# Multiwavelength Vertical Structure in the AU Mic Debris Disk: Characterizing
the Collisional Cascade

David Vizgan11affiliation: Astronomy Department and Van Vleck Observatory, Wesleyan University, 96 Foss Hill Drive, Middletown, CT 06459, USA 22affiliation: Department of Astronomy, University of Illinois at Urbana-Champaign, 1002 West Green St., Urbana, IL 61801, USA 33affiliation: The Cosmic Dawn Center 44affiliation: National Space Institute, DTU Space, Technical University of Denmark, 2800 Kgs. Lyngby, Denmark A. Meredith Hughes11affiliation: Astronomy Department and Van Vleck Observatory, Wesleyan University, 96 Foss Hill Drive, Middletown, CT 06459, USA Evan S. Carter11affiliation: Astronomy Department and Van Vleck Observatory, Wesleyan University, 96 Foss Hill Drive, Middletown, CT 06459, USA Kevin M. Flaherty55affiliation: Department of Astronomy and Department of Physics, Williams College, Williamstown, MA 01267, USA Margaret Pan66affiliation: Center for Astrophysics | Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA Eugene Chiang77affiliation: Department of Astronomy, University of California at Berkeley, Campbell Hall, Berkeley, CA 94720-3411, USA 88affiliation: Department of Earth and Planetary Science, University of California at Berkeley, McCone Hall, Berkeley, CA 94720-3411, USA Hilke Schlichting99affiliation: University of California, Los Angeles, Los Angeles, CA, USA David J. Wilner66affiliation: Center for Astrophysics | Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA Sean M. Andrews66affiliation: Center for Astrophysics | Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA John M. Carpenter1010affiliation: Joint ALMA Observatory, Alonso de Córdova 3107 Vitacura, Santiago, Chile Attila Moór1111affiliation: Konkoly Observatory, Research Centre for Astronomy and Earth Sciences, Eötvös Loránd Research Network (ELKH), Konkoly Thege Miklós út 15-17., H-1121 Budapest, Hungary Meredith A. MacGregor1212affiliation: Department of Astrophysical and Planetary Sciences, University of Colorado, 2000 Colorado Avenue, Boulder, CO 80309, USA

###### Abstract

Debris disks are scaled-up analogs of the Kuiper Belt in which dust is
generated by collisions between planetesimals. In the “collisional cascade”
model of debris disks, dust lost to radiation pressure and winds is constantly
replenished by grinding collisions between planetesimals. The model assumes
that collisions are destructive and involve large velocities; this assumption
has not been tested beyond our Solar System. We present 0$\farcs$25
($\approx$2.4 au) resolution observations of the $\lambda$ = 450 $\mu$m dust
continuum emission from the debris disk around the nearby M dwarf AU
Microscopii with the Atacama Large Millimeter/submillimeter Array. We use
parametric models to describe the disk structure, and an MCMC algorithm to
explore the posterior distributions of the model parameters; we fit the
structure of the disk to both our data and archival $\lambda=1.3$ mm data
(Daley et al., 2019), from which we obtain two aspect ratio measurements at
1.3 mm ($h_{1300}$ = 0.025${}^{+0.008}_{-0.002}$) and at 450 $\mu$m ($h_{450}$
= 0.019${}^{+0.006}_{-0.001}$), as well as the grain size distribution index
$q=$ 3.03 $\pm$ 0.02. Contextualizing our aspect ratio measurements within the
modeling framework laid out in Pan & Schlichting (2012), we derive a power law
index of velocity dispersion as a function of grain size $p=0.28\pm 0.06$ for
the AU Mic debris disk. This result implies that smaller bodies are more
easily disrupted than larger bodies by collisions, which is inconsistent with
the strength regime usually assumed for such small bodies. Possible
explanations for this discrepancy are discussed.

###### Subject headings:

debris disks, stars: individual, AU Mic; submillimeter, planetary systems

##  1\. Introduction

Debris disks form from dust produced by collisions between planetesimals
during the late stages of the planet formation process (Hughes et al., 2018,
and references therein). They are detected around 15-20% of main sequence
stars, although since current sensitivity limits are at least an order of
magnitude above what would be required to detect our Solar System’s Kuiper
Belt, this incidence should be considered a lower limit (Matthews et al.,
2014; Montesinos et al., 2016; Sibthorpe et al., 2018). Debris disks are often
described as “left over” from planet formation, but this description downplays
the dynamic, evolving nature of the bodies within the disk. While stellar
processes, such as radiation pressure and stellar winds, remove dust and other
small particles from a debris disk (Wyatt & Dent, 2002; Chen et al., 2006),
collisional processes continually replenish the dust (Wyatt, 2008; Matthews et
al., 2014; Hughes et al., 2018). The combination of processes is known as a
“collisional cascade” (Dohnanyi, 1969), where the objects in the disks, from
the larger asteroids and comets to the smaller dust grains, undergo inelastic
collisions. These interactions fragment larger planetesimals, transfer mass to
smaller planetesimals, and replenish the dust which is then removed by
radiation pressure and stellar winds.

The theoretical description of the collisional cascade is parameterized by two
power laws: the velocity dispersion $v$ as a function of grain size $a$, which
has power law index $p$, and the grain size distribution $\frac{dN}{da}$ as a
function of grain size, which has power law index $q$:

$v(a)\propto a^{p}$ (1) $\frac{dN}{da}\propto a^{-q}$ (2)

The grain size distribution index $q$ has been shown to be remarkably
insensitive to parameters which describe the fragmentation process e.g. size
and number of the largest fragments, size distribution of fragments (Williams
& Wetherill, 1994) and found to be constant in the case of self-similar
fragmentation outcomes (Tanaka et al., 1996).

The parameters $p$ and $q$ are described within the context of a “gravity” and
“strength” regime, referring to whether gravitational forces or internal
strengths of grains determine the outcome of collisions between bodies (Benz &
Asphaug, 1999); typically, small objects are bound by material strength, while
larger planetesimals are bound by self-gravity (Löhne et al., 2008).

This parametric framework has been applied to the Kuiper Belt to describe its
collisional evolution (Bernstein et al., 2004; Pan & Sari, 2005) and more
generally to the collisional evolution of debris disks (Löhne et al., 2008;
Wyatt, 2008). Building upon this work, Pan & Schlichting (2012) investigated
the grain-size-dependent velocity dispersion within the collisional cascade
framework and found self-consistent steady state solutions for $p$ and $q$
within both the strength and gravity regimes. They pointed out that observing
the scale heights of debris disks could constrain the mass and number of large
bodies stirring the cascade as well as the internal strengths of colliding
bodies.

The vertical structure of a debris disk encodes information about the
dynamical state of the collisional cascade. The scale height $H(r)$ is
typically the standard deviation of the Gaussian profile of dust density as a
function of height above the midplane. It is defined at a given radius, and is
generally assumed to increase linearly with distance $r$ from the star, with a
constant “aspect ratio” $h$, such that $h=H(r)/r$. However, the observed scale
height varies as a function of wavelength, and factors other than the
dynamical state of the collisional cascade can influence the measured scale
height, particularly at short (optical and infrared, or OIR) wavelengths.

Thébault (2009) showed that all debris disks have a natural minimum scale
height at OIR wavelengths due to radiation pressure, stellar winds, and mutual
grain collisions (which can also be altered by the gas content; see Olofsson
et al., 2022). Measurements of the scale height at OIR wavelengths therefore
cannot be used to probe the dynamical conditions within a disk. However,
millimeter-wavelength observations trace larger grains that are resistant to
the effects of radiation pressure and stellar winds (for review see Hughes et
al., 2018, and references therein), which means that measurements of the scale
height at millimeter wavelengths can be used to measure the dynamical
excitation of the dust grains.

Converting a scale height measurement into a velocity dispersion involves
assuming an equipartition between eccentricity and inclination induced by
dynamical interactions between dust grains and planetesimals in the disk, so
that orbital inclination $i$ is $\sim$ $\sqrt{2}h$ for a sufficiently small
$h$ (Wyatt & Dent, 2002). This aspect ratio, when measured at a particular
wavelength, is directly proportional to the velocity dispersion for grains
with sizes comparable to the wavelength of observation (see §4.2), which is,
in turn, proportional to $a^{p}$. Thus, scale height measurements of different
grain size populations (i.e. at different wavelengths) within a debris disk
can describe the velocity dispersion of those bodies across a range of grain
sizes, allowing us to measure the velocity dispersion power law index $p$.

Within the millimeter-wavelength regime, the scale height can still vary with
wavelength, since different wavelengths of light can probe different dust
grain sizes. This feature results from a balance between the grain size
distribution, which tends to tilt heavily towards smaller grains that dominate
the surface area, and the falloff of grain emission efficiency with wavelength
for wavelengths longer than the size of the dust grain. Taken together, these
two trends result in the rule of thumb that the smallest grain that can emit
efficiently will dominate the observation, and that the smallest grain that
can emit efficiently is one that is comparable in size to the wavelength of
light. With observations of a disk at multiple wavelengths, therefore, we can
derive values for $p$ and $q$ and ultimately parameterize the collisional
cascade of a disk via interpretation within the framework described in Pan &
Schlichting (2012).

The power-law index for grain size distribution, $q$, has been inferred via
modeling for many debris disks beyond our own Solar System (e.g. Löhne et al.,
2012; Ricci et al., 2015; MacGregor et al., 2016; Marshall et al., 2017; White
et al., 2018; Hengst et al., 2020; Arnold et al., 2021; Norfolk et al., 2021);
nearly all of these systems are consistent with the theoretical steady-state
grain size distribution index $q=3.5$ derived by Dohnanyi (1969). Generally, a
velocity dispersion of 1 km/s is prescribed to all bodies within the Kuiper
Belt (Leinhardt et al., 2008), which suggests $p=0$ locally (as assumed by
Dohnanyi, 1969); it is important to note that non-negligible differences in
collisional velocities have been measured in various dynamical populations of
Kuiper Belt objects (e.g. Abedin et al., 2021). Nonetheless, measurements of
$p$ have never been made outside of the Solar System.

