The planet Mars has two moons, phobos and delmos.
$(i)$ phobos has a period $7$ hours, $39$ minutes and an orbital radius of $9.4 \times 10^{3} \;km .$ Calculate the mass of mars.
$(ii)$ Assume that earth and mars move in circular orbits around the sun. with the martian orbit being $1.52$ times the orbital radius of the earth. What is the length of the martian year in days?
The planet Mars has two moons, phobos and delmos.
$(i)$ phobos has a period $7$ hours, $39$ minutes and an orbital radius of $9.4 \times 10^{3} \;km .$ Calculate the mass of mars.
$(ii)$ Assume that earth and mars move in circular orbits around the sun. with the martian orbit being $1.52$ times the orbital radius of the earth. What is the length of the martian year in days?
$T^{2}=\frac{4 \pi^{2}}{G M_{m}} R^{3}$
$M _{m}=\frac{4 \pi^{2}}{G} \frac{R^{3}}{T^{2}}$
$=\frac{4 \times(3.14)^{2} \times(9.4)^{3} \times 10^{18}}{6.67 \times 10^{-11} \times(459 \times 60)^{2}}$
$M _{m}=\frac{4 \times(3.14)^{2} \times(9.4)^{3} \times 10^{18}}{6.67 \times(4.59 \times 6)^{2} \times 10^{-5}}$
$=6.48 \times 10^{23} kg$
$(ii)$ Once again Kepler's third law comes to our aid.
$\frac{T_{M}^{2}}{T_{E}^{2}}=\frac{R_{M S}^{3}}{R_{E S}^{3}}$
where $R_{\text {us }}$ is the mars -sun distance and $R_{E S}$ is the earth-sun distance.
$\therefore T_{M}=(1.52)^{3 / 2} \times 365$
$=684$ days
We note that the orbits of all planets except Mercury, Mars and Pluto* are very close to being circular. For example, the ratio of the semi-minor to semi-major axis for our Earth is, $b / a=0.99986$
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