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Solar radiation on Mars
Solar Energy Vol. 45, No. 6, pp. 353-363, 1990 Printed in the U.S.A.
SOLAR
003&092x/90 $3.00 + .oo Copyright 0 1990 Pergamon Press plc
RADIATION
ON MARS
JOSEPH APPELBAUM* and DENNISJ. FLOOD National Aeronauticsand SpaceAdministration,LewisResearchCenter,Cleveland,OH 44135,U.S.A.
Abstract-Detailed information on solar radiation characteristics on Mars are necessary for effective design of future planned solar energy systems operating on the surface of Mars. In this paper we present a procedure and solar radiation related data from which the diurnally, hourly and daily variation of the global, direct beam and diffuse insolation on Mars are calculated. The radiation data are based on measured optical depth of the Martian atmospherederivedfrom imagestakenof the sunwith a specialdiodeon the Vikingcameras;
and computation basedon multiple wavelengthand multiple scatteringof the solarradiation.
(refs. 6 and 7) of the Martian atmosphere allows an estimate of the absorption and scattering out of the NASA, through its Project Pathfinder, has put in place a wide-ranging set of advanced technology programs beam. These estimateswere derived from imagestaken of the Sun and Phobos with a special diode on the to address future needs of manned spaceexploration. Included in the mission setsunder study is the estab- Viking lander cameras. Earth-terrestrial insolation data are accumulated lishment of outposts on the surface of Mars. The Surface Power program in Pathfinder is aimed at providing over many yearsat different locations around the world ultralightweight photovoltaic array technology for such and are given as long-term average values. The optical an application. Detailed information on solar radiation depth data for Mars are derived from less than two data on the Martian surface is necessaryto allow more Mars years. Consequently, the calculated insolation, accurate estimates of photovoltaic power system size in the present paper, corresponds to short-term data. and mass in system analysis and trade-off studies of Furthermore, the measured opacities (optical depth) relevant technology options. Of major concern are the and the calculated insolation pertain to just two lodust storms, which have been observed to occur on cations on the Mars planet; Viking lander 1 (VL 1) is local as well as on global scales, and their effect on located at 22.3”N latitude and 47.9”W longitude, and solar array output. In general, the assumption has been Viking lander 2 (VL2) is located at 47.7”N latitude that global storms would reduce solar array output es- and 225.7 ’ W longitude. However, the similarity in the sentially to zero, becausethe opacity of the atmosphere properties of the dust suspended above the two landing may rise to values ranging from 3 to 9, depending on sites suggeststhat they are also representative of ones the severityof the storm. Furthermore, such high values at other locations, at least, at latitudes not too far from of opacity may persist for long periods of time such the lander’s sites. Data from lander VLl may be used that the requirement for energy storage quickly be- for latitudes 40”N to 40”s and data from lander VL2 comes much too large to be practical. As shown in refs. for higher latitudes. The Martian atmosphere consists 1 and 2 (and is believed to be published for the first mainly of suspended dust particles, the amounts of time to the best of our knowledge), there is still an which vary daily, seasonally,and annually, depending appreciable large diffuse component, even at high opa- on local and global storm intensities and their duration. The optical depth values given in the section entitled citites, so that solar array operation is still possible. Calculation of solar radiation incident on the top Optical Depth are assumed to be constant throughout of Martian atmosphere and on the Martian surface has the day. Large values of optical depth correspond to been previously published (refs. 3-5) taking into ac- global storms, i.e., days with low insolation (dark days). The albedo of the Martian surfacevariesin the range count the direct beam component only of the solar of about 0.1 to 0.4. The irradiances derived in the secradiation. The introduction of the diffuse component tion entitled Solar Radiation correspond to 0.1 albedo, in this paper became possiblewith the normalized net IIux function described in the paper. This paper is an but can be also used for other values of albedo, to the first approximation. extension of the study in refs. 1 and 2. In this paper a normalized net solar flux function As on the planet Earth, the solar radiation on the is introduced from which, together with the variation surface of Mars is composed of two components: (i) of the opacities, characteristics of the solar radiation direct beam and (ii) diffuse component. The direct beam is affected by scattering and absorption along the on the Martian surface are calculated. This includes, path from the top of the Martian atmosphere to the among others, the diurnal and hourly variation of the Martian surface. Measurement of the optical depth global, beam and diffuse radiation on the horizontal surface. The results are presented in a seriesof figures and tables. The solar radiation data and the procedure * This work was done while the author was a National presented in this paper can be used for the calculation Research Council-NASA Research Associate; on sabbatical leave from Tel-Aviv University. of any desired solar radiation quantity in engineering 1. INTRODU~ION
353
354
J. APPELBAUM and D. J. FLOOD
design. New information about Mars may be forthcoming in the future from new analysis of previously collected data, from new Earth-based observation, or from future flight missions. The Mars solar radiation data will thus be updated accordingly.
