M076 · A moving balance

Lagrange Points & Three-Body Channels

If a probe has enough energy to cross from one body's region to the other's, will it actually cross? Not necessarily; enough energy only makes the crossing possible. At that energy, the region the probe cannot enter opens a narrow neck between the two areas. Whether the probe actually passes through depends on where its full trajectory takes it.

Place a massless probe near two bodies that orbit each other. Launch it near a Lagrange point, open the narrow neck in the zero-velocity boundary, and compare the same path in the 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 $\text{Ready}$

Set the starting state and release it

Change the starting state and follow the motion under 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\,

The two primaries move on circular orbits, and the massless probe stays in their plane. The separation, total mass, gravitational constant and angular speed are all one, so one orbit of the primaries takes two pi time units. The axes are centred on the barycentre: the horizontal axis points toward the secondary, and the transverse axis completes a right-handed plane. All starting values are set in the rotating frame.

The orbit is integrated with fourth-order Runge–Kutta at a step of 0.004 or 0.002. The run stops at time thirty, or earlier if any evaluated stage would enter a 0.06-unit exclusion sphere or go beyond radius six from the barycentre. The exclusion spheres and body markers are schematic sizes, not physical radii. Motion is not continued through a collision, and close-encounter accuracy is not claimed.

The separate map, drawn in the rotating frame, shades the regions the probe cannot reach with its starting Jacobi constant. An open neck allows the probe to pass but does not guarantee that it will. Stability at the triangular points uses the linear criterion in the plane, which does not vouch for an arbitrary path of finite amplitude. Halo orbits, station keeping and ephemeris models are not included.

Drag the gold handle to set the starting position and the green handle to set the starting velocity in the rotating frame; the green arrow is drawn at a fixed 0.4 length units per unit of velocity. Both handles appear at time zero. The trajectories, boundaries and exports all come from the same numerical record.

Model assumptions

This is the planar, circular restricted three-body model. It does not compute halo orbits and is not a tool for mission navigation.

The dots replay a saved massless trajectory in the current frozen orbit. Removing mass is a separate, prescribed sequence.

The orbit changes as mass is removed

Separation ratioPeriod ratio

A mass ledger, including what escapes

Donor changeCompanion gainEscaped mass

Fast and thermal radius responses are local hypotheses; a sign does not determine a transfer timescale or a common-envelope fate.

Geometry, ledgers and numerical limits

The closed surface is computed from the synchronous circular point-mass Roche potential. When the requested star overfills it, the drawing stops at the inner critical surface: an open overflow is not replaced by a larger closed sphere. Ballistic probes have no pressure, self-gravity, shocks or back reaction.

Every selected mass ratio is directly integrated with angular refinement. The approximate radius fit remains a comparison. A response near the fit/numerical interval is left unclassified; passing a mass or angular-momentum ledger alone does not certify local dynamics.

The transfer illustration holds the supplied companion radius at $R_a=0.080a_0$ . If it leaves its own lobe, the donor-only classification is withheld. Its expansion and thermal adjustment are not evolved.

The envelope preset pays only the original model’s gravitational envelope cost from a chosen fraction of released orbital energy. Internal/recombination/radiation energy and fluid ejecta are unresolved. A native hydrogen cut supplies a radius, not a relaxed remnant; direct surface overlap and failure to fit a Roche lobe are separate events, neither a merger probability.

$\Omega^2=G(M_d+M_a)/a^3$ defines the frozen orbital units. The mass-removal parameter carries no elapsed time or predicted mass-transfer rate.

Native structure attribution: our gravity/core-cut aggregates from Thoul & Chidester, MESA Summer School 2022 (CC BY 4.0) and Jared Goldberg, MESA School Leuven 2025 (CC BY-SA 4.0). Derived structure aggregates retain their respective terms; this is not a new MESA run.

Read the course and reproduce the historical Beta Lyrae estimate

Start with the essentials

Focus question: Why are some Lagrange points stable and others unstable?
One-sentence intuition: An open neck means the probe is allowed to pass through, not that it will.

Core mathematical model

Equations of motion in the rotating frame

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

The Coriolis terms depend on the probe's velocity.

Effective potential

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

In this rotating frame the two main bodies stay fixed.

Where the probe can go

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

Entering a forbidden region would require the speed squared to be negative, which is impossible.

Stability condition for the triangular points

27\mu(1-\mu)<1

This tests linear stability in the plane only. It does not guarantee what larger orbits will do.

Common difficulties

Gravity at a Lagrange point

Typical misconceptionThere is no gravity at a Lagrange point.

Better mental modelGravity is still there; it supplies the acceleration the orbit needs. The balance with centrifugal force appears only 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 the forbidden region change together.
02
Release near a balance point
Compare Near the triangular point with Nudge the saddle.
What to observe: A probe that starts close to a balance point may stay nearby or leave quickly.
03
Change the frame
Pause, switch frames and replay.
What to observe: The same recorded trajectory looks different in each frame.

Start with the essentials

Focus question: Does a small mass loss help the donor detach?
One-sentence intuition: Compare stellar and lobe radius derivatives under the same mass/angular-momentum prescription; keep the local result separate from a common-envelope fate.

Core mathematical model

Orbital angular momentum

J_o=M_dM_a\sqrt{Ga\over M_d+M_a}

This circular point-mass relation connects the mass ledger, escaped angular momentum and separation. Orbital energy is not a conserved total stellar energy.

A local overflow test

d\ln(R_d/R_L)=(\zeta_*-\zeta_L)d\ln M_d

The donor loses mass, so the final differential is negative. A stellar radius response above the lobe response locally reduces overfill; near a fit or numerical threshold we leave it unclassified.

A conditional envelope endpoint

E_g=\alpha_{\rm CE}\left({GM_cM_a\over2a_f}-{GM_dM_a\over2a_i}\right)

Only the native gravitational envelope cost is included. A core-cut radius and companion radius must then be checked for direct overlap and individual Roche fit.

Common difficulties

A radius fit can give the wrong threshold sign

Typical misconceptionA sub-percent radius approximation guarantees the response derivative.

Better mental modelA derivative error can reverse a near-neutral test. The retained conservative polytrope example does exactly that, so the production mode computes volume derivatives and withholds near-threshold labels.

An energy equation is not common-envelope hydrodynamics

Typical misconceptionA final separation predicts ejecta, survival and merger probability.

Better mental modelThe scalar budget omits pressure, shocks, transport and a relaxed remnant radius. Its supplied-radius geometry is a conditional check, not a fluid fate.

Run the experiment

01
Make a closed surface smaller
Pause the lighter-star preset and drag the gold surface handle inward.
What to observe: The computed surface retreats inside the passage. Only a deliberately launched massless probe remains; it is not a sustained overflow.
02
Compare fast and slow responses
Choose the two-response-times preset, then compare its fast and thermal labels below the orbit.
What to observe: The two assumed stellar exponents can lie on opposite sides of the same lobe response. That distinction does not supply a transfer timescale.
03
Spend less orbital energy
Choose the compact-pair preset and reduce the energy fraction from one toward one tenth.
What to observe: The energy endpoint shrinks. Failure to fit a lobe appears before direct overlap for the native four-solar-mass cut; neither event is a computed merger.

Lagrange Points & Three-Body Channels