Rich example

## Problem

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 $r=414.0$.