Biele et al. Progress in Earth and Planetary Science
https://doi.org/10.1186/s40645-019-0291-0(2019) 6:48

Progress in Earth andPlanetary Science

RESEARCH ARTICLE

Open Access

Effects of dust layers on thermal emission
from airless bodies
Jens Biele1* , Ekkehard Kührt2, Hiroki Senshu3, Naoya Sakatani4, Kazunori Ogawa5, Maximilian Hamm2,
Matthias Grott2, Tatsuaki Okada4 and Takehiko Arai6

Abstract
We have investigated the influence of thin thermally opaque dust layers on the thermal emission of rocks andregolith and determined the thermal response of these dust-covered surfaces to diurnal insolation cycles. Results
are computed for Hayabusa2’s target asteroid (162173) Ryugu, which was observed by thermal infrared instrumentson the orbiter and in situ. We show that even a very thin (10..100 μm) fine-grained porous dust layer with thermal
inertia of 25 J m−2 K−1 s−1/2 can have a significant influence on surface temperatures and alter the apparent thermal
inertia of the underlying material derived under the simplified assumption of a homogenous half space by morethan 20%. The masking of the underlying material is complete at about 1 diurnal skin depth, corresponding to ~ 10 mm
on Ryugu. Between 0.1 and 1 diurnal skin depths, we find a thermal lag smaller than what would be predicted for asurface consisting of dust only.
If a dust cover were present on Ryugu, this should be clearly visible in the data returned by the orbiter’s thermal infrared
imager (TIR) and the MASCOT lander’s radiometer (MARA), which observed a single boulder at the landing site.
However, this appears not to be the case, and dust seems to play a minor role in the thermal emission fromthe asteroid.
Keywords: Thermophysical models, Surfaces, Small bodies, Solar system, Thermal inertia, Dust cover, Layering

Introduction
The Hayabusa2 spacecraft (Watanabe et al. 2017) arrived
at its target asteroid (162173) Ryugu in June 2018 andcharacterized the asteroid’s surface using the spacecraft
camera suite (ONC), near infrared spectrometer (NIRS3),
and thermal infrared imager (TIR). The MASCOT lander(Ho et al. 2017) explored the surface of Ryugu on Oct 3,
2018, and the MASCOT radiometer MARA determined
local surface brightness temperatures over a full diurnal
cycle (Grott et al. 2019). MARA observed a single boulder
significantly larger than the MARA field of view of about
10 cm diameter. Images obtained by the MASCOT camera
MasCam (Jaumann et al. 2017) provide context for
MARA observations. MasCam images have a pixel resolution of 0.15 mm at best and close to 0.3 × 0.7 mm2 inside
the MARA field of view. Thermal conductivity of theboulder in the MARA field of view was determined to beclose to 0.1 W m−1 K−1, much lower than anticipated for* Correspondence: jens.biele@dlr.de1
DLR – German Aerospace Center, RB-MUSC, 51147 Cologne, Germany
Full list of author information is available at the end of the article

competent rock (Grott et al. 2019). Here, we investigate
whether the low determined conductivity could be theresult of a thin dust layer masking the thermal signatureof a much more thermally conductive boulder.

Background
Vertical layering of thermal or physical properties has
been extensively studied for the case of Mars, and it was
found that layering can significantly influence the shapeof the diurnal (and seasonal) surface temperature curves(Putzig and Mellon 2007).
For Mars, modeling of heterogeneous surfaces (Jakosky1979) provided qualitative explanations of deviations froma homogeneous surface model, and a dust over duricrustmodel has been suggested already by Ditteon 1982. To
investigate the effects of layering, Mellon and Putzig 2007
employed a modified numerical model of a layered subsurface (see (Mellon et al. 2004)). They used this layered
thermal model to compute effective temperatures for
idealized layered surfaces and derived apparent thermal inertia to examine its variations. In an effort to© The Author(s). 2019 Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0
International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, andreproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to
the Creative Commons license, and indicate if changes were made.

