A Fat Dead Star

Hello all! My name is Dusty Madison. I do research on pulsars, specifically how they can be used to detect extremely low-frequency gravitational waves from things like supermassive black hole binaries. I hope to use this space to talk about neat new work on pulsars, gravitational waves, and any other astronomical topic I can trick enough people into thinking I'm marginally qualified to discuss.

In this first post of mine, I want to talk about a paper that's a few years old now that many of you are possibly already familiar with: A two-solar-mass neutron star measured using Shapiro delay, or Demorest et al. (2010). This paper came out just after I started working on pulsar timing. I have always found it to be a crazy beautiful paper with profound scientific implications that really deftly shows how damn cool pulsar science can be. It's a modern classic in my field (as evidenced by the 1000+ citations). I think discussing it will be a good warm up for me. Full disclosure: I know most of the authors of this paper personally through my work in NANOGrav and the IPTA, but I didn't know them when I first read this paper and realized how cool it was.

So the reason this paper is awesome is because the subject is a truly remarkable millisecond pulsar/white dwarf binary system. The pulsar has a rotational period of 3.1508076534271(6) milliseconds. The orbital period of the system is 8.6866194196(2) days. Pulsar papers are neat places where you'll see this many significant digits and you can take them seriously. The eccentricity of the orbit is 0.00000130(4)---a nearly perfect circle. What really enables the extra-special science you can do with this system though is the inclination angle is 89.17(2) degrees. This system is almost perfectly edge-on to the line of sight making it ideal for measuring the Shapiro delay.

From Figure 1 in Demorest et al. (2010)

The Shapiro delay is a general relativistic delay that the clock-like pulsations of radio waves from the pulsar experience as they traverse the gravitational potential of the white dwarf companion. Schematically, the pulses have to fall into the gravitational well and climb back out before getting to Earth and this takes a little bit of time. How big the delay is depends on the geometrical configuration of the orbit (it's biggest when the white dwarf lies between the pulsar and the Earth) and the mass of the companion (bigger mass, bigger delay).

With this nearly edge-on binary, if you carefully monitor the Shapiro delay throughout the orbital period, you can very precisely measure the white dwarf mass. Once you know the companion mass to high precision, you can directly infer the neutron star mass. Demorest et al. did this. The figure above shows the Shapiro delay they measured (in microseconds) as a function of orbital phase. The white dwarf mass they infer from the shape of this Shapiro delay curve is 0.500(6) solar masses. This yields a neutron star mass of 1.97(4) solar masses.

This was the largest high-precision mass of a neutron star ever measured and it was a big deal. It was a big deal because general relativity dictates that neutron stars have an absolute maximum mass (see TOV limit) and what that maximum mass is depends on the equation of state of nuclear matter at super-nuclear densities. Now, it's nearly impossible to learn anything in a laboratory about the behavior of matter at super-nuclear densities, especially in the degenerate conditions of a neutron star, so measurements of the sort done by Demorest et al. are essentially the only way to gain insight into the nuclear physics in these regimes of parameter space.

Figure 3 from Demorest et al. (2010)

In the figure on the left, Demorest et al. depict the mass-radius relations anticipated for a wide variety of theoretical equations of state. The red horizontal bar represents the new lower-bound on the maximum neutron star mass that this neutron star (code name J1614-2230) yielded. Many of the proposed equations of state say you can have a neutron star even heavier than 2 solar masses. They are safe. However, many of the curves predict maximum neutron star masses well below 2 solar masses. This single mass measurement basically kills those theories. I love it.

Sadly, many of those equations of state that can't support a 2 solar mass neutron star predict really cool exotic phases of matter in neutron star interiors--things like pion condensates, equilibrium populations of hyperons, or deconfined quark matter. Now, these things can't be completely ruled out. Equations of state have tunable parameters that can be tweaked a bit so that they yield maximum neutron star masses compatible with the 2 solar mass neutron star. But the fact is, the 2 solar mass neutron star painted some of these equations of state into a corner and spurred a lot of activity in nuclear theory circles. I think that's awesome.

The Symplectic Integrator (pt 1): Traditional vs Variational Integrators

To start us off, this is more of an in-depth research post, something I had been meaning to write up for a while, describing some recent work that Leo and I have done with our friend Chad Galley at Caltech, and Alec Turner, an undergraduate student I worked with at McGill University.

