In order for a Kuiper belt object to become a short period comet that enters the inner solar system, the eccentricity of its orbit has to change from about 0 to pretty close to 1. Suppose this magically happens, and the orbit of a Kuiper belt object changes from a nearly circular orbit with radius 35 A. Suppose further that the new orbit has aphelion point farthest from the Sun of 35 A. Let's start with the picture.
David Hammen wrote People are using planetary equations coupled with geometric integration techniques You could also try what I call a simple finite-step simulation using Newton's laws to operate on object masses, positions, velocities and accelerations.
I'm not sure if this falls within what David calls "geometric integration techniques". My point is that you can do it without incorporating the planetary equations. These disadvantages can be overcome by using other techniques.
You do not need to be an expert in numerical methods to use the simple Leapfrog Integration technique described in detail in Feynman Lectures vol I to model Newtonian Precession in Solar system orbits over periods of up to a few centuries.
By running simulations at various time steps e. Another advantage over analytical methods which produce long-term average results is that you can examine behaviours at shorter time-scales.
For example if you graph perihelion direction vs. It is also a lot of fun and very easy once you have written the basic code to play "what if? That means that we can do cycles of computation per second. That corresponds to a computation time of seconds or about two minutes.
Thus it takes only two minutes to follow Jupiter around the sun, with all the perturbations of all the planets correct to one part in a billion, by this method!
But you do have to think carefully about what you can reliably infer from the simulations - for example if your time-step is longer than a few hundred seconds the simulation will indicate precession in the opposite direction to that which really occurs i.Ever heard of a space penguin?
by Gemma Lavender, 4 December which can be seen interacting with its smaller elliptical companion NGC , looks just like a penguin guarding its egg. where the orbits of NGC ’s stars have become incredibly scrambled due to the gravitational tidal interactions with NGC , warping its once.
The Moon's orbit is elliptical (oval) and varies from circular by about 10%. Its perigee is about , kilometers and the apogee about , kilometers.
Ignoring the details, you can see that this conserved quantity is intimately related to the elliptical orbits. In my opinion, this is the lesson that is generalizable (and hence the answer to the question of "why"): Look for conserved quantities and if they exist, they will let you solve the problem.
ORBITS IN SPHERICAL POTENTIALS 21 a) Harmonic Potential: This is just like a spherically symmetric harmonic oscillator, giving closed elliptical orbits centered on the origin, Λ(r) = 1 2r 2.
() For larger k the orbit looks like the “rosette” shown in Figure So instead of moving in closed elliptical orbits, the planets, etc. in our solar system actually movie in orbits that are open helices from the point of view of the galaxy.
some orbits are truly 3D in a different way: They precess in interesting ways.
Around the earth, Looks like your planets rotate their orbital plane so its always. These worlds are smaller than Pluto and travel in elliptical trajectories around the Sun.
An artist's rendering of Sedna, which looks reddish in color in telescope images. NASA / JPL-Caltech.