zondag 24 maart 2013

Horseshoe orbit

This week I spend quite some time reading, correcting and grading assignments for a course called planetary sciences (As a PhD student you also have to do numb making work, it builds character...they say ;)). The assignment was about dynamics in space and in particularly the first question made me go back in time. The story starts a long, long time ago (well a few years but that doesn't sound exciting) in a city far far close by.

I was a student learning the marvelous secrets of astrodynamics. Especially the three-body dynamics  (no not that situation you are think of) caught my attention. In a three body system (like the Sun-Earth-satellite, asteroid or an English teapot, but now we are sliding in a different discussion), all sort of fancy motions occur due to the interaction of both the attraction of the Sun and the attraction of the Earth. With a few simple assumptions, equilibrium locations can be found where asteroids will be drawn to. A sort of graveyard of old rocky asteroids. For the Sun-Jupiter system these asteroids are called the Trojans and Greeks (I told you this was starting as a good story, now it is almost becoming a myth).

As a student, I was fascinated (and I still am) about the concept that such simple physics (three-body dynamics) could be observed in reality. I wanted to try it myself and see if I could generate Trojans and Greeks, in the end I stumbled on a horseshoe (maybe from a wooden one, who knows).

As a real God Khaos (we are staying in the Greek mythology), I created a Sun and a Earth. I gave it both masses equal to our own Sun and Earth. As a proper god I gave both objects one rule to obey. I stole it from (the mortal) Newton (another personality trait of a god), he wrote down the following words a few decades ago:

"Any particle of matter in the universe attracts any other with a force varying directly as the product of the masses and inversely as the square of the distance between them" - Sir I. Newton (1687)

But something went wrong. Both bodies kept crashing into each other, wherever I put them. In my frustration, I pushed the Earth (giving it speed (momentum, because it has mass)) and it begane to orbit the Sun (even gods make mistakes in the beginning and then violence solves the problem). I looked up the ephemeris of our own Earth and gave my Earth the same momentum and position (our Earth has a slightly eccentric orbit, which can be noticed later on).

Now I only needed a third body, an asteroid that orbits close to Earth and I found one in the news. That week, scientists found a stalking asteroid with a 'horseshoe orbit' (ahh, you have to love the press for making science a badly written B-movie). However I found my asteroid (2010 SO16). What was this horseshoe orbit and more importantly would it also appear in my newly created univers? I tracked down the observations of the asteroid and found its state-vector (very important for guys like me, because then we can predict the future using our glass globe (or in my case an Adams-Bashford 15 multistep integrator)).

This is what I saw in my univers:

The red orbit being Earth and the blue, asteroid 2010 SO16. Nice 'almost' circular orbits going round-and-round. Only no 'horseshoe' orbit. Was my univers broken?

Thinking about in what way did I broke my univers, I gave the reference frame a pull (again violence, you have to love being a deity) such that it rotated in the same rotation as my Earth. 50 years long (for a being that is immortal, 50 years is only 3.2 seconds if he has the latest Macbook Pro, with quad core computing power) I watched my spinning univers, and magic unfolded in front of my eyes:

The red orbit became a point (well a small line, but that is due to the small eccentric orbit) and the asteroid (starting at the black square) was oscillating in small circles towards the Earth (after 50 years, arriving at the black asterix). With my previous experience I was afraid that the asteroid would crash into the Earth, but a few years later the asteroid was moving away from Earth, without having touched it. Excited by these observations I let decades run by (in only a few seconds, thanks to good Fortran 77 programming...I mean divine fingers or so!) and a 'horseshoe' figure appeared in my univers:

An even spectacular image was obtained when I changed my vantage point and saw the 'horseshoe' orbit in 3D (it works in the movies, so why not in science):

Earth and the asteroid seen in a beautiful orbital dance of planets. Science did it again and they lived happily ever after...

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