Astronomy · Evidence for planets around other stars

Stellar radial motion and astrometric shifts

View the same stellar reflex orbit along the line of sight and across the sky, retaining the geometry and model conditions needed to infer a companion.

Continue in the approved chapter order. Necessary vocabulary, quantities and assumptions are explained in the lesson.

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After reading, you can

Explain stellar reflex motion through a shared centre of mass.
Distinguish radial-velocity semi-amplitude from a full spatial speed.
Compare astrometric and spectroscopic constraints without assuming they exclude each other.

A companion changes the host star’s motion too

In an isolated two-body system, the star and companion both move around their common centre of mass. The star’s smaller component orbit is often called reflex motion. The mass-weighted relation from the gravity chapter applies: a more massive component has a smaller barycentric orbital scale.

A distant observer may be unable to resolve the companion separately yet still measure changes in the star’s spectrum or position. These are indirect constraints on motion. Connecting them to a companion requires a dynamical model, timing and checks on alternative causes.

Separate motion of the system’s centre from motion within the system. A roughly steady systemic line-of-sight velocity is not the same quantity as an orbital variation about it. Likewise, a sky position can change through overall motion and parallax as well as the component orbit.

The original model below isolates a circular stellar component orbit. It contains no real target, noise series, instrument sensitivity or claim that an arbitrary sinusoid has a planetary cause.

Declare the sky plane and the positive radial direction

Choose coordinates X and Y in the sky plane, with +Z pointing away from the observer. Inclination i is the tilt of the orbital plane from the sky plane: i = 0 degrees is face-on and i = 90 degrees is edge-on. This convention must accompany numerical inclinations.

Let the circular stellar orbit have radius a_star and period P. Write θ = 2πt/P, choosing phase zero explicitly. Its coordinates are X = a_star cos θ, Y = a_star sin θ cos i and Z = a_star sin θ sin i.

The actual circular speed is v_star = 2πa_star/P. Differentiating Z gives radial velocity v_rad = v_star cos θ sin i, with positive values away from the observer. The orbital radial-velocity semi-amplitude is K = v_star sin i.

Semi-amplitude means half the maximum-minus-minimum orbital radial velocity. For this circle the extremes are +K and −K about the systemic value, and peak-to-trough is 2K. K is a projected component scale rather than the full three-dimensional orbital speed for every inclination.

Intermediate inclinations provide both types of motion

At i = 0 degrees, the circular orbit lies in the sky plane. Its line-of-sight velocity contribution is zero in this model, while the projected path remains a circle. At i = 90 degrees, the path projects to a line and the radial-velocity semi-amplitude equals the circular speed.

Edge-on projection does not eliminate astrometric motion: the star still moves back and forth along that projected line. At an intermediate angle, both a sky-plane orbit and a radial-velocity signal remain. The methods are complementary projections rather than mutually exclusive choices.

For i = 30 degrees, K/v_star = 0.5 and the projected semiaxes are a_star and approximately 0.866025a_star. At phase θ = 0 the star is at X/a_star = 1, Y/a_star = 0 and has v_rad/v_star = +0.5. A quarter period later it has X/a_star = 0, Y/a_star ≈ 0.866025 and zero radial velocity.

The original figure uses common coordinate and velocity scales for all three inclinations and matches their phases. Body markers do not encode stellar radii or masses. A projected ellipse, a line and a velocity curve must be read with their different axis quantities.

Original circular stellar reflex model. Inclination is to the sky plane; projected semiaxes are divided by a_star.
i	K / v_star	Projected semiaxes
0°	0	1 and 1
30°	0.5	1 and 0.866025
90°	1	1 and 0; a line track

One orbit, two complementary projections

Scroll sideways to see the full diagram. Its explanation is below.

Original ideal isolated circular stellar reflex orbit, not an observed system. Inclination i is measured from the sky plane: 0 degrees is face-on, 30 degrees intermediate, and 90 degrees edge-on. X and Y lie in the sky plane; +Z points away from the observer. Let θ = 2πt/P. Normalized coordinates are X/a_star = cos θ, Y/a_star = sin θ cos i, and Z/a_star = sin θ sin i. The normalized radial velocity is v_rad/v_star = cos θ sin i, derived from Z with v_star = 2πa_star/P. Positive radial velocity is away. Sky axes have identical X and Y scales shared across all three cases. Velocity panels share time t/P from 0 to 1 and a zero-centred velocity scale. Phase markers match: A means t/P = 0 or 1, B = 1/4, C = 1/2, and D = 3/4. Face-on gives a projected circle and zero orbital radial velocity. At 30 degrees, projected semiaxes divided by a_star are 1 and about 0.866025; K/v_star = 0.5. Edge-on gives a projected line with X/a_star from −1 to +1, retaining astrometric motion, and K/v_star = 1. B and D have the same projected line-centre position at different times; their velocity markers occur at their distinct phases. Complete five-phase rows for each inclination accompany this figure. One consistent optional arbitrary-unit dynamical realization has M_star = 1000 m₀, M_p = 1 m₀, relative semimajor axis 1 L₀, G = (1/1001) L₀³/(m₀ T₀²), total gravitational parameter 1 L₀³/T₀², P = 2π T₀, a_star = (1/1001) L₀, and v_star = (1/1001) L₀/T₀. Marker size represents neither physical stellar radius nor mass. These are normalized stellar-position tracers rather than instrument precision or photocentre measurements; real proper motion, parallax, other companions, light contributions and calibration require their own model.

