# Swing Amplification and the Gaia Phase Spirals

Lawrence M. Widrow
Department of Physics, Engineering Physics and Astronomy, Queen’s University,
Kingston, K7L 3X5, Canada E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

We explore the interplay between in-plane and vertical dynamics in stellar
discs within the framework of the shearing box approximation. Julian and
Toomre used the shearing sheet to show that leading density waves are
amplified as they swing into a trailing ones. We extend their formalism into
the dimension perpendicular to the disc and obtain explicit solutions for the
response of a disc to an impulsive, external excitation. An excitation that is
is symmetric about the mid plane produces a density/breathing wave as well as
two-armed phase spirals in the vertical phase space plane. On the other hand,
an excitation that is antisymmetric about the mid plane leads to a bending
wave and single-armed phase spirals. In either case, self-gravity plays a
crucial role in driving the evolution of the disturbance and determining the
amplitude and pitch angle of the ensuing spirals. We also show that when the
disc is excited by a co-rotating cloud, it develops stationary phase spirals
in the wake of the cloud. The results call into question simple kinematic
arguments that have been used to determine the age of the phase spirals seen
in the Gaia survey.

###### keywords:

Galaxy:kinematics and dynamics - Solar Neighborhood - Galaxy: disc - Galaxy:
structure

††pubyear: 2020††pagerange: Swing Amplification and the Gaia Phase
Spirals–LABEL:lastpage

##  1 Introduction

One of the most intriguing discoveries from Gaia Data Release 2 (Gaia
Collaboration et al., 2018a, b) is the existence of spirals in the vertical,
or $z-w$, phase space distribution function (DF) of Solar Neighbourhood stars
(Antoja et al., 2018). The phase spirals are easily seen in maps of the $z-w$
DF once the smooth background distribution has been removed. They have a
fractional density contrast of a few percent and display a rich morphology
that depends on the properties of the stars under consideration such as
Galactocentric radius, Galactic azimuth, angular momentum, and epicyclic
energy (Laporte et al., 2019; Widmark, 2019; Bland-Hawthorn et al., 2019; Li &
Shen, 2020; Hunt et al., 2022; Frankel et al., 2022; Antoja et al., 2022). For
example the spirals tend to be two-armed in the inner galaxy and one-armed in
the outer galaxy (Hunt et al., 2022). They also appear in $z-w$ maps of the
mean azimuthal and radial velocities.

The most natural explanation for the spirals is that they are disturbances in
the $z-w$ DF from some past event or events that have undergone phase mixing
(Tremaine, 1999). For example, if a local patch of the disc experiences a
"kick" perpendicular to the mid plane, stars will be displaced in the
$w$-direction. The perturbed DF will then shear into a one-armed spiral since
stars with low vertical energy rotate about the origin of the $z-w$ plane at a
higher frequency than stars with high vertical energy. On the other hand, one
might imagine a breathing mode perturbation where the DF is squeezed in $z$
and/or stretched in $w$. Over time, this perturbation will shear into a two-
armed spiral. In either case, if the evolution in phase space is purely
kinematic, then the pitch angle of the phase spiral will depend on the time
since the initial perturbation and the anharmonicity of the vertical
potential. Indeed, the simplest approach to understanding the Gaia phase
spirals is to model the DF as test particles in a fixed one-dimensional
potential, introduce an ad hoc perturbation, and evolve the system until a
spiral pattern matching the one seen in the data is reached. This approach
leads to an estimate of $300-900\,{\rm Myr}$ for the age of the spiral (Antoja
et al., 2018). When the test-particle analysis is extended to three-
dimensions, it can help elucidate the origin of the mean $v_{\phi}$ and
$v_{R}$ spirals (Binney & Schönrich, 2018; Darling & Widrow, 2019).

Similar estimates for the age of the spirals can be obtained by transforming
the data into action-angle-frequency or $(J_{z},\,\theta_{z},\,\Omega_{z})$
coordinates. If the spirals are created by a single event and if the potential
is time-independent, then $J_{z}$ will be constant and
$\theta_{z}=\Omega_{z}t+\theta_{0}$ along a star’s orbit. Thus, in the
$\theta_{z}-\Omega_{z}$ plane, the spirals should appear as parallel ridges
whose slope is proportional to the inverse age of the spiral (Li & Widrow,
2021; Frankel et al., 2022; Li & Widrow, 2023; Tremaine et al., 2022). Note
that both methods require a model for the background gravitational potential.
An alternative approach is to use the shape of the spirals to constrain the
gravitational potential (Widmark et al., 2021a; Widmark et al., 2021b).

A promising candidate for the origin of the spirals is the passage of a dwarf
galaxy or dark matter subhalo through the Galactic disc with the Sagittarius
dwarf galaxy (Sgr) considered a prime suspect (Laporte et al., 2019; Bennett &
Bovy, 2021; Bennett et al., 2022). Sgr is the nearest known neighbor to the
Milky Way (Ibata et al., 1994) and has an orbit that has likely taken it
through the mid plane of the Galaxy several times over the last few Gyr
(Johnston et al., 1995). Purcell et al. (2011) argued that Sgr was crucial in
shaping the Milky Way’s bar and spiral arms while Gómez et al. (2012)
suggested that it could have also generated the vertical bending and breathing
waves seen in both pre-Gaia surveys and Gaia (Widrow et al., 2012; Williams et
al., 2013; Carlin et al., 2013; Xu et al., 2015; Bennett & Bovy, 2019).
Several groups have found vertical phase spirals in high-resolution
simulations of a Sgr-Milky Way encounter though none of these have managed to
reproduce the morphology of the spirals found in the Gaia data (Laporte et
al., 2019; Bennett & Bovy, 2021). Simulations have also been used to explore
other origins of the phase spirals such as the vertical waves generated by a
buckling event in the Galactic bar (Khoperskov et al., 2019). It is worth
noting the spirals are subtle (few percent) phase space features at a scale of
$200\,{\rm pc}$ by $10\,{\rm km\,s}^{-1}$ and therefore push the limits of the
resolution simulations.

The general conclusion from these investigations is that the simple picture of
a kinematic spiral generated from a single event is incomplete, if not
incorrect. For example, the transition from two-armed to one-armed spirals as
one moves out in Galactocentric radius may require multiple events (Hunt et
al., 2022). In addition, when the spirals are transformed from $z-w$ to
$\theta_{z}-\Omega_{z}$ coordinates, they appear as curved rather than
parallel bands (Frankel et al., 2022; Tremaine et al., 2022). Finally the
phase spirals found in simulations are not as tightly wound as the ones seen
in the data (Laporte et al., 2019; Bennett & Bovy, 2021; Bennett et al.,
2022). These results may indicate that self-gravity is essential for modelling
the evolution of the spirals. This point was stressed in Darling & Widrow
(2019) who compared N-body simulations of a test-particle disc with fully
self-consistent ones. In both cases, a bend at the solar circle was introduced
into the disc. The spirals that developed in the test-particle case were easy
to detect and had the expected pitch angle. On the other hand, the spirals in
the live disc were less tightly wound and more difficult to discern.

Recently, Tremaine et al. (2022) proposed an alternative scenario in which
phase spirals are generated by a continual sequence of weak perturbations and
erased by phase space diffusion due to the graininess of the gravitational
potential. In this picture, the pitch angle of the spirals reflects the
diffusion time scale rather than the elapsed time from an initial
perturbation. As a proof of concept, they presented simulations in which test
particles in a fixed potential were subjected to a stochastic sequence of
kicks and were able to reproduce key features of the Gaia spirals.

In this paper we explore the connection between in-plane perturbations and
phase spirals within the framework of the shearing box. In their classic paper
on swing amplification, Julian & Toomre (1966) (hereafter, JT66) followed the
evolution of a plane wave perturbation in a razor-thin disc by integrating the
linearized equations for the surface density and gravitational potential. They
found that in a marginally stable disc (Toomre parameter $Q\ga 1$) with
surface density $\Sigma_{0}$ and epicyclic frequency $\kappa$, waves with
wavelength close to $\lambda_{\rm crit}=4\pi^{2}G\Sigma_{0}/\kappa^{2}$ were
amplified by one or two orders of magnitude as they swung from leading to
trailing. Our first task will be to extend the JT66 formalism into the
dimension perpendicular to the disc, that is, to go from a shearing sheet to a
shearing box. The shearing box approximation has been used in a variety of
problems in theoretical astrophysics such as the study of accretion discs and,
most notably, the magnetorotational instability (Hawley et al., 1995). There,
the evolution of the system is driven by gasdynamics and magnetic fields. In
our case, the system is collisionless and the evolution is driven entirely by
gravity.

