The Martian twilight

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 86, NO. A7, PAGES 5833-5838, JULY 1, 1981

The Martian Twilight RALPH KAHN AND RICHARD GOODY

Center for Earth and Planetary Physics, Harvard University, Cambridge, Massachusetts 02138

JAMES POLLACK

Space Science Division, NASA Ames Research Center, Moffett Field, California 94035

The changing sky brightness during the Martian twilight as measured by the Viking lander cameras is shown to be consistent with data obtained from sky brightness measurements. An exponential distribution of dust with a scale height of 10 kin, equal to the atmospheric scale height, is consistent with the shape of the light curve. Multiple scattering resulting from the forward scattering peak of large particles makes a major contribution to the intensity of the twilight. The spectral distribution of light in the twilight sky may require slightly different optical properties for the scattering particles at high levels from those of the aerosol at lower levels.

1. INTRODUCTION

The twilight measurements on the Viking lander permit us to study the vertical distribution of scattering properties in the Martian atmosphere. In effect, the combination of the planetary shadow and the decreasing number density of particles with height forms the rays of the sun into a beam which scans to progressively higher altitudes as the evening twilight wears on (see the schematic representation of the source function in Figure 1).

In this paper we will discuss the Twilight Rescan Experi- ment performed with the Viking lander cameras. This experiment is a natural outgrowth of earlier work on the terrestrial twilight [Rozenberg, 1966; Volz and Goody, 1962]. The term 'twilight' refers to the time of day when the disk of the sun is just below the local horizon, while the sky overhead is illuminated by direct sunlight. We shall refer to the succession of events as they occur during the evening twilight; the morn- ing and evening twilights are geometrically equivalent. We have introduced the Twilight Rescan Experiment, together with the related Viking lander camera atmospheric optics experiments, in the work by Pollack et al. [1977, hereafter referred to as paper 1]. In paper 1 a model based upon single scattering was compared with the data. In ihis paper we show that multiple scattering plays a dominant role in the Martian twilight and the 9onclusions must be reconsidered.

Our overall conclusion is that, with a minor exception, the Martian twilight can be described quantitatively in terms of dust from the lower atmosphere with a reasonable height distribution. Thus we are not led to new conclusions about the physical state of the Martian atmosphere itself. Nevertheless, the twilight is a cl!•ssical problem of atmospheric optics, and a completely new set of data on a planet other than earth de- serves the effort to confirm that our theories are soundly based.

The analysis presented here is for the VL 1 sol 41 P.M. Twi- light Rescan, which we have selected as the best and most straightforward to ahalyze. Six complete Twilight Rescans are available. These occurred during a period in which the total amount of dust (as measured by solar extinction experiments) was increasing. Sol 41 is the earliest and is the occasion of the most transpa.rent atmosphere by at least a factor of 2. Since multiple scattering is a troublesome feature of the analysis,

Copyright ̧ 1981 by the American Geophysical Union.

Paper number 1A0337. 0148-0227/81/001A-0337501.00

we regard the results for sol 41 as the most tractable. More- over, on sol 41, solar attentuation measurements were made an hour or two before sunset, and a daytime sky brightness measurement was made a few hours earlier. The particle properties deduced from these experiments are reported by Pollack et al. [1979, hereafter referred to as paper 2]. We believe the other Rescans to be of sufficiently less value as not to justify the labor of reduction at this time.

The twilight geometry is illustrated in Figure 1 for the case of zero azimuth (A -- 0). Quantities are defined as follows:

•1 azimuthal angle of the camera direction measured in the plane tangent to the geoid at the location of the camera (•1 -- 0 is defined as the line of intersection of this plane with the plane containing S, O, and D. A very small change in •1 can take place during a twilight run. Although of minor significance, this effect is incor- porated precisely into our calculations.);

B a point effectively outside the atmosphere in the direction of the sun;

C intersection of the line of sight with the shadow edge (this is the location of the lowest primary scattering volume);

CD line of sight of the camera; D location of the camera; O center of the planet; R radius of the planet, equal to 3395 km; S location of the sunset point, where the incident solar

ray is tangent to the planetary surface; SC shadow edge of the planet (all points above SC are di-

rectly illuminated, while all points below receive only sca•ttered light);

Z height above the surface; Zo •]osest approach of the ray BG to the surface; Z• height of point C above the surface, i.e., the height of

the shadow in the line of sight; 8 solar depression angle at the camera (proportional to

}he time elapsed after sunset); e camera elevation angle;

*t* angular radius of the solar aureole; 0 angle of scatter of light from the direct solar beam into

the camera.

