Intermediate scale lunar roughness

I c ~ v s 11, 289-319 (1969)

Intermediate Scale Lunar Roughness J. A. B A S T I N Department of Physics, Queen Mary College, London AlqD D. O. G O U G H 1 Goddard Institute for Space Studies and Courant Institute of Mathematical Sciences, New York Received September 2, 1969 A model surface consisting of parallel troughs has been examined in order to assess the importance of roughness in accounting for the properties of lunar thermal radiation. Insolation, emission from the surface, reabsorption of emitted radiation, and conduction are all considered. Brightness temperatures both in the midinfrared and microwave region are computed for eclipse and lunation conditions, not only as a function of lunar phase, latitude, longitude, and direction of observation, but also for a variety of trough dimensions. All those features of the observed thermal radiation which cannot be accounted for on the basis of a plane homogeneous model are listed and the extent to which they can be accounted for by the proposed model is considered. In particular, a model for which the width and height of the raised portions are both equal to a quarter of the trough interval gives good agreement with the directional effects observed for lunar daytime radiation in the 10-14/x wavelength band. In addition a number of other anomalies, including some already accounted for in the literature by other causes, receive more or less good explanations on the basis of roughness. I. INTRODUCTION

of the model have been studied in some detail (Jaeger, 1953a; Wesselink, 1948; K r o t i k o v a n d Shchuko, 1963) a n d fit the observed m i c r o w a v e a n d infrared observations fairly well. A detailed comparison, however, reveals m a n y discrepancies which are so large t h a t t h e y c a n n o t be explained a w a y b y e x p e r i m e n t a l error. A n y such discrepancy we shall refer t o as a t h e r m a l a n o m a l y . A t t e m p t s to a c c o u n t for t h e m h a v e followed four m a i n a p p r o a c h e s :

F r o m E a r t h - b a s e d p h o t o g r a p h y we k n o w t h a t on a scale of several kilometers t h e Moon's surface is relatively smooth, whereas p h o t o m e t r i c observations a t visual w a v e l e n g t h s show t h a t on a submfllim e t e r or m i c r o n scale the terrain is e x t r e m e l y rough. This c o n t r a s t was n o t e d a n d r e m a r k e d on b y Galilei (1632), b u t only r e c e n t l y has i n f o r m a t i o n been obt a i n e d a b o u t the degree o f roughness on a n i n t e r m e d i a t e scale. The purpose o f this p a p e r is to examine theoretically the effect o f c e n t i m e t e r scale roughness on the t h e r m a l b e h a v i o r of a model o f the l u n a r surface. The first discussions of the t h e r m a l b e h a v i o r of the Moon e m p l o y e d a semiinfinite h o m o g e n e o u s m e d i u m b o u n d e d a b o v e b y a horizontal plane to represent the Moon's u p p e r layers. The properties 1 Now at Institute of Theoretical Astronomy, University of Cambridge, England. 289

(i) The l u n a r surface is stratified in a t h i n u p p e r layer w i t h the t h e r m a l c o n d u c t i v i t y increasing downwards. (ii) There are horizontal variations in t h e albedos a n d the t h e r m a l properties of l u n a r rock. (iii) The t h e r m a l coefficients, in particular the t h e r m a l c o n d u c t i v i t y , v a r y with t e m p e r a t u r e . (iv) Centimeter a n d millimeter scale roughness h a v e i m p o r t a n t t h e r m a l effects.

290

J . A . BASTIN AND D. O. GOUGH

The stratified model was the first to be considered (Piddington and Minnett, 1949; Jaeger and Harper, 1950; Jaeger, 1953b; Lettau, 1962) and gave results only slightly better than those of the homogeneous model. The second approach has been investigated by Fudali (1966), and by Allen and Ney (1969), especially in connection with hot spots. The third approach has received considerable attention (e.g., Muncey, 1958, 1963 ; Clegg et al., 1966; Linsky, 1966; Winter, 1967; Winter and Saari, 1969); in the more recent work the temperature dependence is accounted for by radiative exchange across the interparticulate voids in the lunar rock, or by direct radiative transfer within the rock particles. In this way at least two prominent anomalies have been satisfactorily explained. The centimeter and millimeter roughness idea has already been applied to a number of aspects of the subject but quantitative comparison with observation is sparse. In this paper a more detailed analysis of a specific rough model is presented, applying it to as m a n y as possible of the outstanding problems which still remain. II.

P R E V I O U S STUDIES OF L U N A R ROUGHNESS

Although measurements of the lunar thermal radiation have been made for nearly a century, the first results to show the effects of surface roughness were those obtained by Pettit and Nicholson (1930) using the 100-inch Mount Wilson reflector with a vacuum thermocouple as detector. Measurements were made of the total radiation in the broad band 8-14 ~. In this wavelength range the radiation can be regarded as having been emitted from the surface. Pettit and Nicholson were able to resolve areas sufficiently small to make detailed measurements of the temperature distribution across the disk possible. At full moon, a scan across the equator showed t h a t the thermal energy flux E received from a point on the Moon at which the Earths' zenith angle is $ obeyed the relation E ~ cos3/2~. (2.1)

This is not the result one would anticipate if the lunar surface were smooth. Except near the edge of the insolated area the heat flux emitted almost balances the incident solar radiation (see, for example, Wesselink, 1948) and one would expect E oc cos~.

(2.2)

To explain the observed relation Pettit and l~icholson proposed a rough model consisting of a plane horizontal insulating surface partially covered with insulating spheres set so far apart that, for the values of ~ considered, no sphere stood in the shadow of another. Radiative heat exchange between the fiat surface and the spheres was ignored. The flux radiated by the model is E=E0[cos~+~(

2 cos~

1)1, (2.3)

where ~ is the fraction of the surface covered by spheres and E0, the energy flux t h a t would be received if the spheres were absent. The parameter ~ was computed by equating the formula (2.3) with the observed result at ~ = cos-1(0.6) and a value of 0.32 was obtained. The model provides a good qualitative explanation of the difference between the prediction (2.2) and the empirical law (2.1), but the approximations made are too severe to expect an accurate description. Indeed, the model was chosen for analytical simplicity and was in no way intended to represent the exact topography of the lunar surface. Pettit and Nicholson also noticed t h a t the radiation from the subsolar point varied markedly with direction, having a maximum at full moon when the directions of the Sun and the observer are almost the same. This property also was attributed to roughness. I t was noted further t h a t the subsolar point temperature appeared somewhat less when the Moon was waxing than when it was waning. More recently, other deviations from the properties of the smooth homogeneous model have been found. Sinton (1961, 1962) noted t h a t there is an apparent decrease towards the limb of the thermal

I N T E R M E D I A T E SCALE L U N A R R O U G H N E S S

inertia measured during eclipse, and suggested that it might be a result of surface roughness. ShorthiU (1962) has also invoked roughness to explain this and Pettit and Nieholson's results. In a short note Gear and Bastin (1962) cited, in addition to the evidence given above, that the apparent temperature variation of the center of the lunar disk during lunation is not as great as one would expect from the low value of the thermal inertia deduced from the eclipse data, and that this too might be attributed to roughness. They proposed a corrugated model of the lunar surface with rectangular indentations (Fig. 1), b u t no calculations of its thermal behavior were presented. Provided the roughness is on a scale sufficiently large for conduction to be unimportant during insolation, the surface temperature depends only on the shape of the surface. It was not necessary, therefore, for Pettit and ~icholson to specify the scale of their model, b u t they do refer to "bowl-shaped craters" and "vertical mountain sides", hinting, perhaps, that the scale is of the order of a few kilometers or more. A similar scale is suggested b y Sinton (1962), who uses the terms "valleys" and "peaks". B u t Gear and Bastin (1962) pointed out that the average slope of the lunar surface deduced from photographs taken from Earth is insufficient to account for the infrared observations (see also Smith, 1967), and therefore proposed that the indentations are much smaller than anything that can be resolved in the photographs. The infrared observations during insolation tell us little about the scale, b u t measurements at millimeter wavelengths, which give us information about the temperature distribution beneath the surface, can provide an upper bound: Bastin, Clegg, Gear, Jones, and Platt (1964) made measurements at 1.5 mm and found that the poleward darkening near full moon was much greater than would be expected not only from a smooth homogeneous model b u t also from a rough model with a characteristic scale of a meter or more. Similar darkening has also been found in more accurate measurements at longer wavelengths (Salomono~

291

vich and Losovskii, 1962; Salomonovich, 1967) and b y measurements at 1.2 mm (Bastin and Gear, 1967; Clegg, Newstead, and Bastin, 1969). Indeed, Salomonovich (1962) states that the brightness temperature at 0.8, 2.0, and 3.2 cm varies with latitude ~b as cosl/~b whereas the smooth layer yields approximately cosll4~b (Krotikov and Shehuko, 1963; Gear, 1965). The measurements of Gary, Stacey, and Drake (1965) also show the same effect, though these seem to deviate less from the smooth model. Bastin, Gear, Jones, Smith, and Wright (1964) noticed that at 1.5-mm wavelength the lunar brightness temperatures at phases ~r/2 and 3~/2 are lower than the smooth model predicts and suggested that surface roughness could account for the discrepancy. More accurate measurements b y Gary et al. (1965) and b y Low and Davidson (1965) showed the same effect. In addition to the global properties described above, there are smaller scale inhomogeneities in the thermal radiation which may have been produced b y local variations in the topography. For example, Shorthill, Borough, and Conley (1960), and Sinton (1961) found that the crater Tycho was 40-60°K warmer than its surroundings during total eclipse. Saari and Shorthill (1965) found a number of regions which were warmer than their surroundings during eclipse yet which were not necessarily coincident with any obvious visible feature. They were named "hot spots" though the term had previously been used b y Shorthill (1962) mainly in connection with craters. Bastin (1965) first suggested that these warm areas could be more rough than their surroundings and briefly described the cooling of the corrugated model of Gear and Bastin (1962) to support the idea. A much more thorough study of the cooling of a slightly different model with infinitely deep crevices (Fig. 1) had been made b y Winter (1965), though he did not at that time relate the results to the Moon. He found that the apparent effective temperature, as would be measured b y an observer overhead who could not resolve the crevices, was always greater than that

