First we measure the angular separation between Jupiter and its moons. As the moons orbit around Jupiter, the angular separation will also change. We need to use the maximum value, which is called its elongation.
The elongation depends on how far Earth is from Jupiter, of course, because both planets are moving around the Sun. The further away, the smaller.
Distance of Jupiter from Earth, for this example is 91,257,852 km or approximately ~91000000 km. This can be obtained using the parallax method in a previous page.
To determine the size of Jupiter, we can measure the angular diameter of Jupiter. Together with the distance, we can use this to calculate Jupiter's radius and volume.
$$ r = D \tan \frac{\theta}{2} $$We also measure the time taken for the moon to return to the same position with respect to Jupiter. This is the orbital period T of moon.
| Moon | Angular Separation (degrees) | Period (days) | Period (seconds) |
|---|---|---|---|
| Io | 0.266 | 1.76 | 152,064 |
| Europa | 0.423 | 3.53 | 304992 |
| Ganymede | 0.674 | 7.16 | 618,624 |
| Callisto | 1.185 | 16.69 | 1,442,016 |
Angular diameter of Jupiter for this example
0.09 degrees.
Distance of Callisto using angle
$$r = 91000000 \tan(1.185) = 1882344 \mathrm{km} $$
$$T = 1442016 \mathrm{s}$$
$$ G = 6.67 \times 10^{-11} $$Mass of Jupiter:
$$ M = \frac{4 \pi^2 r^3 }{(GT^2)} $$
$$ M = 1.9 \times 10^{27} \mathrm{kg} $$
Angular radius of Jupiter 0.045 Deg.
$$ \mathrm{radius} = 91,000,000 \tan (0.045) $$ $$= 71471 \mathrm{km} = 71.47 \times 10^6 \mathrm{m} $$
Volume of Jupiter
$$ V = \frac{4}{3} \pi r^3 $$ $$V = 1.53 \times 10^{24}\mathrm{m^3} $$Density of Jupiter
$$ \rho = M/V $$
$$ = 1242 \mathrm{kg/m^3} $$Now we have the average density of Jupiter, we can try to guess what it's made of. We can compare with densities of various materials on Earth:
Density of
granite stone
2600 kg/m3
Aluminium
2700 kg/m3
Iron
7200 kg/m3
Water
1000 kg/m3
Just by comparing the densities of materials we can see Jupiter is not made of stone or metal, or any solid. Jupiter is slightly more dense than water. Probably compressed gas.
We can do something similar to calculate the density of Saturn.
If we repeat the measurements and calculations for Saturn and its moons, obtaining the volume, mass and density, we find that Saturn is less dense than water. In this way astronomers, even before radar or space probes were invented, could determine the nature of the outer planets and that Jupiter and Saturn are made of compressed light gases, not rock or metal.
And they did all this just by carefully measuring angles.
By attaching spectrometers to telescopes, they determined that the gases of Jupiter and Saturn are mostly hydrogen and helium. We will look at blackbody, emission transmission and reflection spectra in a later section.