zaterdag 22 juni 2013

A submarine voyage into the gravity field of Earth

Plate tectonics is a theory that is only 60 years old. Before this time, people were seeing Earth as a solid un-deformable piece of rock. Imaging you being a geophysicist in those days (ok, if you are currently not a geophysicist this is even more difficult, but in comparison try to explain the wonders of the internet to your grandparents.), having trouble to explain most of the things that where observed.

In those times a civil engineer, educated in Delft (my university :) ), went on a submarine voyage, measuring the gravity field of Earth. Wow, they should make a movie about this (I think they even did)! With his own designed pendulum equipment, the Dutch engineer could measure the gravity field with an accuracy up to a few mGal (which is very impressive, just trust me). Who is this brave, 2 meter tall (which I like because I am the same height), submarine-sailing hero? Born on 30 July 1887 in The Hague, he was named Felix Andries Vening Meinesz.

I am currently working in a project to describe and explain the measurements of Vening Meinesz during one of his famous voyages. The project is making me feel like the Indiana Jones of the Geophysicists, going through old notebooks, carefully reading all the details and trying to get a sense of what he did and when he did it (ok without the whip and the cowboy hat, but still). 

Today, I give you a small sneak preview of my work. It is not finished, but still I think it is cool. The part of his voyage we look at is when he sailed over the Walvis Ridge (yes, it is not Whale Ridge, but the Dutch or South African Walvis), where he did several measurements.

The Walvis Ridge is a large sea mount chain, west of South Africa. It has a great effect on escaped eddies from the Indian Ocean (maybe I will explain this in later posts). The Ridge is formed by hotspot vulcanism (see my post about Hawaii), still today forming the island Tristan da Cunha. After visiting South America, Vening Meinesz, onboard the submarine Hr. Ms. K18, sailed to Cape Town, visiting Tristan da Cunha and crossing the Walvis Ridge. On his way he did 2-4 measurements a day, observing the variations in the gravity field due to masses like Walvis Ridge. This is what he saw:

I made a plot of his measurements and subtracted the ellipsoidal shape of the Earth, this results in the free-air anomaly (deviations from the main signal, you can see so much more!!). In the top figure are the measurements of Vening Meinesz (interpolated), with the black error bars. In red is a state-of-the-art satellite gravity field, having a much smaller resolution and thus dampening the signal, removing high-wavelength features. Features that Vening Meinesz could observe in the submarine, because he was much closer to the source. The bottom plot shows the bathymetry of that part of the Atlantic ocean, clearly showing the elevation difference of the Walvis Ridge. The red diamonds are the locations of the submarine during the measurements. The captain really tried to sail close to the ocean floor. I say,  "Kuddos for him and his sonar-crew, not hitting anything!" (ok, at some points they are below the ocean floor, but blame this on the uncertainty of the bathymetry model, not on the skills of the crew of Hr. Ms. K 18). 

Still today, Vening Meinesz's measurements are very impressive, obtained only by watching the swinging of three pendulums. 

woensdag 12 juni 2013

Destroying one of my childhood's fascinations

This week I received a package from the USA, containing my own Crookes radiometer (only a few bucks at (Yes, I am a geek, but the good well developed kind ;) )). This device is a small mill with black and white vanes in a vacuum pumped light bulb which turns around when you put it in sunlight (any light works!). It looks like this (pictures speak a thousand words, well in my case a few dozen)

My physics teacher used this device to explain that light particles have momentum (he should have known better), and due to the different reaction with the white and black colored vanes, the mill would turn. During my whole life, I was really sure about that this explanation was the effect causing the light-mill in vacuum to turn. Until this week.

My old master supervisor (not that he is old, but I am no longer a master student, he is my colleague now) entered my office room with a cup of coffee (this is of course essential for the story) and saw the Crookes radiometer. After making a comment about me doing Disneyland physics, he pointed at the radiometer and asked: “Do you know why the vanes of the radiometer turn?”. During my master we always had good discussions about physics, which I always lost, but now I could tell him that I knew. “Wrong”, was his response, “Just calculated the acceleration of the vanes due to the light particles” and he stepped out of my office, leaving me behind in total confusion. He always did this during my master research, instead of explaining the principle, he encouraged myself to come up with the answer (I know, this is more pedagogic correct). 

As a good scientist does, I consulted the encyclopedia, or in this era of technology, Wikipedia. The Crookes radiometer page has an excellent documentation about this device. It is NOT radiation pressure due to electromagnetic radiation (fancy words for light, heat and other things you can not see), which is far to small to cause the huge (well, I think they are tiny) vanes to rotate. They give four historical explanations of which two are to small to explain the movement. The other two have to be combined to fully explain the motion. This is real science!!! Initial theories are disputed by better ones, and even in the end there is not one explanation but multiple.

Ok, but what are the effects that cause the radiometer to turn. Even Einstein thought this was a great exercise and did some calculations (well if Einstein worked on it, the problem should be very complex). It sort of is. Air molecules in the near-vacuum (vacuum conditions can not be obtained, this will even stop the radiometer from turning) environment exchanging different amounts of momentum with the vanes due to different temperatures. Just read it here. In the end light has something to do with the motion, it is however not the pressure of the photons.

woensdag 5 juni 2013

Trying to figure out spacetime bending

A few years ago I was obsessed about Einstein's theory of gravity. I wanted to know everything about it, so I browsed a lot on the internet, trying to find information about the topic (I could have gone to the library, however I was still in the frame of mind that everything on the internet is correct). So one day I came across the following website: Bending Spacetime in the Basement. Wow, can one do this at home?!?