292

J. A. BASTIN

AND

D. O. GOUGH

1 I

WINTER 196,5

GEAR & BASTIN 1962

/

j

J

.f

J BU~IL 1967

ROEtOF

1968

FIo. 1. Rough models whose ~hermal properties have been studied prior to this paper. of the smooth model with the same thermal conductivity. The difference between the models decreased with increasing zenith angle of observation and actually reversed sign near ~/2. In a later paper Winter (1966) regarded the hot spots as regions of greater roughness; he compared his model with the observations of the crater Aristarchus by Saari and Shorthill (1963) and obtained agreement with crevices about 1 mm wide and separated by about 1 cm. More recently, a rough model with spherical craters has been studied by Buhl (1967) and Buhl, Welch, and Rea (1968). Radiative energy interchange across the craters was taken into account but conduction was either ignored or treated approximately. The authors computed the variation of apparent temperature towards the limb during the lunar daytime and found t h a t it agreed well with the observations if half the surface was craters with a depth-to-diameter ratio of about 0.5. They suggested t h a t the most probable diameter of the craters is a few millimeters. Buhl (1967) also estimated t h a t the surface of a hemispherical crater is about 30°K hotter than its surroundings during an eclipse and t h a t deeper craters remain even hotter, and suggested t h a t the hot spots are

regions with deeper or more numerous craters. He pointed out t h a t such regions would exhibit only small microwave anomalies whereas ff the hot spots were smooth regions of higher thermal inertia than their surroundings there would be a considerable effect on the microwave measurements. His own microwave measurements of Tycho and Copernicus detected no anomaly. Roelof (1968) has considered the thermal behavior of single cubical rocks of high conductivity (4.5 × 10-3 cal cm-lsec -1 °K -1) with edges from 0.3 to 300 cm resting on a medium of lower conductivity with a prescribed surface temperature. This is the first three-dimensional thermal diffusion calculation to be related to the lunar surface. Roelof concluded t h a t the effect of small rocks (assumed isothermal) would be to produce apparent temperature enhancements of about 15°K during eclipse; but otherwise the influence of isolated rocks of any size on Earth-based measurements is probably small compared to t h a t produced by "roughness" of the surface and t h a t only local measurements from spacecraft are likely to be appreciably affected. I t is clear from this review t h a t there is

I N T E R M E D I A T E SCALE L U N A R R O U G H N E S S

considerable evidence from the observations of the thermal radiation from the lunar crust to support the idea that there is significant centimeter or millimeter scale roughness. In addition, several other entirely independent approaches lead to the same conclusion. These will now be enumerated and briefly described.

1. Radar Reflection The diffuse component of the returned radar power (e.g., Evans and Hagfors, 1964) gives a measure of the roughness at the scale of the radar wavelength employed. The results show a mean slope of about 0.02 on a meter scale increasing to 0.4 or greater at centimeter scales, the increase being particularly sharp at about 1 cm. Using time delay and Doppler frequency change caused b y libration as positional discriminants, the radar echo intensity can be measured for localized areas. Pettengill and Henry (1962) have shown that the crater Tycho reflects about eight times the radiation reflected b y its surroundings. Although this may partly result from exposed rock of high density it seems likely that it is due in part to increased roughness of the crater surface (Evans and Hagfors, 1964).

2. Photometric and Polarization Data in the Visual Wavelength Range There is little doubt that the visual photometric properties of the Moon cannot be reproduced b y experiments using smooth rock samples (e.g., van Diggelen, 1959). Materials or material aggregates whose photometric functions resemble that of the Moon have been variously described as having (a) a surface with holes with vertical walls and sharp edges (Orlova, 1956), (b) a surface two-thirds of which is occupied b y deep porous wells (van Diggelen, 1959), (c) a porous surface with open interconnected spaces (Hapke, 1963; Hapke and Van Horn, 1963). At visual wavelengths the polarization of radiation scattered from the lunar surface and from various rough surfaces have been shown to be similar (e.g., Dollfus, 1962). This data

293

is evidence for roughness on any scale greater than the wavelength of the light.

3. Micro'meteorite Statistics The available data about micrometeorite flux densities and the size and shape of the craters which the micrometeorites can produce would lead one to expect considerable surface roughness on a centimeter and millimeter scale. Although the Ranger space vehicle photographs seem to indicate that topographical features on a scale of one to hundreds of meters are frequently a result of selenological disturbances, possibly of a lava flow nature, all b u t the most recently formed will have surfaces that have been severely influenced b y meteoritic bombardment at centimeter and millimeter scales. There is considerable evidence both from terrestrial craters (Baldwin, 1963) and from Ranger photographs (Baldwin, 1965) that the depth-todiameter ratio of craters increases as the diameter decreases, so it might be expected that smaller craters be relatively deeper.

4. Photographs from Soft Landing Vehicles Photographs from Luna 9 and the Surveyor and Apollo spacecraft confirmed in a direct w a y the high value of the roughness at centimeter and millimeter scales; distances less than about 1 mm could not be resolved.

5. Laboratory Experiments In addition to laboratory investigations of crater formation b y impact other attempts have been made to simulate the lunar surface, all of which result in the production of intermediate scale roughness: Fielder et al. (1967) have examined the effect of the boiling off of gases occluded in molten materials when at high temperatures they are suddenly exposed to a vacuum and subsequently allowed to cool, and consider the resulting samples to resemble lunar lava flow. Mills (1969) has experimented with fluidized beds, principally to provide an alternative explanation for crater formation (see Murray, Spiegel, and Theys, 1969). The work of Hapke and Van Horn, who produced "fairy castle"

294

J . A . B A S T I N A N D D . O. G O U G H

s t r u c t u r e s b y sifting small particles in v a c u o has a l r e a d y been m e n t i o n e d . III. THE ROUGH MODEL T h e m o d e l surface a d o p t e d in this w o r k is a mosaic of regions of a horizontal p l a n e into each o f which is cut a regular a r r a y of parallel troughs h a v i n g vertical walls a n d

horizontal bases (Fig. 2). T h e t r o u g h s h a v e t h e s a m e d e p t h d, w i d t h a, a n d are spaced w i t h i n t e r v a l b. All these dimensions are small c o m p a r e d to the linear dimension of a n y region; t h e shapes a n d sizes o f the regions a n d t h e a z i m u t h a l orientations o f the t r o u g h s are a s s u m e d to be r a n d o m l y distributed. A l t h o u g h the model seems r a t h e r artificial, it was chosen in the h o p e

f

J--~'-b

J

.

J

(ii)

Fro. 2. Perspective diagrams of the model considered in this paper. The upper drawing (i) shows three elements of the mosaic. All the top surfaces lie in one horizontal plane, and the floor sttrfaces of the troughs lie in another. The lower drawing (ii) defines the distances a, b, d and the coordinates y, s, z.

INTERMEDIATE

t h a t it would retain with a fair degree of accuracy most of the thermal properties of any rough surface with steep faces and at the same time he simple to study. The use of vertical trough sides was not merely for computational simplicity: the evidence presented at the end of the previous section suggests t h a t such a model would not be unreasonable. The surface bounds a semi-infinite homogeneous medium in local thermodynamic equilibrium, with density p, specific heat c, and thermal conductivity ]C. Fourier's law of heat conduction is assumed to apply and it is presumed t h a t the medium has no internal heat sources so t h a t the temperature T(r, t) at time t and position r in the interior is determined by 0T

O--t- = KV2 T,

(3.1)

where K is the thermometric conductivity (]c/pc). The boundary condition at the surface is - ] c n . V T = (1 -- al) I1 + (1 - x (I~ -- ~T4),

always outside the medium; the kernel is given by 3~f(r, r') = [(r -- r').n(r')][(r' -- r).n(r)]

k--r1' (3.5)

and the insolation is

/(r) = n.l~(r, 1) ~f

Ii(r) = I(r) q- 31 fs ~f(r, r') Ii(r')

1, --

o,

-.,f12 < ,/, < -/2,

I,/,I

(3.3)

12(r) ----fs ~f(r, r')[(1 -- ~2) aT4(r') (3.4)

where I(r) is the direct radiation received at r from the Sun, and the integration is over all parts of the surface S t h a t can be joined to r by a straight line which remains

(3.7)

where ¢ is the zenith angle of the Sun: ¢ =

(3.2)

dS',

(3.6)

where 1 is a unit vector in the direction of the Sun, f is the solar constant (assumed constant at 0.0332 cal cm-~sec-1), and E is the proportion of the solar radiation reaching the surface. The shadowing function fl(r,l) is zero if a straight line from the surface at r is the direction 1 intersects the surface again (r in shadow) and is unity otherwise. For lunation calculations the finite size of the solar disk was ignored so t h a t

~2)

where n(r) is a unit vector normal to the surface; Ii(r), the total incident radiant energy at the solar temperature received per unit time per unit area of the surface at r; I2, the incident radiation which originated as emission from the surface; al and a2 are the weighted mean albedos for radiation at solar and lunar temperatures, and a is the Stefan Boltzman constant. I t is assumed in addition t h a t all the radiation incident on the surface which is not absorbed is isotropically scattered without change in wavelength and t h a t the surface radiates according to Lambert's law, so t h a t

+ 32 I2(r')] dS',

295

SCALE LUNAR ROUGHNESS

-

2j ;

(3.s)

oJ is the synodical angular velocity of the Moon (2.463 × 10-%ec-1), and j is an integer chosen to keep ¢ in the range (-~, ~). During eclipse • is assumed to be the degree of obscuration of one circle by another, neglecting solar limb-darkening and other complicating effects. Only those cases where the path of the center of the Sun in the sky passes through the position of the Earth's center were considered. To compare the properties of the model with the infrared observations it is convenient to compute the apparent mean temperature of the surface as seen by an observer in the equatorial plane of the Moon at zenith angle ~. In order to do this we first need the flux E~dh emitted from a point r on the surface in the wavelength range (h, ;~ q- d2). Under the assumption of isotropic scattering of unabsorbed incident radiation, it is easy to see t h a t E~(r) = (1 - ~ ) F~[T(r)]

+ ~;~ n.lfl(r, 1) •f~ +

~fs~(r,r')E~(r')dS ',

(3.9)

296

J. A. BASTII~ AND D. O. GOUGH

where a~ is the albedo at wavelength )t,

fd,~ is the solar radiation flux at the Moon in (2,2 + d2) and F;~(T)d,~ is the flux t h a t would be e m i t t e d b y a black b o d y a t t e m p e r a t u r e T. The surface average E;~(k), for an observer in the direction k, can be c o m p u t e d from

i

fs n'kfl(r'k)dS

dO, (3.10)

where 0 is the inclination of the length o f the wells in a p a r t i c u l a r region to a line of latitude ; the a p p a r e n t surface t e m p e r a t u r e T~ is t h e n given b y T~(k) = F~I(E~).

(3.11)

F o r infrared observations A~-5

F~(T) FxI(E) =

e~/~T- 1' X

21n(1 +

A/ASE) '

Rayleigh-Jeans law is a good a p p r o x i m a tion, so t h a t T~(r, k)

~-, f : K,~(r) T(r) × exp(--

f~ K~(r')dr')dr, (3.12)

where r is the position on the surface from which the r a y finally escapes into free space and r is the distance from r. The value of effective absorption coefficient Ka is a p p r o x i m a t e d b y (Troitskii, 1962) K~ = / ' % / 2 ~ v K A-1,

(3.13)

where )~ is measured in centimeters and F ~ 0.5 cm -1. Although ~ is c o n s t a n t in the medium, it is nevertheless a function of r since glancing rays e n t e r and leave the m e d i u m several times before finally escaping into free space. Once again it is necessary to c o m p u t e an average over rays to obtain the effective brightness t e m p e r a t u r e T~(k).

