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Microwave imaging of Mercury's thermal emission: Observations and models

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M icro w a v e im a g in g o f M e r c u r y ’s th e r m a l em issio n : O b serv a tio n s
a n d m o d els
M itchell, David LeRoy, Ph.D .
U niversity of California, Berkeley, 1993
C op yrigh t © 1 9 9 3 b y M itch ell, D a v id LeRoy. A ll righ ts reserved.
UMI
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Microwave Imaging of M ercury’s Therm al Emission:
Observations and Models
by
David LeRoy Mitchell
B.S. (University of Virginia) 1985
M.A. (University of California at Berkeley) 1987
A dissertation subm itted in partial satisfaction of the
requirem ents for the degree of
Doctor of Philosophy
in
Astronomy
in the
GRADUATE DIVISION
of the
U NIVERSITY of CALIFORNIA at BERKELEY
C om m ittee in charge:
Professor Imke de P ater, Chair
Professor W illiam J. Welch
Professor Kinsey A. Anderson
1993
T he dissertation of David LeRoy Mitchell is approved:
__________ y / 2 f l f l
D ate
U
I
pail
i<& 3
D ate
skki
I/\A-
/D a ^
University of California at Berkeley
1993
Microwave Im aging of M ercury’s T herm al Emission:
Observations and Models
Copyright © 1993
by
David LeRoy Mitchell
A b stract
Microwave Imaging of M ercury’s T herm al Emission:
Observations and Models
by
David LeRoy Mitchell
Doctor of Philosophy in A stronomy
University of California at Berkeley
Professor Imke de Pater, Chair
We present images of M ercury’s therm al emission at wavelengths of 0.3,
1.3, 2.0, 3.6, 6.2, 18.0, and 20.5 cm. In addition, images of the linearly polarized
com ponent of this emission were obtained from 2.0 to 20.5 cm. Observations were
perform ed w ith the BIMA millimeter interferom eter at A0.3 cm and w ith the Very
Large A rray at centim eter wavelengths. The therm al images are com pared with a
m odel th a t is based on M ariner 10 observations and lunar analogy. A re-analysis
of the M ariner 10 Infrared R adiom eter d ata shows th a t M ercury’s regolith, like
th a t of the M oon, consists of a therm ally insulating surface layer, w ith a thickness
of a few centim eters, atop a highly compacted region th a t extends to a depth
of a t least 4 m eters. The polarization images reveal an increase in the effective
dielectric constant with wavelength and rm s surface slopes th a t range from 15°
at A2.0 cm to 10° at A6.2 cm.
These trends are caused at least in p art by
w avelength-dependent scattering at the surface boundary, although the density
1
gradient near the surface m ay also be a contributing factor. The observed daynight brightness contrast at each wavelength requires a microwave opacity th a t
is at least a factor of 2-3 lower than the opacity of the lunar regolith.
This
difference is likely due to lower Fe and Ti abundances in M ercury’s regolith.
Residual images obtained by subtracting the best-fit level-surface models from
the d a ta reveal therm al depressions at the poles and along the sunlit side of the
m orning term inator. We show th a t this p attern is due to shadowing by surface
topography. M ercury’s equatorial nighttim e brightness tem perature spectrum
from A3.6 cm to A20.5 cm shows th a t radiative heat transport is im portant within
M ercury’s regolith. Evidence for a turndow n in the spectrum from A6.2 cm to
A20.5 cm suggests th a t the latter wavelength begins to probe completely through
the regolith and into the underlying megaregolith. This possibility does not allow
us to place constraints on M ercury’s lithospheric heat flow.
for m y p aren ts
J o y ce an d D a v id M itch ell, Sr.
whose encouragem ent and support m ade this work possible
T able o f C on ten ts
List of F i g u r e s ................................................................................................................vi
List of T a b le s ..................................................................................................................viii
A c k n o w le d g e m e n ts ....................................................................................................... ix
I. I n t r o d u c t i o n ............................................................................................................. 1
II. O b servation s and R e d u c tio n T e c h n iq u e s .................................................. 6
millimeter wavelength o b s e r v a t i o n s ................................................................... 6
centimeter wavelength o b s e r v a ti o n s ...........................................................
11
p o l a r i z a t i o n .....................................................................................................
14
I II . R e s u l t s ..............................................................................................................
20
m illim eter wavelength i m a g e ........................................................................
20
centimeter wavelength intensity i m a g e s ...................................................
21
centimeter wavelength polarization i m a g e s ...............................................
22
IV . T h erm o p h y sica l and R a d ia tiv e T ransfer M o d e ls ..........................
46
temperature variations in the r e g o l i t h .......................................................
47
the lunar r e g o lith .............................................................................................
48
a model for M ercury’s r e g o l i t h ....................................................................
51
....................................................................................
51
numerical s o l u t i o n ........................................................................................
53
normalization of K 3h0 ....................................................................................
54
microwave p r o p a g a t i o n ................................................................................
56
microwave emissivity and p o la riza tio n .......................................................
61
V . A n a l y s i s .............................................................................................................
72
dielectric constant and surface s l o p e s .......................................................
72
microwave o p a c i t y ........................................................................................
75
boundary conditions
topography and s h a d o w in g ...........................................................................
82
V I. S u m m a r y and F u tu re D i r e c t i o n s .......................................................... 124
R e f e r e n c e s .................................................................................................................. 129
A p p e n d ix 1: H e a t F low fro m In te r n a l S o u r c e s ..................................... 138
lunar heat flow m e a s u r e m e n t s ...................................................................... 139
M ercury’s night-side microwave s p e c t r u m ...................................................141
modeling M ercury’s microwave spectrum
v
...................................................142
L ist o f F ig u res
F ig u r e 1: Observing g e o m e tr y ............................................................................
19
F ig u r e 2: BIMA AO.3 cm intensity i m a g e .......................................................
29
F ig u r e 3: BIMA visibility am plitude of 26 Jan u ary 1988
30
F ig u r e 4: BIMA visibility am plitude of 28 M arch 1 9 9 1 ..............................
31
F ig u r e 5
(a ): VLA Al.3 cm intensity i m a g e .................................................
32
F ig u r e 5
(b ): VLA A2.0 cm intensity i m a g e .................................................
33
F ig u r e 5
(c): VLA A3.6 cm intensity i m a g e .................................................
34
F ig u r e 5
(d ): VLA A6.2 cm intensity i m a g e .................................................
35
F ig u r e 5 (e ): VLA A18.0 cm intensity i m a g e ................................................
36
F ig u r e 5 (f): VLA A20.5 cm intensity i m a g e ...............................................
37
F ig u r e 6
(a ): Blank-sky noise distribution at Al.3 c m .............................
38
F ig u r e 6
(b ): Blank-sky noise distribution at A6.2c m ..............................
39
F ig u r e 7
(a ): VLA A2.0 cm polarization i m a g e ........................................
40
F ig u r e 7
(b ): VLA A3.6 cm polarization i m a g e .........................................
41
F ig u r e 7 (c ): VLA A6.2 cm polarization i m a g e ............................................
42
F ig u r e 7 (d ):
A18.0 cm polarization i m a g e ..............................
43
F ig u r e 7 (e ): VLA A20.5 cm polarization i m a g e ...........................................
44
F ig u r e 8: Smooth-surface p o l a r i z a t i o n ...........................................................
45
F ig u r e 9: Density vs. d e p t h .................................................................................
69
F ig u r e 10: T herm al conductivity vs. d e p t h ...................................................
70
F ig u r e 11: Re-calibration of M ariner 10 therm al i n e r t i a ..............................
71
F ig u r e 12 (a ): Rougli-surface polarization
...............................................
97
F ig u r e 12 (b ): Rough-surface polarization ( c o n t . ) .......................................
98
F ig u r e 13: Effective dielectric constant vs.w a v e le n g th ................................
99
VLA
F ig u r e 14: RMS slopes vs. w a v e le n g th ............................................................... 100
vi
F ig u r e 15: Diffusing rough-surface p o l a r i z a t i o n ............................................. 101
F ig u r e 16: Diffuse fraction vs. wavelength
......................................................102
F ig u r e 17: Polarization and emissivity vs. fractional r a d i u s ........................ 103
F ig u r e 18: Model fits to BIMA visibility of 26 January 1988 .....................
104
F ig u r e 19: Model fits to BIMA visibility of 28 M arch 1 9 9 1 ......................... 105
F ig u r e 20 (a ): Model fits to VLA Al.3 cm i m a g e .........................................106
F ig u r e 20 (b ): Model fits to VLA A2.0 cm i m a g e .........................................107
F ig u r e 20 (c ): Model fits to VLA A3.6 cm i m a g e .........................................108
F ig u r e 20 (d ): Model fits to VLA A6.2 cm i m a g e .........................................109
F ig u r e 20 (e ): Model fits to VLA A18.0 cm i m a g e .........................................110
F ig u r e 20 (f): Model fits to VLA A20.5 cm i m a g e .........................................I l l
F ig u r e 21: Effective specific loss tangent s p e c t r u m ......................................... 112
F ig u r e 22: Specific loss tangents of lunar s a m p l e s ......................................... 113
F ig u r e 23 (a ): Residual VLA Al.3 cm i m a g e .................................................. 114
F ig u r e 23 (b ): Residual VLA A2.0 cm i m a g e .................................................. 115
F ig u r e 23 (c): Residual VLA A3.6 cm i m a g e .................................................. 116
F ig u r e 23 (d ): Residual VLA A6.2 cm i m a g e .................................................. 117
F ig u r e 23 (e ): Residual VLA A18.0 cm i m a g e .................................................. 118
F ig u r e 23 ( f ) : Residual VLA A20.5 cm i m a g e .................................................. 119
F ig u r e 24: Insolation vs. t i m e ...............................................................................120
F ig u r e 25: Elevation of Sun vs. t i m e .................................................................. 121
F ig u r e 26: Shadowing effects at Al.3 cm and A2.0 cm
................................ 122
F ig u r e 27: Shadowing effects at A3.6 cm and A6.2 cm
................................ 123
/J
•
e
«
c //
F ig u r e 28: M ercury’s night-side spectrum ( eR = 1 . 9 ) ......................................147
F ig u r e 29: M ercury’s night-side spectrum ( eR = 2 .7 ) .....................................148
F ig u r e 30: M ercury’s internal heat f lo w ..............................................................149
vii
L ist o f T ables
T able I: Observing G e o m e t r y ............................................................................
16
T able II: C a l ib r a to r s .............................................................................................
17
T able III: Image Statistics
................................................................................
25
T able IV : Specific Heat Polynomial C o e f fic ie n ts ..........................................
66
T able V : Night Side Microwave S p e c tr u m ..........................................................145
viii
A ck n ow led gem en ts
M any people have provided assistance, advice, and support during my years
as a graduate student. This dissertation would not have been possible w ithout
the scientific advice, encouragem ent, and financial support of my advisor, Imke de
P a te r, who has been an inspiration. I am also grateful for stim ulating discussions
w ith Harold Weaver and Jack Welch, whose enthusiasm for this project was
contagious. I wish to thank Dick Plam beck for his incredible analogies (the “bugon-the-windshield” analogy stands out as one of his finest) and John C arlstrom ,
who, w ith chips and salsa in hand, rescued me one cold winter evening when
the alarm was blaring and the interferom eter and my spirits were on the rocks.
Mel W right deserves special thanks for providing technical assistance during the
observations, d a ta reduction, and cross-country skiing.
Bob Lin has been a valuable advisor during my years at the Space Sciences
Laboratory, and beyond. His encouragement and faith in me early in my graduate
career m ade all the difference. Although I didn’t appreciate it at the time, I
m ust thank him for m arking my rough drafts w ithout mercy — it has m ade me a
b e tte r w riter. My comet research at SSL also benefited from the help of Kinsey
Anderson, Asoka Mendis, Chuck Carlson, and Dave Curtis.
I wish to thank my BAD grad, cohorts for helping me to survive the m any
perils of graduate school. Special m ention goes to my long-time officemate and
carillonist-incognito, Joe Shields, and to M ark Dickinson, Bill Reach, A rjun Dey,
Joan N ajita, M ark Hurwitz, Tim Sasseen, and Jim Clemmons. Davin Larson has
been provided m any enlightening discussions (scientific and otherwise), and has
been a housem ate and friend for m any years.
Finally, I wish to thank my fiancee, B renda Hatfield, for her unwavering
support, and for putting up w ith m y long hours at th e office while preparing this
dissertation. I couldn’t have done it w ithout her.
x
I. In tr o d u c tio n
At m illim eter and centim eter wavelengths, radiation from M ercury is dom i­
n a ted by therm al emission from th e crust. This emission is strongly m odulated
by diurnal tem perature variations, which diffuse tens of centim eters into the re­
golith. W ithin this “diurnal layer,” th e diurnally-averaged tem p eratu re increases
w ith depth because of significant radiative heat transfer in the vacuum between
grains.
High daytim e surface tem peratures result in efficient downward heat
tra n sp o rt, while cold nighttim e surface tem peratures inhibit upw ard heat tra n s­
p o rt. Below the diurnal layer, the tem p eratu re is not constant as a function
of longitude because of M ercury’s unique non-uniform insolation p a tte rn , which
results from a large orbital eccentricity and a 3:2 resonance between its orbital
an d rotational periods. At the equator (and below th e diurnal layer), two “hot
longitudes” at 0 and 180 degrees are approxim ately 100 K w arm er th a n “cold
longitudes” 90 degrees away. In this region, the tem perature likely continues to
increase w ith depth as a result of heat generated in the interior by radioactive de­
cay and possibly also by core-m antle differentiation (Siegfied and Solomon 1974,
Schubert 1988).
At microwave frequencies, the electrical skin depth of the regolith is roughly
proportional to wavelength (L£ ~ 10-20 A). Thus, centim eter wavelength obser­
vations probe throughout and b en eath M ercury’s diurnal layer. At wavelengths
near 1 cm, the observed brightness is composed of large am plitude diurnal vari­
ations th a t are superim posed on the hot-cold longitude p attern . At longer wave­
lengths ( £
10 cm) diurnal variations are small, and th e observed brightness
depends only on the orientation of the hot-cold longitude p attern . Observations
of therm al emission variations at these wavelengths m ay be used to constrain the
therm al and electrical properties in th e upper few m eters of M ercury’s crust.
1
A lthough the subsurface therm al radiation is unpolarized, it becomes p ar­
tially linearly polarized as it emerges from the crust because of the discontinuity
in the dielectric constant at the crust-vacuum interface. As viewed from E arth ,
th e polarization is zero at the center of the projected disk, and increases tow ard
the edge to a m axim um value, which depends on the refractive index of the sur­
face m aterial and on the surface roughness. The electric vectors of the polarized
com ponent are oriented radially on th e disk. From a modeling perspective, m ea­
surem ents of th e fractional polarization (the ratio of the polarized intensity to
th e to ta l intensity) are valuable because they do not depend on the therm ophys­
ical properties of the regolith, and thus provide an independent constraint on the
dielectric properties and roughness of the surface.
T he first radio observations of M ercury’s therm al emission were single dish
m easurem ents of the disk-integrated brightness as a function of phase angle.
Single dish observations at wavelengths ranging from 0.3 to 11 cm have been
reviewed by M orrison (1970). From these observations, it was determ ined th a t
M ercury’s surface consists of a loosely packed rock powder, similar to th a t of the
Moon. At only two wavelengths, 0.33 and 3.75 cm, was there sufficient phase
angle coverage to determ ine periodic brightness variations and thus constrain the
therm al and electrical properties of the regolith w ithout relying upon absolute
flux calibrations. Such d a ta are difficult to obtain, since frequent, well-calibrated
observations over a period of several m onths are required. M orrison (1970) points
out th a t the am plitude of brightness variations at A0.33 cm relative to those at
A3.75 cm is inconsistent w ith a homogeneous regolith, which suggests a variation
of density with depth. Subsequently, Cuzzi (1974) combined interferom etric ob­
servations at A3.71 cm with disk averages at wavelengths of 0.33, 6, and 18 cm.
These observations suggest an increase in density with depth from ~ 1 g cm -3
2
in th e top few centim eters to ~ 2 g cm 3 at 2.5 m eters. This is consistent with
density variations in the lunar regolith (C arrier et al. 1973, Keihm and Langseth
1973).
Since these observations were perform ed, the ability to image M ercury at
radio wavelengths has improved dram atically with the advent of millim eter in­
terferom eter arrays, the expansion of centim eter arrays, and im provem ents in
detector efficiency.
Imaging is a significant improvement over disk averages,
since brightness variations across the disk (especially from day to night) m ay be
used to constrain the therm al and electrical properties of the regolith, thus elim­
inating the need for extensive observations over m onths and greatly reducing the
dependence on absolute flux calibrations. In addition, it is more straightforw ard
to interp ret relative brightness variations since they are largely independent of
the microwave emissivity, which is expected to vary with wavelength because of
scattering effects. Imaging also provides inform ation th a t cannot be obtained
through single dish observations. W ith sufficient resolution, it is possible to in­
vestigate variations in the physical properties over the surface and the effects of
shadowing caused by surface topography. T he ability to image polarized emission
m akes it possible to constrain the regolith dielectric constant and surface slopes,
and to determ ine the im portance of diffuse scattering at the surface boundary.
Finally, high quality surface tem perature m easurem ents obtained by the M ariner
10 Infrared Radiom eter (Chase et al. 1976) m ay now be combined with microwave
images at several wavelengths to constrain a tem perature- and depth- dependent
m odel of the regolith, allowing a much m ore detailed comparison to the lunar
regolith th a n has previously been possible.
M ercury was first imaged at radio wavelengths by C hapm an (1986), who ob­
tained A2 cm and A6 cm intensity and polarization images with the VLA (Very
3
Large Array) while M ercury was near inferior conjunction. The hot-cold longi­
tude p a tte rn was clearly observed, providing a direct confirmation of M ercury’s
3:2 spin-orbit resonance. However, diurnal brightness variations could not be
fully exploited because only a small fraction of the sunlit hemisphere was visible.
From polarization d a ta at A6 cm, C hapm an derived a dielectric constant of 2.1
± 0.2 (uncorrected for surface roughness).
Burns et al. (1987) and Ledlow et al. (1992) have also imaged M ercury
w ith the VLA at wavelengths of 2 and 6 cm. Using a therm ophysical model
similar to th a t developed by Golden (1977), they found th a t lunar-like values
for the therm al and electrical properties of M ercury’s regolith produced model
images th a t were consistent w ith their observations. In addition, they found
subsurface radiative heat conduction to be im portant and internal heat sources
to be negligible.
