Parallel ipython notebooks and the H test

The CPU on the nice shiny new server I log in to is really not much faster than that of the ratty old laptop I have in front of me. The server has more memory, and more disk space, but CPU-wise it just has more not faster. For that matter even my laptop has two cores. So if I have some heavy-duty computing task, I'd better find a way to make it use multiple cores in parallel. Some tasks are just plain hard to parallelize (solving ordinary differential equations, for example), but it turns up fairly often that I'm doing something embarrassingly parallel: there's a for loop somewhere that just does the same thing to lots of different pieces of input. If only it were easy to hand each piece to a different core! Well, there are various tools for doing this sort of thing, but most of them apply to scripts or programs that run non-interactively. It turns out that ipython offers tools for interactive parallel computing. I'm going to explain how I use them, by working through a test problem, checking some statistics on a periodicity test (the H test).

Visualizing the new pet

I recently wrote about a new object I am studying: a millisecond pulsar with two white dwarf companions. There is lots more I want to say about it, but I think it would be nice to show what it looks like, or at least, to show a video I made trying to make visible what's going on:

Edited to note that Blogger's YouTube embedding is distinctly flaky; video is here. 

This video shows the orbital motions in the triple system. The orbits are drawn to scale, showing the actual motions of the two stars (red and yellow) and the pulsar (white). The first ten seconds are played relatively slowly, showing the motion around the inner orbit, then we speed up to see the motion around the outer orbit. For a sense of the time scale, an "MJD" is a modified Julian day, so a single day long. The larger left panel shows all three bodies, with trails marking the motion of the outer companion and the center of mass of the inner system. The inset in the top right zooms in on the inner system, showing the pulsar and the companion, with trails marking their orbits. The dots that appear on the orbits mark moments when we have observations of the system, color-coded by telescope; it should be clear that we have quite thorough coverage of both orbits. Each measurement tells us the distance to the pulsar to within a kilometer, so that we can measure the tiny deviations of these orbits from perfect Keplerian ellipses, allowing us to reconstruct the orbit.

There's a little more to it than that.

New pet: PSR J0337+17

I did my PhD thesis on PSR J1023+0038, a millisecond pulsar that is at a fascinating point in its evolution. (In fact there have been developments since the thesis was submitted; more about that later.) But during a moment of procrastination, I got involved with a new and fascinating system. The name, unedifying as usual, is PSR J0337+17, and it is unique in that the pulsar has not just one white dwarf companion but two.

Flywheel energy storage

In the quest for something better to run our cars on than gasoline, one of the proposals is flywheels. In fact, for a while there were flywheel-powered buses running in Switzerland and Belgium. On one level, it makes a lot of sense: you're storing energy as mechanical motion, and we're pretty good at transmitting mechanical motion from place to place. On another level it scares the living daylights out of me: a car in motion uses tens of kilowatts, so the car must be able to store hundreds of kilowatt-hours. If you let all those loose at once bad things will happen: 100 g of TNT going off releases about a hundred kilowatt-hours. Fortunately it's hard to get gasoline to do this, but a flywheel is just itching to dump all its energy. Batteries are a little scary too, to be honest. But anyway, that's all a digression: I want to talk about some really staggering examples of flywheel energy storage: pulsars and black holes.

Weighted Poisson Uncertainties

I recently ran across a rather awkward mathematical problem. I'm trying to make a histogram of photon arrival phases, complete with an uncertainty on the number of photons in each bin. Normally this is done by just taking the square root of the number of photons, which is at least approximately right based on Poisson statistics. But in my problem — data from the Fermi space telescope, which is not very good at localizing low-energy gamma rays — the photons are weighted: for each photon I have a probability that it really came from the source. So the values in the histogram should be the total probability. But what should the uncertainty be? The short version is: the square root of the sum of the squares of the weights.

ETA: This is in the literature, without justification as far as I can tell. See below.

Talking about radio astronomy

McGill has a modest observatory on the roof, with a 14-inch (optical) telescope. It was installed quite some time ago, then left to molder for a few years. This annoyed me, just on principle, so I talked my way into rehabilitating it as best I could, with my radio astronomer's skills. Even with it working again, it was just used by the occasional grad student who wanted to take her friends up to see Jupiter, or the surface of the moon, or whatever. Fortunately, we recently acquired an enthusiastic and organized post-doc who pushed hard and set up a public outreach program. Typical nights start with a public lecture, then we take folks outside to two portable telescopes and the bigger one on the roof to look at the stars. Of course, we can't predict the weather, so we have a few demos we can do inside - comet-making, liquid nitrogen, a muon detector, but really the appeal is going out and looking up at the sky. But especially in the summer, we need to entertain the public until it gets dark. Since it's once a month, we have a perennial need for speakers. So I volunteered, to give a one-hour talk on radio astronomy for the general public. I had never given a one-hour talk before, and radio astronomy doesn't produce as many pretty pictures as I would like, but I think it came out pretty well. And we had a videographer who recorded the whole thing, so if you're curious, here it is.

