# Similarity between the kinematic viscosity of quark-gluon plasma
and liquids at the viscosity minimum

K. Trachenko<EMAIL_ADDRESS>School of Physics and Astronomy, Queen
Mary University of London, Mile End Road, London, E1 4NS, UK  V. V. Brazhkin
<EMAIL_ADDRESS>Institute for High Pressure Physics, RAS, 108840,
Troitsk, Moscow, Russia  M. Baggioli<EMAIL_ADDRESS>Instituto de
Fisica Teorica UAM/CSIC, c/ Nicolas Cabrera 13-15, Cantoblanco, 28049 Madrid,
Spain

###### Abstract

Recently, it has been found that the kinematic viscosity of liquids at the
minimum, $\nu_{m}$, can be expressed in terms of fundamental physical
constants, giving $\nu_{m}$ on the order of $10^{-7}~{}{\rm m^{2}/s}$. Here,
we show that the kinematic viscosity of quark-gluon plasma (QGP) has the same
value and support this finding by experimental data and theoretical
estimations. We discuss the implications of this result for understanding the
QGP including the similarity of its shear flow with that in liquids at the
viscosity minimum and propose that the QGP is close to the dynamical
crossover. Our results provide quantitative evidence for the universality of
the shear diffusivity proposed earlier.

††preprint: IFT-UAM/CSIC-20-49

Introduction. The quark-gluon plasma (QGP) is a state of matter emerging at
temperature of about 1012 K and above the deconfinement QCD transition ($\sim
175$ MeV) Casalderrey-Solana  _et al._ (2011). It is produced by highly
energetic collisions Adler  _et al._ (2003); Back  _et al._ (2005a); Adams
_et al._ (2005) and can be thought as a plasma made of free quarks and gluons.
The inter-particle interactions in QGP are strong and are not possible to
treat using conventional theoretical methods such as perturbation theory.

Flow and viscosity are the properties of QGP which have probably been
discussed most. Due to the low value of viscosity to entropy ratio, extracted
using the momentum anisotropy of the particle emission and related to strong
coupling, the QGP is referred to as the “perfect fluid” Shuryak (2017, 2009);
Schäfer and Teaney (2009); Song  _et al._ (2011). The ratio in QGP is close to
the conjectured universal minimum Huot  _et al._ (2007),
$\frac{\eta}{s}\,\geq\,\frac{1}{4\,\pi}\,\frac{\hbar}{k_{\rm B}}$ Policastro
_et al._ (2001); Son and Starinets (2007).

In condensed matter physics, predicting liquid viscosity from theory and
without modelling has not been possible for the same reason related to strong
interactions. Liquid viscosity strongly depends on temperature and pressure.
Viscosity is additionally strongly system-dependent and is governed by the
activation energy barrier for molecular rearrangements, $U$, which in turn is
related to the inter-molecular interactions and structure. This relationship
in complicated in general, and no universal way to predict $U$ and viscosity
from first principles exists. This is appreciated outside the realm of
condensed matter physics Weinberg  _et al._ (1983). Tractable theoretical
models describe the dilute gas limit of fluids where perturbation theory
applies, but not dense liquids of interest here Chapman  _et al._ (1990). The
same problem of strong interactions or, phrased differently, the absence of a
small parameter, were believed to disallow a possibility of calculating liquid
thermodynamic properties in general form Landau and Lifshitz (2013). For
example, the theoretical calculation and understanding of the liquid energy
and heat capacity has remained a long-standing problem Granato (2002) which
started to lift only recently when new understanding of phonons in liquids
came in Trachenko and Brazhkin (2015).

Despite these complications, there is one particular regime of liquid dynamics
where viscosity can be calculated in general form and, moreover, expressed in
terms of fundamental physical constants. Recently, it has been found Trachenko
and Brazhkin (2019) that the kinematic viscosity at its minimum, $\nu_{m}$, is

$\nu_{m}=\frac{1}{4\pi}\frac{\hbar}{\sqrt{m_{e}m}}$ (1)

where $m_{e}$ and $m$ are electron and molecule masses.

The kinematic viscosity $\nu$ is equivalent to the diffusion constant of the
shear diffusion mode $D_{s}$ (transverse momentum diffusivity), and we will be
referring to these properties inter-changibly in this paper depending on the
context.

Eq. (1) follows from writing viscosity at the minimum in terms of two UV
cutoff parameters, the interatomic separation and Debye vibration period, and
subsequently using fundamental relations such as Bohr radius and Rydberg
energy setting these UV parameters in condensed matter phases.

For atomic hydrogen where $m$ is given by the proton mass $m_{p}$, (1) results
in the fundamental minimal kinematic viscosity as

$\nu_{m}=\frac{1}{4\pi}\frac{\hbar}{\sqrt{m_{e}m_{p}}}\approx
10^{-7}~{}{\rm\frac{m^{2}}{s}}$ (2)

We calculated Trachenko and Brazhkin (2019) the kinematic viscosity
$\nu=\eta/\rho$ using the experimental dynamic viscosity $\eta$ and density
$\rho$ for several liquids from Ref. nis  and show $\nu$ in Fig. 1). We
observe that the experimental minima, $\nu_{m}^{exp}$, are in the range of
about

$\nu_{m}^{exp}=(0.5-2)\cdot 10^{-7}~{}{\rm\frac{m^{2}}{s}}$ (3)

and agree with Eq. (1) by a factor 0.5-3 for different liquids Trachenko and
Brazhkin (2019).

Figure 1: Experimental kinematic viscosity $\nu=\frac{\eta}{\rho}$ of noble,
molecular and network liquids showing minima. The experimental data of $\eta$
and $\rho$ are from Ref. nis . $\nu$ for H2, H2O and CH4 are shown for
pressure $P=50$ MPa, 100 MPa and 20 MPa, respectively. $\nu$ for He, Ne, Ar
and N2 are shown at two pressures each: 20 and 100 MPa for He, 50 and 300 MPa
for Ne, 20 and 100 MPa for Ar and 10 and 500 MPa for N2. The minimum at higher
pressure is above the minimum at lower pressure for each fluid. The dashed
line shows the kinematic viscosity of the QGP calculated in this work. Note
that the temperature of QGP is about $10^{9}-10^{10}$ times larger than the
temperature of most liquid viscosity minima.

Here, we show that the kinematic viscosity of the QGP has a value similar to
(2). We use experimental data as well as theoretical estimations to back up
this result. We discuss the implications of the similarity between liquids and
QGP from the point of view of shear flow and particle dynamics and propose
that the QGP is close the dynamical crossover. Our results provide
quantitative evidence for the universality of the shear diffusivity proposed
earlier

Kinematic viscosity. We make two preliminary observations which will become
useful for the subsequent discussion. First, we recall the dynamics of
particles at the minimum of the liquid viscosity, where the kinematic
viscosity is given by Eq. (1). This minimum is due to the dynamical crossover
Trachenko and Brazhkin (2015, 2019) separating (a) the liquid-like dynamics
combining oscillatory and diffusive components of molecular motion where
viscosity decreases with temperature and (b) the ballistic gas-like motion
where the viscosity increases with temperature. The crossover between these
two regimes implies that viscosity has a minimum. These minima are
experimentally seen in liquids, as illustrated in Fig. 1. At the crossover
where molecules lose the oscillatory component of motion and continuously move
diffusively over distances comparable to an interatomic separation, the
viscosity can be evaluated by assuming that the particle mean free path $L$ is
approximately equal to the inter-particle distance $a$. This results in Eq.
(1), in agreement with the experimental viscosity minima Trachenko and
Brazhkin (2019). This mechanism will become useful in the discussion below. We
note that viscosity minimum also appears in a different mechanism involving
relativistic hydrodynamics and considering the effects of short-wavelength
hydrodynamic fluctuations to the classical viscosity estimate Kovtun  _et al._
(2011). The minimum in this picture is related to the breakdown of the
hydrodynamics expansion and Israel-Stewart relaxation time Romatschke (2010).
A minimum for $\eta/s$ was also discussed in holographic models where it
corresponds to a transition between thermal gas background and a big black
hole solution at high temperature Cremonini  _et al._ (2012) and in certain
nuclear matter models Fang:2014wza; Deng:2016pqu.

Second, we observe that the minimal viscosity in liquids (1) is consistent
with the uncertainty relation. As discussed above, the viscosity at the
minimum corresponds to $L=a$ and can be written as $\eta=\rho
va=\frac{m}{a^{3}}va=\frac{pa}{a^{3}}$, where $p$ is the particle momentum and
$\rho=\frac{m}{a^{3}}$ is density. Combining this with the uncertainty
relation $pa\geq\hbar$ gives $\eta\geq\frac{\hbar}{a^{3}}$, or

$\nu_{m}\geq\frac{\hbar}{m}$ (4)

Therefore, the uncertainty relation gives a weaker bound on $\nu$ as compared
to $\nu_{m}$ in (1): the lower bound in (4) is smaller than $\nu_{m}$ by a
factor $\frac{1}{4\pi}\left(\frac{m}{m_{e}}\right)^{\frac{1}{2}}$, which is in
the range 5–23 for liquids shown in Fig. 1. For atomic hydrogen, Eq. (4) gives
the lower bound as

$\nu_{m}\geq\frac{\hbar}{m_{p}}$ (5)

The bound (5) is smaller than (2) by the factor
$\frac{1}{4\pi}\left(\frac{m_{p}}{m_{e}}\right)^{\frac{1}{2}}\approx 3$.