Their best-fit value for x (the ratio of radiative to contact
conductivity at 350 K) of 1.3 is slightly outside the range of 0.4-1.0 derived by
Cuzzi (1974). However, Ledlow et al. note th a t param eter ambiguities call into
question the uniqueness of their model. One such ambiguity involves radiative
heat conduction and internal heat sources, both of which cause an increase in the
average tem perature with depth. This ambiguity is compounded by uncertainties
in the regolith opacity and microwave emissivity, which are required to relate the
observed radio intensity to physical tem perature gradients in the regolith. These
param eters are particularly difficult to separate when d a ta at only two closelyspaced wavelengths are available (see Appendix 1).
More recently, we have undertaken a comprehensive program to image M er­
cury at several wavelengths spanning nearly two orders of m agnitude. We ob­
4
served M ercury at AO.3 cm with th e BIMA millimeter interferom eter array 1, and
at wavelengths of 1.3, 2.0, 3.6, 6.2, 18.0, and 20.5 cm with the VLA2. M ercury
was near m axim um elongation on all observing dates, so th a t both the day and
night hemispheres were visible. This allows us to take full advantage of M ercury’s
large diurnal tem perature variations.
In this paper, we present our observations at each wavelength and discuss the
reduction techniques th a t are required to produce images from the interferom eter
d ata. We pay particular attention to sources of error and to the reliability of the
images. We have independently developed a detailed therm ophysical model th a t
includes tem perature- and depth- dependent therm al and electrical properties,
and also allows for departures from a sm ooth surface. We utilize the M ariner 10
surface tem perature m easurem ents, together with lunar and terrestrial analogies,
to constrain the therm ophysical properties of M ercury’s regolith. From the th er­
m al polarization images we constrain the dielectric constant and the rm s surface
slope at each wavelength. By comparing the results with rad ar studies, we are
able to consider the effects of diffuse scattering at the surface boundary. Next,
we model the day-night brightness contrasts observed in the intensity images
to derive the microwave opacity at each wavelength. The results are com pared
w ith laboratory m easurem ents of lunar and terrestrial m aterials and with lunar
regolith opacities th a t are derived by similar microwave remote-sensing tech­
niques. Finally, we discuss the effects of shadowing by surface topography, which
are revealed by subtracting the best-fit model images from the data.
1 T h e BIM A array of th e H at Creek R adio O bservatory is operated by th e U niversity of
C alifornia a t Berkeley, th e U niversity of Illinois, and th e U niversity of M aryland, w ith
su p p o rt from th e N ational Science Foundation.
2 T h e VLA is a facility of th e N ational R adio A stronom y Observatory, which is o p erated
by A ssociated U niversities, Inc., u n d e r co n tract w ith th e N ational Science F oundation.
5
II. O b servatio n s an d R e d u c tio n T echniques
m illim eter wavelength observations
We observed M ercury from 26 to 29 January 1988, and on 28 M arch 1991,
using the BIMA millim eter interferom eter at the Hat Creek Radio Observatory.
At the tim e of these observations, the interferom eter consisted of three antennas
th a t could be moved along a T -shaped track with dimensions of 300 m eters E-W
and 180 m eters N-S. The observing dates were chosen so th a t M ercury was near
m axim um elongation and thus displayed both the day and night hemispheres.
T he viewing geometry is given in Table I and shown in Figure 1.
T herm al continuum emission from M ercury was observed at a frequency of
86 GHz in the lower sideband and 89 GHz in the upper sideband. In the digital
correlator m ode used, each sideband consisted of four 80 MHz sections th a t were
placed end on end for a to tal bandw idth of 320 MHz. Since the antennas are
sensitive to only one linear polarization, the observed signal would norm ally
be distorted as a result of the changing orientation of the linearly polarized
com ponent across M ercury’s disk. This situation is further complicated because
the receivers ro tate with respect to the sky coordinates during the course of
the observations, since the BIMA antennas employ altitude-azim uth m ounts.
Therefore, we placed a quarter-w ave plate in front of each receiver to convert
the incident linear polarization to circular polarization, thus providing uniform
sensitivity across the disk at all hour angles of observation.
During a 50-second integration, each pair of antennas measures a point in
th e spatial Fourier transform of th e source, which is known as the visibility
function.
(The visibility is a complex quantity, with both an am plitude and
a phase.) During the course of an ~ 8-hour observation, different p arts of the
6
visibility function are m easured as the antennas track th e source, and th e antenna
separations projected onto the plane of the sky vary. Additional sampling of the
visibility function is obtained by moving one or m ore of the antennas to different
positions along the T -shaped track and observing for another day.
Im aging M ercury w ith a 3-element interferom eter is difficult because the
aspect of the planet changes rapidly. At elongation, M ercury presents a ~ 7 ” disk.
Since there is no emission on larger spatial scales, the Fourier sampling theorem
insures th a t th e continuous visibility function can be completely specified by
discrete m easurem ents th a t are separated by an interval no larger than:
Abmax — —
(1)
where b is the projected antenna separation in wavelengths and 20 is the diam ­
eter of the disk in radians. For a 7” disk, this interval is approxim ately 30,000
wavelengths (100 m eters). We found th a t an image w ith three E-W and two N-S
resolution elem ents could be obtained with only two configurations of three an­
tennas, b u t since one day is required to move the antennas to a new configuration
and then to recalibrate the antenna separations and pointing, these observations
m ust be carried out over a least a three day period.
Observations began on Jan u ary 26 with the antennas in a predom inantly
north -south configuration. T he antennas were moved to an east-west configura­
tion and recalibrated on the 27th, bu t poor observing conditions the next day
forced us to complete the observations on the 29th, resulting in a three day in­
terval between configurations. D uring this period, the illum inated fraction of
the projected disk decreased from 54% to 40% as the aspect of M ercury ro tated
through 16 degrees as viewed from E arth . (The disk also increased in size from
7
7.0” to 7.7” , b u t the visibility coordinates can be scaled to com pensate for this
effect.) Combining these d a ta results in a “sm eared” image because different
p a rts of the m easured visibility function correspond to different aspects of th e
planet. Nevertheless, the sm earing occurs on a scale (~ 1 ” at the center of the
disk) th a t is sm aller th a n the resolution of 3.5” N-S and 2” E-W (FW H M ), so
we combined and inverse transform ed the visibility d a ta to produce an image.
T he resulting image is distorted not only by sm earing effects b u t also be­
cause th e visibility d a ta represent only a discrete sample of the continuous Fourier
transform of the source. This “dirty” image is a convolution of the tru e image
w ith th e synthesized beam , which is the Fourier transform of the sampling func­
tion (the locus of projected antenna separations). W ith sufficient sampling, the
synthesized beam is dom inated by an approxim ately G aussian central core w ith
a w idth (corresponding to the resolution of the image) th a t is inversely pro p o r­
tional to the m axim um projected antenna separation. Inevitably, th e synthesized
beam also contains positive and negative “sidelobes” away from the central core.
Since the sampling function is known, it is possible to remove (at least
partially) the sidelobe response from the dirty image to produce a “clean” im age,
which ideally is a convolution of th e tru e image w ith only the G aussian central
core of the synthesized beam . Normally, the technique we use for this purpose is
th e planetary CLEAN m ethod (de P a te r and Dickel 1986, de P a te r 1990), which
is an extension of the stan d ard CLEAN algorithm (e.g., Hogbom 1974, Clark
1980) th a t takes advantage of the fact th a t an image of a planet can usually be
represented by relatively small brightness variations superim posed on a uniformly
bright disk. F irst, a uniform disk is convolved with the synthesized beam and
su b tracted from the dirty image, thereby removing much of the sidelobe response.
N ext, the stan d ard CLEAN m ethod is employed to deconvolve the synthesized
8
beam from the rem aining planetary flux.
Finally, the deconvolved planetary
brightness variations and the uniform disk are each convolved with a “clean”
beam (a G aussian th a t best fits the inner core of the synthesized beam ) and
then added to produce the clean image.
The planetary CLEAN technique is particularly beneficial when there are
m any resolution elements across the disk and when brightness variations are not
too large, in which case the prim ary task of the deconvolution procedure is to
remove the sidelobe response of a uniform disk. This is accurately perform ed in
a single step of the planetary CLEAN m ethod, whereas the stan d ard CLEAN
m ethod, which represents the image as a collection of “point” sources, requires a
large num ber of iterations and generally introduces an artificial m ottling to the
image. The planetary CLEAN m ethod is preferred for the centim eter wavelength
images (see below); however, since the millimeter image consists of only six res­
olution elem ents, and since single-dish observations indicate th a t the day-night
brightness contrast is large at millimeter wavelengths (Epstein et al. 1970), we
employ the stan d ard CLEAN algorithm to minimize the sidelobe response in the
BIMA image.
T he success of any deconvolution procedure is limited by noise in the vis­
ibility d ata, which is caused by receiver noise and also by instrum ental gain
variations and atm ospheric fluctuations th a t occur on time scales too short to be
m onitored by the calibration procedure. Atmospheric phase variations (prim ar­
ily due to dielectric variations in the troposphere at millimeter wavelengths) and
instrum ental gain drifts were m onitored every 30 m inutes by observing a nearby
unresolved quasar (Table II), which should have a constant am plitude and phase
for all projected antenna separations.
9
T he absolute flux scale, which is calibrated primarily against Mars and
Venus, is believed to be accurate to within ~10% . However, since we are con­
cerned only with m easuring relative brightness variations across the disk, we
have m ade no attem pt to refine the flux scale. Instead, we normalize the diskaveraged brightness tem perature, ( T b ) diski to the single-dish m easurem ents of
Epstein et al. (1970), which can be represented by the empirical relation:
( T b ) d is k
=
346 + 149 cos ($ + 16°) + 19 cos (2£e + 16°) K
where $ is the phase angle and
E a rth point.
(2)
is the herm ographic longitude of the sub-
Note th a t this relation has been scaled by a factor of 1.17, in
accordance w ith the argum ents of Ulich et al. (1973), who found system atic
differences between their own m easurem ents and the m easurem ents of Epstein
et al. of the absolute brightness tem peratures of Jupiter, Saturn, M ars, Venus,
and Mercury. Equation 2 gives a value of roughly 280 K during the observations
of 26-29 January, which is used to scale th e BIMA image.
Since the January 1988 observations were designed for basic imaging, only
one of the six antenna pairs exhibited sufficient sensitivity to the day-night bright­
ness contrast to discriminate between various thermophysical models. Since this
was the only observation of its kind at A0.3 cm, we obtained additional d a ta
w ith the BIMA interferom eter on 28 M arch 1991 in order to confirm our earlier
results. The viewing geometry for this observation is given in Table I and illus­
tra te d in Figure 1. The experim ental setup was the same as for the January 1988
observations, except th a t we did not request specialized antenna configurations
for imaging the planet. As before, one of the antenna pairs was in a favorable
configuration for measuring the day-night brightness contrast. Equation 2 gives
10
a value of 258 K on 28 M arch 1991, corresponding to a total flux of 70.5 Jy,
which is used to scale the visibility data.
centimeter wavelength observations
T he centim eter wavelength observations were perform ed with the Very Large
A rray (VLA) during two runs in 1990: 13-15 April and 11-12 August. A sum m ary
of the viewing geom etry is given in Table I. In the VLA continuum mode, two
independently tunable spectral windows are available, each with a bandw idth of
50 MHz. At wavelengths from 1.3 to 6.2 cm, these windows were placed end
on end for a to tal bandw idth of 100 MHz (although the d ata for each window
were recorded separately).
We observed at wavelengths of 18.0 and 20.5 cm
sim ultaneously, with one 50 MHz spectral window centered at each wavelength.
This was done to improve our sensitivity to internal heat sources, which should
be the prim ary cause for an increase in the brightness tem perature at these long
wavelengths (see Appendix 1).
We observed M ercury from A3.6 cm to A20.5 cm with the VLA in its m ost
extended “A” configuration (13-15 April), and at Al.3 cm and A2.0 cm in the
m ore com pact “B” configuration (11-12 A ugust). The VLA consists of 27 an ­
tennas th a t form 351 pairs. These provide very dense sampling of the visibility
function so th a t a high-quality image can be produced from a single 8-hour ob­
servation. During this tim e, M ercury rotates through only 1.8 degrees, which
corresponds to 0.1” at the center of the disk. This is four tim es smaller th an
the highest synthesized resolution of these observations (0.4” at A3.6 cm). In
addition, the dense sampling of the visibility function results in a low sidelobe
response, which is m ore easily removed from the dirty image.
11
A tm ospheric phase fluctuations and instrum ental gain variations were m on­
itored by periodically observing a nearby unresolved quasar (Table II). At wave­
lengths of 1.3 and 2.0 cm, where atm ospheric variations are more rapid, we chose
a calibration interval of 10 m inutes. At the longer wavelengths, the calibration
interval was 20 m inutes. The absolute flux scale was determ ined by observing
a stan d a rd VLA flux calibrator, 3C 48 in April and 3C 286 in A ugust, which is
in tu rn calibrated against 3C 295 at wavelengths > 3.6 cm and NGC 7027 at
shorter wavelengths using the flux scale of Baars et al. (1977). The accuracy of
this procedure, including internal errors, is believed to be 10% at Al.3 cm, 5%
from A2.0 cm to A3.6 cm, and 3% at longer wavelengths.
Applying this absolute flux scale to M ercury is not straightforw ard because
the interferom eter d a ta are prim arily sensitive to brightness variations across the
disk and not to the disk-integrated flux. The reason for this is related to the fact
th a t th e antennas of a radio interferom eter cannot be placed so close, together
as to m easure the innerm ost region of the visibility function. As a result, the
CLEAN procedure generally introduces a small flux offset in the image, even
when the visibility d a ta have been properly calibrated. The innerm ost p a rt of
the visibility function could be obtained from single-dish observations; however,
to a good approxim ation, we can extrapolate from the m easured visibility d a ta
into th e innerm ost region using the visibility function of a uniform disk:
V W = S -d
^ )
;
where S is the disk-integrated flux, and J \ is the
13 = be
(3)
Bessel function of first order
and first kind. In practice, this provides a good approxim ation to the actual
visibility function for (3
0.4. (The first null of J \ occurs at /? = 0.61.) For
12
each w avelength, we chose the antenna configuration (A or B) th a t gave the best
possible spatial resolution while sim ultaneously providing a significant am ount
of d a ta for f3 £ 0.4. We fit the visibility d a ta in this inner region with Equation
3 to determ ine the disk-integrated flux at each wavelength. The results are given
in Table III. T he quoted uncertainties are one stan d ard deviation and include
the uncertainty associated w ith fitting th e visibility d ata as well as the absolute
uncertainty (which dom inates in each case).
T he absolute flux scale is applied to th e images in two steps. F irst, a constant
offset is added to each image so th a t th e m ean flux is zero in the blank-sky
regions adjacent to the planet. The m ean blank-sky flux is determ ined by fitting
a G aussian to the core of the blank-sky noise distribution (see, e.g., Figure 6).
The offset corrections were smaller th an th e m easured rm s in each case. Second,
a m ultiplicative factor is applied so th a t the to tal flux in each image is the sam e
as th a t determ ined by fitting the inner region of the visibility data.
T he sidelobe response was minimized in each image using the planetary
CLEAN m ethod described above. Often, especially at the shorter wavelengths,
significant atm ospheric fluctuations occur on tim e scales th a t are shorter th an
the external calibration interval. W hen there are m any antennas, the complex
a n ten n a gains can be modified to com pensate for these fluctuations by th e “self­
calibration” technique (see, e.g., Cornwell and Fomalont 1989). In a given itera­
tion of self-calibration, the antenna gains are allowed to vary so as to minimize the
difference between the m easured visibility and the visibility predicted by a m odel
of the source. This procedure is over-determ ined since there are m any m ore d a ta
points (antenna pairs) th a n free param eters (individual antenna gains). T he cor­
rected d a ta are then used to produce an improved clean image, which is in tu rn
used as the m odel for the next iteration. The initial source model need not be
13
very accurate. We typically used the clean image produced from the uncorrected
d a ta as the initial model, but we found th a t a uniform disk is also an adequate
first guess. Overall convergence was usually achieved after only two or three
iterations.
T he signal-to-noise ratio (SNR) of the visibility d a ta establishes a m inim um
interval at which atm ospheric variations can be corrected by self-calibration.
This interval ranged from 1 m inute at Al.3 cm to 5 m inutes at A6.2 cm, while
at A18.0 cm and A20.5 cm, it was longer th a n the external calibration interval of
20 m inutes. A tmospheric variations on shorter tim e scales, if present, limit the
effectiveness of the planetary CLEAN procedure and hence the final dynam ic
range of the images. The observations of August 12 were corrupted by large
and rapid atm ospheric phase fluctuations th a t could not be corrected by self­
calibration. Fortunately, we observed at b o th Al.3 cm and A2.0 cm on August
11, and were able to calibrate 2.2 hours of d a ta at Al.3 cm and 3.6 hours at
A2.0 cm.
Because the brightness of M ercury increases approxim ately as the
inverse square of the wavelength, a good SNR was achieved at these wavelengths
despite the shorter integration times. We experienced no such difficulties during
th e April observations; however, instrum ental problems on April 14 lim ited the
observing tim e to 3.6 hours at A6.2 cm. We observed for 8 hours at the rem aining
wavelengths.
polarization
T he VLA receivers separate the incoming radiation into right and left cir­
cular polarization, from which all four Stokes param eters can be derived. The
external gain calibrator (a weakly polarized source) was used to correct for the
in stru m en tal polarization response, while observations of a strongly polarized
14
source with known position angle (3C 138) were used to calibrate the position
angle. Polarization images were obtained at all wavelengths except 1.3 cm, where
there was insufficient d a ta to perform the necessary instrum ental calibrations.
Stokes Q and U images were produced in a m anner very similar to the to tal
intensity (I) images. We used the stan d ard CLEAN procedure to minimize the
sidelobe response in the Q and U images, since the emission is in the shape of a
ring, with both positive and negative intensity. The Q and U visibility d a ta are
form ed from linear combinations of the m easured circular polarization d ata, so
the Q and U images have noise distributions similar to those of the corresponding
I images.
Linear polarization (P ) images are produced by forming the combination:
P = (Q 2 + U2) *
(4)
on a pixel by pixel basis. This combination introduces a bias in the polarization
intensity about equal to the rm s noise level in the Q and U images (Fomalont
1989). We have corrected for this bias. In addition, pixels were blanked in the
P image if either the Q or U intensity was less than twice the rm s noise. The
orientation of the electric field vectors (x) was obtained from:
R adiation polarized parallel to the plane of incidence has a higher Fresnel tra n s­
mission coefficient than the orthogonal polarization, so the electric field vectors
are expected to be oriented radially on the projected disk.