The natural independent variables for this problem are 8, e, •1(= 0), and Zo, which completely define the single scattering

5833

5834 KAHN ET AL.: THE MARTIAN TWILIGHT

••••ource X G function

(B) •• /(B)/•j

o

Fig. 1. The twilight geometry. Symbols are defined in the text. The curve labeled 'source function' is schematic and illustrates the distribution of scattered light along the line of sight. In a typical example with Zl -- 20 km the source function for single scattering peaked 30 km above the shadow edge with a full width at half maximum of 25 km.

geometry. To derive the twilight intensity for particular values of 8 and F, an integration is performed over Zo or an equivalent variable, since Zo defines the paths BG and GD. In prac- tice, we prefer to use, in addition to Zo, the independent variables t9(8, F, A • 0) and Zl(/•, •, A • 0): the former because it specifies the scattering phase function, the latter because it is related to the height at which most primary scattering occurs. A slight change in the azimuth angle allows t9 to be held constant while Z, varies.

We assume that the atmospheric optical properties are a function of Z only (except that we shall consider the possibility of mountains or clouds at the sunset point). This assumption is common to all attempts to interpret twilight data. The Viking camera generated useful data with e varying from about 5 ø to 35 ø for each/•, so we obtain information for each Z, value for a number of combinations of e and & This permits the selection of subsets of the data for which Z, and 19 can be varied independently. For example, we can extract data for a range of Z• values with a constant value of 19, thereby ob- taining information about the vertical distribution of scatter-

same accuracy. This translates into an uncertainty in the position of the shadow edge of about 1 km for the largest Z, values used.

The data were recorded with VL I camera 2. Calibration includes consideration of camera temperature, diode degrada- tion, and vignetting effects. The results reported by Patterson et al. [1977] and additional calibrations in the hands of the Vi- king project indicate an accuracy of ñ 10% in the absolute intensity measurements and ñ5% in relative intensity. These errors do not affect our conclusions in any signficant way. In order to minimize errors due to encoding, the twilight sequence consisted of three successive pictures taken with different gain settings. The overall level of the noise in the data can be judged from the scatter of points with respect to a smooth curve through the data in Figures 5 and 6. It is only of significance for the highest data points and does not influence our conclsuions.

We give the data in terms of observed brightness, normal- ized to that of a Lambert surface in the orbital position of Mars illuminated perpendicularly by the sun (the intensity ratio, Y). Account is taken of the seasonal variation of the Mars- sun distance in this procedure. The twilight sky brightness is sufficiently uniform to allow a linear scheme for interpolation of the data in Z, and 19.

The observed brightness is compared with that calculated from a theoretical model of the twilight. The model assumes a vertical distribution of scatterers which is spherically symme- tric about the planet. Recognition of the presence of discrete clouds or obstacles in the path of the twilight ray will be ex- plained in a later section. We neglect Rayleigh scattering, airglow, and atmospheric refraction. These assumptions will be discussed subsequently.

The model parameters include a vertical distribution of particles which determines the particle number density at each height. We also specify the particle mean radius, the particle size distribution, and the real and imaginary indices of refraction at each height. In selecting the input values we use the results of the solar extinction experiments to constrain the total optical depth of the column, and we use the results of the sky brightness experiment to obtain the particle properties (see paper 2). The optical depth in red light from paper I is 0.37 ñ 0.04. This differs from conclusions based on orbiter data, but, being based upon direct solar measurements, we consider it to be more reliable, since it is based on a model independent ap- plication of Beer's law. The quantitative results in this paper

ers which is independent of the scattering phase function of would not be greatly influenced if it were as small as 0.1. The the particles.

The curves in Figure 2 define the region of the Z•-O plane 5o for which useful data could be obtained during the VL 1 sol I

of the sky with small Z, and small 19 is accessible for observa- tion. Other aspects of the experimental design are discussed in - paper 1.

_

2. DATA REDUCTION AND ANALYSIS

41 P.M. Twilight Rescan Experiment with A -- 0. As time in- creases (increasing 8), we probe a region of the sky at increasing Z• and 19 values. If we repeat the geometric calculation with A -- 15 ø, the area of the Z•-19 plane covered is smaller than is shown in Figure 2, and, in particular, less of the region

The relationship between the camera direction and the sun position is given by the Viking lander ephemeris. This permits us to correct for lander tilt to within a few tenths of a degree and to determine the position of the local horizon with the

z 20

m 10

I I O0 10 20 30 40 50 60 70 80 Z ,• (km)

Fig. 2. Geometrical relationships between Z•,/•, •, and 19 for A -- 0.

