11institutetext: Department of Astronomy, Faculty of Mathematics, University
of Belgrade, Studentski trg 16, 11000 Belgrade, Serbia
11email<EMAIL_ADDRESS>22institutetext: Institute of Astronomy,
Charles University, V Holešovičkách 2, CZ-180 00 Prague 8, Czech Republic
33institutetext: Institute for Astronomy, University of Hawaii, Honolulu HI,
96822, USA

# Low thermal conductivity of the super-fast rotator (499998) 2011 PT

Marco Fenucci 11 Bojan Novaković 11 David Vokrouhlický 22 Robert J. Weryk 33

(Received — / Accepted —)

###### Abstract

Context. Asteroids up to a few tens of meters in diameter may spin very fast,
completing an entire rotation in a period of few minutes. These small and fast
rotating bodies are thought to be monolithic objects, since the weak
gravitational force due to their small size is not strong enough to counteract
the large centripetal force caused by the fast rotation. This argument makes
the rubble-pile structure not feasible for such objects. Additionally, it is
not clear whether the fast spin prevents dust and small particles (regolith)
to be kept on their surface.

Aims. We aim to develop a model to constrain the thermal conductivity of the
surface of the small, fast-rotating near-Earth asteroids. This model may
suggest whether the presence of regolith is likely or not.

Methods. Our approach is based on the comparison between the measured
Yarkovsky drift and a predicted value using a theoretical model, which depends
on the orbital, physical and thermal parameters of the object. The necessary
parameters are either deduced from statistical distribution derived for near-
Earth asteroids population or determined from observations with associated
uncertainty. With that information available, we perform Monte Carlo
simulations, producing a probability density distribution for the thermal
conductivity.

Results. Applying our model to the super-fast rotator asteroid (499998) 2011
PT, we find that the measured Yarkovsky drift can be achieved only if the
thermal conductivity $K$ of the surface is low. The resulting probability
density function for the conductivity is bimodal, with two most likely values
being around 0.0001 and 0.005 W m-1 K-1. Based on this, we find that the
probability of $K$ being smaller than 0.1 W m-1 K-1 is at least 95 per cent.
This low thermal conductivity could be a clue that the surface of 2011 PT is
covered with a thermal insulating layer, composed by a regolith-like material
similar to lunar dust.

###### Key Words.:

minor planets, asteroids: general - minor planets, asteroids: individual:
(499998) 2011 PT - methods: statistical

##  1 Introduction

Knowledge of the surface properties of asteroids provides insights into the
nature of the surface materials and structures, which are, in turn,
fundamental for different reasons. These properties are, for instance,
important in modeling the formation of regolith-like materials (Delbó et al.,
2014), the weakening and degradation of boulders on asteroids due to thermal
shocks (Molaro et al., 2017), and space weathering processes (Brunetto et al.,
2015). Moreover, surface properties are also crucial for planning of
spacecraft interaction with the surface of an asteroid, including deflection
missions (Bruck Syal et al., 2016).

The surface thermal inertia informs us about porosity and cohesion, with low
values being indicative of the presence of a thermal insulting layer (see Alí-
Lagoa et al., 2020, and references therein), consistent with a dusty, porous
regolith, while high values are specific for rocky-like materials. Thermal
inertia depends on density $\rho$, heat capacity $C$ and thermal conductivity
$K$. However, while density and heat capacity typically vary within a factor
of a few, uncertainty in thermal conductivity spans over a range of more than
four orders of magnitude (Delbó et al., 2015). Therefore, in this respect, the
thermal conductivity is the most important parameter to constrain.

Before in-situ exploration of asteroids, it was thought that only very large
objects could be covered by regolith (see Murdoch et al., 2015, and references
therein). However, the visit of the Galileo and NEAR-Shoemaker NASA missions
to asteroids (243) Ida, (433) Eros, and (951) Gaspra revealed that even
smaller asteroids can actually retain regolith-like materials on their
surfaces. Furthermore, with the visit of the JAXA Hayabusa mission to asteroid
(25143) Itokawa in 2005, it became evident from its in-situ observations and
measurements that, despite the low gravitational environment, even sub-
kilometer bodies can retain gravel particles, boulders or regolith on their
surfaces (Miyamoto et al., 2007). More recently, data collected in-situ on
asteroids (101955) Bennu and (162173) Ryugu by the OSIRIS-REx and the Hayabusa
2 missions, respectively, did not reveal fine dust on their surfaces, but
still showed the presence of grain particles and boulders of different sizes
(Lauretta et al., 2019a; Michikami et al., 2019; Morota et al., 2020; Sugita
et al., 2020; Susorney et al., 2020).

These findings changed our perspective on the ability of kilometer and sub-
kilometer sized asteroids to preserve regolith-like materials on their
surfaces. This new perspective raises the question if even smaller objects,
from a few tens to a few hundreds of meters, can also be covered by a dust
layer. The extremely low-gravity environment on such bodies would suggest that
a dust layer is not likely. If we further restrict our considerations to small
super-fast rotators, such a possibility seems even less probable.

Pravec & Harris (2000) found that objects larger than about 150 meters very
rarely have rotation period smaller than about 2.2 hr. This barrier can be
explained by the fact that, for diameters between about 150 meters and 10
kilometers, every Solar System body has a rubble-pile internal structure
(Walsh, 2018), which is not gravitationally strong enough to counteract the
centripetal force due to the fast rotation, thus causing a break-up of the
object. On the other hand, there is no such limit for the spin period of
smaller objects, so that they are allowed (and often seen) to rotate very
fast, even with periods of few minutes. Asteroids smaller than 150 meters in
diameter are therefore considered to be mostly monolithic objects (see, e.g.,
Pravec & Harris, 2000; Whiteley et al., 2002; Polishook, 2013), with strong
internal forces that hold the body intact even at very fast spin rates. Figure
1 shows the measurements of the spin period of near-Earth objects (NEOs), as a
function of the diameter of the body, taken from the Asteroid Lightcurve
Database111http://alcdef.org/ (LCDB) developed by Warner et al. (2009). We can
clearly see the spin barrier for diameters larger than 150 meters, while going
down to a few tens of meters the spin periods can be as short as few to few
tens of minutes (possibly even shorter, but typical observation scenarios
would not detect them). Such fast spin rates, coupled with the very low
gravitational environment due to the small size, are expected to cause the
ejection of material on the surface, thus preventing regolith grains to be
retained.

Figure 1: Measured spin periods for near-Earth objects as of September 2020
and taken from the Asteroid Lightcurve Database (see Warner et al., 2009).

Determining values of thermal parameters for asteroids is a challenge, since
direct measurements from Earth can not be performed. Consequently, reliable
thermal inertia (conductivity) estimates are currently available for a
relatively small number of asteroids (see, e.g., Delbó et al., 2015; Harris &
Drube, 2016; Marciniak et al., 2019). Most often, thermal inertia is derived
from analysis of observations in the infrared band. Unless space-borne, such
data about very small objects are rare.

The dynamics of bodies up to about 30 kilometers in diameter is known to be
influenced by thermal effects, caused by solar radiation. Objects are heated
up by the Sun, so that they re-radiate away the energy in the thermal
waveband, creating a small thrust and causing a drift in semi-major axis,
which is perceived over decades: this physical phenomenon is known as
Yarkovsky effect (Rubincam, 1995, 1998; Farinella et al., 1998; Bottke et al.,
2006; Vokrouhlický et al., 2015). The Yarkovsky effect perturbs the
trajectory, but it is generated by a thermal force that depends on material
properties and the internal structure of asteroids. Therefore, the Yarkovsky
effect is a phenomenon where orbital dynamics is linked to asteroid
composition and physical properties.

The detection of the Yarkovsky effect has become available as part of orbit
determination (i.e., the osculating orbital elements of epoch) on a regular
basis, in cases where high precision astrometry observations are available. So
far this applies uniquely to near-Earth asteroids. An alternative approach to
estimate the Yarkovsky effect induced drift in the orbital semi-major axis has
been formulated for the first time by Nesvorný & Bottke (2004), where the
authors used backward orbit propagation of the Karin cluster members and
monitored convergence of the secular angles. The same method has been later
applied to several other young asteroid families. However, this approach
provides less accurate results than the direct orbit-determination mentioned
above. Additionally, data about small and fast rotating asteroids, relevant in
context of our paper, is not available for the main belt families.

Depending on other available information, measurements of the Yarkovsky effect
allow the physical properties of small objects to be constrained. Chesley et
al. (2003) were able to deduce an interval of values for the thermal
conductivity of (6489) Golevka, compatible with the existence of thin
regolith. Additional examples include Farnocchia et al. (2013), where the
authors were able to constrain the values of the thermal inertia of six NEOs
for which the obliquity was known. Similarly, using thermal infrared
observations it is possible to estimate the thermal inertia of an asteroid
using a suitable thermophysical model (see Delbó et al., 2015, and references
therein). Combining these measurements with the Yarkovsky drift estimation, it
is possible to break the degeneracy between thermal inertia and density, and
to directly derive the bulk density of the object (Chesley et al., 2014;
Farnocchia et al., 2014; Mommert et al., 2014a, b; Rozitis et al., 2014; Reddy
et al., 2016).

