Wag the Dog, Part 2

Long ago, at almost the beginning of this blog, I wrote a post with an attempt at analyzing the goofy wagging behavior of my family's two dogs. I failed to come up with a satisfying answer then (in my defense, chemo) but my friend Chris pointed out that I would have had an easier time using Lagrangian mechanics.

Marika and I have a dog of our own now, Lorna, and she does the exact same vigorous wagging I described before. Marika even gave her the nickname "Stubby Wigglekins", due to her stubbornness with me, and her crazy wagging for Marika. I'm missing the two of them here in France, but with a bit of mathematical modeling, maybe I can get a temporary Lorna stand-in!

We'll consider the dog in three parts: body, butt, and tail. The two joints connecting these parts act like springs, applying a force to bring them straight. The magnitude of the force depends on how much they're bent:

Lagrangian mechanics are entirely equivalent to Newton's approach, but are often easier to work with in cases where energy is conserved. The technique involves first defining the difference between kinetic and potential energy in the system:

The advantage of Lagrangian mechanics is that we're allowed to parameterize the system in whatever terms we want, like the angle of the joints rather than their x/y coordinates. For each joint, we can use the usual results for a spring:

where I is the rotational inertia, 𝜅 is the strength of the spring, and the dot indicates a time derivative. Summing the results from the two joints and plugging in gives

Hamilton's principle says that the integral of this in time should be stationary in the parameters 𝜃1, 𝜃2, and their derivatives. Luckily, we don't actually need to do the integral, and can instead just plug into the general equations this leads to. For each 𝜃, we can write the equation

Applying this to the Lagrangian from earlier,

Much simpler than what I was doing last time, and all because we're working in terms of the angles, and not the x/y position! I decided to put together another HTML5 doodad to experiment with:

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Amplitude 1

Amplitude 2

Stiffness 1

Stiffness 2

Tail Length

Fun to play with, but no substitute for the original!

Zap!

Happy Diagnosis Day! It's been 8 years since that fateful day when doctors discovered a 3 cm mass perched on my pineal gland. In that time, I graduated from college, went off to graduate school, married an amazing woman, got my PhD, and moved to France for a post-doc! In honor of the occasion, today I wanted to discuss radiation.

Part of my treatment included proton radiation, which I've talked about before (long before). While that exposure was medicinal, it's still best to limit overall radiation exposure to skin to about 50 millisievert (mSv) per year. For a couple years after my treatment, I avoided the TSA body scanners, opting for the manual screening. However, during a recent visit (to Michigan, not France) my friend Kevin wondered, How much radiation exposure do you get from flying in the upper atmosphere, compared to the security scans?

It turns out the TSA has actually used two different types of scanner in airports: x-ray backscatter, and millimeter wave. Up until 2012, the TSA mainly used x-ray backscatter machines, but switched to millimeter wave after manufacturer disputes. The EU actually banned x-ray backscatter in 2011 due to health risks.

Normal x-ray machines work by measuring the fraction of x-rays that pass through a material. Denser substances, like bone, absorb more of the x-rays and appear as white areas on the resulting photographic negative. Backscatter machines instead measure the rays that are reflected from an object, giving an image of the outer surfaces. The trouble with this is, x-rays are a form of ionizing radiation:

By Spazturtle - Own work, CC BY-SA 4.0, Link

Ionizing radiation means that it carries enough energy to damage cells, potentially resulting in cancer. This is why (in the EU anyway) these machines are no longer in use.

The alternative full-body scanning device is the millimeter-wave scanner, which use waves with a frequency of

On the chart above, this is just below the visible range, well into the non-ionizing region. Like the x-ray backscatter, these machines form an image by bouncing the waves off your body, and interpreting the reflected signal.

So, on to the exposure issue: The point Kevin brought up is that our sun puts out damaging ultraviolet light, and there are even more powerful cosmic rays coming from elsewhere in the universe. Typically, the Earth's atmosphere protects us from those sources, but by going high in the air, we strip away some of those protections.

The American Association of Physicists in Medicine released a report in 2013 comparing the exposure levels for a person of my size (5'10" and 160 lbs) on the ground, in the air, and in a scanner. The way they present their results though is a bit confusing (2.84 hour flight?). Instead, let's do some unit conversions to find how long, on the ground and in the air, it takes to equal the 11.1 nanosieverts they measured from an x-ray backscatter machine (which, remember, is no longer in use). On the ground, a person my size gets about 3.11 millisieverts per year, which means it takes 113 seconds to equal one scan. In the air, it's even less than that, 12.1 seconds!

There are lots of other issues with airport scanning machines, including privacy, and other possible health risks associated with radiation exposure. From the perspective of standard exposure limits though, you're a lot better off staying in the machine than going in the plane!

Dimensional Dilemma

Just a short post, since I got an apartment earlier this week, and there's still lots of set up to do!

While planning for this move to France, I had to figure out how to fit everything I want to have the next 2 years into airline-approved bags, and I found their size limit for checked luggage interesting. I would have assumed the limit would be volume:

where x, y, and z are the length, width, and height. The trouble with this though, is that it would allow narrow objects with enormous length, like a long pole (and you don't want to get a physicist started on fitting poles into things). Instead, airlines place a limit on the sum of the dimensions:

What's interesting is that this also places a limit on the volume of the object. We get the maximum volume when x = y = z, so

It seems like we're getting extra information here: With only one expression we can limit both the volume and the maximum size. It's not entirely clear to me where this information comes from, but my guess is that it's due to the fact that the linear limit defines a coordinate system, in this case Cartesian. We could imagine a different limit, where items were required to fit inside a sphere of radius r. This would also limit the volume:

Depending on how you choose the linear measurement, you can get different maximum volumes. Makes me wonder whether airlines initially used the volume limit, only to be inundated with pole-vaulters...

