Finding the Mass of a Planet with a Satellite Question 110S: Short Answer

Finding the mass of a planet with a satellite Question R2 g M  110S: Short Answer G The equation How are masses known? 2 3 GMT Data for planets and the Sun, including their masses, are provided in R  2 does this in one step, but it is less obvious how the reference books and also on this CD-ROM (see File 30T 'Planetary 4 orbit data'). How can their masses be known? calculation works. Questions We can find the mass of Earth ME by placing a known mass m at a GMT 2 known distance r from its centre and measuring its weight F with a 1. Rearrange R3  to make M its subject. spring balance. 4 2 F  r 2 Then…. M  . E Gm But we cannot make such measurements at the surface of the Sun or other planets. There we must use the motion of a satellite to calculate the mass. The following relationship is shown for a geostationary 2. Oberon, a moon of Uranus, has a periodic time of 323 hours 5 satellite in section 11.2 of the Advancing Physics A2 student’s book. and its mean distance from the planet is 5.82 x 10 km. GMT 2 Calculate the mass of Uranus. R3  where R is the mean orbit radius and T the orbit 4 2 period. This was derived by noting that the centripetal force on a satellite is provided solely by the gravitational force of attraction. The equation therefore applies to all satellites. Notice that M can be found provided 3. Earth’s Moon has a periodic time of 27.3 days, and its mean we know R, T and G. Astronomers can measure R and T remotely, and orbit radius is 3.8 x 105 km. What do these figures suggest for we already know G. Earth’s mass? A simpler way to see what is going on is to notice that given radius R of orbit and time T, you can at once calculate the orbital speed v. 2R v  T 4. In 1978 a moon of Pluto was discovered, called Charon. That Then you can calculate the centripetal acceleration under gravity moon’s mean orbit radius is 1.9 x 107 m and its orbit period v2 6.4 days. g  R Use these data to calculate the mass of Pluto. Knowing g, you can find the mass M of the planet from the law of gravitation: GM g  R2 so 5. Earth orbits the Sun every 365 days, with a mean orbit radius of 1.5 x 108 km. What do these data suggest as a value for the mass of the Sun?

6. Can we use a similar method to find the mass of a moon (which has no natural satellite)? Explain your answer.