In the 17th century, Kepler proposed his ‘law of harmonies’ for planetary motion. This law suggested that the ratio of the square of the orbital period T to the cube of the mean radius R has the same value for all the planets that orbit the Sun.
Mathematically his ‘law of harmonies’ can be written
where K is a constant.
The table shows data for Earth and Mars.
Planet | T / s | R / m |
Earth | 3.16 × 107 | 1.50 × 1011 |
Mars | 5.93 × 107 | 2.28 × 1011 |
Show that, for Earth and for Mars, K has a value of about 3.0 × 10–19 s2 m–3.
Jupiter is the most massive planet in the solar system and has many moons. Kepler’s law of harmonies applies to the orbiting moons. However, the value of K for the moons is not the same as the value that applies to planets orbiting the Sun.
Ganymede is Jupiter’s largest moon. Ganymede has an orbital radius of 1.07 × 109 m and an orbital period of 172 hours.
Another moon, Io, has an orbital radius of 4.22 × 108 m.
Calculate the orbital period TI of Io about Jupiter.
TI= ............................
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