Fireworks

In honor of Canada Day (and for my neighbours to the south, the fourth of July) here's a video:

While this might look like a meteor, and in fact it is asteroidal material falling to Earth, it's actually some extremely expensive fireworks. It's the Hayabusa spacecraft returning to Earth after visiting asteroid 25143 Itokawa, and the material it brought back was safely encapsulated.

Lucky imaging

We have a telescope on the roof of the physics building. It's a fairly nice fourteen-inch Schmidt-Cassegrain telescope, though in fairly rough shape. Unfortunately, the physics building (and necessarily the telescope) is in the middle of downtown Montreal, which is a terrible place to look at the sky: bright lights, clouds, urban heat island, et cetera. So we have a telescope that collects lots of light but doesn't produce very sharp images. My attempt to work around this is described below.

Breaking out of iTunes

Since I got my "new" hand-me-down smart phone, I've been listening to a lot of podcasts. The phone is smart enough to use only my home wifi to update and download episodes, and the player keeps track of which ones I've read. This is so useful, in contrast to the music player, that I found it works better to keep my audiobooks on my home server and write a quick hack to serve them up as podcast feeds. But mostly I use podcasts like radio programs. Unfortunately, some podcasts are available only through iTunes. This makes sense if they cost money, but, for example, the NASA Lunar Science Institute has a free podcast which is only available through iTunes.

Fortunately it seems that the way these places get the data to iTunes is by serving up a standard RSS feed, which iTunes then wraps up in its proprietary glop. But Michael Sitarzewski helpfully put together a script that can extract the location of the original RSS from iTunes. So, for instance, you can subscribe to the NLSI podcast here. Very nice.

El Radar

I had the opportunity to visit the great radio telescope at Arecibo ("El Radar") last year. It's an astonishing machine, built more like a stadium than a telescope. Sadly, my digital camera died on that trip, so I don't have much in the way of pictures, but I came across this neat youtube clip (above). It's a segment from a BBC program, talking about dark matter and the role of Arecibo in understanding it. I thought it made a nice view of the telescope. I especially like that in some of the audio you can hear the coquì, little tree frogs that provide a constant chorus. (Of course the telescope is also in Contact and some James Bond film — surprisingly the latter provides a better view of the actual telescope.)

I'll talk a bit about the telescope and my own visit, with a few photos, below the jump.

Local Maxima and social annealing

Dan's Data has a new post/magazine article up, about local maxima. Frustratingly, there's no comment section, but I've been reading some interesting things that seem relevant, so I'll post them here. (TL;DR of Dan's post: socially we seem to get stuck in local maxima instead of finding the real best way to do things.)

Testing whether a signal is broad-band

Radio pulsars are generally broad-band sources — you can hear them over a very wide range of radio frequencies, from about a hundred megahertz up to, in a few cases, tens of gigahertz. Their emission does change with frequency, decreasing as a power law, but over reasonable bandwidths we expect to see their signal in all frequency channels.


Radio-frequency interference ("RFI"), on the other hand, is very frequently narrow-band, appearing in just one, or a few frequency channels. In fact, one way we try to manage RFI is by using really wide bandwidths, so that there's so much power from the pulsar signal that narrow-band RFI is drowned out. Unfortunately, all too often the RFI is so strong that even a narrow-band signal can dominate the total power. And since it's narrow-band, dedispersion doesn't smear it out like it would a broad-band interference spike. So narrow-band RFI is one of the kinds of interference that is particularly hard to sift out from pulsar candidates.

