TE Enriched Sample (E)

Orbital Motions under Gravity 3 60 C Kepler’s third law: Period and semi-major axis Kepler’s third law states that special case r Fig. 3.8 A circular orbit is a special case. For any planet, the square of its orbital period T is proportional to the cube of the semi-major axis a of its orbit, i.e. T 2 Ä a 3 . We may also write T 2 = ka 3 where k is a proportionality constant. Kepler’s third law relates the orbital periods and semi-major axes of different planets. It tells us that a planet farther away from the Sun takes a longer time to complete its orbit once. In general, for two planets orbiting the same star, having periods T 1 and T 2 , and orbits of semi-major axes a 1 and a 2 , We shall determine the constant k in the next section when we discuss orbital motion with Newton’s law of gravitation. The Moon moves in an elliptical orbit around the Earth with a period of 27.3 days (d) and a semi-major axis of 384 000 km. Find the period of an artificial satellite orbiting around the Earth in a circular orbit at a distance of 10 000 km from the Earth’s centre. Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Applying Kepler’s third law to the Moon (body 1) and the satellite (body 2), . . d T T a a T T a a 27 3 384 000 10000 0 1147 / / 1 2 2 1 2 3 2 1 1 2 3 2 3 2 # = = = = J L KK J L K J L K d N P OO N P O N P OO n The period is 0.115 days . For a circular orbit, the semi-major axis a is equal to the radius r of the circle. Bodies orbiting around the Earth Example 3.1 T T a a 1 2 2 1 2 3 = J L KK J L KK N P OO N P OO A circular orbit can be considered as a special case of an elliptical orbit. For a circular orbit, the two foci coincide at the centre, and the semi-major axis becomes the radius. As mentioned, Kepler’s laws also apply to other bodies orbiting under a gravitational force. See the following example. Teaching notes Remind Ss that the two plants must orbit around the same star. Also, T 1 and T 2 must be in same unit of time; a 1 and a 2 must be in same unit of length. Teaching notes To help Ss remember T is squared (not cubed), Ts may outline the derivation for circular cases first, noting the time dimension in GM / r 2 = ω 2 r or v 2 / r (see p.65). Leave the details till next section. Q&A (Fig. 3.8) Q: Imagine an elliptical orbit around the star, whose major axis has the same length as the diameter of the circular one. Which orbit has a longer period? A: The same. (Ref: DSE 2020 Q1.2) Teaching notes Remind Ss that the smaller the orbit, the shorter the period. Sample © United Prime Educational Publishing (HK) Limited, Pearson Education Asia Limited 2023 All rights reserved; no part of this publication may be reproduced, photocopied, recorded or otherwise, without the prior written permission of the Publishers.