© J R Stockton, ≥ 2010-03-07

# \$LAG-PTS.HTM.

include1.js & include4.js
and wants styles-a.css.

In hand; appendix for Gravity 4. To be gr4-app1.htm or gravity#.htm?

## Computable Lagrange Plot

Not for MS IE ≤ 8+.

This shows how a two-body system with its Lagrange points rotates, and what happens as mass is moved from one body to another.

### Operation

The results agree with Full Calculator, although the calculation was written independently.

Ignore Eggs.

### Diagram

Masses, angle
M12 Distancesfrom BC
L123 Distancesfrom BC
L45 Distancesfrom a mass
L45 DistancesBC, line, orbit
Plot Size pixels Mass Ratio M1 / M2 Eggs
Separation × Size   Paths Sepns
Eccentricity       Turn /step
Steps Rate ms
Traverse Rotation Both

The plot is "Plot Size" pixels square. With "Traverse", the mass ratio varies non-linearly to the given value from its reciprocal. For circular orbits, "Separation" is the distance between the bodies; for elliptical orbits, it may be the latus rectum.

Eccentricity : circle 0.0, ellipse, parabola 1.0, hyperbola. Curves for eccentricity near or over 1.0 may not be attempted.

For Eccentricity=0.0, the units of Turn are radians per step. In general, it may well represent angular momentum.

NOTE : Don't entirely trust the case of non-zero eccentricity?

The brown discs represent the masses M1 & M2. Their sizes R1 & R2 are scaled assuming equal density (the size scale is arbitrary; at default plot size, real Solar System masses are smaller in proportion to the distances). The barycentre (BC) is marked by a small black yellow-centred ring.

The optional non-dashed curves show the paths of the masses and points around the barycentre. The optional dashed circles have radius equal to the separation of the masses. The blue ones are centred on the masses, and thus cross at L4 & L5. The red one is centred on the barycentre.

The five red "+" signs show the Lagrange points. The angles of the triangles are multiples of 30°

The label pairs L2/L3 and L4/5 exchange when the mass ratio crosses unity.

NOTE : The barycentre is at (0, 0). For simplicity, rotation is achieved by rotating the coordinates, which is why the text co-rotates.

### Pending/Suggestions

• Change of background and other colours
• Show elliptic paths?
• Verify elliptical orbits correctly timed? Improve extrapolation? Are there simple forms for R(t) and θ(t) ?
• Change phase to suit hyperbolic ? How ??