Just recently, on the arxiv, a paper came out that attempts to address the problem of testing whether a candidate is broad-band:

Multimoment Radio Transient Detection
Laura Spitler, Jim Cordes, Shami Chatterjee, Julia Stone
We present a multimoment technique for signal classification and apply it to the detection of fast radio transients in incoherently dedispersed data. Specifically, we define a spectral modulation index in terms of the fractional variation in intensity across a spectrum. A signal whose intensity is distributed evenly across the entire band has a much lower modulation index than a spectrum with the same intensity localized in a single channel. We are interested in broadband pulses and use the modulation index to excise narrowband radio frequency interference (RFI) by applying a modulation index threshold above which candidate events are removed. The technique is tested both with simulations and using data from sources of known radio pulses (RRAT J1928+15 and giant pulses from the Crab pulsar). We find that our technique is effective at eliminating not only narrowband RFI but also spurious signals from bright, real pulses that are dedispersed at incorrect dispersion measures. The method is generalized to coherent dedispersion, image cubes, and astrophysical narrowband signals that are steady in time. We suggest that the modulation index, along with other statistics using higher-order moments, should be incorporated into signal detection pipelines to characterize and classify signals.

This looks very promising, but there's some testing I wish they'd done. More on the subject below the jump.

The Core

I've worked with data from X-ray satellites before, and among the many messy things one has to deal with in real data were blocks of time marked SAA. I knew this stood for "South Atlantic Anomaly", but I had only the vague idea that it was a part of the sky that was geomagnetically inconvenient, so that I had to trim it out of my data. The other day I came across a fascinating BBC documentary, titled "The Core":

This documentary talks about the Earth's core and how we're studying it, from seismology to diamond anvils to huge liquid-sodium dynamo experiments. It makes very interesting watching, but I particularly liked that they used the South Atlantic Anomaly as a hook: it caused problems with certain instruments on Hubble, and the documentary is framed as an investigation into why. (More below the jump if you're not worried about spoilers.)

Least squares and timing noise, part 2

Simulated time series

In my previous post I described a new paper about fitting pulsar parameters in the presence of timing noise using a general least-squares method. It seems like a good approach, but I'd like to look at it in more detail. So: python to the rescue!

Least-squares and timing noise

Figure 4 from the paper: residuals and spectrum

Working with long-term pulsar timing data sets is a nuisance because of so-called "timing noise". Not only is this noise above and beyond the usual observational uncertainties, perhaps because it is torque noise, it tends to be strongest at low frequencies (it is very "red"). Often so much so that leakage from the very lowest frequencies dominates at all analysis frequencies. Various people, myself included, have tried various approaches for dealing with this noise, but a recent arxiv paper shows real promise:

Pulsar timing analysis in the presence of correlated noise

W. Coles, G. Hobbs, D. J. Champion, R. N. Manchester, J. P. W. Verbiest

Pulsar timing observations are usually analysed with least-square-fitting procedures under the assumption that the timing residuals are uncorrelated (statistically "white"). Pulsar observers are well aware that this assumption often breaks down and causes severe errors in estimating the parameters of the timing model and their uncertainties. Ad hoc methods for minimizing these errors have been developed, but we show that they are far from optimal. Compensation for temporal correlation can be done optimally if the covariance matrix of the residuals is known using a linear transformation that whitens both the residuals and the timing model. We adopt a transformation based on the Cholesky decomposition of the covariance matrix, but the transformation is not unique. We show how to estimate the covariance matrix with sufficient accuracy to optimize the pulsar timing analysis. We also show how to apply this procedure to estimate the spectrum of any time series with a steep red power-law spectrum, including those with irregular sampling and variable error bars, which are otherwise very difficult to analyse.

I'd like to look at it in more detail and try some of the techniques on test data.

Science fiction has no imagination, part 1

Every so often I come across something that makes me think that the supposedly imaginative field of science fiction can't hold a candle to reality for weirdness. Today's installment is an arxiv paper in which the authors are seriously discussing quantum teleportation as a way to combine signals from telescopes to form an interferometer:

Longer-Baseline Telescopes Using Quantum Repeaters

Daniel Gottesman, Thomas Jennewein, Sarah Croke

We present an approach to building interferometric telescopes using ideas of quantum information. Current optical interferometers have limited baseline lengths, and thus limited resolution, because of noise and loss of signal due to the transmission of photons between the telescopes. The technology of quantum repeaters has the potential to eliminate this limit, allowing in principle interferometers with arbitrarily long baselines.