In this paper we developed a statistical method to set constraints on the
thermal conductivity of near-Earth asteroids. Our approach is based on the
comparison between the measured Yarkovsky drift and the theoretically
predicted one. Using suitable distributions for the parameters that determine
the magnitude of the drift, we perform a Monte Carlo simulation, and produce a
probability density function for the thermal conductivity $K$. This
probability informs us about the most likely values of $K$, and it can be used
to rule out some other values. We applied our model to the asteroid (499998)
2011 PT, a super-fast rotating near-Earth object with a diameter of about 35
meters. Interestingly, we found that the measured semi-major axis drift is
large enough such that it can be achieved only for low values of thermal
conductivity, with two most likely values being around 0.0001 and 0.005 W m-1
K-1, suggesting the presence of a thermal insulating layer on the surface of
2011 PT.

##  2 Methods

The semi-major axis drift $da/dt$ due to the Yarkovsky effect for a spherical
body placed on a circular orbit around the Sun can be expressed through
analytical formulas, and it is given by the sum of the seasonal and the
diurnal effects (see Appendix A). It depends on the orbital parameters, and on
the physical and thermal characteristics of the body, i.e. the semi-major axis
$a$, the diameter $D$, the density $\rho$, the thermal conductivity $K$, the
heat capacity $C$, the obliquity $\gamma$, the rotation period $P$, the
absorption coefficient $\alpha$, and the emissivity $\varepsilon$. Among these
parameters, the thermal conductivity $K$ is the most uncertain parameter,
since it strongly depends on the composition of the surface and on the
porosity, and it can vary by several orders of magnitude. If a measurement
$(da/dt)_{\text{m}}$ of the Yarkovsky effect is available from astrometry
(see, e. g., Farnocchia et al., 2013; Del Vigna et al., 2018; Greenberg et
al., 2020), solving the model vs observed drift-rate relation

$\left(\frac{da}{dt}\right)(a,D,\rho,K,C,\gamma,P,\alpha,\varepsilon)=\bigg{(}\frac{da}{dt}\bigg{)}_{\text{m}},$
(1)

for $K$, allows us to estimate the thermal conductivity, which can be examined
statistically using a Monte Carlo method. To this purpose, in Eq. (1) the
measured parameters are either fixed to a specific value if their uncertainty
is negligible, or they can be modeled assuming a Gaussian error distribution.
The parameters for which measurements are not available, or are subject of
large uncertainties, can be modeled using a population-based probability
density function (PDF).

###  2.1 Probabilistic model of the parameters of 2011 PT

Asteroid (499998) 2011 PT is a near-Earth object discovered by the Pan-STARRS
1 survey telescope on August 1, 2011. With a semi-major axis of $a=1.3123$ au
and an eccentricity of about $e=0.2147$, it falls in the Amor group, meaning
that its perihelion distance $q=a(1-e)$ satisfies $1.017\text{ au}<q<1.3\text{
au}.$ Hereafter we describe the probabilistic model of the orbital and
physical parameters of 2011 PT, used for the computation of the semi-major
axis induced drift from the theoretical model of the Yarkovsky effect, i.e.
the left hand side of Eq. (1).

####  2.1.1 Absolute magnitude determination

The absolute magnitude $H$ does not play a role in the analytical model of the
Yarkovsky effect of Eq. (1) (see also Appendix A), however it is related to
the size of the object and it is needed to model the albedo distribution (see
Sec. 2.1.2). The JPL Small-Body Database222https://ssd.jpl.nasa.gov/sbdb.cgi
(SBDB) reports a value of $H=23.9$ mag, but it is known that orbital catalogs
all suffer uncertainties in $H$, including systematic effects (Pravec et al.,
2012). To overcome this issue and improve the accuracy of the results, we
remeasured all available Pan-STARRS images and recomputed an absolute
magnitude of $H=24.07\pm 0.42$ mag using the phase angle computed from the
orbital geometry at each observation and the corrections to $V$ band as given
by Denneau et al. (2013). We note that the available data does not allow for a
more accurate determination of $H$.

####  2.1.2 Albedo distribution

Although the albedo does not enter directly in the formulation of the
Yarkovsky effect as given by Eq. (1), it is related to both the density and
the diameter of the object. Indeed, the value $p_{V}$ of the albedo gives a
first indication of the taxonomic type, which determines the density, and it
provides an estimation of the diameter by the conversion formula (see, e.g.,
Bowell et al., 1989; Pravec & Harris, 2007)

$D=\frac{1329\text{ km}}{\sqrt{p_{V}}}10^{-H/5},$ (2)

where $H$ is the absolute magnitude. A PDF of the albedo of a specific near-
Earth object can be constructed making use of the NEOs population model by
Granvik et al. (2018) and the NEOs albedo distribution by Morbidelli et al.
(2020).

Given the orbital elements $(a,e,i)$ and the absolute magnitude $H$, the
population model by Granvik et al. (2018) provides the probability for a NEO
to come from a specific source region, which are namely: the $\nu_{6}$ secular
resonance, the 3:1, 5:2 and 2:1 Jupiter mean-motion resonances, the Hungaria
region, the Phocaea region, and the Jupiter Family Comets (JFC). For the sake
of notation, we denote this probability with $P_{s}(a,e,i,H)$, where the
subscript $s$ is used to denote a specific source route. Numerical values
corresponding to (499998) 2011 PT are given in Table 1.

Table 1: The probability to come from each source region according to the model of Granvik et al. (2018), computed for the orbital elements and absolute magnitude corresponding to the object (499998) 2011 PT. Source Region | $P_{s}(a,e,i,H)$
---|---
$\nu_{6}$ | 0.67623
3:1 | 0.08397
5:2 | 0.00000
Hungaria | 0.23854
Phocaea | 0.00013
2:1 | 0.00113
JFC | 0.00000

In the NEOs albedo distribution by Morbidelli et al. (2020), objects are
binned in three albedo categories:

  * •

category 1: $p_{V}\leq 0.1$;

  * •

category 2: $0.1<p_{V}\leq 0.3$;

  * •

category 3: $0.3<p_{V}$.

For each source region, the model provides the fraction of delivered bodies
belonging to each albedo category $c_{1},c_{2},c_{3}$. For the sake of
notation, we denote the fractions with $p_{s}(c_{i}),i=1,2,3$ where $s$
indicates again the source region. The exact numerical values are reported in
Morbidelli et al. (2020). The Hungaria region mostly provides high albedo
objects, in agreement with the fact that this part of the main belt contains a
large number of E-type asteroids (DeMeo & Carry, 2013, 2014). The inner belt
regions, i.e. the $\nu_{6}$ secular resonance and the 3:1 Jupiter mean motion
resonance, mostly provide asteroids in the S-complex. The only exception is
the Phocaea region which supplies an approximately equal number of dark
(category 1) and bright asteroids (categories 2 and 3). A larger fraction of
dark asteroids coming from the Phocaea region compared to the rest of the
inner main belt is probably due to a dark asteroid family located therein
(Novaković et al., 2017). Nevertheless, the probability for 2011 PT to
originate in this region is very small. The mid and outer belt regions, i.e.
the 5:2 and 2:1 Jupiter mean motion resonances, account for a higher
percentage of C-complex asteroids, according to the fact that primordial
carbonaceous bodies are more abundant at larger semi-major axis values (DeMeo
& Carry, 2013, 2014).

For each escape route, we define an albedo probability function $p_{s}(p_{V})$
assigning a uniform density in the first and second categories, and an
exponentially decaying density in the third category, as done in Morbidelli et
al. (2020):

$p_{s}(p_{V})=\begin{cases}\displaystyle\frac{p_{s}(c_{1})}{0.1},&p_{V}\leq
0.1,\\\\[8.61108pt] \displaystyle\frac{p_{s}(c_{2})}{0.2},&0.1<p_{V}\leq
0.3,\\\\[8.61108pt] \displaystyle
p_{s}(c_{3})\frac{2.6^{-\frac{p_{V}-0.3}{0.1}}}{\int_{0.3}^{1}2.6^{-\frac{x-0.3}{0.1}}dx},&p_{V}>0.3.\end{cases}$
(3)

Given the probabilities $P_{s}(a,e,i,H)$ for the object to come from each
source region, the albedo PDF $p(p_{V})$ for a specific NEO is defined by

$p(p_{V})=\sum_{s=1}^{7}P_{s}(a,e,i,H)\,p_{s}(p_{V}).$ (4)

The resulting albedo PDF for 2011 PT is shown in the top left panel of Fig. 2.

Figure 2: Probability density function for the albedo $p_{V}$ (top left
panel), the density $\rho$ (top right panel), the diameter $D$ (bottom left
panel), and the obliquity $\gamma$ (bottom right panel), computed for asteroid
(499998) 2011 PT.

####  2.1.3 Density distribution

In order to obtain the PDF of the density of 2011 PT, we first divide
asteroids in three main complexes: 1) the C-complex, composed by dark
carbonaceous objects, which includes C-, P-, D-, B-, and T-types, 2) the
S-complex, composed by stony objects, which gathers together S-, Q-, V-, A-,
and R-types, and 3) the X-complex, composed by objects with different
compositions, namely the intermediate albedo M-type, and the high albedo
E-type.

