SETI’s Sky

It was once asked on the SETI@home forums whether the SETI programme really scans a third of the sky, as it scans between 0° and 40° north. A small amount of maths shows that that range really is a third of the sky.

A translucent blue sphere with the range 0 degrees to 40 degress in translucent red

Arecibo is based in Puerto Rico. Its latitude is approximately 20° north, and Arecibo can scan the range of roughly 20° north or south of this. This 0° to 40° north range is highlighted on the translucent sphere above.

To calculate the area, first we divide it into a number of strips. For a strip a latitude θ° of width δθ°, the following inequalities bound its area, A:

two pi cosine, open brackets, theta plus delta theta, close brackets, delta theta is less than or equal to A, is less than or equal to two pi cosine theta delta theta

We then sum the area of these strips to give the total area, T, of the region from θmin to θmax, and take the limit as δθ tends to zero:

limit as delta theta tends of zero of the sum from theta min to theta max of two pi cosine, open brackets, theta plus delta theta, close brackets, delta theta is less than or equal to T, is less than or equal to limit as delta theta tends of zero of the sum from theta min to theta max of two pi cosine theta delta theta

In the limit the summations turn to integrals, and as these are both the same we get:

T equals the integral from theta min to theta max of two pi cosine theta d theta

So:

T equals two pi times, open bracket, sine theta max minus sin theta min, close bracket

After plugging in θmin=0° and θmax=40°, and dividing by 4π (the total surface area of the sphere) we find that the region covers about 0.32 of the sphere - about a third.

This article was last edited on 14th April 2007. The author can be contacted using the form below.