As in JT66, we solve the linearized collisionless Boltzmann equation for a
disc that is perturbed by an external excitation. The approach has some
commonalities with the formalism developed in Banik et al. (2022) who also
considered the response of an isothermal slab to an external potential in
linear theory. However, their analysis did not include self-gravity, epicyclic
motions, or shear, all of which play important roles in the our analysis. We
also build on early studies of self-gravitating modes in plane-symmetric
systems by Kalnajs (1973); Araki (1985); Mathur (1990); Weinberg (1991);
Widrow & Bonner (2015).

Though this work focuses on the $z-w$ phase spirals, the shearing box
machinery can be used to investigate more general questions about the
interplay between in-plane and vertical dynamics. In particular, one can study
the relationship between in-plane disturbances such as spiral arms and the
vertical waves seen throughout the Milky Way’s disc (Widrow et al., 2012;
Carlin et al., 2013; Williams et al., 2013; Yanny & Gardner, 2013; Xu et al.,
2015; Schönrich & Dehnen, 2018; Bennett & Bovy, 2019; Widmark et al., 2022).
The connection between vertical breathing waves and spiral arms was
investigated with N-body simulations by Debattista (2014); Ghosh et al.
(2022); Kumar et al. (2022) and in linear perturbation theory by Monari et al.
(2015, 2016). These studies focused on moments of the DF rather than the DF
itself. And though the latter treated the full 3D geometry of the disc, it
didn’t include self-gravity of the perturbations. Our treatment has the
advantages of analytic methods while still including self-gravity. The price
we pay is that the treatment of epicyclic motion and shear are only
approximate.

An outline of the paper is as follows. In Section 2, we present the formalism
for calculating the response of a disc to external excitations within a
shearing box. In Section 3, we consider the case where a single wave with a
well-defined wave vector is excited impulsively. In particular, we study
perturbations that generate either breathing waves or bending waves. In
Section 4, we compute the stationary response of the disc to a co-rotating
mass. We discuss the implications of our results, limitations of the shearing
box approximation, and avenues for extending this work in Section 5. We
conclude with a summary of our results in Section 6.

##  2 Shearing box equations

###  2.1 particle orbits

Shearing box coordinates are a local Cartesian approximation to cylindrical
coordinates in a rotating frame. They were devised by Hill (1878) to study the
three-body problem and used in early studies of galactic dynamics by Goldreich
& Lynden-Bell (1965), JT66, and Goldreich & Tremaine (1978). A more recent
discussion of the shearing sheet in the context of galactic dynamics and swing
amplification can be found in Fuchs (2001). Here we follow the pedagogical
flow and notation of Binney (2020) (hereafter B20) who provided a particularly
clear and accessible treatment of the JT66 formalism for the shearing sheet.

Let $(R,\,\phi,\,z)$ be inertial cylindrical coordinates for a rotating
stellar disc and consider a patch of the disc centered on $R=R_{0}$,
$\phi=\Omega t$ and $z=0$ where $\Omega$ is the angular frequency of a
circular orbit at $R=R_{0}$. The shearing box coordinates are $x=R-R_{0}$,
$y=R_{0}(\phi-\Omega t)$, and $z$ and the Lagrangian is given by

${\cal
L}=\frac{1}{2}\left(\dot{x}^{2}+\left(1+\frac{x}{R_{0}}\right)^{2}\left(\Omega
R_{0}+\dot{y}\right)^{2}+\dot{z}^{2}\right)-\Phi_{0}(x,\,z).$ (1)

Formally, we assume ${\bf x}\ll R_{0}$ and $\dot{\bf x}\ll R_{0}\Omega$ though
these inequalities are only marginally satisfied for the size of the patch
that we will consider.

The relationships between $\dot{\bf x}$ and the conjugate momenta ${\bf p}$
are given by

$\dot{x}=p_{x}\equiv u,$ (2)
$\dot{y}=\frac{p_{y}}{\left(1+x/R_{0}\right)^{2}}-R_{0}\Omega\equiv v-2Ax,$
(3)

and

$\dot{z}=p_{Z}\equiv w.$ (4)

The quantities $(u,v,w)$ correspond to the radial, azimuthal, and vertical
components of a particle’s velocity relative to the local circular orbit and
are thus the shearing box analogues to the $(U,V,W)$ velocity components often
used to study the dynamics of the solar neighborhood. Since the potential is
independent of $y$, $p_{y}$ is a constant of motion. It is however, ${\cal
O}(R_{0}\Omega)$. Following B20, we introduce $\Delta_{y}\equiv
p_{y}-R_{0}\Omega$, which is the same order as $\dot{\bf x}$.

The Hamiltonian is given by

$H=\frac{1}{2}\left(p_{x}^{2}+\frac{p_{y}^{2}}{\left(1+x/R_{0}\right)^{2}}+p_{z}^{2}\right)-\Omega
R_{0}p_{y}+\Phi_{0}(x,\,z).$ (5)

We assume that the potential is additively separable in $x$ and $z$ and write

$\Phi_{0}(x,\,z)=\xi(x)+\chi(z).$ (6)

Expanding $\xi$ in a Taylor series about $x=0$ we find

$\xi(x)=R_{0}\Omega^{2}x+\frac{1}{2}\left(\Omega^{2}-4A\Omega\right)x^{2},$
(7)

where $A$ is Oort’s first constant and, without loss of generality, we take
$\Phi({\bf x})=0$.

To quadratic order in small quantities, The Hamiltonian can be written as

$H=H_{x}(p_{x},x)+H_{y}(\Delta_{y})+H_{z}(z,p_{z})+\mbox{constant}$ (8)

where
$H_{x}\equiv\frac{1}{2}\left(p_{x}^{2}+\kappa^{2}\left(x-\bar{x}\right)^{2}\right)$
and $H_{z}=\frac{1}{2}p_{z}^{2}+\chi(z)$ are constants of motion and
$\bar{x}\equiv 2\Omega\Delta_{y}/\kappa^{2}$ is the shearing box analogue of
the guiding radius. (See B20 for a more detailed calculation.) In general, a
particle will move along an elliptical orbit about the point
$(x,\,y)=(\bar{x},\,y_{0}-2A\bar{x})$ and execute anharmonic oscillations in
$z$ about the mid plane. We can therefore write

$x(t)=X\cos{\theta_{r}}+\bar{x}$ (9)

where $\theta_{r}=\kappa t+\theta_{0}$ and $X$ and $\theta_{0},$ are
constants. It follows that

$p_{x}=\dot{x}=-\kappa X\sin{\theta_{r}},$ (10)

and

$\dot{y}=-2A\bar{x}-2\Omega X\cos{\theta_{r}},$ (11)

or equivalently

$v=2B\left(x-\bar{x}\right)$ (12)

where $B$ is Oort’s second constant. In addition, we have

$y(t)=y_{0}-2A\bar{x}t-\frac{2\Omega}{\kappa}X\sin{\theta_{r}}.$ (13)

###  2.2 plane wave perturbations

We consider a density perturbation of the form

$\rho_{1}({\bf x},\,t)=e^{i{\bf k}_{p}\cdot{\bf
x}_{p}}\tilde{\rho_{1}}(z,\,t)$ (14)

where ${\bf x}_{p}$ and ${\bf k}_{p}$ are the position vector and wavenumber
in the plane of the disc. Here and throughout, the over-tilde denotes
coefficient of $e^{i{\bf k}_{p}\cdot{\bf x}_{p}}$. This perturbation
represents a spiral wave in the plane of the disc. If the shearing box is
centered on the corotation radius of the wave, then ${\bf x}_{p}\cdot{\bf
k}_{p}$ must be constant for particles on circular orbits and therefore

$k_{x}(t_{0})x+k_{y}y_{0}=k_{x}(t)x+k_{y}y(t).$ (15)

For circular orbits, $x$ is constant, $y(t)-y_{0}=-2A(t-t_{0})x$, and
therefore

$k_{x}(t)=k_{x0}+2k_{y}A(t-t_{0})$ (16)

where $k_{x0}=k_{x}(t_{0})$. Without loss of generality, we can set $t_{0}=0$.
Then

$k_{p}=k_{y}\left(1+4A^{2}t^{2}+\alpha^{2}+4At\alpha\right)^{1/2}=k_{y}/\beta(t)$
(17)

where $\alpha\equiv k_{x0}/k_{y}$ and $\beta\equiv k_{y}/k_{p}$. For a single
mode, we can further set $k_{x0}=0$ so that $t=0$ corresponds to the time when
wave crests are aligned with the $y$ axis.

