Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 338 / 1 / a / ii

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- Solution ii

Solution

Let

n, w, u

be orthonormal unit vectors north, west and upward. A direction of altitude

γ

and westward azimuth

η

is

x = cos γ cos η n + cos γ sin η w + sin γ u .

The north celestial pole and the upper equatorial meridian direction are

p = cos θ n + sin θ u, e = - sin θ n + cos θ u .

In the equatorial coordinate system, the same unit vector is

x = sin δ p + cos δ (cos h e + sin h w)

. Taking dot products with

p, w, e

respectively gives

sin δ cos δ sin h cos δ cos h = sin γ sin θ + cos γ cos θ cos η, = sin η cos γ, = sin γ cos θ - cos γ sin θ cos η .

This is an orthogonal transformation between two bases, so it preserves the length of the direction vector. Recover

h

from both its sine and cosine to keep the correct quadrant. At the zenith, azimuth itself is undefined; the vector equations remain meaningful by continuity. Using the more usual eastward azimuth changes the sign of the second equation.

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