How to Weigh a Planet

Research
Useful links
Useful definitions
Link to daily images of Jupiter and its four biggest moons

Estimating the mass of Jupiter

You are going to estimate the mass of Jupiter by observing the motion of four of its moons over the next few weeks. The four moons you will be observing are the same ones that enabled Galileo to verify Copernicus' theory that the planets move around the Sun, rather than the other way round - Io, Europa, Ganymede and Callisto.

Each school has been allocated a moon to focus on:

• Rusheymead Secondary School - Europa
• St Joseph's College - Ganymede
• Callisto - Northolt High School

and you can all also use the data for Io, to make a comparison with your other moon.

One way to do this is to make observations with a telescope - and if you have the equipment to do this and the weather is good enough, we hope that you will at least make a few direct observations. However, given that you may not have adequate equipment and / or the weather may not good enough, you will also be provided with a link which will show the position of the four moons and Jupiter each day. You will need to:

• record the exact position of the moon you are looking at each day
• use your data to draw a graph of position against time
• use your graph to calculate the mass of Jupiter

What you will see is a picture of a large mass (Jupiter) and four small masses (Io, Europa, Ganymede and Callisto) which appear to move forwards and backwards in a line (as in the first diagram on the right). Why do you think we don't see the roughly circular orbits they are actually travelling in (as in the far right diagram)?

The equation you need for the calculation is:

In this equation, M is the mass of Jupiter, a is the length of the semi-major axis, and p is the period of the moon's orbit. You are going to estimate a and p, and from them, estimate M. We can make things more straight-forward by assuming the orbits are circular, in which case the semi-major axis is simply equal to the radius of the orbit. We need to be careful with the units in which measure a and p, if this equation is to work, however. If a is in units of the mean Earth-Sun distance, known as the astronomical unit (or 1AU), and p is in Earth years, then M will be in units of the mass of the Sun.

You are measuring the perpendicular distance of the moons from the line of sight between Jupiter and the Earth. These distances should give a curve which looks like the graph of y = sin x if you have enough measurements. Draw your graph, and fit a curve to it. The amplitude of the curve is equal to the radius of the moon's orbit, and the period of the curve is equal to the period of the orbit. You will need to think about the units in which your measurements are made, and convert them to the units discussed above.

It is likely that the estimates you get for the mass of Jupiter won't be the same for each moon, and won't be the same as the other schools. Think about reasons why this is likely to be the case.

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Research

You should also do some research into the history linked with this topic:

• Find out more about people like Copernicus, Galileo, Tycho Brahe, and Kepler. What did they discover or find out?
• What invention was it that gave Galileo the opportunity to make the discoveries he did?
• What space missions have been sent to Jupiter and its moons?

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Links to webpages you may find useful

• Animate your own planet and see Kepler's 3rd Law demonstrated
• Animation to help you understand Kepler's Laws
• Just about everything you need! - including poetry, history, pictures, music, maths, astronomy, mythology and much more
• An account by an American high school student of how he went about finding the mass of Jupiter from observations on the moons
• A question and answer on how to use observations on a moon of Jupiter to find its mass

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Useful information and definitions

Astronomical units

Because the distances they have to deal with are so huge, astronomers do not generally use kilometres. Light-years are used for distances between stars and galaxies, but these are too big to be useful in the solar system. The AU is a commonly used unit between the kilometre and the light-year.

1 Astronomical Unit = 149 598 000 kilometres which is approximately 150 000 000 km = 1.5 x 10⁸ km

For comparison, 1 light year = 9.5 x 10¹² km, and the distance between the Sun and Pluto is 5 913 000 000 km = 5.913 x 10⁹ km = 39.48 AU which is about 5.5 light hours.

Mass of the Sun

The mass of the Sun is 1.9891 x 10³⁰ kg. Your result for the mass of Jupiter will be in units of Sun masses, so you will need to convert it to kg.

Sine curve

The diagram shows a sine curve (y = sin x), with the amplitude and period marked on it.

Semi-major axis

In the diagrams on the left, the length of AO = OB is the semi-major axis in both cases. (CO = OD is the semi-minor axis).