Data for the Model

w h e r e A = 3.741 × 10 -5, x = 1.439 ()tin cm) and f;~ can be neglected. One can also estimate the effective t e m p e r a t u r e T e from i n t e g r a t e d emission b y setting a ~ = al, f ~ = f , and F~(T)= aT 4. T h e microwave brightness t e m p e r a t u r e T~ is o b t a i n e d b y solving the equation of radiative transfer along a p a t h in the direction k. Reflection a t the surface was ignored. I n this region o f the spectrum the

The values of the surface and bulk physical p a r a m e t e r s a d o p t e d in this s t u d y are presented in Table I t o g e t h e r with the references from which t h e y were deduced. I t will be noticed that, e x c e p t for the specific heat, t h e y are s o m e w h a t higher t h a n the references suggest. The n o r m a l l y accepted values are in most cases o b t a i n e d b y comparing observations of the Moon with the plane homogeneous model with

TABLE I PHYSICAL CONSTANTS ADOPTED Symbol

Description

p

Density

v

Specific heat

al

Albedo for incoming solar radiation Albedo for outgoing lunar thermal radiation

a2

Value adopted

References

Hapke (1963), Hapkc and Van Horn (1963), Salisbury and Adler (1967) Garstang (1958), Glaser and Wechsler (1965), Sinton (1962) Allen (1962), Harris (1961) Burns and Lyon (1962), Murcray (1965), Van Tassel and Simon (1964)

1.0 gm cm -a 0.2 cal gm -1 see -1 °K-I 0.12 0.14

297

I N T E R M E D I A T E SCALE L U N A R R O U G H N E S S

TABLE II PARAMETERS :FOR THE STANDARD ROUGH MODEL

Symbol

Description

]c

Conductivity

d a b

Trough depth Trough width Trough interval

References J a e g e r (1953a), L o w (1965), T r o i t s k i i (1964, 1965), W e s s e l i n k (1948)

constant physical properties (hereafter referred to as the plane model). If, as is the thesis of this paper, the plane model is considerably in error, the values deduced from it will not be relevant. We should distinguish between the parameters appropriate to intermediate scale roughness (microscopic values, which are those which should be used in our calculations) and the macroscopic values deduced from observations which do not resolve the individual roughness elements. As an example, the mean apparent (macroscopic) value of al deduced from Earth-based observation will be lower than the value pertinent to our model since an incoming photon scattered from the surface of one of the troughs might be reabsorbed. In all cases we have adopted microscopic values so t h a t the predicted apparent macroscopic values deduced from our model agree reasonably well with the references cited in Table I. Although our analysis has treated the parameters in Table I as constants, we have examined the dependence of the properties of the model on variations of the trough shape and size as well as variations in conductivity. As a basis for our work we have, however, considered in detail the properties of a model whose parameters are given in Table I I and will be referred to as the standard model. Various other values of/c, a, b, d have been considered, and these will be discussed when a comparison between the calculations and the observations is made.

Smooth Models We are not suggesting t h a t the lunar thermal anomalies can be accounted for

---

Value adopted 5.56 × 10 -6 c a l c m -1 s e c -1 ° K - 1

4 cm 4 cm 8 cm

only by roughness. In an attempt to give an objective appraisal of some of the other mechanisms for producing them we have investigated not only the plane homogeneous model but also various models with variable conductivity. For comparison with observation we have used the results of these calculations together with those reported by Jaeger (1953a), Saari and Shorthill (1963), and by Linsky (1966).

Method of Solution By assuming t h a t each region of parallel troughs is large compared to the trough dimensions and separation, it is justified to ignore the edge effects from its boundaries. Each region can be examined without regard to its neighbors and the conduction problem becomes two-dimensional and periodic in the horizontal direction. The shadowing problem remains three-dimensional and the position of the shadow depends on the azimuthal orientation of the troughs; for this reason one must consider a distribution of orientations over which the apparent infrared and microwave brightness temperatures must be averaged. The heat conduction equation was solved numerically as an initial value problem in a single trough interval, applying periodic boundary conditions in the horizontal and integrating until the solution became periodic in time. The domain of integration was truncated at a depth D, about ten to t w e n t y times the depth of the trough and always sufficiently great for the temperature variations at t h a t depth to be less t h a n 1% during a lunation. Beyond a few trough depths beneath the surface the temperature distribution is almost

298

J. A. BASTIN AND D. O. GOUGH

independent of the horizontal coordinate and the one-dimensional approximation: aT a2 T at -- K az 2

(3.14)

was used in order to economize on computing time. Here z is the vertical coordinate measured downwards from the upper surface (Fig. 2). The boundary condition at the upper surface can be reduced to a condition on the bounding curve C of the cross section by integrating the kernel in Eqs. (3.3) and (3.4) in the longitudinal coordinate y (see Fig. 2) yielding

I i ( s ) = I(s) + ~1 J~c K(s,

8')

sistently using centered second-order operators to approximate spatial derivatives and employing an explicit time formulation as follows: I f T~ is the temperature at time t, at the grid point (i,j) on a regularly spaced orthogonal spatial grid with interval h and with axes horizontal and vertical, equations of the type

aT~ot t:t~ = Fij(Tr,,)

(3.18)

were represented by T~ +1 : T ~ + 1At[Fii(T~ra) -~ -Fij(Tz*m) ],

I1(~') ds', (3.15)

I2( s) = f c K(s, s')[(1 - as) aT4(s ') + a2 I2(s')] ds',

(3.16)

where

r(r,r')dy and s is a distance parameter on the curve C (Fig. 2). Similar integrations can be performed in the expressions for E~ and E~. Assuming t h a t the Sun remains always in the Moon's equatorial plane one can show t h a t for a trough at latitude ~bwhose length is orientated at an angle 0 south of east

T'~u - T~o = 2hk-a[T~l( Trm) - (1 -- ~1)

× .(T~I) 4] and Eq. (1) by

{cos¢cos¢

for horizontal faces n-I = | l s i n 0 sin ¢ + cos 0 cos ¢ sin ¢1 for the illuminated vertical face. The boundary condition used at the base of the domain was aT a~ = 0 at z = D.

where At = t n + l - t,. In the interior Fit is simply the five-point Laplacian difference formula (or three-point where the onedimensional approximation is made) multiplied by ~; on the upper and lower boundaries Fit was computed by introducing image points outside the domain and eliminating the dependent variable at these points with the use of Eq. (3.1). I t is not obvious how this is best done. For example, on an upper horizontal boundary at (i, 1) Eq. (3.2) might be formally represented by

(3.17)

Alternatively one might match the solution to a downward propagating sinusoidal thermal wave of frequency oJ, but it was found t h a t the condition (3.17), which is insulating, led to a more rapid approach to the time-periodic solution. Difference equations were constructed to represent the differential system con-

a T il

n tffit, = ~h-2(T~° + Ti~ + T in- l l

-~- T n + l l -- 4T~l ).

The temperature at the image point (i,0) can be eliminated between these equations to yield Eq. (3.18) with F i l ~- Kh-2(2T~2 -{- T n _ l l -~- T n + l l - - 4T~1 )

+ 2(pch)-l[I~l(Trm) - (1 -- al) × ~(T~I) 4] - B71 - 2(1 - ~l)(pch) -1 ~(TT~) 4.

But it was found to be preferable to estimate the radiation loss term at time

INTERMEDIATE

SCALE

½(tn + tn+l) instead of tn. Thus, if we write

Tn+ il 1

-

-

n __ T~I - - At < B~I

+

( 1 -- al)(pch) -1

× a {(T~I)4 + 4(T}~l)s T~I+1

LUNAR

299

ROUGHNESS

yielded apparent effective temperatures of the standard model which were different b y less t han ½°//o, except when the angle of observation was less t h a n 10 ° above the horizon when differences of up to 10~/o were obtained.

Properties of the Model

we obtain, neglecting terms quadratic in ( T~

~ - T~'~)IT~I,

Fil =

Bin1-- 2(1 -- ~l)(pch) -1 a(Tnl) 4 1 + 4(1 -- c¢i)(pch) -1 a A t ( T ~ ) a" (3.19)

B y so throwing the radiation loss slightly out of phase with the incident radiation and the interior conduction the numerical scheme became considerably more stable. At convex corners a similar procedure was adopted using two image points. At the concave corners vertical and horizontal gradients from which the Laplacian is formed were estimated by averaging the actual gradient with t h a t which would be required to balance the appropriate component of the incident radiation: thus for the b o tto m left corner at (i,j), say, Fij is of the same form as F~1 in Eq. (3.19) but with n

-2

3

~

n

a_Tn

4T}~)

1

n

2(#ch) -1

1 v n i h n × [2Iij(Tzm) + ~I~j(Tlr~)

--

(I - -

al) a(T~)~],

where I~ and I/~ are the vertical and horizontal components of the total radiation flux incident at (i,j). The results presented in this paper were obtained on a grid with dimensions 12 × 96 and generally with a time step such t h a t Kh-2At,~0.24 ( A t = 2.5 × 10-4 lunar days for the standard model), which was just below the limit of stability, though the relaxation to the time-periodic state was computed with a coarser grid, to save time. For eclipses a shorter time step was used, to ensure adequate resolution. Test runs using a grid of dimensions 20 × 160

The temperature distribution in the model depends on the size and the shape of the indentations. To characterize the size it is convenient to consider the two thermal length scales L1 = (2K/oJ)1/2, which is the penetration depth of a thermal wave of frequency ~o, and L2 = k/(aTs3), which is the temperature scale height at a black surface of temperature T s during free radiative cooling. For the models considered here, L 1 z 5 cm and L 2 ~ 4 × 10-6/Ts a c m (Ts in °K). During most of the day L2 is of the order of 1 mm and we can distinguish three regimes for the standard rough model: Regime A RegimeB Regi m eC

a >> L1 LI>>a>>L2 L2>>a.

I n Regime A the central regions of the ridges receive little information about the temperature of the boundary and so remain at an almost constant temperature. In Regimes B and C significant information can propagate across the ridges in a time short compared to ~o-1 so t h a t the temperature distribution within the ridges depends principally on the instantaneous temperature on the boundary. Where a >> L2 conduction at the surface does not significantly affect the energy balance, so the temperature distribution on the surface depends only on the surface geometry. Conduction is important in determining the surface temperature only in Regime C, when the ridges are more or less isothermal, and when the surface behaves almost as though it were smooth but with a modified albedo. Of course, during the night or during eclipse, conduction is important in determining surface temperatures in all three regimes. We have focused our attention mainly on intermediate scale roughness : Regime B.