We now calculate the experimental value of kinematic viscosity of QGP,
$\nu_{\rm QGP}^{exp}=\frac{\eta_{\rm QGP}}{\rho}$, where $\eta_{\rm QGP}$ is
dynamic viscosity and $\rho$ is density. The parameters we use for our
estimations are listed in Table 1. The density $\rho$ can be estimated in
several ways. First, we write $\rho=\frac{1}{c^{2}}\frac{E}{V}$, where
$\frac{E}{V}$ is energy density (see Table 1). This gives $\rho\approx 5\cdot
10^{18}$ kg/m3. Second, the QGP is a relativistic charged fluid described by
relativistic hydrodynamics Kovtun (2012), where the diffusion constant of the
shear diffusion mode $D_{s}$ (transverse momentum diffusivity), is given by:

$\nu\,\equiv\,D_{s}\,=\,\frac{\eta}{\chi_{\pi\pi}}\,,\quad\chi_{\pi\pi}\,=\,s\,T\,+\,\mu\,q,$
(6)

where $\chi_{\pi\pi}$ is momentum susceptibility, $\mu$ is the chemical
potential, $q$ is the charge density and we set $c=1$.

In the part of the QCD phase diagram where the QGP is observed, the quark
baryonic chemical potential is small, $\mu/T\ll 1$, and the second term in
$\chi_{\pi\pi}$ can be neglected. This can be verified using relativistic
hydrodynamics Kovtun (2012), together with the experimental value for $\eta/s$
given by Song  _et al._ (2011):

$\frac{\eta}{s}\Big{|}_{\rm
QGP}\,\sim\,\frac{3}{4\,\pi}\,\frac{\hbar}{k_{B}}\,\,=\,3\,\frac{\eta}{s}\Big{|}_{\rm{KSS}}$
(7)

where $\frac{\eta}{s}\Big{|}_{\rm{KSS}}$ is the KSS bound Policastro  _et al._
(2001); Son and Starinets (2007).

In the limit of small chemical potential $\mu/T\ll 1$ and neglecting the
pressure contribution, we write $\chi_{\pi\pi}\sim sT$, resulting in:

$\frac{\eta}{\rho}\,\sim\,\frac{\eta}{s\,T}$ (8)

and, restoring the speed of light $c$, find

$\rho\,c^{2}\,\sim\,s\,T\,\sim\,\frac{k_{B}\,T}{\hbar}\,\eta$ (9)

which gives $\rho\approx 1.5\cdot 10^{18}$ kg/m3, similar to the first
estimation.

We note that Eq. (9) contains the Planckian relaxation time
$\tau_{\rm{Pl}}=\frac{\hbar}{k_{B}T}$ which we will discuss later in the
paper.

Finally, we can assume that $\rho$ is approximately given by the proton
density $\rho=\frac{m_{p}}{a_{p}^{3}}$, where $m_{p}$ and $a_{p}$ are proton
mass and size. This is consistent with the observation that the energy density
in QGP is not far from the energy density inside nucleons Back  _et al._
(2005b) and so the proton density can be used in order-of-magnitude
estimations (all our evaluations corresponds to order-of-magnitude
estimations). Using the values in Table 1, this estimate gives $\rho\approx
3\cdot 10^{18}$ kg/m3. This is close to the previous estimations, and we will
use it in our subsequent calculations.

Then, $\nu_{\rm QGP}^{exp}$ is

$\nu_{\rm QGP}^{exp}\approx 10^{-7}{\rm\frac{m^{2}}{s}}$ (10)

as for liquids at the viscosity minimum in (2) and (3).

Given that the dynamic viscosity $\eta$ and the density of QGP are about 16
orders of magnitude larger than those values in liquids, the similarity of
$\nu$ is striking.

$E/V$ |  $1$ GeV/fm3 Back  _et al._ (2005b)
---|---
$\eta$ |  $5\cdot 10^{11}$ Pa$\cdot$ s Schäfer and Teaney (2009)
$m_{p}$ |  $1.67\cdot 10^{-27}$ kg
$a_{p}$ |  $0.84\cdot 10^{-15}$ m
$a$ |  $0.5\cdot 10^{-15}$ m Pasechnik and Šumbera (2017)
$T_{\rm QGP}$ |  $2\cdot 10^{12}$ K Schäfer and Teaney (2009)
Table 1: Parameters used to estimate the properties of QGP.

We now provide a crude theoretical estimation of $\nu$ in QGP. We assume that,
as far particle dynamics is concerned, the QGP is similar to liquids at the
dynamical crossover where $L$ is equal to the inter-particle spacing. This not
a trivial assumption because, as discussed earlier and elaborated on in more
detail below, this regime corresponds to a very particular regime of dynamics
where liquid viscosity is at its minimum. The inter-parton distance, $a=0.5$
fm Pasechnik and Šumbera (2017), is close to $a_{p}$, hence we use $L=a_{p}$.
We write $\eta=\rho va$, where $v$ is the average velocity of QGP particles
involved in setting the viscosity. Approximating the density $\rho$ by
$m_{p}/a_{p}^{3}$ as discussed earlier gives $\eta=p/a_{p}^{2}$, where
$p=m_{p}v$ is interpreted as the momentum of the proton moving with speed $v$.
Estimating $p$ as $p\approx\hbar/a_{p}$ from the uncertainty relation gives
$\eta=\hbar/a_{p}^{3}$ and the theoretical value of kinematic viscosity of
QGP, $\nu_{\rm QGP}^{th}=\frac{\eta}{\rho}$ as

$\nu_{\rm QGP}^{th}=\frac{\hbar}{m_{p}}\approx 10^{-7}~{}{\rm\frac{m^{2}}{s}}$
(11)

as in (10).

Interestingly, the same result follows from an estimation of the diffusion
constant that features in the mean-square displacement of particle motion,
$D$. We note that $D$ is different from $D_{s}$ in Eq. (6). $D$ coincides with
$\nu$ and $D_{s}$ in the gas-like regime of particle dynamics at high
temperature where the same momentum transfer mechanism governs both Navier-
Stokes and diffusion equations Frenkel (1984) ($D$ and $\nu$ are different in
the liquid-like regime at low temperature, where viscosity is $\propto 1/D$).
At low temperature where liquid dynamics combines oscillatory motion and
diffusive jumps between quasi-equilibrium positions and where liquid
relaxation time $\tau$ is the time between these jumps, where
$D\approx\frac{a^{2}}{\tau}$ Frenkel (1984). As temperature increases and
$\tau$ becomes comparable to the shortest timescale in the system $\tau_{0}$
(in liquids $\tau_{0}$ is on the order of Debye vibration period of about 0.1
ps), the oscillatory component of particle motion is lost and particles start
moving continuously, corresponding to the dynamical crossover discussed above
Trachenko and Brazhkin (2015). At the crossover,
$D\approx\frac{a^{2}}{\tau_{0}}$. For QGP, $\tau_{0}$ can be evaluated using
the Planckian relaxation time Zaanen (2019):

$\tau_{\rm{Pl}}=\frac{\hbar}{k_{B}T}$ (12)

The Planckian time $\tau_{\rm{Pl}}$ was related to several fundamental
physical phenomena, including the linear resistivity of strange metals Bruin
_et al._ (2013), universal bounds on quantum chaos Maldacena  _et al._ (2016),
bounds on diffusion Hartnoll (2015); Mousatov and Hartnoll (2020); Behnia and
Kapitulnik (2019), SYK model Patel and Sachdev (2019), magic bilayer graphene
Cao  _et al._ (2019), black holes Sachdev (2015) and holography Sachdev
(2010). We note that $\tau_{\rm{Pl}}$ can appear in transport properties using
semi-classical microscopic physical arguments.111We thank Jan Zaanen for
pointing this out. This is different to the AdS-CFT approach where this
timescale emerges from emergent IR criticality Blake and Donos (2015) and
which contrasts experimental results Zhang  _et al._ (2019) of Planckian
transport in high-temperature thermal conductivity where phonons behave
classically.

Using (12) and approximating the inter-parton distance $a$ by $a_{p}$ as
before gives

$D=\frac{a_{p}^{2}}{\hbar}\,k_{\rm B}\,T_{\rm QGP}$ (13)

Using the temperature of QGP (see Table 1) gives $D\approx 10^{-7}{\rm
m^{2}/s}$ as in (11).

A very small relaxation time $\tau_{\rm Pl}=\frac{\hbar}{k_{\rm B}T_{\rm
QGP}}\approx 0.4\cdot 10^{-23}$ s interestingly contrasts with the large
experimental viscosity of QGP, $\eta_{\rm QGP}=5\cdot 10^{11}$ Pa$\cdot$ s.
This is different from liquids where this viscosity corresponds to the
relaxation time $\tau_{l}=50-500$ s, where we used the Maxwell relation
$\eta=G\tau$ and a typical high-frequency shear modulus $G=1-10$ GPa in
liquids. $\tau_{l}$ is close to that of the solid glass just below the liquid-
glass transition and is about 15-16 orders of magnitude larger than the
shortest time scale of the system given by the Debye vibration period on the
order of $0.1$ ps. Applying the Maxwell relation to QGP, $\eta_{\rm
QGP}=G\tau_{\rm Pl}$ gives $G\approx 10^{35}$ Pa. Combining it with $G=\rho
v^{2}$ (where we neglected pressure in the relativistic case as before) and
using $\rho$ from above gives the transverse speed of sound $v$ close to the
speed of light. Hence, in condensed matter terms, the QGP is an ultra-dense
matter with relativistic speed of excitations but familiar kinematic viscosity
close to that in liquids at the minimum and dynamic viscosity close to the
system at the liquid-glass transition.

Discussion. We now discuss the implications of these results. We first note
that kinematic viscosity of liquids at the minimum in Eqs. (2) and (3) does
not contain the electron charge setting the energy of electromagnetic
interactions and viscosity (the charge cancels out, as does the inter-particle
separation) and depends on the particle mass only Trachenko and Brazhkin
(2019). This implies that $\nu_{m}$ applies to systems with different types of
inter-particle interactions and distances. This is consistent with $\nu$
estimated from the uncertainty relation in Eqs. (4) and (5) which depend on
particle mass only. For this reason, the crude estimation of QGP viscosity in
Eq. (11), where we used the uncertainty relation and the nucleon density to
approximate the QGP density, gives the same result as for liquid hydrogen at
the minimum in Eq. (5). This intriguingly implies that useful insights into
dynamical and transport properties of the QGP can be gained using the concepts
from condensed matter systems such as ordinary liquids, despite different
interactions and different underlying fundamental theory such as QCD in QGP.

