AC01 · Measuring the depth of the sky

Stellar Parallax and Proper Motion

Can two stars beside each other on the sky be far apart in space? They can be far apart. A nearby star shifts more against the distant sky as Earth orbits the Sun. Measuring that yearly wobble gives its distance; a steady drift across the sky is its proper motion.

Drag a star nearer and watch Earth view it from different sides of its orbit. Separate the annual wobble from steady motion using a fitted sequence of simulated positions.

True distance $20.0\,\mathrm{pc}$

Fitted parallax $49.6\,\mathrm{mas}$

Parallax uncertainty $0.907\,\mathrm{mas}$

Fitted eastward motion $18.5\,\mathrm{mas\,yr^{-1}}$

Give the sky some depth

Distance

20

r\;[\mathrm{pc}]

Ecliptic latitude

38

\beta\;[\mathrm{deg}]

Eastward proper motion

18

\mu_{\lambda *}\;[\mathrm{mas\,yr^{-1}}]

Measurement uncertainty

Position uncertainty

4

\sigma\;[\mathrm{mas}]

Drag the glowing star sideways to change its distance, or vertically to change its ecliptic latitude.

How the observations are made

Each setting generates the same twenty-five noise draws at equally spaced dates over 2.4 years. The two coordinate errors within an epoch have correlation minus 0.35; different epochs are independent. A generalized least-squares fit solves two starting positions, two proper motions and one parallax.

Read from geometry to the Gaia catalogue

Model assumptions

The observer follows a circular one-au orbit in the ecliptic plane. Angular motion uses a first-order tangent-plane, single-star model with constant proper motion. The spatial drawing compresses stellar distances and enlarges stars; the angle plot uses milliarcseconds. Twenty-five synthetic two-coordinate observations have known Gaussian noise and a fixed within-epoch correlation; they are not Gaia epoch observations. The separate tutorial uses a frozen Gaia DR3 catalogue sample. Aberration, light deflection, eccentricity, radial acceleration, binaries and the Gaia scanning law are omitted.

East–north angular position

Eastward position over time

Fit quality and model limits

Error ellipses show one standard deviation along their principal axes. The synthetic uncertainty is known, so the parameter covariance is not rescaled by the residual scatter. The three-dimensional drawing compresses distances; the angular plots use the tangent-plane model. Real Gaia catalogue errors and the separate public-data tutorial have their own provenance.

Start with the essentials

Focus question: What separates distance from a star’s own motion?
One-sentence intuition: Parallax repeats annually; constant proper motion accumulates steadily.

Core mathematical model

The geometric distance

\varpi\,[\mathrm{mas}]=\frac{1000}{r\,[\mathrm{pc}]}

Parallax is the angular semimajor axis for an observer baseline of one astronomical unit. Opposite points of Earth’s orbit can separate apparent positions by twice this angle.

Fit the repeated and accumulating parts together

x(t)=x_0+\mu_x t-\varpi\sin(2\pi t),\qquad y(t)=y_0+\mu_y t+\varpi\sin\beta\cos(2\pi t)

Time is in years, tangent positions and parallax in milliarcseconds, and proper motions in milliarcseconds per year. The star has ecliptic longitude zero and latitude beta; the circular observer orbit starts on the positive x axis.

Common difficulties

The whole wobble is twice the parallax

Typical misconceptionThe star moves one parallax between observations six months apart.

Better mental modelParallax is the semimajor axis. At favorable orbital phases, observations six months apart span both sides of the ellipse and differ by twice that value.

An ellipse can drift across the sky

Typical misconceptionThe star must return to exactly the same place every year.

Better mental modelThe periodic parallax repeats, but proper motion carries the whole pattern onward. The fit uses both at once.

Run the experiment

01
Move the star farther away
Choose A nearby moving star, then drag the glowing star to the right.
What to observe: The parallax falls as distance grows. The synthetic positions retain the same noise draws, making the smaller wobble harder to distinguish.
02
Look along the orbital plane
Choose A line on the ecliptic, then A circle at the pole.
What to observe: The projected orbit is a line in the ecliptic and a circle at its pole. Intermediate latitudes give ellipses.
03
Separate the steady drift
Keep the distance fixed and move Eastward proper motion through zero. Export the observations.
What to observe: The track changes from a moving loop to a closed ellipse. The exported record contains the very points that were fitted.

Stellar Parallax and Proper Motion

The sky position draws a moving loop

A nearby star moves against the distant sky

Core mathematical model

The geometric distance

Fit the repeated and accumulating parts together

Common difficulties

The whole wobble is twice the parallax

An ellipse can drift across the sky

Run the experiment

Move the star farther away

Look along the orbital plane

Separate the steady drift

Related experiments and articles

Measuring the sky

Stellar positions and distances