As for any research, the first question you need to answer for yourself is: "how much time do I realistically have and what happens if I don't finish this research in time?"
Organization
It seems like your scope is quite wide. From my experience, I would recommend organizing your research ideas into incremental steps. Each step feeds into the next step of your research. I'll emphasize that the first step of your research should be simple, e.g. "load the ephemeris in my Python script and query it."
A rule of thumb is the $\pi$-rule: take your first time estimate at completing some task, and multiply it by $\pi$. If it's your first time working on this problem, multiply by $\pi^2$. For example, if you think it's one day of work, and you've never done this before, expect more like 9.8 days.
Preemptively think about how you'll present the results of your research. Will these be plots? If so, plan the title of the plots even before you start any technical work. What kind of context will be needed for those plots?
Research ideas
The orbit of Mercury is an interesting topic!
Luckily for any mission designer and spacecraft navigator, NASA publishes the position of all the planets, and most moons and a number of asteroids. This data is available for free: https://ssd.jpl.nasa.gov/?ephemerides . You can either query the HORIZONS database with a specific time and "observer," or you can download one of the BSP ephemeris files (the latest version is de438.bsp I believe, but de405.bsp is still very commonly used despite being a few years old).
Here are some miscellaneous ideas:
- Compare the true position of Mercury as computed by NASA and a celestial dynamics simulation where you ignore relativistic effects and the mass of any other object in the solar system.
- Then, use a pre-existing software for N-body dynamics and compute the orbit of Mercury assuming two or three of the most influential masses (I would guess Sun, Venus and Jupiter, but check the maximum gravitational pull each planet could produce on Mercury if it was as close as it could be to it).
- Finally, extend this other planets.
Gravity assists
Two variables determine the usefulness of gravity assists. First, the mass of the object which you'll use, and second the radius of periapse passage around that planet. This is further explained on this NASA page, but I would also recommend searching for Dr. Kate Davis who teaches interplanetary mission design at the University of Colorado at Boulder. I think there's a PDF handout online which explains the calculations.
The point of a gravity assist maneuver is the transfer some of the energy of the planet to your spacecraft. Hence, the more mass a planet has, the more energy it has, and therefore the more you can "harvest." (In theory, you also slightly slow down the planet, but in practice of course the numbers of negligible). Therefore, a gravity assist around Mercury would not be all that interesting, but one around the Sun would be! And has been used.
Best of luck!