Second, the similarity of the kinematic viscosity of liquids at the minimum
and $\nu_{\rm QGP}^{exp}$ implies similar flow dynamics. In the non-
relativistic case, the shear velocity field is governed by Navier-Stokes
equation $\rho\frac{Dv}{Dt}=\eta\nabla^{2}v$, which depends on
$\nu=\frac{\eta}{\rho}$ only. $\nu$ also features in the Reynolds number,
which governs the dynamical similarity of flows. In the relativistic
hydrodynamics relevant to the QGP, the dynamics of shear modes comes from the
conservation of the stress energy tensor Kovtun (2012):
$\partial_{\mu}T^{\mu\nu}=0$, resulting in the diffusive motion for the shear
modes as $\omega_{s}=-iD_{s}k^{2}$, with the difference that in the
relativistic case $D_{s}=\frac{\eta}{\epsilon+p}$ rather than
$D_{s}=\frac{\eta}{\rho}$. As discussed above, the two diffusion constants are
approximately similar in the range of QGP parameters. This points to the
universality of $D_{s}$ in terms of its applicability to both relativistic and
non-relativistic systems as discussed earlier Hartnoll (2015). We will revisit
this point below.

Third, in deriving (11) we assumed that the mean free path of QGP particles is
about the same as the inter-particle distance. From the condensed matter
perspective, this corresponds to a particular regime of particle dynamics of
liquids and a particular line on the phase diagram as discussed above. In this
regime, the system is outside the low-temperature regime where particles have
a combined oscillatory and diffusive motion and where relaxation time and
viscosity decrease with temperature and vary by $16$ orders of magnitude
(relaxation time varies between $10^{3}$ s at the glass transition and about
$0.1$ ps at high temperature). The system is also outside the gas-like regime
where the mean free part exceeds inter-particle separation and viscosity
increases with temperature and becomes infinite in the ideal-gas limit Schäfer
and Teaney (2009). Instead, the liquid is finely tuned to be in between these
two regimes and at the crossover between the liquid-like and gas-like motion
(see the minima in Fig. 1) where the oscillatory motion is just lost and
particle mean free path is comparable to the inter-atomic separation. In terms
of particle dynamics, the crossover corresponds to the Frenkel line on the
phase diagram Trachenko and Brazhkin (2015); Brazhkin  _et al._ (2013);
Brazhkin and Trachenko (2012) separating gas-like and liquid-like states of
liquids and supercritical fluids. Although the crossover of particle dynamics
is responsible for the viscosity minimum seen in Fig. 1, the temperature and
pressure of the viscosity minimum may depend on the path taken on the phase
diagram Trachenko and Brazhkin (2015).

Having made the assumption that QGP particles are at the dynamical crossover,
we found that (a) the calculated $\nu_{\rm QGP}^{th}$ in (11) and $D_{s}$ in
(13) are close to the experimental value of $\nu_{\rm QGP}^{exp}$ in (10), and
(b) these values of kinematic viscosity of QGP are close to both experimental
and theoretical values of kinematic viscosity in liquids at the minimum
$\nu_{m}$. This similarity gives an insight into the dynamics of QGP particles
at experimental conditions and implies that they are interestingly close to
the  dynamical crossover between the liquid and gas-like states. Furthermore,
we can use the analogy with liquids and predict that (a) the currently
measured “almost perfect fluidity” of QGP Schäfer and Teaney (2009) is not an
inherent property of the QGP but depends on temperature (energy) as in liquids
and (b) future experiments at higher energy will lift the system from the
dynamical crossover into the gas-like regime, corresponding to the increase of
liquid viscosity in Fig. 1, and will detect a viscous response consistent with
gas-like dynamics.

Fourth and finally, it was earlier suggested Hartnoll (2015) that $\nu$ or
transverse momentum diffusivity $D_{s}$ is a universal property in a sense
that it applies to both relativistic and non-relativistic cases, generalizing
the relativistic KSS bound Policastro  _et al._ (2001) commonly used to
discuss the properties of QGP and other systems as mentioned earlier. The
quantitative similarity of $\nu_{m}$ of two vastly different systems found
here supports this view and adds numerical evidence to it.

Acknowledgement. We thank K. Behnia, A. Buchel, S. Cremonini, S.Hartnoll, K.
Landsteiner, P. Romantschke, K. Schalm and J. Zaanen for fruitful discussions
and interesting comments. K. T. thanks EPSRC for support. M. B. acknowledges
the support of the Spanish MINECO’s “Centro de Excelencia Severo Ochoa”
Programme under grant SEV-2012-0249.

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# Main Belt Asteroids with WISE/NEOWISE: Near-Infrared Albedos

Joseph R. Masiero11affiliation: Jet Propulsion Laboratory/Caltech, 4800 Oak
Grove Dr., MS 183-601, Pasadena, CA 91109<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>, T.
Grav22affiliation: Planetary Science Institute, Tucson, AZ<EMAIL_ADDRESS>, A.
K. Mainzer11affiliation: Jet Propulsion Laboratory/Caltech, 4800 Oak Grove
Dr., MS 183-601, Pasadena, CA 91109<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>, C. R.
Nugent11affiliation: Jet Propulsion Laboratory/Caltech, 4800 Oak Grove Dr., MS
183-601, Pasadena, CA 91109<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>, J. M.
Bauer11affiliation: Jet Propulsion Laboratory/Caltech, 4800 Oak Grove Dr., MS
183-601, Pasadena, CA 91109<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>33affiliation:
Infrared Processing and Analysis Center, Caltech, Pasadena, CA , R.
Stevenson11affiliation: Jet Propulsion Laboratory/Caltech, 4800 Oak Grove Dr.,
MS 183-601, Pasadena, CA 91109<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>, S.
Sonnett11affiliation: Jet Propulsion Laboratory/Caltech, 4800 Oak Grove Dr.,
MS 183-601, Pasadena, CA 91109<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We present revised near-infrared albedo fits of $2835$ Main Belt asteroids
observed by WISE/NEOWISE over the course of its fully cryogenic survey in
2010. These fits are derived from reflected-light near-infrared images taken
simultaneously with thermal emission measurements, allowing for more accurate
measurements of the near-infrared albedos than is possible for visible albedo
measurements. As our sample requires reflected light measurements, it
undersamples small, low albedo asteroids, as well as those with blue spectral
slopes across the wavelengths investigated. We find that the Main Belt
separates into three distinct groups of $6\%$, $16\%$, and $40\%$ reflectance
at $3.4~{}\mu$m. Conversely, the $4.6~{}\mu$m albedo distribution spans the
full range of possible values with no clear grouping. Asteroid families show a
narrow distribution of $3.4~{}\mu$m albedos within each family that map to one
of the three observed groupings, with the (221) Eos family being the sole
family associated with the $16\%$ reflectance $3.4~{}\mu$m albedo group. We
show that near-infrared albedos derived from simultaneous thermal emission and
reflected light measurements are an important indicator of asteroid taxonomy
and can identify interesting targets for spectroscopic followup.

##  1 Introduction

The Wide-field Infrared Survey Explorer (WISE, Wright et al., 2010) performed
an all-sky survey in the thermal infrared, imaging each field of view
simultaneously in four infrared wavelengths during the fully cryogenic portion
of the mission and in the two shortest wavelengths when the mission continued
as the Near-Earth Object WISE survey (NEOWISE, Mainzer et al., 2011a). The
four WISE bandpasses are referred to as W1, W2, W3, and W4, and cover the
wavelength ranges of $3.1-3.8~{}\mu$m, $4.1-5.2~{}\mu$m, $7.6-16.3~{}\mu$m,
and $19.8-23.4~{}\mu$m respectively, with photometric central wavelengths of
$3.4~{}\mu$m, $4.6~{}\mu$m, $12~{}\mu$m, and $22~{}\mu$m respectively (Wright
et al., 2010).

The single-frame WISE/NEOWISE data allow us to investigate the thermal
emission and reflectance properties of the minor planets of the Solar system.
Due to their proximity to the Sun, near-Earth objects (objects with perihelia
$q<1.3~{}$AU) are typically dominated by thermal emission in W2 (and
occasionally even in W1), while for more distant objects W2 measures a
combination of reflected and emitted light and W1 is dominated by reflected
light. For all minor planets observed by NEOWISE, bands W3 and W4 are
dominated by thermal emission.

Measurements of the thermal emission from asteroids can be used to determine
the diameter of these bodies through the application of thermal models such as
the Near Earth Asteroid Thermal Model (NEATM, Harris, 1998). NEATM assumes
that the asteroid is a non-rotating sphere with no emission from the night
side, and a variable beaming parameter ($\eta$) is used to account for
variability in thermophysical properties and phase effects. NEATM provides a
rapid method of determining diameter from thermal emission data that is
reliable to $\sim 10\%$ when the beaming parameter can be fit (Mainzer et al.,
2011b). Visible light measurements available from the Minor Planet Center
(MPC)111http://www.minorplanetcenter.net can then be combined with these
models to constrain the geometric albedo at visible wavelengths ($p_{V}$).
However, as these data are not simultaneous with the thermal infrared
measurements, uncertainties due to rotation phase, observing geometry, and
photometric phase behavior instill significant systematic errors in $p_{V}$
determinations. The preliminary asteroid thermal fits presented in Mainzer et
al. (2011e, 2012); Masiero et al. (2011, 2012a); Grav et al. (2011a, b) for
the near-Earth objects, Main Belt asteroids, Hildas, and Jupiter Trojans
account for these uncertainties in the determination of the optical $H$
magnitude resulting in a larger relative error on albedo than is found for
diameter.

For objects that were observed by NEOWISE in both thermal emission and near-
infrared (NIR) reflected light, we can simultaneously constrain the diameter
as well as the NIR albedo. As these data were taken at the same time and
observing conditions as the thermal data used to model the diameter, no
assumptions are needed regarding the photometric phase behavior of these
objects, and light curve changes from rotation or viewing geometry do not
contribute to the uncertainty. These NIR albedos will thus be a more precise
indicator of the surface properties than the visible albedos.

