Radial and transverse motion

A velocity has a direction as well as a magnitude

Relative to an observer and target direction at a specified epoch, radial velocity is the component along the line of sight. Transverse velocity lies in the perpendicular plane. These components refer to one declared coordinate frame and origin; they are not automatically frame-independent properties of the source.

Choose positive radial velocity away from the observer, so a negative value means approach in this convention. Speed is the nonnegative magnitude of a velocity vector. An approaching source can have a negative radial component while retaining a positive total speed.

The transverse plane has two orthogonal directions. To combine component measurements, use compatible units and axes. In particular, an unweighted derivative of right ascension is not automatically the corresponding orthogonal angular component: the usual tangential component includes cos δ, where δ is declination.

Here we use two local orthogonal sky axes directly. The numerical example concerns nonrelativistic relative motion at one epoch. It is not a reconstruction of a real star’s past or future trajectory.

Angular drift alone lacks a physical length scale

For a local angular rate μ measured in radians per second and geometric distance d in metres, transverse speed is v_t = μd in the instantaneous angular-motion relation. Angles in this formula must use radians rather than treating arcseconds as though they were unitless numerical radians.

With proper motion in arcseconds per Julian year and distance in parsecs, the numerical conversion is v_t in km/s = 4.74047046 × μ × d. One Julian year is 365.25 days of 86,400 SI seconds; the factor comes from one au per Julian year expressed in kilometres per second.

The annual-parallax convention d/pc = 1/[p/arcsec] gives the equivalent numerical form v_t in km/s = 4.74047046 × μ/(p), when μ is in mas per Julian year and p in mas. The angular unit must be the same in numerator and denominator.

These equations condition on an appropriate distance or parallax estimate and a proper motion separated from annual observer-driven motion. They do not license unrestricted inversion of a noisy or negative fitted parallax, nor conversion of an arbitrary apparent image displacement into a source velocity.

Use orthogonal components before taking the magnitude

Stipulate proper-motion components (+60, +80) mas per Julian year and a positive precise parallax p = 20 mas. The angular-rate magnitude is √(60² + 80²) = 100 mas per Julian year, and the conventional distance is 50 pc.

Applying the conversion to each component gives transverse velocities about +14.2214 and +18.9619 km/s. Their magnitude is about 23.7024 km/s. The two component signs describe directions along the chosen sky axes, while this magnitude is nonnegative.

Add an independently stipulated signed radial component −20 km/s in the same frame. Orthogonality gives total speed √(14.2214² + 18.9619² + 20²) ≈ 31.0129 km/s, using unrounded components for the calculation.

The negative radial sign states approach; it does not reduce the total speed by subtracting 20 from the transverse magnitude. Radial and transverse velocities point in perpendicular directions, so their magnitudes combine through the sum of squares.

A spectrum supplies a conditional motion constraint

A calibrated displacement of spectral features can constrain radial motion through an appropriate light-Doppler model. The wavelength reference, instrumental calibration, observer motion and spectral-line formation conditions must be specified. A measured shift is evidence to interpret rather than an automatic full spatial velocity.

At sufficiently low speeds, the first-order kinematic wavelength shift is approximately the radial velocity divided by the speed of light. Approaching motion produces a blueward contribution under the chosen convention. Higher precision requires the relevant relativistic and observational corrections.

The exact longitudinal special-relativistic formula assumes collinear source and observer motion. It should not be applied as though an independently known transverse component were absent. Gravitational and cosmological shifts also require their own models rather than a universal conversion of every redshift into this local radial velocity.

The spectrum constrains a line-of-sight contribution after those corrections. Angular data and distance supply transverse information. The three-dimensional interpretation becomes possible by connecting complementary measurements consistently, rather than expecting one spectral shift to determine all components.

Shared astrometric inputs can correlate the inferred quantities

Proper motion and parallax can come from one astrometric fit, so their uncertainties may be correlated. In the numerical form v_t proportional to μ/p, increasing μ increases the result, while increasing p decreases it. The covariance therefore affects the propagated uncertainty with a sign determined by those sensitivities.

For an optional original uncertainty illustration, give the 100 mas-per-year magnitude standard uncertainty 1 mas per year and p = 20 mas standard uncertainty 0.2 mas. If their correlation is +0.5 and a first-order treatment is adequate, relative transverse-speed standard uncertainty is 0.01, giving about 0.2370 km/s.

Treating those same inputs as independent would instead give about 0.3352 km/s. Positive correlation has reduced this ratio’s uncertainty in the stipulated example; correlation does not always increase or always decrease every output uncertainty. No hard-bound or coverage probability is assigned.

A complete result retains the frame, epoch, component convention, calibration and joint uncertainties. The next lesson uses a different observational combination—received brightness and calibrated luminosity—to infer distance while exposing its own degeneracies and corrections.

Worked examples

Try the reasoning

1. Does a proper motion of 100 mas per year imply one unique transverse speed without distance?

   Show the answer and reasoning

   No.

   A physical length scale is required to convert angular rate to speed.

2. Does negative radial velocity mean negative total speed?

   Show the answer and reasoning

   No.

   The radial sign encodes direction; total speed is the magnitude of the full velocity vector.

3. Does a spectral shift alone determine the full spatial velocity?

   Show the answer and reasoning

   No.

   Transverse components need angular-motion and distance information, while the shift itself needs a suitable physical model.

4. Must positive input correlation always increase propagated uncertainty?

   Show the answer and reasoning

   No.

   Its effect depends on the signs and sizes of the output’s sensitivities to those inputs.

What to carry forward

• Radial and transverse components need one consistent frame, epoch and sign convention.
• Distance converts an angular drift into physical transverse velocity.
• Spectral corrections and joint astrometric uncertainty remain part of any combined spatial-motion inference.