Visit to the Roger Babson Anti-Gravity monument at Tufts

In the early 20th century lived an eccentric businessman named Roger Babson. He was a sort of Donald Trump of his time, writing books, founding a college, and running for president. Instead of hating Muslims and Mexicans, what Babson really hated was gravity, and declared it to be mankind's foremost enemy. 

He founded an organization devoted to defeating gravity's grip on humanity, part of which involved organizing the yearly Gravity Research Foundation essay contest, which has been won by the likes of Stephen Hawking and entered by the likes of me. He left a series of monuments around New England dedicated to his crusade. The closest one to me is at Tufts (yes, Tufts), and I visited it this evening. There used to be a yearly tradition of digging up the monument and carrying it to see if gravity had been defeated yet, but that was discontinued.

I plan to write a longer article about this history of his contest and his foundation, and for now you can find other articles about it online. I wrote an article about the history of the contest on PhysicsForums!

The text of the monument reads:

"This monument has been erected by the Gravity Research Foundation, Roger W. Babson founder. It is to remind students of the blessings forthcoming when a semi-insulator is discovered in order to harness gravity as a free power and reduce airplane accidents. 1961"

Tiglath-Pileser's Hunting Omission

This post is on a different topic than usual post-doc ergo propter hoc content.

In the discussions or conclusions of academic papers, you will often find a description of work that the author had done but chose not to include in the paper. The reason is often because it would make the paper too long without adding much interest, or is a bit too tangential to keep the paper on topic. These are slightly different than the "future work will involve..." statements that often come at the end of papers. Including these references to omitted work can inform the reader that due-diligence was performed by the investigator, and also makes the author sound smarter and more impressive.

I was reading an account of the conquests of Tiglath-Pileser I, king of Assyria around 1100 BC. These accounts were inscribed on clay tablets in cuneiform script in the Akkadian language, and were found in the ruins of Assur, in what is now Iraq.

This is actually the equally ferocious Tiglath-Pileser III, who has a better picture than Tiglath-Pileser I.

Assyria was a dominant military power in the Middle East in the late Bronze Age and early Iron Age, before being supplanted by the Babylonians. They are known for their use of chariots in battle, their ferocious descriptions of their conquests, and the scattering of the ten "lost tribes" of Israel after conquering the Northern Kingdom. Tiglath-Pileser I ruled at the end of the Bronze Age, before a period of tumult involving the invasion of the Sea Peoples and the collapse of the Mediterranean Bronze age. Tiglath-Pileser I went on a rampage Eastwards, conquering literally dozens of kingdoms in several pitched battles, looting the region and defiling the temples of the defeated. He writes, in typical Assyrian fashion:

The city of Khunutsa, their stronghold, I overthrew like a heap of stubble. With their mighty troops in the city and on the hills I fought fiercely. I defeated them; their fighting men in the middle of the forests I scattered like chaff. I cut off their heads as if they were carrion; their carcasses filled the valleys and (covered) the heights of the mountains. I captured this city; their gods, their wealth, and their valuables I carried off, and burnt with fire. Three of their great castles, which were built of brick, and the entire city I destroyed and overthrew, and converted into heaps and mounds, and upon the site I laid down large stones; and I made tablets of copper, and I wrote on them an account of the countries which I had taken by the help of my Lord Ashur, and about the taking of this city, and the building of its castle; and upon it I built a house of brick, and I set up within it these copper tablets.

There are many such descriptions. However, after he finished his descriptions of all the armies he defeated and cities he destroyed, he left the following note:

I have omitted many hunting expeditions which were not connected with my warlike achievements. In pursuing after the game I traversed the easy tracts in my chariots, and the difficult tracts on foot. I demolished the wild animals throughout my territories.

In mentioning these omitted hunting achievements, he saves the reader the time that would have gone into taking in all this out-of-scope extraneous information, but uses the stated omission to bolster his credibility as a martial conqueror. I believe that this is one of the first published omissions to other relevant work, setting the stage for scholarly literature for the next three millennia.

This may not be the first reference to a deliberate omission; I haven't thoroughly scoured the Egyptian and Akkadian records.

Physics Shower Thoughts Part II: Cosmic Radiation Pressure and Relativistic Spacecraft

This is the second instalment of "Physics Shower Thoughts," an exploration of an idea that I found interesting that ultimately may not have monumental consequences in the grand scheme of our understanding of the universe. In this case I was inspired by a question on reddit that was answered by another physics blogger, VeryLittle of Quarks and Coffee. The question was about the limits of acceleration given special relativity, and my shower thought was about the role of blue-shifted radiation pressure in limiting that acceleration.

