Lagrange Laboratory Assignment
This laboratory experiment explores the nature of Lagrange points and related
phenomena.
First, you will simulate different orbits of a small asteroid in a toy solar
system model consisting of a Sun-like star and a single planet. This is a
"restricted 3-body" gravitational system, in which only the star and planet are
of significant mass. Such a situation enables tests of the Lagrange
equilibrium points
(see Wikipedia)
that obtain in simple but common cases (circular orbit, small secondary mass).
The simulation is in VPython and can be run in the same Glowscript
environment used for University Physics courses. Ask your instructor for help
if you are unfamiliar with this.
Simulation-based tasks:
- Try putting the asteroid at each of the 5 Lagrange points by setting the
initial position and velocity of the "rock" object accordingly. Let each run
go for a while to study its behavior. Which points appear to be stable?
Do these results agree with theoretical predictions? Explain.
- Experiment with different time step sizes (1 day, 0.1 day, etc.).
Does this affect your results? Explain why or why not.
- Try different values of the planet/star mass ratio (0.001, 0.01, 0.03, etc.). Do these affect the stability of the "stable" Lagrange points? Compare your results to those described in the textbook or in the article below.
- Experiment with reducing the initial star and planet velocities below the circular speed. What effect does this have?
- Test the resonance effect, e.g., with P_asteroid = P_planet / 2, or P_planet * 2 / 3, etc. How does the stability of such orbits compare to others without an obvious resonance (e.g., P_asteroid = 0.732 * P_planet)?
The remainder of the lab is a pencil-and-paper exercise (an old homework
assignment).
- Following a similar procedure to this L2
derivation, derive the position of the L1 point for the Sun - Earth
system. (This is not the point where the gravitational attraction of
the Sun and Earth on a third body is the same! The moon's orbit lies further
from the Earth than this point!) As for L2, the L1 calculation is made more
tractable with
a binomial
expansion.
- The L1 and L2 points define the near and far edges of a region called the
Hill sphere, within
which stable orbits exist. Use the L1 and L2 formulae from the previous step
to calculate which planet in our Solar System has the
largest Hill sphere, and give the numerical result. Use a textbook or
online resources to find the data you need. Then explain WHY this planet has
the largest Hill sphere.
- Use the same formulae to calculate which planet in our Solar System has
the smallest Hill sphere (for simplicity, exclude "dwarf planets").
Explain why THIS answer is correct.
- Calculate and report which moon's mean orbit radius is the largest
fraction of its planet's Hill radius. This is an empirical check on how close
stable moon orbits can get to the edges of the Hill sphere.
Some references:
ASTR 414 Course Page