15
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T able II
Calibrators
Date
A (cm)
C alibrator
Flux (Jy)
1/26/88
0.3
2145+067
4.1
1/29/88
0.3
2145+067
4.1
3/28/91
0.3
3C 454
8.6
8/1 1 /9 0
1.3
1058+015
5.1
8/1 1 /9 0
2.0
1058+015
4.5
4 /1 3 /9 0
3.6
0238+166
2.7
4 /1 4 /9 0
6.2
0238+166
2.0
4/1 5 /9 0
18.0
0238+166
1.2
4/1 5 /9 0
20.5
0238+166
1.1
17
F ig u re C a p tio n
F ig u re 1
T he aspect of M ercury on 26 Jan u ary 1988 (A), 28 M arch 1991 (B), 11 A ugust
1990 (C ), and 13-15 April 1990 (D). T he diam eter of each disk is scaled to th a t
of disk A, which is 7” . T he m orning term inator is shown by th e bold line, and
the direction of the sun is indicated by the arrow. Both m eridians and latitu d e
circles are separated by 30°. A few m eridians are labeled to the right of the
m eridian and above the equator.
18
F ig u re 1
o
19
III. R esu lts
m illim eter wavelength image
T he clean A0.3 cm image is shown in Figure 2 with a contour interval of
34 K (10% of m axim um ). The lowest contour is approxim ately twice the rm s of
15 K, as m easured in the blank-sky regions adjacent to the planet. In principle,
given the to tal on-source integration tim e of ~3.4 hours per baseline, it should
have been possible to achieve an rm s of 0.7 K, which is m uch smaller than the
m easured value. Thus, the dynamic range of this image is limited by sidelobe
response th a t was not completely removed. The standard CLEAN procedure was
only partially successful because with only six antenna pairs, the sampling of the
visibility function was very sparse, resulting in a large sidelobe response th a t was
difficult to remove from the dirty image. This situation was likely exasperated
by dielectric fluctuations in the troposphere th a t might have occurred between
calibration cycles.
T he clean image clearly exhibits a large day-night brightness contrast, but a
quantitative interpretation is problematic because of smearing effects and incom­
plete removal of the sidelobe response. However, both of these difficulties can be
avoided by Fourier transform ing separate model images for each of the observing
dates and com paring directly with the visibility d ata. In Figure 3 the visibility
am plitude for one of the antenna pairs is plotted as a function of M ercury’s hour
angle during the observations of January 26. These d ata represent a cut through
the spatial Fourier transform of the planet.
M ercury’s visibility function is similar to th a t of a uniform disk. Deviations
in the m easured visibility function from Equation 3 contain inform ation about
the day-night brightness contrast, and hence the therm al and electrical properties
20
of the regolith. For comparison, Equation 3 is shown by the solid curve in Figure
3, dem onstrating th a t the m easured visibility is significantly different from th a t
of a uniform disk. This m ethod of analysis is equivalent to measuring relative
brightness variations across the disk and is therefore independent of the absolute
flux scale.
The m easured visibility am plitude as a function of (3 from the M arch 1991
observations is shown in Figure 4 and compared to the expected signal of a
uniform disk. For (3 £ 0.55 the m easured visibility deviates significantly from
th a t of a uniform disk, b u t for smaller values of (3, the uniform disk visibility
provides a good fit to the m easurem ents.
centim eter wavelength total intensity images
The centim eter images are shown in Figure
5. T he appearance of each
image is dom inated by the hot-cold longitude p attern , although diurnal variations
have a significant effect at the shorter wavelengths. At Al.3 cm and A2.0 cm, a
hot longitude (180°) was near the center of the disk and very nearly coincident
with the m orning term inator. The day-night contrast is m ore pronounced at
Al.3 cm because this wavelength probes shallower layers th a n does A2.0 cm. At
wavelengths of 3.6 cm and longer, a cold longitude (270°) was nearest to the
center of the disk, which largely accounts for the different appearance of these
images. As expected, the day-night contrast is greater at A3.6 cm than it is at
A6.2 cm. The A18.0 cm and A20.5 cm images appear different from the A6.2 cm
image prim arily because of the lower resolutions.
Table III gives the statistical properties of each image. The rm s noise, as
m easured in the blank sky regions adjacent to Mercury, generally contains a con­
tribution from artifacts introduced during image restoration, as described above.
21
For comparison, we have calculated the theoretical noise for each image (Crane
and Napier 1989) under the assum ption th a t all artifacts have been removed (see
Table III). T he theoretical noise limit was nearly achieved at wavelengths of 6.2
cm and longer, but at the shorter wavelengths the m easured noise is higher.
Not only do artifacts increase the noise level, but they can also modify the
noise distribution, which is otherwise expected to be Gaussian (Fomalont 1989).
Figure 6 (a) shows the noise distribution in the worst case (Al.3 cm). Although
the central region of this distribution is approximately Gaussian, there is a signif­
icant wing th a t extends from approxim ately 2 to 7 times the rm s. Therefore, one
m ust be cautious when interpreting structure on the disk at Al.3 cm, especially
near the resolution lim it. Figure 6 (b) shows the noise distribution in the best
case (A6.2 cm). This distribution is approxim ately Gaussian, as expected, since
the noise level is near the theoretical limit.
Note th a t relative brightness variations across the disk are much b e tte r de­
term ined th a n the absolute brightness at any one position (com pare columns 3
and 6 of Table III). Relative uncertainties are as small as 1.6 K, while absolute
uncertainties range from 11 to 38 K. At a wavelength of 1.3 cm, the relative
uncertainty is m ore th an an order of m agnitude smaller th an the absolute uncer­
tainty. T hus, it is m uch b etter to utilize brightness variations across the disk in
order to constrain therm ophysical models.
centim eter wavelength polarization images
The P images are shown in Figure 7. As expected, the electric field vectors
are oriented radially on the disk. Note th a t the polarized emission is not circu­
larly sym m etric about the center of the disk because the polarized intensity is
proportional to the total intensity, which varies across the disk. The dependence
22
of the polarization on the total intensity (and hence the therm ophysical prop­
erties of the regolith) can be removed by forming fractional polarization images
(F = y ).
A pixel on the F image was blanked if the corresponding P pixel
was blanked or if the I pixel intensity was less th an twice the rm s noise. The
F images are circularly symmetric about the center of th e disk, so they were
averaged in concentric annuli, spaced at half th e synthesized resolution, in order
to maximize the SNR. Blanked pixels were not included in the averages. The
fractional polarization, as a function of radius on the projected disk, is shown in
Figure 8. T he error bars represent the rm s variations in the annular averaging
process, and are thus affected both by noise in the d a ta and by real variations in
the polarization.
For com parison, the curves in Figure 8 represent the expected fractional
polarization of a sm ooth surface, which depends only on the “effective” dielec­
tric constant of the regolith and the viewing angle between th e surface norm al
and the observer (or, equivalently, radius on the projected disk). We define the
e/ /
effective dielectric constant (eR ) to be th a t of a homogeneous, non-scattering,
sm ooth-surfaced m edium which reproduces the observed polarization. The model
is constructed in such a way as to mimic the steps used in calculating the d a ta
points displayed in Figure 8. F irst, th e fractional polarization is calculated from
the stan d ard Fresnel formulas (Jackson 1975) as a function of projected radius
in order to form a fractional polarization disk. This and a uniform disk of unit
am plitude are each convolved with the synthesized beam. Next, the convolved
polarization disk is divided by the convolved unit-am plitude disk and then aver­
aged in concentric annuli spaced at one half of the synthesized resolution. Note
th a t if eR is allowed to vary with wavelength, this smooth-surface model accu­
rately fits the d a ta in the inner region of the projected disk at all wavelengths.
23
However, near the lim b of the planet, the m easured polarization from A2.0 cm
to A6.2 cm falls below th a t of a sm ooth surface. At A18.0 cm and A20.5 cm, a
sm ooth surface adequately predicts th e m easured polarization at the resolution
of these images.
24
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F igu re C ap tion s
F ig u re 2
Im age of M ercury’s therm al emission at A0.3 cm, which was obtained with the
BIMA millim eter interferom eter from 26 to 29 January 1988, when the planet
was near greatest eastern elongation. The viewing geom etry is given in Table
la n d shown in Figure 1. The contour interval is 34 K, which is 10% of the m ax­
im um intensity of 339 K. (Dashed contours are negative.) T he lowest contour
is approxim ately twice the rm s of 15 K. The grey scale is linear from the m in­
im um (white) to the m axim um (darkest) intensity. The synthesized resolution
(G aussian, FW HM ) is shown in the inset. This image has been scaled so th at
the disk-integrated flux (61.9 Jy) agrees with previous single-dish m easurem ents
(see text).
F ig u re 3
T he visibility am plitude m easured by one of the BIMA antenna pairs as a func­
tion of M ercury’s hour angle during the observations of 26 Jan u ary 1988. The
d a ta have been scaled so th a t the disk-integrated flux (as determ ined from Fig­
ure 2) agrees w ith previous single-dish measurements (see text).
These d ata
represent a cut through the spatial Fourier transform of the planet. The m ea­
surem ents differ significantly from the Fourier transform of a uniform disk (solid
curve), which is indicative of a strong day-night brightness contrast.
F ig u re 4
The visibility am plitude m easured by one of the BIMA antenna pairs on 28 M arch
1991 as a function of /3 , which is proportional to the antenna separation projected
onto the plane of the sky (see Equation 3). The d a ta have been scaled so th at
the disk-integrated flux ( V (/3 = 0) = 70.5 J y ) agrees with previous single-dish
26
m easurem ents (see text'!. The solid curve is the Fourier transform of a uniform
disk, as in Figure 3.
F ig u re 5
Im ages of M ercury’s therm al emission at wavelengths of 1.3 (a), 2.0 (b), 3.6
(c), 6.2 (d), 18.0 (e), and 20.5 (f) cm, which were obtained with the VLA in
April and August of 1990 when the planet was near greatest eastern elongation.
T he viewing geom etry is sum m arized in Table I and shown in Figure 1. For all
images, the contour interval is 10% of the m axim um intensity, except for the
lowest contour, which is either 2% (a-d) or 4% (e-f). The image statistics (e.g.,
noise level, m axim um intensity) are given in Table III. The grey scale is linear
from the m inim um (white) to the m axim um (darkest) intensity. The synthesized
beam (Gaussian, FW HM ) is shown in the insets.
F ig u re 6
(a ) T he histogram is the noise distribution at Al.3 cm in th e blank-sky regions
adjacent to the planet. The curve is a Gaussian which best fits the inner core of
the distribution. T he wing extending from 3 to 10 K in the m easured distribution
corresponds prim arily to the “flames” extending from the edge of the disk in
Figure 5 (a). This wing is caused by rapid atm ospheric fluctuations th a t do not
allow complete removal of the interferom eter sidelobe response (see text). The
blank-sky noise distributions at A2.0 cm and A3.6 cm have similar wings, except
th a t they are less pronounced.
(b ) The blank-sky noise distribution at A6.2 cm (histogram ) and the best-fit
G aussian (curve). The distribution is approxim ately Gaussian, and the m easured
rm s (2.9 K) is near the theoretical limit. This indicates th a t the interferom eter
sidelobe response has been almost completely removed from the A6.2 cm image
27
(Figure 5 [d]). The blank-sky noise distributions at A18.0 cm and A20.5 cm are
similar.
F igu re 7
Images of the linearly polarized component of M ercury’s therm al emission at
wavelengths of 2.0 (a), 3.6 (b), 6.2 (c), 18.0 (d), and 20.5 (e) cm. These corre­
spond to the total intensity images shown in Figure 5 (b-f). Electric field vectors,
each with a length proportional to the polarized intensity, are superim posed on
a linear grey scale. W hite areas are blanked pixels, which are below twice the
rm s (see text), while the grey scale limits are shown at right.
F igu re 8
T he d a ta points are the m easured fractional polarization as a function of radius
on the projected disk at wavelengths of 2.0, 3.6, 6.2, and 18.0 cm. These are
obtained by dividing the polarization images (Figure 7) by th e corresponding
total intensity images (Figure 5) and averaging in concentric annuli spaced at half
the synthesized resolution. T he horizontal dashed line is the m inim um detectable
polarization fraction (at twice the rm s). The vertical dashed line is the lim b of
the planet. The curves represent a sm ooth surface model calculation (see text),
which depends only on the effective dielectric constant of the surface m aterial
(e^7). T he values for e ” are 1.7, 2.0, 2.2, and 2.3 at wavelengths of 2.0, 3.6, 6.2,
and 18.0 cm, respectively.
28
LO
LO
(„ )
O
" a
LO
a A n e ia H
29
Relative
LO
RA (")
F ig u r e 2
F ig u re 3
CO
00
03
(h rs)
o
Hour Angle
CO
C\2
o
C\3
LO
t- h
o^—i
LO
(Xf) a p n ^ i d u i y A n i q i s i A
30
o
F ig u re 4
03
O
00
o
CO
d
z>
o
00
oa
CO
o
SCL
LO
o
o
CO
o
C\2
o
LO
o
^
o
o
CO
c\}
o
(Xf) apn^iiduiv A nW siA
31
o
o
•
F igu re 5 (a)
CD
I
I
O
C\2
CD
CD
^
C\2
O
C\2
I
(n )
O 0Q
9 A I^ I9 H
32
^
CD
I
I
Relative
02
RA (")
I
F ig u re 5 (b )
1—
i—
1—
i—
1—
i—
1—
i
1—
i
1
n
to
Relative
RA (")
i—
CO
^
c\2
(„ )
O
o a a
C\2
S A rp e ia y
33
CO
F igu re 5 (c)
CD
I
I
O
C\2
CD
0
^
W
O
W
I
(n) 09Q S A I ^ P H
34
^
I
CD
I
Relative
02
RA (")
I
F ig u re 5 (d )
CO
I
I
o
C\2
CO
CO
^
W
O
W
I
( n )
a
9 A I^ P H
35
^
CO
I
I
Relative
C\2
RA (")
I
F igu re 5 (e)
Relative
RA (")
i
S3«
00
^
O
(„ )
0 8 Q
00
S A I^ B P H
36
F ig u re 5 (f)
00
8
Relative
RA (")
-I
<D
rQ -I
0
—J----- 1----- 1----- 1----- 1----- 1___ i___ I___ i___ t o
00
^
i
O
(„ )
O SQ
I___ i
i
i
QQ
00
S A n ie p H
37
F igure 6 (a)
O
CO
o
1500
in
o
o
o
o
o
m
ja q rn n j^
38
o
Pixel Value (K)
CO
F igure 6 (b )
C\2
CD
O
1500
O
O
O
O
O
O
to
ja q u in jy j
39
O
Pixel Value (K)
O
Relative
RA (")
F igu re 7 (a)
40
F ig u re 7 (b )
1
i
1
i
1
i
1
i
1
i
1
ri
co
Relative
RA (")
i
0
^
CV
O
W
(„) oaa
41
^
CD
F ig u re 7 (c)
'O' o
o
o
o
o
CO
C\2
T-H
^
I
I
I
I
I
I
|
I
n
co
Relative
RA (")
1
o
CD
^
C\2
(„ )
O
3 9 (3
C\2
9A T}B 18H
42
CD
F igu re 7 (d )
1
i
CO
r
iPf.;
cO
□
O
RA (")
o
Relative
CO
i
i
i
i
i
i
i
J
I
L
CO
_1
I
L
CO
CO
(„)
° s a
9A n ^ i 8 H
43
F igu re 7 (e )
O
C\2
LO
LO
00
Relative
RA ('')
□
-
^
CO
GO
CO
(„ )
O SQ
a A T ^ B ia H
44
sec
F ig u re 8
o
Radius
(arc
CO
-T "
C\2
O
c\i
LO
CD
LO
LO
45
lO
IV . T h e rm o p h y sic a l and R a d ia tiv e T ran sfer M o d els
T he observed images are com pared w ith m odel images th a t are generated
by first solving the therm al diffusion equation to obtain the tem p eratu re w ithin
the regolith as a function of depth and tim e, and th en integrating the radiative
transfer equation to determ ine the therm al microwave emission at the surface. A
realistic m odel requires several param eters to describe th e therm al and electrical
properties of the regolith m aterial and the variation of these properties w ith depth
and tem perature. Such a model exists for the lu n ar regolith as a result of detailed
inform ation from in siiu heat probe experim ents and retu rn ed samples (G ary and
K eihm 1978). M uch less inform ation is available to develop a model for M ercury’s
regolith, so we rely largely on a lunar analogy since m eteoroid bom bardm ent is
responsible for producing the regoliths of b o th M ercury and the Moon. The
higher surface gravity on M ercury could influence the regolith stru ctu re, so an
im p o rtan t constraint on this m odel is provided by m easurem ents of M ercury’s
night-side surface tem perature by the M ariner 10 Infrared R adiom eter (Chase
ei al. 1976).
T he propagation of microwaves through the regolith is affected by scattering
to an extent th a t depends on the observing wavelength, the particle size distri­
bution, an d the n atu re of possible subsurface inhomogeneities, such as stratified
layers. A lthough m ore sophisticated m ethods have been developed (e.g., Tsang
and Kong 1975; Keihm 1982), we adopt the approach of Gary and Keihm (1978)
and incorporate the scattering opacity into an effective absorption coefficient.
T he radiative transfer equation is then integrated as if there were no scattering
by assum ing a blackbody source function. Basically, this m ethod allows the de­
term ination of relative brightness variations at the expense of overestim ating the
microwave emissivity by an unknow n factor. However, since we are concerned
46
only w ith m odeling relative brightness variations, we absorb this error into an
arb itra ry scaling factor for the model images.
Scattering at the surface boundary is treated as a separate problem by re­
placing an ideal spherical surface by a distribution of random ly tilted facets, each
of a size m uch larger than the observing wavelength. The transm ission coeffi­
cients for each facet are determ ined from the stan d ard Fresnel form ulas, except
th a t an allowance is m ade for p artial diffusion by wavelength-scale roughness.
T he observable properties of this rough surface (polarization and limb darken­
ing) are determ ined by taking appropriately weighted averages over the facets.
These effects are constrained by th e polarization images in com bination with
previous ra d a r observations.
temperature variations in the regolith
D iurnal tem p eratu re variations in th e subsurface layers are determ ined by
considering the balance between insolation, conduction of heat through the sub­
surface, and reradiation outw ard. We seek solutions of the one-dimensional th er­
m al diffusion equation:
d
(
dT\
T* ( * & )
dT
=
<6 >
with physically realistic expressions for the density (p), specific heat ( C ) , and
therm al conductivity ( K ).