How tempo2 does its fitting

Pulsars can be difficult objects to study: for example, their radio pulses can randomly change in brightness, turn off, turn on, change shape, and we really don't know why. Nevertheless there has been some excellent science done by studying those very radio pulses. The trick has mostly been to simply not care how bright they are or what shape they have and focus on when they arrive. Since this comes from the rotation of the pulsar, this tends to be very regular. After all, for a ball 10 km across, with more mass than the sun, smoothed to within a millimeter by its own gravity, it takes an awful lot to change how fast it's spinning. What's more, time is the quantity we can make the best measurements of - world time standards drift by something like microseconds over decades, which is something like one part in ten to the fourteen. So pulsar timing is a powerful technique, that can measure pulsar positions and distances, spin-down rates and braking indices, and binary orbits. The standard software has been tempo, which is written in FORTRAN and has certain limitations. A new tool has recently appeared, tempo2, written in C++ and boasting good handling of timing effects down to the nanosecond level. Unfortunately the documentation on this tool is so far somewhat limited, so I've been figuring out how it works. I'd like to describe it, as best I understand it, here. This particular post will talk about how a timing solution is fit to a set of pulse arrival times.

Millisecond pulsar X-rays

Neutron stars are so tiny and so dense that the natural temperature scale for them has them glowing in the X-rays. What's more, they serve as powerful accelerators of electrons, which then naturally produce X-rays. So it turns out that X-ray telescopes provide a very interesting view of pulsars. In light of this, when we discovered my pet source, J1023, we took an X-ray observation of it. It's taken us considerable time to analyze the results and put together a paper describing them, but I think the result will be a valuable contribution to the literature. (More importantly for me, it should make a chapter of my PhD thesis.) The result is the cumbersomely-named "X-ray Variability and Evidence for Pulsations from the Unique Radio Pulsar/X-ray Binary Transition Object FIRST J102347.6+003841".

Much of the paper is devoted to details of data analysis, which I will spare you. But I think the gist is interesting, and not too hard to summarize. 

CASCA 2010 compact objects session 1

Orbital eccentricity in numerical simulations of binary black holes (Harald P. Pfeiffer)


Eccentricity in black-hole binaries is radiated away faster than the semi-major axis, so that they become circular fairly rapidly. Thus, for example, when a binary makes its way into the LIGO band the eccentricity is down around 10^-6. For numerical GR this means you have to be able to measure - for that matter, define - eccentricity, and you need to be able to produce near-zero eccentricities. To define it, you often construct some function that measures deviation from a circular inspiral, but this is difficult because of (among other things) coordinate dependence. It turns out it works better to define eccentricity in terms of its  gravitational wave effects. You also have to work on periastron advance (for which third-order post-Newtonian models are not good enough). How is very low eccentricity achieved? The problem is in constructing your initial data: coordinate location, sizes, velocities; these need to satisfy complicated PDEs to satisfy the constraints, and it's not clear what goal values you should choose to get circular orbits. The approach is to simulate a few orbits, then fix the initial condition as if it were Newtonian; this roughly works, so you iterate the process until you get a nice low eccentricity. When you go to precessing binaries (i.e. rotating black holes) spin-orbit coupling complicates your life, but you can still extract an eccentricity measurement and initial condition corrections.

Rayachaudhuri's Equation in Regge Calculus (Parandis Khavari)


These equations are important in the proof of singularity theorems, lensing, collapse, and cracking under self-gravity. They concern expansion and shear of fluids in self-gravitation. Absent vorticity they imply collapse into a singularity in a finite amount of time. Regge calculus is finite-element GR that fixes flat geometry within simplices; curvature is concentrated on n-2-dimensional sub-simplices ("bones") and can be described by the deficit angle. Geodesics in the Regge calculus are straight lines within simplices; where they meet faces, the angle on the entrance edge matches the angle on the exit edge, but it's a little tricky since there's no unique way to assign the deficit angle to the n-1 simplices around a particular bone. This work is about expansion of geodesics in (2+1)-dimensions. She obtains expressions for shear and expansion of lensing. Remaining problems include that Rayachaudhuri's equation has no unique discrete representation.

Numerical simulations of precessing binary black holes (Abdul Mroue)


Gravitational wave detection will only be possible if we have accurate banks of template gravitational waveforms. The three main categories are inspiral, merger, and ringdown. To build these templates we need some combination of post-Newtonian models and numerical relativity. We expect an event in the LIGO range at a rate <1/year, but advanced LIGO should have ~0.5/day. So building a template bank is crucial. The parameter space (for zero eccentricity) is given by the mass ratio and the two spins. A 15-orbit binary takes ~10^5 CPU-hours (plus a great number of grad student-hours). Less than 100 waveforms are available from all groups worldwide. So far very little work has been done on systems with generic spins. The spin has very significant effects on the system evolution. This work has two major approaches: make BBH runs easier (i.e. reduce the person-hours) by automating the initial setup and transition between regimes, and make BBH runs faster, primarily by making their simulation code run on GPU supercomputers (possibly an order of magnitude speedup). Currently running on his desktop's GPU.