The behavior of the NIR region of an asteroid’s reflectance spectrum can be
used as a probe of the composition of the surface. Spectra of asteroids in the
NIR have been used for taxonomic classification (DeMeo et al., 2009), to
constrain surface mineralogy (Gaffey et al., 2002; Reddy et al., 2012a), and
to search for water in the Solar system (Rivkin & Emery, 2010; Campins et al.,
2010). Near-infrared albedo measurements have also been used to identify
candidate metal-rich objects in the NEO population (Harris & Drube, 2014).
Mainzer et al. (2011d) use the ratio of the NIR and visible albedos as a proxy
for spectral slope, and show a correspondence between this ratio and various
taxonomic classifications. This relation was used by Mainzer et al. (2011e) to
determine preliminary classifications for NEOs, while Grav et al. (2011b) and
Grav et al. (2012) expanded upon this technique to taxonomically classify
Hilda and Jupiter Trojan asteroids.

Masiero et al. (2011) presented NIR albedo measurements of Main Belt asteroids
assuming that the albedos at the W1 and W2 wavelengths were identical.
However, for objects that have a sufficient number of detections of reflected
light in multiple NIR bands we can independently constrain each albedo
($p_{W1}$ and $p_{W2}$ for the W1 and W2 bandpasses respectively). In this
work, we present new thermal model fits of the NIR albedos of Main Belt
asteroids (MBAs), allowing $p_{W1}$ and $p_{W2}$ to vary independently. These
albedos allow us to better distinguish different MBA compositional classes.
They are also particularly useful for investigations of collisional families
seen in the Main Belt, which show strongly correlated physical properties
within each family.

##  2 Data and Revised Thermal Fits

To fit for NIR albedos of Main Belt asteroids, we use data from WISE/NEOWISE
all-sky single exposure source table, which are available for download from
the Infrared Science Archive (IRSA222http://irsa.ipac.caltech.edu Cutri et
al., 2012). We extract photometric measurements of all asteroids observed by
WISE following the technique described in Masiero et al. (2011) and Mainzer et
al. (2011a). In particular, we use the NEOWISE observations reported to the
MPC and included in the MPC’s minor planet observation database as the final
validated list of reliable NEOWISE detections of Solar system objects.

For objects with WISE detections in all four bands, we follow the fitting
technique described in Grav et al. (2012) to independently determine the
albedos in bands W1 and W2. This technique uses a faceted sphere with a
temperature distribution drawn from the NEATM model to calculate the predicted
visible and infrared magnitudes for each object. Diameter, beaming parameter,
and visible, W1, and W2 albedos are all varied until a best-fit is found.
Monte carlo simulations of the data using the measurement errors then provide
a constraint on the uncertainty of each parameter. We require at least three
detections in each band above SNR=4 to use that band in our fit. Main Belt
asteroids are typically closer to the Sun at the time of observation than the
Trojans and Hildas discussed in Grav et al. (2012), and thus the measured W2
flux can have a larger contribution of thermal emission for MBAs. Flux in the
W1 band is typically dominated by reflected light for MBAs observed by WISE,
although low albedo objects ($p_{V}<0.1$) at heliocentric distances of
$R_{\sun}<2.5~{}$AU can be thermally dominated in W1 as well.

In order to ensure reliable fits for W2 albedos ($p_{W2}$), we require that
the beaming parameter be fit by the model. The beaming parameter is a variable
in the NEATM fit that consolidates uncertainties in the model due to viewing
geometries and surface thermophysical parameters, and can be characterized as
an enhancement of the thermal emission in the direction of the Sun. Changes in
thermal properties or phase angles will lead to a range of possible beaming
parameters for MBAs. In order to fit the beaming parameter, our model requires
detections of thermal emission in two bands. We also require that the fraction
of flux in W2 from reflected light be at least $10\%$ of the total flux
measured to fit $p_{W2}$. While this should be sufficient to constrain the W2
albedo in most cases, uncertainty in the beaming parameter can lead to large
uncertainties in $p_{W2}$. To fit W1 albedo ($p_{W1}$), we followed the same
procedure for $p_{W2}$, except now requiring the W1 reflected light to be at
least $50\%$ of the total flux. All objects that fulfilled the above
requirements had optically measured magnitudes available in the literature,
and thus allowed us to fit a visible albedo as well.

We present our updated thermal model fits for all objects satisfying the above
constraints in Table 1, where we give the object’s name in MPC-packed format,
absolute $H_{V}$ magnitude and $G$ photometric slope parameter from the MPC
orbit file, associated family from Masiero et al. (2013) (or “…” if the object
is not associated to a family), and our best fit and associated uncertainty on
diameter, beaming parameter ($\eta$), $p_{W1}$, and $p_{W2}$ if the latter
could be constrained (“…” otherwise). Objects with two epochs of coverage have
each epoch listed separately. All errors are statistical and do not include
the systematic errors of $\sim 10\%$ on diameter and $\sim 20\%$ on visible
albedo (cf. Mainzer et al., 2011b, c; Masiero et al., 2011). Systematic errors
will increase the absolute error on the fitted quantities, but do not affect
relative comparisons within our sample, which is the main goal of this paper.
This table contains $3080$ fits of $2835$ unique objects: $709$ fits of $679$
unique objects with constrained $p_{W1}$ and $p_{W2}$; $2371$ fits of $2219$
unique objects with only $p_{W1}$ constrained; $63$ objects that had one epoch
where both NIR albedos were constrained and one epoch where only $p_{W1}$
could be fit.

We note that some of the fits for diameter and beaming parameter (and thus
albedo) are different from those presented in Masiero et al. (2011). Fits from
NEATM using the same data set will give different values for the diameter as
the beaming parameter is varied. In this case, by independently considering
$p_{W1}$ and $p_{W2}$, as opposed to averaging over both for a single value of
$p_{IR}$, the calculated contribution of thermal flux in W2 will vary, which
will result in a refined value of $\eta$ and therefore diameter. For the
majority of cases, diameters are consistent to within $10\%$ of the previous
value, visible albedos are consistent within $20\%$, and infrared albedo and
beaming parameters are consistent within $15\%$. Revised beaming parameters
tend to be $\sim 5\%$ smaller, making diameters $\sim 3\%$ smaller and visible
albedos $\sim 6\%$ larger. W1 infrared albedos tend to increase $\sim 8\%$,
however we are not necessarily comparing similar quantities as the previous
fits assumed W1 and W2 albedos were equal, allowing W2 measurements to alter
the best-fit value.

Table 1: Revised thermal fits of Main Belt asteroids. Table 1 is published in
its entirety in the electronic edition; a portion is shown here for guidance
regarding its form and content

Name | diameter (km) | $\eta$ | $p_{V}$ | $p_{W1}$ | $p_{W2}$ | $H_{V}$ | G | Family
---|---|---|---|---|---|---|---|---
00005 |  108.29$\pm$ 3.70 |  0.87$\pm$0.10 |  0.27$\pm$0.03 |  0.37$\pm$0.03 |  0.20$\pm$0.10 | 6.9 | 0.15 | 00005
00006 |  195.64$\pm$ 5.44 |  0.91$\pm$0.09 |  0.24$\pm$0.04 |  0.35$\pm$0.03 |  0.48$\pm$0.32 | 5.7 | 0.24 | …
00008 |  147.49$\pm$ 1.03 |  0.81$\pm$0.01 |  0.23$\pm$0.04 |  0.38$\pm$0.04 | … | 6.4 | 0.28 | 00008
00009 |  183.01$\pm$ 0.39 |  0.86$\pm$0.01 |  0.16$\pm$0.03 |  0.33$\pm$0.01 | … | 6.3 | 0.17 | …
00009 |  184.16$\pm$ 0.90 |  0.78$\pm$0.01 |  0.16$\pm$0.01 |  0.34$\pm$0.04 | … | 6.3 | 0.17 | …
00011 |  142.89$\pm$ 1.01 |  0.78$\pm$0.02 |  0.19$\pm$0.02 |  0.35$\pm$0.03 | … | 6.6 | 0.15 | …
00012 |  115.09$\pm$ 1.20 |  0.84$\pm$0.02 |  0.16$\pm$0.03 |  0.32$\pm$0.02 | … | 7.3 | 0.22 | 00012
00014 |  140.76$\pm$ 8.41 |  0.84$\pm$0.15 |  0.27$\pm$0.04 |  0.39$\pm$0.06 |  0.19$\pm$0.16 | 6.3 | 0.15 | …
00015 |  231.69$\pm$ 2.23 |  0.79$\pm$0.04 |  0.25$\pm$0.04 |  0.40$\pm$0.04 | … | 5.3 | 0.23 | 00015
00017 |  84.90$\pm$ 2.03 |  0.77$\pm$0.04 |  0.19$\pm$0.03 |  0.37$\pm$0.08 | … | 7.8 | 0.15 | …

The criteria we apply to our fits, discussed above, result in selection
effects in our sample that conspire to under-represent the lowest albedo
asteroids, as these objects are more likely to fall below our detectability
threshold. This bias is a fundamental result of requiring reflected light
observations in W1 and/or W2, but does not affect population surveys based on
emission in W3 or W4. Our sample requirements also drive us to only use the
data from the fully cryogenic portion of the WISE survey. It is possible to
use fully cryogenic data to constrain the diameter and beaming parameter, and
later measurements from either the NEOWISE post-cryo survey or the recently
restarted NEOWISE mission (Mainzer et al., 2014) during a brighter apparition
to constrain the NIR albedo properties (cf. Grav et al., 2012), although this
technique requires a different method of handling that addresses difference in
viewing geometry. We will apply this technique to Main Belt asteroids in
future work.