The setup is thus: an object (let's call it a spaceship) under the influence of a constant force in its reference frame will accelerate. The acceleration will seem uniform but as it gets very fast relative to its initial rest frame, its velocity will plateau asymptotically towards light speed. In our universe, however, there is a cosmic microwave background that will be blueshifted in front of us as we move in one direction, and redshifted behind us. The radiation pressure from this background will become stronger and counter the force that our spaceship is producing. This may limit the ability of our spaceship to accelerate. So, to what extent does this matter?

It's a good picture, ok.

To figure this out, we need to work out two things: the blue-shifted blackbody radiation pressure, and the relativistic acceleration under special relativity. The shifting of the blackbody spectrum can be considered with varying degrees of complexity, including a Lorentz transformation of the entire spectrum and a mixing of polarizations. However, I am interested in a simple solution and an order-of-magnitude estimate, so I will be using a simpler method: I will consider the Stefan-Boltzmann law given a relativistic Doppler shift of the peak frequency of the blackbody distribution.

Incident electromagnetic power can be converted to a radiation force by dividing by the speed of light. The Stefan-Boltzmann law gives us areal power density, so we can multiply by the cross-sectional area as well to from power density to pressure to force:

$F_{r}=\frac{A}{c}\sigma T^4$

The Stefan-Boltzmann constant $\sigma$ is a product of several powers of Boltzmann and Planck's constants as well as the speed of light, but its value in SI units is easy to remember: 5-6-7-8, or 5.67x10$^-8$ Watts per meter squared per kelvin fourthed.

According to Wien's displacement law, the temperature is proportional to the peak frequency of the distribution, which I'll call f. In our universe at this time, f is about 160 GHz. The proportionality constant is based on numerically minimizing the Planck spectrum, and in terms of frequency Wien's law can be simply expressed:

$T=\alpha f$

The proportionality constant for frequency is actually very close to two times the constants of Pythagoras* and Boltzmann divided by that of Planck, or in SI units about 5.9x10$^{10}$ Hertz per kelvin.

Moving toward a source, the frequency experiences a Doppler shift and is transformed into f', and if the motion is fast enough we should take into account the full relativistic Doppler shift:

$f'(v)=f\sqrt{\frac{1+\frac{v}{c}}{1-\frac{v}{c}}}$

The radiation hitting us from behind is redshifted, and I will deem it small enough to ignore. Plugging this all together, the transformed cosmic radiative force on our spaceship is:

Now we consider the relativistic acceleration of this spacecraft. The way to incorporate special relativity into Newton's law of motion is to remember that the change in momentum is the product of force and time. Momentum is the product of mass, velocity, and the Lorentz factor. If we just have a constant force then we can find the acceleration as a function of time:

If you plot this vs time for some values of F and m you will find that it increases and asymptotically approaches light speed. Somewhat coincidentally, if you set F/m to be Earth's gravity, it will take about a year to get close to light speed.

Accelerating under Earth-gravity-equivalent for three years. To convert seconds to years, remember that pi seconds is a nanocentury.

If we then incorporate our radiation pressure into the force side of the equation, it gets a bit more complicated.

This does not have a closed form expression for the velocity, but if I solve it numerically I find.... that it increases and asymptotically approaches light speed. For the values I use, it is essentially indistinguishable from the radiation-free solution. However, if I calculate values for long enough times with high enough numerical precision, the radiation solution does converge on a sliiiiiightly subluminal value (99 point a bunch of 9's percent light speed). So, what's going on?

This is going on.

Basically when the spaceship is going fast enough, the radiation force becomes strong enough to counter the accelerating force, and the net force is zero. This velocity is easier to calculate, we just solve the transformed radiation pressure equal to constant F, and solve for v. To simplify notation, I have introduced the variable pressure P=F/A:

Just the general form of this expression is $(1+x-2\sqrt{x})/(1-x)$, which is monotone decreasing, passing zero when x=1. In our scenario, that corresponds to $f^{4}/P=\alpha^{4}c/\sigma$, which I guess is the frequency and pressure required to have zero acceleration from the rest frame.

So, let's plug in some realistic values for this critical velocity and see what happens. Using the space shuttle as a framework for a spaceship, it has a mass of two million kg, a thrust of 12.5 meganewtons, a cross-sectional area of roughly 200 square meters. This puts its maximum velocity at 99.99999992% the speed of light, given constant thrust.

Now there are many other factors I didn't take into account. I could use a more rigorous transformation of the blackbody radiation. The length-contraction of the spaceship might change the incidence of the radiation, compounding the effect (it's kind of funny that shape becomes important again, not for aerodynamics but for electrodynamics). The mass could decreases as fuel is ejected from the rocket (maintain fuel supply for this journey could perhaps be attained with a Bussard ramjet, but that's beyond the scope of this article), which increases the acceleration. And it would take so long to accelerate to these high speeds that the cosmic background radiation could become more significantly redshifted due to the intervening cosmic expansion.