First take a good look at the website. He's got a point by stating that he is bending spacetime. It unfortunately has nothing to do with warp-drives and wormholes (Which was what I was looking for), but his experiment is quite cool nonetheless. It somehow proves (I will discuss this later on) that the theory of gravitation found by Newton, not only works for planet and stars, but also for smaller everyday objects. Everything attracts each other.

Impressed by the simplicity of the experiment and the fact that it is about gravitation (I think this is one of the most interesting forces in physics), I showed it to my fellow classmates (I was still studying to become an Aerospace Engineer) and it resulted in heated discussion. Was it really a gravitational pull of the masses that caused the motion in the experiment, or were different forces in play? Maybe it was just tension in the string or a possible magnetic torque. It could even have been a draft (wind) in the basement of the experiment. I always believed (believed in the sense of a good educated guess) it was gravitation at work, but how could I prove this?

My current job is to model the effect of gravitation on the surface of the Earth. The above problem should therefore be a walk in the park for me. I start by assuming that the video coverage is genuine and reports exactly what was going on during the experiment. I also trust our basement spacetime bender and all its statements on his website (again, everything on the internet is true). This enables me to obtain the timing records of the motion, which I can use in my proof.

From the above two stills of the video we can deduce that the full motion - from rest to first contact with the cubic stones - takes about 4:30 min (±15s, he made a time lapse out of it, so the time-resolution of the movie is quite coarse). This gives us an observation with which we can test our models.

So let's make a model that describes this motion. Starting with the law of gravitation of two particles (let's assume that the stones and round-metal objects are point masses. This is what we call modeling, start simple) invented by Newton (He did not know that mass actually bends space and time, but who cares (Einstein did!)):

(Actually this depicts Newton's law of gravitation and his third law of motion) Here, F is the force experienced by the numbered particle, G is the universal gravitational constant, m is the mass of the numbered particle and d is the distance between the two particles. I know equations in a blog will lower its popularity, but hey, it's a physics blog. To clarify, I made a sketch of the situation, and because of symmetry, I only sketched the right part of the rotation device.

Here, M1 is the fixed stone and M2 the round metal object. M2 can only move along the dashed line, s. Half the length of the rotation device is denoted by r and the angle between the initial and current state is depicted by phi (yes, the strange wriggly shape in the bottom corner. It's Greek!). The whole problem can be modeled as a 1D-motion. To find out whether it's only gravitation that moves the rotation device, we calculate to what extend the gravitational pull between the two bodies will affect their movement. The gravitational pull between the two bodies works along the line, d (illustrated in red).

The motion of M2 from its initial resting state can be modeled using a mathematical trick I learned at Aerospace Engineering, namely the state equation and the Euler integrator (I know, this is not the best one, but if you just reduce the timesteps, it will work (hopefully)).
If we can find a relation for double dotted s (as this is the only unknown in this scheme, it really is!), the motion of M2 can be calculated. Double dotted s only depends on the gravitational pull between the masses in the experiment (let's ignore the giant ant, and assume symmetry (which is not entirely correct, but let's start with that and ignore the attraction of the other M1 mass)). So after some proper geometric brain-crunching we obtain a solution for the acceleration felt by M2 (I leave this open for you to do. Hint, it is a tiny-tiny acceleration).

So now it only remains to find values for the geometry of the problem and the masses used: M2 is a lead sinker weight of 169 grams, the rotation bar is 30 cm long, so r = 15 cm, and M1 (the pavement stone, or in Dutch "kinderkopje", which is a bit of an odd name) is 2 kilogram. The difficult bit is to determine the difference between the initial position of the rotation device and the time of impact (in other words what is the angle between the two positions). Or just look at the following figure:

Let's say (wet finger approach (WFA)) this angle is 15 degree ±5 degrees (with this added uncertainty it becomes science ;)). This scientifically approved estimation of the geometry enables us to test the model and find out the duration of the movement. I used the following Matlab script:

clear all;close all;clc;
% The spacetime bending experiment
m1 = 2;
m2 = 0.169;
G = 6.673e-11;
r = 0.15;
% start Euler integration
dt = 1;
tnew = 0;
phinew = pi/2-pi*(15/180);
snew = r*phinew;
sdotnew = 0;
while r*cos(phinew)>0   % M2 hits M1
    % use new values for the calulation
    t = tnew;
    phi = phinew;
    sn_1 = snew;
    sdot_n = sdotnew;
    % Force (gravitation)
    d = sqrt(2.*r^2.*(1-cos(pi/2-phi)));
    alpha = pi/4 + phi/2;
    Fz = G*(m1*m2)./d.^2;
    Falong = Fz.*sin(alpha);
    sddot = Falong./m2;
    % State equation integration
    snew = sn_1 + sdot_n*dt;
    sdotnew = sdot_n + sddot*dt;
    tnew = t + dt;
    phinew = snew/r;
    % plot the results in a figure
    hold on
hold off
axis([0 0.4 0 0.4])

The duration of the motion calculated in the simulation is 12.5 ± 5.7 minutes. Looking back at the 4.5 minutes it took in the video, this suggests that the initial angle was 10 degrees (if it is only gravitational attraction that is playing a role). But the overal exercise tells us, that it could be that gravitational attraction between the two masses is the main contender for the motion. Isn't that cool!