Questions: Do we expect spin-orbit alignment? Maybe; what happens in the final orbits is totally unclear. [No comment on whether binary evolution is likely to produce aligned binary systems.]

Modelling Gravitational Lens Systems with Genetic Algorithms and Particle Swarm Methods (Adam Rogers)


Lensing has been observed on many cosmological structures; it's interesting because on the one hand it depends on the mass of the lens, and on the other hand it provides an important magnification effect. The goal of this research is to try to reconstruct the un-lensed image of the source. They assume a thin lens. The lensing equation is clearly nonlinear, as shown by multiple lensing. Rather than solve a complicated nonlinear equation, one can simply raytrace past a lens shape. One can combine this information into a mapping matrix; in this formalism, finding the source pixel intensities is a linear least-squares problem. Unfortunately the matrix sizes are comparable to the number of pixels [so you need sparsity], so you need to use a small PSF, i.e. optical data. With some cleverness one can reformulate the problem into a deconvolution problem that never needs to construct the huge matrices. On simulated data it's quite effective at recovering even highly distorted images when you know the lens parameters. Finding the lens parameters requires a fitting procedure, for which he's using genetic algorithms and particle swarms. Particle swarm optimizers attract each particle to its local best and the global best according to a spring force. Of the two the particle swarm optimizer is a little faster, while the GA is much more thorough in searching the parameter space (e.g. local loses one of the two 180-degree options for ellipse orientation).

Questions: Have you applied your models to real data? Yes, but it's old data that's already been partially processed. What is he optimizing? "chi-squared between the model and the data"

New properties od teh 35-day cycle of Hercules X-1 (Denis Leahy)


RXTE ASM data of Her X-1, plus PCA data when available. Her X-1 is 6.6 kpc and high galactic latitude; it's a neutron star with a 2.5 Msun A7 companion. Over 35 days (many binary orbits) you get brightness variations. One model is a twisted disk that occults the NS; its shadow on the companion star changes on the same cycle, which explains the optical variations. ASM data shows that the cycle length varies substantially - 34-38 days - and this variation is correlated with the flux. The turn-on appears to occur uniformly in orbital phase. They have 1.58 Ms of PCA data in total. One idea for explaining some of the irregularities is that the impact point on the disk is not the outer edge, since it's not flat; other models include an uneven disk surface or blobs in the accretion stream, but these don't match the data well.

Questions: Why is the disk twisted? Heating by the central source  produces a torque that increases as the disk gets out-of-plane up to the point where it starts shadowing itself.

CASCA 2010: Stars 3 session

Variable stars in the Hyades Cluster (Jayme Derrah)

To establish distance scales you need to measure variable stars at known distances; the Hyades cluster is the closest open cluster to the Earth. Unfortunately it's just out of range for accurate parallax measurements, so distances are measured using the "convergent point method". Variable stars of interest re eclipsing binaries and pulsational variables (though few new examples of the latter are expected). The project uses the Baker-Nunn patrol camera, which has a large enough field of view to include the whole cluster and reaches a limiting magnitude of 19.5 in two minutes; on the other hand there's only one filter, a light pollution blocker. Follow-up of discovered variables will use B and V.

Type 1A Supernova Progenitor Diversity (Ashley J. Ruiter)

Sub-Chandrasekhar-mass WDs should be looked at as SN1A progenitors. SN1A light curves are driven by the amount of nickel produced, so we can use them as standard candles almost without any idea of what the progenitors look like. Theories are CO WD mergers and CO WD collapse triggered by companion Roche lobe overflow. Simulations predict that the former ought to produce NSes instead, while models (including this work) have difficulty producing enough of the latter. In particular, she built a population synthesis code and found an order of magnitude too few overflow models. The idea is that perhaps allowing sub-Chandrasekhar-mass collapse can help. More detailed models, in particular using helium accretion, allow collapse to occur with 0.8-0.9 Msun: you get detonation in a shell of accreted helium which triggers the explosion. She built simulated light curves based on such explosions and found reasonable agreement with observed light curves [but how hard is this, if nickel mass is all you see?]. From a population synthesis point of view, though, they do provide enough. It's also theoretically nice because it explains the observed variety in light curves: it's a function of mass at collapse.

