For the problem below, the expectation is that you submit a standalone HTML file (any images should be embedded) back to GitHub. Data for this problem is provided in the starting repository, which you should accept by following the below link. Don’t forget that you are working in groups, which were posted in class. The first person to accept the assignment should form an easily identifiable group name (likely using the names of your group members). Others in the group should also accept the assignment, and then they can just join the existing group. If you accidentally join the wrong group, I need to fix it manually! So try not to join the wrong group, but if it happens just let me know.
Accept AssignmentThe Problem: Jovian Orbits
Though careful observation, you have observed the exact positions of Jupiter and one of its moons over the past two weeks. While in practice these observations would likely generally have been done using local coordinates, it is also not difficult to imagine that the planet and moon positions could have been compared to the background stars to precisely determine their right ascension and declination. The files jupiter.csv and moon.csv in the repository contain the positions of both objects over the time duration in terms of these equatorial coordinates. Your task in this problem will be to use those provided positions and times to estimate the mass of Jupiter.
I’ll provide you with some scaffolding to get you started and moving in a good direction.
Pay attention to the format of the right ascension and declination in the table! Spaces separate the hours/minutes/seconds or degrees/arcminutes/arcseconds. You’ll likely have to do some manipulations to get it into more workable decimal values.
Taking the difference between the moon’s angular position and Jupiter’s angular position will get you the moon’s position relative to the planet. If you visualize this in a plot, you should see a fairly fully closed ellipse. Note that this ellipse is just what we see from Earth, it is not the actual elliptical orbit that the moon follows around Jupiter. However, owing to the fact that we are viewing it mostly from the side, and because the orbits are largely circular, we can use the semi-major axis of this viewed ellipse to be representative of the semi-major axis of the actual orbit. So fitting an ellipse to the viewed positions and extracting the semi-major axis will be helpful.
Your measured semi-major axis fit will be in degrees, since it is viewed from the Earth (and computed from degrees). You’ll need to convert that to an actual physical distance. You can use the average distance to Jupiter as provided in the CSV to help convert.
In order to eventually get the mass, you need to also know the period of the orbit. The easiest way to compute this is to realize that either coordinate difference (RA or DEC) will be varying over time. So if you plot one of them (or both) vs time, then you can read off how much time elapses between successive peaks (or troughs). I’d highly suggest plotting against the Julian Day (JD) for the time, as then the difference between peaks will just be in days.
As we’ll see in class later, if you know the semi-major axis of an orbit and the period of an orbit, you can compute the mass of the objects involved. The equation to do so looks like:
\frac{\left(a_{AU}\right)^3}{\left(p_{yr}\right)^2} = (M_1 + M_2)_\odot
where a_{AU} is the semi-major axis in astronomic units, p_{yr} is the period of the orbit in years, and the total mass is in solar mass units (fraction of the Sun’s mass). For most objects orbiting a central body (like a moon around a planet or a planet around the Sun), the mass of the central object is huge compared to the orbiting object, so the combined mass is really approximately the mass of the larger object.
Making sure your units are correct, you can use your two values computed earlier to estimate the mass of Jupiter! Look up the actual mass as well. What percentage were you off?