In this midterm we will attempt at writing a program that simulates the solar system. We start with the Earth-Sun system we studied in homework 5 and study elliptical orbits and their properties. We test also elliptical orbits and their dependence on powers \( \beta \) of \( r^{\beta} \). We will test other aspects of the Earth-Sun system and link these to the theoretical discussion on two-body problems with central forces.
Thereafter we will add Jupiter to our system before we move on to including all planets of the solar system and we will attempt at making a code which simulates the solar system.
The relevant reading background is
In our lectures (see also Taylor chapter 8) we have studied an attractive potential
$$ V(r)=-\alpha/r, $$where the quantity \( r \) is the absolute value of the relative position and \( \alpha \) is a constant.
When we rewrote the equations of motion in polar coordinates, we found the analytical solution to the radial equation of motion
$$ r(\phi) = \frac{c}{1+\epsilon\cos{(\phi)}}, $$where \( c=L^2/\mu\alpha \), with the reduced mass \( \mu \) and the angular momentum \( L \), as discussed during the lectures. With the transformation of a two-body problem to the center-of-mass frame, the actual equations look like an effective one-body problem.
The quantity \( \epsilon \) is what we called the eccentricity. Since we will mainly study bounded orbits, we have \( 0 \le \epsilon < 1 \). For the Earth, the orbit is indeed close to circular and at perihelion (the closest distance to the Sun), the Earth's center is about 0.98329 astronomical units (AU) or 147,098,070 km from the Sun's center. For Earth, the orbital eccentricity is \( \epsilon\approx 0.0167 \). The outer planets have more elliptical orbits. For example, Mars has its perihelion at 206,655,215 km and its apehelion at 249,232,432 km.
In this part we will limit ourselves to the Earth-Sun system we studied in homework 5. You can reuse your code with either the Velocity-Verlet or the Euler-Cromer algorithms from homework 5.
This means also that \( \alpha=GM_{\odot}M_{\mathrm{Earth}} \). We will use \( \alpha \) as a shorthand in the equations here. Keep in mind that in homework 5 you scaled \( GM_{\odot}=4\pi^2 \) in your code.
The exercises here are all based on you analyzing the results from your codes from homework 5. Homework 8 is also useful.
As a reminder, we list the equations we studied in homework 5. Newton's law of gravitation is given by a force \( F_G \) (we assume this is the force acting on Earth from the Sun)
$$ F_G=-\frac{GM_{\odot}M_{\mathrm{Earth}}}{r^2}, $$where \( M_{\odot} \) is the mass of the Sun and \( M_{\mathrm{Earth}} \) is the mass of the Earth. The gravitational constant is \( G \) and \( r \) is the distance between the Earth and the Sun. We assumed that the Sun has a mass which is much larger than that of the Earth. We could therefore safely neglect the motion of the Sun.
In homework 5 we assumed that the orbit of the Earth around the Sun was co-planar, and we took this to be the \( xy \)-plane. Using Newton's second law of motion we got the following equations
$$ \frac{d^2x}{dt^2}=-\frac{F_{G,x}}{M_{\mathrm{Earth}}}, $$and
$$ \frac{d^2y}{dt^2}=-\frac{F_{G,y}}{M_{\mathrm{Earth}}}, $$where \( F_{G,x} \) and \( F_{G,y} \) are the \( x \) and \( y \) components of the gravitational force.
You can obviously set \( \alpha=GM_{\odot}M_{\mathrm{Earth}} \).
Our final aim is to write a code which includes the known planets of the solar system.
We will, as before, use so-called astronomical units when rewriting our equations. Using astronomical units (AU as abbreviation)it means that one astronomical unit of length, known as 1 AU, is the average distance between the Sun and Earth, that is \( 1 \) AU = \( 1.5\times 10^{11} \) m. It can also be convenient to use years instead of seconds since years match better the time evolution of the solar system. The mass of the Sun is \( M_{\mathrm{sun}}=M_{\odot}=2\times 10^{30} \) kg. The masses of all relevant planets and their distances from the sun are listed in here in kg and AU.
Pluto is no longer considered a planet, but we add it here for historical reasons. It is optional in this midterm project to include Pluto.