As noted above, previous microwave observations
have shown th a t M ercury’s surface is covered by a regolith, w ith a thickness of
at least ~ 2 m eters, th a t is similar to the lunar regolith. Subsequently, M ariner
10 Infrared R adiom eter m easurem ents showed th a t M ercury’s surface therm al
inertia is com parable to the M oon’s (Chase et al. 1976). In addition, optical
47
photom etry and polarim etry (Veverka et a I. 1988) and rad ar observations (Pettengill 1978) have shown th a t the surface textures of M ercury and the Moon are
quite sim ilar over a broad range of size scales. Therefore, we make use of detailed
knowledge of the lunar regolith from returned samples and in situ experim ents
to develop a depth-dependent model for M ercury’s regolith.
the lunar regolith
The variation of density with depth in the lunar regolith has been inferred
from astro n au t bootprints, vehicle tracks, boulder tracks, and returned core sam ­
ples (C arrier et al. 1991). These studies indicate th a t the density increases rapidly
from ~ 1 to ~ 1.8 g cm -3 over the top 20 cm and then remains essentially constant
to a d epth of a t least 3 m eters. A variety of expressions for the density as a func­
tion of depth have been proposed (C arrier et al. 1973, Olhoeft and Strangway
1975, C arrier et al. 1991), all of which fall between the dashed curves in Figure
9. Regardless of th e exact form of the density profile, C arrier et al. (1991) point
out th a t the density gradient is much steeper th a n would be required to support
the weight of overlying layers and conclude th a t some process, presum ably m ete­
oroid im pacts, m ust stir the top few centimeters while com pacting deeper layers
by vibration a n d /o r stress.
The specific heats of lunar samples are uniform and very close to those
of com m on terrestrial rocks (Cremers 1974). Golden (1977) has developed an
em pirical expression for the specific heat as a function of tem perature th a t ade­
quately fits m easurem ents of lunar and terrestrial samples and also accounts for
48
the asym ptotic behavior of C ( T ) at tem peratures in excess of 350 K:
c0 + c \ T +
C(T) =
<(
c2T
2 + c 3T 3 + C4 T 4
r
/
c 5 + c6 | l - ex p ^
T-T M
Tioo36Qj j
;
;
T < T 350
T > T‘’ 350
3I
(? )
where Tloo = 100 K and T3B0 = 350 K. The coefficients c, are given in Table IV.
We adopt this expression for the specific heat of M ercury’s regolith.
T he therm al conductivity of particulate m aterial can depend strongly on
tem perature when the atm ospheric pressure is less th an about 1 0-3 bar, so th a t
gas molecules do not participate in heat conduction (see Wechsler and Glaser
1965). T he atm ospheric pressures on the Moon and M ercury are m any orders
of m agnitude smaller than this (Vaniman et al. 1991; H unten et al. 1988). The
tem perature dependence occurs because of radiative heat tran sp o rt in the spaces
between grains, which can be significant com pared with phonon conduction along
grain contacts. The therm al diffusion equation can be modified to take this into
account by adding a radiative term to the therm al conductivity (Crem ers 1974,
and references therein):
K = Kr.
{\ £ 350
- )/ '
(8 )
where K c represents the “contact” phonon conductivity, and x is the ratio of
radiative to phonon conductivity at 350 K. Both of these param eters depend on
the particle size distribution, particle shapes, the m anner in which the particles
are packed, and the nature of the therm al contacts between particles. Crudely,
we expect K c and x t° vary with density, so as reference values we define K 360 (p)
49
5
to be th e to tal therm al conductivity at 350 K, and K 3h0 to be the value of K 3S0
at the surface.
Equation 8 has been shown to adequately describe the therm al conductivities
of lunar soil samples in vacuum at densities ranging from 1 to 2 g cm -3 (Cremers
1974). T he values of K 3B0 and x f°r these measurem ents fall within boxes 1 and
5 of Figure
10 . The therm al conductivity exhibits only a m odest increase as
the density varies from the m inim um to the m axim um values attainable in the
laboratory. This behavior is in stark contrast to the therm al conductivity profile
th a t was determ ined in situ by the Apollo 15 and 17 heat probe experiments.
The density and therm al conductivity profiles over the top 15 cm are con­
strained by the surface cooling rate after sunset. Surface tem perature d a ta from
the Apollo 15 and 17 heat probes show th a t this region consists of essentially two
layers: a thin, low density layer with K 360 « 100-200 (cgs units) atop a highly
com pacted layer with K 360 « 1000 (Keihm and Langseth 1973; Figures 9 and
10 , boxes 1 and 2). The transition between the two layers m ay be discontinuous.
The Apollo 15 and 17 core sample d a ta show th a t the compacted layer extends
to a depth of at least 230 cm (Figure 9, boxes 3 and 4). The therm al conduc­
tivity below 15 cm was m easured by sensors attached to deeper sections of the
heat probes. These d a ta show th a t K 350 rem ains roughly constant at a value
of ~ 1 0 0 0 from 35 to at least 230 cm (Langseth et al. 1976; Figure 10, boxes 3
and 4). Presum ably, the lunar sample measurem ents at high densities (p £ 1.5
g cm - 3 ) underestim ate the in situ therm al conductivity because the techniques
used to com pact the lunar samples in the laboratory were not equivalent to the
effects of m eteoroid bom bardm ent on the Moon over geologic timescales a n d /o r
the process of collecting and handling the lunar samples destroyed sintered grain
contacts (if such contacts were present).
50
a model fo r M e r c u r y ’s regolith
M uch less inform ation is available to develop a model for M ercury’s regolith,
so we consider three possibilities: (a) a constant density and therm al conductiv­
ity, (b ) a rapid but continuous increase in the density and therm al conductivity
w ith depth, and (c) a two-layer model th a t exhibits an ab ru p t increase to m axi­
m um density and therm al conductivity at a depth of 2 cm. These are shown as
heavy lines in Figures 9 and 10 . Model a corresponds to a homogeneous regolith
and is included for com parison w ith similar models adopted by previous workers.
T he rates at which K 3S0 and x vary w ith density in model b are chosen to be
consistent w ith laboratory m easurem ents of returned lunar samples. In contrast,
m odel c is designed to approxim ate the structure of the lunar regolith as inferred
from th e in situ heat probe data. We assum e th a t the large increase in K 360 at
the transition between the top layer and the com pacted region results solely from
enhanced phonon conduction. This results in a sharp decrease in x
a depth of
2 cm. Each model is normalized by adjusting K 3i0 so as to reproduce M ariner
10 surface tem perature m easurem ents at a locus of points on M ercury’s night
hem isphere. This procedure will be described below, after we consider boundary
conditions and a m ethod for solving Equation 6 .
boundary conditions
T he therm al diffusion equation m ust be supplem ented by two boundary
conditions. At the surface, total energy conservation provides one of these:
( i ^ ) (1~^6)sin+W~
Jo
=
W bT *
~
(9)
T he two term s on the right side of the equation are the (infrared) energy rad i­
ated by the surface and the energy conducted into the subsurface, where em is
51
the infrared surface emissivity, T a is the surface tem p eratu re, K is th e th erm al
conductivity, and
is the surface tem p eratu re gradient. On the left side of
th e equation, Jo is a constant flux originating from internal heat sources. This
flux is negligible com pared w ith o th er term s in the equation and is included here
only for com pleteness. T he first te rm on th e left is the insolation, where L© is th e
solar lum inosity, r is the sun-planet distance, and Ab is the bolom etric albedo.
T he angle ip is th e elevation of th e sun as seen from a given location on the
surface. The “clipped” sine function, sin+ (ip), is zero (instead of negative) when
the sun is below th e horizon. We adopt constant values of 0.12 for th e bolom etric
albedo (Veverka et al. 1988) and 0.90 for th e infrared emissivity (see, e.g., Chase
et al. 1976). T he precise values of these two param eters are not critical, since
they affect th e surface tem p eratu re through only the fourth root.
T he second b oundary condition is simply one of constant heat flow at an
“equilibrium d e p th ” w here th e am plitude of diurnal tem p eratu re variations has
a tte n u ate d to < 1 K and can be regarded as zero.
dT
dx
where ^ \ d and K<i are the tem p eratu re gradient and therm al conductivity at
the equilibrium depth. We neglect Jo in this analysis because its m agnitude is
expected to be only ~ 2 0 erg cm -2 s -1 (Schubert et al. 1988), which would have
a negligible effect on th e day-night brightness contrast. We consider non-zero
values for Jo in A ppendix 1. T he equilibrium depth is determ ined by adjusting
the depth grid until several of th e deepest grid points exhibit negligible diurnal
variations (A T <C 1 K). A nother necessary criterion is th a t th e choice of an
equilibrium dep th should not p e rtu rb th e tem p eratu re stru ctu re in shallower
62
regions beyond the desired num erical accuracy. D iurnal tem perature variations
in our m odel are ~ 1 K at a depth of 90 cm and less th an the num erical accuracy
of 0.1 K at depths greater th an 135 cm. To be conservative, we added four grid
points below 135 cm to extend the m axim um depth to 230 cm. Equation 10 is
applied to the deepest two levels of this grid. These considerations ensure th a t
th e m odel brightness at any observing wavelength is independent of our choice
of equilibrium depth.
numerical solution
T he tem perature profile m ust be determ ined numerically because of the non­
linear expressions for the therm al param eters and surface boundary condition.
F irst, Equations 6 , 9, and lOare w ritten in dimensionless form by defining a
characteristic length, tim e, and tem perature:
=
2
; r = S2-1 :
’
■
;
T„ =
(
^'
(11>
where L t is the therm al skin depth, 27rf2_1 is one diurnal period, C 350 is the
specific heat at 350 K ( 0.2 cal g -1 K - 1 ), and a is the orbital sem i-m ajor axis.
T he resulting dimensionless equations are solved numerically using the CrankNicholson iteration scheme (Press et al. 1988). We chose a grid in which the
d ep th interval increases geometrically over 40 steps from 0.13 cm at the surface
to 30 cm at the equilibrium depth. This allows high resolution near the surface,
where density and tem p eratu re gradients are large, w ithout including a large
num ber of grid points below 2-3 L t , where th e density is constant and tem per­
a tu re variations have been significantly diffused and atten u ated . The tim e grid
consists of 800 evenly spaced steps per diurnal period. This resolution is about
four tim es higher th a n th a t adopted by previous workers (Golden 1977, Ledlow
53
et al. 1992), b u t we found it necessary in order to accurately model the rapid
tem perature variations at sunrise and sunset. Sensitivity tests dem onstrate th at
the m odel images generated using this grid are insensitive to our choices of depth
and tim e resolution to well within the resolution and accuracy of the data.
From an initially uniform tem perature distribution, the Crank-Nicholson
m ethod typically converges after 3-4 diurnal periods. There are three separate
convergence criteria th a t m ust be satisfied: ( 1 ) the solution m ust be periodic, ( 2 )
during one diurnal period the surface m ust radiate th e same am ount of energy
it receives from the sun, and (3) there m ust be zero net divergence of heat at
all depths. These criteria are explicitly built into the algorithm , and the third
is used to drive the solution towards convergence. Iteration is term inated when
each criterion is satisfied to within a numerical accuracy th a t corresponds to
an error in the tem perature structure of 0.1 K, which is more th an an order of
m agnitude sm aller th an the noise levels of the observed images.
normalization of K 350
It is well known (see, e.g., Jaeger 1953) th a t if the density, specific heat,
and therm al conductivity are constants, then Equations 6 , 9, and 10 describe a
family of solutions th a t are specified by a single “therm al param eter” , 7 :
7
( 12 )
=
/(f)’
w here I = ( K p C ) 1 is the therm al inertia. M ercury is a slow ro tato r (7 « 220
1 ), so the surface is nearly in instantaneous radiative equilibrium during the
day, while after sunset, the surface cools at a rate th a t depends only on the
therm al inertia. Since our model includes variations of the therm al param eters
54
w ith depth and tem perature, we define / 3B0 to be the surface therm al inertia at
a tem p eratu re of 350 K. In this case, the rate at which the surface cools during
§
th e night depends on / 3B0 and on the variation of the therm al param eters with
depth.
Figure 11 shows the results of fitting the M ariner 10 surface tem perature
m easurem ents 3 at a locus of points on M ercury’s night hemisphere by adjusting
K 3S0 in models a, b, and c. Note th a t only the conductivity in the surface layer
of m odel c is allowed to vary, while the conductivity in the com pacted region
rem ains fixed. In contrast, the conductivity throughout the regolith is allowed to
vary in models a and b, while the profile shape rem ains fixed. Since the density
and specific heat have been specified in each model, this procedure is equivalent
to adjusting / 3B0, which is plotted as a function of both longitude and local time
in Figure 11.
A part from local anomalies, the therm al inertia of model a increases by 50%
as the longitude decreases from 360° to 180°. This trend is represented by the
dashed fine, which is a least-squares fit to the d ata (excluding the large anomaly
at 235°). Using a similar model, Chase et al. (1976) also found such an increase
and a ttrib u te d it to variations in the surface rock coverage. While this is likely
the case for local anomalies (since similar features occur on the Moon and are
known to be correlated with surface rock coverage), a system atic increase in the
surface rock coverage over 180° in longitude seems much less likely.
Alternatively, we propose th a t the system atic variation of / 3B0 in model a
is actually an apparent increase with local tim e th a t results from a deficiency in
the model. Models in which the density and therm al conductivity increase with
3 T h ese d a ta are given in Table II of Chase et al. (1976).
55
depth, as expected, produce flatter surface therm al inertia curves4. Even the
m odest variation of p and K 3S0 in model b reduces the net change in the surface
therm al inertia to about 20 %. A larger increase in p a n d /o r K 360 w ith depth
would be required to further flatten the curve.
The large, discontinuous variation of p and K 3B0 in model c further reduces
net change in the surface therm al inertia to about 10 % (as defined by a linear fit
to / 350). It is possible to fine tune model c by adjusting the thickness of th e lowdensity surface layer, the conductivity of the com pacted layer, and the n atu re of
the transition between the two regions, so as to virtually elim inate system atic
<y
_
variations in / 350. However, we will not consider such modifications because it
is impossible to arrive at a unique model. We believe th a t model c is sufficiently
close to the “correct” model because it produces a relatively small net change
§
in / 350 across M ercury’s night hemisphere, and the norm alization of K 360 to the
M ariner 10 d a ta results in a therm al conductivity in the surface layer th a t agrees
with laboratory m easurem ents of lunar samples at low density and with in situ
estim ates of the therm al conductivity in th e upper 2 cm of th e M oon’s regolith.
Therefore, we adopt model c for M ercury’s regolith and discard models a and b.
microwave propagation
T he propagation of radiation through the subsurface depends on the dielec­
tric and scattering properties of the regolith. Assuming a plane parallel m edium
in local therm odynam ic equilibrium, the equation of radiative transfer is:
+ p n vI v = p n vS v
;
kv
= Kavha + < ca
(13)
4 Basically, th e increase in th erm al in e rtia w ith tim e is replaced by an increase w ith d ep th .
T h e m ap p in g betw een tim e and d e p th occurs because of th e finite tim e required for
te m p e ra tu re variations to diffuse dow nw ard.
56
where /„ is the intensity at frequency
ds is a differential p a th length along
the direction of propagation, p is the density, and
kv
is the m ass extinction
coefficient, which accounts for b o th tru e absorption (k “6s) and scattering (k*c0)
losses. The source function, S v, is given by:
.„sca
S v — (1
u}V) B V( T ) + wv$ v
;
u}v —
^ 1/
v
T*
a
w here u>v is the scattering albedo, B v is the Planck function, and
(14)
is the
scattering phase integral, which accounts for scattering of radiation from all
directions (&') into the direction of propagation (k):
$ u( k , s ) =
j
4>v ( k , k ' ) I u( k ' , s ) d n '
(15)
47T
where <j>v is the scattering phase function.
At microwave frequencies, the mass absorption coefficient,
is usually
expressed as:
„aba
27TA
/i ^ t a n A
pX
(16)
where en is the real p a rt of the complex dielectric constant, and the loss tangent,
ta n A , is the ratio of the im aginary p art to the real p a rt. At a given frequency,
the complex dielectric constant of a powder depends on its density. For m ost
lunar and terrestrial rock powders, this dependence is accurately represented by
the Rayleigh mixing formula:
p \e + 2 j
p0 \ e 0 + 2 ‘
57
V
1
where p and e are the density and complex dielectric constant of the powder, and
po and eo are those of the paren t rock. This form ula has been verified for common
terrestrial powders by Cam pbell and Ulrichs (1969), and for a variety of lunar
sam ples by Gold et al. (1976). Over the range of densities usually encountered,
the loss tangent is simply proportional to density. T hus, the specific loss tangent,
L v = p - 1 ta n A , is useful for com paring different samples. Since our observations
cannot separate pure absorption losses from scattering losses, we incorporate the
la tte r into an effective specific loss tangent, L ^ J , as follows:
<■//
— \— Lv
ejf
ta n A f
Lv = —
(l-ta/„)
;
1
<-,Q\
(18)
where it is assum ed for simplicity th a t scattering losses are also proportional to
density.
T he com plete radiative transfer problem , including scattering, has been
solved by T sang and Kong (1975) for a half-space m edium w ith random di­
electric fluctuations and some specific forms of the tem p eratu re profile. They
found th a t the principal effect of scattering was to reduce th e norm al emissivity
by up to 20 % relative to the case of no scattering, and m ore significantly, th a t
the m agnitude of the emissivity was independent of the tem p eratu re profile to
w ithin a fraction of a percent. This is an im p o rtan t result, because otherwise,
scattering effects would modify th e day-night brightness contrast. However, this
m odel was designed to study the propagation of microwaves in terrestrial ice,
and is therefore subject to two lim itations in the context of the current prob­
lem. F irst, the tem p eratu re in the model profiles varies by only 5-15%, whereas
tem p eratu re variations in the regoliths of th e Moon and M ercury are as large
as 50% and 75%, respectively. Second, dielectric fluctuations in the model are
58
sm all ( ,$ 5%) and occur w ith a characteristic correlation length of 2 m m . In
contrast, rock fragm ents in the lunar regolith cause ~ 100 % dielectric fluctuations
an d have a broad size distribution.
Keihm (1982) has studied th e effects of scattering by rock fragm ents on the
microwave brightness of the Moon. Using dielectric properties and size distri­
butions th a t were based on Surveyor and Apollo d ata, he found th a t subsurface
scattering reduced the norm al emissivity by factors of only 1-3%.
However,
Keihm chose to isolate scattering effects by assuming an isotherm al regolith, so
it is not certain th a t the emissivity of a lunar-like regolith is independent of the
tem p eratu re profile. Nevertheless, since the effect of subsurface scattering on
the emissivity of a lunar-like regolith is expected to be small, we extrapolate the
result of T sang and Kong by assum ing th a t the source function can be replaced
by the Planck function, S v = B„, thereby ignoring the effects of tem perature
gradients on the scattering phase integral. Basically, this approach allows the
determ ination of relative brightness variations at the expense of overestim ating
the emissivity by an unknow n factor, which is simply absorbed into an arb itrary
scaling factor for the m odel images.