AU Microscopii, also known in the literature as GJ 803 and HD 197481, is a
prime target for the first measurement of $p$ via multiple resolved scale
height measurements at widely separated millimeter wavelengths. It is an M3IVe
star (Daley et al., 2019) located $9.725\pm 0.005$ pc away from our Solar
System (Gaia Collaboration et al., 2018) that hosts one of the biggest,
brightest, and most edge-on debris disks in the night sky. It is a member of
the $\beta$ Pictoris Moving Group and has an age of 22 $\pm$ 3 million years
(Mamajek & Bell, 2014). AU Mic has a mass of 0.50 $\pm$ 0.03 M⊙ (Plavchan et
al., 2020), a luminosity of 0.09 L⊙, and an effective temperature of 3700
$\pm$ 100 K (Plavchan et al., 2009). AU Mic is also home to two Hot Neptunes
at separations of 0.0645 $\pm$ 0.0013 and 0.1101 $\pm$ 0.0022 au from the star
(Plavchan et al., 2020; Martioli et al., 2020), both of which may be able to
induce radio emission in the star’s corona (Kavanagh et al., 2021). There is
also some preliminary evidence for a possible third planet orbiting between AU
Mic b and c (Wittrock et al., 2022).

AU Mic has been observed across the electromagnetic spectrum, from scattered
light (Liu, 2004; Krist et al., 2005; Augereau & Beust, 2006; Graham et al.,
2007; Schneider et al., 2014; Schüppler et al., 2015; Grady et al., 2020), to
the infrared (e.g. Moshir & et al., 1990; Liu et al., 2004; Matthews et al.,
2015), to millimeter wavelengths (Wilner et al., 2012; MacGregor et al., 2013,
2016; Daley et al., 2019). The star’s violent flaring activity has even been
detected in the ultraviolet (Robinson et al., 2001; Redfield et al., 2002;
Roberge et al., 2005) and in x-ray (Pye et al., 2015) as well as in the
millimeter (Daley et al., 2019; MacGregor et al., 2020). Boccaletti et al.
(2015) detected fast-moving features in scattered light within the debris
disk. These features, sustained over several years (Boccaletti et al., 2018),
were attributed by Chiang & Fung (2017) to dust clouds produced by “dust
avalanches” blown out by the star’s stellar wind, and attributed by Sezestre
et al. (2017) to sequential dust releases from an unseen parent body orbiting
AU Mic. Recent modelling work of AU Mic’s stellar wind predicts violent space
weather and extreme coronal mass ejections from the star (Alvarado-Gómez et
al., 2022); the star’s recently measured magnetic field (Kochukhov & Reiners,
2020; Klein et al., 2021) might also be responsible for the turbulent and
violent coronal ejections (e.g. Kavanagh et al., 2021) from the star.

The AU Microscopii debris disk is almost perfectly edge-on, with an
inclination very close to 90$\degree$ (Daley et al., 2019, and references
therein). Because the AU Mic debris disk is already resolved vertically at one
wavelength, resolving the vertical structure at another wavelength (and
therefore another grain size) allows us to parameterize the disk’s collisional
cascade via calculation of the power law indices $p$ and $q$. MacGregor et al.
(2016) measured an upper limit on the grain-size distribution power law index
$q<3.31$ for the disk using 9 mm observations from the Very Large Array. Daley
et al. (2019) used ALMA observations at 1.3 mm to resolve the vertical
structure of the disk. Their work measured a disk aspect ratio of
$h=0.031^{+0.005}_{-0.004}$ at 1.3 mm and used it to infer an upper limit of
1.8 M⊕ on the total mass of perturbing bodies between radii of $\sim$20-40 au.

This work presents new 450 $\mu$m ALMA observations of the AU Mic debris disk,
at 0$\farcs$25 ($\approx$2.4 au) resolution – the same as the 1.3 mm data from
Daley et al. (2019). We model the debris disk structure, combining
measurements of the scale height and the flux of the disk at both 1.3 mm and
450 $\mu$m to calculate $p$ and $q$ for the AU Mic disk. The observations of
the AU Mic debris disk are presented in Section 2, along with the data
reduction and post-processing methods. The main results of the debris disk
observations are presented in Section 3. Our analysis of the visibilities and
measurement of the 1.3 mm / 450 $\mu$m scale height ratio is presented in
Section 4, and the results of this work are discussed in Section 5. This work
concludes with a brief summary of the main results in Section 6.

##  2\. Observations

###  2.1. 450 $\mu$m disk observations

AU Mic was observed with the Atacama Large Millimeter/submillimeter Array
(ALMA) in the Atacama Desert, Chile, as part of project 2016.1.00878.S (PI: A.
M. Hughes). The system was observed in five executions of one scheduling
block; once on 2017 April 22, twice on 2017 April 23, and twice on 2017 April
26. The weather during observations was excellent, with precipitable water
vapor $<$ 1 mm for all execution blocks. This information, along with beam
sizes and calibration sources, is presented in Table 1. The observations
utilized ALMA’s Band 9 ($\lambda=450$ $\mu$m) receivers and 45 12m antennas.
The correlator was configured to maximum continuum sensitivity by employing
four spectral windows each with a bandwidth of 1.875 GHz. Each window was
divided into 128 channels, each with a width of 15.6 MHz (or $\sim$ 21 km/s).

The primary beam of ALMA’s 12 m antennas is modeled as a 2D Gaussian with a
FWHM (full-width half-maximum) equal to $1.13\lambda/D$ (as described in the
ALMA Technical Handbook for Cycle 7) with a peak response normalized to unity.
Because the width of the primary beam at $\lambda=450\ \mu$m is comparable to
the width of the disk, the observations were conducted in a three-pointing
mosaic, with position offsets of (+3$\farcs$0, -2$\farcs$3) and (-3$\farcs$0,
+2$\farcs$3) in right ascension and declination, centered on the position of
AU Mic and oriented along the previously measured position angle of
128.4$\degree$ (MacGregor et al., 2013).

The four spectral windows were tuned to central frequencies of 676, 678, 680,
and 682 GHz. The baseline lengths span between 12 m and 455 m; the 455 m
baseline traces an angular scale of 0.2” and a spatial scale of 2 au.
Calibration and imaging of the visibilities were conducted via the CASA
software package (McMullin et al., 2007). Standard ALMA reduction scripts were
applied to the data; gain calibration was carried out via a combination of
water vapor radiometry and on-sky gain calibrators, listed in Table 1. Flux
calibration was carried out using quasars with fluxes bootstrapped from the
ALMA calibrator list. The ALMA technical handbook for Cycle 7 quotes a 20%
systematic flux calibration uncertatinty at a wavelength of 450 $\mu$m. The
statistical weights of the visibilities were recalculated and applied to the
data by using the variance around each baseline (see Flaherty et al., 2017).
For computational efficiency, we average the data into spectral bandwidths of
1.875 GHz and within a time interval of 30 seconds.

Parameters | 2017 Apr 22 | 2017 Apr 23 | 2017 Apr 23 | 2017 Apr 26 | 2017 Apr 26
---|---|---|---|---|---
Precipitable water vapor (mm) | 0.36 | 0.58 | 0.65 | 0.32 | 0.35
On-source time (min) | 100 | 90 | 90 | 94 | 75
Flux calibrator | J1924-2914 | J1924-2914 | J1733-1304 | J1924-2914 | J1924-2914
Bandpass calibrator | J1924-2914 | J1924-2914 | J2232-1143 | J1924-2914 | J2232-1143
Gain calibrator | J1957-3845 | J1957-3845 | J1957-3845 | J1957-3845 | J1957-3845

Table 1Observational parameters for the five execution blocks. Forty-five 12 m
antennas, spanning baselines of 12 to 455 m, were used for the observations.

The CASA task tclean was used to image the data using standard Fourier
transform methods. A bright submillimeter source — most likely a galaxy — was
detected to the East of the AU Mic debris disk (we note that the offset of the
galaxy from the star is so large that it is not visible within the boundaries
of Figure 1). In order to subtract this galaxy from the visibilities we used a
mix of CASA and the Python package galario (Tazzari et al., 2018). We first
used the CASA task tclean to create a naturally weighted image. We used the
elliptical Gaussian fitting tool in the CASA task viewer to fit a 2-D Gaussian
at the position of the galaxy. The fit yielded a major axis FWHM = 448 mas, a
minor axis FWHM = 365 mas, an integrated flux of 1.59 mJy, a position angle of
175.7$\degree$, as well as a position of $\alpha$ = 20h 45m 10.28s, $\delta$ =
-31$\degree$ 20’ 33.5”. We then used galario to subtract this 2-D Gaussian
from each of the twelve visibility files.

Because the data is a three pointing mosaic, we multiplied each part of our
mosaicked model galaxy by a 2D Gaussian with a FWHM $=$ 9$\farcs$29 at its
respective pointing. This primary beam correction was necessary in order to
replicate the effects of the primary beam in the ALMA data, which has a
different effect on the galaxy for each of three pointings.

###  2.2. Stellar variability

As discussed in the previous section, AU Mic is a young M dwarf, and
consequently a highly active star, with a history of flaring at mm wavelengths
(MacGregor et al., 2013; Daley et al., 2019). As such, it is important to know
whether for these new 450 $\mu$m data, ALMA observed the star as it flared.
Carter (2018), who previously used this same $\lambda=450\ \mu$m data to model
the disk around AU Mic, used the CASA task gaussfit to fit a Gaussian to the
central peak in the image domain for the first and second half of each
execution block, finding that the stellar flux varied from a minimum of
$1.0\pm 0.3$ mJy in the first block to a maximum of $3.0\pm 0.4$ mJy in the
third block, but did not deviate by more than 3$\sigma$ from the weighted mean
of the 10 values over the course of the 5-execution observation. The flaring
activity of AU Mic can occur over short (i.e. between 1 second and 1 minute)
timescales (MacGregor et al., 2020); still, these short stellar flares should
not affect the disk modelling substantially due to the high angular resolution
of the data.