KAHN ET AL.: THE MARTIAN TWILIGHT 5835

assumed particle properties influence the phase function shown in Figure 4. Much of Figure 4 is directly established from empirical data. The strength of the forward scattering peak is model dependent; its importance is considered when we discuss mutliple scattering.

Our procedure involves integration of the extinction along the paths of a sequence of solar rays BG to produce source functions for single scattering along the line of sight of the camera. We use a Mie scattering program to generate cross sections and phase functions for the particles. The essential purpose of this procedure is to infer the phase function for small scattering angles for which there are no direct sky brightness measurements. As discussed in paper 1, we can neglect the effect of particle shape on the scattering properties at such angles. Finally, we integrate the source function along paths GD coincident with the line of sight to evaluate the singly scattered intensity at the camera location. Multiple scattering will be discussed later.

We attempt to devise models which reproduce the shape of the observed intensity curves for each color. We typically plot the intensity ratio (Y) as a function of Z• for a fixed value of 0. The ratio of observed to model intensity for Z• equal to 20 km is designated by M; thus if M > 1, the model intensity is lower than the observed intensity. One value is obtained for each color, and we distinguish these with subscripts, Mb, M s, Mr for the blue, green, and red light, respectively. We seek a model for which Mb, s,r = 1 for all Z, values.

We now discuss the importance of each of the assumptions contained in our model.

Atmospheric Refraction and the Finite Size of the Sun

We calculate the deflection of the shadow edge from its un- perturbed height Z, due to atmospheric refraction. This problem has been solved in general by Goody [1963]. For a CO2 atmosphere of 7-mbar surface pressure with a scale height of 10 km, we calculate the asymptotic angle of deviation of a graz- ing ray to be 4.76 x 10 -3 deg. For the largest Z, values treated here this corresponds to a height uncertainty of less than 60 m. At Mars the solar disk is about 0.25 ø in size. For the largest Z, values the shadow edge will be spread by about 3 km. Both of these are small in comparison with the full width of the source function distribution (Figure 1). Rayleigh Scattering

The optical depth for a 7-mbar CO2 atmosphere due to Rayleigh scattering is about 1.2 x 10 -3 at 0.49 microns. This is more than 2 orders of magnitude smaller than the lowest atmospheric optical depth measured at either lander site throughout the Viking mission (paper 2) and is at least an order of magnitude smaller than the uncertainty in the measured optical depth; assuming that the atmosphere is well mixed, the Rayleigh contribution to the atmospheric extinction and source functions can safely be neglected for all Z, values, even when allowance is made for the differences between Rayleigh and dust phase functions.

Airglow We can estimate an upper bound to the magnitude of the

Martian airglow by calculating the total flux of solar UV photons with wavelengths less than 1700 A at Mars and by assuming that each one is converted into a visible photon. From terrestrial experience, 1700 A is a reasonable upper limit for the

wavelength of photons which may be converted into visible light by atmospheric ptotochemical processes. At earth this flux of photons gives a sky brightness of 1 megarayl$igh (MR) [Hinteregger, 1970], corresponding to about 0.4 MR at Mars. According to the data given by Huck et al. [1975], this airglow contribution is far below the detection limit of the camera.

We must also consider resonant scattering of solar visible photons, which makes an important contribution to the terrestrial airglow. The concentration of species which could produce this scattering in the Martian atmosphere is now known. Consider sodium, which is a particularly efficient scatterer of visible light in the airglow on earth. This light will only affect the green passband of the Viking lander camera. We follow Chamberlain [1961], adjusting his numerical values to allow for the different distances of earth and of Mars from the sun. If we select a column abundance for sodium of 4 x 109 cm -•, a value representative of the atmosphere of earth, we obtain a maximum sky brightness of 1.6 MR, which is more than an order of magnitude below the camera detection limit.

In summary, none of the likely sources of airglow based on terrestrial experience is likely to produce a component of sky brightness large enough to be detected by the camera.

