AST101: Our Corner of the Universe Lab 8: Measuring the Mass of Jupiter Name: Student number (SUID): Lab section number: 1 Introduction Objectives In a previous lab, we measured the mass of the Earth with a pendulum and found it to be 6 10 24 kg. In this lab we will use Kepler s and Newton s laws to measure the mass of Jupiter and see how it compares to the mass of the Earth. Materials A chart showing the motion of the moons of Jupiter, a ruler and a calculator. 2 The Galilean Moons The Galilean moons are the four largest moons of Jupiter and were discovered in 1610 by Galileo Galilei. They are named Io, Europa, Ganymede and Callisto after the lovers of Zeus, the Greek equivalent of the Roman god Jupiter. The Galilean Moons are among the most massive objects in the Solar system, comparable in size to the planet Mercury, with a radius larger than any of the dwarf planets. Figure 1 shows the Galilean moons as seen through a telescope. Galileo observed these moons orbiting Jupiter, providing evidence for the Copernican model of the solar system, rather than the prevailing Earth-centric model of the Solar system. Figure 1: The four Galilean moons of Jupiter seen through an optical telescope. 1
Io, which has a mass of 8.93 10 22 kg is the closest moon to Jupiter, with a semi-major axis of 421, 800 km. Table 1 shows the masses and semi-major axes of the Galilean moons relative to those of Io. Moon Mass Semi-major axis Io 1 M Io 1 a Io Europa 0.54 M Io 1.6 a Io Ganymede 1.7 M Io 2.5 a Io Callisto 1.2 M Io 4.5 a Io Table 1: The masses and semi-major axes of the Galilean moons, relative to M Io = 8.93 10 22 kg and a Io = 421, 800 km. Remember that Newton s law of gravitation tells us that the force of gravity scales as the mass of the moon, divided by the inverse square of the distance from Jupiter: F = G M JupiterM Moon r 2. (1) If we double the mass of the moon, the force goes up by a factor of 2, but if the moon is twice as far from Jupiter, then force goes down by a factor of 2 2 = 4. First we will rank the Galilean moons by the strength of the force of gravity between the moon and Jupiter. Let s consider Europa, which is 0.54 times as massive as Io and has a semi-major axis 1.6 times that of Io. (For this lab, we will assume that the orbits of the Galilean moons are circular so that r in Newton s law is given by the semi-major axis a.) This means that the force of gravity between Europa and Jupiter is reduced by a factor of 0.54 = 0.21 (2) 1.62 relative to that between Io and Jupiter. So the force between Jupiter and Europa is smaller than that between Jupiter and Io. Using the space below, calculate the strength of the gravitational force between Jupiter and the other two moons (Ganymede and Callisto), relative to the force between Jupiter and Io, and so rank the moons by the strength of the force between Jupiter and each moon from largest to smallest. You do not need to know G or the mass of Jupiter to do this. Largest Smallest 2
3 The Motion of the Galilean Moons Kepler s third law tells us that the square of the period of a planet s (or a moon s or a satellite s) orbit is proportional to the cube of the semi-major axis, or P 2 = Ka 3, (3) where P is the period in years, a is the semi-major axis in A.U. and K is a constant of proportionality. We have seen in a previous lab how this constant K depends on the mass of the object that the planet is orbiting. Newton later showed that all of Kepler s empirical laws could be derived from his own, more fundamental, laws of motion as set forth in his book PhilosophiæNaturalis Principia Mathematica. Newton also showed that the constant of proportionality in the above formula depends upon the masses of the bodies involved P 2 = a 3 /M, (4) where P is the period in years, a is the semi-major axis in A.U. and M is measured in solar masses. The mass of the Sun (written 1 M ) is 1.99 10 33 kg. Remember that a planet s mass does not affect the orbital period of a planet and so M in equation (4) is the mass of the body being orbited. In the above form, we can use Kepler s Third Law to determine the mass M, if we know P and a. In particular, the mass of a planet can be determined from the orbital motion of its moons. In order to apply Newton s revision of Kepler s Third Law, we must know the semi-major axis (in A.U.) as well as the period of one of Jupiter s moons. We will determine each of these from the chart provided on the last page of this lab. We must first make sure we understand what information is contained in the chart. If we observe Jupiter through the telescope at midnight, on October 10, what will we see? In the space below, draw and label a picture of the telescopic view. Be sure you have the correct relative distance of the moons from Jupiter. Draw and label the same situation as viewed from above Jupiter s North Pole. 3
Make another two sketches showing what you would see on October 19. First draw the telescope view Draw and label the same situation as viewed from above Jupiter s North Pole. 4 Measuring the Mass of Jupiter Follow the procedure below to measure the mass of Jupiter: 1. First, obtain the semi-major axes of Callisto and Ganymede. Measure the distance between the extreme points east and west of Jupiter in the motion of Callisto and Ganymede as shown on the chart. Divide by 2 to obtain the semi-major axes with respect to the scale of the chart. All measurements should be made to the nearest tenth of a millimeter. Callisto: a = mm, Ganymede: a = mm. 2. To convert your measurement of the semi-major axis to A.U., measure Jupiter s diameter in mm. Divide Jupiter s diameter at it s equator (9.5 10 4 A.U.) by this measurement to obtain the appropriate scale factor: Scale factor = 9.5 10 4 A.U. Diameter in mm = 3. Multiply the semi-major axis as measured in the scale of the chart by this scale factor to obtain the semi-major axis in A.U. Callisto: a = A.U., Ganymede: a = A.U. = 4
4. Next we will find the period of Callisto and Ganymede orbit s about Jupiter. Again, referring to the chart, measure the distance between the maxima of the greatest eastern or western elongation from Jupiter. Convert this number from days to years, to obtain the period P. Callisto: P = years, Ganymede: P = years. 5. Using your measurements and equation (4) find the mass Jupiter (in solar masses) from both Callisto s and Ganymede s motion. M Jupiter using Callisto s motion: = M, M Jupiter using Ganymede s motion: = M. 6. What is the average of your measurements for Jupiters mass? 7. How much more massive is Jupiter than the Earth, according to your measurements? 5