##  3\. Results

The 450 $\mu$m emission from the disk around AU Mic is shown in Figure 1. The
RMS noise in the image is 240 $\mu$Jy beam -1. By fitting a polygon to the
2$\sigma$ contours of the flux via the CASA task viewer, we measure an
integrated flux density of $15.3\pm 1.1$ mJy. Fluxes at the position of the
star and within the ansae of the disk are also measured using the fitting tool
in the CASA task viewer. The peak intensity at the position of the star is
$1.6\pm 0.2$ mJy beam-1, the northeast ansa of the disk has a peak intensity
of $1.6\pm 0.2$ mJy beam-1, and the southwest ansa of the disk has a peak
intensity of $1.4\pm 0.2$ mJy beam-1. While MacGregor et al. (2013) found
marginally significant evidence for asymmetry in flux at the ansae of the AU
Mic debris disks system, the limb asymmetry at 450 $\mu$m is well within the
uncertainties, and therefore can be considered to be insignificant. This is in
line with Daley et al. (2019), which also did not find significant flux
asymmetry between the ansae of the disk.

Figure 1.— A naturally weighted image of the 450 $\mu$m ALMA observations of
the AU Microscopii debris disk. Contours are [-2, 2, 4, 6, 8]$\times\sigma$,
where $\sigma=$ the RMS noise $=$ 240 $\mu$Jy beam -1. The synthesized beam
dimensions, represented by the gray hatched ellipse in the lower left corner,
are 0.33” x 0.25” with a position angle of 73.7°.

Before fitting any models to the disk, we can begin to characterize the
vertical structure of the AU Mic disk. Figure 2 plots several metrics related
to the disk structure. We first define a disk midplane from the position of
the star at a position angle of 128.48$\degree$ (see §4). We then fit
Gaussians perpendicular to the disk midplane, as a function of offset along
the disk midplane. The top plot shows the spine flux density, which is the
peak flux density of the Gaussian at each offset along the major axis of the
disk. The middle plot shows the elevation, or distance of the disk’s spine
from the disk midplane. The bottom plot shows the beam-deconvolved full-width
at half-maximum (FWHM) of each Gaussian offset along the major axis. The
concept of these graphs is taken from Boccaletti et al. (2015), who plotted
both elevation from the midplane of the disk and intensity of the disk as a
function of projected separation from the star in order to quantify fast-
moving features within the AU Mic debris disk.

Like Daley et al. (2019), we do not find any features at the location of the
fast-moving features detected by Boccaletti et al. (2015). Furthermore, the
bulk of the disk mass appears to reside within a single plane, and there do
not appear to be any significant warps or deviations from this midplane. The
disk limbs show no significant asymmetries at this wavelength. Most
importantly, the vertical structure of the disk is resolved. The beam-
subtracted FWHM is smaller at 450 $\mu$m than at 1.3 mm (see Daley et al.,
2019), which suggests that the scale height at 450 $\mu$m might be lower than
at 1.3 mm. In Section 4 we simultaneously model the high-resolution 1.3 mm
data (from ALMA project 2012.1.00198.S, PI: A. M. Hughes) and 450 $\mu$m data,
including the ratio of the scale height at the two frequencies.

Figure 2.— From top to bottom, as a function of projected separation from the
star: (a) spine flux density, (b) elevation from the midplane of the disk, and
(c) the beam-deconvolved disk FWHM. This projection confirms that we detect no
significant deviations from radial and vertical symmetry. The top plot shows
that, as expected, the brightness of the disk increases as a function of
distance from the central star. The middle two plots confirm the results of
Daley et al. (2019), in that neither warps nor fast-moving features are
detected in this data. Most importantly, the bottom plot shows that the disk
is marginally resolved at $\lambda=450\ \mu$m, with a smaller scale height
than measured by Daley et al. (2019) at $\lambda=1.3$ mm. Missing information
in the bottom plot indicates that the disk is unresolved at those separations
from the star.

##  4\. Analysis

###  4.1. Modeling formalism

To derive a best-fit model and explore the parameter space for posterior
probabilities, we employed the Python package emcee (Foreman-Mackey et al.,
2013) which implements the affine-invariant MCMC (Monte Carlo Markov Chain)
algorithm described in Goodman & Weare (2010). We used a Gaussian log-
likelihood function ($\ln$ $\zeta=$ $-\chi^{2}/2$) to compare a given model
disk with the data. The model debris disks were created using the modeling and
raytracing code described in Flaherty et al. (2015). Here we briefly recap the
functional form and assumptions of the model.

The dust temperature is derived by setting the flux received by the disk from
its host star equal to the flux emitted by the disk and assuming zero-albedo
dust grains, such that the system is in blackbody equilibrium:

$T_{\rm dust}(r)=\left(\frac{L_{\star}}{16\pi\sigma
r^{2}}\right)^{\frac{1}{4}}$ (3)

where $L_{\star}$ is the bolometric luminosity of the system’s host star,
$\sigma$ is the Stefan-Boltzmann constant, and $r$ is distance from the star.
The radial surface density across the disk between the inner radius of the
disk ($R_{\rm in}$) and the outer radius ($R_{\rm out}$) is described by a
power law:

$\Sigma(r)=\Sigma_{c}r^{\gamma}$ (4)

where

$\Sigma_{c}=\frac{M_{\rm disk}(\gamma+2)}{2\pi(R_{\rm out}^{\gamma+2}-R_{\rm
in}^{\gamma+2})}$ (5)

$\gamma$ is the power law index which describes the change in surface density
with radius, and $\Sigma_{c}$ is the surface density normalization constant.

The vertical dust density distribution is modelled as a Gaussian function
whose standard deviation characterizes the vertical structure of the disk.
Recall that the aspect ratio of the disk $h$ is defined as

$h=\frac{H(r)}{r}$ (6)

where $H(r)$ is the scale height. Combining the previous equation with our
equation for the scale height of the disk yields the dust volume density as a
function of distance from the star $r$ and height above the disk midplane $z$:

$\rho_{D}(r,z)=\frac{\Sigma(r)\
\exp\left[-\frac{z}{H(r)}\right]^{2}}{\sqrt{\pi}H(r)}$ (7)

We also model the dust opacity as a power law (Beckwith et al., 1990):

$\kappa_{\nu}=10\left(\frac{\nu}{\rm{10^{12}}\ Hz}\right)^{\beta}\ \rm{cm^{2}\
g^{-1}}$ (8)

and by combining the last two equations we can describe a dust absorption
coefficient $K_{\nu}$ as

$K_{\nu}=\kappa_{\nu}\rho_{D};$ (9)

integrating $K_{\nu}$ along the line of sight from each point in the disk to
Earth yields an expression for the optical depth of the disk $\tau_{\nu}$. We
note here that the dust opacity is not trivial for this project; please see §5
for further discussion.

Our final run consisted of 50 walkers and 4000 steps. We followed up our run
with an autocorrelation analysis, estimating an autocorrelation length of 200
steps. Though the autocorrelation time was still rising by the end of each
chain, after 10 autocorrelation lengths all parameters had levelled off such
that the fractional error in the mean was no more than a few percent, and the
standard error of the mean was stable; we thus removed the first 2000 steps
from our run.

Parameters | Description | Median | Best-fit | Priors
---|---|---|---|---
$\log_{10}M_{\rm disk}$ [M⊙] | Disk dust mass | -8.09${}^{+0.02}_{-0.02}$ | -8.10 | (-10, 3)
$\gamma$ | Surface density index | 2.2${}^{+0.2}_{-0.7}$ | 2.1 | (-5, 5)
$R_{\rm in}$ [au] | Disk inner radius | 22.1${}^{+1.0}_{-0.4}$ | 21.8 | (0, 1000)
$R_{\rm out,\ 1300}$ [au] | Outer radius @ 1.3 mm | 42.0${}^{+0.7}_{-0.2}$ | 42.0 | (0, 1000)
$R_{\rm out,\ 450}$ [au] | Outer radius @ 450 $\mu$m | 38.3${}^{+0.8}_{-0.3}$ | 38.0 | (0, 1000)
$i$ [$\degree$] | Disk inclination | 88.51${}^{+0.44}_{-0.05}$ | 88.51 | (0, 90)
$PA$ [$\degree$] | Disk position angle | 128.48${}^{+0.05}_{-0.02}$ | 128.48 | (0, 180)
$f_{1300,\ \rm Mar}$ [$\mu$Jy] | Central flux density @ 1.3 mm on March 2017 | 376${}^{+7}_{-6}$ | 375 | (0, 1 Jy)
$f_{1300,\ \rm Aug}$ [$\mu$Jy] | Central flux density @ 1.3 mm on August 2017 | 171${}^{+20}_{-25}$ | 132 | (0, 1 Jy)
$f_{1300,\ \rm Jun}$ [$\mu$Jy] | Central flux density @ 1.3 mm, June 2017 | 222${}^{+14}_{-11}$ | 221 | (0, 1 Jy)
$f_{450}$ [mJy] | Central flux density @ 450 $\mu$m | 1.25${}^{+0.08}_{-0.11}$ | 1.27 | (0, 1 Jy)
$h_{1300}$ | Aspect ratio @ 1.3 mm | 0.025${}^{+0.008}_{-0.002}$ | 0.023 | (0, 5)
$h_{1300}/h_{450}$ | Scale height ratio @ 1.3 mm and 450 $\mu$m | 1.35${}^{+0.09}_{-0.09}$ | 1.34 | (0, 4)
$\beta$ | Dust opacity index | 0.06${}^{+0.03}_{-0.03}$ | 0.06 | (-5, 5)

Table 2Results of the MCMC model fitting to both the 450 $\mu$m (this work)
and 1.3 mm (Daley et al., 2019) data of AU Microscopii. We note that ”central
flux density“ refers to the flux density of the central point source (i.e. the
star) in the dataset.

We initialized our MCMC run with fourteen free parameters: dust mass of the
disk ($M_{\rm disk}$), inner radius ($R_{\rm in}$), disk width at both 1.3 mm
and 450 $\mu$m ($R_{\rm{out},1300}$ and $R_{\rm{out},450}$), inclination
($i$), position angle (PA), surface density power law exponent ($\gamma$),
flux density of the central point source for the 450 $\mu$m data ($f_{450}$)
and for each of the observing sessions at 1.3 mm ($f_{1300,\ \rm{Mar}}$,
$f_{1300,\ \rm{Aug}}$, $f_{1300,\ \rm{Jun}}$), the aspect ratio of the disk at
1.3 mm ($h_{1300}$), the ratio of scale heights at 1.3 mm and 450 $\mu$m
($h_{1300}/h_{450})$, and the dust opacity power law index ($\beta$).