For the S-complex we take as a reference the density of S-type, since they are
the most common and numerous one (DeMeo & Carry, 2014; DeMeo et al., 2015),
which is reported to be $\rho=2720\pm 540\text{ kg m}^{-3}$ (Carry, 2012).

On the other hand, C-type asteroids are the most common in the C-complex,
hence we assume their density to model this population. Carry (2012) reports a
density of $\rho=1330\pm 580\text{ kg m}^{-3}$ for the C-type, which however
seems to be a bit high if compared to the precise measurements performed on
asteroids (101955) Bennu and (162173) Ryugu. (101955) Bennu is a B-type
asteroid with a composition similar to the aqueously altered CM carbonaceous
chondrite, and its density is $1190\pm 13$ kg m-3 (Scheeres et al., 2019),
while (162173) Ryugu is a Cb-type asteroid with a composition similar to
carbonaceous chondrites, and a density of $1190\pm 20$ kg m-3 (Watanabe et
al., 2019). These two precise measurements suggest that the value of the
density of C-type asteroids proposed in Carry (2012) may be too large, and
this deviation might be due to large errors introduced by the method used to
produce the estimates. The difference may also be because asteroids tend to
have larger density with increasing diameter (see, e.g., Carry, 2012; Milani
et al., 2014). For this reason, we assume a lower value for the C-complex,
setting the density to $\rho=1200\pm 300\text{ kg m}^{-3}$.

As previously said, the X-complex shows a different variety of densities and
compositions, however the E-type, which is connected to Aubrite meteorites, is
the most common type delivered by the Hungaria region (Ćuk et al., 2014;
Binzel et al., 2019), which is one of the main near-Earth objects source
regions (Granvik et al., 2018). Hence, we assume the density of E-type
asteroids as a reference for the X-complex. Carry (2012) reports only one
measurement with an accuracy better than 20% for this asteroid type, and it is
equal to $2600\pm 200$ kg m-3. However, drawing a statistical picture from one
single measurement may not be correct: for this reason we established other
values. To this purpose, we took into account the grain density of the
connected Aubrite meteorites, which is reported to be $\rho=3200\pm 80\text{
kg m}^{-3}$ (Ostrowski & Bryson, 2019), and we corrected it taking into
account the porosity. The porosity distribution is deduced from the values
reported in Carry (2012): we took all the asteroids with porosity larger than
zero and relative accuracy better than 50% in the density measurements. Then,
we fitted these data with a gamma distribution, see Fig. 3.

Figure 3: Fit of the asteroid porosity with a gamma distribution, using data
from Carry (2012).

Assuming a Gaussian distribution for the density of Aubrite, we corrected it
with the porosity using the formula

$\rho=\rho_{\text{m}}\,\bigg{(}1-\frac{P(\%)}{100}\bigg{)},$ (5)

from Carry (2012), where $P(\%)$ is the porosity and $\rho_{\text{m}}$ is the
density of the meteorite. In this manner, we found a density of $\rho=2350\pm
520\text{ kg m}^{-3}$, which we use for the purpose of this work.

The three taxonomic complexes could be also linked to the three albedo
categories defined by Eq. (3). In this respect, we assume that category 1,
which refers to dark asteroids, corresponds to the C-complex, category 2
corresponds to the S-complex, since it refers to moderate albedo values, while
category 3 is associated to the X-complex, containing higher albedo objects.
This can be used to construct the PDF for the density of 2011 PT.

To generate a distribution of the density, we first generate a random value of
the albedo according to the distribution defined by Eq. (4). Then, based on
the generated albedo we determine the albedo category and the associated
asteroid complex. Finally, we randomly generate a value of the density
according to the density distribution of the corresponding asteroid complex,
that is assumed to be log-normal (see, e.g., Spoto et al., 2014; Tardioli et
al., 2017). The PDF for $\rho$ obtained with this procedure, smoothed using
the kernel density estimation, is reported in the top right panel of Fig. 2.
As expected, for 2011 PT higher density solutions ($>2000\leavevmode\nobreak\
\text{kg m}^{-3}$) are much more likely than the low ones
($<1500\leavevmode\nobreak\ \text{kg m}^{-3}$).

####  2.1.4 Size distribution

The size of an object is estimated using its absolute magnitude $H$ and its
albedo $p_{V}$ by the conversion formula of Eq. (2). Since we determined an
absolute magnitude of $H=24.07\pm 0.42$ mag, the diameter is mostly affected
by the uncertainty on the albedo, and it can vary even of an order of
magnitude if the range of $p_{V}$ is large.

The distribution of the diameter $D$ is obtained again by conversion of the
albedo distribution of Eq. (4): random values of $p_{V}$ are generated first,
and then converted in diameter using Eq. (2). The uncertainty in the
determination of the absolute magnitude $H$ is taken into account assuming a
Gaussian distribution of the errors. The PDF for the diameter of 2011 PT
obtained with this procedure, smoothed using the kernel density estimation, is
reported in the bottom left panel of Fig. 2. The most likely size suggested by
this model is about 35 meters.

####  2.1.5 Obliquity distribution

The obliquity of an asteroid can be determined from photometric measurements
using the lightcurve inversion method (Kaasalainen & Torppa, 2001; Kaasalainen
et al., 2001, 2002; Ďurech et al., 2015), or from radar observations (Hudson,
1994; Hudson et al., 2000; Magri et al., 2007). If measurements are available,
it is possible to model this parameter using a Gaussian distribution, with
mean corresponding to the nominal value and standard deviation corresponding
to the error. On the other hand, if it is unknown, which is the most usual
situation, we have to rely on general properties of the NEOs population.

It is known that in the NEOs population there is an excess of retrograde
rotators (La Spina et al., 2004), caused by the transport mechanism of objects
from the main belt, driven by the Yarkovsky effect (Bottke et al., 2002;
Granvik et al., 2018). A first attempt to derive an obliquity distribution for
near-Earth asteroids has been made by Farnocchia et al. (2013), where the
authors used a four-bin probability density function to match the observed
obliquities. A more sophisticated approach has been used in Tardioli et al.
(2017), where different distributions have been tested. The best fit solution
is obtained for a quadratic distribution of $\cos\gamma$, namely

$p(\cos\gamma)=a\cos^{2}\gamma+b\cos\gamma+c,$ (6)

where the parameters are $a=1.12,\,b=-0.32,\,c=0.13$. The PDF of Eq. (6) is
shown in Fig. 2, bottom right panel.

####  2.1.6 Fixed parameters

#### Semi-major axis

The uncertainty in the semi-major axis $a$ of 2011 PT determined from
astrometry, as reported in the JPL SBDB, is about $9.3\times 10^{-10}$ au:
small changes of $a$ within this uncertainty interval produce negligible
changes in the predicted Yarkovsky drift, therefore this parameter is kept
fixed to the nominal value.

#### Heat capacity

The heat capacity $C$ depends on the physical characteristics of the object,
however its value may vary within a factor of several (Delbó et al., 2015).
Typical assumed values for km-size rocky or regolith-covered main belt
asteroids are in the range 600-700 J kg-1 K-1, while for iron-rich objects the
value is smaller, of the order of 500 J kg-1 K-1 (see e.g Farinella et al.,
1998). However, the heat capacity increases for increasing temperature, and
its value may be higher for near-Earth asteroids, supposedly not exceeding
$\sim$1200 J kg-1 K-1. In Ostrowski & Bryson (2019), the authors reported the
measured heat capacities for different type of meteorites. From their data, we
can draw the conclusion that $C$ is larger than 500 J kg-1 K-1 for stony
meteorites, while it is smaller than this threshold value if there is a high
component of metallic material. However, it is not clear how to produce a
distribution from the typical assumed value for km-size asteroid. Moreover,
data on meteorites are not enough to produce a clear statistical distribution,
they might be biased, and it is not clear how to correct their value when we
consider small asteroids. For all the above reasons, we keep fixed the value
of the heat capacity $C$, and produce the results for a few different
reasonable values.

#### Emissivity and absorption coefficient

Ostrowski & Bryson (2019) reported the measurements of the emissivity for 61
meteorites, and all the objects but one have an emissivity in between 0.9 and
1, with an average value of 0.984. Since we do not expect the Yarkovsky effect
to vary much with respect to this parameter, we fix $\varepsilon$ to the mean
value computed with meteorite measurements. The absorption coefficient is set
to $\alpha=1$ in all our calculations.

####  2.1.7 Parameters modeled with errors in measurements

#### Rotation period

Kikwaya Eluo & Hergenrother (2015) determined the rotation period of 2011 PT
using photometric observations, and it results to be $P=0.17\pm 0.05\text{
hr}$, corresponding to about 11 minutes. Although the small uncertainty might
produce very small variations on the predicted Yarkovsky drift, we take it
into account by assuming a Gaussian distribution of the errors.

