Worksheet 3, Problem 1: A Day in the Life of a Martian

The problem: What is the difference between sidereal and solar day on Mars if Mars has the same rotation period and orbits at 1.5 AU.

What we know: We know a solar day on Mars is how long it takes for the sun to make a complete "rotation" in the sky and return to the same point. Because the Sun is not a fixed point in the sky (due to the fact that Earth and Mars are rotating it) a solar day will differ from a sidereal day, which is how long it takes for a fixed point (typically a star) to reach its same location in the sky.

Image from: http://ottawa-rasc.ca/articles/earl_mike/Satellite_Tracking/Dishes/Satellite_Dishes.html

We also know from Kepler's Law: $$P^2 \: \alpha \:  a^3$$
and from the problem: $$P_e= 1 \: year$$ $$a_e= 1\: AU$$ $$a_m= \frac{3}{2} \: AU$$ (where $P_e$ and $P_m$ is the rotation period of the Earth and Mars respectively and $a_e$ and $a_m$ are the distance between the Sun and Earth and Mars respectively).

Solve:
From Kepler's Law we can set up the ratio: $$\frac{P_e^2}{P_m^2} = \frac{a_e^3}{a_m^3}$$ and isolate the rotation period of Mars: $$P_m =\sqrt{\frac{P_e^2a_m^3}{a_e^3}}$$ plugging in the provided values give: $$P_m =\sqrt{\frac{1^2\cdot \frac{3}{2}^3}{1^3}}$$ which simplifies to: $$P_m=\left( \frac{3}{2} \right)^{\frac{3}{2}} \approx \frac{9}{5}\: Earth \:years$$ when converting to days we get $$\frac{9}{5}\: Earth \:years \cdot 365 \: days \approx 660 \: days\: in \:a \:Martian\: year$$ But this is not enough, we want the difference between the two days in minutes per day, not the length of a Martian year. To find the difference we can use the fact that Mars traverses an entire orbit in a 660 days $$\frac{360\,^{\circ}}{660 \:days} \approx \frac{0.55\,^{\circ}}{day}$$ to show that a sidereal day is about half a degree less than a solar day. Then we can convert to days per minute: $$\frac{360\,^{\circ}}{660 \:days}\cdot \frac{1 \:hr}{15\,^{\circ}} \cdot \frac{60 \: min}{1 \: hr} \approx 2 \frac{min}{day}$$

Conclusion: with the given values, the difference between sidereal and solar day is about 2 minutes. This means that Mars's sidereal day is 2 minutes shorter than its solar day.

Acknowledgements: Many thanks to Barra for working on this worksheet with me.