We initially fit the 450 $\mu$m data alone. However, we found that the MCMC
chains diverged into two families of solutions with different inclinations.
One family had a scale height that matched the results at 1.3 mm, but a
significantly lower inclination, which was not commensurate with what we know
about the viewing geometry. The other family of solutions, which included the
global best-fit, had a comparable inclination to the 1.3 mm results, but a
lower scale height. We realized that fitting the two wavelengths
simultaneously was the best way to study the relative scale heights while
constraining the inclination to be the same at both wavelengths. The viewing
geometry and dust properties within the AU Mic debris disk ($M_{\rm disk}$,
$PA$, $i$, $\beta$) are therefore modeled to be identical at both 1.3 mm and
450 $\mu$m, while the radial and vertical structure ($h_{1300}$, $h_{450}$,
$R_{\rm{out},1300}$ $R_{\rm{out},450}$) are allowed to vary at each
wavelength.

Figure 3.— A visualization of the degeneracy between the AU Mic debris disk’s
aspect ratio at Band 6 (top histogram) and inclination (right histogram). As
the modeled inclination increases, so does the aspect ratio of the disk. This
plot was made using the Python package corner (Foreman-Mackey, 2016).

All of the disk parameters were sampled linearly except for $M_{\rm disk}$,
where we sampled $\log$ $M_{\rm disk}$ instead. We also bounded the parameters
with priors; for instance, we place upper bounds on the angular parameters $i$
and $PA$ of $90\degree$ and $180\degree$ respectively. We note that the
lopsided posteriors in Figure 6 for both $i$ and $PA$ towards the upper bounds
are not reflections of the prior boundary and are physical; please see §4.1 in
Daley et al. (2019) for further discussion of this point, including an
investigation showing that relaxing the boundary on the prior and allowing it
to reflect across the boundary does not change the posterior distribution for
these parameters.

Other parameters were initialized at non-zero values, such as the aspect ratio
of the disk. The best-fit parameters are obtained from the model disk with the
lowest corresponding $\chi^{2}$ value. The model corresponding to the best-fit
parameters is compared with the data in Figures 4 and 5, while the best-fit
parameters for the AU Mic debris disk are presented in Table 2.

Figure 4.— From left to right: (1) 450 $\mu$m ALMA observations of the AU Mic
system. RMS noise is 240 $\mu$Jy beam -1. The restoring beam has the same
dimensions as that of Figure 1 and is represented by the hatched ellipse in
the bottom left corner. The contours represent integer multiples of 2$\sigma$.
(2) Full-resolution image of the modelled debris disk, with the central flux
set to zero. (3) Sky-projected image of the best-fit model disk after sampling
with the same baseline lengths and orientations as the ALMA observations. The
aspect ratio of the model disk is $h=0.019^{+0.006}_{-0.001}$. (4) The
residual generated by subtracting the model disk from the data in the
visibility domain. No significant residuals exist along the major axis of the
debris disk, indicating good agreement between data and model. Figure 5.— From
left to right: (1) 1.3 mm ALMA observations of the AU Mic system, from Daley
et al. (2019). RMS noise is 15 $\mu$Jy beam -1, and the contours represent
integer multiples of 3$\sigma$. (2) Full-resolution image of the modelled
debris disk, with the central flux set to zero. (3) Sky-projected image of the
best-fit model disk after sampling with the same baseline lengths and
orientations as the ALMA observations. Compared to the model shown in Figure
4, this one has an aspect ratio $h=0.025^{+0.008}_{-0.002}$ and a different
central flux due to stellar variability during observations (see §3). Since
both models share the same inclination, it is evident that the scale height at
this wavelength must be slightly larger than at 450 $\mu$m. (4) The residual
generated by subtracting the model disk from the data in the visibility
domain. No significant residuals exist along the major axis of the debris
disk.

A primary beam correction was applied to the model disk before using galario
to compare the model disk to the data visibilities. Because the 450 $\mu$m
data are mosaicked, three primary beam corrections are applied to the model,
each corresponding to the respective location of the mosaic field. The 1.3 mm
data are not mosaicked and therefore only require a single primary beam
correction applied to the phase center for each execution block. After
creating the model disk image, we used galario to convert it into model
visibilities sampled at the same baseline lengths and orientations as the
data, which were then subtracted from the data visibilities and Fourier
transformed to yield residual images. We found during the modeling process
that the results were somewhat sensitive to the assumed size of the primary
beam, due to the contrast in source vs. primary beam size at the two
wavelengths. Our investigation revealed that the primary beam size would need
to deviate by $\sim$10-15% from the value given in the ALMA technical handbook
to significantly alter the results; deviations at the level of 3% from the
nominal FWHM did not produce any significant differences in the posterior
distributions for the outer radius or scale height ratio. We were not able to
determine an uncertainty on the primary beam size either from the technical
handbook or from a helpdesk ticket.

The data, best-fit model, and residual at $\lambda=450$ $\mu$m are shown in
Figure 4, and those at $\lambda=1.3$ mm are shown in Figure 5. At both
wavelengths, the best-fit model subtracts cleanly from the data in the
visibility domain such that no significant residuals exist in the image (and
in fact, the noise appears lower in the residual images than in the data
images due to imaging artifacts caused by the CLEAN algorithm, which does not
tend to work well on long, narrow flux distributions). From the data, we
retrieve a scale height ratio $h_{1300}/h_{450}=1.35^{+0.09}_{-0.09}$, and and
an aspect ratio of $h_{1300}=0.025^{+0.008}_{-0.002}$ at 1.3 mm; from the
posterior distributions of these two parameters, we thus retrieve an aspect
ratio of $h_{450}=0.019^{+0.006}_{-0.001}$ for the data at 450 $\mu$m.

We remark here that for an edge-on ($i=90\degree$) disk, there exists a
natural degeneracy between inclination and scale height. To create a model
that best resembles the data, the algorithm navigates a fine line between
increasing the scale height (i.e. puffiness) of the disk and decreasing the
inclination. However, increasing the scale height will thicken the disk at the
ansae, while increasing the inclination mostly thickens the disk along the
minor axis. This behavior has been described in the literature (e.g. Graham et
al., 2007; Daley et al., 2019), but is mitigated somewhat by combining the two
bands and requiring that they have the same inclination. The Python package
corner (Foreman-Mackey, 2016) is used to visualize degeneracies within the
parameter space. Like Daley et al. (2019), we find degeneracy between the
scale height and inclination of the disk; this degeneracy is presented in
Figure 3. Nevertheless, a scale height ratio of 1 is excluded at the
$3.9\sigma$ level in the posterior distribution.

Figure 6.— Kernel density estimates of the posterior probability distributions
for the MCMC modeling run. The central line represents the median, and the
outer dashed lines represent 16th and 84th percentiles.

###  4.2. The collisional cascade indices

The measurements made in the previous two sections allow us to characterize
the collisional cascade of the AU Mic debris disk within the framework
originally presented by Dohnanyi (1969) and modified by Pan & Schlichting
(2012) to incorporate grain-size dependent velocity dispersions. The
interparticle relative velocity (Wyatt & Dent, 2002) is proportional to the
aspect ratio $h$, i.e.

$\langle v_{\rm rel}\rangle\approx v_{\rm
Kep}(r)\sqrt{\bar{i}^{2}+1.25\bar{e}^{2}}\approx 2.355\sqrt{3}v_{\rm Kep}(r)h$
(10)

Furthermore, the grain size is proportional to wavelength of observation via
Equation 1. The most efficient emission occurs when $\lambda=2\pi a$ (see §
5.1.5 for a discussion of this assumption). We therefore can solve for $p$ by
setting Equation 10 equal to Equation 1 and taking the proportion of aspect
ratios at two separate wavelengths. This yields the relationship

$\frac{h}{h_{0}}=\left(\frac{\lambda}{\lambda_{0}}\right)^{p}$ (11)

where the aspect ratio measurement $h$ corresponds to wavelength $\lambda$,
while the aspect ratio measurement $h_{0}$ corresponds to wavelength
$\lambda_{0}$. The exponent $p$ is then the power law index that describes the
velocity dispersion as a function of grain size. Now that the aspect ratios at
both wavelengths have been modelled in the previous section, we assume that
the grain size is equal to the wavelength of observation and calculate $p$ via
the equation

$\frac{h_{1300}}{h_{450}}=1.35^{+0.09}_{-0.09}=\left(\frac{1300\ \rm{\mu
m}}{450\ \rm{\mu m}}\right)^{p}.$ (12)

Rearranging this equation to solve for the velocity dispersion index yields a
value for $p=0.28^{+0.06}_{-0.06}$. To derive $q$ for the disk, we rearrange
Equation 13 from Draine (2006),

$\beta=(q-3)\beta_{s}$ (13)

to get

$q=3+\frac{\beta}{\beta_{s}}.$ (14)

where $\beta_{s}$ is the absorption cross-section power-law index. Using our
measured value of $\beta=$ 0.06 $\pm$ 0.03 and assuming $\beta_{s}=$ 1.8 $\pm$
0.2 (Draine, 2006), we determine a value of $q=3.03\pm 0.02$. Because this
measurement is taken from forward modeling, and not the image domain (where
the line of sight for an edge-on disk passes through many pixels with
different radii and therefore temperatures), this value is more reliable
because it accounts for local deviations in temperature and therefore in
$\alpha_{\rm PL}$. The power law indices calculated for the AU Mic debris
disk, $p=0.28\pm 0.06$ and $q=3.03\pm 0.02$, are the first model-dependent
constraints placed on the internal strengths of colliding bodies in a disk
beyond the Kuiper Belt.

##  5\. Discussion

###  5.1. Comparison of AU Mic disk structure with values from the literature

In this section we compare our results to other measurements and models of the
AU Mic disk.