300

J.A.

B A S T I N A N D D. O. GOUGH

320 i O-O"~ 0 0 ~3~

300 80

J

1-280 ..-'260 ~0

,380 e=3°°

~ 0 -., f 3 o o

...,...---280 ~260 -240 320~~ e-6o"

1~ J 360 l

320

~26o

~

~240 30C) ~

•320 -3~

300~ 3 2 0

B = 90"

320 ~ / 300 ~2~ ~240 FzG. 3. I s o t h e r m s for regions o f t h e s t a n d a r d r o u g h m o d e l a t l a t i t u d e ~b = 60°N w h e n t h e S u n ' s z e n i t h angle ~ is zero. T h e angle 8 is t h e local selonographic o r i e n t a t i o n o f t h e l e n g t h o f t h e t r o u g h (y direction) m e a s u r e d clockwise f r o m east. T h e m a p s for 0 = 120 ° a n d 150 ° are a l m o s t i d e n t i c a l t o t h o s e for 0 = 60 ° a n d 30 °, a n d are n o t shown.

I N T E R M E D I A T E SCALE L U N A R ROUGHNESS

301

= 180"

T$ = 9 6 ° K 100 100

-' ~

..----.----.120

-

8o1

-

140

80

(~=-120"

Ts = 9 0 " ~ / v

|0o 120

140

280

2624 ~

380

Ts = 2 7 8 ° K

' ~°t~

~

\~1

2,0

!

I/

/~oo

,~

5~ ~ ~ ~ 2 2 2 240

302

J . A. BASTII~ A N D D . O. G O U G H

280 380 .280 ~280

Ts=282°K' ~ l 240 r"

~260

I100 Ts =[09°K ~

f

~

240 100 120 40

~---~--160 ..-180 -200

FIG. 4. Isotherms for a region of the standard rough model at latitude ~b= 60 ° with troughs aligned to the meridian (8 = 90°) for different zenith angles ~ of the Sun. The surface temperatures Ts of the plane model of the same material and at the same location are also shown. I s o t h e r m s of the s t a n d a r d model at latitude 60 ° a n d at m i d d a y are depicted in Fig. 3 for various orientations 8. I t will be noticed t h a t the surface t e m p e r a t u r e varies considerably, especially w h e n 8 = 0 ° w h e n the Sun's radiation, p r o j e c t e d onto the cross-sectional plane, is most glancing. Figure 4 shows the t e m p e r a t u r e distribution for 8 = 90 ° as a function of phase ~b. I n the first p a r t of the morning and the l a t t e r p a r t of the a f t e r n o o n the glancing insolation preferentially heats the u p p e r regions of the surface which reradiates relatively efficiently. The e n e r g y propag a t e d d o w n w a r d b e n e a t h the lower horizontal faces is considerably less t h a n the n e t e n e r g y received b y the plane model. N e a r m i d d a y when shadowing is less severe, the b o t t o m of the well can be m a i n t a i n e d at a high t e m p e r a t u r e because

m u c h o f the r a d i a t i o n from it is reabsorbed, a n d the t o t a l e n e r g y flux into the m e d i u m is greater. Figure 5 shows the isotherms during eclipse cooling. D u r i n g the night, the low surface temp e r a t u r e s m a k e L2 similar to L1 a n d now t h e r e are only two regimes to distinguish. W h e n a >> L1 t e m p e r a t u r e gradients p e r p e n d i c u l a r t o the surface are m u c h greater t h a n the surface gradients a n d the e n e r g y balance a t the surface is d e t e r m i n e d primarily b y t h e g e o m e t r y a n d n o t b y the scale o f the model. T h e plot against size o f the excess over the plane model a t midnight of the effective t e m p e r a t u r e a n d a p p a r e n t surface t e m p e r a t u r e s a t 4 F a n d 14 F, w h e n viewed from o v e r h e a d (Fig. 6), illustrates this point. I t is m a i n l y because the r o u g h model has considerable surface t e m p e r a t u r e variations t h a t it appears

380 r

360

Ts=397°K

.------~- 300

26O

280 55 rain Ts =264°K

~ i :

3

~

~

~320 f 3 0 0

2OO 180min Ts =/78°K

3OO 320

320 - 300

5~-0rni~0

380

T ~

~ 300 '-- 320 FIG. 5. I s o t h e r m s d u r i n g a n d a f t e r eclipse for a region o f t h e s t a n d a r d r o u g h m o d e l w i t h 8 = 90 ° a t t h e c e n t e r o f t h e d i s k (~b = 0 °, ¢ = 0 ° in t h e m i d d l e o f t h e eclipse). T h e eclipse is one for w h i c h t h e t r a j e c t o r y o f t h e S u n ' s c e n t e r in t h e l u n a r s k y p a s s e s t h r o u g h t h e p o s i t i o n o f t h e E a r t h ' s c e n t e r ( m a x i m u m eclipse d u r a t i o n ) . T i m e is m e a s u r e d f r o m t h e i n s t a n t a t w h i c h o b s c u r a t i o n o f t h e Sun a t t h e disk c e n t e r begins. T h e last m a p is a t a t i m e 30 m i n a f t e r t h e eclipse a t t h e disk c e n t e r h a s e n d e d . T h e surface t e m p e r a t u r e s Ts o f t h e p l a n e m o d e l are also shown.

304

J. A. BASTIN AND D. O. GOUGH

S

• 4/J.

20 14/~

O-O:)

Z~T °K

I0

0

J

I I0

0

I 0

I 20

I

I 30

cm

FIG. 6. D e p e n d e n c e o f t h e m i d n i g h t b r i g h t n e s s t e m p e r a t u r e a t t h e c e n t e r o f t h e disk on w a v e l e n g t h a n d on t h e scale o f roughness. E x c e s s e s o v e r t h e p r e d i c t i o n s o f t h e p l a n e m o d e l o f t h e o v e r h e a d b r i g h t n e s s t e m p e r a t u r e s o f regions o f r o u g h m o d e l s w i t h 0 = 90 ° are p l o t t e d as a f u n c t i o n o f a, for )t = 4 F, )~ = 14/~ a n d for i n t e g r a t e d r a d i a t i o n . T h e s h a p e s o f t h e r o u g h m o d e l s are all t h e s a m e as t h a t of the s t a n d a r d model (a = b/2 = d).

hotter, since in averaging infrared and integrated emission greater weight is given to areas of high temperatures.

The radiation is beamed quite strongly in the vertical so it does not necessarily follow that the rough model is losing heat

I

i

25O

TQK

20O

150

0"

I

I

30 °

60"

q,

~0°

FIG. 7. S t e a d y t e m p e r a t u r e below t h e p e n e t r a t i o n o f t h e t h e r m a l w a v e as a f u n c t i o n o f l a t i t u d e . O p e n circles are t h e values c a l c u l a t e d for t h e p l a n e h o m o g e n e o u s m o d e l , closed circles are for t h e s t a n d a r d r o u g h model.

INTERMEDIATE SCALE LUNAR ROUGHNESS

faster. During the day the emitted radiation is beamed approximately towards the Sun.

To compare the total energy balance of different models, integrated over a lunar cycle, one need only look at the temperature T~ at great depths. I t was found that for ¢ = 0 and for a in the range 2-25 cm, in the model with a - - - d = b / 2 , Too remained almost constant and close to that of the plane model. At higher latitudes there is less direct insolation into the wells and T~ decreases below that of the plane model (Fig. 7). The shape of the indentations is important too, deep wells being more efficient at trapping radiation and so producing higher values of T~. For example, T~ for the model with 2a = b -~ d = 4 cm at ¢ = 0 was about 8°K higher than for the smooth model, whereas for the model with a = d = b/3 = 4 cm it was 2°K lower. I V . COMPARISON WITH OBSERVATION

All those thermal observations which cannot be accounted for b y the theory of the plane model will now be considered. They will be compared with the prediction of the present rough model and also with any other models which have been used to account for them. We shall classify the thermal anomalies according to the following three criteria: (i) surface-subsurface (ii) daytime-nighttime-eclipse (iii) global-localized Not all the possible categories have welldocumented anomalies. Those anomalies which will be discussed in this section are first listed below :

Surface, daytime, global (1) Poleward and equatorial scans at full moon show less limb-darkening than would be expected from the plane model. (2) Subsolar point measurements give a directional variation of apparent temperature, with the maximum at about full moon.

305

Surface, daytime, localized (3) Regions of high visual albedo have temperatures below average. (4) Surface isotherms show a correlation with lunar terrain.

Surface, nighttime, global (5) The midnight brightness temperature decreases with increasing wavelength in the range 8-20 ~. (6) Temperatures during the early part of the lunar night do not drop so rapidly as the predictions of the plane model that yields the correct midnight temperature. (7) Nighttime brightness temperatures are higher than those predicted b y the plane model that best explains the eclipse observations.

Surface, nighttime, localized (8) Over the lunar surface, the predawn temperature varies from 70 ° to 150°K. (9) The rayed craters remain hotter than their surroundings for several days after lunar sunset.

Surface, eclipse, global (10) During eclipse, the temperature decreases first less rapidly, then more rapidly, and finally less rapidly than would be expected from the plane model. (11) Cooling during eclipse is more rapid near the limb than at the disk center.

Surface, eclipse, localized (12) There are a number of anomalous "hot spot" regions that remain up to 60°K hotter than their local surroundings during eclipse. (13) H o t spots are located preferentially towards the center of the lunar disk. (14) H o t spots are located at regions of high radar return. (15) The more intense hot spots have an infrared brightness temperature that varies rapidly with wavelength. (16) The maria remain warmer than the upland areas during eclipse.

Subsurface, daytime, global (17) The mean microwave temperature is greater than the mean infrared temperature, and increases with wavelength.

306

J.A.

BASTII~ A N D D. O. OOUOH

(18) Microwave poleward darkening is more p r o n o u n c e d t h a n would be e x p e c t e d from the plane model. (19) The increase in microwave t e m p e r a t u r e after lunar dawn is less rapid t h a n t h a t predicted b y the plane model. 1. Poleward and Equatorial Scans at Full Moon

Figures 8 and 9 show a p p a r e n t surface t e m p e r a t u r e s deduced e x p e r i m e n t a l l y at 8-14 /x b y Saari and Shorthill t o g e t h e r with the predictions at 11/x from various models. The results h a v e in all cases been normalized so t h a t the t e m p e r a t u r e at the disk center is unity. All the rough models give b e t t e r agreem e n t with e x p e r i m e n t t h a n the simple s m o o t h model. P a r t i c u l a r l y good agree-

ment

is f o u n d for the

model h a v i n g

b - a = d = b/4 = 4 cm. T h e e x p e r i m e n t a l

results t h u s f a v o r a flat surface o f which a relatively small area is covered b y blocks, r a t h e r t h a n a surface covered b y crevices or cracks. The models with t e m p e r a t u r e d e p e n d e n t c o n d u c t i v i t y a n d the various layered models predict results which differ b y less t h a n 1% from those predicted b y the plane model. F o r this reason no a t t e m p t has been m a d e to represent t h e m on the diagrams b u t it is clear t h a t t h e y c a n n o t explain the e x p e r i m e n t a l results. 2. Anisotropy of the Brightness Temperature at the Subsolar Point

The a n i s o t r o p y of the surface radiation e m i t t e d from the subsolar point observed b y P e t t i t a n d Nicholson (1930) has been

S

i.8 o.i C,~) -r(o)

/

0"4

0.2

I 90 S

60"

| 30 °

0°

I

I

30"

60 ° ~

90" N

FIG. 8. Poleward variation of the meridian brightness temperature at 11 g. The open circles refer to the measurements of Saari and Shorthill (1967a). The thick continuous curve indicates the values obtained for the plane model. The thin continuous curve was obtained from the standard rough model, and the broken curve from the rough model with b -- a = d = b/4 = 4 cm.