#### Yarkovsky drift measurement

The Yarkovsky drift has been determined by Greenberg et al. (2020) with a
nominal value and an uncertainty of

$\bigg{(}\frac{da}{dt}\bigg{)}_{\text{m}}=(-88.44\pm 14.6)\times 10^{-4}\text{
au My}^{-1},$

which we used to perform the Monte Carlo simulation. This parameter is also
modeled assuming a Gaussian distribution of the errors. Note that the JPL SBDB
provides an automatic solution of the non-gravitational transverse
acceleration $A_{2}=(-2.237\pm 0.302)\times 10^{-13}$ au d-2, corresponding to
$(da/dt)=(-86.9\pm 11.7)\times 10^{-4}$ au My-1, while Del Vigna et al. (2018)
report a value of $(da/dt)=(-91.3\pm 14.44)\times 10^{-4}$ au My-1, both
nicely consistent with the value given by Greenberg et al. (2020).

###  2.2 Monte Carlo simulation

We generate samples

$\displaystyle\\{(D_{h},\rho_{h})\\}_{h=1,\dots,n_{p_{V}}},$
$\displaystyle\\{\gamma_{j}\\}_{j=1,\dots n_{\gamma}},$
$\displaystyle\\{P_{k}\\}_{k=1,\dots n_{P}},$
$\displaystyle\\{(da/dt)_{i}\\}_{i=1,\dots n_{\text{Yarko}}},$

for the parameters modeled with a distribution, following the procedure
described in the previous section, where
$n_{p_{V}},n_{\gamma},n_{P},n_{\text{Yarko}}$ denote the number of points in
each corresponding sample. Diameter and density are generated in pairs using
the same values of albedo, therefore taking into account their correlation. In
this manner we avoid to produce non-physical combinations such as small
diameter (obtained by converting a high albedo) and low density (obtained by
converting a low albedo). For a given Yarkovsky drift $(da/dt)_{i}$ and a
possible combination of parameters $(D_{h},\rho_{h},\gamma_{j},P_{k})$, we
numerically invert the relation

$\left(\frac{da}{dt}\right)(a,D_{h},\rho_{h},K,C,\gamma_{j},P_{k},\alpha,\varepsilon)=\bigg{(}\frac{da}{dt}\bigg{)}_{i},$
(7)

computing the values of the thermal conductivity $K$ for which the measured
drift is achieved. In this manner, for each combination of parameters, we
obtain

$K_{1}^{(h,j,k,i)},\dots,K_{n}^{(h,j,k,i)},$ (8)

where $n$ is the number of solutions of Eq. (7). Collecting all the solutions
together, we obtain a distribution for the possible values of the thermal
conductivity $K$ of the specific object, from which we can compute the
corresponding probability density function using the kernel density estimator.

##  3 Results

###  3.1 Preliminary constraints on the thermal conductivity

A first clue that the thermal conductivity of 2011 PT might be small is given
by simply computing the Yarkovsky drift for the nominal values of the
parameters. For instance, fixing the diameter to $D=35$ meters, and the heat
capacity to $C=680$ J kg-1 K-1, we computed the maximum negative Yarkovsky
drift (thus selecting $\gamma=180\degr$) as a function of the density $\rho$
and the thermal conductivity $K$. The resulting drift values, together with
the three level curves corresponding to the measured nominal value and the
1-$\sigma$ levels, are reported in Fig. 4.

Figure 4: The estimated maximum negative Yarkovsky drift for 2011 PT, obtained
for $C=680$ J kg-1 K-1, $D=35$ m, and $P=0.17$ h. The red level curves
correspond to the nominal Yarkovsky drift (level curve in the between), and
the drift obtained taking into account the uncertainties (external level
curves).

Since here we are computing the maximum negative Yarkovsky drift, from the
plot of Fig. 4 we can conclude that the values of $(\rho,K)$, residing inside
the area determined by the external level curve at 1-$\sigma$ level, are all
compatible with the measurements of the drift. For all the values of the
density we obtained a small value of thermal conductivity $K$, which is always
less than 0.3 W m-1 K-1, and decreases with increasing density. For instance,
for $\rho=1500$ kg m-3, we have

$0.00001\text{ W m${}^{-1}$ K${}^{-1}$}\lesssim K\lesssim 0.074\text{ W
m${}^{-1}$ K${}^{-1}$},$

while, doubling the density to $\rho=3000$ kg m-3, we obtain

$0.00005\text{ W m${}^{-1}$ K${}^{-1}$}\lesssim K\lesssim 0.0055\text{ W
m${}^{-1}$ K${}^{-1}$}.$

These considerations already suggest a low thermal conductivity of the
surface, which has to be analyzed in a more depth taking into account the
uncertainties in the parameters involved, and by performing the Monte Carlo
simulations explained in the previous section.

We note that the fast rotation of (499998) 2011 PT is a key element of the
solution. This is because the maximum drift-rate $da/dt$ is obtained when the
diurnal $F$-value is maximum in Eq. (14). The maximum occurs when
$\Theta_{\text{d}}\simeq 1$, namely when the corresponding diurnal thermal
parameter is of the order of unity. Since $\Theta_{\text{d}}$ depends on the
ratio $K/P$ (see Eq. 18), the optimal conductivity $K$ scales as $K\propto P$.
Therefore, the whole pattern seen in Fig. 4 would slide toward larger
conductivity values in the ordinate for more slowly rotating bodies. In the
case of the very fast-rotating asteroid 2011 PT, $K$ is constrained to values
smaller than $\sim$0.1 W m-1 K-1.

###  3.2 Results of the Monte Carlo simulations

To better quantify the low-thermal conductivity of the surface of (499998)
2011 PT, we performed a Monte Carlo simulation with the method described in
Sec. 2. We generated samples
$\\{(D_{h},\rho_{h})\\},\,\\{\gamma_{j}\\},\,\\{P_{k}\\},\,\\{da/dt\\}_{i}$ of
50000 points for diameter and density, and 10000 points for obliquity,
rotation period, and Yarkovsky drift, respectively. To perform the Monte Carlo
experiment and generate the results, we randomly chose 1 million combinations
of the parameters. The output distribution for the thermal conductivity $K$
was then fitted using the kernel density estimation, so that to obtain a
smooth probability density function. We ran different simulations for some
values of the heat capacity $C$, namely

$C=500,\,680,\,800,\,1000,\,1200\text{ J kg${}^{-1}$ K${}^{-1}$}.$

Using these values we explore the expected range of $C$ for near-Earth
objects, from which we can extrapolate an eventual trend. The results of the
Monte Carlo simulations are shown in Fig. 5, together with the corresponding
distribution obtained for the thermal inertia

$\Gamma=\sqrt{\rho KC}.$ (9)

Figure 5: Derived thermal parameters of 2011 PT obtained from the Monte Carlo
simulations, for different values of the heat capacity $C$. The plots on the
first column show the distribution of the thermal conductivity $K$, while the
plots on the second column show the distribution of the thermal inertia
$\Gamma$. The thermal inertia is obtained through $\Gamma=\sqrt{\rho KC}$,
where $\rho$ is the corresponding value of the density used to invert Eq. (7).

From the plots we can notice that we always obtained a bimodal distribution,
with a first peak in the interval

$0.00001\text{ W m${}^{-1}$ K${}^{-1}$}<K<0.001\text{ W m${}^{-1}$
K${}^{-1}$},$

and the second one in the interval

$0.001\text{ W m${}^{-1}$ K${}^{-1}$}<K<0.1\text{ W m${}^{-1}$ K${}^{-1}$}.$

The distributions of the thermal inertia $\Gamma$ are also bimodal, with a
first peak in the interval

$5\text{ J m${}^{-2}$ K${}^{-1}$ s${}^{-1/2}$}<\Gamma<20\text{ J m${}^{-2}$
K${}^{-1}$ s${}^{-1/2}$},$

and the second one in the interval

$75\text{ J m${}^{-2}$ K${}^{-1}$ s${}^{-1/2}$}<\Gamma<150\text{ J m${}^{-2}$
K${}^{-1}$ s${}^{-1/2}$}.$

Moreover, as the heat capacity $C$ increases, a third peak started to grow at
large values of thermal conductivity, maintaining however a small probability.
The bimodal distribution is explained by the fact that, for each set of
parameters
$\\{(D_{h},\rho_{h})\\},\,\\{\gamma_{j}\\},\,\\{P_{k}\\},\,\\{(da/dt)_{i}\\}$,
there are always two low thermal conductivity solutions of Eq. (1), as can be
easily noticed from Fig. 4. Furthermore, since the input distributions all
have only one most likely value, most of the combinations of input parameters
selected during the Monte Carlo simulations are gathered nearby a most likely
one, and this produces two well distinct peaks of low $K$. On the other hand,
the third peak with small probability is caused by other high thermal
conductivity solutions of Eq. (1) appearing at high heat capacity, for large
diameter (around $\sim$60 meters) and small density (around $\sim$1100 kg
m-3). These combinations of density and diameter are generated with small
probability, since they correspond to the low albedo case.

To give quantitative boundaries for the most likely values of thermal
conductivity, we computed the probabilities

$\begin{split}P_{1}&=P(0.00001<K<0.001),\\\ P_{2}&=P(0.001<K<0.1),\end{split}$
(10)

for the different simulations, as well as their sum and the two local maxima
$K_{1},K_{2}$. The results are reported in Table 3.

Table 2: The probabilities $P_{1},\,P_{2}$ of Eq. (10), the corresponding sum,
and the locations of the two local maxima $K_{1},\,K_{2}$, for the different
Monte Carlo simulations.