For general orbits we have

$\displaystyle{\bf k}_{p}\cdot{\bf x}_{p}$
$\displaystyle=k_{x0}\left(\bar{x}+X\cos{\theta_{r}}\right)$
$\displaystyle+k_{y}\left[y_{0}+2X\left(At\cos{\theta_{r}}-\frac{\Omega}{\kappa}\sin{\theta_{r}}\right)\right].$
(18)

and therefore

${\bf k}_{p}\cdot{\bf x}_{p}|_{t^{\prime}}={\bf k}_{p}\cdot{\bf
x}_{p}|_{t}+\psi(t^{\prime})-\psi(t).$ (19)

where

$\psi(t)=k_{x0}X\cos{\theta_{r}}+2k_{y}X\left(At\cos{\theta_{r}}-\left(\Omega/\kappa\right)\sin{\theta_{r}}\right).$
(20)

###  2.3 equilibrium model

By Jeans theorem, $f_{0}$ can be written as a function of the integrals of
motion $H_{x}$, $\Delta_{y}$, and $H_{z}$ (Binney & Tremaine, 2008). Here, we
assume that it is independent of $\Delta_{y}$ and separable in $H_{x}$ and
$H_{z}$. Following B20, we further assume that the in-plane factor of the DF
is given by the Maxwell-Boltzmann distribution. For the vertical factor, we
use the DF for the lowered isothermal plane (Weinberg, 1991). Putting these
together, we have

$f_{0}({\bf x},\,{\bf
p})=\frac{\Omega\Sigma_{0}}{\left(2\pi\right)^{3/2}\kappa
z_{0}\sigma_{x}^{2}\sigma_{z}}e^{-H_{x}/\sigma_{x}^{2}}F_{z}(H_{z})$ (21)

where $\Sigma_{0}$ is the surface density, $z_{0}\equiv\sigma_{z}^{2}/\pi
G\Sigma_{0}$ is the characteristic thickness of the system, and

$F_{z}(H_{z})=\begin{cases}N_{z}\left(e^{-H_{z}/\sigma_{z}^{2}}-e^{-E_{0}/\sigma_{z}^{2}}\right)&0<H_{z}<E_{0}\\\
0&\mbox{otherwise}.\end{cases}$ (22)

The constant $N_{z}$ is defined so that

$\frac{1}{\sqrt{8\pi}\sigma_{z}z_{0}}\int dzdp_{z}F_{z}(H_{z})=1$ (23)

and

$\Sigma_{0}=\int d^{3}{\bf p}\,dzf_{0}({\bf x},\,{\bf p}).$ (24)

Note that our definition of the in-plane DF differs from the one in B20 since
we use $p_{y}$ as the azimuthal velocity coordinate rather than $v$. The
vertical potential $\Phi_{z}$ and density are determined by solving Poisson’s
equation

$\frac{d^{2}\Phi_{z}}{d^{2}z}=4\pi G\rho_{0}(z)$ (25)

where

$\displaystyle\rho_{0}(z)$ $\displaystyle=\int d^{3}{\bf p}f_{0}$ (26)
$\displaystyle=\frac{N_{z}\Sigma_{0}}{2z_{0}}\left(e^{-\chi_{z}(z)/\sigma_{z}^{2}}\,{\rm
erf}(t)-\frac{2}{\sqrt{\pi}}te^{-E_{0}/\sigma_{z}^{2}}\right)$ (27)

with $t=\left(E_{0}-\chi(z)\right)^{1/2}/\sigma_{z}$. Note that for a self-
consistent model, that is, one where the potential is determined entirely from
the disc itself, we recover the result for the isothermal plane in the limit
$E_{0}\to\infty$:

$\rho(z)=\frac{\Sigma_{0}}{2z_{0}}{\rm sech}^{2}(z/z_{0})$ (28)
$\chi(z)=2\sigma_{z}^{2}\log{{\rm cosh}(z/z_{0})}$ (29)

and $N_{z}=1$ (Spitzer, 1942; Camm, 1950). The lowered isothermal plane is the
one-dimensional analogue of the lowered isothermal sphere or King model (King,
1966) and provides a model in which the density is identically zero for $|z|$
greater than some finite truncation length.

###  2.4 gravitational potential

The contribution to the gravitational potential from the density perturbation
in equation 14 can be written

$\Phi_{1}({\bf x},\,t)=e^{i{\bf k}_{p}\cdot{\bf x}_{p}}\tilde{\Phi}_{1}(z,t)$
(30)

where

$\frac{\partial^{2}\tilde{\Phi}_{1}}{\partial
z^{2}}-k_{p}^{2}\tilde{\Phi}_{1}=4\pi G\tilde{\rho}_{1}(z,\,t)~{}.$ (31)

The solution is given by

$\tilde{\Phi}_{1}(z,t)=-\frac{2\pi G}{k_{p}}P(z,t)$ (32)

where the Green’s function integral

$P(z,t)=\int_{-\infty}^{\infty}\tilde{\rho}_{1}(\zeta,t)e^{-k_{p}|z-\zeta|}\,d\zeta.$
(33)

has dimensions of surface density. The result for a razor-thin disc,

$\tilde{\Phi}_{1}(z,t)=-\frac{2\pi G}{k_{p}}\Sigma_{1}(t)e^{-k_{p}|z|},$ (34)

is recovered by setting
$\tilde{\rho}_{1}(\zeta,t)=\tilde{\Sigma}_{1}(t)\delta(\zeta)$. Similarly, the
$z$-derivative of the potential is given by

$\frac{\partial\tilde{\Phi}_{1}}{\partial z}=2\pi GQ(z,t)$ (35)

where

$Q(z,t)\equiv\int_{-\infty}^{\infty}\tilde{\rho}(\zeta,t)e^{-k_{p}|z-\zeta|}{\rm
sgn}(z-\zeta)d\zeta.$ (36)

###  2.5 linearized distribution function

In linear theory, we write $f({\bf x},\,{\bf
p},\,t)=f_{0}(z,\,p_{z})+f_{1}({\bf x},\,{\bf p},\,t)$ and
$H=H_{0}+\Phi_{1}({\bf x},\,t)$. The collisionless Boltzmann equation is then

$\frac{df_{1}}{dt}\equiv\frac{\partial f_{1}}{\partial
t}+\left[f_{1},\,H_{0},\right]=\left[\Phi_{1},\,f_{0}\right]$ (37)

where $\left[~{},~{}\right]$ are the usual Poisson brackets (Binney &
Tremaine, 2008). This equation admits the following integral expression for
$f_{1}$:

$f_{1}({\bf x},{\bf p},t)=J_{p}+J_{z}$ (38)

where

$J_{p}({\bf x},\,{\bf
p})\equiv\int^{t}_{t_{i}}dt^{\prime}\,\frac{\partial\Phi_{1}}{\partial{\bf
x_{p}}}\cdot\frac{\partial f_{0}}{\partial{\bf
p}_{p}}=i\int^{t}_{t_{i}}dt^{\prime}\,{\bf k}_{p}\cdot\frac{\partial
f_{0}}{\partial{\bf p}_{p}}\,\Phi_{1}$ (39)

and

$\displaystyle J_{z}({\bf x},\,{\bf p})$
$\displaystyle\equiv\int^{t}_{t_{i}}dt^{\prime}\,\frac{\partial\Phi_{1}}{\partial
z}\frac{\partial f_{0}}{\partial p_{z}}.$ (40)

The lower bounds for the integrals represent an initial time when
perturbations are first introduced while the integrands are evaluated along
unperturbed orbits as given in Section 2.1. For $J_{p}$, we use the fact that

$\frac{\partial f_{0}}{\partial p_{y}}=\frac{\partial
f_{0}}{\partial\Delta_{y}}=\frac{2\Omega}{\kappa^{2}}\frac{\partial
f_{0}}{\partial\bar{x}}$ (41)

to find

${\bf k_{p}}\cdot\frac{\partial f_{0}}{\partial{\bf
p}_{p}}=-\frac{f_{0}}{\sigma_{x}^{2}}\left(k_{x}p_{x}-2\Omega
k_{y}(x-\bar{x})\right).$ (42)

Since this expression is evaluated along an unperturbed orbit, we have

$\frac{\bf k_{p}}{k_{p}}\cdot\frac{\partial f_{0}}{\partial{\bf
p}_{p}}=\frac{f_{0}\beta}{\sigma_{x}^{2}}\left(\left(\alpha+2At^{\prime}\right)\kappa
X\sin{\theta_{r}}+2\Omega X\cos{\theta_{r}}\right).$ (43)

For a plane wave perturbation

$\rho_{1}({\bf x}^{\prime},t^{\prime})=e^{i{\bf k}_{p}^{\prime}\cdot{\bf
x}_{p}^{\prime}}\tilde{\rho}_{1}(z^{\prime},\,t^{\prime})=e^{i{\bf
k}_{p}\cdot{\bf
x}_{p}}e^{i\left(\psi^{\prime}-\psi\right)}\,\tilde{\rho}_{1}(z^{\prime},t^{\prime})$
(44)

and therefore

$\displaystyle\tilde{J}_{p}({\bf x},{\bf p},\,t)$
$\displaystyle=-\frac{iG\Sigma_{0}\Omega}{\left(2\pi\right)^{1/2}\kappa
z_{0}\sigma_{x}^{4}\sigma_{z}}e^{-H_{x}/\sigma_{x}^{2}}F_{z}(H_{z})$
$\displaystyle\times\int^{t}_{t_{i}}dt^{\prime}\,\beta^{\prime}e^{i(\psi^{\prime}-\psi)}P(z^{\prime},t^{\prime})$
$\displaystyle\times\left(\left(\alpha+2At^{\prime}\right)\kappa\,X\sin{\theta_{r}}+2\Omega\,X\cos{\theta_{r}}\right).$
(45)

We recover equation 38 of B20 by setting $\alpha=0$,
$\tilde{\rho}_{1}=\tilde{\Sigma}_{1}\delta(z)$, and
$F_{z}=\sqrt{8\pi}z_{0}\sigma_{z}\delta(z)\delta(p_{z})$ and then integrating
over $z$ and $w$.