This work stemmed out of previous work that Chad, Leo and I had done together extending Chad's gorgeous work on nonconservative actions. I won't discuss the nonconservative action formalism in this post, but I will provide a brief summary next week in part 2 in order to describe our use of it for the "Slimplectic" Integrator.

Traditional vs Symplectic Integration

Long-term numerical integration of physical systems are extremely important when studying the stability and evolution of planetary systems, or other astrophysical N-body systems. Traditional integration methods, such as explicit Euler or classical Runge-Kutta schemes, are unstable over thousands or millions of dynamical times, leading to errors in the constants of motion, e.g. energy or momenta, that tend to grow linearly with time. 

In a classic paper of computational astrophysics, Jack Wisdom and Matt Holman described a "symplectic map" for N-body dynamics which allowed for long-term integrations of (conservative) N-body orbits to occur, while preserving physical properties like the constants of motion. Symplectic integrators can be built by partitioning a system's Hamiltonian and then applying the BCH theorem in order to derive mappings that approximate the motion up to a given order.

Traditional integrators, which can be built to minimize the local "error" in the integration-step, can perform comparatively better over short integrations, however in the long term, the non-linear error resulting from the drifting of "constants" of motion make them far worse than symplectic methods.

Solar system evolution stability using first-order traditional and variational (symplectic) integrators. (Hairer, Lubich & Wanner 2006)

Plots of the energy and angular momentum drift from another set of integrations comparing a 4th order symplectic integrator (SI4) to 4th order Runge-Kutta (RK4) taken from Kinoshita (1991).

In order to understand why methods like symplectic integrators are able to preserve the physical constants of motion so well it is helpful to utilize the framework of Variational Integrators and Noether's Theorem.

Noether's Theorem

Of course, since we are discussing the preservation of "constants" of motion, any explanation we have should make full use of Noether's theorem. It is one of the most powerful and beautiful theorems in mathematical physics, and describes the relationship between constants of motion and the symmetries of the system.

Noether's theorem is one of the most powerful in mathematical physics.

More precisely, Noether's theorem relates (differentiable) symmetries of the action that describes the system with quantities that will be conserved by the Euler-Lagrange equations that are derived by extremizing that action.

This means that if we can perform continuous transformations to the system that leave the action invariant, then there will be a constant of motion related to that symmetry which is preserved as the system evolves. Thus, time-translation symmetry leads to energy conservation, translational symmetry leads to momentum conservation, and rotational symmetry leads to angular momentum conservation.

For traditional integration methods, like Runge-Kutta, the focus is on approximating the equation of motion. The differential equation of motion is usually discretized into a form that can be implemented directly on a computer through, for example, finite differencing. Thus, even if the original equation of motion is the Euler-Lagrange equation stemming from some particular action, the actual discrete equation of motion that is solved by the computer does not come from the action, and will not know about its symmetries, so Noether's theorem does not apply. In essence this is why traditional integration methods are not able to prevent long-term drift of the so-called "constants" of motion.

Standard integration methods (e.g. Runge-Kutta) focus on discretizing the equations of motion, leading to a disconnect with the original action. This prevents Noether's theorem from applying, and, without the appropriately constant "constants" of motion, long-term instability occurs.

Variational Integration

A simple way to understand how these integrators work is through the framework of variational integrators (see e.g. the fantastic Marsden & West 2001). Symplectic integrators can generally be written (locally) as variational integrators, and I find this approach, championed by the late, great Jerry Marsden at Caltech, to be much more intuitive than the normal symplectic approach. 

Variational Integrators work by discretizing the action integral itself, rather than the equations of motion. By varying an action that is already discretized, discrete equations of motion are obtained which can be exactly numerically implemented, which are indeed the Euler-Lagrange equations of an action that is closely related to the physical one. 

But how do you discretize the action integral? Exactly how you would discretize any integral, through numerical quadrature! By applying well-known numerical quadratures of a given accuracy order, and then varying the discrete action integral, we can obtain discrete equations of motion that are also accurate up to the same order. 

For Variational Integrators, numerical quadrature rules applied to the action integral yield integrators of the same order after the discretized action is varied. Numerical quadrature is much easier to explain to people than the BCH theorem. 

As long as the discretized action possesses the same symmetries as the original action, the discrete Euler-Lagrange equations we obtain from them will be subject to Noether's theorem, conserving the appropriate quantities (in fact Noether's theorem also tells us how to find the form for the associated conserved momenta, which may be slightly different than in the physical system due to the discretization).

