P

Diagram for finding distance to an outer planet.

planet's Orb it ne

Diagram for finding distance to an outer planet.

the outer planet apart. The Earth is at E1 and E2, respectively, when these are made. The angles ^ and are directly determined. The angles 01 and 02 are known, as well as the distance x. (If the Earth's orbit were circular, then 01 = 02.) We then know — 01 and — 02, and can find d1 and d2, and, finally, r. The advantage of this method is that each point in the planet's orbit can be traced, with the observations overlapping in time.

We have already encountered Kepler's laws (Fig. 22.9) when we discussed the orbits of binary stars. After all, orbiting planets and orbiting stars must obey the same laws of physics.

The first law has to do with what types of paths the planetary orbits can be:

(1) The planets move in elliptical orbits with the Sun at one focus.

The second law, a consequence of angular momentum conservation, states:

(2) A line from the Sun to a planet sweeps out equal areas in equal times.

The Earth's orbit is not very eccentric, but we can still notice the effects of the Earth moving faster at some times than at others. By coincidence, the Earth is closest to the Sun just after the beginning of the year. Thus, it is slightly closer during the (northern) winter than during the (northern) summer. The result is that the number of days from the beginning of fall to the beginning of planet's Orb it

(a) Kepler's second law. Each shaded triangle has the same area. (b) Kepler's third law. In the time the inner planet moves from P| to P' the outer planet moves from P2 to P'2

(a) Kepler's second law. Each shaded triangle has the same area. (b) Kepler's third law. In the time the inner planet moves from P| to P' the outer planet moves from P2 to P'2

spring is less than the number of days from the beginning of spring to the beginning of fall. This shows up in our calendar in two ways. First, the beginning of spring is usually on the 20th of March, while the beginning of fall is usually on the 22nd of September. Second, the shortest month, February, is in the middle of the winter. The third law states:

(3) The square of the period of an orbit (measured in years) is equal to the cube of the semi-major axis of the orbit (measured in AU).

This follows from the inverse square law for gravity. (Actually, Newton deduced the inverse square law by seeing what gravitational law was necessary to give Kepler's third law. See Problem 22.5.)

There is nothing in Kepler's laws which tells us how far each planet should be from the Sun.

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