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 Incoming links: Projected separation

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 316 / 1 / iii / Solution Created 2026-09-25 Updated 2026-10-14

The projected separation is

R_{sky} = x^{2} + y^{2} = a cos^{2} f + cos^{2} I sin^{2} f = a 1 - sin^{2} I sin^{2} f .

Rotating the projected coordinates through

- Ω

gives components

a cos f

along the nodal line and

a cos I sin f

perpendicular to it. The position angle is therefore most safely written with the quadrant-preserving function

ϕ = Ω + o p er a t or nam e a t an 2 (cos I sin f, cos f),

or, wherever the corresponding tangent is finite,

tan (ϕ - Ω) = cos I tan f .

Figure 1.
Inclined circular orbit and its projection on the plane of the sky
. The three-dimensional view labels the line of nodes and a planet on an inclined circular orbit. Projection along the line of sight turns the circle into an ellipse whose major axis is the nodal line. Details.

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