Biele et al. Progress in Earth and Planetary Science(2019) 6:48

estimate depth to ground ice for Phoenix Lander sitecharacterization, Titus et al. 2006 fit TES brightness
temperature observations to two-layer thermal-model
results, yielding estimates of ground-ice depth wellwithin a seasonal skin depth in all three of theirstudy regions; they found a best-fit model with a
58 mm-thick layer of sand-like material (thermal
inertia Γ = 216 J m−2 K−1 s−1/2) over a substrate withhigh thermal inertia (700 J m−2 K−1 s−1/2).
Ruff and Christensen 2002 studied Martian dust (grain
size ~ 40 μm similar to our model dust) but are mostly
concerned developing the Martian dust cover index DCI
which discriminates dust-covered (typical thicknesses ofcentimeters to meters) regions from dust-free regions
and correlates well with albedo and TI. They note thatwhile albedo is sensitive for the first few micrometers,
DCI is sensitive to first tens of micrometers, and TI issensitive to the first few centimeters on Mars. They
speculate (no data or calculations are shown to support
this) that an intermediate dust cover of few 10s of micrometers to a few millimeters may show up in DCI but
not strongly in TI, not fully obscuring, in the TI sense,
the underlying materials.
So, for the quantitative application of two-layer
models, for Mars, mostly a high TI layer over a low TI
subsurface with a maximum Γ contrast of ~ 10:1, usuallymuch less, has been studied (Vasavada et al. 2017;
Edwards et al. 2018). For the high-over-low Γ layering, it
is found that generally ground temperatures rise laterand more slowly in the morning, peak later, and fall off
more slowly (and with a more linear trend) in the afternoon and evening afternoon relative to a homogenous
surface with intermediate Γ (Vasavada et al. 2017).
The perhaps most interesting work for a general
two-layer case on Mars on apparent thermal inertia
has been presented by Putzig and Mellon 2007. They
allow both high-over-low and low-over-high Γ layeringand find some regions with evidence for the latter(e.g., dust/duricrust, 1–4 mm thick, Γ 56/889) as do
Audouard et al. 2014.
For the Moon and Mercury, thermal conductivity anddensity profiles have been derived from Apollo-era data
(e.g., (Keihm and Langseth Jr 1973, Mitchell and De
Pater 1994)). In particular, only the e-folding length forthe profiles have been determined from observed brightness temperatures (Vasavada et al. 1999, Vasavada et al.
2012, Hayne et al. 2017). For asteroids, telescopic
observations indicate that vertical layering may also bepresent. In particular, thermal conductivity, but also
density, seem to increase as a function of depth (Harrisand Drube 2016). However, methods employed andresults obtained for the Moon and Mercury cannot bedirectly applied here. Instead, we expand on the work of,e.g., Paige et al. 1994.

Page 2 of 11

Methods/Experimental
In the following, we will report results for regolith thermophysical properties using the surface thermal inertia,
but we want to point out that there are a number ofcomplications which need to be kept in mind whenusing this quantity. Thermal inertia is defined asΓ¼

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
k ρcP

ð1Þ

where k is thermal conductivity, ρ bulk density, and cP
specific heat capacity at constant (zero) pressure. However, using thermal inertia as defined by Eq. (1) only
makes physical sense if k, ρ, and cP are constant, while itis well known that both k and cP depend on temperature(Sakatani et al. 2017; Biele and Grott 2018). Therefore, Γneeds to be interpreted with caution, and it is commonly
evaluated at a representative surface temperature to
mediate the above limitation.
It is also important to note that while (at given albedo,
emissivity, insolation) the surface temperature only
depends on Γ provided that k, ρ, and cP are constant, thetemperature-depth profile is determined by the productρcP as well as k. Therefore, the same Γ can result indifferent temperature profiles, and layered surfaces
should be described by k, ρ, and cP rather than Γ. Therefore, while using Γ to discuss results, all calculations
carried out here use k, ρ, and cP as variables.
In order to approximate the surface as a half space,considered particle sizes have to be smaller than thediurnal skin depth (Sakatani and Ogawa 2017). The
thermal skin depth is the e-folding depth of the subsurface diurnal or seasonal thermal wave. The diurnal
temperature curve probes the subsurface to a depth ofthe order of a few diurnal thermal skin depth, i.e., a few
centimeters for Ryugu (Table 2).
We will study the case of a simple yet instructive
2-layer model, in which we assume a homogenous
fine-grained material of very low thermal inertia Γ tocover a homogeneous subsurface of much higher Γ. In themodeling, the temperature dependence of thermophysicalproperties as well as lateral heterogeneity and roughness
effects will be ignored.
The general setup of our model is shown in Fig. 1,
and two cases will in general be discussed in the following: first, we will consider the effect of a thin dust
layer on the thermal emission of solid bedrock with athermal conductivity appropriate for carbonaceouschondrites (left hand side of Fig. 1), and second, we
will consider the influence of dust covering either
porous rock or broken up regolith (right hand side ofFig. 1). While the thermophysical properties (Table 1)of the individual layers will be kept constant in our
analysis, the thickness of the dust layer δ will bevaried.

Biele et al. Progress in Earth and Planetary Science(2019) 6:48

Page 3 of 11

Fig. 1 Left: Two-layer model of an irradiated surface consisting of competent bedrock covered by a thin dust layer. For the rock, a thermal inertiaof 1183 J m−2 K−1 s−1/2 will be assumed, while the dust layer has a thermal inertia of 25 J m−2 K−1 s−1/2 (compare Table 1). Average dust thickness
is δ. Right: same as left, but for a dust-covered porous rock or coarse regolith with thermal inertia 300 J m−2 K−1 s−1/2S ð1−AÞ
∂T 
4
cos
ð
θ
Þ
¼
εσ
T
−k
ð
0
Þ
B
r 2h
∂x x¼0