Read the paired quantities at matching phases

The complete table below pairs the same five phases for each stipulated inclination. X, Y and Z are divided by the physical stellar orbit radius a_star; radial velocity is divided by the circular stellar speed v_star. These normalized quantities are dimensionless, while t/P is elapsed time divided by period.

Positive Z and radial velocity point away from the observer. At the edge-on quarter phases, B and D share a projected sky position but have different line-of-sight positions. The table keeps those quantities distinct even when their projected markers overlap. Values rounded below include exact zero for negligible trigonometric roundoff.

Complete original circular stellar reflex projections. One phase convention, sky plane and positive-away radial direction apply to all fifteen rows; no observed system is represented.
Inclination i	t / P	X / a_star	Y / a_star	Z / a_star	v_rad / v_star
0°	0	1	0	0	0
0°	0.25	0	1	0	0
0°	0.5	-1	0	0	0
0°	0.75	0	-1	0	0
0°	1	1	0	0	0
30°	0	1	0	0	0.5
30°	0.25	0	0.866025404	0.5	0
30°	0.5	-1	0	0	-0.5
30°	0.75	0	-0.866025404	-0.5	0
30°	1	1	0	0	0.5
90°	0	1	0	0	1
90°	0.25	0	0	1	0
90°	0.5	-1	0	0	-1
90°	0.75	0	0	-1	0
90°	1	1	0	0	1

The observed record includes more than the orbital signal

Spectroscopy estimates radial motion through shifts of stellar spectral features under a stated wavelength calibration and Doppler convention. This uses light propagation, not the sound-wave formula for motion through a material medium. Instrument drift and the observer’s own motion must be accounted for appropriately.

Changing stellar surface activity or line shapes can affect measured spectral shifts. A dynamical interpretation is strengthened by tests involving multiple features, activity information, calibration and consistent timing. A periodic-looking spectral quantity alone is not uniquely a planet.

Astrometry measures angular positions relative to a reference frame. For a distant small-angle circular model, the angular semimajor scale is approximately a_star/d radians, where d is the distance in the same physical length units. It is an angle rather than a physical orbital radius.

An astrometric solution includes reference position, overall proper motion, parallax, orbital motion and relevant instrumental effects. If an unresolved companion contributes light, the measured centre of light can differ from the position of the host star alone. The measured tracer must match the model.

Amplitude becomes a mass constraint through dynamics and geometry

A larger companion can produce a larger stellar reflex orbit, but radial amplitude also depends on period, eccentricity, orientation and the mass of the host. The general Keplerian radial-velocity relation retains the total mass rather than silently discarding the companion mass.

When the companion is much lighter than the host, radial velocities commonly constrain a quantity expressed approximately as companion mass times sin i. That shorthand is a restricted approximation. At unknown inclination it does not supply a unique true mass, and it is not exact inversion for arbitrary mass ratios.

Astrometric geometry, transits or other complementary evidence can help constrain inclination and the physical orbit. Combining them requires shared parameters and uncertainties to be handled together. A transit favours nearly edge-on geometry but does not make every measured inclination exactly 90 degrees.

Eccentric or multi-companion systems need more than the single circular sinusoid used here. The later parameter lesson develops the full mass function and period–orbit relation. First, the next lesson contrasts resolved companion light with gravitational magnification of an entirely different background source.

Worked examples

Recover the projected amplitude

For the synthetic circle at i = 30 degrees, what are K and peak-to-trough velocity relative to v_star?

sin(30°) = 0.5.
K = 0.5v_star.
The maximum-minus-minimum orbital velocity is 2K = v_star.

The semi-amplitude is half the circular stellar speed; peak-to-trough equals that speed in this particular geometry.

Read the limiting projected path

Does the edge-on circular model have no astrometric motion?

At i = 90°, Y = a_star sin θ cos i = 0.
X = a_star cos θ still varies between −a_star and +a_star.
The two-dimensional projected ellipse degenerates to a line, rather than a stationary point.

No. Its projected motion follows a line while the radial-velocity amplitude is maximal for the same circular speed.

Try the reasoning

What does i = 0 degrees mean in this lesson?
Show the answer and reasoning
A face-on orbital plane in the sky plane.
The convention is the angle of the orbital plane to the sky plane; it is stated explicitly with the projections.
Can radial velocity and astrometry both constrain a circular orbit at an intermediate inclination?
Show the answer and reasoning
Yes.
The line-of-sight component and sky-plane track are both nonzero.
Is K always the full spatial orbital speed?
Show the answer and reasoning
No.
For the circle K = v_star sin i; it is a semi-amplitude of a projected velocity.
Does one periodic spectral signal by itself uniquely identify a planet?
Show the answer and reasoning
No.
Activity, line-profile behaviour, calibration and alternative dynamical models require appropriate tests.

What to carry forward

Both bodies participate in common-centre motion.
Radial velocities and astrometry observe complementary projections with explicit sign and inclination conventions.
Mass inference requires dynamics, host properties, geometry and tests of competing signal sources.

References

Teaching resources and research supporting this lesson’s explanations and examples.