Question

a) Using the data on Mars' moon Phobos, (its mass and distance from the planet) determine using the general form of Kepler’s Third Law (a la Newton) the mass of Mars. 

b) List in point form information about Venus, Earth and Mars. Compare and contrast their characteristics from the perspective of the habitability for life both past and present on these planets. 

Answer 1

a) The mass of Phobos is $m=1.07\cdot10^{16}\,\mathrm{kg}.$ The radius of orbit is $9377\,\mathrm{km}.$ To obtain the mass of a planet we should know the period of rotation that is equal to $7^h39^m = 27540^s.$ According to the third Kepler's Law

$\dfrac{T^2}{R^3} = \dfrac{4\pi}{G(M+m)}$ , therefore the mass of Mars is $M = \dfrac{4\pi^2R^3}{T^2G}-m = \dfrac{4\pi^2\cdot(9377\cdot10^3)^3}{(27540)^2\cdot6.67\cdot10^{-11}} - 1.07\cdot10^{16} = 6.4\cdot10^{23}\,\mathrm{kg}.$

We may note that the mass of Phobos is very small compared to the mass of Mars.

b) Venus: $R = 6051\,\mathrm{km}, M =4.9\cdot10^{24} \,\mathrm{kg} = 0.8M_{\text{Earth}}, T = 737\,\mathrm{K}, P_{\text{Atm}} = 91\,\mathrm{atm}.$

Earth: $R = 6378\,\mathrm{km}, M = 6\cdot10^{24}\,\mathrm{kg}, T = 287\mathrm{K}, P_{\text{Atm}} = 1\,\mathrm{atm}.$

Mars: $R = 3397\,\mathrm{km}, M = 6.4\cdot10^{23}\,\mathrm{kg} = 0.1M_{\text{Earth}}, T = 210\mathrm{K}, P_{\text{Atm}} = 6\cdot10^{-3}\,\mathrm{atm}.$

We may see that Venus has the mass and radius similar to the parameters of Earth. But the temperature is significantly larger and the atmosphere is very thick and possesses the large pressure, so the planet is very inconvenient for life and the liquid water can't exist on the surface of Venus. The climate is hot due to the greenhouse effect. Nevertheless, scientists make attempts to detect some kinds of life in the layers of atmosphere, not on the surface.

The conditions on Mars are slightly more convenient for hypothetical life. The temperature on the surface is below 0, but some kinds of living organisms can live in such conditions. But there is too low concentration of oxygen in atmosphere and the pressure is very low, so large organisms should have some artificial source of air. Also there is very weak radiation belt near Mars, so the living organisms will suffer from UV radiation, micrometeoroids and particles with high energy.