A similar calculation leads to the following expression for $J_{z}$:

$\displaystyle\tilde{J}_{z}({\bf x},{\bf p},\,t)$
$\displaystyle=-\frac{G\Sigma_{0}\Omega
N_{z}}{\sqrt{2\pi}\kappa\sigma_{x}^{2}\sigma_{z}^{3}}e^{-H_{x}/\sigma_{x}^{2}}e^{-H_{z}/\sigma_{z}^{2}}$
$\displaystyle\times\int^{t}_{t_{i}}dt^{\prime}\,w^{\prime}e^{i\left(\psi^{\prime}-\psi\right)}Q(z^{\prime},t^{\prime}).$
(46)

###  2.6 vertical phase space DF

We obtain the DF in the $z-w$ plane by integrating $f_{1}$ over ${\bf p}_{p}$.
Following equation 38 we write

$f_{1z}(z,\,p_{z})=e^{i{\bf k}_{p}\cdot{\bf x}_{p}}\left(\tilde{\cal
J}_{p}+\tilde{\cal J}_{z}\right)$ (47)

where $\tilde{\cal J}_{p,z}\equiv\int d^{2}{\bf p}_{p}\tilde{J}_{p,z}$. To
carry out the integral we use the following change of variables from B20.
Since $dp_{y}=d\Delta_{y}=\left(\kappa^{2}/2\Omega\right)d\bar{x}$, we have
$dp_{x}dp_{y}=\left(\kappa/2\Omega\right)dU_{x}^{\prime}dU_{y}^{\prime}$ where

$U_{x}^{\prime}=p_{x}=-\kappa X\sin{\theta_{r}}$ (48)

and

$U_{y}^{\prime}=\kappa\left(x-\bar{x}\right)=\kappa X\cos{\theta_{r}}.$ (49)

In the $(U_{x}^{\prime},\,U_{y}^{\prime})$ system, $\theta_{r}$ is a polar
angle. We can therefore use $(U_{x},\,U_{y})$ coordinates that are obtained
from $(U_{x}^{\prime},\,U_{y}^{\prime})$ by rotating $\theta_{r}$ into
$\theta_{0}$. The result is

$\displaystyle\tilde{\cal J}_{p}(z,w,t)$
$\displaystyle=-\frac{iG\Sigma_{0}F_{z}(H_{z})}{\sqrt{2\pi}z_{0}\sigma_{x}^{4}\sigma_{z}}\int^{t}_{t_{i}}dt^{\prime}\beta^{\prime}P(z^{\prime},t^{\prime})$
$\displaystyle\times\int d^{2}U{\bf c}\cdot{\bf
U}e^{-U^{2}/2\sigma_{x}^{2}+2i{\bf b}\cdot{\bf U}}$ (50)

where the vectors ${\bf b}$ and ${\bf c}$ are defined in B20 and the Appendix.
The integral over ${\bf U}$ can be done analytically to yield

$\displaystyle\tilde{\cal J}_{p}(z,w,t)$
$\displaystyle=\frac{\sqrt{8\pi}G\Sigma_{0}F_{z}(H_{z})}{z_{0}\sigma_{z}}$
(51) $\displaystyle\int^{t}_{t_{i}}dt^{\prime}\,\beta^{\prime}{\bf c}\cdot{\bf
b}e^{-2\sigma_{x}^{2}b^{2}}P(z^{\prime},t^{\prime}).$ (52)

This expression can be written in terms of the Toomre parameter

$Q\equiv\frac{\kappa\sigma_{x}}{3.36G\Sigma_{0}}$ (53)

and the critical wavenumber for axisymmetric perturbations

$k_{\rm crit}\equiv\frac{\kappa^{2}}{2\pi G\Sigma_{0}}.$ (54)

The result is

$\tilde{\cal J}_{p}(z,w,t)=\frac{\kappa
F_{z}(H_{z})}{\sqrt{8\pi}z_{0}\sigma_{z}}\int^{t}_{t_{i}}dt^{\prime}K_{p}(t,t^{\prime})P(z^{\prime},t^{\prime})$
(55)

where

$K_{p}(t,t^{\prime})=4\beta^{\prime}{\bf c}\cdot\hat{\bf
b}\exp(-0.572Q^{2}\hat{b}^{2})$ (56)

and $\hat{\bf b}=(\kappa/k_{\rm crit}){\bf b}$. This equation is an example of
a Volterra integral and extends the well-known result from JT66 into the
dimension perpendicular to the mid plane of the disc. The function $K_{p}$,
which B20 referred to as the JT kernel, is the same as in the case of a razor
thin disc. The difference here is that it is multiplied by the Green’s
function integral, which is also a function of $t^{\prime}$. Thus, the
effective kernel is $K_{p}P$.

A similar calculation leads to

$\tilde{\cal
J}_{z}=-\frac{N_{z}e^{-H_{z}/\sigma_{z}^{2}}}{\sqrt{8\pi}z_{0}^{2}\sigma_{z}}\int^{t}_{t_{i}}dt^{\prime}p_{z}^{\prime}K_{z}(t,t^{\prime})Q(z^{\prime},t^{\prime})$
(57)

where

$K_{z}(t,t^{\prime})=\exp(-0.572Q^{2}\hat{b}^{2}).$ (58)

The integral of this term over $z$ and $w$ is zero and therefore it doesn’t
contribute directly to $\Sigma_{1}$. In short, ${\cal J}_{p}$ describes the
redistribution of mass in the plane of the disc while ${\cal J}_{z}$ describes
the redistribution of mass in the $z-w$ plane.

###  2.7 physical parameters

For definiteness we set physical quantities for our calculations as follows.
We take the angular frequency of the shearing box to be
$\Omega=V_{c}/R_{0}=(230{\rm km\,s}^{-1})/8\,{\rm kpc}\simeq 28.8\,{\rm
km\,s}^{-1}\,{\rm kpc}^{-1}$ and Oort’s first constant to be $A=\Omega/2\simeq
14.4\,{\rm km\,s}^{-1}\,{\rm kpc}^{-1}$. The choice of $A=-B=\Omega/2$
corresponds to a Mestel disc, which has a flat rotation curve. The epicycle
frequency is then $\kappa=\sqrt{2}\Omega\simeq 41\,{\rm km\,s}^{-1}\,{\rm
kpc}^{-1}$. Following B20, we use $\kappa t/\pi$ as a dimensionless time
variable when plotting the time evolution of various quantities. For
reference, $\pi/\kappa\simeq 77\,{\rm Myr}$. We assume a surface density for
the equilibrium system of $\Sigma_{0}=7\times 10^{7}\,M_{\odot}{\rm
kpc}^{-2}$, which gives a volume density in the mid plane of $\rho_{0}\simeq
0.16\left(\sigma_{z}/15\,{\rm km\,s}^{-1}\right)^{-2}M_{\odot}\,{\rm
pc}^{-3}$. With these parameters, the critical wavelength if $\lambda_{\rm
crit}\simeq 8.2\,{\rm kpc}$. These values are roughly consistent with values
for the Solar neighbourhood. Of course, a more realistic model would include a
multi-component disc along with other contributions to the gravitational
potential such as the gas disc and dark halo.

##  3 impulsive excitations

###  3.1 breathing wave excitation

We first consider the response of the disc to a plane-wave impulsive
excitation. We begin by assuming that the excitation is localized in the mid
plane and symmetric in $z$. The density in equations 33 and 36 is then given
by

$\tilde{\rho}_{1}(z,t)=\frac{\Sigma_{e}}{\kappa}\delta\left(t-t_{i}\right)\delta(z)+\tilde{\rho}_{s}(z,t).$
(59)

where $\rho_{s}$ is the density perturbation in the disc itself, i.e., the
self-gravity term. Since this is a single wave with well defined $k_{y}$, we
can set $k_{x}(t=0)=k_{x0}=0$. In the absence of self-gravity, the vertical DF
given by

$\displaystyle\tilde{f}_{1z}(z,w,t)=\Sigma_{e}e^{-k_{p}|z_{i}|}$
$\displaystyle~{}~{}\times\left(K_{p}(t,t_{i})F_{z}(H_{z})-\frac{w_{i}N_{z}}{\kappa
h_{z}}{\rm sgn}(z_{i})K_{z}(t,t_{i})e^{-E_{z}/\sigma_{z}^{2}}\right).$ (60)

Note that $z_{i}=z(t_{i})$ and $w_{i}=w(t_{i})$ are implicit functions of $z$
and $w$.

