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.



I have been involved with the drift scan survey for quite some time, but it is coming to an end: observations finished in 2007, and Jason Boyles was just running the last beams through (I'm guessing the disk had been forgotten on a shelf somewhere) and looking at the candidates, and he found a lovely bright millisecond pulsar. Since he was finishing his thesis, he handed off the pulsar to the rest of the group.

Initial timing made it really clear that there was a 1.6-day orbit, but deviations showed up pretty quickly. Scott Ransom got all excited and sent around an email saying that maybe we had a triple system, but Ingrid Stairs - famously meticulous - scoffed. A few days of careful timing later and she was ready to agree: we had a triple system. This was astrophysically fascinating, but observationally challenging. In order to observe a pulsar you really want to have a good model predicting the orbits, so you can account for all the Doppler shifts and fold the signal at the appropriate period of the pulsar. But a triple system doesn't have a nice clean orbital model the way a binary does. So while you can fake it by just adding the delays of two orbits together, the prediction goes wrong very quickly. So in order to avoid losing track of the exact number of turns, we started observing the pulsar almost every day. Marten van Kerkwijk improved the situation by noticing that we could do better than just adding delays by making the center of mass of the inner orbit move around the outer orbit. Once he had implemented this, we weren't in danger of losing phase any more, but his model still failed to account for gravitational interactions between the orbits.

All this is mostly second-hand, because I had a thesis to write too. So while I was on the emails I wasn't following them too closely. But then I got to the point of assembling papers to form a thesis. This takes a lot of painstaking work, much of which is frankly boring. So, procrastination. I came across an email bemoaning the fact that we didn't have a timing model for the system, and I thought, wait a minute, can't I do that the dumb way? By which I meant: never mind all the clever mathematics Newton used to go from his theory of gravity to get Kepler's laws. Never mind Kepler's laws; they don't work right in this system anyway. Let's just apply Newton's laws directly.

This was a project I was kind of familiar with, actually. In high school, I had come across a throwaway line in a Larry Niven novel that described five planets orbiting their center of mass in a pentagon, and I wondered, "is that really stable?" So I wrote a code to simulate the motion of five objects under Newton's laws, and concluded that it really wasn't stable. The math of Newton's laws really isn't that complicated, and (though I didn't have access to it back then) there is plenty of code out there to efficiently and accurately simulate motion given the laws - it's just a differential equation solver.

So, thought I, why not try just simulating Newton's laws to predict the motion of the triple system? I couldn't abandon Kepler's laws entirely, of course, because they were the most comprehensible way to describe the initial conditions. But I immediately extracted positions, velocities, and masses and then used math no more complicated than $GMm/r^2$ and $F=ma$. This gave me orbits. To get pulse arrival times, to first order, each pulse is just delayed by the distance of the pulsar behind the plane of the sky. So I got all this working and then asked for some real data to work with. Since the collaboration was at loose ends, they sent me some. And to my astonishment, the data lined up beautifully. Not perfectly; there were some systematics left. But I was already doing better than the Keplerian model.

Blue is the Keplerian model;
green is the Newtonian.
The vertical scale is microseconds.

Needless to say, we got excited. And I soon figured out a couple of minor effects I had ignored; for example, the pulsar suffers from time dilation due to its motion and due to its position in the gravitational field of the two companions. Accounting for this variable slowing was the key remaining ingredient, and my model agreed with the data leaving no visible systematics and with a scatter close to what the instruments were claiming as an uncertainty. In other words, the model accurately described the system motion.

What was most amazing was what came out of the model fit. In order to specify positions, velocities, and masses for the three objects, I had to specify all kinds of properties of the orbits that are not usually measurable. For example, normally you can easily measure the projected semi-major axis - that is, the part of the orbital motion that moves the pulsar towards or away from us - but the actual semi-major axis can't be measured, since you can always replace a system with a wider system that is more massive and seen more face-on without changing any of the measurements. But in the triple system, the orbital interactions depend on these parameters, so fitting for the orbital interactions allows us to measure them.

Remember I said we were taking observations every day? In fact, it's a bright pulsar and we have an incredibly intense campaign of observations with Arecibo, Green Bank, and Westerbork. So I have thirty thousand microsecond-level measurements spanning a year and a half. And fitting a model to this massive data set, it turns out, constrains all the system parameters to an amazing degree: we know the pulsar and white dwarf masses to four decimal places, for example (the pulsar mass is 1.4375(12) solar masses, sadly not large). Scott claims these are some of the best-measured masses outside the Solar System but I'd prefer to do a bit more reading before making that claim. Nevertheless we have fantastic measurements of almost all the system's parameters.

It took a while, but we finally got a paper together and written and published in Nature (and the arXiv of course), describing this unique system. There has been more work, and in fact I'm writing at least two more papers about the system, but I think I'll save discussion of that for later posts.