## R

If we measure the time delay, and we know the planet's radius, R, then we can solve equation (24.1) for O. This does not give a point uniquely. There is a whole ring of points that all have the c

S (a) HST image of Mars at closest approach to Earth. (b) Space-based image of Mars. [(a) STScI/NASA; (b) NASA]

S (a) HST image of Mars at closest approach to Earth. (b) Space-based image of Mars. [(a) STScI/NASA; (b) NASA]

same 0. As viewed from the radio telescope, lines of constant time delay appear as concentric rings about the closest point.

We now look at the Doppler shift. If a point on the equator moves with a speed v0, then a point at latitude p moves with speed (see Problem 24.2)

Now we view this point from the above the pole, assuming the line from the point to the pole makes an angle 0 with the line from the pole to the telescope. The Doppler shift depends on the radial velocity, vr(0, p), which is given by vr (0, p) = v(p) sin 0

As seen from the telescope, lines of constant Doppler shift form concentric rings about the point on the equator that is just appearing from the back side, and the point on the equator that is just about to disappear.

By combining time delay and Doppler shift data, we limit the source of the echo to two possible points. These are the two points where the time delay circle intersects the Doppler shift circle. The remaining ambiguity in the location of the feature can be removed by observing at a different time where the feature's location and Doppler shift have changed. More recently, orbiting spacecraft have also been used for higher resolution radar mapping on Venus and Mars.

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