####  5.1.1 Viewing geometry

When fitting the disk structure to both the 1.3 mm and 450 $\mu$m data, we
assumed a common viewing geometry for both wavelengths. The measured position
angle of 128.48${}^{+0.05}_{-0.02}$ degrees is in strong agreement with recent
measurements of the disk’s position angle; for example, Cycle 0 observations
of the disk with ALMA (MacGregor et al., 2013) found a disk position angle of
128.41 $\pm$ 0.13$\degree$, and Daley et al. (2019) found a position angle of
128.50 $\pm$ 0.07$\degree$ which places our measurements well within 1$\sigma$
of their work.

The inclination of the disk is less trivial, with values in the literature
reaching as high as 89.4$\degree$ (Krist et al., 2005) or assumed to be
89.5$\degree$ (MacGregor et al., 2013; Schüppler et al., 2015). Our
measurement of the inclination is 88.51${}^{+0.44}_{-0.05}$ degrees is in
strong agreement with the inclination measurement of Daley et al. (2019),
$i=88.5^{+0.3}_{-0.2}$. Since the 450 $\mu$m data are less vertically extended
than the 1.3 mm data, they provide a more stringent constraint on the disk
inclination. Assuming a common geometry thus restricts the 1.3 mm data to a
region of parameter space with higher $i$ and larger $h$ than in Daley et al.
(2019).

####  5.1.2 Radial structure

The inner radius of the disk at 450 $\mu$m is 22.1${}^{+1.0}_{-0.4}$ au. There
is tentative evidence in the literature for the presence of an inner ring of
debris (e.g. Matthews et al., 2015); Daley et al. (2019) modelled as being at
a radius of 11.9${}^{+1.7}_{-1.8}$ au. If we consider the AU Mic system to be
analogous to the Kuiper Belt, this inner ring of debris could be considered
analogous to our Solar System’s asteroid belt which lies between the orbits of
Mars and Jupiter. We do not include a ring in our fit to the data, as given
the lower signal-to-noise ratio (SNR) in the 450 $\mu$m data compared to the
1.3 mm data, we would not expect to detect it. The inner radius is also much
farther than the two planets which orbit AU Mic at semi-major axes of
$a=0.0645\pm 0.0013$ au (Plavchan et al., 2020) and $1.1101\pm 0.0022$ au
(Martioli et al., 2020) respectively. This could imply the presence of another
planet in the system in order to account for the location of the inner disk,
and which could potentially stir the collisional content of the disk.

Recent work by Pearce et al. (2022) found, after adopting an inner disk radius
for the AU Mic system of 28.7 $\pm$ 0.3 au, that a planet with a mass of 0.44
$\pm$ 0.03 MJup and semi-major axis of 21.9 $\pm$ 0.3 au would be able to
“sculpt” the inner part of the AU Mic debris disk. Since our model yields a
different inner radius than in Pearce et al. (2022), we recalculated this
planet mass and radius using their publicly available
code111https://github.com/TimDPearce/SculptingPlanet, finding that a planet
with a minimum mass of 0.34 $\pm$ 0.03 $M_{\rm Jup}$ and maximum semimajor
axis of 17.0${}^{+0.7}_{-0.3}$ au would be able to truncate the inner part of
the disk that is modeled in this work.

It is not trivial to assume a common radial geometry, and we ultimately allow
the model to have a different outer radius for each data set. We do not vary
the inner radius with wavelength since our signal-to-noise in the 450 $\mu$m
data is lower than 1.3 mm and therefore cannot provide insight into the low-
SNR features that tentatively indicated an inner ring in that data set. We
define the outer radius ratio to be analogous to our scale height ratio; we
find that $\Delta R_{1300}/\Delta R_{450}=1.23^{+0.02}_{-0.03}$; in other
words, the outer radius of AU Mic’s debris disk is over 20% larger at 1.3 mm
than at 450 $\mu$m.

The outer radius at 1.3 mm for our model is measured to be
42.0${}^{+0.7}_{-0.2}$ au, which deviates slightly above 1$\sigma$ from the
best-fit of 41.5${}^{+0.4}_{-0.5}$ au measured by Daley et al. (2019) and by
just under 3$\sigma$ from the outer radius of $40.3\pm 0.4$ au measured by
MacGregor et al. (2013). Our results at 1.3 mm for the outer radius differ
from previous measurements because we are able to constrain our model with the
450 $\mu$m data set, partially assuming a common geometry at both wavelengths.
The outer radius at 450 $\mu$m, on the other hand, is measured to be
38.3${}^{+0.8}_{-0.3}$ au, which is significantly lower than at 1.3 mm. We did
not attempt to constrain the shape of the outer edge at either wavelength,
since the lack of residuals for the best-fit model is consistent with an
abrupt cutoff.

Naively one would expect that the smaller grains would exist farther away from
the star than larger grains. Several works (e.g., Schneider et al., 2014;
Ahmic et al., 2009; Wilner et al., 2011, 2012) describe ways in which the
outer radius of a debris disk can differ between millimeter and OIR
wavelengths, primarily due to the “halo” of small grains blown out by stellar
winds and radiation pressure. We see the opposite however; the larger grains
appear to lie farther from the star than the smaller grains.

Another possibility to explain the difference in radii at the two wavelengths
is gas drag, similar to a modified version of the effect discussed in Powell
et al. (2017). However, the direction of the gas drag effect should predict a
smaller outer radius at longer wavelengths, which is the opposite of the
direction we observe. On the other hand, stellar winds moderated by giant
stellar outbursts as described in Chiang & Fung (2017) could induce ram
pressure that would make collisionally created submillimeter grains somewhat
more eccentric than collisionally created millimeter grains at birth, which
would predict a difference in radii in the direction that we see.

####  5.1.3 Vertical structure

The disk appears thinner at 450 $\mu$m than at 1.3 mm in the image domain, and
this result is confirmed via modeling in the visibility domain. Since the disk
at 1.3 mm is 35% puffier than at 450 $\mu$m, these results imply that the 450
$\mu$m-sized grains in the disk have a smaller velocity dispersion than the
1.3 mm-sized grains, and indicates that there is likely some damping of
velocities within the disk.

An aspect ratio at $\lambda=450$ $\mu$m of 0.019${}^{+0.006}_{-0.001}$ at a
radius of 40 au implies a scale height at 40 au of 0.76${}^{+0.24}_{-0.04}$
au. The scale height represents the standard deviation of a Gaussian
distribution of dust particles; it is thus possible, alternatively, to
describe the vertical structure of the disk by measuring the full-width at
half maximum (FWHM) of this distribution. The relationship between the FWHM
and the standard deviation $\sigma$ is ${\rm FWHM}=2\sqrt{2\ln 2}\sigma$, so
that the FWHM of the disk at 450 $\mu$m is 1.79${}^{+0.56}_{-0.09}$ au. The
scale height at 40 au for the 1.3 mm dust emission is 1.00${}^{+0.32}_{-0.08}$
au, and the FWHM is 2.35${}^{+0.75}_{-0.19}$ au. Because our modelled scale
height at 1.3 mm is slightly lower than what was modelled by Daley et al.
(2019), our FWHM is also slightly lower.

It is important to note that when examining how the half-width at half-maximum
(HWHM) of the disk changes as a function of wavelength, discrepancies between
modelled FWHM and apparent FWHM appear across the literature for AU Mic
observations due to differences in assumed or fitted inclination, and so for
this comparison only models of the AU Mic debris disk will be discussed. There
seems to be some evidence (see Table 3) that the FWHM, and hence the scale
height, decreases as a function of wavelength in the OIR regime. The main
outlier is Schüppler et al. (2015), who modelled the dust production and dust
dynamics from collisions in the AU Mic debris disk, and attempted to fit the
resulting collisional models to many different SED points from scattered light
(Fitzgerald et al., 2007; Schneider et al., 2014) to $\lambda=$ 1.3 mm
(MacGregor et al., 2013); this work was not intended for fitting images, and
the approximate FWHM given in Table 3 results from an approximation relating
their reported eccentricities to inclinations in equipartition, which may be
considered an overinterpretation. Alternatively, it is possible that scale
height (and ultimately the velocity dispersion of grains within the disk) may
fluctuate rather than monotonically increase/decrease as a function of
wavelength. More observations of the disk at other wavelengths are necessary
to determine whether there is a global trend of disk scale height with
wavelength.

Model reference |  $\lambda$ ($\mu$m) |  $i$ ($\degree$) | FWHM (au)
---|---|---|---
Daley et al. (2019) | 1350 | 88.5 | 2.9${}^{+0.5}_{-0.4}$
This work (2022) | 1350 | 88.51 | 2.35${}^{+0.75}_{-0.19}$
This work (2022) | 450 | 88.5 | 1.79${}^{+0.56}_{-0.09}$
Schüppler et al. (2015) | $>100$ | 89.5 | 0.4
Schüppler et al. (2015) | $<70$ | 89.5 | 4
Metchev et al. (2005) | 0.647 - 1.63 | 89 | 1.6
Krist et al. (2005) | 0.430 - 0.833 | 89.4 | 1.8

Table 3The full-widths at half-maximum (FWHMs) of several models of the AU Mic
debris disk, measured at a distance of 40 au from the star, across a wide
range of wavelengths. All values for FWHMs are derived from Table 4 in Daley
et al. (2019). There appears to be marginal evidence for an increase in scale
height of the disk as a function of wavelength, though the measured variations
in scale height with wavelength are complex. Modeling methods employed by
Schüppler et al. (2015) are different enough from other studies (and not
intended for direct comparison with vertical structure measured from images)
that they may not provide a good comparison for the FWHM of the disk.

####  5.1.4 Dust mass and opacity

The dust mass that we derive depends strongly on the assumed mass opacity. We
measure a dust mass of (2.7 $\pm$ $0.1)\times 10^{-3}\rm{M}_{\earth}$ within
the AU Mic debris disk. This dust mass is over three times smaller than the
measurement of (9.28 $\pm$ 0.13) $\times$ $10^{-3}$ M⊕ made by Daley et al.
(2019) for an opacity of 2.3 cm2 g-1. It is also smaller than the dust mass of
7 $\times$ 1025 g (approximately 1.2 $\times$ 10-2 M⊕) reported by MacGregor
et al. (2013) by adopting an opacity of 2.7 cm2 g-1 and assuming a dust
temperature at 35-45 au of 25 K, and smaller than the dust mass of $\sim$ 0.01
M⊕ calculated by Strubbe & Chiang (2006) by fitting a thermal spectrum to an
assumed steady-state collisional cascade.

