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

The unit binary separation and unit mean motion imply through Kepler third law thatThe centre of mass conditions put $M_1$ at $x_1=-{\mu }_2$ and $M_2$ at $x_2={\mu }_1$. Therefore

In a frame with unit angular velocity, transforming the acceleration introduces the Coriolis acceleration and centrifugal acceleration. Radiation reduces the attraction of $M_1$ by the factor $1-{\beta }_1$, soWith the effective potentialits Cartesian components are exactlyThis is the photogravitational restricted three-body problem.

Multiply the three equations by $\dot{x},\dot{y},\dot{z}$ and add. The Coriolis acceleration does no work because its two terms cancel, leavingEquivalently, the radiation-modified Jacobi constant is

DefineAt an equilibrium point, the velocity and acceleration vanish. Direct differentiation givesandThus the equilibrium conditions are

At a Triangular Lagrange point, $y\ne 0$, so $A=1$. The $x$ equation then givesSubstitution into $A=1$ and ${\mu }_1+{\mu }_2=1$ yieldsIntersecting these two circles givesAs ${\beta }_1$ rises from zero to one, $s$ falls from one to zero. The two points move along the unit circle about $M_2$, from the classical equilateral positions toward $M_1$, where they coalesce when the attraction of $M_1$ is fully cancelled.

Put $q_i=1-{\beta }_i$. Repeating part (v) with radiation from both bodies givesThe two triangular points are intersections of circles with these radii and centre separation one. Iftheir coordinates areFor ordinary outward radiation pressure, $0\le {\beta }_i\le 1$, the non-collinear points exist precisely when the strict triangle inequalityholds. Equality merges the two points on the line of centres.

The same absorption and re-emission that produce radial radiation pressure also produce Poynting–Robertson drag. This velocity-dependent force removes specific orbital energy and angular momentum, destroys the exact Jacobi constant, and turns the formal equilibrium points into slowly drifting configurations. Stellar-wind drag can provide a similar correction.