If the electrical param eters vary gradually on scales com parable to the
w avelength 5 and if the source function is given by the Planck function, then
the solution to E quation 13 can be w ritten (Golden 1977):
5 A ctually, th e electrical p a ra m e te rs likely do not vary gradually on th e scale of th e wave­
len g th (see F ig u re 10). N evertheless, we a d o p t E q uation 19 as it sta n d s and consider the
effects of ra p id d ielectric variations separately.
59
oo
p y x ) k v ( x ) ax
= [i-b,w ]
-T .m
o V 1 _ e «l ^ ) “
p
where T B is the brightness tem perature of radiation with polarization
P
emerging
at an angle 6 to the surface norm al (toward the observer). The factor [l —i2P(#)]
is the microwave emissivity, where R P(0) is the surface reflection coefficient for
polarization
P.
T he emissivity will be discussed in more detail below, when we
consider polarization effects.
T he subsurface brightness tem perature, T b(x), is given by:
T b(x)
Tr N
e s p i a l - 1
Tr = ~
K
(20)
where T ( x ) is the physical (kinetic) tem perature of the regolith. At microwave
frequencies T r <C T , so th a t T b(x) ~ T ( x ) . If we ignore shadowing, the lowest
surface tem perature attained in our model is about 75 K during th e night at
high latitudes (but not at the poles, where the large disk of the Sun always peeks
above the horizon). At the highest frequency of these observations (89 GHz) and
a physical tem perature of 75 K, T b differs from T by about 3%.
In order to generate a model I image, the regolith tem perature as a function
of depth and tim e is determ ined for a grid of surface elements in both longitude
and latitude. (L atitude elements alone are insufficient because of M ercury’s nonuniform insolation p attern .) Since the obliquity of M ercury’s orbit is nearly zero,
sym m etry allows us to restrict the calculations to one quarter of the p lan et’s sur­
face (0-180° longitude, 0-90° latitude). This region is divided into a grid of 25
60
longitudes and 8 latitudes, which gives resolutions of 7.2 and 11.25 degrees, re­
spectively. Finer meshes consisting of up to 50 longitudes and 10 latitudes were
tested but did not significantly improve the accuracy of model images at the
resolution of the data. Once T ( x ) is determ ined at each grid point, the bright­
ness tem perature as viewed from E a rth is calculated by integrating Equation 19
through the regolith. The resulting grid of brightness tem peratures is interpo­
lated and m apped into geocentric right ascension and declination coordinates
to form a model image, which is then convolved with a Gaussian beam to the
resolution of the data.
microwave emissivity and polarization
In addition to lowering the norm al emissivity, scattering also modifies the
angular dependences of the emissivity and fractional polarization.
As a first
approxim ation, R P(9) is given by the Fresnel transm ission coefficients (Jackson
1975) for a sharp interface between a semi-infinite, homogeneous m edium and
free space. Assuming th a t the subsurface therm al radiation is unpolarized, the
total emissivity, E ( 9 ) , is given by:
E W
= 1 -
\ r ±(6) - 1 ^ ( 0 )
(2 1 )
while the fractional polarization ( F ) is given by:
F W
-
W ) —
{
}
Deviations in R P(9) from the Fresnel formulas might occur because: ( 1 ) the den­
sity gradient near the surface corresponds to a gradient in the dielectric constant
th a t is expected to occur on a scale th a t is neither small nor large relative to
61
centim eter wavelengths, ( 2 ) subsurface scattering reduces the m agnitude of the
emissivity, (3) scattering at the surface boundary by wavelength-scale structures
partially diffuses the em ergent radiation, and (4) surface slopes th a t are large
com pared to the observing wavelength b u t small com pared to th e im age reso­
lution modify the variation of th e net emissivity w ith projected radius on the
planet. We consider each of these effects in tu rn .
If the dielectric gradient occurs on a scale m uch longer th an the wavelength,
then approxim ate treatm ents can be used (Wait 1981), which am ount to grad­
ually bending the radiation through the dielectric gradient (w ithout reflection
loss) according to Snell’s law, and calculating the Fresnel coefficients using the
dielectric constant at the surface. (This is the approach of Golden [1977], which
we have adopted.) Alternatively, if the dielectric gradient occurs on a scale m uch
shorter th an the wavelength, then it is a good approxim ation to tre a t th e gradient
as a discontinuity, and the Fresnel coefficients are calculated using the dielectric
constant of the homogeneous (com pacted) m edium below the dielectric gradient.
However, th e dielectric gradient near the surface of the Moon (and probably
M ercury) occurs over a few centim eters, which is com parable to th e wavelengths
of these observations. As a result, R P(0) is expected to be interm ediate between
the surface value and the value below the dielectric gradient to an extent th a t
depends on the wavelength.
Subsurface scattering contributes to the total extinction (Equation 13) and
reduces the microwave emissivity, as discussed above. Scattering at th e surface
boundary by wavelength-scale structures partially diffuses the em ergent rad ia­
tion, thus reducing its fractional polarization. The effectiveness of scattering at
a given wavelength depends on the abundance of rock fragm ents, surface stru c­
tures, and inhomogeneities w ith sizes on the order of the wavelength. R ad ar
62
observations of M ercury indicate th a t scattering is im portant at centim eter wave­
lengths. H arm on and O stro (1988) estim ate th a t diffuse scattering accounts for
27% of the to tal ra d a r cross section of M ercury at a wavelength of 13 cm. For
regolith m aterial, scattering should be m ore im portant at shorter wavelengths,
since there are generally m any m ore scattering centers on smaller scales. This is
sup p o rted by microwave ra d a r echoes from M ercury, which exhibit larger diffuse
com ponents at wavelengths near 3 cm th an at longer wavelengths (Clark et al.
1988).
T he emissivity and polarization will also differ from th a t of a sm ooth sur­
face as a result of surface structures on scales m uch larger th an the observing
wavelength. Large scale “slopes” (
10A) m ay be thought of as dividing a given
region into a distribution of facets, each w ith its own tilt relative to the sm oothsurface norm al. Surface slopes have only a small effect near the center of the
projected disk (Golden 1977), b u t near th e limb they increase the microwave
emissivity and decrease the fractional polarization. For viewing angles £
70°,
th e Fresnel coefficients depend sensitively on the viewing angle. Facets near the
limb th a t are tilted tow ard the observer have a much higher emissivity (and pro­
jected area) th a n those tilted away from the observer. Thus, therm al emission
near the limb is dom inated by facets th a t are tilted tow ard the observer. These
facets have a lower fractional polarization th an a sm ooth spherical surface would
have a t the sam e position. Pettengill (1978) has sum m arized ra d a r determ ina­
tions of the rm s surface slopes on Mercury, which are 6 .0 , 8.7, and 9.7 degrees
at wavelengths of 70, 13, and 3.8 cm, respectively. These are very sim ilar to
lu n a r slopes, which continue to increase at shorter wavelengths, w ith a value of
33 degrees at AO.86 cm (Beckm ann and Klem perer 1965).
63
Ill order to m odel variations of the emissivity and fractional polarization over
th e disk, we consider a surface elem ent (i.e., an image pixel) to be com posed of
m any facets, each w ith a size m uch larger th an th e observing wavelength. The
em issivity and polarization of each facet as viewed from E a rth are calculated
from the stan d a rd Fresnel form ulas using an effective dielectric constant,
,
which is expected to differ from the tru e dielectric constant, e n , because of the
effects described above. The emissivity and polarization of the surface elem ent is
th en determ ined by averaging over th e facets. For simplicity, we ignore m u tu al
shadow ing of the facets. Similar models have been utilized by Cuzzi (1974) and
Golden (1977), while a m ore sophisticated approach, which includes th e effects
of shadowing, has been developed by Shin and Kong (1982).
We assum e th a t the facet tilts, «r, relative to th e m ean surface norm al have
a G aussian distribution:
p(ff) =
\ / f '8 -1 e x p ( “ & )
w here s is the rm s surface slope.
(23)
(T he norm alization of this distribution is
approxim ately correct for s £ 20 degrees.) The azim uthal orientation, /?, of each
tilted facet is taken to be random : p{/3) = (27r)- 1 . For an element w ith a m ean
surface norm al at an angle 6 to the line of sight, the emissivity is given by the
m ean of the emissivities of the constituent facets, each weighted by its projected
area.
For a large num ber of facets, this operation can be represented by an
64
integral:
2
f d(3 f dap(/3)p(<r)E(8f ) cos+(9f )
(E(8)) = 0 J
f ----------------------------------
(24)
J d p J d a p(P) p(a) cos+(8f )
o
o
where 8f(a,P',8) is the angle between an individual facet’s surface norm al and
the line of sight. T he “clipped” cosine function, cos +( 6 f ) , is zero (instead of
negative) when Of > 90°, in which case the facet is not visible from E arth .
T he fractional polarization of a surface element ( ( jF(0 ))) is determ ined by
averaging the fractional polarizations of th e constituent facets, each weighted by
its emissivity and projected area, and projected onto the m ean plane of incidence.
In addition, we include a correction for p artial diffusion of the em ergent radiation
a t the surface boundary by wavelength-scale structure.
^
7r
2ir
2
I dP I d a p ( P ) p { a ) F { 0 f ) E ( 0 f ) cos ( 0f ) |cos(<£>)|
<m >
= d - * ) ^ -------------------- ^
-----------------------------
(25)
where S is the fraction of the emergent radiation th a t is diffused, and <p is the
angle between the plane of incidence for an individual facet and the m ean plane
of incidence. Note th a t this accounts for p artial cancellation of the polarization
due to the random azim uthal orientations of the facets, and th a t the electric
vector of the polarized component rem ains radially oriented on the projected
disk. Once the fractional polarization is determ ined as a function of 0, a model
F im age is constructed in the same m anner as the sm ooth-surface polarization
m odel described in section III.
65
Table IV
Specific Heat Polynomial Coefficients*
coefficient
value
Co
-5 .5 3 9 X 10 ~ 3
Cl
5.084 X 10 “ 4
C2
3.587 x 10 ~ 6
C3
-1 .7 6 1 x 10 “ 8
c4
2.308 x 10 - 11
C5
2.029 x 10 " 1
C6
3.830 x 10 ~ 2
* from Golden (1977)
66
F ig u re C a p tio n s
F ig u re 9
The heavy lines are the density as a function of depth for three models of M er­
cury’s regolith (a, b, and c). Models b and c coincide above 2 cm and below
50 cm. T he dashed curves represent the range of lunar densities as inferred from
astro n au t bootprints, vehicle tracks, boulder tracks, and core tube samples (Olhoeft and Strangw ay 1975). The shaded boxes num bered 1 and 2 show the range
of densities inferred from in situ m easurem ents of the surface cooling rate after
sunset (K eihm and Langseth 1973). Boxes 3 and 4 show the range of densities
inferred from th e Apollo 15 and 17 core samples (Langseth et al. 1976).
F ig u r e 10
T he heavy lines show the total therm al conductivity (K 350, left panel) and the
ratio of radiative to phonon conductivity ( x , right panel) as a function of depth
for three models of M ercury’s regolith (a, b, and c) at a reference tem perature
of 350 K. Boxes 1 and 5 encompass laboratory m easurem ents of returned lunar
samples (C rem ers 1974). Shaded boxes 1 and 2 show the range of therm al con­
ductivities inferred from in situ m easurem ents of the surface cooling ra te after
sunset (K eihm and Langseth 1973). Shaded boxes 3 and 4 show the range of
therm al conductivities m easured by the Apollo 15 and 17 heat probes (Langseth
et al. 1976). T he much lower therm al conductivities of returned lunar samples
(dashed box 5) could result from disruption of the packing arrangem ent a n d /o r
destruction of sintered grain contacts during collection and handling.
F ig u re 11
T he surface therm al inertia of M ercury’s regolith at a tem perature of 350 K versus
Herm ographic longitude (bottom scale) and local tim e (top scale), as determ ined
67
by fitting M ariner 10 Infrared Radiom eter m easurem ents of M ercury’s night-side
surface tem p eratu re w ith the three models shown in Figures 9 and 10 . Models a
and b require system atic increases of 50% and 20%, respectively, in th e surface
therm al inertia over 180° of longitude, which we consider to be unlikely (see
tex t). T he surface therm al inertia of model c exhibits the smallest system atic
variation ( 10 %) as a function of local time.
68
F ig u r e 9
-
(cm)
1
-
Depth
10
100
-
1.5
1
n
D en sity (g c m
69
)
F ig u re 10
h -H -H
O
O
(u io ) i^ d a a
70
F ig u re 11
o
CD
<d\
o
o
CO
o
z>
C\2
o
o
o
CO
o
C\2
CO
CO
o
CO
LO
CO
CO
LO
cvi
o
71
(degrees)
C\2
Longitude
Local Time
(Mercury
h o u rs)
CO
V . A n alysis
In this section, we determ ine the electrical properties of M ercury’s regolith
by com paring our observations with model images generated with the therm al,
•
•
■
eSS
radiative tran sp o rt, and facet models described above. First, we constrain eR ,
s , and 8 at each wavelength by fitting the fractional polarization images w ith the
facet model. These param eters constrain the variation of the microwave emissiv­
ity over the projected disk of Mercury. Next, we determ ine the effective specific
e/f
loss tangent ( L u ) a t each wavelength by modeling th e relative diurnal brightness
variations across M ercury’s disk. T he opacities we derive for M ercury’s regolith
are com pared with lunar microwave opacities, which have been determ ined by
rem ote microwave techniques and by laboratory m easurem ents of returned lunar
samples. Finally, we present the residual images obtained by subtracting the
best-fit models from the data. These images are not dom inated by noise bu t
ra th e r exhibit the therm al effects of shadowing due to topography.
dielectric constant and surface slopes
As a limiting case, we first model the fractional polarization under the as­
sum ption th a t there is no diffuse scattering at th e surface boundary (5 = 0 ).
Figure 12 shows the results of adjusting the effective dielectric constant (panel
A) and the rm s surface slope (panel B) to best fit the polarization d ata. Note
th a t the fractional polarization within ~ 3 ” of the disk center depends almost enej f
tirely on eR , while the fractional polarization near the limb depends sensitively
on s. This makes it possible to determ ine these param eters simultaneously. The
effective dielectric constant and the rm s surface slope are plotted as a function
of the observing wavelength in Figures 13 and 14, respectively. The effective
dielectric constant clearly increases as the wavelength increases from 2 to 6 cm.
72
The rm s slope appears to decrease over this range, as might be expected, bu t
the d a ta are also consistent w ith a constant rm s slope.
An increase in the effective dielectric constant with wavelength could be
caused by an increase in density w ith depth th a t is gradual relative to the shortest
wavelengths and sharpens relative to the longer wavelengths. For com parison,
we provide a density scale in Figure 13 th a t is derived from the Rayleigh mixing
form ula (E quation 17) assuming e0 = 6.5, and po — 2.8 g cm - 3 , which are
typical of lunar rocks and basaltic terrestrial rocks. If the density gradient alone
ef f
determ ines the variation of eR w ith wavelength, then a value of 1.7 at A = 2 cm
places an upper limit to the average surface density of about 0.8 g cm - 3 , which
corresponds to a porosity of m ore th an 70% and is among the lowest surface
densities estim ated for the Moon (C arrier et al. 1973). Similarly, a value of 2.3
at A = 18 cm provides a lower limit to the m axim um com pacted density of 1.2
g cm - 3 . T he m axim um densities of lunar fines samples are in the range 1 .6 - 2.1
g cm - 3 .
A lthough such low densities for M ercury’s regolith are not excluded, the
values for the effective dielectric constant th a t we derive from the therm al po­
larization d a ta are significantly smaller th an those derived by rad ar techniques
(see Figure 13, open symbols). Furtherm ore, the radar d ata show no significant
variation in the effective dielectric constant from 3.8 to 70 cm. Similar differences
between passive therm al and rad ar determ inations of the effective dielectric con­
stan t of the Moon have been noted by Hagfors (1970). These discrepancies likely
e/ /
result from the fact th a t the rad ar m ethod of determining eR and s is based
on the coherently reflected echo component (Pettengill 1978), whereas th e pas­
sive therm al polarization m ethod is significantly affected by diffuse scattering at
the surface boundary, which reduces the polarization of the em ergent radiation,
73
thus biasing the effective dielectric constant to smaller values. Scattering should
be m ore im p o rtan t at short wavelengths, which could account for the observed
ef f
variation of eR with wavelength.
To investigate the im portance of diffuse scattering at the surface, we adopt
the ra d a r values of the effective dielectric constant (2.7) and rm s surface slope (9.7
degrees), and refit the observed polarization at each wavelength by adjusting the
diffuse fraction of the em ergent radiation (5). Figure 15 shows the results of this
procedure. Note th a t the best fits (solid lines) are almost indistinguishable from
those in Figure 12. We find th a t scattering m ust diffuse ~20% of the em ergent
radiation at A18.0 cm, 30% at A6.2 cm, 45% at A3.6 cm , and 60% at A2.0 cm
(Figure 16), which indicates th a t surface structures become progressively m ore
abundant from decim eter to centim eter scales6. This tren d is in accord with
ra d a r observations of M ercury and the Moon, which have similar ra d a r scattering
properties. Taking ra d a r observations of M ercury and the Moon together, the
diffuse portion of the returned echo power increases gradually from 20% to 27%
as the wavelength decreases from 68 to 13 cm (H arm on and O stro 1988), then
increases sharply to about 85% at A0.86 cm (Lynn et al. 1964).
ef f
Together, eR , s, and 6 constrain the angular dependence of the microwave
emissivity, which we apply to the model I images in place of the Fresnel form ulas.
In Figure
17, the emissivity and polarization as derived from the facet model
(solid curves) are com pared w ith the Fresnel formulas (dashed curves). Since we
have ignored the effects of subsurface scattering and the rapid dielectric gradient
near the surface, the facet model provides only an upper lim it to th e actual
emissivity. To w ithin a m ultiplicative constant, the rough surface emissivity is
6 T h u s, th e q uarter-w ave p lates were probably unnecessary for our AO.3 cm observations.
74
nearly identical to the Fresnel emissivity except in the outer 5% (~ 0 .2 ” ) of the
p la n et’s projected radius. T hus, only the outerm ost resolution elem ent of the
m odel I images differs significantly from a sm ooth surface calculation.
microwave opacity
ej f
T he effective specific loss tangent ( L v ) is constrained at each wavelength
by modeling the day-night brightness contrast. Basically, the tem p eratu re in­
creases w ith depth on the night side and decreases with depth on the day side.
eff
As L v increases, shallower depths are probed, and the day-night brightness
contrast increases. This effect is m ore pronounced at short wavelengths, which
probe shallower layers where tem p eratu re gradients are larger. The procedure
for constraining the effective specific loss tangent consists of generating a series
of m odel images th a t differ only in the value of L v . At a wavelength of 0.3
cm , we calculate the spatial Fourier transform of each model image and generate
synthetic visibility curves for direct com parison w ith the interferom eter d ata. At
the centim eter wavelengths, the models are subtracted directly from the observed
images by a procedure th a t allows for a scaling factor and slight positional offsets
an d expansions.