With regard to the dust opacity, it is not entirely physical to vary the dust
opacity index $\beta$ without altering the dust opacity constant $\kappa$ as
well, although their dependence on one other is complex and ambiguous. As
Equation 8 states, we assume a normalization constant of 0.1 cm2 g-1 for our
model regardless of the value of $\beta$. However, the effect of this
assumption returns a lower dust mass of the disk than previous measurements.
It was necessary to allow variations in $\beta$ in order to fit the flux
precisely at both wavelengths. We left the opacity normalization unchanged at
a frequency of 100 GHz because our main focus is on the spatial distribution
of flux rather than the conversion from total flux to mass (which is a
separate though no less interesting problem), and allowing $\beta$ and $M_{\rm
disk}$ to vary ensures that the flux will be normalized at both wavelengths.

####  5.1.5 Modeling assumptions

There are a number of assumptions that we make in our modeling process that
should be highlighted; deviations from any of these assumptions could alter
the interpretation of our observational results. We assume that the grain size
is approximately equal to the wavelength, i.e. $\lambda\approx 2\pi a$. This
assumption rests on two other simplifying assumptions: (1) that the grain size
distribution follows a smooth power law index $q$, with values comparable to a
Dohnanyi distribution so that the number of particles (and therefore emitting
surface area) is strongly weighted towards smaller particles, and (2) that the
emission efficiency of a dust grain falls off as $1/\lambda$ for wavelengths
longer than $\approx 2\pi a$. The net effect of these two power law
distributions is that the smallest grain that can emit efficiently (namely,
one with size comparable to the wavelength of observation) will dominate the
emission at a given wavelength. However, one can imagine a situation where one
grain size dominates the total grain size distribution within the disk, which
would yield a weaker scaling relation and hence a larger value for $p$.
Alternatively, an optically thicker disk would return a lower value on $p$,
since the wavelength scaling relation would become less sensitive to particle
size.

We utilize Equation 13 from Draine (2006) as a crude approximation of the
grain size distribution at submillimeter wavelengths; to retrieve a more
accurate grain size distribution index for the disk (at sub-mm and mm
wavelengths), it is necessary to fully model the dust and grain composition
within the disk. Recently, Arnold et al. (2021) attempted to model the grain
composition of the AU Mic disk at scattered light wavelengths, but found that
it was possible to model the spectrum of the disk with many different
compositions.

There are additional uncertainties associated with our derivation of $q$. For
instance, when calculating $q$ for our disk, we assume a dust opacity power
law index from Draine (2006) of $\beta_{s}=1.8\pm 0.2$. It is critical to note
that Draine (2006) used Equation 2 to explain why protoplanetary disk opacity
spectral indices $\beta$ were lower than the expected opacity spectral indices
for interstellar medium $\beta_{\rm ISM}=1.7$. However, here we are taking our
measured $\beta$ and assuming a value for $\beta_{\rm ISM}$ to retrieve the
grain size distribution index, effectively the inverse of the work done in
Draine (2006).

Furthermore, $\beta_{s}$ is sensitive to the composition of the debris disk.
Löhne (2020) demonstrated a link between the millimeter spectral index
$\alpha_{\rm mm}$ and the grain size distribution index $q$, which implies
that the size distribution of AU Mic can vary with the assumed disk
composition. By using data from MacGregor et al. (2016) to calculate the
Planck and millimeter spectral indices, Löhne (2020) showed that if AU Mic’s
disk was composed primarily of water ice, $q$ would be close to 3, whereas a
composition of amorphous carbon or pyroxene would return a value greater than
the Dohnanyi index $q=3.5$. But for the most part, Löhne (2020) retrieves
values for $q$ from his analysis that are larger than what would be retrieved
via the cruder approximation from Draine (2006).

There is also an inherent inconsistency in assuming a constant value of q
throughout the disk while also measuring different scale heights for two
different grain sizes (because if you have different scale heights for
different grain sizes, then the grain size distribution must change with
height above the midplane). Fundamentally, we cannot tell whether the scale
height is different for the two wavelengths because one size of grain is not
present above a certain height (which we interpret as a difference in velocity
of grains seen at two wavelengths and thus is reflected by $p$), or whether
the grains are present but the composition changes in such a way as to make
them fainter at one wavelength than another (which would mimic as a change in
$q$). We have a theoretical prediction that favors the former scenario (which
is why we favor that explanation in our analysis), but we do not have reason a
priori to expect a change in composition with scale height. However, the
current data cannot rule out either scenario.

Finally, central assumptions of Pan & Schlichting (2012) are that the dust is
in a steady state and is not subject to radial transport. Given the observed
fast-moving features in scattered light, there are reasons to expect some non-
steady-state processes in the AU Mic disk. As discussed in § 5.1.2, the
likelihood of stellar eruptions and the known flaring behavior of AU Mic both
suggest variations in radiation and stellar wind rates that could perturb the
disk. Estimates of the stellar wind mass loss rate in the AU Mic system range
from $\sim 5-600$ times the solar wind value (Augereau & Beust, 2006; Kavanagh
et al., 2021; Alvarado-Gómez et al., 2022). These effects should be more
pronounced on the small grains than on the large grains, but the difference in
outer radii between the 450 $\mu$m and 1.3 mm images does suggest that stellar
winds could be playing a role in shaping the millimeter morphology of the
disk. In addition, a transport-dominated disk would be expected to exhibit a
flatter size distribution than a typical disk (Reidemeister et al., 2011;
Wyatt et al., 2011), which could explain why the measured value of $q$ in the
AU Mic system is unusually flat compared with other debris disks (MacGregor et
al., 2016).

On the other hand, radial transport would only significantly affect the
interpretation of our results if the transport timescale is short compared
with the collision timescale for the grains. Millimeter grains are probably
too large for Poynting-Robertson drag to be relevant, but stellar wind drag
could be relevant in this size range. We can roughly estimate the transport
timescale by figuring out how long it takes for the mass of solar wind
particles landing on a grain to be equal to the grain’s original mass
(assuming that the stellar wind is confined within about 1 Sr):

$m_{\mathrm{dust}}\approx\dot{M_{*}}\Omega_{\mathrm{dust}}t_{\mathrm{rad}}$
(15)

where $m_{\mathrm{dust}}$ is the mass of a dust grain, $\dot{M_{*}}$ is the
stellar wind mass loss per time, $\Omega_{\mathrm{dust}}$ is the solid angle
subtended by a dust grain, and $t_{\mathrm{rad}}$ is the radial transport
timescale. This relationship yields a transport timescale of a few Myr for 1
mm-size dust grains and a stellar wind mass loss rate of 10$\times$ the solar
value, which is long compared with the expected 100,000-yr collision timescale
expected at a distance of 40 au from a 0.5 M☉ star in a disk with a vertical
optical depth of order $10^{-3}$. Since the latter timescale is for collisions
between similar-size particles, it is a firm upper bound on the timescale for
catastrophic collisions. A more careful calculation conducted by Strubbe &
Chiang (2006) yields drift timescales that are even longer than this rough
estimate for similar parameters. It is also important to note that the upper
end of the estimated stellar wind rates for the AU Mic system are larger by a
factor of 60, which shortens the timescale to a value more comparable with the
age of the system. While the transport timescale does approach the age of the
system, as long as the collisional timescale is shorter than the transport
timescale, the collisional cascade would still be expected to operate in the
standard (non-transport-dominated) way.

###  5.2. The AU Mic Collisional Cascade

In § 4 we derive values for the velocity dispersion power law index, $p=$ 0.28
$\pm$ 0.06, and the grain size distribution power law index, $q=$ 3.03 $\pm$
0.02, via modeling the AU Mic debris disk. Here, we describe the AU Mic
collisional cascade within the context of theoretical models for collisional
cascades, and to compare our results with measurements of $p$ and $q$ within
the Kuiper Belt. We note here that the millimeter and sub-millimeter data
employed in this paper constrain the collisional cascade to $\approx$
millimeter-sized debris particles. As we discuss shortly, these measurements
could fluctuate due to variations (e.g. Fraser, 2009; Belyaev & Rafikov, 2011)
in the overall size distribution of debris within the disk, and that our
comparison to the Kuiper Belt in particular requires some extrapolation.

####  5.2.1 Comparison with collisional cascade models

A pioneering study by Dohnanyi (1969) was the first to characterize how the
grain size distribution would equilibrate in the context of a steady-state
collisional cascade, finding analytically that the grain size distribution
index $q=3.5$, for disks under steady-state conditions and a few further
simplifying assumptions. Other analytical (e.g. Williams & Wetherill, 1994;
Wyatt et al., 2007) and numerical models (e.g. O’Brien & Greenberg, 2003;
Krivov, 2007) recovered the Dohnanyi solution for steady-state collisions
within a debris disk.

Pan & Schlichting (2012) extended this work to consider how the velocity
dispersion would equilibrate for grains of different sizes. Similarly to the
cascade equations in § 1, they parameterize the collisional cascade indices
$p$ and $q$ via Equations 1 and 2. They analytically described several
mechanisms that would damp the motion of bodies within the collisional cascade
and hence affect the measured indices. They present a complete formalism for
steady-state collisional cascades; the model presented in Dohnanyi (1969), for
example, is a consequence of setting $p=0$ in the strength regime (i.e. no
velocity damping), and the Dohnanyi grain size distribution index $q=3.5$
results from setting the energy per unit mass $Q^{*}(a)$ to be constant i.e.
independent of grain size $a$.