as long as on the Earth, corresponding to the greater length of the Martian year (Table 1). Furthermore, they are distinctly unequal in duration as a result of the appreciable eccentricity of the Martian orbit. For that reason, the Martian year is not divided into months. Table 1 gives the duration of the Martian seasons in terrestrial and Martian days (a Martian day 2. OFHCAL DEPTH a sol, 1 sol = 24.65 hr). Areocentric longitudes Ls The most direct and probably most reliable esti= 0” and 180” correspond to the spring and fall equimates of opacity are those derived from Viking lander nox for the northern hemisphere, respectively, and L, imaging of the Sun. Figures 1 and 2 show the seasonal = 90” and 270’ correspond to northern and southern variation of the normal incidence of the optical depth summer solstices,respectively. at the Viking lander locations VLl and VL2, respectively. The season is indicated by the value of L,, ar3. GLOBAL AND LOCAL DUST STORMS eocentric longitude of the Sun, measured in the orbital The intensity of Martian global and local dust plane of the planet from its vernal equinox, L, = 0”. Figures 1 and 2 were derived from references by Pol- storms [ 8,9] is defined in terms of opacity of the dust lack [ 6,7] and Zurek [ 8 ] and were discretized for each it raises. Global dust storms are those which obscure 5” of L, value. As mentioned before, the optical depth planetary-scale sections of the Martian surface for is assumed to remain constant throughout the day. many Martian days (~01s))whereas local dust storms Opacities are minimum during the northern spring (L, are lessintense, and form and dissipate in a few days = O”-90”) and summer (L, = 90”-180”), and max- or less. From a photovoltaic system design point of imum during southern spring (L, = 180”-270”) and view, the intensity, frequency, and duration of these summer (L, = 270°-360”), the seasonsduring which storms may be viewed as partially cloudy and cloudy most local and major dust storms occur. When dust days for which additional energy storage in the phostorms are not present, the optical depth is typically tovoltaic system must be taken into account. The about 0.5. Two global dust storms occurred during the characteristicsofglobal and local dust storms are listed periods of each observation as indicated by the high below. values of the optical depth (they are lower bound values). 3.1 Global dust storms Mars has seasonscomparable to those of Earth. 1. One, or occasionally two global dust storms of planetary scale may occur each Martian year. The However, the seasonsare on the average about twice 3.5
2.5
0
60
120 180 240 AREOCENTRJCLONGITUDE. Ls, DEG
300
360
Fig. 1. Opticaldepth asmeasuredfor Viking Lander VLl as function of areocentric longitude.
Solarradiation on Mars
355
2.5
0
60
120 180 AREOCENTRICLONtITlJDE,
L,,
240 DEG
300
360
Fig. 2. Opticaldepth asmeasuredfor Viking Lander VL2 asfunction of areocentriclongitude. duration may vary from 35 to 70 days or more. 3. Local dust storms last a few days. Although global dust storms do not occur every 4. The opacity of local dust storms may be assumed year, their occurrence is fairly frequent. greater than 1. 2. Global dust storms begin near perihelion, when insolation is maximum (southern spring and sum4. SOLAR RADIATION AT THE TOP OF mer) in the southern mid-latitude. MARS ATMOSPHERE 3. The first global dust storm observed by VL ( 1977) spread from a latitude of 40”s to a latitude 48”N The variation of the solar radiation at the top of the Mars atmosphere is governed by the location of in about 5 to 6 days. 4. The opacity during the global dust storm is greater Mars in its orbit and by the solar zenith angle, and than 1. is of direct beam radiation. The beam irradiance, in W/m’, is given by: 3.2 Local dust storms 1. Local dust storms occur at almost all latitudes and throughout the year. However, they have been observedto occur most frequently in the approximate latitude belt 10” to 20”N and 20” to 4O”S, with where S is the solar constant at the mean Sun-Earth more dust clouds seenin the south than in the north, distance of 1 AU, i.e., S = 1371 W/m’; r is the inthe maioritv - _ of which occurred during southern stantaneous Sun-Mars distance in AU (heliocentric distance) given by ref. 10: spring. 2. Basedon Viking orbiter observations,it is estimated r= 41 -e2) that approximately 100 local storms occur in a given (2) 1 + e cos 0 Martian year. Table 1. Mars seasonalduration Areocentric longitude of the sun, Ls
Duration
to to to to
90” 180" 270" 360"
or
the
season
Mars
Martian days 0 90 180 270
of
0"
Spring Summer Autumn Winter
Autumn Winter Spring Summer
194 178 143 154 669
Terrestrial days 199 183 147 158 687
J. APPELBAUM and D. J. FLOOD
356 800
600
0
60
120 AREOCENTRIC
180 LONGITUDE.
240 L,.
300
360
DEG
Fig. 3. Beam irradiance at the top of Mars atmosphere as function of areocentric longitude.
