TE Enriched Sample (E)

Kepler’s laws of planetary motion 61 3.1 D Orbital motion around the Sun Kepler’s third law can be reduced to a simpler form when applied to orbital motion around the Sun. First, applying it to the Earth, we have T E 2 = ka E 3 where T E and a E are the period and semi-major axis of the Earth’s orbit. For another planet orbiting the Sun, T 2 = ka 3 with the constant k . Dividing the two equations, we have T T a a E E 2 3 = J L K d N P O n If the semi-major axes are given in AU and the orbital periods are given in years (y), a E = 1 AU and T E = 1 year, and T 2 = a 3 It is important to note that the above equation only applies to • orbital motion around the Sun, and • the unit of T is the year while the unit of a is AU. Table 3.1 shows the data for planets in the solar system. Note that the value of T 2 is almost equal to the that of a 3 for each planet, and the graph of T 2 versus a 3 gives a straight line of a slope equal to one (Fig. 3.9). planet T / y T 2 / y 2 a / AU a 3 / AU 3 Mercury 0.241 0.058 0.387 0.058 Venus 0.615 0.38 0.723 0.38 Earth 1.00 1.0 1.00 1.0 Mars 1.88 3.5 1.52 3.5 Jupiter 11.9 141 5.20 141 Saturn 29.5 870 9.54 870 Uranus 84.0 7100 19.2 7100 Neptune 165 27 000 30.1 27 000 Table 3.1 Kepler’s third law applied to the planets in the solar system ( MC: DSE 2020 Q1.2) 0 0.1 1 10 100 1000 10 000 100 000 Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune 100 000 10 000 1000 100 10 1 0.1 T 2 / y 2 a 3 / AU 3 Fig. 3.9 A graph of T 2 against a 3 for various planets Teaching notes The direct application of T 2 = a 3 is not required in the HKDSE syllabus. However, it is good to let Ss to know such a short- cut when they calculate the orbital motion around the Sun. Teaching notes Remind Ss that ‘year’ is actually the ‘the Earth year’ in Table 3.1, the time for the Earth to revolve around the Sun once. 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.