We then solve the one-dimensional heat conduction
equation (Delbo et al. 2015), which is given by:∂T ðt; xÞ
∂
∂T
¼
k ðxÞ
ρðxÞcp ðxÞ
∂t
∂x
∂x

ð2Þ

prescribing surface insolation:

ð3Þ

as the upper boundary condition at x = 0. Here, T is
kinetic temperature, σB the Stefan-Boltzmann constant, t
is time, x is depth, S is the solar constant, A is the bondalbedo, rh is the heliocentric distance in units of AU, andTable 1 Parameters used in this study (Wada et al. 2018; Watanabe 2019)
Property

Symbol

Value

Units

Comment

Solar constantS

1366.1

W/m2

Solar irradiance at 1 AU

Stefan-Boltzmann constantσ

5.67 × 10−8

W/m2/K4

Pole orientation

RA

96.40101[°]

Right ascension J2000

DEC− 66.39948[°]

Declination J2000

Rotation periodP

7.632622[h]ObliquityI

171.6428[°]

Orbital period

T

473.9

[d]

Absolute time

Oct 03, 2018

Season parameter

Lp

153.5913[°]

Retrograde spin!

Orbit parameterse0.19034[]

Subsolar latitude − 5.277°

Perihelion distanceq0.96315[AU]

Heliocentric distancerh

1.2447[AU]

Geographical latitudeϕ− 34.60[°]

MASCOT observation site

Specific heat capacity

cP

700

[J kg−1 K−1]

K ((Biele and Grott 2018), average temperature 215 K)

Emissivity

ε

0.90

[]

Hemispherical NIR emissivity

Bond albedo

A

0.0178[]Grain density

ρg

2100

[kg m−3]

Bedrock density

ρb

2000

[kg m−3](1)Distance to Sun on Oct 3, 2018

Low porosity < 5%

−3

Regolith/porous rocks bulk density

ρreg

1480

[kg m ]

Dust bulk density

ρdust

366

[kg m−3]

Corresponding porosity is 83%

Bedrock heat conductivity

kb

1.0

[W K m ]Resulting Γ = 1183 J m−2 K−1 s−1/2

Regolith heat conductivity

kreg

0.0866[W K−1 m−1]

Resulting Γ = 299.5 J m−2 K−1 s−1/2−1Resulting Γ = 25.31 J m−2 K−1 s−1/2

Dust heat conductivity
(1)

kdust

0.0025−1

Corresponding porosity is ~ 30%−1−1[W K m ]Angle between asteroid’s vernal equinox—the ascending node of its equator and orbit—and perihelion, measured in the orbital plane. Orbital elements are2018 average osculating elliptical elements

Biele et al. Progress in Earth and Planetary Science(2019) 6:48

θ is the local solar zenith angle, where cos θ = 0 during
the night. As for the lower boundary condition, weprescribe zero heat flux at a depth of three annual skindepths, and T(t,x) = 0 is chosen as the initial condition
at t = 0.
The employed algorithm for solving Eqs. (2) and (3) is
described in (Kührt and Giese 1989) and provides T(x,t)
for independent k, ρ, and cp without explicitly using
thermal inertia. In our model, depth-dependence ismodeled as a step function increase of k und ρ at a given
depth x = δ, and we verified that this does not result in
any numerical problems. A finite element method fromthe Fortran NAG library is used for solving the differential equation. Spatial discretization is chosen such thatconvergence of the results is reached, and at least 10
grid points are located in the dust layer, whereas at least
40 grid points are distributed below. The grid size
increases quadratically with depth, where the maximum
spacing is limited to be smaller than 3 times the orbital
thermal skin depths (i.e., up to 9 m). We found that the
precise form of the depth grid is not important, as longas spatial and temporal convergence is achieved.
It is well known (e.g., Pelivan 2018) that the numberof rotations (or orbits) needed for convergence is a function of Γ, the daily insolation and the choice of initialconditions. To achieve convergence, calculations were
performed for at least half an orbital period, startingfrom aphelion. Heliocentric distance is changed stepwise, at midnight. The convergence in depth and space
was then tested, and it was found that using half thenormally used depth step size does not result in asignificant change of the obtained temperature profiles.
We therefore estimate that the systematic numerical uncertainty of the surface temperatures is smaller than 0.1 K.
The global and orbital parameters used in this study
are summarized in Table 1, taken from Wada et al. 2018
and recent Hayabusa2 mission results ((Sugita et al.
2019, Watanabe et al. 2019)).
For the material parameters, we use realistic values
either observed for Ryugu or based on measurements ofcarbonaceous chondrites and the lunar regolith. For thegrain or bedrock density, we use an average of the values
reported for porous CI and CM meteorites (Consolmagno
et al. 2008), while the porosity of porous rock or coarse
regolith is consistent with either the estimated rock porosity (~ 30%) from MARA measurements (Grott et al.
2019) or a random close packing of polydisperse, slightly
irregular, 1 cm diameter particles, giving the same thermal
conductivity than the porous rock.
For the dust fraction, a thermal inertia Γ ≈ 25 to 31 J
m−2 K−1 s−1/2 has been assumed, representative, e.g., of
the uppermost lunar surface with grain sizes < 70 μm
and a porosity of 83% (Hapke and Sato 2016). Bothlayers have been treated, in the model, as continuous