T he BIMA interferom eter d a ta are com pared with synthetic visibility curves
•
•
•
•
eff
«
in Figures 18 and 19. The visibility on 26 Jan u ary 1988 is sensitive to L v during
the first half of the ru n (hour angle < 0 ), bu t not during th e second half. This
allows us to normalize the model curves at positive hour angles and then fit the
d a ta at negative hour angles by adjusting the effective specific loss tan g en t— a
procedure th a t is equivalent to modeling relative brightness variations across the
disk. Evenly spaced values for L JS of 0.0026, 0.0052, 0.0078, 0.0104, and 0.0130
were chosen to bracket the d a ta at negative hour angles. The d a ta are best fit
75
w ith L v
fn 0.0078, with an uncertainty of roughly ±0.0026. These values are
shown by the middle three curves. A similar procedure is utilized for the M arch
1991 d a ta set, where the model curves are normalized at /? < 0.4. T he d a ta are
generally consistent w ith L ^ f = 0.0078 ± 0.0026 (middle three curves).
A lthough acceptable fits to the visibility d a ta can be obtained (given the
m easurem ent uncertainties), no single curve has an optim al shape for fitting the
entire d a ta sets. Small positional offsets a n d /o r expansions, such as those con­
sidered below for the centim eter images, do not significantly affect the shape of
the model visibility am plitude, and thus cannot improve the fits. System atic
differences between the model curves and the d ata might be caused by shadow­
ing effects, which decrease the brightness along the sunlit side of the morning
term inator and at the poles (see below). However, based on the results at cen­
tim eter wavelengths, we believe th a t shadowing effects do not significantly bias
the determ ination of the effective specific loss tangent at A0.3 cm.
At the centim eter wavelengths we subtract model images directly from the
observed images with a least-squares procedure th a t simultaneously adjusts a
scale factor, positional offsets (in right ascension and declination), and an expan­
sion factor in order to minimize the residuals. The scale factor absorbs uncertain­
ties in the absolute flux scale and in the m agnitude of the microwave emissivity,
while the positional offsets correct ephemeris errors. Typical values for th e scale
correction and offset are 5% and 0.1” , respectively. The offsets are w ithin the
uncertainties of the JP L FK 5 ephemeris (Standish 1992), which has been used
for these observations.
In contrast, the expansion corrections (typically 1.005) are much larger th a n
can be accounted for by ephemeris uncertainties. There rem ain three possible
76
causes for the expansions. F irst, an expansion of 0.5% am ounts to only ~ 2 0 % of a
0 . 1 ” im age pixel and could be a consequence of “graininess” in the observed and
m odel images. Second, atm ospheric fluctuations can cause errors in removing
the sidelobe response from the interferom eter images, as discussed above. These
errors can take the form of “flames” extending from th e limb of th e planet (see
Figures 5 [a] and 6 [a]), which bias the model subtraction procedure. Finally, it
is possible th a t the expansions are compensating for errors in the model emis­
sivity near the limb. In any case, such errors are circularly sym m etric about the
disk center and confined to the outerm ost resolution element of the images, and
are thus only weakly coupled to the determ ination of the day-night brightness
contrast.
T he results of model subtraction at each wavelength are shown in Figure
eff
20 . W hen L v
is too small, the model brightness is too low on the day side
and too high on the night side. This situation results in positive residuals on
the day side and negative residuals on the night side. The opposite effect occurs
ef f
when L v is too large. T he top right panel at each wavelength shows the residual
image w ith the value of L 'l’ th a t best reproduces the observed day-night contrast.
The diurnal layer is sufficiently transparent at A18.0 cm and A20.5 cm th a t the
therm al images are insensitive to
over a very broad range. It is not possible
eff
to place useful constraints on L v at these wavelengths.
T he contour intervals in Figure 20 were chosen to be approxim ately three
times the rm s noise at each wavelength (except at A18.0 cm and A20.5 cm, where
the contour interval is only twice the rm s). Recall th a t the noise distributions
at Al.3 cm, A2.0 cm, and A3.6 cm have non-Gaussian tails, so it is probably
inappropriate to interpret unresolved features at the “three-sigm a” level in term s
of anom alous therm ophysical a n d /o r electrical properties of the regolith. We
77
note th a t unresolved three-sigm a features on the A2.0 cm residual image do not
in general have corresponding features on the Al.3 cm image, even though such
features should be more prom inent at shorter wavelengths. The best-fit residual
images do, however, exhibit features th a t are composed of several resolution
elements and occur at m ore th an one wavelength. These will be discussed in
m ore detail below.
The derived values of
are plotted as a function of the observing fre­
quency in Figure 2 1 . The error bars do not correspond to formal statistical
•
ef f
uncertainties, b u t rath er indicate the deviation from the best-fit value of L v at
which a significant day-night m ism atch becomes apparent from inspection of the
visibility and residual contour plots. For reference, note th a t the model curves in
Figures 18 and 19, and the residual images in Figure 20 are presented for values
of L ^ f at the best-fit value, and at ± 1 and ± 2 times the error bars depicted in
Figure 21. (The error bars at A18.0 cm and A20.5 cm would exceed the bounds
of the plot.)
Laboratory m easurem ents of lunar fines and terrestrial basalts are presented
in Figure 21 for comparison with the values we derive for M ercury’s regolith. An
average of the 450 MHz loss tangents of 21 lunar samples (Gold et al. 1976) is
shown by the triangle. The error bars indicate dispersion in the m easurem ents
th a t is prim arily caused by mineralogical differences among the samples. Each
dashed line connects one of the 450 MHz measurem ents with m easurem ents of the
same sample at 9 GHz (B assett and Shackelford 1972; diamond symbols). These
dashed lines may indicate frequency dependence of the loss tangent, although
no m easurem ents exist at interm ediate frequencies, and only two samples have
been m easured at 9 GHz. The star symbols connected by a solid line are farinfrared m easurem ents of an individual lunar sample (Clegg et al. 1972). The
78
slope of this line is well established by several m easurem ents at interm ediate
frequencies. Finally, the E a rth symbols ( 0 ) represent averages of 7 terrestrial
basalts at 450 MHz and 35 GHz (Cam pbell and Ulrichs 1969). If we ignore
the lower dashed line, which represents an unusually small loss tangent, then
the rem aining m easurem ents appear to define the average lunar loss tangent
spectrum from 450 MHz to 300 GHz.
ef f
T he derived values of L v for M ercury are apparently low by factors of 2-3
com pared with expected lunar values, bu t the actual difference is likely to be
even larger. G ary and Keihm (1978) pointed out th a t laboratory m easurem ents
of the effective loss tangents of lunar samples underestim ate scattering losses
at wavelengths longward of ~ 1 m m because the samples were sieved prior to
m easurem ent in order to remove particles larger th an 1 m m in size. Scattering
losses in the lunar regolith, with the naturally occurring distribution of rock frag­
m ents, are expected to be significant at centim eter wavelengths. Keihm (1982)
has estim ated th a t scattering losses account for 10-65% of
at wavelengths
ranging from 3 to 13 cm, depending on the abundance and size distribution of
rock fragm ents. Thus, the effective specific loss tangent of the lunar regolith at
centim eter wavelengths could be more th an a factor of two larger th an shown in
Figure 21.
The comparatively small value of
for M ercury’s regolith could be due
to mineralogical differences between the surfaces of M ercury and the Moon. The
principle loss m echanism at centim eter wavelengths is believed to be transcon­
duction (Linsky 1966), which is the m otion of trapped im purity ions in response
to applied electric fields. At millimeter wavelengths, the long-wavelength tails of
far-infrared resonances m ay also contribute to the absorption (Clegg et al. 1972).
Transconduction losses depend sensitively on the abundance of im purity ions.
79
Linsky noted th a t the loss tangents of several terrestrial m aterials increase by
1-2 orders of m agnitude when the abundance of im purity ions increases by only
a few percent.
Gold et al. (1976) found th a t the 450 MHz loss tangents of lunar fines in­
crease w ith the abundance of ilm enite, which is the m ost common titanium bearing m ineral on the Moon (ideal formula: FeTiOs). This correlation is re­
produced in Figure 22, where th e various symbols indicate the Apollo mission
from which each sample was collected (see caption). Not included in the figure
are laboratory m easurem ents of lunar rocks, which follow the same tren d b u t
span larger ranges in ilm enite content and loss tangent th a n do th e fines. One
of these (sam ple 60025) has an ilmenite content of 0.5% and a 450 MHz specific
loss tangent of 0.0006.
Ilm enite is also opaque at optical wavelengths and is largely responsible
for th e albedo differences between the lunar m aria and highlands (Pieters 1978,
W ilhelms 1984). One would therefore expect the microwave opacity of the m aria
to be larger th a n th a t of the highlands. Evidence for this trend can be seen
in Figure 22, where highland and m are samples are shown with open and filled
sym bols, respectively.
Although the Apollo samples were collected from only
a tiny fraction of the M oon’s surface, ground-based microwave images of the
M oon at wavelengths of 0.3 and 3.55 cm provide evidence th a t loss tangent
differences between m aria and highlands exist on a global scale.
G ary et al.
(1965) and Keihm and G ary (1979) found a system atic p a tte rn of microwave
brightness anomalies on images of the full Moon at wavelengths of 0.3 and 3.55
cm. M aria were observed to be brighter th an highlands by 2.6 K at A0.3 cm and
2.8 K at A3.55 cm. Since the physical tem p eratu re decreases w ith depth on the
80
sunlit hem isphere, these anomalies can be attrib u te d at least in p a rt to a higher
microwave opacity in the m aria (Keihm and G ary 1979).
M ercury’s surface is system atically brighter th an the lunar surface and ex­
hibits m uch smaller albedo contrasts between its sm ooth plains and cratered
terrain (S trom 1984, Spudis and Guest 1988). From M ariner 10 photom etric
observations, Hapke et al. (1975) found th a t M ercury’s sm ooth plains (prim arily
in the vicinity of the Caloris basin) range from brighter th a n m ost of the lu­
n a r m aria to as bright as the lunar highlands, while M ercury’s intercrater plains
(highlands) are 60% brighter th an the lunar highlands. In addition, fresh-rayed
craters on M ercury are twice as bright and spectrally bluer th an their lunar coun­
te rp a rts. All of these observations can be explained if M ercury’s surface is low
in Fe and Ti relative to the Moon (Hapke et al. 1975). Since the abundances of
Ti and ilm enite are strongly correlated, one would then expect the microwave
opacity of M ercury’s regolith to be com paratively low. O ur m easurem ents of the
effective specific loss tangent of M ercury’s regolith show th a t this is in fact the
case.
A low titan iu m abundance in M ercury’s regolith may indicate th a t, unlike
the lu n ar m aria, M ercury’s sm ooth and intercrater plains are not volcanic in
origin, b u t rath er arise from the ejecta and im pact melt from basin-forming im ­
pacts (W ilhelms 1976, Oberbeck et al. 1977). However, several lines of evidence
indicate th a t M ercury’s sm ooth and intercrater plains m ust be volcanic in origin
(S trom 1984): ( 1 ) There appear to be an insufficient num ber of im pact basins
to account for the plains m aterial. (2) Age differences exist between some basins
an d nearby plains. (3) M ercury’s large average density, strong m agnetic field,
and global p a tte rn of lobate scarps strongly suggest th a t M ercury underw ent a
81
period of differentiation, large-scale melting of the m antle, and global expan­
sion, which should have resulted in extensive volcanism. Strom (1984) suggested
th a t th e low Fe and Ti abundances on the surface of M ercury could result from
differentiation of Fe and Ti to deeper levels th an on the Moon, as a result of
m ore extensive lithospheric melting and a larger surface gravity. Volcanic source
regions on M ercury would then be depleted in Fe and Ti relative to the lunar
case.
topography and shadowing
T he best-fit residual images from Al.3 cm to A6.2 cm are not dom inated
by noise, b u t ra th e r exhibit a distinctive p a tte rn of negative residuals in polar
regions (which sometimes extend along the term inator), and positive residuals in
equatorial regions. To examine the residual p a tte rn in more detail, we assum e
th a t topography causes a negligible microwave brightness anomaly, relative to a
level surface, in equatorial regions on the night hemisphere. (This will be ju sti­
fied below.) T he best-fit model images are rescaled to fit the observed night-side
equatorial brightness while holding the previously determ ined positional offsets
and expansion factor constant. Since the microwave opacity has been optimized
to reproduce the observed day-night contrast, this procedure also provides agree­
m ent on m uch of the day side.
T he rescaled residual images are shown in Figure 23. The residual p a tte rn
now consists of a therm al depression along the sunlit side of th e m orning te r­
m inator, which is weakest at the equator and strengthens toward the poles. At
higher latitudes the depressions extend into th e night hem isphere, a tren d th a t is
especially prom inent at A3.6 cm and A6.2 cm. The overall strengths of the th e r­
m al depressions depend on the observing wavelength and on the aspect of the
82
planet. The depressions weaken from Al.3 cm to A2.0 cm and from A3.6 cm to
A6.2 cm, b u t the strength of the equatorial depression at A3.6 cm is com parable
th a t at Al.3 cm. All of these trends can be understood in term s of shadowing,
once M ercury’s unique insolation p a tte rn is taken into account.
One feature th a t is difficult to explain by shadowing effects is the apparent
longitudinal shift of the therm al depression at A2.0 cm relative to th e correspond­
ing depression at Al.3 cm. (The therm al depressions at A3.6 cm and A6.2 cm
nearly coincide.)
However, the longitude of a depression m axim um depends
som ew hat on the value of L ™ . If we choose a slightly smaller value of if™ (but
still w ithin the uncertainties), the day residuals become slightly positive and the
night residuals become slightly negative. In this way, it is possible to shift the
therm al depression at A2.0 cm to coincide with the Al.3 cm depression.
In th e following, we dem onstrate th a t this p a tte rn is likely caused by shadow­
ing, which has not been accounted for in our level-surface model. Thermophysical
models th a t explicitly include the effects of surface topography have been devel­
oped by B uhl et al. (1968a, 1968b), by Spencer (1990, and references therein),
and m ost recently by Paige et al. (1992). Using these studies as a guide, we
approxim ate shadowing in our level-surface model by artificially inhibiting solar
illum ination for a fraction of the surface. In this way, we are able to account for
the general morphology and m agnitude of the observed microwave residuals.
Shadowing causes both “transient” and “perm anent” effects. The transient
effect occurs predom inantly in equatorial regions where topographic lows (e.g.,
crater floors) and hillsides are alternately in shadow and sunlight as the day
progresses. Buhl et al. (1968a, 1968b) have studied the heating and cooling of
equatorial, splierical-section craters on the Moon. During the day, vei'y large
83
DAY
surface tem perature contrasts (A Ta
~ 150 K) can be m aintained inside the
crater, with sunlit regions w arm er and shadowed regions cooler than nearby level
areas. These tem perature contrasts are greatest ju st after sunrise and ju st before
sunset. At night, craters cool less efficiently th an level regions because the crater
interior does not radiate into 2-n steradians, bu t instead radiates some energy
back to itself. T he surface tem perature contrast between the crater interior and
nearby level areas depends on the crater’s depth-to-diam eter ratio, bu t is much
N IG H T
smaller th a n the daytim e contrast (A T s
~ 10 K). The anomalous cooling
of lunar craters during eclipses has been observed at infrared wavelengths by
Shorthill and Saari (1965). Observed eclipse anomalies are typically ~10-30 K,
b u t therm al anomalies are expected to be smaller during the lunar night because
of the m ore gradual decrease in solar illumination and the longer cooling tim e
(B uhl et al. 1968b).
These effects should be even larger on Mercury, where the equatorial sur­
face tem perature extrem es (~100-700 K) are m uch larger th an on the Moon
(~100-400 K ). To estim ate the m agnitude of the shadowing effect on Mercury,
we consider two reference longitudes at the equator: 165° for the observations of
11 A ugust 1990, and 265° for the observations of 13-14 April 1990. These two
longitudes are near the m idpoints of the equatorial therm al depressions observed
at Al.3 cm and A3.6 cm, respectively.
The insolation at these two longitudes is very different because of M ercury’s
non-uniform insolation p attern , which is responsible for the hot-cold longitude
p a tte rn in the I images (Figure 5), as discussed above. The total insolation at
165° is m ore th an twice th a t at 265°, as illustrated in Figure 24. The inflection
84
from 1 to 3 Phr (M ercury hours7) after sunrise at 265° occurs as the Sun executes
a retrograde loop a t perihelion. As viewed from M ercury’s surface, th e m otion
of th e Sun is prim arily radial during these loops. T he Sun is directly overhead
4 ??hr after sunrise at 165°, b u t the insolation peaks during th e retrograde loop
centered a t 6.7 (Jhr. T he shaded regions of Figure 24 indicate th e insolation th a t
is received by a level surface while th e Sun is w ithin 15° of th e horizon. This
accounts for a m uch larger fraction of th e to ta l insolation a t 265° (16%) th a n at
165° (1%).
T he elevation of the Sun during th e m orning, as viewed from each of the
reference longitudes, is shown in Figure 25. D uring th e observations of 11 A ugust
1990, th e Sun h ad been above th e horizon for 0.6 (?hr at a longitude of 165°. In
co n trast, during th e observations of 13 April 1990, th e Sun had been above the
horizon for 3.4 (?hr at a longitude of 265°. D uring m uch of this tim e, the Sun
rem ained alm ost motionless at an elevation of only 5°.
T he microwave brightness anomalies in equatorial regions are expected to
be largest n ear th e sunlit side of the m orning (or evening) term in ato r, where
te m p e ratu re contrasts w ithin craters are largest and where th e fraction of the
surface area in shadow (as viewed from E a rth ) is also largest. To estim ate the
microwave brightness contrast between sunlit and shaded regions at each refer­
ence longitude, we solve twice for the tem p eratu re profile. In th e first calculation,
we utilize th e insolation expected for a level surface, as we have done u p to this
point, to provide an estim ate of the tem p eratu re profile in sunlit regions. The
second calculation proceeds exactly as the first until sunrise of the final ite ra ­
tion, when we artificially tu rn off th e insolation and allow th erm al diffusion to
7 O n e M ercu ry h o u r is equivalent to one d iu rn a l perio d (176 days) divided by 24, w hich is
slightly longer th a n a week.
85
proceed. This provides an estim ate of the tem p eratu re profile in shaded regions.