Extended Layers and Obstacles at the Sunset Point

A layer of dust in the line of sight with a horizontal extent of tens of kilometers or more, or an obstacle (e.g., a mountain or a cloud) at the sunset point can both cause inflections in a plot of intensity (Y) against Z,, for a fixed scattering angle (0). Since we can select data at constant 0 or constant Z,, it is in principle possible to distinguish between the two causes. The record for sol 41 P.M. showed no significant inflections. On this occasion, therefore, there was no evidence in favor of dust layers in the line of sight nor obstacles at the sunset point. In order to make this statement quantitative we have performed numerical experiments with models of dust layers containing one third of the total atmospheric opacity concentrated near to 40-km altitude. The layer thickness was varied, •lnd we concluded that a 10-km layer was marginally detectable. Nar- rower layers, or layers containing less dust, are not inconsistent with our data.

Multiple Scattering in the Twilight

In paper 1 we attempted to fit the observed data to a single scattering model, i.e., using source functions der•,ved from the attenuated direct solar beam incident upon scatterers in the line of sight. For any aerosol distribution compatible with the optical depth and aerosol properties deduced from other Vi- king observations the intensity calculated from this model is far smaller than is observed, showing that secondary and higher-order scattering are of major importance. We shall nevertheless conclude that the scale height of the scatterers deduced on the basis of a single scattering model is approxi- mately correct.

The problem of multiple scattering in the complex geometry of the twilight is well known (see Rozenberg [1966] for an account of this phenomenon in the terrestrial twilight). Unlike the terrestrial case, the major source of atmospheric opacity in the Martian sky is dust with a very large forward peak in the scattering phase function. Viking observations of the daytime sky brightness show the sun to be surrounded by a bright aureole. The scattering phase function deduced from these observations is presented in papers 1 and 2.

https://www.researchgate.net/publication/236530385_The_extreme_ultraviolet_solar_spectrum_and_its_variation_during_a_solar_cycle?el=1_x_8&enrichId=rgreq-14e0eb99-5434-4bd0-bde8-6cbba9fd65f3&enrichSource=Y292ZXJQYWdlOzQ2OTAyOTk7QVM6MTAzNzg2MTk0NDA3NDMzQDE0MDE3NTU5NjU2ODQ=

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0.20

0.9 Z•

0,6 o.]4ho

/ø/ /o\ o • OlO

o.o.P - / z

o ' ' ' ' ' ' ' ' ' g 14 16 18 20 22 24 26 28 • 32 0.06

SCATTERING ANGLE, 0.04• 0.02• 0.•

Scale height, km Mr 20 8.2 l 0 23.2

5 232.7

Fig. 3. Sky brightness at 32 ø divided by its value at scattering angle t9, for two values of Z!.

In order to treat the problem we must distinguish between multiple scattering involving radiation reaching the line of sight which has not been deflected more than a few degrees from the direct solar beam (the aureole component) and radiation which is scattered at large angles (the diffuse component). The diffuse component may originate from single scattering at large scattering angles or by many scatterings from the aureole. The origin is not important, for• whatever it may be, a combination of the curved geometry of the surface with the vertical distribution of scatterers gives rise to a source function in an extended layer whose maximum lies between the geometric shadow edge and the ground [Rozenberg, 1966].

Consider first the diffuse component. A rough calculation shows that if it is evenly distributed over a few steradians, multiply scattered light would not be detected in the Viking camera twilight observations. We may confirm this conclusion by a less precise but perhaps more convincing treatment, used by Rozenberg.

The layered character and low altitude of the diffuse source function imply that the twilight intensity will vary only slowly with the scattering angle (0), other parameters being held constant. Thus if we examine the data for intensity ratio (Y) as a

0.8

0.6 Q

0.5 -

0.4 -

0.3 -

0.2 -

03 -

O0 10 2:0 30 40 50 60 70 80 D• (deg)

Fig. 4. Weighted integral of the phase function in terms of the maximum scattering angle. After Pollack et al. [1979].

5 10 ]5 20 25 30 35 40 45 50 Zi (kin)

Fig. 5. C/iiculations of the light curve for single scattering models for three different dust scale heights compared to the sol 41 P.M. measurements (open squares). Calculations are for the red color band and t9 = 20 ø. Optical data are taken from paper 3 and are de. scribed in the text. In the case of the theoretical curves the intensity [atio has been multiplied by Mr, chosen to normalize against the observed data for Z! = 20 km.

function of 0, at fixed Zi, we can use the intensity at large 0 to place an upper limit on the multiple scattering for small 0.