For objects with a sufficient number of detections in W1 but below our W2
sensitivity limit or our threshold for the fraction of reflected light in W2,
we determine only the W1 albedo. These objects are either too small to reflect
a detectable amount of light in W2, may be dominated by thermal emission in W2
(common for low albedo objects, $p_{W1}<0.1$), or may have a blue spectral
slope over the $3-5~{}\mu$m range and thus “drop out” of W2. Each of these
scenarios will have a different implication for interpreting the distribution
of W2 albedos, most notably that our data are least sensitive to smaller,
lower albedo objects, as well as objects with blue $p_{V}$-$p_{W2}$ or
$p_{W1}$-$p_{W2}$ spectral slopes. Interpretation of the distribution of NIR
albedos or spectral slopes, particularly as a function of taxonomy or size,
must thus be made with the appropriate caveats. We also explore stacking of
the predicted positions of these object in W2 to recover drop-out objects in
future work.

##  3 Discussion

###  3.1 Albedo comparisons

Figure 1 shows $p_{W1}$ and $p_{W2}$ for all Main Belt asteroids with
sufficient data to constrain these parameters, compared to the fitted
diameter. The W1 band is more sensitive than the W2 band (single-frame
$5\sigma$ sensitivity of $\sim 0.22~{}$mJy vs $\sim 0.31~{}$mJy respectively,
Cutri et al., 2012), and W1 detections are less frequently contaminated by
thermal emission, meaning we are able to measure $p_{W1}$ for more asteroids
than have $p_{W2}$ measurements ($2835$ vs $679$). Both data sets show a
strong bias against small, low-albedo asteroids, as is expected for data that
require measurement of a reflected light component. A further bias against
dark objects in $p_{W2}$ due to rising thermal emission overtaking the small
reflected light component is also present. From the data available we see no
evidence for a non-uniform distribution of $p_{W2}$, in contrast to $p_{W1}$
which shows three significant albedo clumps at $p_{W1}\sim 0.06$, $p_{W1}\sim
0.16$, and $p_{W1}\sim 0.4$.

Figure 1: (a) W1 infrared albedo ($p_{W1}$) compared to fitted diameter, where
color also indicates $p_{W1}$ (as used in Figure 7). (b) W2 infrared albedo
($p_{W2}$) compared to diameter. As the W1 and W2 detections are a measurement
of reflected light they are strongly biased by albedo. The dearth of small,
low albedo objects in this plot that are observed in Masiero et al. (2011) is
an artifact of this bias.

Visible albedos for over $136,000$ Main Belt asteroids were presented in
Masiero et al. (2011) and Masiero et al. (2012a). These measurements were
based on the conversion of apparent visible magnitudes from a wide range of
predominantly ground-based surveys to absolute $H_{V}$ magnitudes when the
orbit was determined by the MPC. Absolute magnitude is then converted to a
predicted apparent magnitude during the epoch of the WISE/NEOWISE
observations, often after assuming a photometric $G$ parameter (cf. Bowell et
al., 1989) and assuming rotational variations are averaged over during the set
of thermal infrared observations. These conversions and assumptions will
instill additional uncertainty in the $p_{V}$ determinations beyond what would
be expected from uncertainties in the flux measurements and from thermal
modeling. As a result, the fractional error on $p_{V}$ is typically $50-100\%$
larger than the fractional error on diameter from thermal model fits.

We compare the NIR albedos presented here to these visible albedos in Figures
2 and 3. For the majority of objects, $p_{W1}$ traces $p_{V}$ and can thus be
used as an analog when $p_{V}$ is not available. The uncertainties of the
$p_{W1}$ measurements in our data are smaller than the errors on $p_{V}$, and
thus act as a better constraint of the surface properties. The relationship
between $p_{W2}$ and both $p_{V}$ and $p_{W1}$ is less distinct, and varies
over a large range of values for objects spanning high and low $p_{V}$ and
$p_{W1}$ albedos.

Comparing the $p_{V}$ distribution to Figure 10 from Masiero et al. (2011), we
see that our sample contains significantly fewer low albedo objects than would
be expected from a random sample of all Main Belt asteroids. The lack of low
albedo objects is due primarily to the observational selection effect
imprinted on our dataset by the requirement that the objects be detected in W1
and/or W2 in reflected light. This bias will increase as albedo decreases,
preferentially selecting objects with higher albedos. A survey with deeper
sensitivity in these wavelengths would allow us to probe smaller sizes at all
albedos, but would still be subject to the same observational biases.

Figure 2: W2 albedo vs W1 albedo for $679$ Main Belt asteroids. The color of
the points indicate their visible albedo following the colormap presented in
Masiero et al. (2011) and shown in the colorbar, while the size of the point
traces the fraction of the W2 flux that was due to reflected light. The dotted
line shows a 1-to-1 correspondence; objects below the line will have a blue
spectral slope from W1 to W2, while objects above the line will have a red
slope.

Figure 3: Comparisons (top) and histograms (bottom) of asteroid albedos at
visible ($p_{V}$), W1 ($p_{W1}$), and W2 ($p_{W2}$) wavelengths. Blue
points/lines show objects with all three albedos fit by the thermal model
($679$ objects), while green crosses/dashed lines show objects with only
visible and W1 fits ($2835$ objects). Plots of $p_{W2}$ do not include green
crosses as these objects do not have this parameter constrained by the model
fits. Dotted black lines in the comparison plots indicate a 1-to-1 relation.
While visible and W1 albedos show clear clumping, W2 albedos show no
separation within our measurement errors.

Following Grav et al. (2012), we can use our albedo measurements as a proxy
for spectral slope from visible wavelengths through the NIR. Objects that fall
above the 1-to-1 relationship in the top portion of Figure 3 will have a red
spectral slope across the wavelengths plotted, while objects below this
relation will have a blue spectral slope. As $p_{W2}$ is the most poorly
probed of the three parameters, there will be inherent detection biases
against blue spectral slopes from objects that “drop out” and fall below our
W2 detection threshold. For this reason objects with the bluest slopes,
particularly low-albedo objects, will be under-represented in our fits of
$p_{W2}$.

To better compare the spectral slope information, in Figure 4 we show the
difference in albedo between $p_{V}$, $p_{W1}$, and $p_{W2}$ normalized to the
measured $p_{W1}$ value. Objects with positive values have a red spectral
slope, while objects with negative values have a blue slope. High albedo
objects ($p_{W1}>0.1$) tend to show red slopes from visible to W1 wavelengths,
and then blue slopes between W1 and W2. This behavior is similar to what is
observed for Eucrite meteorites at these wavelengths (Reddy et al., 2012b),
and what would be expected from extrapolating a typical S-type asteroid
spectrum (DeMeo et al., 2009). High albedo objects without a measured $p_{W2}$
albedo show similar visible-W1 slopes to those with a measured $p_{W2}$.

Low albedo objects ($p_{W1}<0.1$) behave quite differently from their high-
albedo counterparts. While slightly red from the visible to W1, these objects
show a wide range of visible-W2 and W1-W2 slopes, from neutral in color to
very red. An important caveat to this is shown by the objects without $p_{W2}$
fits, which have slopes ranging from moderately red to significantly blue.
Blue-sloped objects would be much fainter in W2 than W1 and thus would drop
out from detection or be dominated by thermal emission in W2. It is probable
that there is a population of these objects with blue W1-W2 slopes that are
not represented in our plots. Extrapolting from the NIR spectra of low albedo
objects from DeMeo et al. (2009), we associate our objects that have red
visible-W1 slopes with C-type and D-type objects. We can similarly associate
the objects having blue-slopes with B-type asteroids, however we note that
only $\sim 1\%$ of objects studied by DeMeo et al. (2009) were identified as
B-type asteroids, while $\sim 10\%$ of the asteroids in our study have low
albedo and blue spectral slope. From Neese (2010) we find that the majority of
our blue sloped objects that have Bus-DeMeo taxonomic classifications are
identified as B or Ch class objects, the latter of which represents a fraction
of the spectroscopic sample comparable to the fraction of our sample in this
group. Our blue slope may be indicative of the presence of mineralogical
absorption features in the spectra of low albedo objects at the wavelengths
covered by W1.

Figure 4: Spectral slope, normalized to the W1 albedo, over visible-to-W1
wavelengths (left), visible-to-W2 wavelengths (center) and W1-to-W2
wavelengths (right), compared to the W1 albedo. The dotted line shows a
neutral slope; objects with positive slope values have red spectra and objects
with negative values have blue spectra. High albedo objects tend to be red-
sloped from visible to W1, and blue sloped from W1 to W2, while low albedo
objects tend to be flat or red across the whole range. Color/shape of points
is the same as used in Figure 3.

###  3.2 D-type asteroids

Asteroids with D-type taxonomic classifications become increasingly common as
distance from the Sun grows, from the Main Belt through the Jupiter Trojan
population (DeMeo & Carry, 2013). These objects, especially the Jupiter
Trojans, were likely implanted from a more distant reservior during the early
chaotic evolution of the Solar system (Morbidelli et al., 2005) and thus
represent primitive material distinct from objects that formed in the warmer
region of the Main Belt. NIR albedo can be used to probe the distribution of
these objects and differentiate between classes of primitive bodies. Grav et
al. (2012) compare $p_{W1}$ and $p_{W2}$ to distinguish asteroids with D-type
taxonomic classification from those with C- and P-type, and are able to
determine the overall population fraction of D-type objects in the Jupiter
Trojan and Hilda populations. They find that the majority of Jupiter Trojans
are D-type at all sizes, while the Hilda population transitions from a
minority of D-types at diameters $D>40~{}$km to a majority at smaller sizes.

Following Grav et al. (2012), we show in Figure 5 an expanded view of the
objects with lowest infrared albedos. We highlight the region of albedo-space
that is occupied by D-type asteroids in the Trojan and Hilda populations. The
diameter and albedo fits from Grav et al. (2012) rely on the same model and
assumptions as we use here, and so comparisons between the two populations
should only depend on the random error associated with the fits. Only $2\%$ of
all objects for which we measure $p_{W1}$ and $p_{W2}$ fall in this region;
with the exception of (114) Kassandra and (267) Tirza (which are spectrally
classified as T- and D-type objects, respectively), all other candidate D-type
objects are in the outer Main Belt and have diameters between
$10~{}$km$<D<40~{}$km, consistent with the diameter regime where D-types
dominate the Hilda asteroids. One object, (1755) Lorbach, is identified as an
S-type in Neese (2010), but this classification relies on only two optical
colors. For the outer Main Belt, we do not see a significant population of
D-type objects like what is observed in the Hildas and Trojans (DeMeo & Carry,
2013), but this is expected from the lower efficiency of dynamical
implantation compared with the Hilda and Trojan populations Levison et al.
(2009).