where a is the Mars semimajor axis in AU, and e is the Mars eccentricity, i.e., e = 0.093377; and 0 is the true anomaly given by: 0 = L, - 248”
(3)
where L, is the areocentric longitude and 248” is the areocentric longitude of Mars perihelion. The SunMars mean distance in astronomical units (AU) is 1.52369 15; therefore, the mean beam irradiance at the top of the Martian atmosphere is 137 I / 1.52369 15* = 590 W/m*. The instantaneous beam irradiance is givenbyeqns(l)to(3): SPRING SUmER
AUTUMN WINTER
G = 590 [l + ecos(L,
:
G &,, =
G&OS
2
(5)
where z is the zenith angle of the incident solar radiation given by: cos z = sin C#J sin 6 + cos i$ cos 6 cos 0
sin 6 = sin &sin L,
(7)
15
5
% 0 F: z -5 3 -10 z 2 -15 d - -20 -25
0
90
180
270
360
AREOCENTRICLONGITUDE, Ls, DEG
Fig. 4. Variationof solardeclinationangle6,with areocentric longitude, LS.
(6)
where C#J = latitude; 6 = declination angle; w = hour angle measured from the true noon westward. The solar declination angle is given by
10 Y
kg
(4)
and is shown in Fig. 3. The beam irradiance on a horizontal surface is
20 H
- 248”)12 (1 - e2)*
ob
0 = 900
0 = -900
T = 18:00
T = 06:OO
+ 0 = 00 T = 12:OO
Fig. 5. Solar time and hour angle relation.
357
Solar radiation on Mars 600
500
0
12
14
16 SOLAR TINEE.H
18
12
14
16 SOLAR TIR,
Fig. 6. Diurnal variation of beam irradiance on a horizontal surface at top of Mars atmosphere.
where 6, = 24.936” is the Mars obliquity of rotation axis. The variation of the solar declination angle is shown in Fig. 4. The four seasonspertain here to the northern hemisphere, the reverseis true in the southern hemisphere. The ratio of Mars to Earth length of day is 24.65 / 24. It is convenient, for calculation purposes, to define a Mars hour, H, by dividing the Martian day (sol) into 24 hours. Using the same relationship between the Mars solar time T and the hour angle as for the Earth, we write: w = 15T-
0
20
180.
(8)
This is shown in Fig. 5. The final solar radiation results can then be adjusted by the above ratio to correspond to actual (terrestrial) time. Examples of the solar radiation calculation procedure and results in this paper pertain to L, = 69",
18
20
H
Fig. 7. Diurnal variation of hourly beam insolation on a horizontal surface at top of Mars atmosphere in Mars watt hours WH).
120”, 153”, 249’, and 299” at Viking lander VLl location 4 = 22.3”N. Areocentric longitude L, = 69" corresponds to aphelion; L, = 249’ to perihelion; L, = 153” to mean radiation of 590 W/m*; L, = 120” corresponds to the lowest atmosphere opacity of 0.4; and L, = 299” to the highest opacity of 3.25 (Fig. 1). For a given L, and latitude 4, one can calculate the zenith angle z as function of solar time T using eqns (6) to (8). The beam irradiance on a horizontal surface is then determined using eqns (4) and (5). The diurnal variation of the beam it-radianceon a horizontal surface at the top of the Mars atmosphere for the above-mentioned values of L, is shown in Fig. 6. Becauseof symmetry, only the afternoon values are shown in the figure. The sunset hour angle is given by ref. 11: w, = cos-‘( -tan 4 tan 6)
(9)
Table 2. Hourly and daily beam insolation on a horizontal surface at top of Mars atmosphere [VL: $J= 22.3”N]
r
Hourly
Iobh
(Hhr/m2)
L
for
Mars hours
16:OO
14:oo
472 510 551 467 451
461 449
480 386 374
337 363 378
270 264
H ending 17:oo
240 256 254 129 130
at:
la:00 131 138 116 a 10
Daily H,bh,
(1 sol) Whr/m2 I
19:oo
26 24 7 ---
4248
4562 4745 3542 3441
J. APPELBAUM and D. J. FLOOD
358
Table 3. Normalized net flux function f(z, 7) at the Martian surface Optical depth
Zenith
T
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1
0.885 .866 .847 .828 .810 .793 .776 .760 .745 .732 .713 .697 .682 .666 .651 .637 .622 .609 .596 .582 .552 .518 .486 .460 .434 .411 .370 .294 .228
1:: 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.25 2.50 2.75 3.00 3.25 3.50 4.00 5.00 6.00
