SCIENCE · INVESTIGATE AND EXPLAIN

Close both component orbits around a common center

Both members of a binary respond to the same mutual gravitational force, but their accelerations differ inversely with mass. Neither body must remain fixed. In the center-of-mass frame their position vectors are opposite and obey M₁r₁=M₂r₂. The more massive component follows the smaller orbit, while both complete a cycle in the same period.

Learning outcome: Vary secondary mass fraction at fixed relative axis, then vary axis at fixed fraction. Close the radius sum, mass-weighted position and opposite momenta, and identify an observation that distinguishes barycenter from photocenter.

Starting recap

Retain relative versus barycentric geometry, time versus angle, fitted versus instantaneous velocity and complete energy/angular-momentum boundaries.

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Try

At a=4 AU and secondary fraction 0.25, masses are 3 and 1 solar masses, radii 1 and 3 AU, and period 4 years. Speeds are π/2 and 3π/2 AU/year. Position balance is −3×1+1×3=0; opposite momentum magnitudes are each 3π/2.

Open the experiment

Predict

Does the heavier star follow the larger or smaller path around the common center?

Your prediction and reason

Experiment

Vary secondary mass fraction at fixed relative axis, then vary axis at fixed fraction. Close the radius sum, mass-weighted position and opposite momenta, and identify an observation that distinguishes barycenter from photocenter.

Close binary positions and momenta around the common center

Worked teaching model · synthetic values · Original circular binary total mass 4 solar masses, conventional P²=a³/M. Primary axis=a f, secondary axis=a(1−f); both have the same period. Displayed velocities are opposite tangential directions at the shown phase; no mass-luminosity assumption.

Relative circular semimajor axis: 4 AU Secondary fraction of total mass: 0.25 fraction

Reset keeps your prediction, captures and practice history.

The heavier component follows the smaller barycentric orbit while both share one period.

Their relative separation is the sum of the two barycentric radii. Opposite momenta cancel in the center-of-mass frame; equal gravitational force does not imply equal acceleration, speed or radius. Brightness is not used to assign mass.

Original conditional quantities and explicit units
Quantity	Value	Unit
Primary mass	3	solar masses
Secondary mass	1	solar masses
Primary barycentric radius	1	AU
Secondary barycentric radius	3	AU
Common period	4	years
Primary tangential speed	1.570796	AU/year
Secondary tangential speed	4.712389	AU/year
Signed center-of-mass numerator	0	solar mass AU
Signed momentum sum	0	solar mass AU/year

These original models are not measured orbit records or operational mission instructions. Retain the frame, assumptions and missing evidence before interpreting a real system.

The heavier component follows the smaller barycentric orbit while both share one period.

Observation to attach to a capture

Captures preserve recipes and replay their calculated results. They are synthetic model evidence, not physical measurements.

Understand

1. Define the orbital evidence

2. Connect geometry, force and conservation

The original circular binary has total mass 4 solar masses and relative separation a AU. If the secondary mass fraction is f, then M₂=4f and M₁=4(1−f). The primary radius is af and the secondary radius a(1−f), adding to a. The shared period is √(a³/4) years. The fraction labels refer to mass, not flux or a spectral brightness estimate.

Each speed is 2π times its own radius divided by the common period. At the displayed phase the tangential velocities are opposite, so signed total momentum cancels. This is a useful independent check on both mass and speed assignments. Equal force does not mean equal speed or equal radius unless the component masses are equal.

The original diagram shows circular barycentric orbits in their orbital plane. Real visual binaries can be eccentric and projected, and measured image positions can refer to a light-weighted photocenter rather than the mass center. A source-labeled bright spot is not automatically the center of mass. Those additional observations and assumptions must be retained before real mass inference.

3. Work the original case

4. Predict and investigate

Does the heavier star follow the larger or smaller path around the common center?

Apply the evidence to a changed case

At relative a=2 AU and equal masses, find component radii, common period and speeds. Compare the relative orbital axis with either component radius.

Retain the evidence frame, calculations and assumptions before opening public feedback.

Compare a worked response

Each mass is 2 solar masses and each radius 1 AU. Period is √2 years and each speed is √2π AU/year. The relative axis is 2 AU, twice either barycentric radius; equal brightness was not assumed.

Source and boundaries

Primary mechanism reference. Checked 30 October 2026. Original prose, numerical cases and diagrams; source context is not a claim that these fictional records were observed. No actual mission operation or instrument aiming is requested.

Apply

Public practice checks a specific claim. It does not establish independent mastery.

At relative a=2 AU and equal masses, find component radii, common period and speeds. Compare the relative orbital axis with either component radius.

Your explanation and limitations

Continue

Completion requires a saved prediction, at least one capture and a reflection. Presence checks do not grade your reasoning.

Skipping retains experiments and awards no mastery.

Previous: Infer a total mass from the correct orbit and bounded measurements Next: Separate a radial-velocity amplitude from an instantaneous sample

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