Page 4 of 11

matter. Note that we fix the properties of the assumed
upper (dust) layer and do not investigate how theresults converge to the thermal response of the subsurface if both layer’s thermal properties converge.
The bedrock thermal conductivity is a typical ~ 250 K
value for carbonaceous meteorite analogues (Cold Bokkeveld (CM): 0.5 W/m/K, NWA5515 (CK): 1.5 W/m/K(Opeil et al. 2010; Opeil et al. 2012)), while the thermal
conductivity of the coarse regolith and dust have been
calculated using the model of Sakatani et al. 2017, evaluating conductivity at a mean temperature of 300 K. For
the specific heat capacity, values for lunar regolith at
215 K have been assumed, which is close to the averagetemperatures we find in our study. Note that our model
implicitly assumes that the dust layer is optically andthermally thick (opaque), i.e., the layer thickness δ mustbe large compared to the median grain diameter; otherwise, one would need to solve the fully time-dependent
system of the radiative and heat transfer equations (Haleand Hapke 2002). The model is therefore only strictly
applicable for δ > at least 5 monolayers of dust particles;
thus, very small dust layer depths implicitly require veryfine dust for the results to be correct. This can beachieved for C-type dust of grain sizes between 0.1 and70 μm with a porosity varying between 70 and 81%(calculated with Sakatani et al. 2017).
To put the considered layer thicknesses into perspective, we have furthermore calculated the thermalskin depths associated with Ryugu’s orbital and diurnal
periods, where the skin depth is defined byrﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ.
s ¼ kPπρcp

ð4ÞHere, P refers to the orbital and diurnal periods,
respectively, and resulting skin depths for the different
materials considered are summarized in Table 2.
In the following, we will estimate the influence of athin dust cover on the surface temperature, and this will
be quantified by estimating the apparent thermal inertiaof the dust-covered surface, i.e., the thermal inertia thatwould be obtained from the measured temperature
curves under the assumption that the regolith is indeed
homogeneous rather than layered. To derive apparent
Table 2 Diurnal (sdiurnal) and annual (sannual) skin depths forbedrock, coarse regolith, and dust, assuming material
parameters as given in Table 1. “Porous rock” and “coarse
regolith” are synonyms in this paper
Material

sdiurnal [m]sannual (m)Bedrock

0.0790

3.05

Porous rock

0.02701.04

Dust

0.009240.357

Biele et al. Progress in Earth and Planetary Science(2019) 6:48

thermal inertia, four methods are applied, and these will
be evaluated in the following. First, thermal inertia canbe estimated from the maximum daytime temperature,assuming that insolation is known. Second, the minimum nighttime temperature may be used. Third, thephase shift of the maximum temperature with respect tolocal noon can be evaluated. And finally, one can fit (ina least-squares sense) part of (e.g., only daytime or nighttime) or even the entire diurnal temperature curve to ahomogeneous model. The difference between the thermal
inertia of the underlying material and the calculated
apparent thermal inertia then indicates how the dust layerchanges thermal emission of the base material.
To eliminate sampling effects and rounding errors
when estimating thermal inertia from the maximum
daytime temperature and phase shift, the temperature
curves calculated by solving Eqs. 2 and 3 were output at
a resolution of 1° in rotation angle and then fitted in awindow of ± 45° around the maximum by recursive
orthogonal polynomials. The optimal degree of the polynomials was determined according to Akaike’s information criterion (Akaike 1974) and was found to vary
between 6 and 10. Maximum temperature Tmax as wellas its phase shift φ with respect to local noon was thenread off from the polynomial fit. The daily minimum
temperature Tmin was determined by taking the global
minimum of the 1° resolution T(t) curves directly.
For the determination of apparent thermal inertia,
values obtained from the diurnal temperature curves
using the above methods have been compared to thosefor homogenous surfaces, with Γ being varied between
25.3 and 1183 J m−2 K−1 s−1/2. Note that while all quantities derived here depend on insolation and therefore
on latitude and season, we have restricted the present
analysis to the actual spin and orbital parameters of
Ryugu on Aug 3, 2018, and all curves are evaluated at
the geographical latitude of MASCOT’s landing site. The
curves φ (log(Γ)), φ (log(Γ)), and φ (log(Γ)) are strictly
monotonic and have been fit by empirical functions(distorted logistic curves) in log(Γ); root-mean-square
residuals are 0.013°, 0.019 K, and 0.1 K for φ, Tmax, andTmin, respectively, and the functions are physicallyreasonable when Γ tends to zero or infinity. We willneed them later to derive the apparent Γ from layered
model phase lag, minimum and maximum temperatures.
We also define, for later uncertainty estimations, aworst-case modeling temperature bias curve as a piecewise linear curve of temperature bias versus rotation
angle, given by the nodes [− 180, − 94, − 86, + 93, +179]°, ±[0.35, 1.0, 0.1, − 0.1, 0.35] K.
One of the common approaches for the interpretationof thermal infrared data is to use a thermal model todetermine Γ from a best fit to the data. In general, single
layer, homogenous models are used to determine thePage 5 of 11