The variational integrator first discretizes the action before performing the variation to obtain the discrete equations of motion. As long as the discretized action possesses the same symmetry as the original action, the discrete Euler-Lagrange equations of motion know about an action, and will exactly preserve the appropriate constants of motion, via Noether's Theorem. 

There are a few wrinkles here, of course. By performing the discretization in time, we break the time-translation symmetry in an exponentially small way. This is the reason why symplectic/variational integrators do not precisely preserve energy in the same way they preserve the other momenta. However, it can be shown that the energy error is always bounded, resulting in energy error that oscillates around zero, with an envelope that depends on the resolution. 

Next week...

Of course, because variational integration or symplectic methods involve systems where an action or Hamiltonian can be specified, they are, by and large, restricted to conservative problems. Problems that involve friction, drag, tidal dissipation, or other nonconservative effects cannot take advantage of the traditional symplectic approach. 

Next week, in part 2 of this post, I will talk about how we used our new nonconservative action principle to develop a variational integration method that applies to general nonconservative systems. This "Slimplectic" Integrator* has all the long-term computational benefits of symplectic integrators, but it can be applied to nonconservative systems where regular symplectic integrators cannot be used. 

*I call it this because phase-space volume "slims" down for dissipative systems, but is constant for conservative ones.

Welcome to the RemarXiv.

This is our new research blog for encouraging us to discuss and summarize interesting research results, new ideas, or interesting papers that we may come across. This includes both interesting new papers posted to the arXiv (primarily astro-ph or gr-qc) or even older papers that we might find particularly useful in our own research.

The purpose of this blog is help stimulate our own research efforts by forcing us to summarize, for our peers, the work we are doing, and the old or new results in the field. It is not meant to be exhaustive, or terribly in-depth. It is only meant to stimulate our own research and personal interest.

The Rules: Each of us should post at least once a week. This must be related to research, though it does not have to be scientific in content (for example, in the future I may post about the tools and software I am currently using for my scientific workflow). It can be a post summarizing/discussing a paper that we have read, be it old or new, or it can just be an interesting new idea that we would like to post for stimulating discussion. We may use this space to discuss previous results, or summarize some of our own work for a more general audience. 


Thus far, the RemarXivists are:

Dave Tsang: CTC Fellow at the University of Maryland. Dave works on astrophysical dynamics in a variety of contexts, from black hole accretion, to exoplanetary dynamics. His current research focuses on N-body and disk-planet interactions for exoplanets, and on neutron star physics during gravitational wave induced inspiral of compact binaries.

Leo C. Stein: Postdoctoral Researcher at Caltech. Leo's research interests are studying and testing general relativity and other theories of gravity from an astrophysical standpoint.  He has investigated how “almost-general-relativity” theories can affect gravitational observables. An important observation which would be able to distinguish between GR and almost-GR is the inspiral rate in a compact binary system, detected either through radio pulsar timing or directly with gravitational waves.




George Pappas: Postdoctoral Research Associate at Ole Miss. George is an expert in General Relativity, focusing on the strong field regime. He works on compact objects and the spacetime around them in General Relativity and in alternative theories of gravity

Dusty Madison: Jansky fellow at National Radio Astronomy Observatory (NRAO).  Dusty's research is on pulsars, specifically how they can be used to detect extremely low-frequency gravitational waves from things like supermassive black hole binaries. By pushing precision pulsar timing to its limits with world-class radio telescopes and instrumentation and continually improving data analysis techniques, the pulsar community is poised to detect extremely low-frequency gravitational waves within five to ten years and begin a new era in the study of black holes, gravity, and currently unknown astrophysical phenomena.

Maria (Masha) Okounkova: graduate student in physics at Caltech, Princeton '14 physics undergrad. Works in numerical relativity, currently working on simulating collapse to naked singularities in general relativity and binary black hole simulations in almost-general relativity (with Leo Stein). Advised by Yanbei Chen and Mark Scheel in the TAPIR group, member of SXS collaboration.

Leo's #366papers tweets

As with most new years' resolutions, I was a bit ambitious. I resolved to read #366papers throughout the year—one per day! I kept up for the first few days and tweeted about each paper I read. However I soon became overwhelmed and fell behind.

In discussions with Dave and George, we decided that it's still a nice idea, but 366 is too lofty a goal. Hence we started The RemarXiv.

For posterity, I've collected my #366papers tweets in a Storify which is embedded below.

New RemarXs will come soon!