$C$ | $P_{1}$ | $P_{2}$ | $P_{1}+P_{2}$ | $K_{1}$ | $K_{2}$
---|---|---|---|---|---
500 | 0.50860 | 0.47788 | 0.98649 | $1\times 10^{-4}$ | 0.0076
680 | 0.52367 | 0.46725 | 0.99092 | $7\times 10^{-5}$ | 0.0057
800 | 0.53200 | 0.45785 | 0.98985 | $6\times 10^{-5}$ | 0.0048
1000 | 0.54317 | 0.43511 | 0.97828 | $5\times 10^{-5}$ | 0.0037
1200 | 0.54916 | 0.40935 | 0.95852 | $4\times 10^{-5}$ | 0.0032
333The heat capacity in the first column is expressed in J kg-1 K-1, and the
thermal conductivity in the fifth and sixth columns are expressed in W m-1
K-1.

From these values, and from the plots of Fig. 5, we can conclude that the
probability for the thermal conductivity of being larger than 0.1 W m-1 K-1
was always almost zero, with slightly larger values for larger values of heat
capacity. The highest peak $K_{1}$ lies always in the first interval and it is
around 0.0001 W m-1 K-1, while the lowest one $K_{2}$ belongs to the second,
being of the order of $\sim$0.005 W m-1 K-1. Moreover, we can notice that they
both move to lower values of thermal conductivity as the heat capacity $C$
increases. The probability $P_{1}$ results to be always greater than the
probability $P_{2}$, but only with a few percent in difference. The values of
$P_{1}$ and $P_{2}$, as function of the heat capacity $C$, can be fitted
linearly and the results are show in Fig. 6, together with the 99% confidence
interval region of the fit. As the heat capacity increases, the probability
$P_{1}$ increases, while $P_{2}$ decreases, with a more pronounced slope than
$P_{1}$, which results in a decreasing trend in their sum (see Table 3). This
fact was however expected, due to the presence of the third peak growing at
large thermal conductivity.

Figure 6: Linear fits of the probabilities $P_{1}$ and $P_{2}$ with respect to
the heat capacity $C$. The dashed lines represent the 99% confidence interval
region.

##  4 Discussion

###  4.1 Robustness and limitations of the model

####  4.1.1 Uncertainty due to the orbital eccentricity

The semi-major axis drift due to the Yarkovsky effect described analytically
in Appendix A refers to an object revolving around the Sun on a circular
orbit. However, the Yarkovsky effect may be increased on elliptic orbits
(Spitale & Greenberg, 2001), and it may cause an object to reach the measured
migration rate even for larger values of thermal conductivity. While
significant changes are expected for orbits with high eccentricity, it is
interesting to test the role of moderately small orbital eccentricity in the
case of (499998) 2011 PT.

The osculating semi-major axis drift due to the Yarkovsky effect for an
asteroid placed on an eccentric orbit is given by

$\frac{da}{dt}=\frac{2}{n^{2}a}\,\@vec{f}_{Y}\cdot\@vec{v},$ (11)

where $n$ is the mean motion, $\@vec{v}$ is the heliocentric orbital velocity
and $\@vec{f}_{Y}$ is the instantaneous value of the Yarkovsky acceleration.
The acceleration $\@vec{f}_{Y}$ can be computed analytically, for instance,
using the formulation given in Vokrouhlický et al. (2017), including both the
diurnal and the seasonal effects (again with the simplifying assumption of
linearization of the boundary conditions). To compute the total semi-major
axis drift we average the values of the instantaneous drift rate of Eq. (11)
over an orbital period numerically (rather than analytically as before for the
circular orbit).

To test the effects of the orbital eccentricity on our results, we compared
the mean drift rates obtained for the circular orbit and the eccentric orbit,
performing again a Monte Carlo simulation. Obviously, we chose the
eccentricity $\sim$0.215 appropriate for (499998) 2011 PT. We used the same
distributions of Sec. 2 for the diameter, the density, the obliquity, and the
rotation period. Moreover, we produced a sample of the thermal conductivity
using the distributions obtained in Sec. 3. Using a million combinations of
the parameters, we evaluated the two different drift rates and computed the
relative difference.

We applied this procedure to (499998) 2011 PT, using a fixed heat capacity of
$C=680$ J kg-1 K-1. The distribution of the percent absolute differences had
two peaks, one around 1% and the other one at 10%, and almost all the
combinations gave a relative difference smaller than about 10%. Moreover, the
peak at 1% was found to be corresponding to the lower peak in thermal
conductivity, while the peak at about 10% was associated to the second peak in
$K$, meaning that a higher conductivity produces larger changes in the
Yarkovsky drift when the eccentricity of the orbit is taken into account. The
contribution of the eccentricity to the total semi-major axis drift is
therefore limited, and the circular orbit approximation should not produce
misleading results for the thermal conductivity.

####  4.1.2 Further issues of the Yarkovsky effect modelling

We also briefly discuss further issues related to the Yarkovsky effect model
outlined in Appendix A and later used in Sec. 3. The Yarkovsky effect in real-
world objects is more complicated in several aspects not yet explored above.
These aspects include (i) non-linearity of the boundary conditions for heat
conduction at the surface, (ii) non-isotropy of the thermal emission by the
surface facets (”thermal beaming”), (iii) non-sphericity of the shape. While
we did not test any of them in detail, we note that luckily some of them may
affect the result in opposite ways, thus potentially compensating each other
(at least partially), while others may play a minimal role for (499998) 2011
PT.

As an example we note that the non-linearity effects decrease the
theoretically predicted semi-major axis drift rate $da/dt$ in the relevant
range of thermal conductivity values (see already Rubincam, 1995; Čapek,
2007). The reduction factor ranges between $0.7$ and $0.9$. On the other hand,
thermal beaming effects always increase the semi-major axis drift rate by a
factor ranging between $1.1$ to $1.5$ (see, e.g., Rozitis & Green, 2012). The
two effects thus have a tendency to compensate each other, though obviously in
particular cases some residual factor may remain. At large obliquity, which we
consider the most probable for 2011 PT, the non-sphericity decreases the
predicted semi-major axis drift rate (see, e.g., Vokrouhlický, 1998b). In
fact, at this configuration the body presents smaller cross-section to the
sunlight, and thus processes less thermal energy if compared to a spherical
body. The reduction factor may be similar to that characterising the non-
linearity effects. While we do not have a shape model of 2011 PT, the low
amplitude of the available photometric lightcurve in Kikwaya Eluo &
Hergenrother (2015) suggests a near-spherical shape; or, better, taken the
opposite way there is no suggestion of largely non-spherical shape of 2011 PT.
Thus it is possible that non-sphericity effects in the evaluation of the
Yarkovsky drift rate are only small (modifying the predicted semimajor axis
drift not more than a factor $\sim(0.8-0.9)$; see Fig. 3 in Vokrouhlický,
1998b).

Another set of real-world details that were not included in our simple
analytical model of the Yarkovsky effect has to do with physical parameters.
In particular, they were assumed uniform and constant. In reality, things may
be more complicated. For instance the thermal conductivity, and to a lesser
extend also the heat capacity, may be temperature dependent. This is
especially true for powdered mixtures of minerals (such as the proposed
regolith-like surface), because energy transfer by conduction at the contact
of the grains is efficiently accompanied by radiative transfer through
cavities. Luckily, the orbital eccentricity of (499998) 2011 PT is small, such
that the thermal conditions at perihelion and aphelion do not change
dramatically. For low-enough conductivity they may not exceed the temperature
variations of the daily cycle. For instance the equilibrium subsolar
temperature at pericenter and apocenter are $\sim$320 K and $\sim$390 K,
respectively. Data in Gundlach & Blum (2013) indicate that the conductivity
would typically change in this temperature range by a factor $\sim$1.5-1.7
only. So our assumed constant values should represent a median within such a
range of variation. While the true effects of varying conductivity are not
modeled here, we expect their effect remains small.

Yet another issue is the assumption of homogeneity of the physical parameters.
Consider, for instance, the density $\rho$ discussed in Sec. 2.1.3. The way we
introduced it corresponds to the volume-average (bulk) density. However,
advocating for the existence of the regolith layer, we should ideally adopt a
more complex model of the body: a sphere with a core having the bulk density
and a lower-density layer at the surface. For the fast rotation, the dominant
diurnal effect remains constrained to the skin processes. Therefore the
density in the right hand side of Eqs. (16), (17) and (18) should be the
surface density, rather than the bulk density. Because the surface density is
assumed to be smaller (or at maximum equal) to the bulk density, the same
thermal lag (and thus also dynamical effect) would need larger conductivity.
The scaling is not a priori clear. In the limit of the regolith depth much
larger than the penetration depths of diurnal and seasonal thermal wave
$l_{d}$ and $l_{s}$, $K$ recalibrates proportionally to the ratio of the bulk
density over the surface density (this is because the thermal parameter in
Eqs. 17 and 18 contains a $K\rho$ factor). Since the plausible density ratio
may peak at $\sim$2, the required conductivity values would be skewed to
somewhat larger values. For instance the nicely symmetric solution pattern
seen in Fig. 4 would shift to larger conductivity values for densities above
$\sim$2000 kg m-3. However, even this approach may be too simple when $l_{d}$
and, especially $l_{s}$, become comparable or larger than the regolith depth.
In this case a full-fledge two-layer model would be needed. Vokrouhlický &
Brož (1999) developed this approach for the seasonal component, however a
complete formulation for the diurnal component is not presently available.
Therefore, at this moment we can only conclude with noting that the fully
consistent thermal model with the low-density regolith layer included may
result in a moderate increase of the surface conductivity value.