B oth of these profiles are likely cooler th a n the actual profiles, since we have ig­
nored scattering an d reradiation w ithin craters, which heats shaded regions, and
we have not accounted for enhanced insolation of slopes th a t are tilted tow ard
th e sun. Nevertheless, we adopt these profiles for th e purpose of m aking rough
estim ates.
T he “sunlight” and “shade” tem p eratu re profiles are shown in Figures 26
and 27. T he horizontal axes are given in units of the optical dep th ( r ) at each
of th e observing wavelengths:
t
( x , A)
=
x
—
PK
=
27IV ^ " ( p L u J ) x
-------------- -----—
A
(26)
At microwave frequencies, the depth ( x ) at which the optical depth reaches
unity is often referred to as the electrical skin depth. Using the effective specific
loss tangents derived above and the physical param eters of the com pacted region
(p = 1.8 g cm - 3 , eR = 3), the electrical skin depths are approxim ately 17, 35,
70, an d 120 cm at wavelengths of 1.3, 2.0, 3.6, and 6.2 cm, respectively.
T he brightness anomalies caused by shadowing are expected to be weaker at
A2.0 cm th a n at Al.3 cm , because the layer containing the shade-sun tem p eratu re
co n trast is m ore tran sp aren t at the longer wavelength (Figure 26). If we integrate
th e radiative transfer equation with the tem p eratu re profiles in Figure 26, the
difference in brightness tem peratu re between shaded and sunlit regions is —17 K
a t Al.3 cm and —8 K at A2.0 cm. These brightness contrasts can account for the
observed anomalies (roughly —7 K and —3 K, respectively) if ~40% of the surface
is in shadow. For a solar elevation of 15°, this corresponds to a crater depthto-diam eter ratio of ~ 0 . 1 , if craters cover 100 % of the surface. For com parison,
86
simple craters on M ercury (diam eter ;$ 10 km) are observed to be isom etric
w ith a depth-to-diam eter ratio, including the crater rim , of 0.25 (Pike 1988).
Given this trend, one m ight expect the therm al depressions at A3.6 cm and
A6.2 cm to be even weaker. However, this is not the case because of the very
different insolation histories at 165° and 265° (Figures 24 and 25).
At 165°
the tem p eratu re contrast between shaded and sunlit regions has p enetrated to
a depth of 11 cm during the 0.6 $ h r th a t th e Sun has been above the horizon
(Figure 26). B ut at 265°, the Sun has been above the horizon for 3.4 ? h r, so
th a t inform ation about sunrise has p en etrated to a depth of
(Figure 27).
= 26 cm
Since the shade-sun tem p eratu re contrast affects a much larger
volume of the regolith at 265°, the shadowing effects at A3.6 cm and A6.2 cm
are actually com parable to those a t Al.3 cm and A2.0 cm, despite the fact th a t
the regolith is m ore transparen t at the longer wavelengths. As before, if we
integrate the radiative transfer equation w ith the tem perature profiles in Figure
27, the difference in brightness tem p eratu re between shaded and sunlit regions
is —14 K at A3.6 cm and —8 K at A6.2 cm. T he strengths of the observed
equatorial anomalies are roughly —10 K at b o th wavelengths. At A3.6 cm the
m odel brightness contrast can account for the strength of the observed anom aly
if ~70% of the surface is in shadow. For a solar elevation of 8.5°, this corresponds
to a crater depth-to-diam eter ratio of ~ 0 . 1 , if craters cover 100 % of the surface.
This is sim ilar to the values obtained at Al.3 cm and A2.0 cm. However, the m odel
brightness contrast at A6.2 cm is somewhat sm aller th an the observed anomaly,
which likely reflects the uncertainties in the m odel subtraction procedure and
the crudeness in our estim ation of the shadowing effect.
In equatorial regions of the night hem isphere, tem perature anomalies caused
by topography are expected to be m uch smaller. W ith a crater geom etry th a t
87
approxim ates simple craters on the Moon (and M ercury), Buhl et al. (1965b)
showed th a t the surfaces of lunar craters should be only ~ 5 K w arm er th an sur­
rounding level areas, and th a t these craters cool in such a way as to mimic a level
region w ith an enhanced therm al inertia. Since our therm al model has been cal­
ibrated to reproduce m easurem ents of M ercury’s night side surface tem perature,
we have, at least to a first approxim ation, already com pensated for the effect
of craters on the night side. In contrast, daytim e surface tem peratures do not
depend on the therm al inertia because M ercury is a slow rotator. Therefore, the
day-night contrast in the model corresponds to th a t between a cratered surface
on the night side and a level surface on the day side. For this reason, we have
chosen to normalize the best-fit model images to equatorial regions on the night
hem isphere of the observed images.
The “perm anent” shadowing effect occurs at higher latitudes, where shad­
ows persist throughout the day. As a result, these regions receive less sunlight per
unit surface area than a level surface would at the sam e latitude. This “isolation
deficit” gets larger with increasing latitude and results in negative residuals on
b oth the day and night hemispheres. Since the obliquity of M ercury’s orbit is
nearly zero, it is possible for topographic lows very close to the poles to receive
no insolation at all. Such regions receive heat only by radiation from surrounding
hills and by conduction from below, and can therefore achieve very cold tem per­
atures (Paige et al. 1992). Anomalous rad ar echoes have been observed from the
polar regions of M ercury (H arm on and Slade 1992), which could be caused by
w ater ice in perm anently shadowed areas (Slade et al. 1992).
The perm anent shadowing effect is also expected to vary with longitude
because of M ercury’s non-uniform insolation p attern .
Much depends on the
elevation of the Sun as it executes its retrograde loop at perihelion.
88
In the
vicinity of cold longitudes, the retrograde loop occurs when the Sun is close to
the horizon, whereas in the vicinity of hot longitudes the retrograde loop occurs
when the Sun is near its highest elevation. Consequently, long shadows persist
for a greater fraction of the day at cold longitudes, resulting in a larger insolation
deficit (see Figure 24). This effect accounts for the much stronger polar therm al
depressions observed at A3.6 cm and A6.2 cm com pared w ith those at Al.3 cm
and A2.0 cm.
89
F ig u re C a p tio n s
F ig u re 12
T he d a ta points and the horizontal and vertical dashed lines are as in Figure 8 .
The curves represent the fractional polarization expected from a rough surface
for which there is no diffuse scattering at the surface boundary (see text). The
solid curves represent the best-fit values of the effective dielectric constant and
the rm s surface slope. The dot-dash curves show the effect of varying the effective
dielectric constant (panel A) or the rm s surface slope (panel B) by the am ounts
listed below. For wavelengths of 2.0, 3.6, 6.2, and 18.0 cm, the effective dielectric
constants are: 1.70 ± 0.08, 2.00 ± 0.15, 2.20 ± 0.20, and 2.30 ± 0.40, respectively.
Likewise, the rm s surface slopes are: 15 ± 5, 13 ± 6 , 10 ± 10 , and < 30 degrees.
F ig u re 13
The solid symbols show the effective dielectric constant (left scale) as a function
of observing wavelength based on a rough surface model in which there is no
diffuse scattering at the surface boundary. The points and error bars are based
on th e curves shown in Figure 12. T he open symbols show the effective dielectric
constant th a t is derived from rad ar observations. Assuming a solid density of
2.8 g cm -3 and a solid dielectric constant of 6.5, the effective dielectric constant
can be converted into an effective density (right scale) using the Rayleigh mixing
form ula (E quation 17).
F ig u re 14
The solid symbols show the rm s surface slope as a function of observing wave­
length based on a rough surface model in which there is no diffuse scattering at
the surface boundary. The points and error bars are based on the curves shown
90
in Figure 12. The open symbols show the rm s surface slopes th a t are derived
from ra d a r observations of M ercury (□) and the Moon (A ).
F ig u re 15
The d a ta points and the horizontal and vertical dashed lines are as in Figure 8 .
T he curves represent the fractional polarization expected from a rough surface
w ith diffuse scattering a t the surface boundary. The effective dielectric constant
and the rm s surface slope are based on ra d a r observations, while the diffuse
fraction of the em ergent radiation is varied in order to best fit the d ata. The
solid curves represent the best fits, and the dot-dash curves show the effects of
varying th e diffuse fraction. For wavelengths of 2 .0 , 3.6, 6.2, and 18.0 cm, the
diffuse fractions are: 60 ± 5, 45 ± 7, 30 ± 10, and 20 ± 20 percent, respectively.
F ig u re 16
T he symbols show the fraction of the em ergent radiation th a t is diffusely scat­
tered at the surface boundary as a function of observing wavelength. The points
and error bars are based on the curves shown in Figure 15. The open symbols
show the diffuse fractions of returned rad ar echoes from M ercury (□) and the
M oon (A ).
F ig u r e 17
T he solid curves shows the microwave emissivity and fractional polarization preef f
dieted by the facet model (eR = 2.7, s = 9.7 degrees, S = 0.2,0.6) as a function
of fractional radius on the projected disk of the planet. This is com pared to
the emissivity and fractional polarization of a sm ooth surface (dashed curves:
e/ /
eR = 2.7, s = 0, 8 = 0). Since the effects of subsurface scattering and dielectric
gradients have been ignored, the solid curve represents an upper lim it to the
91
actual emissivity. The horizontal b ar corresponds to approxim ately 0 . 1 ” on the
images.
F ig u r e 18
T he d a ta points are as in Figure 3. T he curves are model calculations th a t differ
only in th e value for th e effective specific loss tangent. From the lowest to the
highest curve near an hour angle of —3, th e specific loss tangents are 0.0026,
0.0052, 0.0078, 0.0104, and 0.0130. The specific loss tangents of retu rn ed lunar
sam ples are ~ 0.0200 at this wavelength.
F ig u re 19
T he d a ta points are as in Figure 4. T he curves are model calculations th a t differ
only in the value for the effective specific loss tangent. From the lowest to th e
highest curve near
= 0.6 (b), th e specific loss tangents are 0.0026, 0.0052,
0.0078, 0.0104, and 0.0130. The specific loss tangents of returned lunar samples
are ~ 0.0200 a t this wavelength.
F ig u r e 20
(a ) Residual therm al images at Al.3 cm th a t are obtained by subtracting a series
of models from the observed image (Figure 5 [a]). In the subtraction procedure,
we minimize the residuals by varying an arb itrary scale factor, as well as allowing
for slight positional offsets and expansions (see text). The models differ only in
the value for the effective specific loss tangent, which is shown at the top of
each panel. T he contour interval is 5 K, which is approxim ately three tim es the
rm s noise a t this wavelength. D ashed contours (w ith grayscale) are negative.
T he bo tto m right panel shows the aspect of M ercury during this observation
(see Figure 1 [c]). T he day-night brightness contrast is best fit with an effective
specific loss tangent of 0.0039 ± 0.0006.
92
(b ) Same as (a), except for a wavelength of 2.0 cm. The contour interval is 5 K
(approxim ately three times the rm s), and the best-fit effective dielectric constant
is 0.0029 ± 0.0006.
(c ) Same as (a), except for a wavelength of 3.6 cm. The contour interval is
10 K (approxim ately three times the rm s), and the best-fit effective specific loss
tangent is 0.0026 ± 0.0006.
(d ) Same as (a), except for a wavelength of 6.2 cm. The contour interval is
10 K (approxim ately three times the rm s), and the best-fit effective specific loss
tan g en t is 0.0026 ± 0.0013.
(e ) Same as (a), except for a wavelength of 18.0 cm. T he contour interval is
15 K (approxim ately twice the rm s). At this wavelength, th e model images are
insensitive to the opacity.
(f ) Same as (a), except for a wavelength of 20.5 cm. The contour interval is
15 K. At this wavelength, the model images are insensitive to the opacity.
F ig u r e 21
T he effective specific loss tangent as a function of frequency. T he solid sym ­
bols w ith error bars are values for M ercury derived from the relative day-night
brightness contrast (Figures 18, 19, and 20 ). The open triangle with error bars
represents an average of 21 lunar samples at 450 MHz (Gold et al. 1976). Each
dashed line connects m easurem ents of an individual lunar sample at 450 MHz
and 9 GHz (B assett and Shackleford 1972; open diam onds). The E a rth symbols
( © ) w ith error bars represent averages of 7 terrestrial basalt samples a t 450
MHz an d 35 GHz (C am pbell and Ulrichs 1969). The stars connected by a solid
line are far-infrared m easurem ents for an individual lunar sample (Clegg et al.
93
1972). The slope of the solid line is well-established by several m easurem ents at
interm ediate frequencies.
F igu re 22
The specific loss tangents of 21 lunar samples as a function of ilmenite content
(adapted from Table 1 of Gold et al. 1976 and Table A9.16 of C arrier et al. 1991).
Filled symbols represent m are samples, while open symbols represent highland
samples. The symbol shapes identify the Apollo mission from which each sample
was collected: circles (11, 12, and 14), triangles (15), squares (16), and stars (17).
The fine represents a least-squares fit to the d ata (excluding the anomalously low
Apollo 15 m are sample). On average, the 450 MHz opacity of highland m aterial
is smaller than th a t of m are m aterial.
F igu re 23
Residual images at wavelengths of 1.3 (a), 2.0 (b), 3.6 (c), 6.2 (d), 18.0 (e), and
20.5 (f) cm th a t are obtained by readjusting the scale factor of the best-fit resid­
ual images of Figure 20 (a-f) so as to fit the night-side equatorial brightness in
an absolute sense. Contour intervals are 5 K (a-b), 10 K (c-d), and 15 K (e-f),
which are roughly three times the rm s (a-d) or twice the rm s (e-f). Dashed con­
tours (with grayscale) are negative. We have chosen to extrapolate the effective
specific loss tangent at A6.2 cm (0.0026) to wavelengths of 18.0 and 20.5 cm (ef). T herm al depressions at both poles and along the sunlit side of the morning
term inator are likely due to shadowing by surface topography (see text).
F ig u re 24
The insolation (in units of the solar constant at E arth) as a function of tim e at
herm ographic coordinates of (165°, 0°) and (265°, 0°). At a longitude of 165°
94
(265°), a level surface receives approxim ately 1% (16%) of its total insolation
while the sun is within 15° of the horizon (shaded regions).
F ig u re 25
The elevation of the Sun vs. tim e after sunrise as viewed from herm ographic
coordinates of (165°, 0°) and (265°, 0°).
The horizontal scale is in units of
“M ercury hours” , which are approxim ately equivalent to E a rth weeks. The solar
symbols ( 0 ) show the elevations of the Sun during the observations of 11 August
1990 (as viewed from 165°) and 13 April 1990 (as viewed from 265°). One E a rth
week after sunrise at 265°, the Sun stops rising and rem ains almost motionless
for two E a rth weeks as it goes through a retrograde loop. In contrast, the Sun
rises w ith a nearly constant angular rate of m otion at 165°.
F ig u re 26
The tem perature as a function of optical depth at herm ographic coordinates of
(165°, 0°) on 11 August 1990. The solid curve shows the therm al profile in
level sunlit regions, while the dotted curve is an estim ate of the therm al pro­
file in shaded regions, which is obtained by delaying sunrise in our level-surface
model. Inform ation about sunrise has propagated to a physical depth of 11 cm
(dashed line) during the 0.6 M ercury hours th a t the Sun has been above the
horizon. T he depression in the microwave brightness caused by shadowed re­
gions is expected to be weaker at A2.0 cm th an at Al.3 cm since th e layer th a t
contains the tem perature anomaly (depth < 1 1 cm) is m ore transparent at the
longer wavelength (com pare upper and lower horizontal scales). Viewed at nor­
m al incidence, these tem perature profiles would produce brightness tem perature
contrasts ( T j ,AD — T b N ) of —17 K and —8 K at wavelengths of 1.3 and 2.0 cm,
respectively.
95
F ig u re 27
T he sam e as Figure 26, except for a longitude of 265° on 13 April 1990. Infor­
m ation about sunrise has propagated to a physical depth of 26 cm (dashed line)
during the 3.4 M ercury hours th a t the Sun has been above the horizon. Viewed
at norm al incidence, the shade and sunlight tem perature profiles produce bright­
ness tem perature contrasts (T™ ADE _ T ^ N ) of —14 K and —8 K at wavelengths
of 3.6 and 6.2 cm, respectively.
96
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V I. S u m m a r y and F u ture D irectio n s
We have im aged M ercury’s therm al em ission at wavelengths of 0.3, 1.3, 2 .0 ,
3.6, 6.2, 18.0, and 20.5 cm w ith the BIMA millimeter interferom eter and the
Very Large Array. Im ages of the linearly polarized component of this emission
were obtained from 2.0 to 20.5 cm. The principle objective of this study is to
constrain the therm al, electrical, and scattering properties in the top 2-3 m eters
of M ercury’s regohth to a level of accuracy th a t allows meaningful comparisons
w ith the M oon, and thus to take advantage of detailed knowledge of the Moon
from extensive rem ote observations, in situ experim ents, and returned samples,
to b e tte r u n derstand Mercury.
This project was designed to take advantage of improvem ents in microwave
im aging capabilities and detector efficiency th a t have taken place since previous
microwave studies of M ercury in the mid-1970’s. Imaging is a significant im ­
provem ent over disk averages, since relative day-to-night brightness variations
across the disk m ay be used to constrain the therm al and electrical properties of
the regolith, thus elim inating the need for absolute flux calibrations, which is the
largest source of experim ental uncertainty in single dish observations. F u rth er­
m ore, it is m ore straightforw ard to interpret relative brightness variations since
they are largely independent of the microwave emissivity, which depends in p art
on poorly constrained scattering effects th a t vary w ith the observing wavelength.
M easurem ents of M ercury’s nighttim e surface tem perature provide one of
the best constraints on regolith therm al param eters (density, specific heat, th er­
m al conductivity). A re-analysis of M ariner 10 Infrared Radiom eter d a ta with
our m odel shows th a t M ercury’s regolith likely consists of a therm ally insulating
surface layer w ith a thickness of a few centimeters overlying a region of much
124
higher therm al conductivity.
This stru ctu re is observed in the lunar regolith
(Langseth et al. 1976) and is believed to be the result of m icrom eteorite bom ­
bardm ent, which stirs and m aintains the top layer a t a low density while com ­
pacting deeper layers by vibration a n d /o r stress. W ith this two-layer regolith
structure, we avoid the system atic increase in th e surface rock coverage over 180
degrees of longitude th a t is otherwise required to account for the M ariner 10
surface tem p eratu re d a ta (Chase et al. 1976). T here rem ain localized th erm al
anomalies th a t are best explained by varying am ounts of surface rock coverage,
although variations in th e thickness of th e surface layer a n d /o r variations in the
degree of regolith com paction could also be contributing factors.