In § 4 we determine values for the velocity dispersion power law index
$p=0.28\pm 0.06$ and the grain-size distribution index $q=3.03\pm 0.02$ by
modeling the radial and vertical density structure of the AU Mic debris disk
at both 450 $\mu$m and 1.3 mm simultaneously. These parameters place AU Mic’s
collisional behavior within the gravity regime per Pan & Schlichting (2012),
where the velocities of small bodies are greater than large bodies, and where
catastrophic collisions significantly damp the velocities of bodies within the
disk. In this regime, $0.20<p<0.25$ and $3.21>q>3$.

Catastrophic collisions in the gravity regime have been described in
theoretical work. For steady-state collisional cascades, several works
(Williams & Wetherill, 1994; Tanaka et al., 1996; Pan & Sari, 2005) retrieve
the Dohnanyi grain-size distribution power law index $q=3.5$. Describing a
semi-steady collisional cascade, Pan & Sari (2005) derive $2.87<q<3.14$ for a
collisional population of objects whose binding energies are dominated by
gravity and whose planetesimal population decreases over time.

Some works have attempted to model the conditions necessary for catastrophic
collisions. Leinhardt & Stewart (2009) conducted full numerical simulations of
catastrophic collisions and found that shear strength is important in
determining collisional outcomes for km-sized objects within the gravity
regime. Leinhardt & Stewart (2012) used numerical methods to explore
collisional outcomes, including catastrophic and super-catastrophic
collisions, between bodies in the gravity regime; however, this work mostly
focused on collisional outcomes in the context of planet formation. Other
works have attempted to model collisions in the context of debris disks;
Gáspár et al. (2012) numerically modelled collisional cascades in debris disks
and found a steeper $q=3.65\pm 0.05$ when accounting for both erosive and
catastrophic collisions within the disk and when considering conditions such
as varying tensile strength of object material; this value is slightly steeper
than the Dohnanyi solution retrieved by other analytical works in the
literature. Future theoretical work could build upon Pan & Schlichting (2012)
by accounting for the effects of grain composition on collisional outcomes and
on the equilibrium conditions of the collisional cascade.

Supporting evidence for catastrophic collisions in the AU Mic debris disk is
not difficult to find. The fast-moving features detected by Boccaletti et al.
(2015) and later confirmed by Boccaletti et al. (2018) have been described as
“dust avalanches” (Chiang & Fung, 2017) where sub-$\mu$m dust particles gain
mass as they shatter other planetesimals in the disk; these outflows are
attributed to the stellar winds. Our observations did not detect these
features at 450 $\mu$m; however, the same destructive collisions which yield
“dust avalanches” may also be responsible for the damping of velocities within
the disk.

It is certainly surprising to place observations nominally probing millimeter
grains within the gravity regime. Previous work has found that the strength-
gravity regime boundary lies at sizes of order hundreds of meters (Benz &
Asphaug, 1999), or perhaps as small as a few meters for extreme models like
loosely bound pebble piles (Krivov et al., 2018), so we would expect
millimeter grains to lie in the strength regime instead. Ultimately, our
measurements of $p$ and $q$ are probing the power law $\gamma$ that is used to
describe the power law index of the critical fragmentation energy,
$Q^{*}\propto a^{\gamma}$. Using the relationship
$\gamma=\frac{21-2p-6q+2qp}{q-1}$ (Kobayashi & Tanaka, 2010; Pan &
Schlichting, 2012), our measured values of $p$ and $q$ imply a value of
$\gamma=1.95$. Fundamentally, this means that it takes less energy to destroy
a small grain than a larger one. While this property is normally associated
with the gravity regime, it doesn’t necessarily require that a grain’s self-
gravity is the physical reason for this, but it is not immediately obvious
what else could explain these results.

One alternative possibility is that the millimeter grains are actually in the
strength regime, but that the size distribution is not a simple power law, and
instead exhibits “ripples” or “waves” as a function of grain size (e.g., Campo
Bagatin et al., 1994; Wyatt et al., 2011). Such ripples occur in response to a
sharp cutoff in the grain size distribution, for example due to radiation
pressure or stellar wind pressure that preferentially removes grains smaller
than a certain size. The relevance of this effect depends on the “wavelength”
of the waves, which equals the size ratio between a body and the smallest
particle that can collisionally destroy it. In order for the effect to
strongly alter the interpretation of the observations, the wavelength would
need to be comparable to (i.e., not much shorter or longer than) the ratio of
the two wavelengths we used to measure the change in brightness and vertical
structure – in this case, $\sim 3$. As a rough estimate, a scale height of a
few percent at a distance of 40 au from a 0.5 M☉ star corresponds to a
collisional velocity of order 100 m s-1, so that the energy per unit mass
delivered by an incoming projectile in a collision would be $10^{8}$ erg g-1.
If the energy per unit mass needed to destroy a grain is $Q^{*}$, and all the
same size grains have the same mass density, then for a catastrophic collision
we require:

$\frac{\mathrm{target~{}size}}{\mathrm{projectile~{}size}}\approx\left(\frac{10^{8}\,\mathrm{erg\,g^{-1}}}{Q^{*}}\right)^{1/3}\
.$ (16)

This ratio is the “wavelength” of the waves. Estimates for $Q^{*}$ vary
widely, up to $10^{7}$ erg g-1 for small ($<1$ km) sizes (e.g., Holsapple,
1993; Benz & Asphaug, 1999), which suggests that the wavelength could
conceivably be comparable to the factor of $\sim$3 between the band
frequencies.

This effect also depends on the amplitude of the waves, however. The amplitude
might be particularly low for AU Mic compared to other systems, because a
sharp cutoff in the size distribution due to radiation pressure is not
expected for this low-luminosity star. AU Mic is so red that the dust grain
size at which the stellar luminosity $L_{*}/c$ balances gravity is small
compared to the peak wavelength of the stellar blackbody, which means that
absorption of AU Mic’s radiation by these grains will be inefficient and they
will not become unbound. Absorption at smaller dust grain sizes is even less
efficient, so no sharp cutoff in the grain size distribution is expected due
to radiation pressure. However, if a stellar wind operates that is strong
enough to cause the transport time to dominate over the collision time, a
sharp grain size cutoff could be introduced.

####  5.2.2 Comparison with the Kuiper Belt

The only other debris disk with a comparably characterized collisional cascade
is the Kuiper Belt. While observations support a rich collisional history in
the Kuiper Belt, a definitive model of the collisional evolution of Kuiper
Belt objects remains an open question (Kenyon & Bromley, 2004; Leinhardt et
al., 2007). Pan & Sari (2005) applied their work to the Kuiper Belt and
determined that Kuiper Belt objects can be described as “strengthless” rubble
piles which are prone to shattering.

Within the Kuiper Belt, the indices $p$ and $q$ that describe the collisional
cascade have been measured across many different size regimes. The velocity
dispersion of Kuiper Belt objects (KBOs) of all observed sizes is about 1 km
s-1 (Leinhardt et al., 2008), which suggests a value for $p=0$, though Abedin
et al. (2021) showed that non-negligible differences in collisional velocities
exist between different populations of Kuiper Belt objects. It is more
complicated to describe the dust grain size distribution of the Kuiper Belt
for two reasons. Astronomers are mainly able to observe the km-sized objects
in the belt as opposed to the dust, since the lower mass of dust in the Kuiper
Belt results in a lower collision rate, so that most of the dust is ejected by
the large planets out of the Solar System, and the rest spirals in towards the
star due to Poynting-Robertson drag (Yamamoto & Mukai, 1998), whereas in
debris disks around distant stars the km-size objects are invisible and only
the dust can be observed. Furthermore, the grain size distribution power law
slope varies dramatically as a function of grain size in the Kuiper Belt,
unlike the smooth power law assumed for debris disks. Nevertheless, a Dohnanyi
grain-size distribution power law index of $q=3.5$, and more generally the
power law relationship $\frac{dN}{da}\propto a^{-q}$, has been shown to agree
broadly with observations of Kuiper Belt objects (e.g. Bernstein et al., 2004;
Kenyon & Bromley, 2004; Schlichting et al., 2013).

It is also very difficult to observe Kuiper Belt dust due to foreground
emission from the zodiacal cloud, and thus dust is typically modelled (e.g.
Poppe, 2016). Within the mm/sub-mm regime, the modeled grain size distribution
index has varied in different works between $q\sim 2.5$ (Moro-Martín &
Malhotra, 2003) to $q<3.5$ (Vitense et al., 2012), to $q=3.5$ (Kuchner &
Stark, 2010), with the caveat that there are likely waves in the size
distribution so that the measured slope $q$ is sensitive to the particle sizes
included in the measurement. The dust populations that can be probed lie
within the $\mu$m regime, and thus are difficult to compare with the mm-sized
dust in this work; these populations can be constrained today using the
Venetia Burney Student Dust Counter (SDC) on the New Horizons spacecraft.

Poppe et al. (2019) combined modelling with dust measurements from the SDC to
constrain the dust grain production in the Kuiper Belt, finding that the
Kuiper Belt emits the most thermal emission within the 50-100 $\mu$m regime,
with sculpting of the belt’s inner radius by Jupiter clearly detectable. This
sculpting feature is also apparent at 500 $\mu$m, though the Kuiper Belt at
this wavelength is less bright; nevertheless, perhaps this is a supporting
argument for the existence of a planet shaping the inner radius of the AU Mic
disk (as argued by Pearce et al., 2022). Bernardoni et al. (2022) further
studied the $\mu$m-sized dust (0.356 $<$ $r_{g}$ $<$ 2.370 $\mu$m) using
detections from the SDC, finding that the overall flux estimates of the Kuiper
Belt agree with predictions from modelling (Poppe, 2016; Poppe et al., 2019).
However, neither of these works calculate the size distribution index for
$\mu$m-sized dust in the Kuiper Belt, as has been done for km-sized objects.