In setting up the equations we can limit ourselves to a co-planar motion and use only the \( x \) and \( y \) coordinates. But you should feel free to extend your equations to three dimensions, it is not very difficult and the data from NASA are all in three dimensions. You find these data at the NASA has an excellent site at https://ssd.jpl.nasa.gov/horizons/app.html#/ site. From there you can extract initial conditions in order to start your differential equation solver. At the above website you need to change from OBSERVER to VECTOR and then write in the planet you are interested in. The generated data contain the \( x \), \( y \) and \( z \) values as well as their corresponding velocities. The velocities are in units of AU per day. Alternatively they can be obtained in terms of km and km/s.
We will start with the three-body problem, still with the Sun kept fixed as the center of mass of the system but including Jupiter (the most massive planet in the solar system, having a mass that is approximately 1000 times smaller than that of the Sun) together with the Earth. This leads to a three-body problem. Without Jupiter, the Earth's motion is stable and unchanging with time. The aim here is to find out first how much Jupiter alters the Earth's motion.
The program you have developed in homework 5 can easily be modified by simply adding the magnitude of the force between the Earth and Jupiter.
This force is given again by (without vector symbols, only as function of the value of the relative distance)
$$ F_{\mathrm{Earth-Jupiter}}=-\frac{GM_{\mathrm{Jupiter}}M_{\mathrm{Earth}}}{r_{\mathrm{Earth-Jupiter}}^2}, $$where \( M_{\mathrm{Jupiter}} \) is the mass of Jupyter and \( M_{\mathrm{Earth}} \) is the mass of the Earth. The gravitational constant is \( G \) and \( r_{\mathrm{Earth-Jupiter}} \) is the distance between Earth and Jupiter.
We assume again that the orbits of the two planets are co-planar, and we take this to be the \( xy \)-plane (you can easily extend the equations to three dimensions, feel free to run your calculations in two or three dimensions).
The following gives you an opportunity to earn five extra credit points on each of the remaining homeworks and ten extra credit points on the midterms and finals. This assignment also covers an aspect of the scientific process that is not taught in most undergraduate programs: scientific writing. Writing scientific reports is how scientist communicate their results to the rest of the field. Knowing how to assemble a well written scientific report will greatly benefit you in you upper level classes, in graduate school, and in the work place.
The full information on extra credits is found at https://github.com/mhjensen/Physics321/blob/master/doc/Homeworks/ExtraCredits/. There you will also find examples on how to write a scientific article. Below you can also find a description on how to gain extra credits by attending scientific talks.
This assignment allows you to gain extra credit points by practicing your scientific writing. For each of the remaining homeworks you can submit the specified section of a scientific report (written about the numerical aspect of the homework) for five extra credit points on the assignment. For the two midterms and the final, submitting a full scientific report covering the numerical analysis problem will be worth ten extra points. For credit the grader must be able to tell that you put effort into the assignment (i.e. well written, well formatted, etc.). If you are unfamiliar with writing scientific reports, see the information here
The following table explains what aspect of a scientific report is due with which homework. You can submit the assignment in any format you like, in the same document as your homework, or in a different one. Remember to cite any external references you use and include a reference list. There are no length requirements, but make sure what you turn in is complete and through.
| HW/Project | Due Date | Extra Credit Assignment |
| HW 3 | 2-8 | Abstract |
| HW 4 | 2-15 | Introduction |
| HW 5 | 2-22 | Methods |
| HW 6 | 3-1 | Results and Discussion |
| Midterm 1 | 3-12 | Full Written Report |
| HW 7 | 3-22 | Abstract |
| HW 8 | 3-29 | Introduction |
| HW 9 | 4-5 | Results and Discussion |
| Midterm 2 | 4-16 | Full Written Report |
| HW 10 | 4-26 | Abstract |
| Final | 4-30 | Full Written Report |
You can also gain extra credits if you attend scientific talks. This is described here.
This opportunity will allow you to earn up to 5 extra credit points on a Homework per week. These points can push you above 100% or help make up for missed exercises. In order to earn all points you must:
Approved talks: Talks given by researchers through the following clubs:
All the material on extra credits is at https://github.com/mhjensen/Physics321/blob/master/doc/Homeworks/ExtraCredits/.