INTERMEDIATE SCALE L U N A R ROUGHNESS

307

b~.lk %. %. 0"8

/

I

0"6

T

(o)

0"4

L.O. 2

I

I

i

I

i

I

I

-90"

-60*

-30"

O"

30"

60"

90"

E

w

FIG. 9. Equatorial temperatures at full moon. The notation is the same as in Fig. 8. confirmed by Geoffrion, Korner, and Sinton (1960) and by Saari and Shorthill ( 1967a). The measured fluxes are plotted against zenith angle $ in the polar diagram Fig. 10, together with the predictions of various models. None of the smooth models have any angular dependence. However, the rough model with b - a - - d = b/4 = 4 cm is in good agreement with the observations. The rough surface thus exhibits a gross emission albedo which depends on direction. The vertical walls of the troughs are cooler t h a n the illuminated horizontal surfaces and so the energy flux decreases as the zenith angle increases from zero and the contribution from the walls increases. In a narrow region at the top, the sidewalls are heated by lateral conduction just beneath the illuminated horizontal surface. At very oblique angles all the contribution from the walls comes from this region and there is a slight increase in the energy flux.

The region is too narrow to be adequately resolved on the grid t h a t was used, so the precise forms of the curves in Fig. 10 near = 90 ° are not to be trusted. Indeed, the increase probably depends quite critically on the geometry of the model and m a y not occur at all for the actual lunar surface. The observation by Pettit and Nicholson t h a t for any zenith angle the apparent brightness temperature is greater when the Moon is waning than when it is waxing has not been confirmed by the more recent observations (Geoffrion, Korner, and Sinton, 1960; Saari and Shorthill, 1967a). The models illustrated in Fig. 10 do in fact give fluxes which are about 1% higher for the waning moon. However such an effect is scarcely larger than the deviations produced by localized variations in the lunar surface and limitations of the observing techniques. Somewhat greater differences are predicted for larger blocks of

~08

J.A.

BAS~NA.ND

D.O. GOUGH

O

\

5 \

\\ ° :° oi/ \ \ °\i o°

', \',

/

/

/ I

FIG. 10. N o r m a l i z e d radiant intensity Fa at 11/z at t h e subsolar point as a function of the zenith angle ~ of observation. Open and filled symbols refer, respectively, to values o b t a i n e d from t h e w a x i n g and w a n i n g m o o n : triangles refer to P e t t i t a n d Nicholson (1930), squares to Geoffrion et al. (1960), and circles to Saari and Shorthill (1967a). The n o t a t i o n for t h e theoretical curves is t h e same as in Fig. 8.

dimensions which are close to the penetration distance L1 and which are small enough for the west face to receive sufficient heat from the east face by conduction to inhibit its cooling, yet not so small t h a t the thermal capacity is insufficient to keep the temperature distribution out of equilibrium. For very large blocks the effect again becomes small. The only other authors to have theoretically derived the subsolar point radiation pattern are Buhl et al. (1968). Of the models they examined, best agreement with observation was obtained with models having a considerable fraction of vertical or almost vertical surfaces (shallow cylinders or hemispheres) and in this respect agrees with our findings. Buhl et al. favor hemispherical craters to their shallow cylinders, but there seems to be no justification for this in the present context, especially since they compare their results

with the early and somewhat inaccurate measurements of Pettit and Nicholson. Indeed, the more recent measurements, which yield a polar diagram more pointed at the zenith, seem to favor the reverse. 3. Correlation of Temperature with Albedo The deviations of the observed points from the predicted curves in Figs. 8 and 9 are strongly correlated with local variations of absorption albedo which are deduced from the photometric function. I f the brightness temperatures are adjusted to the values they would be if all the lunar surface had the same albedo and assuming the actual emission albedos vary in the same way as the albedo for absorption at solar wavelengths (following Troitskii, 1965, and taking due account of the slope of the Planck function at 8-14 ~), the result is in good agreement with the

309

I N T E R M E D I A T E SCALE L U N A R R O U G H N E S S

than Low's measurement at 20 ~. Such a result would be expected from the rough model, whose surface has quite considerable relative temperature variations. Because the dependence of the Planck function on temperature steepens with decreasing wavelength at about 100°K (Wein approximation), the hotter regions contribute relatively less to the emitted radiation as the wavelength increases, and so yield lower apparent brightness temperatures (Fig. 6). The standard model (averaged over 8) has a difference of about 5°K between the brightness temperatures at 11 and 20 ~, and the model with b - a = d = b[3 = 4 cm has a difference of 3°K. I t should be noted that outcrops of comparatively highly conducting rock that are almost completely submerged in the general surface material would also produce surface temperature variations, independent of roughness (see Allen and l~ey, 1969).

theoretical curve. I t should be noticed, however, that the observed albedo varies with lunar phase (Saari and Shorthill, 1967b), suggesting that it is modified b y roughness of the surface.

4. Correlation of Daytime Temperature with Terrain Features Measured brightness temperatures over the illuminated lunar disk are correlated with visual features. As a result of the increased incident flux, ground sloping towards the Sun is hotter than ground sloping away. But, unlike that in many other areas, the temperature variation on crater slopes is not as great as the global variation depicted in Figs. 8 and 9 (Saari and Shorthill, 1963), and this points to greater intermediate scale roughness. This conclusion is not inconsistent with the observation that the brightness tempera, ture of many of the crater floors, after adjustments for albedo, are higher than the surrounding temperatures.

6. Slow Cooling Rates at Sunset

5. Wavelength Variation of the Midnight Brightness Temperature in the 820 /~ Wavelength Range

This anomaly was shown clearly b y the measurements of Murray and Wildey (1964) and was interpreted b y later authors as indicating the temperature dependence of the conductivity of lunar rock. Plane homogeneous models with temperatureindependent conductivity cannot possibly account for the observed results; this can be seen in Fig. 11, which shows also the good agreement obtained from a temperature-dependent plane model.

The measurements of the infrared brightness temperature of the center of the disk at lunar midnight are given in Table III. Because the fluxes are low there is considerable experimental inaccuracy, which explains the diversity in the results, b u t there seems little doubt that the 8-14 /z measurements show a higher temperature TABLE

III

LuNJLR MIDNIGHT BRIGHTNESS TEMPERATUHES

Observer

Date

Wavelength band (~)

Lunar midnight brightness temperature at center of disk (°K)

Petit and Nicholson Petit and Nicholson corrected by Saari Sinton Murray and Wildcy Saari Low

1930 1964 1962 1964 1964 1965

8-14 8-14 8-14 8-14 8-14 20

120 ° 109 ° 122 ° 107 ° 100 ° 99 ° {estimated from average across a disk diameter)

3]0

J . A. B A S T I N AND D. O. GOUGH I

180

I

160

T °K 140

\ ' , .

-

_

120

1(10

9ff

i

120"

i

150°

I

180=

Fro. I 1. Cooling at the disk center during the first half of the lunar night. Filled circles indicate the measurements of Murray and Wildey (1964). The thick continuous curves are the 11-~ brightness temperatures of two plane models; the numbers are the values of the thermal conductivity in cal cm -1 sec -1 °K -1. The thin continuous curve is for the rough model with b = d = 2a = 8 cm. The broken curve is for a smooth model with the temperature dependent conductivity k = 3 × 10-6 (1 ÷ 4 × 10-s T 3) cal cm -1 see -1 °K-1. Since this result provides one o f the chief a r g u m e n t s in f a v o r of t e m p e r a t u r e d e p e n d e n t c o n d u c t i v i t y it is pertinent to see to w h a t e x t e n t r o u g h models m i g h t a c c o u n t for it. The models with shallow indentations favored b y the anomalies a l r e a d y considered differ from the plane model b y too little to explain the observations. A model with deep wells is b e t t e r (Fig. 11), a n d could doubtless be i m p r o v e d b y reducing its scale, b u t there is little other evidence to s u p p o r t it. I t is interesting to note, however, t h a t a plane model with small, densely spaced, deep n a r r o w wells similar to t h a t described b y W i n t e r (1965) t r a n s p o r t s h e a t vertically b y c o n d u c t i o n a n d radiation in a similar

m a n n e r to particulate models with c o n d u c t i v i t y of the form A + B T ~.

7. High Lunar Nighttime Temperatures Measurements of the central disk midn i g h t brightness t e m p e r a t u r e s a n d their i n t e r p r e t a t i o n h a v e an interesting history. The early 8-14/~ m e a s u r e m e n t s p r o d u c e d a value of a b o u t 120°K which, on the basis of the plane homogeneous model, implies a value of the t h e r m a l p a r a m e t e r y = (kpc) -1/~ of 600-700 cm ~ sec 1/2 ca1-1 °K. The eclipse measurements, on the o t h e r hand, i m p l y the considerably higher v a l u e : y = 1000-1200. However, m o s t of the more recent m e a s u r e m e n t s during a lunation listed in Table I I I i m p l y t h a t y ~ 850,

INTERMEDIATE SCALE LUNAR ROUGHNESS

and if we accept Low's measurement a value of about 1000 is obtained. This apparent agreement cannot be reconciled with the idea that the thermal conductivity depends on temperature, which we know is strongly suggested b y other anomalies (Muncey, 1958; Clegg et al., 1966; Linsky, 1966). We should expect a considerably lower value for ~ during eclipse than during the lunar night because the temperatures near the surface are higher. Roughness again offers an explanation for the apparent paradox: because of the temperature variations on the surface the microwave brightness temperature T~ during the night is greater than the effective temperature Te (Fig. 6), and the proportional excess increases as the night progresses, partly because the Planck function steepens with decreasing temperature at constant ~ and partly because the relative temperature fluctuations increase. The brightness temperature therefore drops more slowly than the effective temperature, which is a measure of the heat loss rate, and so an estimate of ~ based on T~ will be too low. This effect is less pronounced during eclipse and m a y even be reversed in the early stages, so the real differences in ~ between eclipse and nighttime are compensated for. For example, the model we have favored above (b-a=d=b/3~-4 cm) can be made to agree with observations if a variable conductivity of about that used for Fig. 10 is used.