We find no objects in the inner Main Belt with albedos consistent with D-type
objects. This is in contrast to the results of DeMeo et al. (2014) who find a
small population of these bodies; however, this difference can be understood
through the selection effects in our survey. Although our sample probes a
large number of objects with semimajor axis $a<2.5~{}$AU only a handful have
low albedo. Inner Main Belt low albedo asteroids are more likely to have
significant thermal emission in W2, so we are not able to determine $p_{W2}$
for these objects and they will not appear in our analysis.

Figure 5: W1 and W2 albedos for all measured objects. The red ellipse marks
the region populated by D-type asteroids as identified in Grav et al. (2012).
This taxonomic classification shows no significant representation in the Main
Belt objects studied here.

###  3.3 Low-$p_{W1}$/High-$p_{W2}$ objects

Figure 5 shows a group of objects with low visible and W1 albedos
($p_{V},p_{W1}<0.1$) but high W2 albedo ($p_{W2}>0.1$), which also appear as
the objects with the reddest W1-W2 slopes in Figure 4. This class of object
does not have an analog in the Jupiter Trojan or Hilda populations (Grav et
al., 2012) where we find parallels to other low-albedo MBA populations.
Objects from this group that have spectroscopic or photometric taxonomic
classifications in PDS are typically designated as C-type or a related
subclass (Neese, 2010). There are occasional objects with other
classifications such as X-, F- and S-, or dual classifications, although often
in these cases the designation is based on only 2 color indices and so is of
low reliability. It is possible that this group could represent a different
class of objects that is not found in the more distant Solar system
populations, or instead could be a failure of the thermal model to converge
for certain objects with low albedos.

In order to test if these objects are a result of a failure of the fitting
routine, we take all objects in our fitted population with $p_{W1}<0.1$ and
compare the set with $p_{W2}\geq 0.1$ to the set with $p_{W2}<0.1$ (referred
to as “low-high” and “low-low” respectively). The low-high and low-low test
sets are approximately the same size (44 vs 48 objects), and have similar
distributions of semimajor axes, eccentricities, inclinations, $p_{V}$ and
$p_{W1}$. The primary difference between these two groups is that the low-high
objects have significantly smaller heliocentric distances at the time of
observation than the low-low objects, resulting in higher subsolar
temperatures. The diameters of the low-high objects are also
characteristically smaller than those of the low-low group, however we cannot
distinguish if this is an actual difference between the groups or is a change
in sensitivity as a result of the low-high objects being closer to the Sun and
telescope at the time of observation, and thus warmer and brighter.

The asteroid (656) Beagle is a particularly interesting case for testing the
differences between these two sets of objects. NEOWISE observed this asteroid
at two different epochs, both while fully cryogenic, with good sensitivity at
all four bands. One epoch of observations results in a NEATM best-fit that
falls into the low-low group, while the other epoch falls into the low-high
group. The low-high epoch data were taken when Beagle was $0.21~{}$ AU closer
to the Sun ($2.82~{}$AU vs $3.03~{}$AU for the low-low case), following the
trend seen for the overall population. The best-fit for NEATM in the low-high
epoch has a beaming parameter of $\eta=1.46$ and a diameter of $D=62~{}$km
while the low-low epoch has best-fit values of $\eta=1.03$ and $D=48~{}$km
which is the reverse of the diameter trend mentioned above. This large
disagreement in diameter is not unexpected given the difference in best-fit
beaming parameter which is inversely proportional to the fourth power of the
subsolar temperature used in the NEATM model, and thus will change the model’s
emitted flux.

The observations used for our fits were visually inspected, as well as
compared to the WISE all-sky atlas of stationary sources, and show no
significant contamination by background stars or galaxies. We note that (656)
Beagle has a large amplitude lightcurve (A$>1~{}$mag) and a period of
$7.035~{}$hours (Menke, 2005). Although large amplitudes can increase
uncertainty in the fits, our data consist of 12 data points over $1$ day and
15 data points over $1.25$ days, so both epochs cover multiple rotations. As
such, light curve variations should be averaged over by our fits, and should
only contribute a small amount to the total uncertainty in the fit.

As a test of our model, we perform a NEATM fit using only bands W1, W2, and W3
as constraints, assuming the W4 measurements are anomalously high, and a fixed
beaming parameter of $\eta=1.0$. When using a fixed beaming parameter we
cannot adequately constrain $p_{W2}$, and so assume it is equal to $p_{W1}$.
For these restricted fits, both epochs converge to diameters that agree to
within $10\%$, but they cannot reproduce the measured magnitudes as well as
the full-fit case. As we are using one fewer constraint but two fewer
variables, this is not surprising. The fits for (656) Beagle given by Masiero
et al. (2011) are nearly identical to these restricted fits, but also cannot
fully reproduce the measured magnitudes, particularly for the low-high epoch.
Restricting our model further and only fitting W1 and W3, we find that both
epochs converge to nearly identical diameters, and visible and infrared
albedos.

We can understand these results by looking at where the best-fit model
deviates from the data. For the low-high epoch, the full NEATM fit cannot
reproduce the W2, W3, and W4 fluxes simultaneously, with the W2 and W4
measurements showing excesses not observed in W3. Our full model finds a best
fit solution allowing W3 and W4 to determine the diameter and beaming which
under-produces flux in W2, but corrects that by increasing $p_{W2}$. If we
ignore the W4 measurements, the W2 and W3 fluxes still cannot be reproduced in
the low-high epoch solely with thermal emission and reflected light without
resorting to extreme changes in $p_{W2}$.

One possible explanation for the disagreement between epochs is that we are
observing significant differences between the thermal emission in the morning
and afternoon hemispheres of the asteroid. If (656) Beagle has a relatively
high thermal inertia, there may be a significant lag to the thermal re-
emission of incident light which is not accounted for in the NEATM model. Our
two epochs of observation are at phase angles of $\alpha\sim 20^{\circ}$, but
on opposite sides of the body. (656) Beagle is on a low-inclination orbit, so
if we assume the rotation pole is oriented perpendicular to the orbital and
ecliptic planes and that the rotation is prograde, then the data from the low-
high epoch would correspond to the afternoon hemisphere and the data from the
low-low epoch would correspond to the morning hemisphere. Future work will
implement a full thermophysical model of this object to test if the W2 and W4
excesses can be explained by a morning/afternoon dichotomy. For all other
objects in the low-high group which were only observed at a single epoch, we
cannot currently differentiate between poor fits to the beaming parameter and
actual excesses in the W2 and/or W4 bands.

An alternate possibility is that these fits are indicative of problems with
the flux measurement of partially saturated sources in the WISE data. Cutri et
al. (2012) discuss the process by which fluxes are measured for saturated
sources through PSF-fitting photometry. Flux measurements are available for
sources many magnitudes above the brightness where the central pixel saturates
through fitting of the PSF wings, however for very bright sources in bands W2
and W3, there appears to be a slight over-estimation of the fluxes. None of
the objects we fit here had W2 magnitudes in this saturated regime, however
the majority of objects with $p_{W1}<0.1$ had W3 magnitudes in this
problematic region.

We correct for saturation estimation issues in our thermal model, however
there is the potential that the error for asteroidal sources cannot be
adequately described by this correction, which was calibrated for stars. The
difference in the spectral energy distributions through the W3 bandpass of
hot, blue stars and cooler, red asteroids potentially could result in
differences deep in the wings of the PSF for each type of source that are not
fully encompassed by the color correction. These subtle changes can have a
significant impact on saturated sources where only the wings are available for
profile-fitting, however there are an insufficient number of well-calibrated,
W3-bright sources with the appropriate spectral energy distribution to correct
for this effect. Although this error may only have a small effect on other
physical parameters within our modeled systematic uncertainties, due to W2’s
position on the Wien’s side of Main Belt asteroid thermal emission for some of
our objects, a small change in W3 can result in a large change in W2 flux, and
thus our interpretation of the W2 albedo. As such, caution is strongly
encouraged in interpreting fits for objects with very bright W3 magnitudes
(W3$<4~{}$mag).

###  3.4 NIR Albedos of Asteroid Families

The distributions of visible albedos for members of each asteroid family have
much narrower spread than the albedo distribution of the Main Belt as a whole
(Masiero et al., 2011) as is expected from a population resulting from the
collisional breakup of a single parent body. As the (4) Vesta family shows a
narrow albedo distribution but originated from a differentiated body, we do
not expect the albedo distributions of other cratering-event families that may
have been partially- or fully-melted to differ significantly from non-
differentiated families. It is possible that families formed from the complete
disruption of a differentiated parent body may show a broader albedo
distribution, though we do not see any evidence for a case like this in our
data. Visible albedo can also be used to improve family membership lists by
rejecting outlier objects that are dynamically similar to the family (Masiero
et al., 2013; Walsh et al., 2013). Using the refined family lists from Masiero
et al. (2013) we investigate the distribution of $p_{W1}$ for families as a
more accurate tracer of the surface properties of these asteroids.

Figure 6 shows the distribution of $p_{W1}$ albedos for the $8$ families where
more than $20$ members had a $p_{W1}$ albedo measurement. Asteroid families
break into three clear groupings, following the three peaks in the albedo
distribution shown in Figure 3. Our dataset depends on reflected light
measurements, so high-albedo families are over-represented in the distribution
compared with the population of all known families, which is dominated by low-
albedo families. The only low NIR-albedo family with more than $20$ measured
objects was (24) Themis, however other families such as (10) Hygiea, (145)
Adeona, (276) Adelheid, (511) Davida, (554) Peraga (equivalent to other lists’
Polana family), and (1306) Scythia also show low NIR albedos, but these
families contain only a small number of objects with measured $p_{W1}$. The
families (4) Vesta, (8) Flora, (15) Eunomia, (208) Lacrimosa, (472) Roma, and
(2595) Gudiachvili all have high $p_{W1}$ and show only a small spread in mean
albedo, while (135) Hertha (equivalent to other lists’ Nysa family) and (254)
Augusta join them at a lower significance level.