To a first approxim ation, the reflection and transm ission of microwaves at
the surface is given by the stan d ard Fresnel formulas for a sm ooth, sharp bou n d ­
ary between a homogeneous m edium and a vacuum . In this case the fractional
polarization depends only on the dielectric constant of th e regolith and th e angle
between the line of sight and the surface norm al. However, the observed polar­
ization deviates m easurably from this model. Polarization d a ta near the limb of
the planet require rm s surface slopes of approxim ately 10 degrees, which are sim ­
ilar to rm s slopes derived from specular rad ar echoes from M ercury (Pettengill
1978). To obtain consistency with estim ates of the regolith dielectric constant
th a t are derived from specular ra d a r echoes, the em ergent therm al radiation m ust
be partially diffused (and depolarized) at the surface boundary. T he fraction of
the em ergent radiation th a t m ust be diffused decreases monotonically from 60%
at an observing wavelength of 2 cm to ~ 20 % at 20 cm. These values are similar
to estim ates of surface diffusion th a t are based on th e ratio of non-specular to
specular ra d a r echo power (Evans 1969). Thus, the observed therm al polariza­
tion as a function of incidence angle and wavelength is consistent w ith a surface
125
th a t is sm ooth and undulating on m eter scales and progressively rougher from
decim eter to centim eter scales.
T he microwave opacity we derive for M ercury’s regolith by modeling the
day-to-night brightness contrast at each wavelength is at least a factor of 2-3
lower th a n typical opacities of retu rn ed lunar samples. This difference is likely
due to m ineralogical differences betw een th e surfaces of M ercury and th e Moon.
Gold et al. (1976) dem onstrated th a t the microwave opacities of lu n ar samples
increase w ith the ilm enite (FeTiC>3 ) content. Hapke et al. (1975) noted th a t
system atic albedo differences between M ercury and the Moon could be explained
by relatively low abundances of Ti and Fe on Mercury. Thus, the low microwave
opacity we derive for M ercury’s regolith is likely a m anifestation of a low ilm enite
abundance on Mercury.
These lu n ar comparisons are based in p art on an extrapolation of the 450
MHz electrical properties of lu n ar samples to the 1.5-86 GHz frequencies of our
observations. This extrapolation is subject to significant uncertainties because
there are only two m easurem ents of the electrical properties of lu n ar samples
a t centim eter wavelengths. The im portance of scattering in a lunar-like regolith
a t centim eter wavelengths has never been addressed by laboratory experim ents,
although m odel calculations show th a t such scattering m ay be im p o rtan t (T sang
and Kong 1975, Keihm 1982), and scattering has been cited as a significant
source of uncertainty in the frequency dependence of the microwave opacity and
emissivity. Significant progress in th e interpretation of microwave observations
of M ercury, the Moon, and other solid surfaces could be m ade w ith laboratory
m easurem ents of the electrical and scattering properties of lunar (a n d /o r lunar
analog) samples at centim eter wavelengths. These samples should have a broad
particle size distribution.
126
Difference m aps obtained by normalizing the best-fit therm al models to the
observed equatorial night-side brightness and subtracting these from the d ata
clearly exhibit the effects of shadowing along the sunlit side of the morning te r­
m inator and at the poles. The therm al depressions are weakest at the equator
and strengthen and extend into the night hemisphere tow ards the poles. This
morphology, as well as the relative strengths of the therm al depressions from
wavelength to wavelength, can be understood in term s of a simple model of shad­
owing. “Transient” shadowing occurs in equatorial regions, where topographic
lows and hillsides are alternately in shadow and sunlight as th e day progresses.
This effect is strongest near the morning term inator, where tem perature con­
trasts between shadowed and sunlit regions are largest and where the fraction
of the surface in shadow (as seen from E arth) is also largest.
“P erm anent”
shadowing occurs at higher latitudes, where shadows persist throughout the day,
resulting in an insolation deficit th a t strengthens tow ard the poles. Shadowing
effects are significantly stronger near cold longitudes, where sunrise and sunset
occur very slowly and allow tem perature anomalies to penetrate much deeper
into the regolith. These results dem onstrate th a t it should be possible to analyze
the residual images w ith a more realistic model of topography and shadowing,
in order to characterize the strengths and morphologies of the observed therm al
depressions in term s of crater geometry and crater density.
Imaging capabilities at millimeter wavelengths will soon improve d ram at­
ically w ith the expansion of the BIMA interferom eter from 3 to 9 antennas.
The greatly increased sampling of the visibility function will allow more reliable
imaging of planets w ith a single configuration of the antennas, thus avoiding the
“sm earing” effects th a t occur when attem pting to image an extended, rapidly
varying object (such as M ercury) with a limited num ber of antennas. O ther
127
im provem ents th a t are planned for the near future, such as the capability to
m easure polarization and to observe at A0.1 cm, will help to characterize M er­
cury’s electrical and scattering properties at short wavelengths. At A0.1 cm, we
m ay begin to see evidence for microwave absorption by the long-wavelength tails
of far-infrared resonances, which need not be correlated with the ilmenite abun­
dance. Polarization at wavelengths of 0.3 and especially 0.1 cm is expected to
be very weak (
5%), b u t could provide inform ation about small scale surface
texture.
128
R eferen ces
B aars, J. W. M., R. Genzel, I. I. K. Pauliny-Toth, and A. Witzel, 1977, The
Absolute Spectrum of Cas A; An A ccurate Flux Density Scale and a Set
of Secondary C alibrators, Astron. A strophys. 61, 99-106.
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137
A p p e n d ix 1: H e a t F lo w fro m I n te r n a l S o u rc e s
Internal heat sources, such as radioactive decay and differentiation, produce
a heat flux through the regolith th a t can in principle be detected by microwave
observations over a sufficiently broad wavelength range. Early theoretical calcu­
lations of M ercury’s therm al evolution (Siegfried and Solomon 1974) indicated
th a t the present-day lithospheric heat flux could be as large as 60 erg cm -2 s - 1 ,
b u t these calculations ignored sub-solidus convection in the m antle. More recent
calculations (Schubert et al. 1988) indicate th a t sub-solidus m antle convection
actually plays an im portant role, resulting in a lithospheric heat flux of only 20
erg cm -2 s- 1 . An experim ental m easurem ent of the heat flux would provide a
valuable constraint on models of M ercury’s form ation and subsequent therm al
evolution.
Below the diurnal layer, the tem p eratu re gradient required to support a heat
flux of Jo is given by:
dx
=
K (x,T)
(27)
K ’
where K ( x , T ) is the to tal therm al conductivity at depth x and tem p eratu re T.
W ith a regolith therm al conductivity of 1000 erg s -1 cm -1 K -1 (see Figure 10 ,
model c), a tem perature gradient of 2 K per m eter is required to support a heat
flux of 20 erg cm -2 s- 1 . This gradient should produce a system atic increase in
the microwave brightness tem perature w ith wavelength, since longer wavelengths
probe deeper, w arm er layers.
In practice, it is quite difficult to determ ine Jo by inverting the microwave
spectrum because such an inversion requires an accurate absolute calibration of
138
the microwave m easurem ents, as well as accurate knowledge of the vertical struc­
tu re of the therm al conductivity and the wavelength dependences of the opacity
and emissivity. Proper treatm ent of the emissivity is particularly troublesome
because it depends on the dielectric gradient near the surface, subsurface scat­
tering by rock chips, and subsurface dielectric boundaries, such as the interface
between the regolith and the megaregolith. The history of attem p ts to invert the
lunar microwave spectrum provides a good illustration of these difficulties.
lunar heat flow m easurem ents
T he first attem pts to m easure the lunar heat flow were based on inversion
of the microwave spectrum at wavelengths ranging from 0.4 to 50 cm. Using
a homogeneous, non-scattering regolith model, Krotikov and Troitsky (1964)
obtained a heat flux of 54 erg cm -2 s- 1 . Subsequently, Tikhonova and Troitsky
(1969) derived a heat flux in the range of 30-40 erg cm -2 s -1 by utilizing a more
realistic (but still crude) regolith model in which the density increases linearly
w ith depth. B oth of these estim ates are well outside the range of 15-20 erg cm -2
s -1 th a t is based on in situ Apollo heat probe measurem ents (Langseth et al.
1976).
Linsky (1966) pointed out th a t radiative heat tran sp o rt within the regolith
is an im portant effect at wavelengths shortw ard of 3.2 cm, since it causes an in­
crease in the m ean tem perature with depth through the diurnal layer as a result
of non-linearity introduced into the therm al diffusion equation. W ith a hom o­
geneous, non-scattering regolith model th a t included radiative heat tran sp o rt,
Linsky derived a heat flux of 14 erg cm - 2 s -1 th a t is rem arkably close to the
range determ ined in situ. However, Linsky assumed th a t the therm al conduc­
tivity thoughout the regolith was the same as the surface value of ~250 erg s -1
139
cm 1 K x. W ith a more realistic subsurface therm al conductivity of 1000 erg
s _1 cm -1 K - 1 , Linsky’s value is too large.
Schloerb et al. (1976) employed a novel interferometric approach at a wave­
length of 49 cm, which took advantage of the changing orientation of the line
of sight through the regolith from the center to the limb of the projected disk
of the Moon. They derived a small negative heat flow (tem perature decreases
w ith depth) and concluded th a t their observations might have probed completely
through the regolith and into the underlying “megaregolith,” a region of large,
unconsolidated blocks (Cooper et al. 1974) th a t is believed to have therm al and
electrical properties more similar to solid rock th an to the com pacted regolith
dust.
Fisher and Staelin (1977) dem onstrated th a t it is possible to model th e lu­
n ar microwave spectrum with a heat flux of zero by considering the wavelengthdependent effects of subsurface scattering on the emissivity. The principle weak­
ness of their model is in the choice of a single characteristic size for the scatterers,
so th a t emissivity effects were naturally strongest at wavelengths near this size.
Nevertheless, they showed th a t scattering effects are potentially im portant and
likely depend on the particle size distribution.
W ith the benefit of ground-truth d ata from the Apollo missions, Keihm
(1984) developed detailed therm al, electrical, and scattering models of the lunar
regolith in an attem pt to evaluate the feasibility of heat flow m apping by an
orbiting microwave radiom eter. Keihm adopted a two-layer regolith model based
on in situ Apollo 15 and 17 heat probe m easurem ents, which is similar to model
c in Figures 9 and 10. Using scatterer size distributions based on Surveyor and
Apollo d ata, he found th a t subsurface scattering effects over m ost of the lunar
140
surface should produce an emissivity increase from A5 cm to A20 cm of only
;$ 1 %. A similar decrease in the emissivity over the same wavelength range
is expected to result from the dielectric gradient near the surface. (Note th a t
these two effects partially cancel each other.) Finally, the effect of the dielectric
interface at the base of the regolith is expected to be negligible in the lunar case
for wavelengths ,$ 30 cm, which is consistent w ith the experim ental results of
Schloerb et al. (1976). Thus, the microwave em issivity of the Moon is expected
to be approxim ately independent of wavelength between 5 and 30 cm.
M e rc u ry ’s night-side microwave spectrum
Instead of analyzing the disk-average spectrum , which is given in the last
column of Table III, a localized spectrum is determ ined at a position on th e night
hem isphere away from the poles, th e term inator, and th e limb in order to avoid
shadowing effects and uncertainties in th e limb emissivity. A convenient location
th a t satisfies these criteria is at herm ographic coordinates of 330° longitude and
obs
0° latitude. The m easured brightness tem peratures (Tfl ) at these coordinates
from A3.6 cm to A20.5 cm are given in th e second column of Table V. These values
represent weighted averages over a beam-sized region centered at the reference
coordinates.
Since the beam size increases from A3.6 cm to A20.5 cm, a correction factor
is determ ined from the best-fit model image at each wavelength th a t relates the
brightness tem perature at the reference coordinates with a hypothetical obser­
vation of infinitely high resolution. Although it would be m ore appropriate to
apply this correction to the model, presentation of th e spectrum is m ore straig h t­
forw ard if it is instead applied to the m easured brightness tem peratures. The
beam correction, A
, is given in the third colum n of Table V.
141
A nother potentially im p o rtan t correction is for the diffuse Galactic contin­
uum emission. An interferom eter, such as the VLA, is sensitive only to brightness
variations w ithin the field of view and thus would not norm ally be sensitive to
the diffuse G alactic background. However, a correction is necessary for a plane­
ta ry observation because the planet blocks out a resolved disk-shaped region of
th e background, which introduces a negative offset to the observed image. D ur­
ing the observations of 13-15 April 1990, M ercury was at Galactic coordinates
of 156° longitude and —37° latitu d e ( l ” , b" ). M easurem ents at frequencies of
408 and 1420 MHz at these coordinates yield a background level and a spec­
tra l index (Reich and Reich 1988), which can be used to estim ate th e brightness
te m p e ratu re of the background at wavelengths from 3.6 to 20 cm:
A T " ( i / aHl) =
( l - 0 K ) x ( ^ ) ‘ ! ‘ +2.8K
(28)
where i/GHz is the observing frequency in GHz. The background levels obtained
from this expression are given in th e fourth column of Table V. The final corrected
brightness tem peratures and the uncertainties are given in the last two columns
an d illustrated in Figure 28. T he spectrum is rem arkably flat. In fact, a constant
brightness te m p eratu re of 400 K provides a good fit to the d ata, although the
uncertainties allow for m ore realistic models of th e microwave spectrum , such as
those shown by th e curves, which are described below.
m odeling M e rc u ry ’s m icrowave spectrum
M odel calculations are perform ed for a hypothetical observation w ith infinite
resolution at the reference coordinates using the ephemeris of 14 April 1990.
T he evolution of the microwave spectrum as a result of therm al diffusion and
M ercury’s slight rotation during 13-15 April is negligible (~ 1 K) com pared with
142
other uncertainties. T he regolith therm al param eters are shown by curve c in
Figures 9 and 10. The m easured effective specific loss tangent of 0.0026 is used at
wavelengths of 3.6 and 6.2 cm, and it is assum ed th a t L
rem ains constant from
A6.2 cm to A20.5 cm. We consider two values for the effective dielectric constant:
the surface value in m odel c (1.9) and the value determ ined by ra d a r observations
(2.7). We ignore variation of th e microwave emissivity w ith wavelength under the
assum ption th a t the regolith scattering properties are similar on M ercury and
the Moon. However, M ercury’s regolith opacity is 2-3 times smaller th an the
M oon’s, so th a t a given wavelength probes at least twice as deep into M ercury’s
regolith. If the regolith thicknesses are similar on M ercury and th e Moon, then
the effects of th e dielectric boundary at the base of M ercury’s regolith could
become apparent at wavelengths as short as ~ 15 cm.
Before we a tte m p t to constrain the heat flux, it is first necessary to deter­
mine the im portance of radiative heat tran sp o rt in the regolith (the param eter
X in E quation 8 ). In the case of th e Moon, radiative heat tra n sp o rt is largely
responsible for the 40-45 K increase in th e m ean tem p eratu re from the surface
to the base of th e diurnal layer (Keihm and Langseth 1973). This m ean tem ­
p eratu re gradient, which corresponds to x — 2-3, is much larger th a n would be
required to tra n sp o rt only the heat flux from internal sources.
e//
Figure 28 shows m odel spectra for eR = 1.9, Jo = 0, and several values
•
of x a t the surface. A x °f zero is clearly excluded by the observed spectrum ,
dem onstrating th a t radiative heat tran sp o rt is im p o rtan t w ithin M ercury’s re­
golith. A x °f unity is marginally consistent w ith all of the d ata, although the
best-fit value appears to decrease from ~1.5 a t wavelengths of 3.6 and 6.2 cm
to ~ 0 .7 at the longer wavelengths. This discrepancy, if real, could indicate th a t
the longer wavelengths are beginning to probe beneath the regolith and into the
143
raegaregolith. If this is th e case, then it is impossible to place useful constraints
on the heat flux from internal sources, since the 3.6-6.2 cm wavelength range
alone provides insufficient sensitivity to Jo.
A value for x ° f order unity is similar to previous estim ates of M ercury’s
radiative heat tran sp o rt (Cuzzi 1974, Ledlow et al. 1992); however, these esti­
m ates all assum e an effective dielectric constant of ~ 2 , which corresponds to a
norm al emissivity of 0.97. T he actual emissivity is likely to be lower. Figure
eSS
29 shows m odel spectra for eR = 2.7 and Jo = 0. This value for the effective
dielectric constant, which corresponds to a norm al emissivity of 0.94, is based
on ra d a r observations and is supported by our analysis of the observed therm al
polarization images (see section V). Note th a t a factor of two increase in x is
required to com pensate for the lower emissivity. At wavelengths of 3.6 and 6.2
cm, the best-fit value of x is ~ 3 , which is similar to lunar values in th e range
of 2-3 th a t are determ ined from in situ heat probe m easurem ents (K eihm and
Langseth 1973).
e/ f
Figure 30 shows m odel spectra for eR = 2.7, x = 1-3, and several values
of the lithospheric heat flux, Jo. The value of x was chosen as a compromise
between the best-fit values at short and long wavelengths. It is possible to obtain
m arginally acceptable fits to the spectrum w ith Jo £ 40 erg cm -2 s- 1 . This upper
lim it is consistent with the current theoretical estim ate of ~ 2 0 erg cm - 2 s -1
(Schubert et al. 1988). However, this fairly tight constraint on Jo is deceptive,
because the emissivity could be significantly lower at wavelengths of 18.0 and
20.5 cm than at shorter wavelengths as a result of the regolith-megaregolitli
interface. Because the brightness tem peratures at A18.0 cm and A20.5 cm may
depend jointly on the thickness of the regolith and on the lithospheric heat flow,
we cannot place constraints on heat flow alone.
144
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F ig u re C a p tio n s
F ig u re 28
T he therm al microwave spectrum on 14 April 1990 at herm ographic coordinates
of 330° longitude and 0° latitude. The d a ta have been corrected for the differing
resolutions from A3.6 cm to A20.5 cm and also for the diffuse Galactic background.
T he solid curves are calculations using the two-layer regolith m odel (curve c in
Figures 9 and 10 ) w ith several values of x at the surface. The regolith electrical
ef f
ef f
param eters are eR = 1 .9 and L„
= 0.0026.
F ig u re 29
The d a ta and m odel curves are th e sam e as Figure 28, except th a t eR has been
increased to 2.7. In order to fit th e observed spectrum , larger values of x are
required to com pensate for the reduced microwave emissivity.
F ig u re 30
T he d a ta are the same as in Figure 28. The solid curves are calculations using
the two-layer regolith model (curve c in Figures 9 and 10) with x = 1-5. The
•
•
e fJ
cjJ
regolith electrical param eters are eR = 2.7 and L v = 0.0026. Each of th e four
m odel curves represents a different value of the net subsurface heat flux (Jo in
Equations 9 and 10). From the b ottom curve to the top, J q = 0, 20, 40, and 60
erg cm -2 s - 1 .
146
F ig u re 28
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