surface thermal inertia, and it is thus essential to understand what thermal inertia one would retrieve if theobserved surface was covered by a fine dust layer.
Having calculated diurnal temperature curves for dustcovered bedrock and porous rock/regolith, respectively,
we have now applied four methods to estimate what wecall apparent thermal inertia Γeff, i.e., the thermal inertia
one would obtain by inverting or best fitting (in aparticular sense) the two-layer model’s temperature datausing a one-layer model. Apparent thermal inertia has
been determined here from phase lag, maximum
temperature, minimum temperature, a least-squares
fit to the entire diurnal temperature curve, and froma least-squares fit to the daytime and nighttime diurnal
temperature curve.
In detail, we proceeded as follows: for the leastsquares fits, an interpolating function for surface temperatures of our homogeneous model for arbitrary
25 < Γ < 1200 J m−2 K−1 s−1/2 was prepared. It interpolates
the calculated curves (for 10 discrete values of log(Γ)) tothe desired log(Γ), spline-interpolation). The X2(Γ), thatis, the sum of the squared deviations between the actual
two-layer temperatures and the interpolated homogeneous temperatures, is minimized to find Γeff. The fit isextended over all rotation angles for the day+night
method, for rotation angles from − 93° to + 93° for the
day only method, and over [− 180..− 94, + 93.. + 179°] forthe night only method. The accuracy of the calculatedΓeff depends mostly on the interpolation for the homogeneous models; comparing spline to pchip (Piecewise Cubic
Hermite Interpolating Polynomial) to a linear
interpolation in log(Γ) as the worst case, relative uncertainties in Γeff are estimated as typically < 1% withsomewhat larger uncertainties for δ = 3..5 mm, where theone- and two-layer curves have their largest discrepancies.
To find a conservative uncertainty estimate of Γeff, we add
a worst-case modeling temperature bias curve to the nominal 2-layer surface temperature and find the corresponding change in Γeff.
For the remaining methods, based on the two-layer
phase lag, maximum and minimum temperature, we justdetermine the intersection of the fit functions in φ, Tmax,
and Tmin of our homogeneous calculations with the twolayer values. As the fit functions are strictly monotonous, this gives a unique value each. Based on the rms
residuals of the fit functions in φ, Tmax, and Tmin of ourhomogeneous calculations plus the uncertainty from ourworst-case modeling temperature bias curve (and its derivative at noon), we estimate the precision of the calculated
apparent thermal inertia as 0.2% (lag, Tmax) to 5% (Tmin).

Results and discussion
To demonstrate the influence of dust on thermal emission, we have calculated example diurnal temperature

Biele et al. Progress in Earth and Planetary Science(2019) 6:48

curves for the cases of dust-covered bedrock and regolithby solving Eqs. 2 and 3, for a dust layer thickness ofδ = 1 mm. They are shown in Fig. 2. As is evident
from the figure, at this dust thickness, the dust layer masks
the thermal emission of the underlying material alreadyalmost completely, and both diurnal curves are hardly
distinguishable. Only small differences in nighttime
temperature and cooling rate after sunset remain, butthese are generally below 10 K. Overall, curves resemble
those of low thermal inertia units, i.e., they exhibit small
phase lag φ, high Tmax, and have a symmetric appearance.
The influence of the bottom layer is most visible shortly
after sunset, when the fast cooling rate suddenly decreases,
givingrise toa “kink”inthetemperature curveandresulting
in night-time temperatures which are significantly higher
than for the homogeneous case. The cooling rate at night issignificantly lower (and almost vanishes for intermediate
dust thicknesses) thaninthehomogeneouscase.
To study the influence of layer thickness on the shapeof the diurnal temperature curve, we have calculated
surface temperatures varying the dust layer thicknesses δ
between 0 and 100 mm. Results of these calculations are
summarized in Fig. 3, where phase lag φ of the maximum daytime temperature with respect to local noon,
the maximum temperature Tmax, as well as the minimum temperature Tmin, are shown as a function of δ.
Here, the red line corresponds to the model of dust covered regolith/porous rock, while the blue line corresponds to the bedrock case. In addition, dashedhorizontal lines indicate φ, Tmax, and Tmin which