Despite the modeling issues discussed above, it is worth mentioning that
Chesley et al. (2014) made a prediction of the bulk density of asteroid
(101955) Bennu by combining a model for the Yarkovsky effect, based on known
physical properties, and the measurements of the semi-major axis drift
obtained from orbit determination. The more accurate bulk density value
obtained during the OSIRIS-REx mission (Scheeres et al., 2019) revealed that
the estimate by Chesley et al. (2014) was indeed correct. This successful
prediction could suggest that the model is accurate enough for the analysis
performed on 2011 PT.

###  4.2 The low thermal conductivity of (499998) 2011 PT

The results obtained with the Monte Carlo simulations suggest that the
Yarkovsky drift determined for (499998) 2011 PT can be achieved preferentially
for small values of thermal conductivity $K$, with two most likely values
corresponding to the local maxima: the first one around 0.0001 W m-1 K-1 and
the second one around 0.005 W m-1 K-1 (see Table 3). This conclusion can be
explained in plain words by the combination of the fast rotation and the
relatively large Yarkovsky drift. Indeed, if the thermal conductivity were
large, the fast rotation would largely spread the thermal gradient on the
surface of the asteroid, and a large semi-major axis drift would be impossible
to achieve. On the other hand, a low thermal conductivity would be able to
keep a fair difference in temperature between the day-side and the night-side
of the asteroid, that would be able to produce larger semi-major axis drifts.
The results obtained here indicate that values of thermal conductivity larger
than 0.1 W m-1 K-1 are very unlikely, and the admitted values are instead
fully compatible with the presence of a thermal insulating layer on the
surface. This suggests that the surface is covered by a regolith-like material
or very thin dust. As a comparison, the values of the thermal conductivity of
lunar dusty regolith, obtained during the Apollo missions, are between 0.0012
W m-1 K-1 and 0.0035 W m-1 K-1 (Cremers & Birkebak, 1971; Cremers & Hsia,
1974), which are comparable with the values of the second peaks reported in
Table 3. Assuming that thermal properties of the surface give indication about
the grain size, Gundlach & Blum (2013) found that the radius of lunar regolith
is estimated to be about 50 $\mu$m, which is compatible with the size
distribution of sample lunar soil (Bevan M. French et al., 1991). According to
this, the value of thermal conductivity of the second peak suggests the
presence of dusty regolith grains with a diameter of several tens of $\mu$m on
the surface of (499998) 2011 PT.

At the same time, the thermal conductivity generally decreases with reducing
particle size, and thus the smaller values obtained for the first peak might
suggest an even smaller size of the grains. However, the dust layer in
practice consists of particles of different sizes, but the thermal properties
of particle size mixtures have not yet been well-studied. Recently, Ryan et
al. (2020) developed a 3D model to study the thermal conductivity of regolith,
by simulating heat flow in randomly packed spheres. These authors found that
results predicted by simpler theoretical models are not reliable when the
particles themselves are made of a material that itself has relatively low
thermal conductivity, that could result in significant underestimation of
particle sizes on asteroid surfaces.

It is worth noting that typical values for the thermal conductivity of bare
rock are around 2-5 W m-1 K-1, however low values of the order of 0.1 W m-1
K-1 might be obtained also for a very large porosity of the material, since
measurements performed on meteorites show that $K$ decreases as the porosity
increases (Opeil et al., 2010, 2012; Ostrowski & Bryson, 2019). Kikwaya Eluo &
Hergenrother (2015), using photometry data, suggested a classification as a
X-type asteroid for (499998) 2011 PT, which would imply a stony composition.
Moreover, the high spin rate rules out the possibility that this object is
actually a pure rubble-pile (Pravec & Harris, 2000), since it is lower than
the cohesionless spin-barrier of 2.2 hr, hence the large porosity hypothesis
does not seem very likely. On the other hand, it is not fully clear what
minimum level of cohesion forces are able to withstand moderate stresses
existing on 2011 PT, what the associated level of maximum porosity is and what
it implies for thermal inertia of the surface. These issues deserve further
analysis in the future.

We also note that the third peak at high thermal conductivity (see Fig. 5,
bottom panels), arising at large heat capacity $C$, would either suggest a
bare rock composition or the presence of irony material on the surface. On the
other hand, the presence of iron on the surface would cause a decrease of the
heat capacity, as shown by measurements performed on meteorites collected in
Ostrowski & Bryson (2019), which is in contradiction with the fact that this
peak grows at large $C$, leaving therefore only the bare rock composition
compatible with the third peak solution. However, the occurrence of a high
thermal conductivity, associated to the third peak, is very unlikely due to
the low probability. According to the above facts, the hypothesis of the
presence of dusty regolith-like material seems the most probable one.

Harris & Drube (2016) analyzed the published values of thermal inertia
$\Gamma$ estimated through thermophysical models (Delbó et al., 2015), and
suggested a possible correlation between the thermal inertia and the rotation
period of NEOs, such that $\Gamma$ is decreasing for increasing spin rates. On
the other hand, Marciniak et al. (2019) found no evidence of this trend at
very slow rotation rates. Assuming that the trend of Harris & Drube (2016) is
valid also for shorter rotation periods, our result would support this
hypothesis, though we underline that data used by Harris & Drube (2016) and
Marciniak et al. (2019) refer to objects with spin period larger than 2 hr,
corresponding to the spin barrier of rubble-piles. Moreover, it is difficult
to draw any reliable conclusion about the trend for super-fast rotators, at
the moment, since there are evidences that such rotators could have also a
high thermal inertia, as the reported estimate for asteroid (54509) YORP
indicates (Delbó et al., 2015).

###  4.3 Regolith production and retention on small bodies

The low thermal conductivity we found for the surface of (499998) 2011 PT
could be explained with the presence of regolith. We discuss here the usual
mechanisms that are capable to produce such material, and give some general
hypotheses of how it may be retained on the surface of a fast rotator.

It is generally thought that regolith can be produced in different ways. One
of the main hypotheses is by micrometeoroid impacts, which produce small
grains on the surface by means of fallback ejecta and by breaking up of
boulders (Hoerz et al., 1975; Horz & Cintala, 1996). However, the velocity of
crater ejecta is much larger than the escape velocity of small asteroids
(Housen & Holsapple, 2011), making this process not very feasible in this
context. A second possibility is by thermal fatigue, proposed by Delbó et al.
(2014). The temperature on the surface of an asteroid follows a diurnal cycle,
and the stress resulting from sudden changes of temperature can produce
damages to the surface material. This mechanism of rock weathering and
cracking without ejection could be a possible explanation for the presence of
regolith on small bodies. Moreover, thermal fatigue is a process which is
independent from the size of the asteroid, as opposite from the micrometeoroid
impacts process. The cracking induced by the constant day-night cycle has been
recently observed on asteroid (101955) Bennu, with in-situ images taken during
the OSIRIS-REx mission (Molaro et al., 2020). Another hypothesis is that the
asteroid itself is active. An active asteroid holds some mechanism that causes
mass shedding and ejection of particles from the surface, such as volatile
releasing, rotational disruption, ice sublimation, and electrostatic lofting
(Jewitt et al., 2015). Particles may be ejected from the surface with low-
enough velocity and consequently re-impact on the asteroid again, contributing
to the cracking of superficial material due to micro-impacts. This process of
particle ejection and fallback has been observed during the OSIRIS-REx mission
(Lauretta et al., 2019b; McMahon et al., 2020), but it is not clear if such
activity is limited to primitive bodies of spectral types within the C-complex
(which would not be applicable for 2011 PT).

To explain the presence of regolith on small and fast rotators, Sánchez &
Scheeres (2020) developed a model to study under which rotational conditions
the surface material is retained, partially lost, or completely lost. The
authors found that the covering material is preferentially lost across some
regions of the body, and the area grows as the spin rate increases.
Nevertheless, regions at high latitudes are able to keep regolith at
arbitrarily high spins. This poses a clue that even small and very fast
rotators can retain regolith-like grains and dust on their surface. However,
it remains to be seen what happens to these polar seed-regions when the
Yarkovsky-O’Keefe-Radzievskii-Paddack (YORP) effect (Vokrouhlický et al.,
2015) causes changes in the rotation rate, and potentially also orientation of
the spin axis in space. A robust retention process of the formed grains on the
surface may be needed.

