Notes from the 2017 Boris Garfinkel Lecture

I very recently had the privilege of attending the entirety of this year's Boris Garfinkel Lecture series, delivered by Konstantin Batygin of Caltech. For my personal benefit (and to share this with some of my friends before Konstantin finally publishes the paper on this later this year), here follow my notes taken during the second one of these lectures.

Schrödinger evolution of self-gravitating disks

$N$-body systems under Newtonian gravity admit a mean-field approximation, first developed by Lagrange, known as the secular theory; the logical limit of this is, of course, a debris disk. Now, existing characterisations of the stability of self-gravitating astrophysical disks are fairly easy to write down (e.g. Toomre's $Q$, or the Knuthson number $\lambda_p \over H$), but offer no insight into time evolution proper. Debris disks in particular are amenable to neither fluid analysis nor analytical treatment. While kinetic theory can be applied (qua the CBE), Konstantin prefers not to seek explicit recourse to numerics.

Consider a parameterisation of a debris disk as a series of rings with semimajor axes

where $\beta$ is the aspect ratio of the disk (defined as the ratio of vertical vs radial scale lengths, and assumed to be much less than unity for a thin disk). It is evident that these rings are spaced geometrically; equivalently, in log-space we define $\rho_j = \log a_j$ such that $\Delta \rho = \log(1 + \beta) \sim \beta$ (for a very thin disk) is constant. Moreover, to truncate the disk, we assume that $\rho_\mathrm{max} \to 2\pi$, and $\rho_\mathrm{min} = 0$, with respect to some choice of units (e.g. solar radii)¹.

Let us further assume that the surface density of this disk goes as

Then each of these rings has a mass

Having performed a sort of averaging into rings in this manner, Lagrange's secular theory tells us how the orbital parameters of these virtual rings evolve over time. In particular, the first-order terms of the Hamiltonian that involve the inclination of each ring (expanded in multipole moments) go as

where $i_j$ is the inclination and $\Omega_j$ the longitude of ascending node of the orbit of the $j^\mathrm{th}$ such ring. The $B_{j,j'}$ are coupling coefficients, representing the gravitational potential; the relevant terms can be written as

with $n_j$ the epicyclic frequency and $b_{3/2}^{(1)}(\alpha) \equiv {2 \over \pi} \int_0^{\pi} {\cos \psi \over \left(1 + \alpha^2 - 2 \alpha \cos \psi\right)^{3/2}} ~ \mathrm{d} \psi$ is Laplace's correction factor^{2 3}. The diagonal terms go as

At this point Konstantin performs real magic by making not one, but two judicious choices of symplectomorphism:

1. Define $p_j = i_j \cos \Omega_j$, $q_j = i_j \sin \Omega_j$; this is a real symplectomorphism, with $p_j$ and $q_j$ being conjugate to each other. The canonical form under pullback is the sum of $\mathrm d q_j \wedge \mathrm d p_j$.
2. Define $\eta_j = {1 \over \sqrt 2} \left(p_j + \mathbb i q_j\right) = {1 \over \sqrt 2} i_j e^{\mathbb i \Omega_j}$; under pullback, the canonical form becomes a sum of $\mathbb i \left( \mathrm d \eta_j \otimes \mathrm d \eta_j^* + \mathrm d \eta_j^* \otimes \mathrm d \eta_j\right)$ (i.e. symmetric and imaginary, so antisymmetric under Hermitian conjugation). Pleasingly, not only do $\eta_j$ and $\eta_j^*$ become canonically conjugate to each other⁴, but the canonical form becomes symmetric (i.e. no formal distinction between canonical coordinates and momenta).

In terms of these variables, the partial Hamiltonians read

In particular, the canonical equation for $\eta_j$ then goes as

where the additional factor of $\mathbb i$ comes from the new structure of the canonical form in these complex coordinates. Writing this out in full gives

recognising that we can define $B_j \sim B_{j,j\pm 1} \sim -B_{jj}/2$ in the limit of small $\beta$. This is obviously suggestive of a limit to a second derivative, which we pursue as

But as Laplace's correction goes as ${1 \over \pi \beta^2}$ and $\Delta \rho \sim \beta$, we obtain $\left(\Delta\rho\right)^2 B_j \sim {n_j \over \pi} {m_j \over M}$. Substititing the expression for $m_j$ above and simplifying with the Keplerian expression for the epicyclic frequency gives ${\beta \over 2}\Sigma_0 \sqrt{G M a_0}$, which is a constant that we will call $\omega_i$. Then we see that we have essentially Schrödinger's equation, with no potential term, and with $\omega_i$ taking the place of Planck's constant, as

With the boundaries as described earlier, this gives the evolution of a particle in a potential well. Assuming that the corresponding time-independent problem is valid, the eigenfunctions are to be cosines (by Konstantin's prescription), as we assume that any variations to the orbital parameters die out as we approach infinity (although the inner boundary condition is harder to motivate).

We can play the same game with the eccentricities, and likewise we can recover a similar Schrödinger-type equation. The partial Hamiltonians for those variables are of the form

where the $\varpi_j$ are the longitudes of periapsis of the ring orbits, $e_j$ their eccentricities, and $A_{j,j'}$ are the analogous coupling constants. Defining $\mu$ in the same manner for $e, \varpi$ as we did $\eta$ for $i, \Omega$ gives another Schrödinger equation with a different constant $\omega_e$.

In terms of this formulation, the general-relativistic correction to Lagrange's formalism (which begins as a correction to the partial Hamiltonians as ${3 G M \over a_j c^2}n_j e_j^2$) winds up being a central potential ${3 G M \over a c^2}\sqrt{GM \over a^3}$, which, in terms of $\rho$, goes as $\exp \left[-5\rho/2\right]$. Fortunately, the eigenfunctions of such a potential well have been extensively characterised during some early work in quantum scattering theory: they are combinations of Bessel functions, which suppress the eigenfunction amplitude near the origin, while recovering oscillatory behaviour as we approach infinity.