Since o = 2k/P, where P is the period of the planet, the period of the relative motion of the two planets, Preb is related to P1 and P2 by

Diagram for finding the distance to an inner

Earth's Orbft

planet.

Now we let one of the planets be the Earth, and express the periods in years. First we look at the Earth plus an inner planet. This means that P1 is the period of the planet and P2 is 1 yr. Equation (22.1) then becomes

Similarly for the Earth and an outer planet, equation (22.1) becomes

In each case Prei is the synodic period and P is the sidereal period.

We now look at how the sizes of various planetary orbits are determined. The technique is different for planets closer to the Sun than the Earth and farther from the Sun than the Earth. Fig. 22.7 shows the situation for a planet closer to the Sun. When the planet is at its greatest elongation, it appears farthest from the Sun. The planet is then at the vertex of a right triangle, as shown in the figure. Since we can measure the angle E between the Sun and the planet, we can use the right triangle to write sin E = r/1 AU

where r is the distance from the planet to the Sun. This equation can be solved for r to give us the distance to the planet, measured in astronomical units.

Methods like this gives us distances in terms of the astronomical unit. Even if we don't know how large the AU is, we can still have all of the distances on the same scale, so we can talk about the relative separations of the planets. The current best measurement of the AU comes from situations like Fig. 22.7. We can now bounce radar signals off planets, such as Venus. By measuring the round-trip time for the radar signal (which travels at the speed of light), we know very precisely how far the planet is from the Earth. The right triangle in Fig. 22.7 gives us cos E = d/1 AU

Since E is measured and d is known from the radar measurements, the value of the astronomical unit can be found. This distance is approximately 150 million kilometers (93 million miles). The exact value is accurate to within a few centimeters.

It is more complicated to find the distance to an outer planet. There are two different methods. The easier one was derived by Copernicus, but is not good for tracing out the full orbit. It just gives the distance of the planet from the Sun at one point in its orbit. Kepler's method of tracing the whole orbit is shown in Fig. 22.8. We make two observations of the planet, one sidereal period of

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