Write an expression for the gravitational potential VgV_{\text{g}}Vg at the surface of a spherical planet of mass MMM and radius rrr.
The table below shows some astrophysical data for two newly discovered planetary bodies, Calypso and Boreas, orbiting a distant star.
| Planet | Mass / kg\text{kg}kg | Radius / m\text{m}m | Mean distance from parent star / m\text{m}m |
|---|---|---|---|
| Calypso | 4.80×10234.80 \times 10^{23}4.80×1023 | 3.40×1063.40 \times 10^{6}3.40×106 | 8.20×10108.20 \times 10^{10}8.20×1010 |
| Boreas | 1.20×10231.20 \times 10^{23}1.20×1023 | 1.80×1061.80 \times 10^{6}1.80×106 | 2.10×10122.10 \times 10^{12}2.10×1012 |
Show that the escape velocity vvv of a gas molecule of mass mmm on the surface of Boreas is given by the equation
v=2GMr v = \sqrt{\frac{2GM}{r}} v=r2GMwhere MMM is the mass of Boreas and rrr is its radius.
Calculate the escape velocity vvv of gas molecules on the surface of Boreas. (Use G=6.67×10−11 N m2 kg−2G = 6.67 \times 10^{-11}\text{ N m}^2\text{ kg}^{-2}G=6.67×10−11 N m2 kg−2)
Explain why Calypso has almost no atmosphere while Boreas retains a dense atmosphere. Use data from the table to support your explanation.
