# Here are a few of my favorite things.

Schematic representation of a chaotic interaction between two binary star systems.  The angle of each vertex indicates quantitatively the energy of each star, at a given time-step (Leigh N. W. C., Geller A. M. 2012, MNRAS, 425, 2369).

## -Chaos in Newtonian Gravity

The gravitational three-body problem has been around for a long time.  Newton himself dabbled with it, albeit briefly.  With hundreds of years of research behind it, one thing has become clear about the general three-body problem:  it is an example of chaos in nature.  Given the positions and velocities of all three particles at some initial time, it is not possible to derive a simple analytic formula to predict their positions and velocities at any subsequent time.  It only gets worse if additional bodies are thrown in to the mix.  My research aims to exploit the chaotic nature of the problem, to create models that predict the outcomes of a statistical ensemble of such small-number gravitational interactions.  One such example includes applying a radioactive decay formalism to the disruption of small systems of stars (Leigh et al. 2016; Ibragimov, Leigh, et al. 2017).  We find that such an approach can indeed accurately describe the physics.  This should perhaps be expected, since both gravity and the strong nuclear force obey an inverse-squared force law, at least at short distances.  You can find those papers here:

Ibragimov T. & Leigh N. W. C. (2017), in preparation.

I study combinatorics in chaotic Newtonian dynamics in various shapes and forms. In one collection of papers, we adopt finite-sized particles and focus on interactions that produce direct collisions between any two stars. Our long-term goal is to construct an equation that gives the probability of a given collision event occurring over the course of the interaction, as a function of the total encounter energy and angular momentum as well as the numbers and properties of the particles.

Why would this be such a useful model to have in our toolbox?  Direct collisions during fewbody interactions have been studied in a variety of contexts and are thought to be crucial for a number of ubiquitous astrophysical processes, including stellar collisions and blue straggler formation in globular clusters (e.g. Leonard 1989; Fregeau et al. 2004; Leigh, Knigge & Sills 2007; Hypki & Giersz 2016, 2017) and even galactic nuclei (e.g. Shara & Shaviv 1974; Davies et al. 1998; Bailey & Davies 1999; Yu 2003; Dale et al. 2009; Leigh et al. 2016), the formation of runaway stars in O/B associations (e.g. Blaauw & Morgan 1954; Perets & Subr 2012; Oh et al. 2015; Ryu, Leigh & Perna 2017a,b,c), the formation of intermediate-mass and supermassive black holes via runaway stellar collisions (e.g. Portegies Zwart et al. 2004; Giersz et al. 2015; Stone, Kuepper & Ostriker 2017), the collisional growth of protoplanetary disks (e.g. Goldreich, Lithwick & Sari 2004; Lithwick & Chiang 2007), the formation of massive elliptical galaxies in galaxy groups and clusters (e.g. Binney & Tremaine 1987; Balland et al. 1998; Trinchieri et al. 2003), and the list goes on.

In our most recent paper, the third in the series, we expanded our method for application to the majority of the available phase space.  In other words, we very seriously improved our model.  After rigorous comparisons to N-body simulations, it performs its intended purpose at the order-of-magnitude level.  We are one short paper away from adapting our model to the remaining parameter space, and fully testing it using numerical scattering simulations.  You can find these papers here:

The chaotic four-body problem has also piqued my curiosity in recent years.  Direct encounters between pairs of binaries in star clusters are the main mode of formation for stable triple star systems.  In the point-particle limit, single-binary interactions can never form stable triples.  At least four stars are needed to get the job done.  We recently developed a statistical mechanics-based method to probabilistically predict the properties of any outcome of the chaotic four-body problem, for any initial total energy and angular momentum for the encounter.  This includes the orbital separations and eccentricities of the inner and outer binaries of any resulting stable triples, as well as the distribution of angles between the inner and outer orbital planes.  Our method can tell you, for any set of initial conditions, what fraction of objects produced in binary-binary interactions will produce stable triples in the active Lidov-Kozai regime!  That's just a fancy term to describe a special triple configuration that allows for internal dynamical evolution.  If the particles have finite sizes, as is the case for real stars, this can drive the inner binary pair to collide and merge.

What does all this mean? With the preceding tools, we can perform the simulations on pen and paper. No more computers. Any limitations of modern technology are overcome, using a clever analytic approach pioneered by J. J. Monaghan in 1976 (Monaghan 1976a,b). You can find my papers on these topics here:

Stone N. C. & Leigh N. W. C. (2017), in preparation.

The Milky Way globular cluster Omega Centauri (STScI/NASA/ESA).

## The origins of galactic globular clusters and their contribution to black hole binary mergers

These behemoths weigh-in at up to a million times the mass of the Sun, and date back almost all the way to the beginning of the Universe. They are home to some of the highest stellar densities in the Universe, with central densities that can reach well over a million solar masses per cubic parsec. If you took the Sun and its closest neighbor Proxima Centauri, the distance between which defines a parsec (pc), and plopped them down right in the core of a dense globular cluster, well over a hundred stars would fall between the pair. Due to these high densities, single and binary stars regularly undergo direct gravitational interactions, carving out an intricate dance in time and space. This regularly serves to swap stellar-mass black holes in to binaries, so that they can later merge in a burst of gravitational waves. Globular clusters are factories for black hole binary mergers. But we are only able to observe these factories in operation now. In order to understand how they were built and the treasures they should be expected to produce, we must somehow rewind their evolution over billions of years to say something about the initial conditions. Only then will we be able to say something about how these beasts birth black hole binaries, and are contributing to the gravitational waves currently being observed by aLIGO.