As VeryLittle pointed out in his reddit post, such a cosmic speed limit already exists for protons: the GZK limit is the speed at which cosmic ray protons see the CMB blueshifted to the point that it interacts with the now-gamma rays. This corresponds to an energy of 8 joules, but the speed is close enough to light that the number of 9's in the percentage doesn't really matter (whereas for the space shuttle it's only 7 decimal 9's). I guess protons are more electroaerodynamic than spaceships.

So, to summarize, the blueshifting of the cosmic microwave background may have an extremely tiny effect on the limiting speed of relativistic spacecraft. However, if I were a relativistic spacecraft engineering, I wouldn't worry about it.

*also known as the square root of two.

Where might Planet Nine be hiding?

On January 20, 2016, the inferred presence of a large planet far beyond Neptune was announced. The finding was a joint collaboration between Mike Brown, who is known for discovering the Kuiper belt objects that helped nix Pluto's planet status, and Konstantin Batgyin who is an expert in orbital simulations. The finding was based on alignments in the orbits of several trans-Neptunian objects, whose alignment can be explained by the presence of a distant planet about ten times the mass of Earth and 30 times the distance of Neptune. It can also be the result of rare events occurring due to  random chance. In March, Brown and Batygin released another paper (initially on ArXiv) trying to figure out where this planet can be found, if it does indeed exist. The paper has since been published  in Astrophysical Journal Letters, a respected journal for short communications about astronomy and astrophysics.

As in many aspects of physics research, even in the absence of positive data one can learn things by asking the right questions about negative results. In this case the question is "Given that we have not yet detected this planet, where could it be?" rather than repeatedly asking "Is it there? No? Ok how about there?" at various points in the sky.

There are a number of telescopes that scan the sky looking for various things (well, that's basically what telescopes do...I'll refer to these as survey telescopes to distinguish them from telescopes like Hubble that look at specific targets when directed). There is one called WISE that is looking for distance planets and nearby brown dwarves (distant from the solar system's perspective and nearby from the galaxy's). So they consider where this planet could be and not be detected by WISE. The same goes for other survey telescopes: one called CRTS, one called Pan-STARRS, one called the Dark Energy Survey. Another group of researchers, in the time since the January announcement, did an analysis of Saturn's orbit given hypothetical perturbations from this planet, and used the fact that Saturn and the asteroids have not been very perturbed to rule out a few more regions.

What remains is a band of sky where Brown and Batygin's analysis shows this planet must be in order to explain the orbital alignments of the trans-Neptunian objects, and various regions of that band are excluded from these various non-detections.

The band of sky where this planet is expected, and all the ruled out regions of that band, leaving only the black areas. This is an image from their paper which I have annotated. I respect Brown and Batygin as scientists but I disapprove of their use of .jpg for a paper figure! And yet here I go using yellow text on white background.

 So, all that remains in the realm of possibility is a medium chunk of sky in the Northern hemisphere, and a tiny one in the Southern hemisphere. Interestingly, Brown gave his first public lecture on the topic at MIT, which I attended. He showed a similar figure, except there was a much larger region in the Southern hemisphere. Since then, it has been ruled out by the Saturn analysis.

Where is this big region on the actual sky? I have approximately traced it out on a sky map, which is shifted 180 degrees compared to the image above. In the Northern hemisphere, the big region encompasses Orion, which is one of the easiest constellations to find!

Very approximate locations. If you think you can do a better job of mapping the graph regions onto the sky map, please show me and I will credit you. Update (July 20 2016): An anonymous commenter has made a better version, with the yellow showing the search regions.

This tells us how to focus our telescope searches to best find this hypothetical planet. In the Northern hemisphere, the best telescope for the job is the Japanese Subaru telescope (named after the Japanese word for the Pleiades, not the car company whose name has the same origin). This may take up to five years, unless they can get more dedicated telescope time, or refine their search with more accurate computations or the discovery of more aligned trans-Neptunian objects.

So, by looking in this region, it could find this new planet, or it could rule out its hypothetical existence. 

The Robophysics session at the APS March Meeting was delightful

I recently returned from the March Meeting of the American Physical Society (APS) in Baltimore. It is the largest annual physics conference, with about 10,000 attendees and 50 simultaneous lecture sessions laden with technical comic sans. Everyone gets 10 minutes to speak and 2 minutes for questions, and then are affirmatively ushered off to stay on schedule. My talk was on Tuesday. There are about 10 research groups in my field and all of them were there, so it was a very relevant session. On Friday afternoon I had seen everything I needed to see and was pretty exhausted, so decided to go to a session called Robophysics: Physics Meets Robotics. I am glad I did.

