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...

zondag 17 maart 2013

Science cat goes chaotic

A few years ago, I was surfing the web to learn anything about Schrodinger's cat. The one in the box, nor dead nor alive. After clicking on several wikipedia pages (as a pre-graduate student is accustomed to do), I came across the following page.

This was not Schrodinger's cat, but another scientist's cat. Vladimir Arnold (1960) used a picture of a cat to show a particular and cool mathematical feature. Why scientists use cats, I don't know, maybe because these creatures have an intrinsic appearance of being wise and distinguished (except when you put them under the shower, then they transform in complete and utter self pity).

The feature he described was a chaotic mapping procedure. He could transform a picture of a cat into  noise (wow, that is useful), and after some time the cat suddenly arose out of the noise in all its glory (it was as if seeing a magician at work). I wanted to see if I could do the same, hide a cat in chaotic noise and get him back. So I obtained the original paper of Arnold and started working on his method.

I, not being a mathematician (or magician), was struggling through the difficult and a bit obscure material. After a few hours I was able to let my cat disappear in noise, however it stayed there and never came back. That night I could not sleep, desperately thinking about my cat and how to get him out of the (now almost becoming the dark and mysterious, and maybe even a bit magical) noise. How did Arnold get his cat out of the noise?

At 3 AM in the night, I stepped out of bed and walked back to my magical setup (well you can argue this, but computer programming feels like doing magic sometimes). The night sky was clear and the moon was lighting up my keyboard. Maybe due to this magical light (or the coffee I had made for myself) I found the reason why my cat would not magically appear from the noise. The result:

First, I was thinking to put music behind this movie, but I think the beauty of the movie speaks for itself. I have put the number of iterations of the process in the top left corner. The cat is ripped appart (sheared) to become noise. A few times the ghost of the cat is turning up in the movie, and in the end the cat is able to reappear in full form.

I am thinking of applications (You may do to, please leave comments ), however I am still not able to think about a good one. But the beauty of this mathematical trick is remarkable, its just magic.

Please try it yourself!

zaterdag 9 maart 2013

Another blog about satellites (but they never get boring)

Satellite? But that is just a flying (we say orbiting) robot in outer space. Why am I so enthusiastic about a bit of metal rubble that is flying 7 km/s at an altitude of a few hundreds of kilometers above the roofs of our houses? Just because I can...

A few days ago Holland was in a high energy state. We broke a record. *sigh* It was the hottest day in March since...well since we measured temperature. The Dutch media had nothing else to report, but this glorious moment in history. Luckily for you guys, there is the DeepEarthScience blog!

So what did I do that day? I was standing on the roof of the tallest building in Delft looking at the city centre and thinking by myself: "This could be a good topic for the blog." The view was awesome:

In the back, a beautiful view of city center Delft in all its glory. And even further away the city of our government Den Haag. In the front-right of the picture you can see our two antennas, which we use for listening to satellites. The one in the front, a Yagi-antenna, we used that day to listen to a weather satellite (NOAA-19, an American satellite that is making pictures of Europe all the time).

This kind of satellites are good for testing your setup, because they are always on and broadcasting their signal. Our satellite sometimes doesn't have its day and is not turned on. This is especially a problem if you want to test some new equipment and the boss is looking (long story, maybe even not completely true, but...well continue with the blog).

This day we needed a satellite that really worked because I had some students to show the setup and (without announcing) ze Big Bozz had brought some people to look at our experiment. The student that operated the ground station, is a great thinker and had foreseen this, so he had chosen a pass of the NOAA-19 weather satellite. It was a high elevation (so high above in the sky) pass and the signal we received was very strong:

The center frequency of this figure is 137.100 MHz. Time runs from bottom to top. The satellite was flying over Africa-Middle East-Europe-Nord pole, drifting a little bit to the west. You can clearly see the doppler effect in the signal, a clear shift in frequency to the left (The jump at the beginning was a fiddle with the radio, not a sudden teleportation of the satellite). The signal is made of several side-bands. These bands are used to store data in the signal. 

The ground station operator (the smart student) said he had an extra treat for us. He was able to decode the signal that was coming from the NOAA-19 satellite. Everybody can do this, who is a little bit handy with radio shack equipment. The signal of the NOAA type satellites are not encrypted, so if you can listen to them you are able to reproduce the following figure yourself:

You can clearly see Holland without any clouds, a beautiful day. The noise in the image is man made. We did not yet set up the automated tracking of the satellite. We have to do this ourselves  so the noise is a mistake from our side. We were busy drinking coffee (engineers, you know :) ), and not paying attention to the azimuth and elevation (location in the sky) of the satellite. So after a while the satellite drifted outside the view of the antenna. This is also clearly visible in the Fourier transform of the signal (fig 2), just find the 3 resembling errors in both figures. However in the end we can say that the setup works.

And I think the last image is an even better view of Delft... 

zondag 3 maart 2013

Determinists are a bunch of hippies

...was the verbal shout from across the room/desk. My colleague's vision on determinists. Are they hippies?

Maybe its because there is an x in his last name, I don't know. But he has a point.

Determinists are people that believe that the universe is governed by solid, deterministic laws. If you say you have free speech, they say it is only input-based output of the brain. Everything is a reaction to a cause. As the famous physicist Albert Einstein (No physics blog can do without him) said: "God doesn't play dice". He meant that there is no stochastic, chance-based effects in the world. The Universe is deterministic.

However quantum mechanical observations disapprove this. In the quantum mechanics field, non-deterministic phenomenon are observed on a daily basis. Take for example the Heisenberg principle. If you know the location of the particle, you don't know its speed (exactly it is momentum, but that is the same as velocity times mass, so we don't have to bother about the weight of the statement). My dad once tried to use this principle when he got pulled over for speeding: "But officer, if you know I was driving on this highway, how did you know my velocity?" Of course, this only works in micro-space, so the assumption of my dad was wrong and the officer question exactly this part of his argumentation with a large fine.

But there are real scientific observations of this principle. Just Google Heisenberg principle experimental observations and indulge yourself in the quantum mechanics.

So where do the hippies come in the picture. To be a determinist you have to follow your gut and believe that in the end with infinite knowledge (a very accurate initial state), the universe can be propagated in the exact way. There is however no experiment to test this hypothesis. You should build two exact universes, let them progress in time and space and see if the follow the same path. This is not possible and thus there remains room for intuition and believe. A determinists believes there is no coincidence and disregards any observation that has non-deterministic phenomenon  just by stating a lack of knowledge. They follow their intuition, like hippies.