Galileo, the Tides, and action at a distance

This is the first post on this blog and its aim is to test the various features of the blog. But it wouldn't be proper to step completely out of character, therefore the topic will be on some past research. Really old to be exact. So the topic will be on Galileo's views on tides and his notions on action at a distance (actually, this is not an entirely uninteresting topic in modern days, although in a different context). Furthermore, the material is taken mainly from Wikipedia, from the references given at the end, combined to cover the specific topic as I intended to present it (the material that was taken exactly as it was from the source is in quotation marks).

So, it is interesting to begin with how Galileo viewed the phenomenon of the tides in the context of the time and with respect the main controversy that he was involved in, i.e., the motion of the Earth around the Sun.

"Cardinal Bellarmine had written in 1615 that the Copernican system could not be defended without "a true physical demonstration that the sun does not circle the earth but the earth circles the sun". Galileo considered his theory of the tides to provide the required physical proof of the motion of the earth. This theory was so important to him that he originally intended to entitle his Dialogue on the Two Main World Systems, the Dialogue on the Ebb and Flow of the Sea. The reference to tides was removed by order of the Inquisition.

For Galileo, the tides were caused by the sloshing back and forth of water in the seas as a point on the Earth's surface sped up and slowed down because of the Earth's rotation on its axis and revolution around the Sun. He circulated his first account of the tides in 1616, addressed to Cardinal Orsini (Discourse on the Tides). His theory gave the first insight into the importance of the shapes of ocean basins in the size and timing of tides; he correctly accounted, for instance, for the negligible tides halfway along the Adriatic Sea compared to those at the ends. As a general account of the cause of tides, however, his theory was a failure.

If this theory were correct, there would be only one high tide per day. Galileo and his contemporaries were aware of this inadequacy because there are two daily high tides at Venice instead of one, about twelve hours apart. Galileo dismissed this anomaly as the result of several secondary causes including the shape of the sea, its depth, and other factors."[1]

"Galileo dismissed the idea, held by his contemporary Johannes Kepler, that the moon caused the tides."[1]

"Discourse on the Tides does not include gravitational forces in its theory to explain the Earth's orbit and does not consider the relation between the ocean and cosmic gravitational forces, like that of the moon. Occurring invisibly, gravity was far too mystic for Galileo's consideration. Galileo did end the Discourse on the Tides with reservations that his theory may be incorrect and the hope that further scientific investigation will confirm his proposal."[2]

This stance towards gravitation and the notion of action at a distance without mediators that Newton introduced was common at the time.

"Descartes' book of 1644 Principia philosophiae* (Principles of philosophy) stated that bodies can act on each other only through contact: a principle that induced people, among them himself, to hypothesize a universal medium as the carrier of interactions such as light and gravity—the aether. Newton was criticized for apparently introducing forces that acted at distance without any medium."[3]

Newton's law of universal attraction:

$$\large\vec{F}=-G\frac{m_1 m_2}{r^2}\hat{r}$$

yes we have tex

Beyond that, the theory was widely accepted and there was also a controversy between Newton and Hook regarding who was the father of the theory. It seems that Halley, Wren, De Moivre, and the Royal Society had some part to play in the controversy until in the end Clairaut settled the matter.[3]

In addition, there is this paper on the ArXiv that discusses the issue, titled: "The reception of Newton's Principia"

"Newton's Principia, when it appeared in 1687, was received with the greatest admiration, not only by the foremost mathematicians and astronomers in Europe, but also by philosophers like Voltaire and Locke and by members of the educated public. In this account I describe some of the controversies that it provoked, and the impact it had during the next century on the development of celestial mechanics, and the theory of gravitation."[5]

So, this was more or less the state of things at that time with respect to the notion of action at a distance and the tides. Let me wish a safe journey to this blog with this first step outside the door in the roads of the Blogoshpere**.


* In Principia Philosophiae one can also find the introduction of Newton's first law of motion, while foundational work on dynamics can be found in Galileo's book Dialogo sopra i due massimi sistemi del mondo (Dialogue on the two main world systems).

** “It's a dangerous business, Frodo, going out your door. You step onto the road, and if you don't keep your feet, there's no knowing where you might be swept off to.” J.R.R. Tolkien, The Lord of the Rings.

References:

[1] Wikipedia: Galileo, Kepler and theories of tides in Galileo Galilei
[2] Wikipedia: Discourse on the Tides
[3] Wikipedia: Historical context in Philosophiæ Naturalis Principia Mathematica
[4] Galileo's Big Mistake
[5] "The reception of Newton's Principia", arXiv:1503.06861 [physics.hist-ph]