20
30
40
50
60
70
80
0.883 .865 .846 .827 .810 .791 .773 .756 .740 .725 .709 .692 .677 .661 .646 .630 .615 .600 .587 .573 .542 .509 .478 .450 .424 .400 .360 .286 .223
0.882 .860 .841 .821 .802 .785 .766 .750 .733 .717 .700 .683 .667 .650 .633 .618 .601 .586 ,571 .558 .522 .492 .462 ,434 .410 .387 .347 .275 .215
0.880 .858 .836 .815 .796 .775 .755 .736 .717 .700 .682 .662 ,646 .629 .612 .597 .581 .568 .551 .537 .501 .469 .440 .414 .390 .367 .330 .258 .200
0.876 .851 .826 .802 .778 .755 .733 .710 .690 .670 .651 .632 .613 .596 .580 .563 .546 .531 .514 .500 .462 .430 .401 .376 .354 .333 .296 .230 .178
0.870 .836 .806 ,778 .752 .725 .700 .675 .650 .628 .604 .585 .567 .546 .530 .512 .494 .480 ,464 .448 .410 ,378 .353 .330 .308 .290 .258 .203 .153
0.857 .813 .774 .740 .708 .677 .646 .616 .587 .560 .539 .518 .498 .478 .460 .441 .424 .408 .393 .378 .343 .316 .293 .273 .254 .240 .212 .166 .130
0.830 .758 .708 .667 .628 .593 .555 .520 .487 .455 .433 .413 .394 .379 .362 .348 .332 .318 .304 .293 .265 .242 .224 .206 .193 ,180 .160 ,.130 ,103
0.755 .640 .562 .502 .452 .414 .383 .360 .336 .317 .300 .288 .273 .262 .251 .240 .232 .224 .217 .208 .190 .174 .158 .150 .140 .132 118 :094 .080
Td = & cos-'( -tan + tan 6).
(10)
It is of interest to calculate the solar beam insolation on a horizontal surface in watt hours per square meter ( Whr/m 2), for a desired period of time between hour angles wr and 02. This is obtained by integrating eqn (5), i.e., =
E
Gob P -2
X s
WI
Z,
10
and the number of Mars daylight hours is
Zobh
angle dw
(sin~sin6+cos~cos6cosw)dw
(11)
85 0.635 .470 .412 .373 .342 .318 .298 .280 .264 .252 .239 .230 .220 ,210 .202 ,195 .188 181 :176 .170 156 :145 .136 128 :120 110 :100 .080 .068
define an hour. The daily solar insolation, Hobhron a horizontal surface, in watt hours per square meter, is often needed. This is obtained by integrating eqn ( 11) over the period from sunrise to sunset. One gets 24**
Hobh = -G a
1
+ cos 4 cos 6 sin w, .
(13)
Table 2 givesthe hourly beam insolation z&h, and the daily beam insolation &bh, on a horizontal surface at the top of the Mars atmosphere for actual terrestrial time. The diurnal variation of the hourly beam insolation in Mars watt hours (WH) is shown in Fig. 7.
or 5. SOLAR
Zobh
RADIATION
ON THE
SURFACE
OF MARS
12* = n
The variation of the solar radiation on the Martian surface is governed by three factors: (i) the Mars-Sun + cos $ cos G(sin w2 - sin wi) . (12) distance, (ii) solar zenith angle, and (iii) the opacity I of the Martian atmosphere. The global solar radiation A commonly used quantity is the hourly insolation, is composed of the direct beam and diffuse compoin watt hours per square meter. In that case(Joand w2 nents. The direct beam irradiance, Gb, on the Martian * Replace 12 by 12.325 in eqns ( 11) and ( 12) to get the insolation with reference to actual (terrestrial) time.
* * Replace 24 by 24.65 in eqn ( 13) to get the insolation with reference to actual (terrestrial) time.
359
Solar radiation on Mars 600
net flux functionf( Z,T) where the parameters are the zenith angle z and the optical depth T. This table pertains to an albedo of 0.1 but can be used for higher albedo values to a first approximation. Using this data we calculated the global solar irradiance. We assumed that the diffuse irradiance is obtained by subtracting the beam from the global irradiance. The solar irradiance components, on a horizontal Martian surface, are related by
500
400
300
Gh = Gbh+ Gdh
(16)
where Gh global irradiance on a horizontal surface; Gbh direct beam irradiance on a horizontal surface; Gd,, diffuse irradiance on a horizontal surface. The diffuse irradiance of the Martian atmosphere may be a result of a different mechanism than that for the Earth atmosphere, nevertheless, to a first approximation, we will apply eqn ( 16) as for Earth-terrestrial calculations.