Forces of Darkness

Another question from my honorary Papou (Greek grandfather): As you
know I am very troubled by Dark Matter ..... it makes sense in my mind sometimes and sometimes not. [...] Somehow, for my understanding, Science needs a very convincing argument and some form of proof that Dark Matter exists. I read whatever comes before me about Dark Matter, it is a compelling theory. If I recall correctly did not Einstein use a "constant" to compensate for his calculations? Was that perhaps to compensate for Dark Matter?

There are actually two forces in physics commonly referred to as "dark", dark matter and also dark energy. The use of dark in the names partly refers to the fact that we don't really understand them. Some observations, which I'll discuss in a moment, don't line up with our current theory of gravity, so dark matter and dark energy were added as a sort of question mark to fill in the gap. One target of the Large Hadron Collider (now that they've found the Higgs boson) is a dark matter candidate called a WIMP (weakly-interacting massive particle).

Dark Matter

My first physics class at Swarthmore was an overview of modern physics, covering special relativity, quantum mechanics, and cosmology. For the last topic, we were introduced to dark matter through the example of galactic rotation curves, which is how I plan to explain it today.

Imagine tying a weight to a piece of rope and swinging it in a circle. The rope pulls taut with a force related to the mass of the weight, m, the speed of its motion, v, and the length of the rope, r,

This is called the centripetal force, because it always points toward the center of the circle.

Based on the amount of light coming from distant galaxies, astronomers can estimate how much mass is contained in different parts of the disk. Via gravity, that mass provides the centripetal force that makes the galaxy rotate, so we can estimate how fast that rotation will be:

where M_enc is the total mass enclosed in radius r. Setting this equal to the rotation equation above gives a velocity of

The trouble is, this doesn't line up with observations – Astronomers have found that galaxies spin significantly faster than the observed mass would imply.

Via Wikipedia

This could be explained by the presence of extra matter that we can't see, hence "dark". Other models have been proposed that are based on space behaving differently at galactic-size scales, but so far none have surpassed the accuracy of assuming extra matter.

Dark Energy

Even using just the matter we see, Einstein calculated that the universe would eventually collapse from the gravitational pull. This disturbed him, since he expected to find a static universe, one that neither expands nor contracts. That led him to add a cosmological constant called Λ (capital lambda) to his equations, which provided a balancing force. Later observations by Hubble showed that the universe was in fact expanding, and accelerating in its rate of expansion! Based on observations from WMAP, a space telescope, scientists estimated the total energy content of the universe (remember E = mc^2):

Via Wikipedia

 

In essence this says that we're not really sure what 95% of the universe is made of! Don't assume that means science is useless though: We can predict the behaviors of astronomical objects with stunning accuracy. All that's missing is an explanation for these behaviors. I always like to think of physics as successive approximations: We have a model that works extremely well, but after we learn a little more, we can make it that much better!

CBCB (Compact Binary Coalescence... and Blues)

Last week, I told you about my work on gravitational waves that went into my PhD. My post-doc is still in the same field, but I'm looking for an entirely different type of wave than before. The continuous waves I studied before were weak, but long lasting. The work I do now is on waves from compact binary coalescences, CBCs. These are among the most powerful waves we expect to see, but last only seconds. All the detections LIGO and Virgo have made so far were from CBCs.

The name "Compact Binary Coalescence" is awfully jargony, so let me take it piece-by-piece:
Compact: A compact object is one with unusually high density. In this case it refers to either a neutron star, or a black hole. How high is the density? One teaspoon of neutron star material weighs 10 million tons, and black holes are even denser!
Binary: A binary system has two objects that orbit their common center of gravity. If one object is significantly bigger than the other, it might look like the small one orbits the big one, but even our Sun has a bit of wobble from the pull of the surrounding planets.
Coalescence: As the objects orbit, they radiate energy in the form of gravitational waves. That energy comes from their gravitational potential, which draws them closer together as it decreases. A closer orbit is also a faster one (think of Mercury), so the pair radiate stronger waves, which further speeds the process. Eventually, the two collide and coalesce into a single object.

I can show you what this type of orbit looks like thanks to the recently (but not yet officially) released LIGO Orrery:

This was inspired by the Kepler Orrery released a few years ago. It shows simulations of all the binary black hole detections LIGO and Virgo have made so far. Below the orbits, you can see a plot of the gravitational wave signal. Each peak of the wave corresponds to a half-orbit of the bodies. As the rate of revolution increases, so does the frequency of the wave.

The signal is divided into three parts
Inspiral: The majority of time is spent in this phase, as the orbit decays and the bodies spiral inward (good name, right?). The wave is sinusoidal with increasing frequency and amplitude.
Merger: When the bodies first collide, the new object is oblong, and takes some time to smooth out. This is the peak wave output.
Ringdown: There are tremendous forces involved in the collision, so the black hole that results from the merger needs to shake off the extra energy. You can't see it in the plots shown here, but immediately after the big burst are a couple more small waves.

I'm working with a group here in Annecy, France that's part of the European gravitational wave collaboration, Virgo. We look for these CBC signals in the data coming from the two LIGO detectors in the US, and the Virgo detector in Italy. The search technique I'm part of is called MBTA, which is not the Massachusetts Bay Transit Authority, but the Multi-Band Template Analysis. The goal is to identify signals as fast as possible, so we can send the coordinates to partner groups observing in the electromagnetic spectrum. This allowed us to show last year that gamma ray bursts are associated with neutron star mergers.

I have a lot to learn in this new pursuit, but everyone at my office and in town have been incredibly welcoming. I've been given an amazing opportunity to live an work in such a beautiful part of the world!