Page 6 of 11

correspond to homogeneous dust, regolith, and bedrock,respectively.
In Fig. 3, φ, Tmax, and Tmin vary smoothly as a functionof δ, and limiting values correspond to those for the
homogeneous case for small layer thicknesses, whilecurves converge to the values corresponding to a homogeneous dust layer for large thicknesses as expected. For
thicknesses greater than about a diurnal skin depth, theupper layer becomes thermally “opaque” and the model
results are indistinguishable from the constant property
case (for a full orbit, δ needs to be greater than aseasonal skin depth, ~ 36 mm, to converge to theconstant property case).
For the phase lag, the strongest change with dust coveroccurs at a layer thickness close to δ = 0.1 mm for dustcovered bedrock and δ = 0.3 mm for dust-covered regolith,
respectively. A peculiar feature is the region between
about 1 mm ≤ δ ≤ 15 mm, where the thermal lag is significantly smaller than the value for homogeneous dust. Thiseffect is discussed below. For Tmax and Tmin, the strongest
influence is seen around δ = 1 mm. Noticeable effects
already occur for extremely small amounts of dust, andthe results converge to pure dust properties for large
thicknesses: using the 10% and 90% amplitudes, thechange of lag angle occurs between layer thicknesses 0.01
and 0.4 mm for dust on bedrock and 0.04 and 0.9 mm fordust on coarse regolith. The change of Tmax occurs
between layer thicknesses 0.07 and 3 mm for dust on
bedrock and 0.2 and 5 mm for dust on coarse regolith andthe change of Tmin occurs between layer thicknesses 0.2

1.0 mm dust on bedrock350

1.0 mm dust on porous rock

350

250

250

T (K)

300

T (K)

300

200

200

150

150
layeredporous rockdust

layeredbedrock
dust

100

100
-150

-100

-50

0

(deg)

50

100

150

-150

-100

-50

0

50

100

150

(deg)

Fig. 2 Left: Example diurnal temperature curve (bold line) for bedrock covered with 1 mm of dust. Right: same as left, but for porous rock/coarse
regolith covered with 1 mm of dust. The surface temperature curves for the homogeneous dust and subsurface material are shownfor comparison

(2019) 6:48

Biele et al. Progress in Earth and Planetary Science

Page 7 of 11

lag (deg)

30
20
10
0
10 -3

10 -2

10 -1

10 0

10 1

10 2

10 -2

10 -1

10 0

10 1

10 2

10 -2

10 -1

10 0

10 1

10 2

Tmax (K)

340
320
300
280
10 -3

Tmin (K)

250

200

150

10 -3

(mm)

Fig. 3 Phase lag φ, maximum diurnal temperature Tmax, and minimum diurnal temperature Tmin as a function of dust thickness δ. The red fit
curve corresponds to the case of dust on coarse regolith, whereas the blue fit curve corresponds to dust on bedrock. Broken horizontal lines
represent φ, Tmax, and Tmin for the homogeneous cases of dust, regolith, and bedrock, respectively

and 9 mm for dust on bedrock and 0.4 and 10 mm fordust on coarse regolith. We see that phase lag is most sensitive to a surface dust layer and minimum temperature isleast sensitive.
Results for apparent thermal inertia are summarized inFig. 4. Already an extremely thin dust layer is found to
significantly alter the retrieved apparent thermal inertia,
which furthermore depends significantly on the applied
inversion method. Out of the tested methods, determination of thermal inertia from lag is particularly sensitive
to dust, whereas the estimation based on minimum
temperature is least affected. However, even in this best
case, apparent thermal inertia starts to significantly deviate (> 7% for the porous rock case, 25% for the bedrockcase) from the background value for dust layer thicknesses
above δ > 0.1 mm, corresponding to only 100th of a diurnal skin depth. Apparent thermal inertia converges to thatof the surface dust once layer thicknesses corresponding
to 1–2 diurnal skin depths (9–18 mm) are reached.

As may be expected, the masking effect is more pronounced for the high thermal inertia bedrock, and a 1 mmdust layer will completely mask the underlying bedrock if
Γeff is based on the phase lag of the maximum diurnal
temperatures. Furthermore, even if the estimate is basedon minimum diurnal temperatures, apparent thermal inertia will still be changed from 1183 to 241 J m−2 K−1 s−1/2.
Similarly, the apparent thermal inertia of porous rock will
be changed from 300 J m−2 K−1 s−1/2 to 41 J m−2 K−1 s−1/2
and 163 J m−2 K−1 s−1/2 for 1 mm dust using thesemethods, respectively. All other methods yield intermediate results with the method based on phase lag
being the most sensitive to dust coverage.
Example diurnal temperature curves and six model
curves for the case δ = 0.4 mm are shown in Fig. 5 for the
bedrock and regolith case, respectively. This corresponds
to the minimum physically realistic layer thickness for the
chosen dust grain size. Temperature curves as calculated
from the two-layer model (true) are shown in blue, andBiele et al. Progress in Earth and Planetary Science(2019) 6:48

Page 8 of 11

Fig. 4 Top: apparent thermal inertia Γeff as a function of dust cover layer thickness δ for bedrock. Γeff has been calculated using various methods(see Methods/Experimental). Top and bottom horizontal dashed lines indicate Γ for the homogeneous rock and dust, respectively. Vertical dottedlines indicate dust layer thickness in units of the diurnal thermal skin depth (9.2 mm) of the dust. Data points are connected by straight lines toguide the eye. Error bars represent the effects of worst-case modeling errors as described in the main text. Bottom: same as top, but for
dust-covered porous rock/coarse regolith

best fitting model curves as derived assuming a homogeneous one-layer model are shown for comparison. As isevident from the figure, all one-layer model curves exhibit
large misfits with respect to the 2-layer model, even if they
are capable of fitting phase lag, maximum, and minimum
temperatures, respectively. Systematic deviations ofseveral tens of K are common, in particular during
daytime and shortly after sunset. The least successfulmodel in reproducing the diurnal temperature curveis the one based on the inversion of the thermal lag, which
is most severely affected by the presence of dust. Also
note that none of the models reproduces the peculiar kinkof the two-layer curve immediately after sunset, which ischaracteristic for our low-over-high thermal inertia
vertical layering.
The result of a phase lag that is even smaller than the
one for pure dust (for dust thicknesses of ~ 0.1 to ~ 1
dust diurnal skin depth) is surprising and it deserves an

explanation. We tried hard to explain the anomaly withnumerical errors: after all, in the relevant δ range, thedifference between noon and maximum temperature ismuch smaller than 0.1 K, the estimated absolute accuracy of our thermal model. The determination of the
maximum, hence the thermal lag (phase of the diurnal
maximum relative to local noon), then solely depends onfitting the model temperature curve with the optimum
degree recursive orthogonal polynomial as describedabove, ± 45° around noon. However, numerical experiments showed that a 0.1 K Gaussian noise on themodel temperatures can only change the retrieved
phase lag by < 0.1 °; the only relevant influence is bya time-varying temperature bias. Analytically, the bias
in phase lag produced by a slope dTbias/dx in themodel temperature is δφ = (dTbias/dx)/d2T/dx2 (x: rotation angle in °, d2T/dx2 curvature of the surface
temperature at the maximum, typically − 0.025 K/°2(2019) 6:48

Biele et al. Progress in Earth and Planetary Science

Page 9 of 11=0.4 mm on regolith
340

320

320

300

300

280

280

260

260

T (K)

T (K)

=0.4 mm on bedrock
340

240

240

220

220

200

200

180

180
160

160
TRUE

from 2 fit to T

from 2 fit to day Tfrom 2 fit to night Tfrom lagfrom Tminfrom Tmax

0

100

140

140
-150

-100

-50

0

50

100

150

rotation angle (deg)

-150

-100

-50

50

150

rotation angle (deg)Fig. 5 Left: temperature as a function of rotation angle for the case of bedrock covered by a 0.4 mm dust layer (true, blue curve) along with
6 best-fitting models calculated under the assumption of a homogeneous one layer model. Right: same as top, but for dust-covered regolith

for the anomalous lag range). To explain the anomaly,e.g., for δ = 3 mm on bedrock, we thus would need adTbias/dx of > 0.05 K/°, that is, 2.2 K over the course of 45°
in rotation angle (1 h in time). This is not consistent with
our analysis of the numerical accuracy of the thermal
model: calculating the case of 3 mm dust on bedrock
both in standard resolution and with much increased
resolution in the time and depth grids, we find abias between − 0.026 and + 0.06 K, and a slope atnoon ± 45° of just 4 × 10−5 K/°. This can bias thephase lag by only 0.002°. Our worst-case modelingtemperature bias curve (see above), which has been used
to calculate the error bars in Fig. 4, has a dTbias/dx
of − 0.0011 K/° and can change the phase lag by 0.05°
at most.
So, we have to conclude that this phase lag anomaly
might be a real, physical effect.
Qualitatively, for δ in the range of 0.1 to 1 diurnal skindepths, the subsurface will store heat at night (compare
the very small cooling rate at night in Fig. 5). After sunrise, the subsurface is still warm and the heat flux fromstored heat adds up to the daytime heat flux from insolation, the combined effect of which is a maximum
temperature significantly earlier than expected for ahomogeneous dust surface with no upwelling heat.
How likely is a dust layer on the surface of a small
body? Due to the low gravity, which is close to only
1.5 × 10−4 m s−2 on Ryugu and corresponds to an escapevelocity of only 0.4 m s−1, small asteroids can hardly keepfines produced during impacts. However, not all fineparticles necessarily reach escape velocity, and lofted
particles may re-accumulate on the surface as so-called
air-fall. On the other hand, seismic accelerations during

impact events may reach 0.1 m s−2 at distances of 200 m
from the impact site (Garcia et al. 2015) and can thus bea factor of one thousand larger than gravity. As the
granular bond number (Scheeres et al. 2010), which isdefined as the ratio between van der Waals forces andthe weight of the particle, is larger than 1 for dust particles smaller than 0.1 m under Ryugu gravity conditions,only dust particles < 0.1 mm are expected to stick during
large impacts. Such particles could for example originate
from micrometeorite impacts, solar wind sputtering, or
thermal fatigue (Delbo et al. 2014; Szalay et al. 2018)
and could cover coarser pebbles or bedrock surfaces. It
therefore seems possible that such fines could be present
on small airless bodies.