600 500
400
300
200 100
0
10
20
30 40 50 60 SUN ZENITH ANGLE. Z. DEG
70
80
(B) EFFECT OF SUN ZENITH ANGLEWITH OPTICAL DEPTHAS A PARMTER.
Fig. 8. Variation of global irradiance with optical depth and sun zenith angle on a horizontal surface. (A) Effect of optical depth with sun zenith angle as a parameter. (B) Effect of sun zenith angle with optical depth as a parameter.
surface normal to the solar rays is related by Beer’s law to the optical depth, 7, of the intervening atmospheric haze: Gb = G,bexp[-Tm(z)]
OPTICAL DEPTH, T
6
2 cc
(A) EFFECTOF OPTICAL DEPTHWITH SUN ZENITH ANGLEAS A PARANETER.
5 Y
600
(14)
where m(z) is the air mass determined by the zenith angle z, and can be approximated, for zenith angles up to about 80”, by 1 m(z) g cos z .
(15)
0 SUN ZENITH ANGLE. Z. DEG
The net solar flux integrated over the solar spectrum on the Martian surface was calculated by Pollack [ 121 based on the multiple wavelength and multiple scattering of the solar radiation. Derived data from this calculation are shown in Table 3 by the normalized
(B) EFFECT OF SUN ZENITH ANGLEWITH OPTICAL DEPTHAS A PARAKTER.
Fig. 9. Variation of beam irradiance with optical depth and sun zenith angle on a horizontal surface. (A) Effect of optical depth with sun zenith angle as a parameter. (B) Effect of sun zenith angle with optical depth as a parameter.
360
J. APPELBAUM and D. J. FLOOD
200
0 i = . E 5 9 s w Y k E
4 0
1
2
3 4 OPTICAL DEPTH. T
5
6
8
10 12 14 SOLARTIME. H
6
(A) EFFECT OF OPTICAL DEPTH WITH SUN ZENITH ANGLE AS A PARAMETER.
16
18
20
Fig. 12. Diurnal variation of global Gh, beam Gbh, and diffuse Cd,,k-radiance on a horizontal Mars surface at Viking Lander VLI.
400
Gbh =
300
G&OS
10
0
20
30 40 50 60 SUN ZENITH ANGLE. Z. DEG
70
80
(B) EFFECT Cf SUN ZENITH ANGLE WITH OPTICAL DEPTH AS A PARAMETER.
Fig. 10. Variation ofdiffuse irradiance with optical depth and sun zenith angle on a horizontal surface. (A) Effect of optical depth with sun zenith angle as a parameter. (B) Effect of sun zenith angle with optical depth as a parameter. The global irradiance by
Gh on a horizontal
surface is given
Gh = GobcosZ F The beam irradiance obtained by 600
"(.I:
Q= 22.3'
(17)
Gbh on a horizontal
N, L, = 153'e T = 0.5.
surface is
G,h = 590 Id/"'
Z exp
(18)
The diffuse irradiance on a horizontal surface is obtained from eqns ( 16 ) to ( 18 ) . Figures 8 to 10 describe the variation of the global, beam and diffuse irradiantes, respectively, on a horizontal Martian surface: and are given in pairs as functions of the optical depth r and zenith angle z. The irradiances were calculated based on Table 3 data and the mean irradiance of 590 W/m’. The variation of the global irradiance Gh, eqn ( 17), is shown in Fig. 8. The beam irradiance Gbh is obtained using eqn ( 18) and is shown in Fig. 9. The beam irradiance shows a sharp decrease with increasing optical depth, and a relatively moderate decrease with increasing zenith angle. The diffuse irradiance Gdh is shown in Fig. 10. The diffuse irradiance shows a sliding maximum with the variation of the zenith angle. The solar radiation (global, beam and diffuse) variation (diurnal, hourly and daily) can be calculated based on the preceeding equations and the f (2,~) data of Table 3. The following examples pertain again to the Viking lander VLl location and areocentric longitudes L, = 69”, 120”, 153”, 249’, and 299”. Daily solar insolation are also given for L, = O", 30", 60", 90”, 150”, 180”, 210”, 240”, 300”, and 330”. For a
500
100
0
4
6
8
10 12 14 SOLARTIME. H
16
18
20
Fig. 11. Diurnal variation of global G, , beam Gbh,and diffuse Gdhirradiance on a horizontal Mars surface at Viking Lender VLI. * The factor 0.9 comes from the expression ( 1 - albedo) in the denominator. For an albedo of 0.1, the denominator is 0.9.