Such a retention phenomenon may be provided by some kind of cohesive forces,
such as van der Waals forces, produced by the direct interaction of solar
plasma with surface material, creating a superficial electric field. Cohesion
may be able to hold fine grains together and stick them to the surface, and
they have been demonstrated to be very high in lunar regolith (Mitchell et
al., 1974). The strength of the cohesive force is represented by the bond
number $B$, which is given by

$B=10^{-5}g_{A}^{-1}d^{-2},$ (12)

where $g_{A}$ is the ambient gravity and $d$ is the grain diameter (Scheeres
et al., 2010; Rozitis et al., 2014). The stability of surface material needs a
bond number larger than ten (Rozitis et al., 2014), hence we can set upper
constraints on the diameter of grains. Assuming the nominal rotation period
and an obliquity of $170\degr$, we computed the ambient gravity $g_{A}$ as a
function of the diameter $D$ of the asteroid and the density $\rho$, using a
spherical model for the shape. Then, we computed the upper limit for the grain
diameter $d$ setting a bond number equal to ten in Eq. (12), obtaining a
function of $R$ and $\rho$. The result is shown in Fig. 7.

Figure 7: The maximum grain size, as a function of the density $\rho$ and the
diameter $D$. Assuming a bond number equal to ten in Eq. (12), the maximum
grain size allowed refers to the value obtained for the maximum ambient
gravity $g_{A}$ over the asteroid surface.

The maximum grain diameter obtained is between 1 and 4 millimeters. Moreover
these bounds are essentially independent of the obliquity $\gamma$ between
$90\degr$ and $180\degr$. These estimates give upper bounds for the size of
particles that can be held on the surface by means of cohesive forces, and
they are well above the $\sim$10-100 $\mu$m size of the lunar regolith
expected from the second peak of thermal conductivity. Therefore, the
hypothesis of the existence of cohesive forces is a reasonable explanation,
which could further be explored using better shape and thermophysical models,
in case that additional appropriate observational data will be acquired.

##  5 Summary and conclusions

In this paper we developed a statistical method to constrain the thermal
conductivity $K$ of (499998) 2011 PT, a near-Earth object of about 35 meters
in diameter which rotates very fast, with a period of 11 minutes. The method
is based on the comparison between the measured Yarkovsky drift and the
theoretically predicted one, which depends on the orbital and physical
parameters of the asteroids. Modeling distributions for the input parameters,
we used a Monte Carlo simulation to produce a distribution of values of the
thermal conductivity, which are compatible with the measured Yarkovsky drift.
When the sample is obtained, a probability density function for $K$ was
computed, and it was used to deduce the most likely values. The method
developed here is very flexible with respect to the known and unknown input
parameters, and it can be easily applied to asteroids with different size and
rotation period. It may be also customized on a case by case basis, in order
to obtain the best possible solution in each individual case.

The obtained probability density functions for the thermal conductivity of
2011 PT always have two local maxima, one around 0.0001 W m-1 K-1, and the
another one around 0.005 W m-1 K-1. These two values are the most likely ones,
with the first one having a slightly higher probability. With the high
probability of at least 95 per cent, we constrained the thermal conductivity
to be in the range 0.00001 W m-1 K-1 ¡ $K$ ¡ 0.1 W m-1 K-1. This is the first
case that the low thermal conductivity solution for a small and super-fast
rotating asteroid has been obtained with such high probability. The low-
thermal conductivity can be interpreted as the presence of regolith-like
material or thin dust on the surface of (499998) 2011 PT, opening many related
scientific questions, and in particular how these super-fast rotators are able
to hold such material on their surface.

To better explore this possibility, however, more observations and
characterizations of small asteroids are needed. A more precise answer can be
obtained by planning space missions to one or more such NEOs, developing
eventually a technique to sample some surface material of fast rotators, which
can be brought back to Earth for an accurate analysis. In this respect, the
proposed extension of the JAXA Hayabusa 2 mission to visit the asteroid444
http://www.hayabusa2.jaxa.jp/enjoy/material/press/Hayabusa2_Press_20200915_ver9_en2.pdf
1998 KY26, which is about 30 meters in diameter and rotates with a period of
about 10 minutes, may provide very interesting insights on the structure and
the composition of very small and fast rotating objects.

###### Acknowledgements.

We thank the anonymous referee for the suggestions that helped us to improve
the manuscript. M. F. and B. N. have been supported by the MSCA-ITN
Stardust-R, Grant Agreement n. 813644 under the European Union H2020 research
and innovation program. The work of D. V. was partially funded by the Czech
Science Foundation (grant 18-06083S). This research has made use of data
and/or services provided by the International Astronomical Union’s Minor
Planet Center. Pan-STARRS is supported by the National Aeronautics and Space
Administration under Grant No. 80NSSC18K0971 issued through the SSO Near Earth
Object Observations Program.

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##  Appendix A Analytical formulation of the Yarkovsky effect

We recall here the analytical formulation of the Yarkovsky effect given in,
for instance, Vokrouhlický (1998b, 1999). We assume to have a spherical body
moving on a circular orbit around the Sun, that rotates along a fixed axis. By
linearizing the boundary condition of the heat diffusion problem, the drift in
semi-major axis $a$ due to the Yarkovsky effect is given by two distinct
components: the seasonal effect

$\bigg{(}\frac{da}{dt}\bigg{)}_{\text{s}}=\hphantom{-}\frac{4\alpha}{9}\frac{\Phi}{\omega_{\text{rev}}}F(R^{\prime}_{\text{s}},\Theta_{\text{s}})\sin^{2}\gamma,$
(13)

and the diurnal effect

$\bigg{(}\frac{da}{dt}\bigg{)}_{\text{d}}=-\frac{8\alpha}{9}\frac{\Phi}{\omega_{\text{rev}}}F(R^{\prime}_{\text{d}},\Theta_{\text{d}})\cos\gamma.$
(14)

In the above equations, $\alpha$ is the absorption coefficient of the surface,
$\Phi$ is the radiation pressure coefficient (e.g., Vokrouhlický et al. 2015),
$\omega_{\text{rev}}$ is the orbital frequency (mean motion), and $\gamma$ is
the spin axis obliquity. $R^{\prime}_{\text{s}}$ and $R^{\prime}_{\text{d}}$
are the scaled (nondimensional) values of the radius $R$, defined as

$R^{\prime}_{\text{s}}=\frac{R}{l_{\text{s}}},\quad
R^{\prime}_{\text{d}}=\frac{R}{l_{\text{d}}},$ (15)

where $l_{\text{s}},\,l_{\text{d}}$ are the penetration depths of the seasonal
and diurnal thermal wave given by

$l_{\text{s}}=\sqrt{\frac{K}{\rho C\omega_{\text{rev}}}},\quad
l_{\text{d}}=\sqrt{\frac{K}{\rho C\omega_{\text{rot}}}}.$ (16)

Note that $l_{\text{s}}$ and $l_{\text{d}}$ depend on the thermal conductivity
$K$, the heat capacity $C$, and the density $\rho$ of the asteroid.
Additionally, the two length scales depend on the respective frequencies: (i)
the spin frequency $\omega_{\text{rot}}$ in the case of the diurnal effect,
and (ii) the orbital frequency $\omega_{\text{rev}}$ in the case of the
seasonal effect. The thermal parameters $\Theta_{\text{s}},\Theta_{\text{d}}$
also depend on the physical and thermal characteristics of the object, and
they are defined as

$\displaystyle\Theta_{\text{s}}=\frac{\sqrt{\rho
KC\omega_{\text{rev}}}}{\varepsilon\sigma T_{\star}^{3}},$ (17)
$\displaystyle\Theta_{\text{d}}=\frac{\sqrt{\rho
KC\omega_{\text{rot}}}}{\varepsilon\sigma T_{\star}^{3}},$ (18)

where $\sigma$ is the Stefan-Boltzmann constant, $\varepsilon$ is the
emissivity and $T_{\star}$ is the subsolar temperature, defined by
$\varepsilon\sigma T_{\star}^{4}=\alpha\mathcal{E}_{\star}$, with
$\mathcal{E}_{\star}$ being the solar radiation flux at the distance of the
body. The function $F$ in Eqs. (13) and (14) depends on both the corresponding
scaled radius and the thermal parameter, and it determines the total magnitude
of the Yarkovsky drift. The definition of $F$ is given by

$F(R^{\prime},\Theta)=-\frac{k_{1}(R^{\prime})\,\Theta}{1+2k_{2}(R^{\prime})\,\Theta+k_{3}(R^{\prime})\,\Theta^{2}}.$
(19)

The coefficients $k_{1},k_{2},k_{3}$ are positive analytical functions of the
scaled radius, and their precise definition can be found, for instance, in
Vokrouhlický (1998a, 1999). It is worth noting that the seasonal component
always produces an inward migration, while the direction of migration for the
diurnal component depends on the obliquity $\gamma$.


# Overview of The MediaEval 2021 Predicting Media Memorability Task

Rukiye Savran Kiziltepe1, Mihai Gabriel Constantin2, Claire-Hélène Demarty3,
Graham Healy4, Camilo Fosco5, Alba G. Seco de Herrera1, Sebastian Halder1,
Bogdan Ionescu2, Ana Matran-Fernandez1, Alan F. Smeaton4, Lorin Sweeney4
1University of Essex, UK
2University Politehnica of Bucharest, Romania
3InterDigital, France
4Dublin City University, Ireland
5Massachusetts Institute of Technology Cambridge, Massachusetts, USA.
<EMAIL_ADDRESS>
###### Abstract.