Figure 1: Amplitude of an impulsively-excited plane wave as a function of time
in the absence of self-gravity. The amplitude has been normalized by the
amplitude of the impluse. Shown are results for discs of various thicknesses
as set by the vertical velocity dispersion where $\sigma_{z}=5-30\,{\rm
km\,s^{-1}}$ in steps of $5\,{\rm km\,s^{-1}}$ for orange, red, purple, cyan,
green, and brown. The black curve is the result for a razor thin disk.

In Fig. 1 we show the evolution of $\Sigma_{1}/\Sigma_{e}$ for the case when
$t_{i}=-1.5\pi/\kappa$, $Q=1.2$, and $k_{y}=k_{\rm crit}/2$. The timescale
between successive peaks corresponds to the period for epicyclic motions,
$2\pi/\kappa$. The figure illustrates the effect the disc’s thickness has on
the effective kernel since, in the absence of self-gravity,
$\Sigma_{1}/\Sigma_{e}$ is proportional to $K_{p}P$. In Fig. 2 we plot the
reduction in the peak value of $\Sigma_{1}$ relative to what the peak value
would be in a razor-thin disc. Toomre (1964) and JT66 suggested that one could
account for finite thickness effects by multiply $K$ by
$(1-\exp{\gamma})/\gamma$ where $\gamma\equiv hk_{y}(1+(2At_{i})^{2})^{1/2}$
and $2h$ is the effective thickness of the disc. This expression can derived
from equations 32 and 33 by taking the disc is a uniform density slab of
thickness $2h$. For the fit in Fig. 2 we use $hk_{y}=0.9z_{0}k_{\rm
crit}/2=0.9\left(0.53Q\sigma_{z}/\sigma_{x}\right)^{2}$ where the numerical
factor of $0.9$ is obtained via chi-by-eye and accounts for the fact the
density in our truncated isothermal model is not uniform.

Figure 2: Reduction in amplitude of a kinematic wave as a function of
$\sigma_{z}$. The blue points give the reduction in amplitude relative the
result for the razor thin disc. The black curve is the phenomenological
formula due to Toomre (1964) and JT66

as discussed in the text.

In Fig. 3 we show the vertical DF at nine different times for the case where
$\sigma_{z}=15\,{\rm km\,s}^{-1}$. The DF winds up with a pitch angle that
increases linearly with time (see below). The time-dependence of the amplitude
of the perturbation is consistent with what we showed in Fig. 1. In
particular, the amplitude reaches local maxima at $\kappa
t/\pi\simeq-1.2,\,1.5,\,3.5$ and local minima at $\kappa t/\pi\simeq 1,3.1$.

As discussed above, the ${\cal J}_{p}$ and ${\cal J}_{z}$ terms in equation 47
involve very different aspects of mass redistribution. The ${\cal J}_{p}$ term
describes mass redistribution in the plane and therefore changes the local
surface density. Conversely, ${\cal J}_{z}$ describes a redistribution of mass
in the $z-w$ plane, but leaves the surface density unchanged. This difference
is illustrated in Fig. 4 where we show separate contributions to the vertical
DF from ${\cal J}_{p}$ and ${\cal J}_{z}$ at two different epochs. For $\kappa
t/\pi=-1$, shortly after the disc has been excited, the contribution from
${\cal J}_{p}$ is proportional to $F_{z}(H_{z})\Phi_{1}$, which is symmetric
in $z$. This contribution changes the shape of the $z-w$ DF while preserving
the $z\to-z$ and $w\to-w$ symmetries. We can therefore view it as a breathing
wave along with mass redistribution in the $x-y$ plane. Over time, the pattern
winds up into a two-armed phase spiral. With ${\cal J}_{z}$, the perturbation
due to the excitation is initially proportional to
$wF_{z}(H_{z})\partial\Phi_{1}/\partial z$. Thus, the amplitude of the
perturbation is maximal along the diagonals of the $z-w$ plane with a sign
that alternates as one circles the origin. This pattern is evident in the
lower middle panel of Fig. 3. The perturbation winds up into a two armed
spiral with alternating positive and negative bands. These points are further
illustrated in the right-most panels where we show the density as a function
of $z$. While both ${\cal J}_{p}$ and ${\cal J}_{z}$ lead to density
perturbations that are symmetric in $z$, the variations with $z$ for the
latter are more prominent due to the geometry of the phase space perturbation.

Figure 3: Vertical ($z-w$) phase space DF when self-gravity is ignored at nine
different epochs for the case where $\sigma_{z}=15{\rm km\,s}^{-1}$. The
numbers in the upper right corners of each panel give the time in units of
$\pi/\kappa$. The numbers in the upper left corners indicate the maximum value
for the phase space density as indicated by the color scale in the sense that
one integrates the map over $z$ and $w$ to obtain the value of
$\Sigma_{1}/\Sigma_{e}$ in Fig.1. Figure 4: Separate contribution to the
vertical DF from the ${\cal J}_{p}$ (left column) and ${\cal J}_{z}$ (right
column). The upper row is for $\kappa t/\pi=1$ while the lower row is for
$\kappa t/\pi=3.4$. The sum of the two contributions yields the total DF as
seen in the corresponding panels of Fig.3. The rightmost column shows the
contributions from ${\cal J}_{p}$ and ${\cal J}_{z}$ to the vertical density
as dotted and dashed lines, respectively. The solid line shows the number
density for the equilibrium model.

In Fig.5 we show the surface density as a function of time for the case when
self-gravity, and hence swing amplification, are included. For a razor thin
disc, the surface density is amplified by a factor of $\sim 35$. The
amplification factor decreases with increasing disc thickness. For example,
when $\sigma=20\,{\rm km\,s}^{-1}$, the peak is only 13% of what it is for the
razor thin case even though the kernel is only reduced by 20%. Nevertheless,
the perturbation is still amplified by a factor of $4.7$.

Figure 5: Surface density a function of time for the case where self-gravity
is included. Physical conditions and line colors as the same as in Fig. 1.

In Fig.6 we plot the vertical DF for the same nine epochs as in Fig.3. The
effects of self-gravity are striking. In the absence of self-gravity, the
contributions from ${\cal J}_{p}$ and ${\cal J}_{z}$ are comparable and so we
have a phase spiral combined with a modulation of the local surface density.
With self-gravity, there is a rapid amplification of the surface density and
compression in the $z-w$ phase space at $t\sim 1.5\pi/\kappa$. This in-plane
compression of the disc sets off new vertical phase spirals so that by $t\sim
4\pi/\kappa$ the spirals are less tightly wound but stronger than in the case
without self gravity.

Figure 6: Vertical DF when self gravity is included. The physical conditions
and epochs are the same as in Fig. 3.

These points are further illustrated in Fig.7 where we show the DF as a
function of $\Omega_{z}$ and $\theta_{z}$. Plots of Gaia data in this space
were made by Li & Widrow (2021); Frankel et al. (2022); Li & Widrow (2023) and
Tremaine et al. (2022). In the absence of self-gravity, the phase spiral is
transformed into parallel, diagonal bands with a slope proportional to the
reciprocal of the age of the spiral. With self-gravity, the bands are steeper
suggesting a younger age. In fact, the slope is related to the time between
the swing-amplification peak and the observation time. In addition, the bands
appear to bend upward in a manner similar to what is seen in Frankel et al.
(2022) and Tremaine et al. (2022).

Figure 7: Vertical phase space DF in the angle-frequency
($\theta_{z}-\nu_{z}$) plane. The left panel shows the total DF for $\kappa
t/\pi=4.1$ (lower-middle panel of Fig. 3 mapped onto the $\theta_{z}-\nu_{z}$
plane. The black line-segment is the expected slope from kinematic phase
mixing. It corresponds to an age of the disturbance of $t-t_{i}=5.6\pi/\kappa$
and lines up with the ridges. The right panel shows the same plot for the case
when self-gravity is included (lower-middle panel of Fig. 6. In this case, the
black line segment corresponds to $t-t_{sw}\simeq 2.2\pi/\kappa$ where
$t_{sw}$ is roughly the time of the swing-amplification peak.

