I am a * dynamicist*. By this I mean that I try to understand and predict the dynamical evolution of a wide variety of astrophysical systems, such as binary black holes, star clusters, and galaxies. To do this I use the tools of

**theoretical physics**(e.g. Hamiltonian mechanics, nonequilibrium statistical mechanics and kinetic theory), but I also employ

**numerical simulations**and am involved with (and continually guided by)

**observational data**.

Below I summarize a few of the scientific projects I have been involved in, focusing on my two current preoccupations: (i) the **secular evolution of galaxies** and (ii) the **formation, evolution and dynamics of binary systems**.

### Secular Evolution of Galaxies

For most of the systems I am interested in, the predominant force is **gravity**. However, because of the similarity between the Newtonian gravitational force and the Coulomb electrical force, many problems I care about have analogues in **plasma physics**. I have made a habit of stealing results from the kinetic theory of plasmas and applying them to various problems concerning secular evolution of galaxies. These have included collective relaxation, spiral instabilities, linear response theory, and bar-halo friction.

**CASE STUDY: Hamilton et. al (2023) Galactic bar resonances with diffusion: an analytic model with implications for bar-dark matter halo dynamical friction, ApJ**

Many galaxies, including our own galaxy the Milky Way, have a ‘

**bar**’ structure at their center— an elongated collection of millions of stars, that gradually rotates as if it were a solid body. Galaxies are also surrounded by massive

**dark matter haloes**. When the rate at which the bar rotates resonates with a dark matter particle’s orbital frequency, the dark matter can suck angular momentum out of the bar, causing it to slow down.

Previous theories of this

**bar-halo interaction**ignored the fact that dark matter particles are not only influenced by the bar, but also experience random ‘diffusive’ forces from other passing dark matter clumps, gas clouds, and so on.

**Some amount of numerical diffusion is also an inevitable consequence of finite-resolution cosmological simulations**. In this paper we

**quantified the impact of diffusion**on the delicate resonant process of bar-halo friction. In the non-diffusive limit we recovered the classic result of Tremaine & Weinberg that the friction vanishes under complete phase-mixing, but we showed that finite diffusion suppresses phase mixing, leading to a finite negative torque.

This work was a

**collaboration with 3 plasma physicists**, none of whom had worked on galactic dynamics before. We realized that the mathematics we needed to solve the problem was precisely that used to understand energetic particle motion in

**tokamak fusion plasmas**. The table below shows where our work fits among various classical studies of wave-particle interactions in both plasma kinetics and galactic dynamics.

### Dynamics of Binaries

Another key interest of mine is the dynamical evolution of **binary systems**, consisting of two objects (stars, black holes, or whatever) orbiting one another. Binaries are the astrophysicist’s harmonic oscillator: one can solve the unperturbed problem exactly (a Keplerian ellipse) and then develop understanding of more complex problems with perturbation theory.

During my PhD – for which I won the 2021 International Astronomical Union PhD Prize – I developed a **general secular theory** for the dynamical evolution of any binary orbiting an arbitrary axisymmetric potential (Hamilton & Rafikov 2019a). The hierarchical three-body problem, and the problem of Oort comets torqued by the Galactic tide, both arise as special cases of this theory. More generally, large-amplitude eccentricity oscillations typified by the **Lidov-Kozai mechanism** – and often invoked to explain e.g. black hole mergers and hot Jupiter formation -are in fact quite general whenever a wide binary orbits an axisymmetric host system, such as a globular cluster (Hamilton & Rafikov 2019b).

Cluster-tide driven eccentricity excitation constitutes a new **merger channel** for the black hole and neutron star binary mergers currently being detected by LIGO/Virgo (Hamilton & Rafikov 2019c). Such binaries are crucially affected by **general relativistic** apsidal precession, and I have built this effect into the formalism (Hamilton & Rafikov 2021). I have also investigated in detail the interplay between secular evolution of binaries and **gravitational wave emission** (Hamilton & Rafikov 2022).

**CASE STUDY: The Mysterious Wide Binaries in the Milky Way**

**Wide binaries**are bound pairs of stars with semimajor axes 1000 AU and above. These rather fragile systems are found in huge abundance in the Milky Way, and have highly unusual orbital properties, including an unexplained

**superthermal eccentricity distribution**: they are, on average, significantly more eccentric than one would expect. Where did this property come from? Is it a consequence of nature (formation process) or nurture (dynamical evolution)?

**(i)**

*. These binaries are so wide that the collective gravitational field of the Galactic disk can modify their eccentricities completely. However, in Hamilton (2022) I employed a simple Galactic tide model, and demonstrated numerically that*

**Nurture****this effect could not be responsible**for producing the observed superthermal distribution. Simply put, unless you start with a superthermal distribution, Galactic tides won’t produce one.

But whenever I gave a talk about that work, someone would ask

*‘Is this a theorem*?’ In other words, was my numerical demonstration just reflecting a general mathematical certainty? With Shaunak Modak, a grad student at Princeton, we showed that the answer is yes: Liouville’s theorem implies that the basic conclusions of Hamilton (2022) are in fact true for

*any*ensemble of binaries in

*any*weak tidal field at

*any*time.

We have since investigated the other major dynamical effect on wide binaries, namely impulsive scattering by passing stars. [Spoiler alert: that can’t produce the superthermal distribution either.] Watch this space…

**(ii)**

*. If dynamical processes are not responsible for the wide binaries’ superthermal distribution, then binaries must be born with it. With Siyao Xu, Hsiang-Chih Hwang and Dong Lai, we proposed a simple model for this formation, in which stars pair up at random following star formation in a turbulent gas cloud. Assuming the turbulent velocity of the gas is imprinted on the stars it formed, we showed that this channel naturally produces a highly superthermal eccentricity distribution.*

**Nature**But the mysteries of wide binaries keep on growing: in Hwang et al. (2022) we showed that

**twin wide binaries**(those where the two stars have very similar masses) are even more eccentric still, perhaps pointing to a circumbinary disk origin.