We get a little tricky in some of the following papers, and develop new methods to constrain or estimate the initial cluster mass, the initial mass-radius relation, the initial stellar mass function, the initial binary fraction and the initial distributions of binary orbital parameters.

This paper also presents a method for calculating the initial total cluster mass for any given globular cluster, using only the observed present-day mass and the slope of its stellar mass function:

The take-away message is that, in order to operate, computers need a user. The user's ignorance thus becomes one of the many limitations inherent to modern computational machines. In the preceding papers, we have begun down the road of removing this user-based ignorance. The initial conditions we find using the present-day observational data can be used as input to simulations for star cluster evolution. This will ensure that your simulations produce clusters consistent with the observed properties of the Milky Way globular cluster population.

If what you really care about are the black holes lurking in globular clusters, and how the initial conditions might affect their merger rates, then check out these papers:

Simulation of the Cosmic Web from the Millennium Simulation Project (Springel et al. 2005, Nature, 435, 629).

## Cosmology

There's really no easy way to initiate a conversation about this next topic without sounding somewhat pedantic, and more than probably a little crazy. I'll just dive right in.

I take great interest in cosmology, particularly from the perspective of a collisional dynamicist. What do I mean by this? Modern cosmology is founded on Einstein's Field Equations with a Cosmological Constant. This immediately implies a field-based approach, applied over many many orders of magnitude in space and time. Now, in most gravitationally-interacting dynamical systems, the field approximation breaks down at very high densities at the inter-particle scale. This marks our entry in to the "collisional" (as opposed to "collisionless" or "field-like") regime, the point at which individual pairs of particles begin to interact strongly, inducing strong deflections to each other's trajectories. In this limit, the cosmological frame of reference must also enter a collisional regime since, after all, the observer is a part of the Universe. Arguably, modern cosmological theory makes no real attempt to self-consistently account for these issues. Instead, the observer approaches a space-time singularity (i.e., the Big Bang) at very early times.

One way to look at this is that the cosmological frame of reference in LambdaCDM is a reasonable approximation at late times only. But what about the very early Universe? Is it possible to self-consistently construct a frame of reference that obeys the laws of physics while simultaneously bridging orders upon orders of magnitude in time and space? And, if so, what might this be telling us?

Cosmology is at a crossroad. On the one hand, the field has a functional model to describe the large-scale structure of the Universe, its history and observations thereof. On the other hand, it points to the presence of both Dark Matter and Dark Energy without offering a reason or explanation for their existence, and leaves us practically defenseless in trying to understand their origins. One way to go about tackling this problem is to try to introduce new physically-motivated models that successfully reproduce all of the currently available observations and empirical data, while offering fresh insights and predictions that will lead us to new physics. But how exactly does one go about this?

One tried-and-true method is to first identify physical inconsistencies in the old model(s), in an attempt to create new models that correct them. As described above, the observer frame of reference in cosmological models is a key concept argued to suffer from inconsistencies.

What if we could turn the model inside-out, and reconstruct it in a non-inertial frame of reference influenced only by gravity? Such frames of reference have survived rigorous empirical scrutiny; they are known to exist in this Universe.

In "A Novel Mechanism for the Distance-Redshift Relation" (Leigh & Graur 2017), we explore one such model, and confront it with some of the available observational data, with a focus on reproducing the observed distance-redshift relation using our model. What we find is rather surprising: to construct the relevant model, one must first assume the existence of an effectively infinite sea of very very long-wavelength particles at very very low energy scales. But, as it turns out, this removes the need for a cosmological redshift in explaining the distance-redshift relation, and replaces it with the more familiar (arguably) gravitational redshift. Not to over-state the significance of our results, since our simple toy model leaves many questions unanswered, but the mechanism proposed here to account for the observed distance-redshift relation could, in principle, contribute to both Dark Energy and Dark Matter. You can find our paper here:

Plot showing the parameter space in the cluster triple fraction-binary fraction-plane (Leigh et al. 2017, in prep). The solid circles show the observed data for several open clusters and the Galactic Field, whereas the open triangles show the data from Milone et al. (2012) and Leigh et al. (2017, in prep). The solid blue line shows the steady-state solution for our choice of input parameters. The blue arrows show flow lines or trajectories for the time evolution of the host cluster, calculated from a suite of numerical scattering experiments.

## Triple Stars and other Stellar Molecules

Consider a cluster of stars, and compare it to a cloud of gas. Both are systems of interacting particles, and both obey the laws of thermodynamics. Both of these systems can be given a temperature, which quantifies the typical relative speeds between interacting particles or stars. In star clusters, a higher temperature is synonymous with a higher velocity dispersion. The basic idea is as follows: the hotter the system, the harder it is for particles to stick together. In the case of a gas, this means that hotter gases tend to have fewer molecules, relative to a colder gas. In star clusters, this means that clusters with higher velocity dispersions tend to have lower binary and triple fractions, relative to clusters with lower velocity dispersions. The fraction of objects that are binaries corresponds to the percentage of unresolved point sources in the cluster that are actually bound pairs.  In a series of papers, my collaborator Aaron M. Geller and I showed that this leads to rich and interesting dynamics in even low-density open star clusters. This challenges the long-standing belief that only globular clusters are "dynamically active". Life now becomes a little more interesting, since most star clusters fall in the open cluster category, in our own Galaxy and abroad. One example of this phenomenon is that ongoing interactions involving binary stars are more likely to be directly interrupted by incoming objects before completion in clusters with lower temperatures (i.e., open clusters). You can check out these papers here: