1
Auxiliary Material for
2
Vesta surface thermal inertia properties map
3
M.T. Capria,1 F. Tosi,1 M.C. De Sanctis,1 F. Capaccioni,1 E. Ammannito,1 A. Frigeri,1 F.
4
Zambon1, S. Fonte1, E. Palomba1, D. Turrini1, T. N. Titus2, S. E. Schroeder,3 M. Toplis,4
5
J.-Y. Li,5 J.-P. Combe,6 C.A. Raymond,7 C.T. Russell8
6
7
1
Istituto di Astrofisica e Planetologia Spaziali, INAF, Rome, Italy
8
2
Astrogeology Science Center, U.S. Geological Survey, Flagstaff, Arizona, USA
9
3
Deutsches Zentrum für Luft und Raumfahrt, Berlin, Germany
10
4
Institut de Recherche en Astrophysique et Planétologie, Observatoire Midi-Pyrénées, Toulouse,
11
France
12
5
Department of Astronomy, University of Maryland at College Park, Maryland, USA
13
6
Bear Fight Institute, Winthrop, Washington, USA
14
7
Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California, USA
15
8
16
California, USA
Institute of Geophysics and Planetary Physics, University of California, Los Angeles,
17
18
19
20
21
1- VIR data
22
Launched on September 27, 2007, Dawn reached Vesta on July 16, 2011. The VIR spectrometer
23
combines two data channels, Visible and Near Infrared, in one instrument. It operates in the
24
spectral range from the UV to the near IR (0.25-5.1 m) with a moderate-to-high spectral
25
resolution and imaging capabilities. While the primary objective of VIR is to determine the
26
surface composition and mineralogy, the long-wavelength component of the spectra (4.5 – 5.1
27
m) can also be used to derive the surface temperature for warm surfaces (>180K).
28
VIR acquired the first resolved hyperspectral images of Vesta on June 30, 2011. During the
29
following Approach Phase, on July 23-24, 2011, 40 hyperspectral cubes of Vesta were acquired
30
at a heliocentric distance of 2.23 AU, with an average resolution of 1305 m per pixel. At that
31
time, the North Pole was permanently in darkness, while the southern polar region, including the
32
Rheasilvia impact basin, was in permanent light. The sub-solar latitude was -26.7°; the phase
33
angle of the data analyzed in this work spanned the interval from 7.9° to 42.7°, and was, for most
34
of the images, higher than 12° and under 35°.
35
We decided to begin our analysis of the surface of Vesta from the data acquired during the
36
Approach phase because they cover (more than once in many cases) most of the illuminated part
37
of the surface (Figure S1), they are homogeneous (they have been acquired in a very restricted
38
time range, two days), and the pixel resolution, 1305 m/pixel, is low enough to smooth out small
39
topographical differences. The idea was to derive, through this analysis, a framework to be
40
further refined through the analysis, in a near future, of the higher resolution data acquired in the
41
other phases of the Dawn mission.
42
At the spatial scale explored during the Approach phase, the observed surface temperature
43
reached a maximum value, around the sub-solar point, of 270 K at a local solar time of about 13
44
h. Figure S2 shows theoretical temperature curves, drawn for different values of thermal inertia,
45
versus local solar time at the time of Dawn observation. The general behavior of surface
46
temperature on a global scale is mainly determined by latitude, season and local solar time.
47
Superimposed on this general trend, smaller scale variations can be due to local illumination
48
conditions and to variations in the thermophysical properties in the first few centimeters of the
49
Vestan soil. At the scale of Figure S2, drawn for a location below the equator, surface
50
temperatures are dominated by latitude, with the equatorial regions warmer than the southern and
51
northern latitudes. By comparing the measured temperatures, reported in Figure S2 as a red
52
asterisk, with theoretical values, we observe that maximum temperature is fairly high and
53
attained shortly after midday, both clear indications of a low thermal inertia.
54
55
56
2 - The thermophysical model
57
To quantify thermal inertia from measured temperatures, a thermophysical model is required,
58
solving the heat conduction equation and providing the temperature as a function of thermal
59
conductivity, albedo, emissivity, density, and specific heat. A finite-differences scheme is used
60
to solve the 1D heat transport equation, applied to each of the layers into which the depth from
61
the surface to 50 m has been subdived:
62
63
𝜕𝑇
𝜌𝑐 𝜕𝑡 = ∇[𝐾 ∙ ∇𝑇]
(1)
64
65
The thickness of the layers is 1 cm close to the surface and is increasing towards the interior of
66
the body. The surface boundary condition is written as follows:
67
68
𝑆(1−𝐴𝑠 )𝜇
𝑟2
𝜕𝑇
= (1 − 𝜀𝜉)𝜀𝜎𝑇 4 + 𝐾 𝜕𝑥 (2)
69
70
In the above equations, ρ is the density, c is the specific heat, T is the temperature, t is time,  is
71
the infrared emissivity, K the thermal conductivity, S is the solar constant, A is the bolometric
72
Bond albedo, r is the heliocentric distance (in AU), µ is the cosine of the local solar zenith angle
73
and σ is the Stefan-Boltzmann constant.
74
The parameter ξ [Lagerros, 1998; Davidsson et al., 2009] has been introduced to characterize the
75
topography (roughness) on sub-pixel scale. Roughness is thought to be the cause of thermal-
76
infrared beaming, the phenomenon for which a surface emits in a non-lambertian way with a
77
tendency to reradiate the absorbed radiation towards the Sun. The beaming is originated by the
78
multiple scattering between the surface elements, increasing the amount of energy absorbed by
79
the surface, and by the fact that surface elements oriented towards the Sun will become much
80
hotter than a flat surface. Sub-pixel roughness can be considered a measure of the increase in
81
surface area compared to the projected flat footprint of a terrain. High values of ξ will increase, if
82
the phase angle of the observation is reasonably low, the computed surface temperature, while
83
low values, indicating a flatter surface, will have an opposite effect. The roughness
84
interpreted, for example, in terms of the ratio between flat and cratered areas:
can be
85
86
𝐴
𝜉 = 1 − 𝐴 𝑓𝑙𝑎𝑡 (3)
𝑠𝑢𝑟𝑓
87
88
Flat surfaces will have
values close to 0, while very irregular surfaces will have values
89
approaching unity. The range of values used in this analysis is comprised between 0.2 and 0.67,
90
corresponding to extreme values for the parameter X:
91
92
𝑋 = (1 − 𝜀𝜉)𝜀 (4)
93
94
The input parameters mainly determining the temperature computation (and consequently the
95
derivation of thermal inertia) are the thermal conductivity, the sub-pixel roughness and the Bond
96
albedo. In our case, the Bond albedo is assumed to be known [Schröder et al., 2013]; albedo data
97
are available for all the latitudes between 90°S and 30°N.
98
The model is applied to a given location (specified by latitude, longitude and the corresponding
99
illumination condition). The body is followed along its orbit from a starting time up to a given
100
UTC date and time; the time interval on which the code is run is chosen in order to stabilize the
101
results of the computation. The model is applied to a detailed shape model of Vesta,
102
reconstructed by R. Gaskell on the basis of FC images [Raymond et al., 2011]. The average
103
triangular plate has an area of 0.2855 km2. Lateral heat conduction can be neglected because the
104
facets of the shape model on which it is applied are always larger than the thermal skin depth (of
105
the order of centimeters).
106
The illumination condition, for each time step and position, is obtained through SPICE [Acton,
107
1996]. A layered terrain, with regolith on the surface and density increasing towards the interior,
108
is assumed. This is, by the way, in agreement with the results described in Vasavada et al.
109
[2011], who found that a graded regolith model is the most suitable to simulate the properties of
110
the lunar surface. Thermal inertia is then derived from thermal conductivity, density and heat
111
capacity at a given temperature. The output of the model is, for any given location and UTC
112
time, surface and interior temperatures and thermal inertia. An example of the output of the
113
model can be seen in Figures S3 and S4, in which theoretical temperatures with respect to local
114
solar angle are shown for a latitude of 20°S and an UTC of July 23, 2011. In the figures, the
115
curves corresponding to three different values of K (see section 3 for details) are plotted, each
116
one with two different values for the sub-pixel roughness, a minimum (continuous curve,
117
obtained with ξ=0.2) and a maximum (dash-dot curve, obtained with ξ=0.67).
118
119
3 - The thermal conductivity
120
It is assumed that the surface of Vesta is covered by particulate materials with different degrees
121
of compaction (and density), ranging from fine dust to coarse regolith (incoherent rocky debris),
122
all characterized by a fairly low thermal inertia. We consider lunar soils analogous to these
123
materials because the meteorites coming from Vesta, HEDs, belong to the same class
124
(achondritic meteorites) as the meteorites that come from the Moon, for lunar soils in situ
125
measurements (heat flow experiments) that are available [Cremers et al., 1974]. In the model,
126
empirical expressions derived from the results of these experiments are used to express the
127
thermal conductivity. Three classes of material (see Figure S5), called lunar dust, dust and
128
regolith, have been considered in the analysis. The expressions, depending on temperature and
129
density, are of the type:
130
131
𝑘𝑐𝑜𝑛𝑑 + 𝑘𝑟𝑎𝑑 T3 (5)
132
The first term takes into account the conductive transfer across the solid particles, while the
133
second term takes into account the radiative heat transfer, sensible at high temperatures, across
134
the empty spaces between the particles. The density is assumed to increase linearly with depth,
135
starting with a minimum value (surface layer) of 1200 kg/m3 for the lunar dust, 1220 kg/m3 for
136
the dust and 1320 kg/m3 for the regolith. Maximum values are, respectively, 1800 kg/m3 for the
137
lunar dust, 1920 kg/m3 for the dust and 1930 kg/m3 for the regolith.
138
139
3 – Derivation of the regional thermal inertia
140
The procedure followed in order to derive the regional thermal inertia values from the Approach
141
phase data is as follows.
142
143
(a) The surface of Vesta covered during the Approach phase is divided in quadrangles of 5°
144
latitude and 10° longitude. The size of the quadrangles has been chosen so that it is always much
145
larger than the instantaneous field of view (IFOV) of a pixel, and in each one of them, enough
146
VIR pixels are found. The area of a quadrangle is, at the equator, of the order of 103 km2.
147
Only observations up to 30°N were considered, in order to avoid the problem of thermal flux
148
enhancement due to hot spots [Rozitis and Green, 2011; Keihm et al., 2012]. The phenomenon,
149
due to the local topography allowing the Sun to be still above the horizon for points with large
150
incidence angle, is seen as pixels on the nightside showing moderate temperature and relatively
151
low uncertainties and is very sensitive when going close to the northern limb, from 40°N on. A
152
similar, less strong, increase in the thermal flux is seen in the pixels observed at local solar times
153
close to the local early morning and late afternoon. To take into account this effect, an accurate
154
physical modeling of directional-dependent roughness would be needed, but this is beyond the
155
scope of the present analysis. Moreover, this phenomenon affects a limited number of pixels that
156
can be easily discarded.
157
158
(b) A retrieved temperature (Tr) represents the average temperature, at the time and date of the
159
observation, measured on an area defined by the instantaneous field of view (IFOV) of a VIR
160
pixel; to each pixel a planetocentric latitude and longitude are associated, derived from the
161
geometry information associated to VIR data. All the Tr values can be assigned to a given
162
quadrangle, on the basis of their latitude and longitude. Figures 6, 7 and 8 show, for three of
163
these quadrangles, the retrieved temperatures plotted versus the phase angle of the observation.
164
These three plots are good representative of the rest of the quadrangles, and show that the phase
165
angle range spanned by the measured points is quite small.
166
In each quadrangle, Tr values are assigned to different local solar time intervals (see Figure S9);
167
to this end, the 24-hour time range at which the rotation period of 5.342 h has been scaled is
168
divided into intervals of half an hour.
169
170
(c) In the following step, the average value of all the retrieved temperatures assigned to a given
171
time interval is computed (see Figure S10). The pixel resolution is low enough that topographical
172
differences are smoothed, hence an average temperature can be considered as representative of
173
the retrieved temperature at that local solar time in a given quadrangle. The standard deviation of
174
each average retrieved temperature point is also evaluated; we discard the few scattered points
175
that on some quadrangles increase the standard deviation to values higher than 5 K.
176
Only for 522 quadrangles there are sufficient observations to provide at least two average points;
177
the quadrangles in which there is only one average point have not been considered in the
178
analysis. Most of the quadrangles contain 2 to 4 average points.
179
180
(d) From the thermal model, temperature curves are computed representing the average
181
temperature, at the observation date, of each quadrangle; to this end, theoretical temperatures are
182
computed and averaged, on the shape model, on an adequate number of locations (3 or 9,
183
depending on the area of the quadrangle) falling in that quadrangle. Only the three classes of
184
materials described above have been considered in this procedure. High thermal inertia values
185
typical of compact material and exposed bedrock have not been considered in this analysis,
186
because these kinds of material are not expected to show up at regional scale. Three families of
187
curves are obtained in this way, each K value corresponding to a set of sub-pixel roughness
188
values, from a minimum value of 0.2 to a maximum value of 0.67. In Figures S3 and S4, two
189
curves per family are plotted: for each of the 3 type of materials taken into account, the curves
190
with the minimum and maximum values of the parameter expressing the sub-pixel roughness are
191
shown, marked by different line types. All the intermediate values of this parameter (in step of
192
0.01) give rise to a family of intermediate curves, not shown in the plots.
193
194
(e) The curves obtained with the average retrieved temperatures are then compared with the
195
theoretical curves, in order to find the best-matching couple of (K, ξ) values (see Figure S10). To
196
this end, a chi-square test is applied, and the pair of K, ξ values, for which this difference is
197
minimal, is considered to be representative of the quadrangle.
198
199
3 – Derivation of the thermal properties map
200
201
Figure S11 shows how a thermal inertia class has been attributed to each quadrangle.
202
In this plot, the theoretical temperature family of curves for the 3 classes of material considered
203
in the analysis are shown. Also shown are 5 different examples of daily curves built with points
204
(diamonds in the plot) representing averaged observed temperatures. The 5 classes that can be
205
distinguished in the plot are the following:
206
207
1- VLH (TI = 10± 5 Jm−2 s−0.5 K−1, ξ=0.67± 0.02) – If the retrieved temperatures fall
208
between the blue dash-dot curve (dust, maximum roughness) and the black dash-dot
209
curve (lunar dust, maximum roughness), the thermal conductivity corresponds to lunar
210
dust coupled with a high subpixel roughness.
211
2- LH (TI = 20 ± 10 Jm−2 s−0.5 K−1, ξ=0.67± 0.02) - If the retrieved temperatures fall
212
between the red dash-dot curve (regolith, maximum roughness) and the blue dash-dot
213
curve (dust, maximum roughness), two classes of material, lunar dust and dust, coupled
214
with a very high subpixel roughness, could be assigned to this area.
215
3- AA (TI = 30± 10 Jm−2 s−0.5 K−1, ξ=0.44± 0.02) – If the retrieved temperatures fall
216
between the black continuous curve (lunar dust, minimum roughness) and the red dash-
217
dot curve (dust, maximum roughness), it is impossible to attribute the area to only one of
218
the 3 classes of material. Subpixel roughness has intermediate values, far from the
219
extreme values.
220
4- HL (TI = 40 ± 10 Jm−2 s−0.5 K−1, ξ=0.2± 0.02) - If the retrieved temperatures fall under
221
the black continuous curve (lunar dust, minimum roughness) and the blue continuous
222
curve (dust, minimum roughness), two classes of material, dust and regolith, coupled
223
with a very low subpixel roughness, could be considered as representative of this area.
224
5- VHL (TI = 50 ± 5 Jm−2 s−0.5 K−1, ξ=0.2± 0.02) - If the retrieved temperatures fall between
225
the blue continuous curve (dust, minimum roughness) and the red continuous curve
226
(regolith, minimum roughness), the thermal conductivity is characteristic of regolith, and
227
is coupled with a very low sub-pixel roughness.
228
229
230
In regards to the attribution of values to the thermal inertia in the different classes, it must be
231
noted that we deal with ranges of values rather than with a single value. This is due to the fact
232
that thermal inertia, depending on temperature, is not a constant for a given location. The value
233
assigned to each class is an average value that takes into account the diurnal and seasonal
234
variability. The uncertainty is lower for the two classes pointing to only one type of material.
235
236
Supplemental references
237
 Acton, C.H. (1996), Ancillary Data Services of NASA's Navigation and Ancillary
238
Information Facility, Planet. Space Sci.,44, 65-70, doi:10.1016/0032-0633(95)00107-7.
239
 Cremers, C. J., and H. S. Hsia (1973), Thermal conductivity and diffusivity of Apollo 15
240
fines at low density, Proc. of the 4th Lunar Conf, 2459-2464.
241
 Davidsson, B. J. R., P. J. Gutierrez, and H. Rickman (2009), Physical properties of
242
morphological units on comet 9P/Tempel 1 derived from near_IR Deep Impact spectra,
243
Icarus, 201, 335-357, doi:10.1016/j.icarus.2008.12.039.Lagerros, J. S. V. (1998),
244
Thermal physics of asteroids. IV Thermal infrared beaming, Astron. Astrph., 332, 1123-
245
1132.
246
 Keihm, S. J. et al. (2012), Interpretation of combined infrared, submillimeter, and
247
millimeter thermal flux data obtained during the Rosetta fly-by of Asteroid (21) Lutetia,
248
Icarus, 221, 395-404, doi:10.1016/j.icarus.2012.08.002.
249
 Raymond, C. A., et al. (2011), The Dawn Topography Investigation, Space Science
250
Reviews, 163, 487-510,doi:10.1007/s11214-011-9863-z.
251
 Rozitis, B., and S. F. Green (2011), Directional characteristics of thermal-infrared beaming
252
from atmosphereless planetary surfaces – a new thermophysical model, MNRAS, 415,
253
2042-2062, doi:10.1111/j.1365-2966.2011.18718.x.
254
255
256
 Schröder, S.E., et al. (2013), Resolved photometry of Vesta reveals physical properties of
crater regolith, Planet. Space Sci., 85, 198-213, doi:10.1016/j.pss.2013.06.009.
Captions
257
Figure S1. The figure shows the coverage of the VIR data belonging to the phase of the mission
258
analyzed in the paper, the Approach phase. The observations were performed on July 23-24,
259
2011. Most of the areas have been covered more than once, at different local solar times.
260
Figure S2. Theoretical temperature curves versus local solar time are drawn at the time of Dawn
261
observations for a location below the equator. Each temperature curve has been drawn for a
262
different value (increasing from top to bottom) of thermal inertia. The vertical dash-dot line
263
marks the position of the local midday. The shift towards the right (afternoon) of maximum daily
264
temperatures with increasing thermal inertia is clearly visible. The red asterisk marks the position
265
(local solar time = 12.7 h) of an average of measured maximum temperatures at the same
266
location as the theoretical curves.
267
268
Figure S3. Theoretical temperature curves for the latitude 20°S at the time of the Approach
269
mission phase are shown for the whole rotation period of Vesta. The color of the curves
270
corresponds to different kind of materials (different thermal conductivity expressions), while the
271
line type corresponds to different values of the parameter ξ, the sub-pixel roughness. Continuous
272
line: ξ=0.2, corresponding to minimum roughness; Dash-dot line: ξ=0.67, corresponding to
273
maximum roughness.
274
Figure S4. Theoretical temperature curves for the latitude 20°S at the time of the Approach
275
mission phase are shown only in the time interval for which retrieved temperatures are available.
276
The color of the curves corresponds to different kind of materials (different thermal conductivity
277
expressions), while the line type corresponds to different values of the parameter ξ, the sub-pixel
278
roughness. Continuous line: ξ=0.2, corresponding to minimum roughness; Dash-dot line: ξ=0.67,
279
corresponding to maximum roughness. It should be noted that in this quadrangle the time
280
interval in which observed temperatures are available is very large; in most of the quadrangles,
281
the time interval ranges between 10 and 14 (local solar time).
282
Figure S5. Thermal conductivity values are plotted versus temperature for each class of material
283
considered in this work. The density is 1200 kg/m3 for the lunar dust, 1220 kg/m3 for the dust
284
and 1320 kg/m3 for the regolith.
285
Figure S6. Observed temperature points versus phase angles for the quadrangle [5°S, 240°E].
286
Figure S7. Observed temperature points versus phase angles for the quadrangle [5°N, 180°E].
287
Figure S8. Observed temperature points versus phase angles for the quadrangle [25°N, 130°E].
288
Figure S9. The plot shows all the observed temperature points (Tr), marked by crosses, falling in
289
a particular quadrangle, at latitude 5°S, 240°E.
290
Figure S10. The plot shows the average of the retrieved temperatures, shown as red asterisks,
291
computed for each time interval of 0.5 h. The time assigned to each asterisk is the average of the
292
local solar times of the retrieved temperature falling in that time interval. The standard deviation
293
is also shown. The retrieved temperatures are compared with the theoretical curves. The color of
294
the curves corresponds to the 3 different types of materials, while the line type corresponds to
295
different values of ξ, the sub-pixel roughness. Continuous line: ξ=0.2, corresponding to
296
minimum roughness; dashed line: ξ=0.44.
297
Figure S11. The plot shows how the thermal inertia can be assigned to each quadrangle, on the
298
basis of the position of the curve obtained from the retrieved temperatures. The curves represent
299
theoretical temperatures, obtained for different values of K and ξ. The different colors identify
300
the class of material (black for lunar dust, blue for dust, red for regolith), while the line type
301
identifies the roughness level. Only the curves with the minimum (continuous line) and
302
maximum (dash-dot line) sub-pixel roughness, corresponding respectively to ξ=0.2 and ξ=0.67,
303
are shown. The points represent instead different examples of curves obtained from averaged
304
retrieved temperatures. In this case, colors represent the 5 classes of thermal properties (VLH,
305
LH, AA, HL, VHL) derived in the analysis.
306
307
308