The (221) Eos family is the only one of the large families to have a moderate
NIR albedo, in between the high- and low-albedo populations, indicating that
this family has surface properties that are rare among the large Main Belt
asteroids. The $p_{W1}$ values for this family confirm the observed moderate
visible albedo as a separate grouping that could not be conclusively
distinguished from the high $p_{V}$ population by Masiero et al. (2013). The
Eos family parent has a K-type spectral taxonomy in the Bus-DeMeo system
(DeMeo et al., 2009). K-type objects are considered ‘end-members’ of the
classification scheme, and have a $1~{}\mu$m absorption feature typically
associated with silicates such as olivine, but are distinct in spectroscopic
principal component space from the majority of S-class objects. Clark et al.
(2009) and Hardersen et al. (2011) associate K-type objects with the parent
body of carbonaceous chondrite meteorites, specifically CO chondrites, while
Mothé-Diniz (2005) show evidence that (221) Eos may have been partially
differentiated. Broz & Morbidelli (2013a) calculate the time since the breakup
of the (221) Eos family as $1.5-1.9~{}$Gyr, making it one of the oldest Main
Belt families with a measured age. These observed properties, when taken
together, paint the Eos family as having a unique evolutionary history that
can be studied using remote observations in combination with hand samples from
the meteorite record to trace the early history of the Solar system.

Figure 6: W1 albedo distributions for $8$ asteroid families with more than
$20$ measured NIR albedos, identified by the family ID given in Masiero et al.
(2013) and Table 2. Families show narrow distributions of albedos correlating
with one of three major albedo groupings.

We note that approximately half of the objects fit for the (298) Baptistina
family had albedos similar to the Eos family, while the remainder appear to be
drawn from the high-albedo group. This result is based on only a small number
of measured Baptistina members, and thus is not conclusive, however if
confirmed would further impede attempts to assign a unique composition to this
family (cf. Reddy et al., 2011) or determine its age and evolution (cf.
Masiero et al., 2012b).

Figure 7 shows the proper orbital eccentricity and inclination of all objects
with measured $p_{W1}$. The Main Belt is split into three regions by proper
semi-major axis ($a$): the inner-Main Belt (IMB, $1.8~{}$AU$<a<2.5~{}$AU), the
middle-Main Belt (MMB, $2.5~{}$AU$<a<2.82~{}$AU), and the outer-Main Belt
(OMB, $2.82~{}$AU$<a<3.6~{}$AU). We show separately the objects that were
associated with an asteroid family by Masiero et al. (2013) and those that are
members of the background population. The (221) Eos family stands out
distinctly in the belt, although objects with similar $p_{W1}$ are present in
the background population in all three regions.

These plots show the clear trend of albedo decreasing with distance from the
Sun, however our observational bias against small, low albedo objects
amplifies this effect. Broz et al. (2013b), Carruba et al. (2013), and Masiero
et al. (2013) observe halos of objects beyond the limits of typical family-
identification techniques, however we do not see evidence for these halos in
the background population in our dataset. Halos are typically associated with
asteroids that have dispersed a large distance from the family center via
Yarkovsky and gravitational forces, which will have smaller diameters than
objects that were above our sensitivity limit for $p_{W1}$ determination. Our
significantly smaller sample size than what is typically used in surveys
investigating family halos may also contribute to their absence in our data.

Figure 7: Proper orbital inclination (incl) vs eccentricity (ecc) for inner-
(left), middle- (center), and outer-Main Belt populations (right), for objects
associated with families (top) and background objects not linked to families
(bottom). Colors of the points map the W1 albedo (from green to black to
magenta for increasing $p_{W1}$), following Figure 1a.

In Table 2 we present the orbital and physical properties for all families
identified in Masiero et al. (2013) that had at least one member with a fitted
NIR albedo. We list the name of the family, average proper orbital elements,
largest (Dmax) and smallest diameter (Dmin) represented in our sample, W1
albedos with standard deviations, and number of family members with data
sufficient to fit. We also provide for reference the mean $p_{V}$ and standard
deviation from Masiero et al. (2013). For cases where only a single body had a
measured $p_{W1}$ (often but not always the parent body of the family), Dmin
is marked with a ‘…’ entry and no standard deviation is given for the mean W1
albedo for families with less than 10 members. Asteroids that have been
incorrectly associated with families may have very different mineralogies and
thus spectral behavior in the NIR, which could make those objects more likely
to fulfill the selection requirements for measured $p_{W1}$. Thus, particular
caution is necessary when dealing with families suffering small number
statistics, especially families with only a single $p_{W1}$-fit object. We
note that the mean $p_{V}$ albedos presented in Masiero et al. (2013) are
based on larger numbers of objects and so will generally be more accurate than
the mean $p_{W1}$ values given here.

It is also possible to use the W1 albedo to further refine family memberships,
particularly for confused cases such as the Nysa-Polana complex. Masiero et
al. (2013) divided this complex into a high albedo component with largest body
(135) Hertha and a low albedo component with largest body (554) Peraga which
is nearly twice the diameter of (142) Polana. We use NIR albedo to reject
objects from the low-albedo family that had moderate visible albedos but W1
albedos characteristic of the high-albedo family. Asteroids (261), (1823),
(2717), and (15112) can thus be rejected as members of the (554) Peraga group
based on W1 albedo. We note that because of a typo in Masiero et al. (2013)
(135) Hertha was mistakenly listed as associated with (554) Peraga instead of
with its own family, which we correct here. Walsh et al. (2013) present
dynamical arguments to divide the (554) Peraga family into two sub-families,
however we are unable to see any distinction between these groups in visible
or W1 albedo.

Table 2: Average orbital and physical properties for asteroid family members
with measured $p_{W1}$. Mean $p_{V}$ values are taken from Masiero et al.
(2013)