Figure 3 shows Y(O)/Y(32 ø) for Zi = 20 and 50 km. If we assume that alLlight at 0 = 32 ø is ['rom the diffuse component, we anticipate that this component is unimportant for Z! -< 20 km, 0 _< 20 ø. Calculations for the aureole component (see below) and the absence of any leveling off at large scattering angle in Figure 3 both indicate that the observed •[6nsity ratio at 0 = 32 ø is a very strong upper limit to the d{ffu• component. For 0 = 20 ø and Z! •< 30 km, we believe that the diffuse component can safely be neglected.

Turning now to the aureole, we consider two models: (1) an analytic model for the case in which the aureole is very nar- i'ow, which yields the total intensity and the contribution from each individual order of scattering and (2) a numerical computation of first- and second-order scattering only, for finite scattering angles.

In the analytic model we consider the solar flux f(G) at G in the direction BG (Figure 1) in the limit •/* --• 0, i.e., for a very narrow aureole. We find

f(G) = foe -•(•'• • [•ooQ•'(B, G)]n/n! =/o e-"(a'ø3(l-'"øQ) r•O

(])

where

Q = -•- P(,/) sin ,/d,/ Here fo is the solar irradiation outside the atmosphere, •(B, G) is the optical path from B to G, C0o is the particle single scattering albedo, •/is the angle of scattering in the aureole, •/* is the assumed limit to the aureole, and n is the order of scattering.

KAHN ET AL.: THE MARTIAN TWILIGHT 5837

0.18• 0.16 Scale height,

km Mr 20 4.3

0.14 I 1 0 10.2 '• 5 23.5

o •_ o.12

i..- O. lO z

,- o. o8 0.06

\.

0.04 '"2 0.02

0.00•) 5 10 15 20 25 30 35 40 45 50 Zl(km)

Fiõ. 6. Same as Fiõure 5 except that secondary scatterins from an 8 ø aureole is included.

In this model the scattered light differs in no way from the incident radiation except that, some having been absorbed, there is less of it. There is, however, more by the factor exp [,(B, G)•0oQ] than in the absence of the aureole.

We shall show that single scattering models fail to account for the observed intensity of twilight by a factor on the order of 20 (M -• 20). The light in the aureole could account for this if ,(B, G)•0oQ •- 3, a value which is not inconsistent with data obtained in paper 2. We may argue as follows:

We wish to evaluate ,(B, G) at the maximum of the source function curve in Figure 1. Provided that ,(B, C) >> 1, as it is for Mars, we anticipate that the maximum will occur when the exponent in (1) is of the order of unity

•-(B, G)(1 - woQ) = 1

For the required value of M (•(B, G)woQ -• 3), woQ -• 0.75. According to paper 2, Wo -• 0.86 for the average conditions observed by the Viking lander, and QOt*) is given by Figure 4. We choose •/* = 8 ø, for which value QOt*) = 0.5 and woQ = 0.43. There are a number of possible explanations for this dis- crepancy. The exponent in (1) may be larger than unity for the maximum source function. If it were twice as large, the required value of woQ would decrease to 0.6. The definition of Q depends upon the assumed value of •/*' for •/* -- 10 ø, Figure 4 gives Q -- 0.57 in blue light. Finally, a stronger forward peak in the phase function is possible. The diffraction peak gives Q -- 0.5 by Babinet's principle, but light can also be transmitted through translucent particles, adding to the forward peak (see, for example, geometric optics calculations of raindrops tabu- lated by Goody [1964]). The dust in the upper atmosphere might differ in this respect from that measured earlier in the day in the troposphere.

Our numerical secondary scattering model calculates the twilight intensity curve with primary plus secondary scattering only, using paper 2 optical data (Wo -- 0.86, Q from Figure 4, and •/* -- 8ø). Note that we have already demonstrated that most of the observed intensity must be associated with orders of scattering higher than the second. The purpose of the

model is to illustrate the effect of the spreading of aureole light on the inversion of the twilight data. A spread of +8 ø corresponds to a vertical displacement of +50 km at the line of sight when Z, = 20 kin. Most of the aureole intensity will not spread to this extent, but the effect of the blurring must be evaluated.

To simplify the calculation, all of the aureole is taken to originate at point F (Figure 1). This underestimates the spreading for some rays and overestimates it for others. It un- doubtedly affects the calculations quantitatively but probably not qualitatively. To eliminate the assumption would require an elaborate computation which we judged to be unjustified by the probable return.