4
6
8
10 12 14 SOLARTIME, H
16
18
20
Fig. 13. Diurnal variation ofglobal Gh, beam G,,, and diffuse Gdhirradiance on a horizontal Mars surface at Viking Lander VLI.
0.65 .40 -50 1.40 3.25
69”
120" 153" 249" 299"
T
Day-L,
diffuse
0.65 .40 .50 1.40 3.25
69" 120" 153" 249" 299"
Hourly
T
beam
global
Day-L,
Hourly
Hourly
173 128 167 244 172
(Whr/m*)
for
270 314 310 125 63
Mars
175 206 190 46 25
hours
H
80 101 75 2 1
18:00
ending
11 15 3 ---
19:oo
at: 1 Daily
3430 3965 3987 1951 1052
(1 sol) insolation, "h? Whr/m*
global
236 331 318 51 2
14:oo
Ibh
191 272 251 27 1
15:oo
(Whr/m2)
for
131 195 167 10 ---
16:OO
Mars
69 106 79 2 --
17:oo
hours
H
21 34 15 ---
18:00
ending
----
19:oo
at:
3 2
1 Daily
1816 2603 2371 323 12
(1 sol) insolation, "bhl Whr/m*
beam
164 127 165 226 151
14:oo
Idh
156 125 159 183 109
15:oo
(Whr/m2)
139 119 143 115 63
16:00
for
Mars
106 101 111 44 25
17:oo
hours
ending
60 67 60 2 1
18:00
H
10 13 3 ---
19:oo
at:
1 Daily
1615 1362 1617 1629 1039
(1 sol) insolation, Hdh* Whr/m*
diffuse
13.70 13.60 12.96 10.95 11.04
1 Daylight
13.70 13.60 12.96 10.95 11.04
1 Daylight hours Td, hr
13.70 13.60 12.96 10.95 11.04
1 Daylight hours -fd$ hr
Table 6. Hourly and daily diffuse insolation on horizontal surface at Mars surface [ VL 1: q5= 22.3ON]
I-
13:oo
insolation
Ih
Table 5. Hourly and daily beam insolation on a horizontal surface at Mars surface [VLI: 6 = 22.3”N]
259 362 354 71 3
13:oo
insolation
431 490 522 315 175
insolation
Table 4. Hourly and daily global insolation on a horizontal surface at Mars surface [ VLI: $J= 22.3”N]
r
118 roe 125 149 94
I Daily mean diffuse irradiance, W/m*
133 191 183 29 1
mean beam irradiance, W/m*
Daily
250 292 308 178 95
Daily mean global irradiance, Wtm*
E 9
0 3
362
J. APPELBAUM
and
D. J. FLOOD
0
60
120 180 240 300 AREOCENTRICLONGITUDE. L,, DEG
360
Fig. 16. Daily global insolation on a horizontal Mars surface at Viking Lander VLI. “12
14
16 SOLAR
TIE,
18
20
H
Fig. 14. Diurnal variation of hourly global insolation on a horizontal surface on Martian surface.
given L, and $, one can calculate the variation of the zenith angle z as function of the Mars solar time T using eqns ( 5 ) to ( 8). Referring to Fig. 1 for the given L,, the optical depth r is determined; with Table 3 and eqns ( 16) to ( 18) one can calculate the solar radiation variation for the given day. The results are shown in Figs. 11 to 13. Becauseof symmetry around 12:00, the graphs in Figs. 12 and 13 are the forenoon or afternoon variation. The figures show clearly that for higher opacities, the diffuse component dominates the solar radiation. The hourly insolation (global, beam and diffuse) on a horizontal surfacecan be calculated based on Figs.
11 to 13 by integrating hourly areas. The daily insolation Hh on a horizontal surface is the summation of the hourly values. The beam insolation, for a desired period of time, can be also calculated by: 12* Ibh =
7
Gob
s
O2(sin $I sin 6 a,
+ cos q5cos 6 cos w)exp[ -r/ (sin C$sin 6 + cos C#J cos 6 cos w)]dw.
(19)
Tables 4 to 6 give the hourly global IJ,, beam I&, and diffuse Idh insolation as well asthe daily global Hh, beam Hbh and diffuse Hdh insolation. Included in the tables are also the number of Martian daylight hours and the daily mean irradiance. For a day (L, = 299) with a relative high opacity, the daily mean global irradiance is still appreciable and is about 30% of that in a clearday. The diurnal variation of the hourly global insolation on a horizontal surface in Mars watt hours (WH) is shown in Fig. 14. The percentage of diffuse and beam insolation for the five analyzed L, days is shown in Fig. 15. The daily global insolation on a horizontal surface on Mars is shown in Fig. 16 for twelve areocentric longitudes covering a Martian year. Using the procedure outlined, one can calculate the variation of the solar radiation for any desired day to use for any engineering system design. 6. CONCLUSIONS
69
120 153 249 AREOCENTRICLONGITUDE. L,, DEG
299
Fig. 15. Percent of diffuse and direct beam insolation on a horizontal Mars surface.