8. Local Variations in Measured Predawn Surface Temperatures We have examined a variety of rough models to see whether roughness might be responsible for the very large temperature variations measured b y Low at 20/~ before lunar dawn. In the central region of the disk, plane models predict lower brightness temperatures than rough models of the same material throughout the whole of the lunar night, and one might suppose that the lowest recorded brightness temperature (70°K) resulted from the observation of a plane surface. However, we could find no rough model which had brightness temperatures as high as 150°K, the highest

311

to be observed, though deep-welled models have vertical brightness temperatures at 20/z in excess of 100°K just before dawn. The possibility that the effective thermal inertia varies b y a factor of 20 or more over the lunar surface will be considered in the discussion of hot spots.

9. Enhanced Crater Temperatures Early in the Lunar Night There is no doubt that several of the rayed craters are maintained at temperatures considerably above their surroundings for several days after lunar sunset. The results published so far (Murray and Wildey, 1964; Murray and Westphal, 1965 ; Saari and Shorthill, 1967a) show that immediately after sunset the excess is normally about 25-30°K and decreases with an e-folding time of about 36 hr. A similar b u t smaller enhancement is found in many of the maria. I t seems hard to explain such a slow decay of the temperature difference b y assuming differences in the temperature dependence of the thermal conductivity alone, though a more sophisticated model in which the conductivity is independently a function also of depth, such as that of Winter and Saari (1969), might succeed. On the other hand, Fig. 11 shows that an explanation invoking increased roughness is certainly possible. In this connection it is worth noting that, as would be expected from a roughness hypothesis, this anomaly has not been observed near the limb. 10. Anomalous Eclipse Cooling A least-squares fit of the experimental data for the lunar disk as a whole (Pettit, 1940 ; Saari and Shorthill, 1963) to a plane homogeneous model with uniform conductivity shows that both at the beginning and in the final stages of cooling the actual rate of cooling during eclipse is less rapid than predicted b y the model. The comparatively slow early cooling at high temperatures is almost certainly a result of the initially high conductivity, whereas the later stages can be explained b y assuming that the conductivity is also an increasing function of depth, as was first

312

J . A. BASTIN AND D. O. GOUGH

suggested by Jaeger and Harper (1950). The type and extent of roughness considered in the previous sections does not appreciably affect this explanation. The shallow model with b - a = d -- b/3 = 4 cm cools somewhat more slowly than the plane model with the same conductivity, but the cooling curve is scarcely distinguishable from t h a t of a plane model with a thermal inertia about 5% greater. 11. R a p i d L i m b Cooling d u r i n g E c l i p s e

Measurements made by Sinton (1962) and Saari and Shorthill (1963) show t h a t during eclipse the drop in brightness temperature near the limb compared to t h a t at the center of the disk is generally greater than would be expected on the basis of the plane model. In regions 2' of arc from the limb (zenith angle of observation 60 ° ) the brightness temperature at the center of totality is about 10°K lower than disk center measurements would suggest. No smooth opaque model can produce such a result, but the rough models we have considered do, good agreement with observation being obtained with the standard model. Observed variations of this effect with position on the disk are high, so little significance can be attached to the agreement with this particular model. 12-15. H o t S o o t s I t does not seem to be possible to account for the more intense hot spots with variations of emissivity alone (Fudali, 1966). Nor can concentrations of internal heat sources provide the explanation, since these would produce even more intense hot spots during the lunar night. There are three other suggestions which have been advanced and which seem more plausible: local increase in thermal inertia (Fudali, 1966), a particulate model with local increase in particle size (Winter, 1967), local increase in degree of roughness. We have examined the behavior of a number of rough models under eclipse conditions in an attempt to account for the hot spot observations, the basic idea being t h a t the hot spot regions have a greater degree of roughness than their

immediate surroundings. The general results of the computations show the following : (i) Rough models have higher brightness temperatures after eclipse cooling than the plane model with the same conductivity. For deep wells near the center of the disk the enhancements can be as high as 60°K during totality. The enhancements decrease towards the limb. (ii) Rough models of the same shape yield enhancements which are independent of size when a is greater than about 10 cm, and which decrease in the centimeter and millimeter region (Fig. 12). Although these findings suggest t h a t an explanation for hot spots in differential roughness may be found, it must be admitted t h a t some modification is necessary to account for the most intense spots. In particular, models with crevices which are sufficiently deep to produce the largest enhancements have too high a surface temperature during the lunar day to be consistent with observation, ~ even when an allowance is made for the measured high albedo of the spots. However, the decrease in the brightness temperature enhancements predicted by roughness near the limb seem to be consistent with the radial distribution of hot spots observed by Saari and Shorthill (1967a), though it may be possible to explain this statistic as a result of the finite angular resolution of the detector. The rayed craters, which are known to be intense hot spots, have been described by both Fudali (1966) and Winter (1966), who have made similar suggestions for their evolution. I t was postulated t h a t some catastrophic event produces a rough surface littered with rocks which ha~e sufficiently high slope to prevent them from being covered by a layer of micrometeoritic debris. This idea is supported by the high radar return intensity (Thompson and Dyce, 1965) from hot spots, suggesting t h a t they are regions of high density or roughness. 2 H u n t et al. (1968) r e p o r t a w e a k h o t s p o t in Mare H u m o r u m w h i c h a p p e a r s h o t t e r t h a n its surroundings during the day.

INTERMEDIATE SCALE LUNAR ROUGHNESS 40

313

~

30

ATOK 20

I0

0 0

i

I

I0

20

30

C] c m .

F r o . 12. E x c e s s c e n t e r o f disk, 11-/~ b r i g h t n e s s t e m p e r a t u r e s o f r o u g h m o d e l s a t t h e m i d p o i n t o f a n eclipse as a f u n c t i o n o f scale. T h e c o m p a r i s o n is m a d e w i t h t h e p l a n e m o d e l o f t h e s a m e m a t e r i a l . T h e s h a p e s o f t h e r o u g h m o d e l s are all t h e s a m e as t h e s t a n d a r d m o d e l (a = / ) / 2 = d).

According to Winter, it is the geometry of the surface which causes the thermal anomaly. Fudali, on the other hand, supposes t h a t the exposed rocks have a high conductivity and it is this which is responsible for the high eclipse temperatures. I t seems likely to us t h a t both these mechanisms operate to reinforce each other, as has been suggested already by Saari and Shorthill (1967a). The case of Tycho deserves discussion since a number of different kinds of observation of this crater have now been made. The recent results of Alien and Ney (1969) indicate very clearly t h a t during the lunar night we receive radiation from at least two types of surface within Tycho with temperatures of about 100 ° and 200°K. This cannot be explained by our simple rough model unless some regions of it have a much higher conductivity than the mean. I t may be possible for the particulate model of Winger and Saari (1969) to produce such high temperature differences if the contact conductivity between the particles were sufficiently low. Meter-size rocks of high conductivity resting on a surface of low conductivity cannot provide an explanation (l%oelof, 12

1968), for in order to remain at a high enough temperature it is necessary either for rocks to be so large t h a t their own heat capacity is sufficient to store the excess energy t h a t is radiated (> 20 meters) or t h a t they are outcrops and thus are thermally bonded to a large accessible reservoir of heat beneath the surface. Allen and Ney find t h a t their measurements are consistent with the assumption t h a t one-tenth of the surface is of high conductivity rock, which is plausible in the light of the Surveyor VII photographs. However, there is some independent evidence t h a t the roughness of the surface is greater than usual: In the first place, if the surface were smooth a 10% area of rock would not account for the high radar return intensity (Pettengill and Thompson, 1968). Secondly, the temperature of Tycho around midday, when compared to the value at dawn, seems to be higher than one would expect on the basis of the plane model or a composite model of two plane models of different conductivity. Figure 13 shows the daytime temperature excesses of the crater over the surroundings deduced from maps of Saari and Shorthill (1967b). Also shown are the excesses over a plane

314

J . A . BASTIN AND D. O. GOUGH I

i

I

-

I

3O A T °K

-20

0

f

-90

=

-60

-10

I

=

-30

•

=

0

I "t"30 °

I +60

'=

"1"90 =

FIG. 13. Difference b e t w e e n t h e 11- F b r i g h t n e s s t e m p e r a t u r e s of t h e c r a t e r T y c h o a n d is s u r r o u n d ings t h r o u g h o u t l u n a r d a y t i m e . T h e t h i n c o n t i n u o u s line is t h a t p r e d i c t e d b y a s m o o t h c o m p o s i t e m o d e l for T y c h o h a v i n g 0.1 o f t h e surface c o n s i s t i n g o f m a t e r i a l w i t h (kpc) -~ = 20 a n d t h e r e m a i n d e r h a v i n g (kpc) - t = 1000, t h e s u r r o u n d i n g s b e i n g e n t i r e l y of m a t e r i a l w i t h (kpc) - j = 1000. T h e t h i c k c o n t i n u o u s line r e p r e s e n t s t h e s a m e m o d e l b u t w i t h a n excess f r a c t i o n of a b o u t 0.15 of t h e a r e a of T y c h o c o v e r e d w i t h blocks w h i c h m a y or m a y n o t b e of t h e h i g h e r c o n d u c t i v i t y m a t e r i a l . T h e c u r v e w a s c o n s t r u c t e d b y a d d i n g t o t h e c u r v e for t h e m o d e l w i t h a s m o o t h surface 0.3 o f t h e difference b e t w e e n t h e s t a n d a r d r o u g h m o d e l a n d t h e p l a n e h o m o g e n e o u s model. T h e filled circles r e p r e s e n t t h e m e a n t e m p e r a t u r e excesses w i t h i n t h e c r a t e r r i m d e d u c e d f r o m t h e c o n t o u r m a p s of Sa~ri a n d S h o r t h i l l (1967b); t h e o p e n circle is a n e x t r a p o l a t i o n t o d a w n o f t h e m e a s u r e m e n t o f A l i e n a n d N e y (1969).

model w i t h (kpc) -1/2 = 1000 of a combination of plane models nine-tenths of which has (kpc)-l/2= 1000 and one-tenth with (kpc)-l/~= 20, which was suggested by Allen and Ney to account for their nighttime measurements. The figure shows good agreement between the composite model and the daytime measurements. The agreement can be improved if the effects of roughness on the radiation are taken into account. We conclude therefore t h a t although the hot spot anomaly in Tycho is likely to be primarily the result of exposed areas of high conductivity, the thermal measurements suggest t h a t the region has a higher degree of roughness than its upland surroundings.