We emphasize that the value for $q$ measured for AU Mic’s debris disk in this
work purely describes mm to sub-mm grain populations. It is possible that, as
has been modelled for the Kuiper Belt (Kuchner & Stark, 2010), the $p$ and $q$
values we measure for the disk are sensitive to waves in the overall
distribution of velocity dispersion and grain size within the AU Mic disk. The
only way to verify this hypothesis is via observations of the disk at many
different wavelengths. Also noteworthy is work by Arnold et al. (2021), who
modelled three different shapes of sub-micron ($\approx$ 0.2 $\mu$m) dust
grains in the AU Mic debris disk and concluded that the value of $q$ measured
by fitting the scattered light spectrum varied dramatically from
$q=3.72^{+0.43}_{-0.28}$ to $q=4.57^{+0.28}_{-0.40}$ (Lomax et al., 2018). To
summarize, the values of $p$ and $q$ that we measure in the AU Mic debris disk
are broadly comparable to those measured for the Kuiper Belt and consistent
with the range predicted by the physics of the collisional cascade, although
the comparison is complicated by our inability to directly compare values for
the same range of grain sizes and by our lack of knowledge of the composition
of the dust grains.

###  5.3. Future work

Evidently, variations must exist in the overall grain size distribution for
the AU Mic system; in order to draw more general conclusions about AU Mic’s
collisional cascade across wider ranges of grain sizes, it is necessary to
make high-resolution, multi-wavelength observations of the vertical structure
of the disk within other portions of the electromagnetic spectrum. While
Fitzgerald et al. (2007) performed multi-wavelength observations of the disk
in the infrared, higher resolutions are necessary to determine whether the
trend of increasing scale height continues for $\mu$m and sub-$\mu$m grains.
Probing the cascade behavior of smaller or larger grains via observations at
other wavelengths will make it possible to determine to what extent AU Mic’s
debris disk can be described by a Dohnanyi size distribution power law (i.e.
$q=3.5$) and to what extent AU Mic is consistent with theoretical predictions
for a steady-state collisional cascade. With regards to AU Mic, Band 3 – or,
eventually, Band 1 – observations of the disk would be able to extend our
cascade description further into millimeter wavelengths. It should also be
possible to use the Square Kilometer Array (SKA) or, eventually, the Next
Generation Very Large Array (ngVLA) to observe at even longer wavelengths.
These observations would allow us to determine both whether waves exist within
the collisional cascade (i.e. the size distribution and velocity dispersion
indices) at millimeter wavelengths, as well as whether the trend of increasing
FWHM extends beyond into the centimeter-wavelength regime.

Some testable predictions include that if the disk is in fact transport-
dominated at the wavelengths probed by our data, we would expect to see a
steepening of the power law index $q$ at longer wavelengths, because even in
the AU Mic system those larger grains would no longer be transport-dominated.
Likewise, for disks around earlier-type stars with lower stellar wind rates,
we would expect to observe $p$ and $q$ values such that $\gamma<0$, indicating
that millimeter grains are indeed in the strength regime.

No measurements of $p$ have been made in other debris disks beyond the Kuiper
Belt. In this sense, it is currently impossible to compare our measurement of
$p$ to the velocity dispersion of bodies in other disks. However, measurements
of $q$ have been made for several debris disks. MacGregor et al. (2016) used
the VLA to observe debris disks around seven nearby stars: HD 377, 49 Ceti, HD
15115, HD 61005, HD 104860, HD 141569, and AU Mic, and determined their grain
size distribution indices $q$. After supplementing this sample with other disk
observations, they derived a weighted mean for $q=3.36\pm 0.02$. Some of these
systems have inclinations that are nearly edge-on: HD 377 has an inclination
$i=85\pm 5\degree$ (Choquet et al., 2016), HD 15115 has an inclination
$i=86.3\pm 0.3\degree$ (Sai et al., 2015), and HD 61005 has an inclination
$i=84.1\pm 0.2$ (Olofsson et al., 2016). These inclinations may be large
enough to directly resolve the vertical structure of these disks, and multi-
wavelength observations of these disks will enable astronomers to directly
compare their collisional cascades with steady-state models and with AU Mic
and the Kuiper Belt.

Other debris disk candidates which are bright and close to edge-on include the
disk around $\beta$ Pictoris (e.g. Smith & Terrile, 1984; Matrà et al., 2019;
Janson et al., 2021) and the recently discovered disk around
BD$+$45$\degree$598 (Hinkley et al., 2021). If these candidates could be
observed at both high resolution, necessary to resolve their vertical
structures, and at multiple wavelengths such that the relationship between
observed wavelengths and disk aspect ratio is significant, then their
collisional cascades could be parameterized in a similar fashion to the
process described in this work. The disk around $\beta$ Pictoris is especially
fascinating in that its vertical structure is more complicated than the
vertical structure of the AU Mic debris disk; for example, Matrà et al. (2019)
modelled the vertical structure of the $\beta$ Pictoris disk’s mm-sized grains
with two Gaussians, since a single Gaussian was unable to reproduce the
vertical distribution of dust within the disk. It is ultimately challenging to
totally contextualize these results with respect to other (extrasolar)
circumstellar disks, since our measurements are the first of their kind.
However, our results provide support for the idea that steady-state
collisional cascades may be ongoing in a debris disk beyond our solar system
(Dohnanyi, 1969; Pan & Schlichting, 2012).

##  6\. Conclusions

This work presents 450 $\mu$m observations of the AU Microscopii debris disk
from ALMA which we use to investigate the collisional physics within the disk.
AU Microscopii’s debris disk is well suited for a parameterization of its
collisional physics (within the framework presented in Pan & Schlichting,
2012) since the disk is viewed edge-on. This quality of the disk enables
measurements of its scale height, and hence the velocity dispersion of the
grains within the disk. The main findings of this work are presented below:

  1. 1.

The morphology of the 450 $\mu$m emission is well-described by an axisymmetric
disk. We do not detect any features consistent with the fast-moving features
described by Boccaletti et al. (2015). The vertical structure at 450 $\mu$m is
spatially resolved. We simultaneously fit a parametric model of the radial and
vertical density structure to both 450 $\mu$m and 1.3 mm ALMA data (described
in Daley et al., 2019) which assumes an identical viewing geometry at both
wavelengths. The inclination of the disk is measured to be
$i=88.51^{+0.44}_{-0.05}$ degrees.

  2. 2.

Between $\lambda=450$ $\mu$m and $\lambda=1.3$ mm, we find a scale height
ratio $h_{1300}/h_{450}=1.35\pm 0.09$. Our measurement of the aspect ratio at
1.3 mm is slightly lower but agrees within the uncertainties of the 1.3 mm-
only fit by Daley et al. (2019). Nevertheless, our results confirm what was
already apparent in the image domain: that the vertical FWHM of the disk is
thinner at 450 $\mu$m than at 1.3 mm.

  3. 3.

The dust opacity index $\beta$ and aspect ratios of the disk can be converted
into parameters that describe the collisional cascade of the disk. Per the
framework described by Pan & Schlichting (2012), which builds on the work of
Dohnanyi (1969) by exploring the effects of grain-sizes on steady-state
equilibrium, we calculate a velocity dispersion power law index $p=0.28\pm
0.06$ by comparing the aspect ratio of the disk height at both wavelengths,
and a grain size distribution power law index $q=3.03\pm 0.02$ from modeling
the dust opacity index $\beta$. These values place AU Mic’s collisional
cascade within the “gravity regime” where collisional damping is caused by
catastrophic collisions, and smaller bodies are more easily destroyed than
larger ones. This result is surprising, since millimeter-size grains would be
expected to be in the strength regime instead. Possible explanations for the
discrepancy include ripples in the grain size distribution (which could
plausibly occur at the necessary “wavelength” in grain size, although the
amplitude is expected to be small because of the relatively low influence of
radiation pressure in the AU Mic system), or a transport-dominated disk due to
the influence of the stellar wind (although the collision rate should be high
compared to the radial transport rate).

  4. 4.

The measured value of $p$, $q$ for the AU Mic debris disk implies that the
velocities of grains traced by the 450 $\mu$m continuum emission are damped by
destructive interactions. However, it is difficult to draw more general
conclusions; more observations of the disk across a wider range of wavelengths
(such as visible light, mid-IR, or longer-$\lambda$ radio) are necessary to
confirm whether the disk’s collisional cascade is globally governed by
catastrophic collisions in the gravity regime. Ultimately, we hope that these
surprising results will spark follow-up work both theoretical and
observational to elucidate how representative (or not) the AU Mic system is
and whether new physical frameworks are necessary to interpret the results.

  5. 5.

Bright, edge-on debris disks are best suited for investigations into their
collisional physics, and future work in parameterizing the collisional
cascades of other extrasolar debris disks will eventually allow us to
contextualize AU Mic’s debris disk and the Kuiper Belt within a broader
framework of collisional physics in debris disks.

##  7\. Acknowledgements

We thank the anonymous referee for insightful comments that improved the
paper. We thank Sasha Krivov, Grant Kennedy, and Luca Matrá for thoughtful
discussion. We thank Cail Daley for sharing code and for constructive
feedback. DV is funded by an Open Study/Research Award from the Fulbright U.S.
Student Program. AMH and DV gratefully acknowledge support from a Cottrell
Scholar Award from the Research Corporation for Science Advancement. MAM
acknowledges the National Aeronautics and Space Administration under award
number 19-ICAR192-0041.

This paper makes use of the following ALMA data: ADS/JAO.ALMA#2016.1.00878.S,
ADS/JAO.ALMA#2012.1.00198.S. ALMA is a partnership of ESO (representing its
member states), NSF (USA) and NINS (Japan), together with NRC (Canada) and NSC
and ASIAA (Taiwan), in cooperation with the Republic of Chile. The Joint ALMA
Observatory is operated by ESO, AUI/NRAO and NAOJ. This research has made use
of NASA’s Astrophysics Data System and Astrophysics Data System Bibliographic
Services.

This work has made use of data from the European Space Agency (ESA) mission
Gaia (https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing
and Analysis Consortium (DPAC,
https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has
b‘een provided by national institutions, in particular the institutions
participating in the Gaia Multilateral Agreement. This research has made use
of the SIMBAD database, operated at CDS, Strasbourg, France.

This research made use of Astropy, a community-developed core Python package
for Astronomy (Astropy Collaboration et al., 2018, 2013). This research made
use of pandas (McKinney, 2010, 2011). This research made use of NumPy (Harris
et al., 2020). This research made use of emcee (Foreman-Mackey et al., 2013).
This research made use of the CASA software package (McMullin et al., 2007).

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