###  3.2 bending wave excitation

Our previous examples focused on breathing waves and two-armed vertical
spirals. The excitation of bending waves and one-armed spirals requires an
external density that is anti-symmetric in $z$. As a toy model for this
process, we take the external mass distribution to be

$\tilde{\rho_{e}}=\frac{\Sigma_{b}}{\kappa\Delta^{2}}\delta(t-t_{i})ze^{-z^{2}/2\Delta^{2}}~{}.$
(61)

Figure 8: Vertical phase space DF when self-gravity is ignored for nine
different epochs. The figure is similar to Fig.3 except that here, the disc is
excited by an anti-symmetric density given by equation 61. The color scale is
centered on zero and its stretch is the same in each panel. Integrating the
absolute value of the DF in each map over $z$ and $w$ yields $0.1\Sigma_{b}$
where $\Sigma_{b}$ is the amplitude of the external excitation in equation 61.

The evolution of the system in the absence of self gravity is shown in Fig. 8.
The initial anti-symmetric perturbation winds up just as one expects. On the
other hand, when self gravity is included (Fig. 9) the DF is amplified during
the early stages of evolution and the rate at which the system phase mixes is
significantly slower. Without self gravity the system undergoes $\sim 3$ phase
wrappings by $t\simeq 5\pi/\kappa$ as compared with just a single phase
wrapping in the self-gravitating case.

The importance of self gravity for an anti-symmetric disturbance may seem
surprising since the total surface density is identically zero. However, one
can think of the external density $\rho_{e}$ as the combination of two
parallel discs, one with positive surface density and the other with negative
surface density. Since the time scale for swing amplification is comparable to
the time scale for mixing between the upper and lower components of the disc,
it is able to amplify these components separately before mixing takes hold.
Only after one or two wrappings is achieved does phase mixing proceed at the
rate expected from pure kinematics.

Figure 9: Vertical DF for a bending perturbation when self-gravity is
included. In this case, the number in the upper left corner of each panel
gives the amplification factor relative to the case where self-gravity is
ignored (Fig. 8).

##  4 Excitation of the disc by a cloud

Next, we consider the disc’s response to a massive cloud on a circular orbit,
that is, a cloud at rest in the shearing box. Following JT66 and B20, we
decompose the density of the cloud into Fourier modes and compute the response
of the disc from each mode using the formalism developed in the previous
section. We assume that the perturbing mass is a Gaussian in ${\bf x}$,

$\rho_{e}({\bf x})=\frac{M}{\left(2\pi\right)^{3/2}\Delta^{3}}e^{-|{\bf
x}|^{2}/2\Delta^{2}}~{},$ (62)

where $\Delta$ and $M$ are the size and mass of the cloud, respectively. The
${\bf x}_{p}$ Fourier transform is then

$\tilde{\rho}_{e}({\bf
k}_{p},\,z)=\frac{M}{\sqrt{2\pi}\Delta}e^{-\Delta^{2}|{\bf
k}_{p}|^{2}/2}e^{-z^{2}/2\Delta^{2}}~{}.$ (63)

As discussed in JT66 and B20, each Fourier mode evolves according to the
equation developed in the previous section. We can therefore replace
$\tilde{\rho}_{1}(\xi,\,t^{\prime})$ in equations 33 and 36 with
$\tilde{\rho_{e}}+\tilde{\mu}_{s}\left({\bf k}_{p};\zeta,t^{\prime}\right)$.
Note that while $\tilde{\mu}_{s}$ is treated like $\tilde{\rho}_{s}$ in our
plane-wave calculations, here it has dimensions of mass per unit length. The
volume density is calculated via an inverse Fourier transform, that is, an
integral over $k_{x}$ and $k_{y}$. Likewise, an inverse Fourier transform is
required to get $f_{1z}$ as a function of $x$ and $y$.

Figure 10: Surface density from a massive cloud that is on a circular orbit at
${\bf x}=0$. The contours, in units $M/{\rm kpc^{2}}$, are as follows: solid
black – $0.2,0.4,0.6$; dashed black – $0.001,0.01$; solid red – $-0.2$; dashed
red – $-0.01,0.001$.

In Fig.10 we show the surface density perturbation generated by the cloud for
the case when $\Delta=0.05\lambda_{\rm crit}$ and $Q=1.2$. The features of the
response have been discussed at length in JT66, Fuchs (2001), and B20. The
ridge that runs from the lower left to the upper right arises from swing-
amplified trailing waves that originated as leading waves and roughly follows
the line $x=y/2At_{sw}$ where $t_{sw}\simeq\pi/\kappa$ is the time of the
primary swing-amplification peak. The structure is a stationary disturbance in
the disc though individual stars are continuously passing through it.

Figure 11: Thumbnail images of the fractional residuals of the $z-w$ DF across
the $x-y$ plane. The positions of the images correspond to the blue stars in
Fig. 10. Images for the lower half of Fig.10 are obtained by letting ${\bf
x}_{p}\to-{\bf x}_{p}$. The fractional residual is found by dividing $f_{1z}$
by the equilibrium DF. The numbers in the upper right corners of each panel
indicate the maximum residual as a percentage under the assumption that
$M=10^{9}\,M_{\odot}$.

In Fig.11 we show the vertical DF at the 15 positions across the disc
indicated by blue stars in Fig.10. These DFs are calculated on a $200\times
200$ grid in the $z-w$ plane and then smoothed using the Scipy routine
ndimage.gaussian_filter with sigma$=8$. We then divide the DFs by the
equilibrium DF to obtain a fractional residual map. The numbers in the upper
right corners of each panel indicate the residual as a percentage assuming
$M=10^{9}\,M_{\odot}$. Evidently, the pattern of perturbations in the $z-w$
plane is strongly dependent on the position within the mid plane. The phase
spirals are most prominent between one and two times $\lambda_{\rm crit}$ from
the perturbing mass and close to the wake produced by swing amplification. We
might have anticipated this from Fig. 6 where we found that the spirals arose
soon after peaks formed in surface density. Elsewhere in the $x-y$ the
perturbation takes the form of a breathing wave.

##  5 Discussion

Toomre (1981) described swing amplification as a conspiracy between shear,
epicyclic motion or shaking, and self-gravity. The formalism presented in
Section 2 allows us to study the actions of an additional conspirator: phase
mixing in the dimension perpendicular to the mid plane. Phase mixing regulates
swing amplification since the effects of self-gravity are diminished once the
system has undergone one or two phase wrappings in the $z-w$ plane. Our
formalism also allows us to explicitly account for the reduction in self-
gravity due to the finite thickness of the disc. We are therefore able to
demonstrate the validity of the phenomenological formula from Toomre (1964)
and JT66.

The key takeaways from Section 3 and 4 are that swing amplification can
enhance phase spirals by amplifying a disturbance in the $z-w$ plane before
phase mixing takes hold. Moreover, stationary phase spirals can form in the
wake of a co-rotating mass. In the usual picture of phase mixing, individual
stars follow the ridges and troughs of the vertical DF as it winds up. Here,
individual stars pass through the spiral in the same way that stars pass in
and out of the spiral arms of a disc galaxy. Taken together, these results
call into question the simple picture of the Gaia phase spirals where their
shapes are determined by kinematic phase mixing.

There are obvious improvements and extensions that we can make to the
calculations presented in this paper. First, external components to the
background potential can be included to account for a thin gas disc and
extended dark halo. Doing so will change the vertical potential and hence rate
at which disturbances in the $z-w$ plane wind up while reducing the effect of
self-gravity. (See, figure 2 of Tremaine et al. (2022)). Second, we might
consider more realistic scenarios for exciting the disc such as a passing
satellite galaxy or dark matter sub halo, as in Section 6 of B20. One expects
that the response of the disc to a passing satellite will be intermediate to
the response from an impulsive excitation and a stationary cloud. In addition,
one might add intermediate-scale masses to stir up the system as a means of
testing the diffusion hypothesis in Tremaine et al. (2022).

It is worth commenting on the computational complexity of our calculations. In
the shearing sheet, the computational complexity for the response to a single
wave of definite $k_{y}$ is ${\cal O}(N_{t}^{2})$ where $N_{t}$, the number of
time steps in the Volterra integral, is ${\cal O}(10^{2})$ for the
calculations presented in this paper. The computational complexity for the
response to a general, time-dependent excitation will then be
$N_{k}^{2}N_{t}^{2}$ where $N_{k}$ is the number of points for each dimension
of the ${\bf x}_{p}-{\bf k}_{p}$ Fourier transform. However, for a stationary
perturbation, such as the co-moving cloud, one can use the time-invariance on
the solution to replace the $k_{x}$ integral with an integral over $t$. The
computational complexity is therefore reduced to $N_{k}N_{t}^{2}$ (B20). In
our calculations, the complexity is increased by a factor of $N_{z}^{2}N_{w}$
where $N_{z}$ and $N_{w}$ and the number of grid points in the $z$ and $w$
directions, respectively. Calculations for the co-rotating cloud are therefore
$N_{z}^{2}N_{w}N_{k}N_{t}^{2}$ and can be done in one to several hours on an 8
processor 3.2 GHz machine. The computation time for a general time-dependent
perturbation will be on the order of one to a few days, long but not
prohibitive. The computation might be improved by using angle action variables
in place of $z$ and $w$ as in Banik et al. (2022). Alternatively, one might
resort to N-body simulations in the shearing box (Fuchs et al., 2005).