16. Enhanced Temperatures Maria during Eclipses

of

Lunar

The infrared measurements of Saari and Shorthill (1967b) show t h a t during eclipse

the maria appear on average 7.5°K warmer than their surroundings. Although the maria appear warmer during the lunar day too, this excess ( ~ 6 ° K ) cannot account for as much as half of the excess observed during eclipse on the basis of albedo variations alone. One possibility is t h a t the maria have more intermediate scale roughness than the upland areas; for example, the standard model predicts a center of disk 11-/~ brightness temperature 5°K higher than the model with b - a = d = b/3 = 4 cm at full moon, but at the center of totality of an eclipse the excess is nearly 10°K. Of course, differences between the thermal inertia of the maria and their surroundings are a very plausible alternative. To check the possibility of greater intermediate scale roughness within the maria, we compared the 8-14 brightness temperatures of maria with those of their surroundings on both sides of the disk throughout the lunar day,

315

INTERMEDIATE SCALE LUNAR ROUGHNESS

using the measurements of Saari and Shorthill. An area centered in Mare Crisium, on the western side, was found always to have a higher brightness temperature than the surrounding uplands. The difference increases throughout the day as the angle of illumination approaches the angle of observation. The reverse trend was observed on the eastern side when regions of maria and upland were compared near Aristarchus. This result is certainly consistent with the beaming of emitted radiation which roughness produces. 17-19 Microwave Brightness Temperatures Muncey (1958) pointed out t h a t the mean, taken over all phases, of the lunar microwave brightwave brightness temperature exceeds the corresponding mean brightness temperature at 8-14 t~. Since the former relates to subsurface thermometric temperatures and the latter depends basically on the surface temperature, Muncey interpreted the result as indicating an increase of conductivity and specific heat with temperature. Linsky (1966) gave a more detailed analysis of the problem, attributing the effect to the presence of radiative conductivity (see also Clegg et al., 1966). Linsky, and also Troitskii (1967) summarized the available microwave data and concluded t h a t the mean brightness temperature increases with wavelength. There is an initial sharp increase with wavelength of the average brightness temperature in the range below 2 cm, which results from the rectifying effect of the temperature-dependent conductivity on the heat input. Beyond this, the more gradual increase with wavelength may be due to a global central heat source. Although this explanation for the initial rapid increase seems satisfactory, it should be pointed out t h a t rough surfaces produce a similar result ; in fact, the standard model predicts an effect of about the right magnitude. I t is interesting t h a t highly conducting outcrops of the type considered in the discussion of Tycho could also account for the result. Because the angular resolution of most microwave telescopes is relatively poor, it is customary to summarize the poleward

darkening by defined by

a

single parameter

m

T~(¢) _ costa ¢,

T~(0)

where T~(¢) is the brightness temperature on the meridian at latitude ¢. We have calculated m for the plane model and various rough models as a function of wavelength )~ and phase ¢. There is some resemblance between the computed and measured values, and the rough models agree marginally better with the observations than the plane model does. All models predict t h a t m has a minimum at ¢ = 0 (full moon) but this is observed only in the 8-mm measurements of Salomonovich (1967). In general the measured values are considerably greater than the models predict. The poor agreement was disappointing because it was hoped t h a t an estimate of the scale of the roughness would have been obtained from this analysis. Perhaps roughness on a somewhat smaller scale than t h a t considered would produce the observed result, but we did not consider the possibility sufficiently likely to warrant the computing time necessary to test it. Compared to the predictions of the plane homogeneous model the microwave temperatures measured at the center of the disk are low just after dawn and just before and after sunset. This shows up most strongly in the 1-mm observations (Bastin et al., 1964; Low and Davidson, 1965). In Fig. 14 we plot the results of Low and Davidson together with the predictions of the plane model and the rough model with b - a = d = b / 3 = 4 cm. Since the value of the constant F in Eq. (3.13) for the absorption coefficient was obtained by comparison with the predictions of the plane model with y = (kpc)-l/2= 350, we are at liberty to choose a different value for our models. We therefore chose t h a t value of Kz which yielded good agreement with observation just before dawn (K~.lmm 2.5 cm). It can be seen t h a t although the rough model is better than the plane model, the improvement is hardly significant. For comparison, the prediction of the particulate model of Winter and Saari is

316

J . A . BASTIN AND D. O. O 0 U O H I

I

1.0

I

•

•

Tx 1~)10.8 T x (0)

~~, oI

0.4

oo

0.2

0

-180 °

n --90 °

)

)

0

90°

¢

180°

F I o . 14. R e l a t i v e c e n t e r o f d i s k b r i g h t n e s s t e m p e r a t u r e s a t 1.2 m m a s a f u n c t i o n o f p h a s e ¢ . T h e filled circles indicate the measurements of Low (1965). The continuous curve was obtained from the r o u g h m o d e l w i t h b -- a = d = b/3 = 4 e r a a n d t h e b r o k e n c u r v e . . . . was obtained from the plane model. The broken curve -----is t h e p r e d i c t i o n o f t h e p a r t i c u l a t e m o d e l o f W i n t e r a n d S a a r i ( 1 9 6 9 ) .

also included in the figure; it is somewhat b e t t er in the lunar morning and a little worse in the afternoon. I t is worth remarking, however, t h a t the observed values for the early part of the morning were deduced from scans, and so m a y not be a true measure of the brightness temperature at the center of the disk. Comparison was also made with the observations at 3.3 mm. We have not displayed the results because t hey are similar to those at 1.2 mm: the rough and plane models predict very similar curves, though th ey were obtained for somewhat different values of the absorption coefficient, the rough model yielding lower apparent temperatures after dawn and around sunset than the plane model, and the rough model yielding higher values t h a n Winter and Saari's model before midday and lower values afterwards. The values of x~ required to produce agreement with observations just before dawn for both the rough and the smooth models were consistent with the linear dependence of KA-1 o n h.

V.

CONCLUSION

Roughness is responsible for the pronounced directional properties of the lunar thermal radiation. The measurements of these properties at infrared wavelengths during the lunar day are consistent with the corresponding predictions of a model with a surface part of which is covered with raised blocks. As well as this more obvious result there are a number of other features of the lunar radiation which m ay be a consequence of roughness. These include the differences between the 10 and 20 ~ midnight brightness temperatures, center-to-limb variations in the rate of eclipse cooling, and the enhancement of the temperatures of r a y craters early in the lunar night. In addition various other phenomena, normally attributed to other causes, such as the enhanced maria temperatures and the increase of mean microwave brightness temperature with wavelength m ay also be to some degree caused by roughness. B ut there are, however, two other properties, namely the slow global cooling at the

INTERMEDIATE SCALE LUNAR ROUGHNESS

beginning of the lunar night and the exact form of the eclipse cooling, which we find cannot be explained even in part by roughness but which already have satisfactory explanations. There is evidence that maria have a somewhat higher degree of intermediate scale roughness than upland areas. It is not impossible that the weaker hot spots are entirely due to roughness. Observations of the thermal radiation from those ray craters which are intense hot spots indicate that the terrain of these craters is rougher than the surroundings and there is strong evidence to suggest the presence of a small fraction of surfaces of thermal conductivity about 10 -3 cal cm-isec -1 °K -I which may well be rocky outcrops. The comparison of microwave observations with our calculations weakly indicates some preference for a rough terrain though the correspondence is not good enough to give any information about its nature. In summary, we feel that roughness satisfactorily accounts for many observed deviations from the predictions of the smooth homogeneous model, though there are a number of other anomalies which can only be adequately explained by other means, principally by considering the conductivity to vary with location as well as depending on both temperature and depth. ACKNOWLEDGMENTS W e t h a n k Dr. J . T o o m r e for helpful discussions. This w o r k was started w h e n b o t h a u t h o r s were visitors at the J o i n t I n s t i t u t e for L a b o r a t o r y Astrophysics of t h e N a t i o n a l B u r e a u of Standards and t h e U n i v e r s i t y of Colorado, and was c o m p l e t e d while D.O.G. held a N A S - N R C Senior P o s t d o c t o r a l R e s i d e n t R e s e a r c h Associateship. REFERENCES ALLEN, C. W. (1962). "Astrophysical Quantities' ', (2nd Edition), p. 145. Athlone Press, London. ALLEN, D. A., AND NEY, E. P. (1969). Lunar thermal anomalies: Infrared observations. Science 164, 419-421. BALDWIN, R. B. {1963). The Measure of t h e Moon. U n i v . of Chicago Press, Chicago, Illinois.

317

BALDWIN, R. B. (1965). The crater d i a m e t e r d e p t h relationship f r o m R a n g e r V I I photographs. A. J . 70, 545-547. BASTIN, J . A. (1965). L u n a r h o t spots. Nature 207, 1381. BASTIN, J . A., CLEGG, P. E., GEAR, A. E., JONES, G. O., AND PLATT, C. M. (1964). L u n a r brightness distribution at 1.5 m m w a v e l e n g t h . Nature 203, 960-961. BASTIN, J . A., AND GEAR, A. E. (1967). L u n a r observations in the w a v e l e n g t h range 1-3 m m . Proc. Roy. Soc. (London) A.$96, 348-353. BASTIN, J. A., GEAR, A. E., JONES, G. O., SMITH, H. J . T., AND WRIGHT, P. J. (1964). Spectroscopy at e x t r e m e I n f r a r e d w a v e l e n g t h s I I I . Astrophysical and atmospheric measurements. Proc. Roy. Soc. (London) A.278, 543-573. BUHL, D. (1967). R a d i a t i o n Anomalies on t h e L u n a r Surface. Space Science Lab. R e p t . Series 8, No. 1, U n i v . of Calif., Berkeley. BUHL, D., WELCH, W. J., AND REA, D. G. (1968a). R e r a d i a t i o n and t h e r m a l emission f r o m illuminated craters on the lunar surface. J. Geophys. Res. 73, 5281-5295. BUHL, D., W E L C H , W . J., A N D REA, D. G. (1968b). A n o m a l o u s cooling of a cratered lunar surface. J. Geophys. Res. 73, 7593-7608. B u r N s , E. A., AND LYON, R. J . P. (1962). E r r o r s in the m e a s u r e m e n t of t h e t e m p e r a t u r e of the Moon. Nature 196, 463-464. CLEGG, P. E., BASTIN, J. A., AND GEAR, A. E . (1966). H e a t transfer in lunar rock. Monthly Notices Roy. Astron. Soc. 1S3, 63-66. CLEGG, P. E., NEWSTEAD, R. A., AND BASTIN, J. A. (1969). Millimetre and s u b m i l l i m e t r e a s t r o n o m y . Phil. Trans. Roy. Soc. 264, 293-305. DOLLFUS, A. (1962). Polarisation studies of planets. I n " P l a n e t s a n d Satellites" (B. Middlehurst, ed.) pp. 343-399. U n i v . of Chicago Press, Chicago, Illinois. EVANS, J. V., AND HAGFORS, T. (1964). On t h e i n t e r p r e t a t i o n of r a d a r reflections from t h e Moon. Icarus 3, 151-160. FIELDER, G. (1961). " S t r u c t u r e of the Moon's Surface." P e r g a m o n Press, N e w York. FIELDER, G., GUEST, J. E., WILSON, L., AND ROGERS, P. S. (1967). N e w d a t a on s i m u l a t e d L u n a r material. Planetary and Space Sci. 15, 1653-1666. FUDALI, R. F. (1966). I m p l i c a t i o n s of t h e n o n u n i f o r m cooling b e h a v i o r of the eclipsed Moon. Icarus 5, 536-544. GALILEI, G. (1632). D i a l o g o . . . d o v e ne i congressi di q u a t t r o giornate si discorre sopra i due massimi sistemi del m o n d o tolemaico, e copernicano; . . . , B a t i s t a Landini, Florence.