This lively fellow made an appearance.

There were about 30 talks spread over two sessions, of which I saw about ten. They generally followed the following format.

1. Here is a video of an animal. Look at how smoothly it moves.
2. Here is our best attempt to make a robot to do that. Look at how terrible it did.
3. Here are several slides of variational calculus to guide us towards a better robot.
4. Here is our new robot. We had an undergrad film it for hours on end to gather data. Enjoy.

This video was played in multiple talks:

There was a variant of the format above, where they filmed animals with high-speed cameras in weird situations to see how they function, for example filming mudskippers pushing themselves up sandy hills. One researcher showed how a centipede rapidly increases its vibration frequency and amplitude when poked, and attempted to make a robot centipede that mimicked this.

The main motivation for the research was to improve the way robots interact with their environment, which currently has to be hard-coded (e.g. for manufacturing robots) and is not at all adaptive, and his holding back the expansion of robotic utility. Much like in nanotechnology, researchers are trying to move forward by drawing inspiration from nature.

I will summarise a few of them. I look a few potato-quality pictures which I might try to supplement by looking up the papers.

One was titled "Is a snake a wave or a particle" and was about the angle that snakes scatter when they slither between two posts. They built a snake robot, had an undergrad gather some data on single-snake diffraction, and found that the snake scattering angle distribution has discrete peaks. Someone in the audience asked how a snake could interact with itself like that, which prompted a brief discussion of the history of quantum mechanics. There were quite a few snake-related talk, including one about dropping flying snakes from the top of a room full of high-speed cameras.

Another was about the propagation of vibrations through a spider web. They showed a high speed video of a spider sensing and attacking a moth in its web, then built a large-scale mechanical spider web full of accelerometers with varying tension to study how the perturbation of the prey reaches the spider. After the talk, somebody asked the question:

"In 30 years when we live in a post-apocalyptic robotic hellscape, and I got caught in a human-sized robotic spider web trap, do you have any recommendations as to how I can try to vibrate the web to avoid detection the longest?" 

 Another talk was inspired by a scene from The Simpsons, where Homer kicks a table in order to vibrate a bowl of dip closer to himself so he could dip his chip without getting up. They managed to invent a mechanism to vibrate a surface to create arbitrary flow fields along the surface and move objects around in precisely controlled ways.

He embiggened the session with his cromulent research.

I only made it to the second session, so I missed the keynote invited lecture in the first. Overall it was a nice coda to week of intensive science exposure.

One of the labs that presented a few talks has a website about this with many videos, which I suggest perusing.

Article on the basics of positron emission tomography (PET).

Read it at PhysicsForums!

https://www.physicsforums.com/insights/basics-positron-emission-tomography-pet/

5phases1cup: Dry Ice in Pluronic Gel

Today a 0.1 mL vial of DNA that I ordered arrived in the lab. It was shipped in a box 100,000 times its size, full of dry ice to keep it cold. Not wanting to let the dry ice go to waste and make my CO2 emissions productive, I decided to drop it in a vial of pluronic gel that another postdoc in my lab prepared. Let's take a look at what happened, and then I'll explain.

A stabilized version can be found here.

Pluronic is an ABA tri-block co-polymer, meaning that it has three regions of repeating chemical units, and the outer two are the same. When it's above a critical temperature, it collapses into a micelle with the out part forming a protective ball around the inner part. These micelles form a network and behave like a gel, but if the temperature is lowered it reverts back to a liquid, sort of like an re-un-boilable egg.

This image cannibalized from a google image search for pluronic.

When I dropped the dry ice in the pluronic, it landed on top of the gel and quickly cooled the region immediately below it, turning it liquid. This wave of cold slowly propagated downward, liquifying the gel in the process. Gravity gradually brought the ice pellet towards the bottom as it cleared out its own passage downward.

The slow but inexorable pull of gravity brings the dry ice pellet down to the bottom of the vial as it clears its own path via a wave of liquefied gel. This is easier to see in the stabilized video.

The CO2 sublimated into bubbles which floated upwards, leaving a sticky pluronic foam at the top which gradually overflowed out of the vial. Eventually, the liquid around the dry ice got so cold that it started undergoing an actual liquid-solid phase transition, encasing the pellet in H2O ice (possibly with pluronic mixed in?), limiting the region from which the bubbles emerged.

That's what's up.

So during this process, we had gel, liquid, gas, and two kinds of ice. Science is cool!