Questions: Where do these helium-rich companions come from? Many are He white dwarfs, which arise naturally in such binaries (lost their hydrogen through Roche lobe overflow). What's that blue line on the graph? It turns out if you start with 0.9 Msun WDs the merger scenario looks a little more plausible. What assumptions go into the pop. synth? Standard IMF, etc. Where does the mass go during binary evolution? Eddington-limited transfer, so probably outflows.

Tau Sco: The Discovery of the Clones (Veronique Petit)

Part of the MiMeS (Magnetic Field in Massive Star) collaboration. There ~35 known magnetic OB stars (i.e. B directly detected, which is hard), but it has long been suspected that B is ubiquitous in massive stars. ("Magnetic fields are to astrophysics as sex is to psychology." - someone) These stars are short-lived but important, and the key complicating parameters are rotation and mass loss; we expect B to substantially affect the wind (as it does in the Sun). Tau Sco, a B 0.5 V star, is the focus of this talk. We can measure B by the Zeeman effect, i.e. circular polarization across a spectral line. If you sample the stellar rotation, you can actually produce B maps. Unlike all other massive magnetic stars which have roughly dipolar fields, tau Sco's field is complex and multipolar. Studies of the wind using UV line profiles show strange inconsistencies between different line traces; there seems to be some correlation between the UV variability as a function of stellar rotation and the B. It's natural to assume that the wind anomalies arise from the complexity of B, but it was hard to test until their recent discovery of two "clones" that have similar wind anomalies and slow rotation. ESPADON shows that these stars are magnetic; sadly it has not yet been possible to sample the stellar rotation densely enough to map the magnetic field. Preliminarily, though, a dipole model suggests that surface B is roughly equal to that of tau Sco. 

Questions: Have people measured this supposed correlation between B and the wind? Yes, but it doesn't really predict tau Sco. How strong are these Bs? ~500 G. Does B of tau Sco vary with time, particularly in flares? We don't know, they had to assume it depended only on rotation. Could you use the Bayesian inference to estimate non-dipolar fields? There's too much uncertainty when you don't know about the stellar rotation.

Tracing Wolf-Rayet wind structures (Alexandre David-Uraz)

This work focuses on WR113. WR stars have very high mass loss rates, high enough that radiation pressure alone can't explain it; we need to understand the physics of clumping. WR113 = CV Ser is a binary system in which the companion can often be seen through the WR wind. It's a double-line binary, so they can in principle get a decent picture of the orbit, but so far the spectrum has been too messy. Once this is accomplished, they should be able to separate the two spectra quite well - average everything in the WR frame, then subtract this and switch to the companion's frame, then back and forth. In any case, the "eclipse" is of depth ~0.6 mag - well, in Lamontagne et al. from the 90s; in a 1963 paper the eclipse seems variable (0.1 mag sometimes and 0.6 mag), and in 1970 they didn't see an eclipse at all. MOST sees a shallow eclipse but variable - in fact, two successive eclipses differ very substantially. The real focus of the research is random variations due to clumping, and indeed they see both photometric and spectroscopic evidence for clumping. There's also the issue of a colliding-wind shock, which shows up in the lines (in particular there's streaming along the shock); with luck this should help establish system geometry. The next step is Fourier analysis for pulsations in star or wind, and wavelet analysis for random clumping. The goal is to link these to spectroscopic data and constrain the clumping.  

Questions: This is a complicated system; is there evidence for an accretion disk? Unclear, but the colliding winds suggest not. Is this the only eclipsing WR star? No, there are others. 

Limb-Darkening and Stellar Atmospheres (Hilding Neilson)

Limb darkening is particularly interesting these days as observations become more constraining. They affect planet parameters inferred from transits (and fitting can constrain limb darkening). Optical interferometry can directly measure it, and microlensing can help as well. Limb darkening tells you something about conditions in stellar atmospheres; Schwarzschild used solar eclipse observations to show that the solar atmosphere is in radiative rather than adiabatic equilibrium. Traditionally, though, people just use empirical limb darkening laws that are pretty crummy; in fact popular parameterizations have fixed points that are more or less independent of the parameters, and fit observations rather badly. The fixed point arises because the models the empirical laws are based on all make the Eddington approximation. Spherical symmetry rather than plane-parallel atmospheres helps spread the fixed point out. Detailed modelling suggests that the true value at the fixed point probes atmosphere physics. Observations suggest that you can make these inference even with the wrong models.