The following data is an extract from a planetary fact sheet published by NASA.
| Distance from sun, \(r\) \((\quantity{10^6}{km})\) | Orbital period, \(P\) \((\mathrm{days})\) | Orbital speed, \(v\) \((\mathrm{km\,s^{-1}})\) | Mean surface temperature, \(T\) \((\mathrm{^\circ C})\) | |
|---|---|---|---|---|
| Mercury | \(57.9\) | \(88\) | \(47.4\) | \(167\) |
| Venus | \(108.2\) | \(224.7\) | \(35\) | \(464\) |
| Earth | \(149.6\) | \(365.2\) | \(29.8\) | \(15\) |
| Mars | \(227.9\) | \(687\) | \(24.1\) | \(-65\) |
| Jupiter | \(778.6\) | \(4331\) | \(13.1\) | \(-110\) |
| Saturn | \(1433.5\) | \(10\,747\) | \(9.7\) | \(-140\) |
| Uranus | \(2872.5\) | \(30\,589\) | \(6.8\) | \(-195\) |
| Neptune | \(4495.1\) | \(59\,800\) | \(5.4\) | \(-200\) |
| Pluto | \(5906.4\) | \(90\,560\) | \(4.7\) | \(-225\) |
Physics suggests that the orbital period, \(P\), and distance from the sun, \(r\), are related by a formula of the form \[P=A \,r^k\] where \(A\) and \(k\) are constants. By plotting a suitable graph, estimate the values of \(A\) and \(k\).
Use your results to predict the orbital period of Ceres, a dwarf planet in the asteroid belt with \(r=414.0\).