The principal result from this model is illustrated by a comparison of Figures 5 and 6, the former for primary scattering only and the latter for primary plus secondary. Both treatments give the same best fit scale height, equal to 10 kin. The secondary scattering model has an intensity greater by a factor 2. If we use Q•0o = 0.4-0.5, as suggested by the paper 2 optical data, in the second term in the analytic model we would predict a factor between 1.7 and 2. The two treatments are therefore consistent as regards primary and secondary scattering. However, as we have already concluded, a larger value of Q•0o is necessary to account for observed intensity from all orders. If a larger value had been used in our numerical calculation, the intensity of the secondary scattering would have in- creased in the same proportion. The more important effect is to increase the intensity from orders higher than the second.

3. DISCUSSION

Most of our conclusions have already appeared in the pre- vious sections. We can find no evidence for a layered structure in the clouds. The data are consistent with a dust scale height of 10 km but less so with 5 or 20 kin. A mixed atmosphere with a constant volume proportion of carbon dioxide to dust is a possibility. Difficulty in representing multiple scattering and uncertainty as to the appropriate optical parameters to use make it hard to interpret the absolute value of the intensity. There appears, however, to be no major inconsistency with the data given in paper 2.

There are some indications of differences between upper and lower atmosphere dust from the optimum brightness ra- tios in blue and red light, M• and Mr. If the paper 2 optical properties are correct for the twilight, there should be no dif- ference between M• and Mr for the same scale height. In fact, in all of our models, M• exceeds Mr by a factor between 1.8 and 2.0.

In order to satisfy the twilight data we are therefore obliged to change the optical properties of the aerosol in the blue relative to that in the red as compared to the sky brightness measurements. This might be done by introducing a component in the upper atmosphere with neutral scattering properties in addition to the familiar reddish dust of the Martian lower atmosphere. A wavelength-dependent change in the phase function is also conceivable.

Small differences between the optical properties of dust in the upper and lower atmospheres are not unexpected. If we make the reasonable assumption that the source of dust is the Martian surface, some differential sedimentation is likely to take place as the dust is mixed upwards. The size distribution of dust particles is probably shifted towards smaller sizes at the higher levels. Moreover, at the higher levels, some ice for- mation may take place altering the surface properties and


therefore the optical properties of the dust. Our general conclusion is therefore that there is no significant inconsistency between the optical properties of the atmosphere required to explain the twilight and those of a well mixed atmosphere containing lower atmospheric dust at all levels.

Our investigation therefore leads us to no unexpected information on the physical state of the Martian atmosphere. It suggests, however, that our understanding of the optics of the twilight is sufficient to explain the course of this complex phenomenon on two quite different planets.

Acknowledgments. We wish to acknowledge the assistance of K. Bilski with some of the computer runs. R. Kahn was supported by NASA grants NSG 7398 and NSG 22-007-228 during different phases of this work. R. Goody was supported by NASA grant NGL 22-007- 228.

The Editor thanks T. Thorpe for his assistance in evaluating this paper.

REFERENCES

Chamberlain, J. W., Physics of the Aurora and Airglow, Academic, New York, 1961.

Goody, R. M., A note of the refraction of a twilight ray, J. Atmos.

Sci., 20, 502-505, 1963. Goody, R. M., Atmospheric Radiation, p. 309, Clarendon, Oxford,

1964. Hinteregger, H. E., The extreme ultraviolet solar spectrum and its var-

iation during a solar cycle, Ann. Geophys., 26, 547-554, 1970. Huck, F. O., H. F. McCall, W. R. Patterson, and G. R. Taylor, The

Viking Mars lander camera, Space Sci. Instrum., 1, 189-241, 1975. Patterson, W. R., F. O. Huck, S. D. Wall, and M. R. Wolf, Calibra-

tion and performance of the Viking lander cameras, J. Geophys. Res., 82, 4391-4411, 1977.

Pollack, J. B., D. Colburn, R. Kahn, J. Hunter, W. Van Camp, C. E. Carlston, and M. Wolf, Properties of aerosols in the Martian Atmo- sphere as inferred from Viking lander imaging data, J. Geophys. Res., 82, 4479-4496, 1977.

Pollack, J. B., D. Colburn, F. M. Flasar, R. Kahn, C. E. Carlston, and D. Pidek, Properties and effects of dust particles suspended in the Martian atmosphere, J. Geophys. Res., 84, 2929-2945, 1979.

Rozenberg, G. V., Twilight, a Study in Atmospheric Optics, Plenum, New York, 1966.

Volz, F. E., and R. M. Goody, The intensity of the twilight and upper atmospheric dust, J. Atmos. Sci., 19, 385-406, 1962.

(Received June 30, 1980; revised February 2, 1981;

accepted February 24, 1981.)

Documents

The Martian twilight