlation lengthens the error ellipse in some directions and shortens it in others. Ignoring it changes the directional weights. The resulting parameter variance can increase or decrease depending on the observation geometry.

Does exactly zero residual chi-square show that a measurement is more precise than its noise specification?

One record cannot establish that. Exactly zero residuals are essentially impossible for independent continuous Gaussian draws. Check whether the record was constructed as a noiseless reference or whether noise was actually added.

Sources and calculation scope

  1. Lindegren et al., Gaia EDR3 — The astrometric solution, arXiv:2012.03380v1

    Sections 3.1, 5.3–5.7; equation 23 and figures 10–12. The fixed single-source model and covariance conventions apply to the basic DR3 astrometry inherited from EDR3.

  2. Lindegren et al., Parallax bias versus magnitude, colour, and position, arXiv:2012.01742v2

    Sections 2.3, 4.1 and 8; figures 2, 5, 21–22 and tables 9–10. The tutorial applies no zero-point correction and does not claim calibrated catalogue distances.

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