Effective design and utilization of solar energy depend to a large extent on adequate knowledge of solar radiation characteristics in the region of solar energy system operation. In this paper we presented a procedure and solar radiation related data from which the diurnally, hourly and daily variation of insolation on Mars were calculated. This includes the global, beam and diffuse insolation on a horizontal surface, from * Replace the 12 by 12.325 in eqn (19) toget the insolation with reference to actual (terrestrial) time.
363
Solar radiation on Mars
which any desired solar radiation quantity can be derived for an engineering design. The global insolation on the surface of Mars was derived based on the normalized net solar flux functionf(z,r); the beam insolation was determined by Beer’s law relating the insolation to the optical depth of the Martian atmosphere; and the diffuse insolation was calculated as the difference between the global and the beam insolation. The optical depths were measured at the two Viking lander locations, but can also be used, to the first approximation, for other locations. One of the most important results of this study is that there is a large diffuse component of the insolation, even at high optical depth, so that solar energy system operation is still possible. In the absence of long-term insolation data on Mars, the data presented in this paper can be used until updated data are available from new analysis or future flight missions. Acknowledgment-We are very grateful to James B. Pollack from the Space Science Division, NASA Ames Research Center for supplying us with the data from which thef(z,T) table was derived. NOMENCLATURE
Radiation
values
Gb beam irradiance on Mars surface
Gh,
Hh
global, beam and diffuse irradiance on Mars horizontal surface beam irradiance at the top of Mars atmosphere Gob G obh beam irradiance on a horizontal surface at the top of Mars atmosphere H Mars hour ( l/24 sol) global, beam and diffuse daily insolation on > Hbh > Hdh Mars horizontal surface H obh daily beam insolation on a horizontal surface at the top of Mars atmosphere hr actual (terrestrial) hour global, beam and diffuse hourly insolation on I,, Ibh, Idh Mars horizontal surface I obh hourly beam insolation on a horizontal surface at the top of Mars atmosphere Gbh,
Gdh
Subscripts b beam values d diffuse values h horizontal values 0 values on top of Mars atmosphere Other values e eccentricity = 0.093377 L, areocentric longitude
airmass Sun-Mars distance solar constant = 1371 W/m2 at the mean Sun-
Earth distanceof 1 astronomicalunit (AU) Mars solar time Mars daylight hours zenith angle declination angle obliquity
true anomaly optical depth latitude hour angle sunset hour angle REFERENCES
1. J. Appelbaum and D. J. Flood, The Mars climate for photovoltaic system operation, Space Power 8 ( 3 ) , 307317 (1989). 2. J. Appelbaum and D. J. Flood, Photovohaic power system operation in the Mars environment, The 24th Intersociety Energy Conversion Engineering Conference, IECEC-89, Washington, D.C., pp. 841-848, Aug 6-l 1 (1989). 3. J. S. Levine, D. R. Kramer, and W. R. Kuhn, Solar radiation incident on Mars and the outer planets: Latitudinal, seasonal,and atmosphericeffects,ICARUS 31, 136-145 (1977). 4. E. Van Hemehijk, The influence of global dust storms on the mean seasonal daily insolation at the Martian surface, Earth, Moon, and Planets 33, 157-162 (1985). 5. E. Van Hemelrijk, The oblateness effect on the mean seasonal daily insulation at the Martian surface during global dust storms, Earth, Moon, and Planets 38, 209-216 (1987). 6. J. B. Pollack, et al., Properties of aerosols in the Martian atmosphere, as inferred from Viking Lander imaging data, Journal of Geophysical Research 82( 28), 4419-4496 (1977). I. J. B. Pollack, et al., Properties and effectsof dust particles suspended in the Martian atmosphere, Journal of Geophysical Research 84(B6), 2929-2945 (1979). 8. R. W. Zurek,Martian greatduststorms:An update,ICARUS 50,288-310 ( 1982). 9. A. R. Peterfreund and H. H. Kieffer, Thermal infrared properties of the Martian atmosphere, 3. Local dust clouds, Journal of Geophysical Research 84(B6), 28532863 (1979). 10. E. V. P. Smith and K. C. Jacobs, Introductory astronomy and astrophysics, W. B. Saunders, Philadelphia ( 1973). 11. J. A. Duffie and W. A. Beckman,Solar engineering of thermal processes, Wiley,New York ( 1980). 12. J. B. Pollack,R. M. Harberle,J. Schaeffer,and H. Lee, Simulation of the general circulation of Martian atmosphere I: Polar processes,Journal ofGeophysical Research 95, B2, 1447-1473 (1990).