Of course, in the end, the shearing box is not a perfect substitute for a
rotating disc. The critical wavelength is $2-3$ times the exponential scale
length of the disc. This means that the surface density of a realistic disc
varies considerably on scales considered in this paper and the separability
assumption for the potential is surely suspect.

##  6 Conclusions

In this work, we presented a formalism to study the response of a small local
patch of a stellar disc to an external perturbation within the framework of
the shearing box approximation. It extended the shearing sheet formalism of
JT66 in the dimension perpendicular to the disc and allowed us to examine what
happens within the disc as it responds to an external perturbation.

In general any disturbance in the vertical structure of a disc leads to $z-w$
spirals as the disturbance undergoes phase mixing. The main result of this
paper is that self-gravity can amplify disturbances in the disc before phase
mixing takes hold. This amplification is strongest for leading waves as they
swing into trailing ones. Perhaps unexpectedly, the process also works with
bending waves. Finally, it is possible to set up stationary phase spirals in
the wake of a co-rotating mass.

A complete understanding of the Gaia phase spirals is still lacking.
Investigations that range from test particles in one dimension to fully self-
consistent N-body simulations have failed to reproduce the structures found in
the data in all their complexity. The shearing box calculations presented in
this work provide an intermediate approach since they include self-gravity and
an approximate form of epicyclic motion and differential rotation or shear.
Thus, despite their obvious limitations, they allow one to explore the effects
of self-gravity and the interplay between in-plane and vertical dynamics.

## acknowledgements

We acknowledge the financial support of the Natural Sciences and Engineering
Research Council of Canada.

##  Appendix A Kernel definitions

With the change of variables described in Section 2.6 we have $H_{x}=U^{2}$.
Furthermore, we can use standard trigonometric identities to write

$\psi(t^{\prime})-\psi(t)=2{\bf b}\cdot{\bf U}$ (64)

where

$b_{x}(t,t^{\prime})\equiv\frac{k_{x}}{\kappa}\left(A(t^{\prime}S^{\prime}-tS)+\frac{\alpha}{2}\left(S^{\prime}-S\right)+(\Omega/\kappa)(C^{\prime}-C)\right)$
(65)

and

$b_{y}(t,t^{\prime})\equiv\frac{k_{y}}{\kappa}\left(A(t^{\prime}C^{\prime}-tC)+\frac{\alpha}{2}\left(C^{\prime}-C\right)-(\Omega/\kappa)(S^{\prime}-S)\right).$
(66)

Likewise, we have

$\left(\alpha+2At^{\prime}\right)\kappa X\sin{\theta_{r}}+2\Omega
X\cos{\theta_{r}}=2{\bf c}\cdot{\bf U}$ (67)

where

$c_{x}^{\prime}\equiv-\left(\frac{\alpha}{2}+At^{\prime}\right)C^{\prime}+\frac{\Omega}{\kappa}S^{\prime}$
(68)

and

$c_{y}^{\prime}\equiv\left(\frac{\alpha}{2}+At^{\prime}\right)S^{\prime}+\frac{\Omega}{\kappa}C^{\prime}~{}.$
(69)

## Data Availability

The data underlying this article were generated by numerical calculations
using original Python code written by the author. The code incorporated
routines from NumPy (Harris et al., 2020) and SciPy (Virtanen et al., 2020).
The data for the figures and the code will be shared on reasonable request to
the author.

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# Context-VQA: Towards Context-Aware and Purposeful Visual Question Answering

Nandita Naik
Stanford University
<EMAIL_ADDRESS>Christopher Potts
Stanford University
<EMAIL_ADDRESS>Elisa Kreiss
Stanford University
<EMAIL_ADDRESS>
###### Abstract

Visual question answering (VQA) has the potential to make the Internet more
accessible in an interactive way, allowing people who cannot see images to ask
questions about them. However, multiple studies have shown that people who are
blind or have low-vision prefer image explanations that incorporate the
context in which an image appears, yet current VQA datasets focus on images in
isolation. We argue that VQA models will not fully succeed at meeting people’s
needs unless they take context into account. To further motivate and analyze
the distinction between different contexts, we introduce Context-VQA, a VQA
dataset that pairs images with contexts, specifically types of websites (e.g.,
a shopping website). We find that the types of questions vary systematically
across contexts. For example, images presented in a travel context garner 2
times more “Where?” questions, and images on social media and news garner 2.8
and 1.8 times more “Who?” questions than the average. We also find that
context effects are especially important when participants can’t see the
image. These results demonstrate that context affects the types of questions
asked and that VQA models should be context-sensitive to better meet people’s
needs, especially in accessibility settings.

##  1 Introduction

Figure 1: We propose the dataset Context-VQA, where images are paired with
various contexts. This example shows how both questions and answers vary
across different contexts. On a social media website, the details about people
within the image are more relevant, while in the context of a shopping
website, people will want to know more specific details about the suitcase.

Images are omnipresent on the Web, appearing in various contexts—for example,
news websites, shopping websites, or social media [6, 16]. This poses an issue
for nonvisual accessibility: How will people who cannot see the image (e.g.,
due to image loading issues or a visual impairment) understand the content
that is presented? In this case, a textual interpretation of the image is
needed. While much recent work has focused on leveraging AI to generate image
descriptions [4, 11], being able to inquire about specific details gives the
user agency over obtaining information that is specifically suited to their
needs (which is, for example, leveraged in interactive accessibility studies
[14]). Visual Question Answering (VQA) models are a promising tool for
providing image-based information on the fly. However, most VQA efforts are
focused on evaluating machine “understanding,” an abstract task which seeks to
test the spatial or object reasoning of machine learning models. The resulting
systems aren’t easily extendable to an accessibility application [2]. The
VizWiz dataset is an exception, where image–question pairs are obtained from
the actual use case of blind and low-vision people asking questions about
information in the visual world they’re taking pictures of [7].

In contrast to traditional machine understanding VQA datasets, we propose
Context-VQA, a dataset which recenters the focus on the use case of image
accessibility and the changing information needs for images occurring in
various sources online. Context-VQA complements VizWiz but is distinct from
it, since the images are part of the general public discourse and not directly
conditioned on a question.

Current VQA tasks assume a one-size-fits-all approach, where each image is
associated with a fixed set of questions. However, recent work suggests that
different pieces of information become relevant depending on the particular
context in which people encountered the image [17, 18]. Consider the question
and answer provided in Figure 1. It is difficult to infer the context from the
image itself, yet the meaning of the image changes depending on the context.
Our Context-VQA dataset presents images in distinct contexts to elicit more
naturalistic question distributions.

The purpose of the Context-VQA dataset is to align previous VQA efforts with
an explicit accessibility goal, and to situate questions and answers within
diverse Web contexts to more closely reflect user needs. Each entry consists
of a question, image, answer, and context, where the question and answer are
both conditioned upon the context. We find that context shapes which questions
become informative and that this contextual sensitivity is even more
pronounced when participants are prompted to ask questions based on a
description of the image and not the ground-truth image itself. This suggests
that specifically in cases of image inaccessibility, the context plays a major
role for VQA tasks.

Our contributions are:

  1. 1.

A context-sensitive VQA dataset, with questions and answers that were written
for images situated in specific Web contexts111Dataset and code will be
released upon publication.. To account for variation in question-writing
strategies, half of the participants were shown the image and half were shown
an AI-generated description of the image as a basis for their questions. The
dataset therefore allows not only an investigation of context-sensitivity, but
also of task formulation. While the image-visible version of the study might
allow for more grounding, the description-only version imitates the user
experience more closely.

  2. 2.

An assessment of how Web context shapes which pieces of information in an
image are of interest. We find that the distribution of question types varies
across contexts in an intuitive manner. These findings underscore that the
context affects the meaning of an image, and motivate context-aware,
purposeful VQA models.

##  2 Related Work

#### Context in Image Descriptions

Making image content accessible through textual descriptions and explanations
requires selecting pieces of visual information that appear most relevant.
This has been a focus of investigation specifically for the usefulness of
image descriptions. Traditionally, generating image descriptions takes a one-
size-fits-all approach, where a single description is used across many
different contexts. However, there is a growing consensus that people’s
information needs for the same image changes depending on the context where
the image appears [18, 17, 14, 10]. For example, while people prefer details
about the color and attributes of the clothing on shopping websites,
information about the relationships between people depicted in the image
become more relevant in the context of social media. Furthermore, users also
want image descriptions to clarify why the image appears on a site [17]. In an
interactive image accessibility study between a BLV and a sighted user, this
context dependence was also observable from the specific questions the BLV
participant asked about the images [14], suggesting that context needs to play
a role not just for image descriptions but also VQA systems.

