Binary-star laboratory

Two stars can orbit rapidly while showing almost no motion towards or away from us. The missing ingredient is our viewing angle. In this experiment, build a circular binary and turn its orbit from edge-on to face-on. Watch what changes in the velocity curves—and what stays the same about the stars.

For more context, Weighing stars explains the centre of mass and how an orbital period constrains mass. Reading starlight explains why shifts in spectral lines reveal motion along our line of sight. The activity below supplies the essential ideas.

Two stars, two velocity curves

Both stars orbit the centre of mass, which lies closer to the more massive star. Neither star is fixed at the centre. Because they take the same time to complete their orbits, the star travelling around the larger circle also moves faster.

A spectrograph measures a star’s radial velocity: the part of its velocity along our line of sight. Positive values here mean motion away from us; negative values mean motion towards us. We subtract the overall motion of the system’s centre of mass. In a circular binary, the two curves then rise and fall in opposite directions. This is the circular limit of the radial-velocity relation developed in Tatum n.d., §18.2, pp. 2–5 (PDF).

Predict first. Keep both masses and their separation fixed. If you turn the orbit face-on, will its period change? Will its radial-velocity signal disappear? Make both predictions before pressing View face-on.

Try a challenge

Can the same velocity measurements hide different true masses?

Suppose you know the period and both radial-velocity curves, but not the inclination or the orbit on the sky. Predict whether these measurements uniquely determine the masses. Compare the two systems below.

System 1

Masses (Suns)

0.6 + 0.2

Separation

1 AU

Inclination

90°

Period

1.118 years

A velocity amplitude

6.66 km/s

B velocity amplitude

19.98 km/s

True total mass

0.8 Suns

System 2

Masses (Suns)

4.8 + 1.6

Separation

2 AU

Inclination

30°

Period

1.118 years

A velocity amplitude

6.66 km/s

B velocity amplitude

19.98 km/s

True total mass

6.4 Suns

Reveal the explanation

These two circular systems have the same period, mass ratio and radial-velocity curves at corresponding phases. Increasing both masses eightfold and the separation twofold keeps the period fixed. The true orbital speeds double, but viewing at 30° instead of 90° halves their line-of-sight projection. The true total masses are 0.8 and 6.4 Suns; the measured combination, total mass × sin³(inclination), is 0.8 Suns for both. Independent inclination or sky-orbit information can distinguish them.

Explore · construct a circular binary, then change your view

Sky view · A: filled dot · B: outlined square

A radius: 1 AU · B radius: 3 AU

The cross marks the centre of mass. Star symbols are not physical sizes.

These are calculated orbits and velocities, not observations. The sky view projects a circular orbit; its scale changes with the selected separation. The velocity axis stays fixed when only inclination changes. At zero inclination the two velocity curves overlap at zero, although the stars still orbit.

Set the stars and your viewing angle

Star A mass: 3 solar massesStar B mass: 1 solar massesSeparation of the two stars: 4 AUOrbital inclination: 60 degreesPosition through the orbit: 0%

0° means face-on; 90° means edge-on. Separation is the distance between the stars in this circular model, not either star's distance from the centre.

Load system 1Load system 2Reset experimentView face-onView edge-on

The 3- and 1-solar-mass stars orbit every 4.000 years. At 60°, their velocity amplitudes are 6.449 and 19.35 km/s. The curves constrain a projected total mass of 2.598 solar masses; the true total is 4.

The default example and table work without JavaScript. If the controls do not activate, you can reload the laboratory or continue with the worked activity below.

Read the velocities in a table

Current model: velocities relative to its centre of mass

Orbit fraction	Time (years)	A (km/s)	B (km/s)
0	0	6.449	-19.35
0.25	1	0	0
0.5	2	-6.449	19.35
0.75	3	0	0
1	4	6.449	-19.35

The barycentre's own radial motion has been subtracted. We model point masses without eccentricity, eclipses, tides or stellar evolution. A selected mass does not specify a star's radius or spectrum. Download the model and read its assumptions and sources.

Read the default system

The initial masses are three and one times the solar reference mass, and the two stars remain four astronomical units apart. One astronomical unit, abbreviated AU here, is about the Earth–Sun distance. A travels one AU from the centre of mass; B travels three AU. Their shared period is about four years.

At an inclination of 60°, the largest radial speed of A is about 6.45 km/s, while B reaches about 19.35 km/s. This largest excursion from the central velocity is the curve’s amplitude. The faster-moving star is the less massive one: the amplitude ratio is one to three, so the mass ratio is three to one.

Move the position slider through one quarter of the orbit. Both radial velocities become zero at the same instant, but neither star stops. At that position their motions are across our line of sight. Compare this momentary zero with the face-on case, where the radial velocities stay zero through the entire orbit.

A viewing angle can hide mass

Press View edge-on. All orbital motion is in a plane containing our line of sight, so the velocity amplitudes reach their largest possible values for these masses and separation. Now reduce the inclination to 30°. Both amplitudes fall to half their edge-on values. The period and true masses have not changed.

When both spectral curves can be measured, their amplitude ratio still gives the mass ratio for any nonzero inclination in this ideal model. Their shared amplitude scale and period determine a different combination: total mass multiplied by the cube of the sine of the inclination. Without another measurement of the inclination, that combination is a lower bound on the total mass, not a unique mass measurement.

Exactly face-on, there is no orbital radial-velocity signal. Its two zero amplitudes have no measurable ratio. Real instruments also have uncertainties, so a tiny signal at a small inclination can be difficult to detect; this noiseless model should not be read as a detectability forecast.

Show the math: reconstruct the measured combination

Let the masses be $M_A$ and $M_B$, their orbital radii $a_A$ and $a_B$, their common period $P$, and their separation $a=a_A+a_B$. Newtonian gravity and the centre-of-mass condition give

$$M_Aa_A=M_Ba_B, \qquad P^2=\frac{4\pi^2a^3}{G(M_A+M_B)}.$$

The second equation applies to the relative orbit; using only $a_A$ would give the wrong total mass. See the derivation in NASA/GSFC 2020, Eqs. 1–4.

For a circle, speed is circumference divided by period. Projection through an inclination $i$ gives the two amplitudes

$$K_A=\frac{2\pi a_A}{P}\sin i, \qquad K_B=\frac{2\pi a_B}{P}\sin i.$$

Adding these equations gives $a\sin i=P(K_A+K_B)/(2\pi)$. Substitution into Kepler’s relation yields

$$(M_A+M_B)\sin^3i =\frac{P(K_A+K_B)^3}{2\pi G}, \qquad \frac{M_B}{M_A}=\frac{K_A}{K_B}.$$

Use consistent units: seconds and metres per second if $G$ is in SI units. The ratio formula requires a nonzero measured signal. With the default masses and $i=60^\circ$, the projected total is $4(\sqrt{3}/2)^3\simeq2.598$ solar masses, while the true total is four.

The plotted curves use $v_A=K_A\cos(2\pi t/P)$ and $v_B=-K_B\cos(2\pi t/P)$. Time zero is our chosen phase origin, not an observed date. An eccentric orbit generally does not give these simple cosine curves.

Try to make two systems look identical

Set A to one solar mass, B to half a solar mass, separation to two AU, and inclination to 90°. Note the period and both amplitudes. Then set A to eight, B to four, separation to four AU, and inclination to 30°. What can a measurement of the two velocity curves alone distinguish?

Answer and explanation

The period and both velocity curves are identical in this ideal model, even though the masses are eight times larger in the second system. Doubling the separation multiplies its cube by eight; increasing the total mass by eight therefore leaves the period unchanged. Each star’s actual orbital speed doubles, but the projection factor falls from one to one-half. The two effects cancel.

An independent inclination or resolved orbit can break this degeneracy. The laboratory’s sky view differs between the two systems, but a spectrograph measuring an unresolved pair does not automatically supply that picture. The plotted velocities alone cannot select one set of true masses.

Use the model within its limits

This is a constructed, isolated Newtonian binary with circular orbits. We do not simulate spectra, eclipses, stellar radii, mass transfer, relativistic effects or measurement noise. Selecting an edge-on view does not establish that the stars eclipse: that would also depend on their sizes and geometry. If only one set of spectral lines is visible, even the two-curve mass-ratio argument is unavailable.

The model also leaves out energy loss through gravitational radiation, so its stars do not spiral inward or merge. Gravitational waves from merging black holes require a different physical calculation and a different kind of detector.

The model uses the exact nominal solar gravitational parameter adopted by IAU 2015 Resolution B3, p. 3 (PDF) and the exact astronomical unit in IAU 2012 Resolution B2, recommendation 1 (PDF). A model year is 365.25 days of 86,400 seconds. These conventions make the calculation reproducible; they do not make the rounded worked example an observation.

For classrooms and clubs

Preparation: read a graph and compare ratios. Objective: distinguish a physical parameter from a projection effect. For a ten-minute activity, keep masses fixed and ask pairs of students to predict the changes at 90°, 30° and 0°. The key answer is that amplitudes change while the period and true masses stay fixed. For a longer investigation, reproduce the two indistinguishable systems above and propose one additional observation that could distinguish them. If controls are unavailable, the static table and the explained example support the same discussion. Print with explanations to include the derivation and solutions.

Continue with why planets stay in orbit, connect mass and temperature on the H–R diagram, or compare a different projection effect in the transit lab.

References

• International Astronomical Union (2012). Resolution B2: On the re-definition of the astronomical unit of length, recommendation 1. XXVIII General Assembly. Resolution PDF.
• Mamajek, E. E., et al. (2015). IAU 2015 Resolution B3 on Recommended Nominal Conversion Constants for Selected Solar and Planetary Properties, p. 3. Stable record; Open PDF.
• NASA/Goddard Space Flight Center (2020). “Deriving Kepler’s Formula for Binary Stars.” Imagine the Universe!, updated 23 September 2020, Eqs. 1–4. Derivation.
• Tatum, J. B. (n.d.). Celestial Mechanics, Chapter 18, “Spectroscopic Binary Stars,” §18.2, pp. 2–5. University of Victoria. Open PDF. Accessed 27 September 2026.