Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 316 / 2 / vi

Put

q_{i} = 1 - β_{i}

. Repeating part (v) with radiation from both bodies gives

r_{1} = q_{1}^{1/3}, r_{2} = q_{2}^{1/3} .

The two triangular points are intersections of circles with these radii and centre separation one. If

X = \frac{r _{1}^{2} - r _{2}^{2} + 1}{2},

their coordinates are

x = X - μ_{2}, y = \pm r_{1}^{2} - X^{2}, z = 0 .

For ordinary outward radiation pressure,

0 \leq β_{i} \leq 1

, the non-collinear points exist precisely when the strict triangle inequality

(1 - β_{1})^{1/3} + (1 - β_{2})^{1/3} > 1

holds. Equality merges the two points on the line of centres.

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