M076 · A moving balance

Lagrange Points & Three-Body Channels

Place a massless probe between two orbiting bodies. Launch near a Lagrange point, open a zero-velocity neck and compare the same path in rotating and inertial frames.

Model time $\text{Ready}$

Jacobi constant $\text{Ready}$

Absolute Jacobi drift $\text{Ready}$

Inner channel $\text{Ready}$

Triangular-point linear stability $\text{Ready}$

Trajectory boundary $\text{Ready}$

PLACE / RELEASE / COMPARE

Change the start. Follow the gravity.

Secondary mass fraction

\mu

0.01\,

Initial horizontal position

x_0

0.51\,

Initial transverse position

y_0

0.87\,

Initial horizontal velocity

v_{x,0}

0.00\,

Initial transverse velocity

v_{y,0}

0.00\,

Use half timestep

b_h

0.00\,

Circular primaries, massless planar probe. Unit separation, total mass, gravity constant and angular speed; one primary orbit takes two pi time units. Barycentric axes: horizontal toward the secondary, transverse completes the right-handed plane. All initial controls are rotating-frame quantities.

Fourth-order Runge–Kutta, step 0.004 or 0.002. Stop before any evaluated stage enters a 0.06-unit exclusion sphere, beyond barycentric radius six, or at time thirty. Exclusion spheres and body markers are schematic, not physical radii. No collision continuation or close-encounter accuracy claim.

The separate rotating-frame map shades locations forbidden by the initial Jacobi constant. Open necks permit transit but do not guarantee it. Triangular-point stability is the linear planar criterion; it does not certify an arbitrary finite-amplitude path. No halo orbits, station keeping or ephemeris model.

The gold handle edits the starting position; the green handle edits rotating-frame velocity at a fixed gain of 0.4 length units per velocity unit. Both appear at time zero. All trajectories, boundaries and exports use the same numerical record.

Planar circular restricted three-body model; no halo orbits or mission navigation.

Start with the essentials

Focus question: What makes a quiet point stable?
One-sentence intuition: An open neck is permission, not a prediction.

Core mathematical model

Rotating-frame acceleration

\ddot x-2\dot y=\partial_x\Omega,\quad\ddot y+2\dot x=\partial_y\Omega

The Coriolis term depends on velocity.

Effective potential

\Omega=\frac{x^2+y^2}{2}+\frac{1-\mu}{r_1}+\frac{\mu}{r_2}

The primaries remain fixed in this rotating frame.

Allowed motion

C=2\Omega-\dot x^2-\dot y^2,\quad 2\Omega\geq C

A forbidden region would require negative squared speed.

Triangular-point criterion

27\mu(1-\mu)<1

This is linear planar stability, not a global orbit guarantee.

Common difficulties

A model boundary

Typical misconceptionA Lagrange point has zero gravity.

Better mental modelGravity supplies the required orbit acceleration; centrifugal balance appears in the rotating frame.

Run the experiment

01
Place a probe
Drag the gold start and green velocity handles.
What to observe: The Jacobi constant and forbidden map change together.
02
Release a balance
Compare the triangular and saddle presets.
What to observe: Nearby starts can remain near or depart rapidly.
03
Change the frame
Pause, switch frames and replay.
What to observe: The same recorded trajectory has a different apparent shape.