318

J.A. BASTIN AND D. O. GOUGH

GARSTANG,R. H. ( 1958). The surface temperature of the Moon. J. Brit. Astron. Assoc. 68, 155-165. GARY, B. (1967). Results o f a radiometrie Moonmapping investigation at 3 m m wavelength. Astrophys. J. 147, 245-254. GARY, B., STACE¥, J., AND DRAKE, F. D. (1965). Radiometric mapping of the Moon at 3 m m wavelength. Astrophys. J. Suppl. Ser. 12, 239-262. GEAR, A. E. (1965). " A Study of Millimetre Wavelength Radiation from the Sun and Moon," pp. 30-60. Thesis, London University. GEAR, A. E., AND BAST:N, J. A. (1962). A corrugated model for the lunar surface. Nature 196, 1305. GEOFFRION, A. R., KORNER, M., AND SINTON, W. M. (1960). Isothermal contours of the Moon. Lowell Obs. Bull. 5, 1-15. GLASER, P. E., AND WECHSLER, A. E. (1965). Small scale structure of the lunar surface. Icarus 4, 104-105. GIBSON, J. E. (1961). Lunar surface characteristics indicated by the March, 1960, eclipse and other observations. Astrophys. J. 1as, 1072-1080. HAPKE, B. W. (1963). A theoretical photometric function for the Lunar Surface. J. Geophys. Res. 68, 4571-4586. HAPKE, B., AND VAN HORN, H. (1963). Photometric studies of complex surfaces with applications to the Moon. J. Geophys. Res. 68, 4545-4570. HARRIS, D. L. (1961). P h o t o m e t r y and colorim e t r y of planets and satellites. I n "Planets and Satellites" (G. P. Kuiper and B. M. Middlehurst, eds.), pp. 272-342. Univ. of Chicago Press, Chicago, Illinois. I~IuNT, G. R., SALISBURY, J. W., AND VINCENT, R. K. (1968). Lunar eclipse: Infrared images and an anomaly of possible internal origin. Science 162, 252-254. JAEGER, J. C. (1953a). Conduction of heat in a solid with periodic boundary conditions, with an application to the surface temperature of the Moon. Proc. Camb. Phil. Soc. 49, 355-359. JAEGER, J. C. (1953b). The surface temperature of Moon. Australian J. Phys. 6, 10-21. JAEGER, J . C., AND HARPER, A. F . A. (1950). Nature of the surface of the Moon. Nature 166, 1026. KROTIKOV, V. D., AND SHCHUKO, O. B. (1963). The heat balance of the lunar surface layer during a lunation. Soy. A s t r o n . - - A . J . 7, 228-232. KROTIKOV, V. D., AND TROITSKH, V. S. (1964). Radio emission and the nature of the Moon.

Soviet Phys.--Uspekhi (Engl. transl.) 6, 841-871. LETTAU, H. H. (1962). A theoretical model of thermal diffusion in non-homogeneous conductors (with applications to conditions in the Moon's Crust). Gerlands Beitrgge zur Geophysilc 71, 257-271. L:NSK¥, J. L. (1966). Models of the lunar surface including temperature-dependent thermal properties. Icarus 5, 606-634. Low, F. J. (1965). Lunar nighttime temperatures measured at 20 microns. Astrophys. J. 142, 806-808. LOW, F. J., AND DAWDSON, A. W. (1965). Lunar observations at a wavelength of 1 millimeter. Astrophys. J. 142, 1278-1282. MILLS, A. A. (1969). Fluidization phenomena and possible implications for the origin of Lunar Craters. Nature 224, 863-866. MUNCE¥, R. W. (1958). Calculations of lunar temperatures. Nature 181, 1458-1459. MUNCE¥, R. W. (1963). Properties of the lunar surface as revealed by thermal radiation. Aust. J. Phys. 16, 24-31. MURCRAY, F. H. (1965). The spectral dependence of lunar emmissivity. J. Geophys. Res. 79, 4959-4962. MURRAY, B. C., AND WESTPHAL, J . A. (1965). Infrared astronomy. Sci. A m . 213 (2), 20-29. MURRAY, B. C., AND WILDEY, R . L. (1964). Surface temperature variations during the lunar nighttime. Astrophys. J. 139, 734750. MURRAY, J . D., SPIEGEL, E . A., AND THEYS, J. (1969). Fluidization on the Moon (?). Comments Astrophys. Space Sci. 1, 165-171. O~LOVA, N. S. (1956). Photometric relief of the lunar surface. Astron. Zh. U.S.S.R. 33, 93-100. PETTENGILL, G. S . , ANn HENRY, J . C. (1962). Enhancement of radar reflectivity associated with the lunar crater Tycho. J. Geophys. Res. 67, 4881-4885. PETTENGILL, G. H., A N D THOMPSON, W. W. (1968). A radar study of the lunar crater Tycho at 3.8 cm and 70 cm wavelengths. Icarus 8, 457-471. PETTIT, E. (1940). Radiation measurements on the eclipsed moon. Astrophys. J. 91, 408-420. PETTIT, E., AND NICHOLSON, S. B. (1930). Lunar radiation and temperatures. Astrophys. J . 71, 102-135. PII)DINGTON, J. H., AND MI~r~rETT, H. C. (1949). Microwave thermal radiation from the Moon. Aust. J. Sci. Res., Series A2, 63-77. ROEI, OF, E. C. (1968). Thermal behavior of rocks on the lunar surface. Icarus 8, 138-159. SAAB:, J. M. (1964). The surface temperature of

INTERMEDIATE SCALE LUNAR ROUGHNESS the Antisolar point of the Moon. Icarus 3, 161-163. S.~a~i, J . M., AND SHORTHI£Z,, R. W. (1963). Isotherms of crater regions on the illuminated and eclipsed Moon. Icarus 2, 115-136. SAARI, J. M., AND SHORTHILL, R. W. {1965). Thermal anomalies on the totally eclipsed Moon of December 19, 1964. Nature 205, 964-965. SAARI, J. M., AND SHORTHILL, R. W. (1967a). Review of lunar infrared observations. I n "Physics of the Moon," Vol. 1S, pp. 57-99. Science and Technology Series, American Astronautical Society. SAARI, J. M., AND SHORTHILL, R. W. (1967b). Isothermal and Isophotic Atlas of the Moon: Contours through a Lunation. NASA Report CR 855. SALOMONOVICH, A. E. (1962). Lunar thermal radio emission in the centimetre band and some characteristics of the surface layer. Sov. Astron.--A. J . 6, 55-61. SALOMONOVICR, A. E. (1967). Measurements of lunar radio brightness distribution and certain properties of its surface layer. Prec. Roy. See. (London) A296, 354-365. SALOMONOVICH, A. E., AND LOSOVSKII, B. YA (1962). Radio-brightness Distribution on the Lunar disc at 0.8 cm. Sov. Astron.--A. J. 6, 833. SALISBURY, J'. W., AND ADLER, J. E. M. (1967). Density of the lunar soil. Nature 214, 156158. SHORTHILL, R. W. (1962). Measurements of Lunar temperature variations during an eclipse and throughout a lunation. Boeing Document DI-82-0196. SHORTHILL, R. W., BOROUGH, H. C., AND CONLE¥, J . M. (1960). Enhanced lunar thermal radiation during a lunar eclipse. Publ. Astron. Soc. Pacific 72, 481-485. SMITH, B. G. (1967). Lunar surface roughness, shadowing, and thermal emission. J. Geophys. 2es. 72, 4059-4067. SINTON, W. M. (1960). Eclipse temperatures of the lunar crater Tycho. Lowell Obs. Bull. 5, 25-26. SINTON, W. M. (1961). Recent radiometric studies of the planets and the Moon. In

319

"Planets and Satellites" (G. P. Kuiper, ed.), Chap. 11. Univ. of Chicago Press, Chicago, Illinois. SI~rON, W. M. (1962). Temperatures OH the lunar surface. I n "Physics and Astronomy of the Moon" (Z. Kopal, ed.), Chap. 11. Academic Press, New York. THOMPSON, T. W., AND DYCE, R. B. (1965). Mapping of lunar radar reflectivity at 70 cm. J. Geophys. Res. 71, 4843-4853. TROITSKII, V. S. (1962). Nature and physical state of the surface layer of the Moon. Soy. Astron.--A. J . 6, 51-54. TROITSKII, V. S. (1964). Certain results of lunar investigation by radiophysical methods. Soviet Astron.---A. J . 8, 76-79. TROITSKII, V. S. (1965). Nonuniformity in the properties of the top layer of the lunar surface. Soviet Astron.--A. J . 8, 576-582. TROITSKII, V. S. (1967). Investigation of the surfaces of the Moon and planets by means of thermal radiation. Prec. Roy. Soc. (London) A296, 366-398. VANDIGGELEN, J. (1959). Photometric properties of lunar crater floors. Rech. Obs. Utrecht 14, No. 2. VAN TASSELL, R. A., AND SIMON, I. (1964). Thermal emission characteristics of mineral dusts. I n " T h e Lunar Surface Layer, Materials and Characteristics" (J. W. Salisbury and P. E. Glaser, ed.), pp. 445-468. Academic Press, New York. WESSELINK, A. J. (1948). H e a t conductivity and nature of the lunar surface material. Bull. Astron. Inst. Netherlands 1O, 351-363. WINTEr, D. F. (1965). "Transient Rad i at i v e Cooling of a Semiinfinite Solid with Parallelwalled Cavities." Boeing Document D1-820449; also Intern. J. Heat Mass Transfer (1966), 9, 527-532. WINTER, D. F. (1966). Note on the nonuniform cooling behavior of the eclipsed Moon. Icarus 5, 551-553. WINTER, D. F. (1967). Transient radiative heat exchange at the surface of the Moon. Icarus 6, 229-235. WINTER, D. F., AND S ~ I , J. M. (1969). A particulate thermophysical model of the lunar soil. Astrophys. J. 156, 1135-1151.