This paper describes the MediaEval 2021 Predicting Media Memorability task,
which is in its 4th edition this year, as the prediction of short-term and
long-term video memorability remains a challenging task. In 2021, two datasets
of videos are used: first, a subset of the TRECVid 2019 Video-to-Text dataset;
second, the Memento10K dataset in order to provide opportunities to explore
cross-dataset generalisation. In addition, an Electroencephalography
(EEG)-based prediction pilot subtask is introduced. In this paper, we outline
the main aspects of the task and describe the datasets, evaluation metrics,
and requirements for participants’ submissions.

††copyright: rightsretained††doi: ††isbn: ††conference: Multimedia Evaluation
Workshop; December 13-15 2021; Online††journalyear: 2021

##  1\. Introduction

Information retrieval and recommendation systems deal with exponential growth
in media platforms such as social networks and media marketing. New methods of
organising and retrieving digital material are needed in order to increase the
usefulness of multimedia events in our daily lives. Memorability, like other
important video properties, such as aesthetics or interestingness, can be
viewed as useful to contribute in the selection of competing videos especially
when developing advertising or instructional material. In advertising,
predicting the memorability of a video is important since multimedia materials
have varying effects on human memory. In addition to advertising, this task
may have an impact on other fields such as film making, education, and content
retrieval.

The Predicting Media Memorability task addresses this problem. The task is
part of the MediaEval benchmark and, following the success of previous
editions (Cohendet et al., 2018; Constantin et al., 2019; García Seco de
Herrera et al., 2020; Savran Kiziltepe et al., 2021), creates a common
benchmarking protocol and provides a ground truth dataset for short-term and
long-term memorability using common definitions.

##  2\. Related Work

The computational study of video memorability is a natural extension of
research into image memorability prediction, which has gained increasing
attention in the years after Isola et al.’s work (Isola et al., 2013). Models
have reached remarkable predictive accuracy for image memorability (Khosla et
al., 2015; Squalli-Houssaini et al., 2018), and we have just begun to see the
application of approaches such as style transfer to enhance image memorability
(Siarohin et al., 2019), demonstrating that we have progressed from simply
measuring memorability to using it as an evaluation aspect.

In contrast, computer science research on visual memorability (VM) is still in
its early stages. Recent studies on video memorability have focused on the
short-term (Newman et al., 2020), but the lack of studies on VM can be
explained by a number of factors. To begin with, there are currently not
enough publicly available data sets for training and testing models.The second
problem is the absence of a standardised definition for VM. In terms of
modeling, previous attempts at VM prediction (Shekhar et al., 2017; Cohendet
et al., 2019) have identified several features that contribute to VM
prediction, including semantic, saliency, and colour features. However, the
work is far from complete, and our ability to propose effective computational
models will aid in meeting the challenge of VM prediction.

The purpose of this task is to contribute to the harmonisation and advancement
of this rapidly growing multimedia field. Additionally, in contrast to prior
work on image memorability prediction in which memorability was tested only a
few minutes after memorisation, we present a dataset containing long-term
memorability annotations. We expect that models trained on this will produce
predictions that are more indicative of long-term memorability, which is
preferred in a wide variety of applications. This year we also distribute an
external dataset for generalisation purposes and propose a new pilot task
which is EEG-based video memorability.

##  3\. Task Description

The Predicting Media Memorability task asks participants to develop automatic
systems that predict short-term and long-term memorability scores from short
videos. Participants were given a dataset of short videos with short-term and
long-term memorability scores, raw annotations, and extracted features.
Participants were assigned three sub-tasks:

  * •

Video-based prediction: Participants are required to generate automatic
systems that predict short-term and long-term memorability scores of new
videos based on the given video dataset and their memorability scores.

  * •

Generalization (optional): Participants will train their system on one of the
two sources of data we provide and will test them on the other source of data.
This is an optional sub-task.

  * •

Electroencephalography (EEG)-based prediction (pilot): Participants are
required to generate automatic systems that predict short-term memorability
scores of new videos based on the given EEG data. This is a pilot sub-task and
details for it can be found in (Sweeney et al., 2021).

##  4\. Collection

This task utilises a subset of the TRECVID 2019 Video-to-Text video dataset
(Awad et al., 2019). The dataset contains Twitter Vine videos where various
actions are performed. This year, the dataset has been expanded and normalised
short-term memorability scores are provided with memory alpha decay values.
Additionally, we open the Memento10K (Newman et al., 2020) dataset to
participants. Apart from traditional video information like metadata and
extracted visual features, part of the data will be accompanied by
Electroencephalography (EEG) recordings that would allow to explore the
physical reactions of users.

A set of pre-extracted features are also distributed as follows:

  * •

image-level features: AlexNetFC7 (Krizhevsky et al., 2012), HOG (Dalal and
Triggs, 2005), HSVHist, RGBHist, LBP (He and Wang, 1990), VGGFC7 (Simonyan and
Zisserman, 2015), DenseNet121 (Huang et al., 2017), ResNet50 (He et al.,
2016), EfficientNet b3 (Tan and Le, 2019);

  * •

video-level feature: C3D (Tran et al., 2015);

  * •

audio-level feature: VGGish (Hershey et al., 2017).

Three frames from each video were used to extract image-level features: the
first, the middle, and the last frame. Additionally, each TRECVid video
includes at least two textual captions summarising the action, whereas
Memento10K includes five. The annotations acquired from participants included
the first and second appearance positions of each target video, as well as
participants’ response times and the keys pressed while watching each video.

###  4.1. TRECVid 2019 Video-to-Text dataset

The TRECVid 2019 Video-to-Text dataset (Awad et al., 2019) contains 6,000
videos. In 2021, three subsets were distributed as part of the MediaEval
Predicting Media Memorability task. The training set contained 588 videos, the
development set 1,116 videos and the test set 500 videos. Each video has two
associated memorability scores indicating its likelihood of being remembered
after two distinct periods of memory retention. Similar to previous editions
of the task (Cohendet et al., 2018; Constantin et al., 2019), memorability was
measured twice using recognition test: a few minutes after the videos were
shown (short-term) and 24-72 hours later (long-term). The videos are released
under Creative Commons licences that allow their redistribution.

The ground truth dataset was generated using a video memorability game
protocol proposed by Cohendet et al. (Cohendet et al., 2019). The memorability
game was formed in two versions. One was made available on Amazon Mechanical
Turk (AMT), and another was made available for general use in three languages:
English, Spanish, and Turkish.

In the video memorability game protocol, participants were expected to watch
180 and 120 videos in short-term and long-term memorisation steps,
respectively. The goal was essentially to press the space bar whenever
participants recognise a previously seen video, which allows for the
determination of which videos they do and do not recognise. The game begins
with the repetition of 40 target videos after a few minutes to accumulate
short-term memorability labels. Regarding the first step’s filler videos, 60
non-vigilance filler videos are shown once. After a few seconds, 20 vigilance
filler videos are repeated to ensure that participants are paying attention to
the task. After 24 to 72 hours, the same individuals are anticipated to return
for the second step, which involves collecting labels for long-term
memorability. This time, 40 target videos chosen at random from the non-
vigilance fillers in the first stage and 80 fillers chosen at random from new
videos are displayed to determine the target videos’ long-term memorability
scores. The percentage of correct recognition for each video is used to
calculate short-term and long-term memorability scores.

###  4.2. Memento10K dataset

The Memento10K dataset (Newman et al., 2020) contains 10,000 three-second
videos depicting in-the-wild scenes, with their associated short-term
memorability scores, memorability decay values, action labels, and 5
accompanying captions. The scores were computed with 90 annotations per video
on average, and the videos were shown to participants without sound. 7,000
videos were released as a training set, and 1,500 were provided for
validation. The last 1,500 videos were used as the test set for scoring
submissions.

##  5\. Submission and Evaluation

As it is with previous tasks, each team is expected to submit both short-term
and long-term memorability predictions. A total of ten runs, five for each,
can be submitted for video-based prediction. In addition, participants can
submit five runs per optional sub-task (generalisation and EEG-based
prediction). All information, including given features, ground truth data,
video sample titles, features extracted from visual material, and even
external data, may be used to build the system. Short-term and long-term
annotation memorability runs must be submitted separately and must not include
each others.

Classic evaluation metrics (including Spearman’s rank correlation) are used to
compare the predicted memorability scores for the videos with the ground truth
memorability scores.

##  6\. Discussion and Outlook

In this paper we introduced the 4th edition of the Predicting Media
Memorability at the MediaEval 2021 Benchmarking initiative. With this task, a
comparative assessment of current state-of-the-art machine learning techniques
to predict short- and long-term memorability can be conducted. A dataset
containing short videos is distributed with memorability annotations and
external data is provided for generalisation purposes. Moreover, EEG
annotations are also provided for a pilot study. Related information has also
been made available to participants so they can refine their strategies. The
2021 MediaEval workshop proceedings presents details on the participants’
approaches to the task including methodologies used and findings.

###### Acknowledgements.

MGC and BI’s contribution is supported under project AI4Media, a European
Excellence Centre for Media, Society and Democracy, H2020 ICT-48-2020, grant
#951911. The work of RSK is partially funded by the Turkish Ministry of
National Education. This work was part-funded by NIST Award No. 60NANB19D155
and by Science Foundation Ireland under grant number SFI/12/RC/2289_P2.

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