#### Visual Question Answering Datasets

VQA tasks probe vision–language models for their alignment between the two
domains. VQA-v2 [5] emphasizes open-ended, free-form questions. Visual7W [19]
requires the model to ground its answer in specific regions of the image.
CLEVR [9] and GQA [8] test spatial reasoning abilities. OK-VQA [13] requires
outside knowledge to answer questions. These prior datasets are focused on
evaluating machine “understanding,” an abstract task which tests spatial and
logical reasoning [2]. In contrast, Context-VQA seeks to prioritize the type
of questions that are most relevant to people based on contextually situated
images.

The Visual Dialog [3] dataset established the task of “visual chatbots,”
models that can hold a conversation about an image and answer follow-up
questions. Similarly, the questions in Context-VQA were generated from giving
study participants image descriptions and asking them to write follow-up
questions. Where Context-VQA diverges from Visual Dialog, however, is the
incorporation of context into the dataset, which moves away from a one-size-
fits-all approach and towards reflecting the variety of questions asked for
contextually situated images.

VizWiz [7], similarly to Context-VQA, is also designed to reflect image
accessibility needs but for a distinct goal. All the images and the questions
in VizWiz come from blind people who had a question about something in their
environment, took a picture, and uploaded it to the VizWiz app along with
their question, where crowdsourced human workers then provided an answer [7].
For instance, someone might take an image of the error message on their
rebooting computer screen or a clothing item of which they want to know the
color. In VizWiz, the images are taken to explicitly answer a single question.
However, the images that need to be made accessible online are often intended
to meet multiple information needs in varied contexts. Thus, Context-VQA
addresses a separate image accessibility purpose than VizWiz and complements
that effort.

##  3 Dataset Construction

The goal for the Context-VQA dataset is to allow for a direct comparison of
how questions for images change based on image context. To do so, we collected
naturalistic images from six different types of websites, which were selected
based on prior work [18]. In a norming study, participants were presented with
an image and, for each of the six contexts, rated how likely it was that the
image appeared in the given context. We used this study to select plausible
image-context pairs that were further annotated with questions (Experiment 1)
and answers (Experiment 2). All human-subject experiments were run on the
crowdsourcing platform Prolific under an IRB protocol.

Figure 2: Distribution of question types across contexts for the description-
only study (top) and the image+description study (bottom), where error bars
show variance. The distinction between contexts is more pronounced in the
description-only study, while in the image+description study, the question
type proportions are more similar across contexts. Particularly in the
description-only study, Social Media and News have more _who_ questions,
Travel has more _where_ questions, and Shopping has more _what_ questions.

###  3.1 Materials

The contexts needed to be specific enough to be informative, yet broad enough
to account for a range of images. Drawing inspiration from [18], we selected
six contexts: Shopping, Travel, Social Media, Health, News, and Science
Magazines. These contexts also overlapped (e.g., an image of clothes could
appear in Social Media and Shopping).

The images were sampled by hand, prioritizing high-quality digital images from
diverse websites. Images were sourced in the Shopping category from clothes
and dorm shopping websites; images in the Travel category from travel blogs
and hotel websites; images in the Health category from health and fitness
blogs; images on Social Media from Twitter, Pinterest, and Instagram; images
in the Science category from Popular Mechanics and Popular Science; and images
in the News category from the New York Times and San Francisco Chronicle. We
selected seven images from each source context, resulting in 42 images. Based
on the image-context norming study results, we selected 35 images for further
annotation, which were rated to plausibly appear in up to 3 distinct contexts.

###  3.2 Experiment 1: Question Generation

Our first study aimed to generate questions conditioned on the image and
context.

Task There were two versions of the question generation task: description-only
and image+description. In the description-only task, participants were given a
context and an image description generated by BLIP-2 [12]. In the
image+description version, participants were additionally shown the image
itself. As an annotation quality check, we first asked them to write a
justification for why the image might appear in the given context—low-effort
or incomprehensible justifications resulted in data exclusion. They were then
asked to suggest two follow-up questions that a person who only had access to
the image description would ask. Prior work has noted that having the
questioner not see the image could help remove visual priming bias–for
example, questioners will often write, “Is there a dog in this image?” only if
there is a dog in the image [3]. The description-only condition also mimics
the real-life condition, where people wouldn’t see the image directly and ask
follow-up questions based on a short description.

We ran six trials. Each participant saw six different, randomly selected
image–context pairs, and we ensured that no participant saw the same image or
context repeated twice.

Participants We recruited 55 participants over Prolific. They took an average
of 8.5 minutes to complete the trials and were paid $13.52 per hour on
average.

Post-processing We collected 1,320 questions across both studies. We excluded
participants who expressed in the post-questionnaire that they might have done
the study incorrectly, or whose written justifications for why the image
appeared in the context was less than 30 characters, resulting in 1,096
questions. We also manually filtered questions that were inappropriate or not
questions, leaving 1,032 questions. For each of the 65 unique image-context
pairs, we selected two questions at random from each study for further answer
annotation.

###  3.3 Experiment 2: Answer Generation

For the questions generated in Experiment 1, we collected answer annotations.

Task Each participant saw six unique image–context pairs, associated with a
question, which were randomly sampled from our dataset of questions. They were
asked to justify the occurrence of the image in the context and answer the
question based on the image. They also had an option to indicate if the
question was unanswerable.

Participants We recruited 100 participants over Prolific. Participants took an
average of 8.5 minutes to complete and were paid an average of $13.57 per
hour.

Post-processing We collected 1,260 answers for 202 questions. We excluded
participants who expressed in the post-questionnaire that they might have done
the study incorrectly, or whose written justifications for why the image
appeared in the context was less than 30 characters. After exclusions, the
total number of answers collected was 568, averaging 3 question-answers per
image-context pair, 8 of which were rated unanswerable by at least 3 people.

##  4 Dataset Analysis

Based on prior research, we hypothesized that the context images are presented
in shapes the questions participants will ask. To test this hypothesis, we
investigated whether participants chose different question types dependent on
the context. For analysis, we selected the question types of _what_ , _who_ ,
_where_ , _why_ , _is_ , _how_ , and _when_ , where _is_ questions represent
all questions requiring only binary yes/no responses (i.e., _are_ , _does_ ,
_was_ , and _is_). We prompted GPT-4 [15] to label all questions with their
question type, and manually verified the labels. _When_ questions only
occurred once, so we omitted them in the analysis.

Figure 2 shows the question type distribution across contexts for all
questions collected from the description-only study (top) and the
image+description study (bottom). For both study conditions, _what_ and binary
response questions (_is_) occur most frequently. We observe clear effects of
context, especially in the description-only study. For instance, questions
asked for images that appear in the shopping context are more likely to be
_what_ questions ($p<0.001$) and less likely to be _where_ questions
($p<0.01$) compared to other contexts.222All statistical tests are generalized
linear model analyses. Questions for images appearing on social media,
however, are the least likely to be _what_ questions ($p<0.001$) but are the
most likely to be _who_ questions ($p<0.001$). These results can help provide
insight into where a model must be especially robust, and they highlight the
importance of systems that can adjust to contextually changing user needs.

While some context effects are reflected in the image+description study (e.g.,
in the _where_ and _how_ distributions), they are overall less pronounced than
in the description-only study. We hypothesize this is because the questioners
have less information in the description-only study to inform their questions.
These results suggest that context-sensitivity in VQA needs to play an
especially important role when the image itself isn’t visible to the end user,
and is therefore particularly relevant for the image accessibility setting.
Image+description VQA study setups might provide misleading results, since the
presence of the image dampens the context effect.

Prior work suggests that when questioners don’t see the image, they ask longer
and more open-ended questions, which necessitates more detailed answers [3].
The average response length for our answers, across question types and study
conditions, ended up being 11.03 words. This shows that the answers in our
dataset are meaningful and not restricted to binary responses. This is an
uncommonly long average response length for VQA datasets–for a point of
comparison, Visual Dialog [3] reported an average answer length of 2.9 words,
which was still higher than VQA [1] (1.1 words), and Visual 7W [19] (2.0
words). We hypothesize the main contributors to answer length are expressions
of uncertainty or additional details.

##  5 Conclusion

We present the Context-VQA dataset and use it to show how context changes the
distribution of question types for VQA models. Unlike previous VQA datasets,
Context-VQA is context-sensitive and reflects the distribution of question
types that people will ask in different contexts. We also show initial
evidence that the questions vary across contexts, especially when the image
can’t be seen.

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