Conclusions
We have calculated the influence of dust layers on thesurface temperature on airless bodies by solving the heat
conduction equation for a simple two-layer model.
Results indicate that already a very thin layer of dust on
top of ordinary millimeter- to centimeter-sized regolith,porous rock, or compact bedrock has a significant systematic influence on the observed surface temperatures.
When interpreting these temperatures in terms ofhomogeneous thermophysical properties, the derived
apparent thermal inertia is neither representative for thesurface nor the subsurface layer for a wide range of dust
layer thicknesses. Masking of subsurface thermal properties starts already at dust thicknesses as low as 0.01
diurnal skin depths, while the dust becomes thermallyopaque for layer thicknesses in excess of two skindepths. This is qualitatively a similar result as reported
by Paige et al. 1994 and Putzig and Mellon 2007, butBiele et al. Progress in Earth and Planetary Science(2019) 6:48

our results suggest that significant and nearly complete
masking occurs already at much smaller layer thicknesses for our layer model.
This effect occurs for all the methods applied toestimate thermal inertia using homogeneous models.
While using the thermal phase lag of the maximum
daytime temperature to arrive at apparent Γ results inthe most erroneous thermal inertia estimates, allmethods suffer from the principle problem that evensmall dust layers significantly alter the thermal characteristics of the surface, which cannot be adequately modeled using a single homogeneous layer.
A dust-covered surface should be readily detectablein the diurnal temperature curve, which would show
the characteristic kink of the temperature curve
shortly after sunset for intermediate δ. Furthermore,
the large fitting residuals when trying to invert thedata using a one-layer model should also indicate that
two layer models are required to properly fit the data.
Then, data should be inverted using a thermal modelwith depth-dependent thermophysical properties without relying on thermal inertia if indications for adust-covered surface exist. Such a model will requireat least two layers and 5 free parameters, i.e., kdust,
kreg , ρcp,dust, ρcp,reg , as well as the dust layer thickness
δ. If a priori knowledge is available as, e.g., for theMoon, some of these parameters may be fixed to reduce the complexity of the inversion (Vasavada et al.
1999), but this is generally lacking for asteroids.
Therefore, all five parameters are required since theconcept of thermal inertia breaks down if thermophysical properties are not invariant with depth. However,
the overall accuracy and uniqueness of such an inversion with noise has yet to be determined.
High-resolution images returned from the surface of
Ryugu show no indications for any appreciable dust layeron the surface of the rock observed by MASCOT(Jaumann, R. et al.: In-situ investigation of asteroid(162173) Ryugu by the Mobile Asteroid Surface Scout(MASCOT) Camera (MASCam), submitted to Science,2019). The diurnal temperature curve as observed by the
MASCOT radiometer MARA can be well explained by ahomogeneous Γ ≈ 280 J m−2 K−1 s−1/2. As there is also nokink discernible at sunset (Grott et al. 2019), this indicates that dust plays an insignificant role in the thermal
emission from the asteroid.
Acknowledgements
Organization of stimulating international scientific exchange in the Hayabusa2
IRSG (International Regolith Science Group) by JAXA is gratefully acknowledged.
Authors’ contributions
JB proposed the topic, conceived and designed the study, and analyzed thesimulated data sets. EK carried out the thermal model numerical calculations.
MG and MH added to the scientific discussion of results. HS and TA checked
the calculations with their own codes and provided important comments. NS,

Page 10 of 11

KO, and TO discussed models and provided background on the TIR instrument.
All authors read and approved the final manuscript.
Funding
KO was supported by the JSPS Core-to-Core Program “International Network
of Planetary Sciences”. NS and TO are supported by the JSPS Kakenhi No.
JP17H06459 (Aqua Planetology).
Availability of data and materials
Simulation data is available from the corresponding author upon reasonable
request.
Competing interests
The authors declare that they have no competing interest.
Author details
1
DLR – German Aerospace Center, RB-MUSC, 51147 Cologne, Germany. 2DLR
– German Aerospace Center, Institute for Planetary Research, 12489 Berlin,
Germany. 3Planetary Exploration Research Center, Chiba Institute ofTechnology, Chiba 275-0016, Japan. 4Institute of Space and Astronautical
Science, Japan Aerospace Exploration Agency, 3-1-1 Yoshinodai, Chuo-ku,
Sagamihara, Kanagawa 252-5210, Japan. 5Department of Planetology, Kobe
University, 1-1 Rokkodaicho, Nada, Kobe, Hyogo 657-8501, Japan. 6Division of
System and Information Engineering, Ashikaga University, 286-1 Omae-cho,Ashikaga-shi, Tochigi 326-8558, Japan.
Received: 24 September 2018 Accepted: 27 May 2019

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