Family | semimajor axis | eccentricity | inclination |  Dmax |  Dmin | $<p_{V}>$ | $<p_{W1}>$ | Sample
---|---|---|---|---|---|---|---|---
| (AU) |  | (deg) | (km) | (km) |  |  | Size
00434 | 1.937 | 0.077 | 20.947 | 7.62 | 3.01 |  0.725 $\pm$ 0.172 | 0.736 | 3
00254 | 2.197 | 0.122 | 4.202 | 11.85 | 5.71 |  0.298 $\pm$ 0.105 | 0.420 | 7
00008 | 2.244 | 0.141 | 5.251 | 155.74 | 3.86 |  0.291 $\pm$ 0.091 |  0.435 $\pm$ 0.074 | 70
00298 | 2.256 | 0.148 | 5.723 | 12.32 | 5.36 |  0.146 $\pm$ 0.034 | 0.284 | 9
00163 | 2.325 | 0.215 | 5.008 | 4.78 | … |  0.053 $\pm$ 0.016 | 0.401 | 1
00587 | 2.338 | 0.222 | 23.901 | 12.23 | 4.41 |  0.310 $\pm$ 0.090 | 0.421 | 4
01646 | 2.353 | 0.102 | 8.002 | 12.47 | 11.57 |  0.204 $\pm$ 0.081 | 0.221 | 2
00004 | 2.366 | 0.101 | 6.518 | 109.51 | 4.53 |  0.357 $\pm$ 0.110 |  0.466 $\pm$ 0.129 | 22
00012 | 2.390 | 0.185 | 8.853 | 126.64 | 7.62 |  0.066 $\pm$ 0.021 | 0.318 | 2
00135 | 2.405 | 0.178 | 2.455 | 82.15 | 3.77 |  0.280 $\pm$ 0.088 |  0.372 $\pm$ 0.071 | 13
00302 | 2.407 | 0.110 | 1.576 | 6.11 | … |  0.062 $\pm$ 0.021 | 0.422 | 1
00554 | 2.418 | 0.157 | 3.049 | 55.18 | 11.75 |  0.061 $\pm$ 0.021 |  0.054 $\pm$ 0.012 | 13
00752 | 2.463 | 0.091 | 5.049 | 60.85 | … |  0.053 $\pm$ 0.014 | 0.048 | 1
13698 | 2.469 | 0.118 | 6.534 | 5.31 | … |  0.367 $\pm$ 0.098 | 0.412 | 1
01658 | 2.560 | 0.172 | 7.749 | 13.81 | 5.97 |  0.255 $\pm$ 0.074 | 0.343 | 3
00472 | 2.562 | 0.094 | 15.009 | 47.04 | 6.25 |  0.261 $\pm$ 0.079 |  0.363 $\pm$ 0.051 | 39
00005 | 2.576 | 0.198 | 4.514 | 113.00 | … |  0.240 $\pm$ 0.105 | 0.365 | 1
00606 | 2.587 | 0.179 | 9.631 | 39.53 | … |  0.117 $\pm$ 0.028 | 0.137 | 1
05079 | 2.601 | 0.247 | 11.730 | 14.76 | … |  0.068 $\pm$ 0.020 | 0.047 | 1
00404 | 2.628 | 0.229 | 13.062 | 105.41 | 73.07 |  0.060 $\pm$ 0.025 | 0.059 | 2
00015 | 2.630 | 0.149 | 13.181 | 299.21 | 4.45 |  0.263 $\pm$ 0.084 |  0.382 $\pm$ 0.073 | 126
00569 | 2.634 | 0.175 | 2.659 | 13.18 | … |  0.054 $\pm$ 0.016 | 0.064 | 1
00145 | 2.676 | 0.170 | 11.642 | 132.59 | 14.70 |  0.062 $\pm$ 0.018 | 0.055 | 8
00410 | 2.727 | 0.253 | 8.824 | 27.28 | 8.49 |  0.085 $\pm$ 0.028 | 0.075 | 3
00539 | 2.739 | 0.164 | 8.274 | 56.04 | … |  0.061 $\pm$ 0.023 | 0.039 | 1
00396 | 2.742 | 0.168 | 3.497 | 37.29 | … |  0.093 $\pm$ 0.024 | 0.115 | 1
00808 | 2.744 | 0.132 | 4.902 | 37.68 | 9.51 |  0.232 $\pm$ 0.071 | 0.380 | 2
00363 | 2.750 | 0.045 | 5.480 | 19.34 | … |  0.068 $\pm$ 0.018 | 0.079 | 1
00128 | 2.750 | 0.088 | 5.181 | 193.08 | … |  0.075 $\pm$ 0.024 | 0.071 | 1
01734 | 2.769 | 0.194 | 7.951 | 23.82 | 8.72 |  0.056 $\pm$ 0.017 | 0.074 | 6
00847 | 2.777 | 0.070 | 3.742 | 30.08 | 8.78 |  0.218 $\pm$ 0.075 | 0.359 | 5
00272 | 2.783 | 0.048 | 4.232 | 25.67 | 21.09 |  0.047 $\pm$ 0.014 | 0.109 | 3
00322 | 2.783 | 0.198 | 9.521 | 73.15 | … |  0.078 $\pm$ 0.027 | 0.193 | 1
01128 | 2.788 | 0.048 | 0.659 | 48.63 | … |  0.048 $\pm$ 0.013 | 0.046 | 1
02595 | 2.791 | 0.132 | 9.068 | 14.62 | 6.59 |  0.262 $\pm$ 0.075 |  0.343 $\pm$ 0.059 | 21
01668 | 2.806 | 0.178 | 4.152 | 25.83 | … |  0.052 $\pm$ 0.014 | 0.048 | 1
03985 | 2.851 | 0.123 | 15.052 | 22.11 | 16.19 |  0.176 $\pm$ 0.058 | 0.213 | 3
00081 | 2.854 | 0.180 | 8.233 | 123.96 | … |  0.056 $\pm$ 0.016 | 0.053 | 1
00208 | 2.875 | 0.048 | 2.129 | 49.99 | 10.36 |  0.237 $\pm$ 0.063 |  0.335 $\pm$ 0.032 | 59
00845 | 2.940 | 0.036 | 11.999 | 58.53 | … |  0.061 $\pm$ 0.017 | 0.055 | 1
00179 | 2.972 | 0.076 | 9.027 | 74.58 | … |  0.223 $\pm$ 0.069 | 0.326 | 1
00816 | 3.004 | 0.145 | 13.168 | 50.09 | … |  0.051 $\pm$ 0.026 | 0.054 | 1
00221 | 3.020 | 0.077 | 10.181 | 95.62 | 10.03 |  0.158 $\pm$ 0.048 |  0.190 $\pm$ 0.065 | 186
00283 | 3.070 | 0.109 | 8.917 | 145.55 | 17.68 |  0.048 $\pm$ 0.019 | 0.084 | 2
02621 | 3.086 | 0.128 | 12.128 | 47.92 | … |  0.081 $\pm$ 0.029 | 0.066 | 1
01113 | 3.112 | 0.137 | 14.093 | 48.37 | … |  0.074 $\pm$ 0.031 | 0.350 | 1
00780 | 3.117 | 0.070 | 18.186 | 114.26 | … |  0.056 $\pm$ 0.018 | 0.060 | 1
01040 | 3.122 | 0.197 | 16.728 | 22.67 | 7.56 |  0.225 $\pm$ 0.075 | 0.348 | 4
00511 | 3.138 | 0.192 | 14.439 | 285.84 | 23.46 |  0.065 $\pm$ 0.026 | 0.076 | 8
00024 | 3.142 | 0.153 | 1.457 | 193.54 | 17.43 |  0.068 $\pm$ 0.021 |  0.073 $\pm$ 0.012 | 95
00928 | 3.143 | 0.193 | 16.359 | 62.54 | 24.29 |  0.075 $\pm$ 0.038 | 0.057 | 2
00010 | 3.143 | 0.130 | 5.514 | 153.58 | 16.69 |  0.070 $\pm$ 0.023 |  0.079 $\pm$ 0.035 | 17
03330 | 3.154 | 0.199 | 10.149 | 15.49 | … |  0.044 $\pm$ 0.015 | 0.053 | 1
00490 | 3.165 | 0.061 | 9.323 | 79.87 | … |  0.069 $\pm$ 0.022 | 0.050 | 1
00778 | 3.169 | 0.262 | 14.282 | 19.36 | … |  0.066 $\pm$ 0.020 | 0.070 | 1
01306 | 3.170 | 0.091 | 16.448 | 72.24 | 15.54 |  0.061 $\pm$ 0.021 |  0.107 $\pm$ 0.056 | 14
00031 | 3.177 | 0.196 | 26.445 | 281.98 | 17.31 |  0.057 $\pm$ 0.016 | 0.068 | 3
00618 | 3.189 | 0.058 | 15.879 | 131.23 | … |  0.056 $\pm$ 0.018 | 0.063 | 1
00702 | 3.190 | 0.021 | 21.598 | 196.47 | 26.15 |  0.066 $\pm$ 0.022 | 0.071 | 3
00276 | 3.190 | 0.072 | 22.189 | 100.36 | 21.98 |  0.068 $\pm$ 0.022 |  0.073 $\pm$ 0.009 | 11
01303 | 3.215 | 0.126 | 19.023 | 102.43 | 28.58 |  0.049 $\pm$ 0.017 | 0.069 | 2
00087 | 3.485 | 0.054 | 9.846 | 288.38 | … |  0.057 $\pm$ 0.017 | 0.082 | 1

##  4 Conclusions

We present revised thermal model fits for Main Belt asteroids, allowing for
the albedo in each of the near-infrared reflected wavelengths to be fit
independently. The $3.4~{}\mu$m and $4.6~{}\mu$m spectral regions covered by
the WISE/NEOWISE W1 and W2 bandpasses are poorly probed in ground-based
spectroscopy but can be used to provide insight into asteroid mineralogical
composition by constraining spectral slope. In total we present $3080$ fits of
$p_{W1}$ and/or $p_{W2}$ for $2835$ unique Main Belt objects.

The MBA population has three distinct peaks in our observed $p_{W1}$
distribution at $p_{W1}\sim 0.06$, $p_{W1}\sim 0.16$, and $p_{W1}\sim 0.4$.
The high and low $p_{W1}$ peaks correspond to the high and low visible albedo
groups observed previously, while the moderate $p_{W1}$ peak corresponds to an
intermediate visible albedo that is blended with the high $p_{V}$ objects in
visible albedo distributions. The distribution of albedos we measure have a
larger fraction of high-albedo objects than what was observed for the MBA
visible albedo distribution, however this is an effect of the biases in our
sample selection.

Asteroid families have narrow $p_{W1}$ distributions corresponding to one of
the three observed $p_{W1}$ peaks. The (221) Eos family represents the only
significant concentration of objects near the peak at $p_{W1}\sim 0.16$,
although other objects with this albedo that are not related to asteroid
families are scattered throughout the entire Main Belt region. This family
also corresponds to an unusual ‘end member’ taxonomic classification, K-type,
that has been suggested to correspond to a partially differentiated parent or
olivine-rich mineralogy. NIR albedo measurements provide a way to rapidly
search the known population for candidate K-type objects in the Main Belt, and
are a powerful tool that acts as a proxy for asteroid taxonomic type.

Our results show that the majority of high albedo objects, believed to have
surface compositions dominated by silicates and similar to ordinary chondrite
meteorites, show an overall reddening from visible to W1 wavelengths similar
to what is seen in the NIR. The spectra become blue from W1 to W2, which is
also seen in some meteorite populations, particularly the Eucrites. This
overall picture is consistent with a primarily-silicate dominated composition.
Objects with moderate infrared albedos show similar behavior across the
wavelengths probed here, although the lower albedo value at W1 may indicate
subtle differences in composition from the high albedo population or even a
mix of different mineralogies.

The low albedo objects in our sample show a much wider range of behavior in
these spectral regions. Many object show red slopes across all wavelengths
consistent with the NIR spectral behavior of C/D/P-type objects. However
approximately $10\%$ of our population show a blue slope from visible to W1,
even in spite of the biases against blue-sloped, low-albedo objects in our
sample. These objects are associated with B and Ch spectral taxonomies. The
blue visible-to-W1 spectral slope in the Ch class objects may be indicative of
a significant absorption feature at W1 wavelengths from minerals such as
carbonates.

The fits presented here are based on reflected light, and thus our sample will
not accurately represent the true distribution of $p_{W1}$ or $p_{W2}$. Small,
low albedo asteroids as well as objects with blue NIR spectral slopes are more
likely to be undetected in the W1 and/or W2 wavelengths and thus
underrepresented in our population distributions. A larger survey with greater
sensitivity in these spectral regions is required to extend these results to a
population comparable to the one with measured diameters and visible albedos.

## Acknowledgments

JM was partially supported by a NASA Planetary Geology and Geophysics grant.
CN, RS, and SS were supported by an appointment to the NASA Postdoctoral
Program at JPL, administered by Oak Ridge Associated Universities through a
contract with NASA. We thank the referee for the helpful comments that greatly
improved this manuscript. This publication makes use of data products from the
Wide-field Infrared Survey Explorer, which is a joint project of the
University of California, Los Angeles, and the Jet Propulsion
Laboratory/California Institute of Technology, funded by the National
Aeronautics and Space Administration. This publication also makes use of data
products from NEOWISE, which is a project of the Jet Propulsion
Laboratory/California Institute of Technology, funded by the Planetary Science
Division of the National Aeronautics and Space Administration. This research
has made use of the NASA/IPAC Infrared Science Archive, which is operated by
the Jet Propulsion Laboratory, California Institute of Technology, under
contract with the National Aeronautics and Space Administration.

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