SOLAR
003&092x/90 $3.00 + .oo Copyright 0 1990 Pergamon Press plc
RADIATION
ON MARS
JOSEPH APPELBAUM* and DENNISJ. FLOOD National Aeronauticsand SpaceAdministration,LewisResearchCenter,Cleveland,OH 44135,U.S.A.
Abstract-Detailed information on solar radiation characteristics on Mars are necessary for effective design of future planned solar energy systems operating on the surface of Mars. In this paper we present a procedure and solar radiation related data from which the diurnally, hourly and daily variation of the global, direct beam and diffuse insolation on Mars are calculated. The radiation data are based on measured optical depth of the Martian atmospherederivedfrom imagestakenof the sunwith a specialdiodeon the Vikingcameras;
and computation basedon multiple wavelengthand multiple scatteringof the solarradiation.
(refs. 6 and 7) of the Martian atmosphere allows an estimate of the absorption and scattering out of the NASA, through its Project Pathfinder, has put in place a wide-ranging set of advanced technology programs beam. These estimateswere derived from imagestaken of the Sun and Phobos with a special diode on the to address future needs of manned spaceexploration. Included in the mission setsunder study is the estab- Viking lander cameras. Earth-terrestrial insolation data are accumulated lishment of outposts on the surface of Mars. The Surface Power program in Pathfinder is aimed at providing over many yearsat different locations around the world ultralightweight photovoltaic array technology for such and are given as long-term average values. The optical an application. Detailed information on solar radiation depth data for Mars are derived from less than two data on the Martian surface is necessaryto allow more Mars years. Consequently, the calculated insolation, accurate estimates of photovoltaic power system size in the present paper, corresponds to short-term data. and mass in system analysis and trade-off studies of Furthermore, the measured opacities (optical depth) relevant technology options. Of major concern are the and the calculated insolation pertain to just two lodust storms, which have been observed to occur on cations on the Mars planet; Viking lander 1 (VL 1) is local as well as on global scales, and their effect on located at 22.3”N latitude and 47.9”W longitude, and solar array output. In general, the assumption has been Viking lander 2 (VL2) is located at 47.7”N latitude that global storms would reduce solar array output es- and 225.7 ’ W longitude. However, the similarity in the sentially to zero, becausethe opacity of the atmosphere properties of the dust suspended above the two landing may rise to values ranging from 3 to 9, depending on sites suggeststhat they are also representative of ones the severityof the storm. Furthermore, such high values at other locations, at least, at latitudes not too far from of opacity may persist for long periods of time such the lander’s sites. Data from lander VLl may be used that the requirement for energy storage quickly be- for latitudes 40”N to 40”s and data from lander VL2 comes much too large to be practical. As shown in refs. for higher latitudes. The Martian atmosphere consists 1 and 2 (and is believed to be published for the first mainly of suspended dust particles, the amounts of time to the best of our knowledge), there is still an which vary daily, seasonally,and annually, depending appreciable large diffuse component, even at high opa- on local and global storm intensities and their duration. The optical depth values given in the section entitled citites, so that solar array operation is still possible. Calculation of solar radiation incident on the top Optical Depth are assumed to be constant throughout of Martian atmosphere and on the Martian surface has the day. Large values of optical depth correspond to been previously published (refs. 3-5) taking into ac- global storms, i.e., days with low insolation (dark days). The albedo of the Martian surfacevariesin the range count the direct beam component only of the solar of about 0.1 to 0.4. The irradiances derived in the secradiation. The introduction of the diffuse component tion entitled Solar Radiation correspond to 0.1 albedo, in this paper became possiblewith the normalized net IIux function described in the paper. This paper is an but can be also used for other values of albedo, to the first approximation. extension of the study in refs. 1 and 2. In this paper a normalized net solar flux function As on the planet Earth, the solar radiation on the is introduced from which, together with the variation surface of Mars is composed of two components: (i) of the opacities, characteristics of the solar radiation direct beam and (ii) diffuse component. The direct beam is affected by scattering and absorption along the on the Martian surface are calculated. This includes, path from the top of the Martian atmosphere to the among others, the diurnal and hourly variation of the Martian surface. Measurement of the optical depth global, beam and diffuse radiation on the horizontal surface. The results are presented in a seriesof figures and tables. The solar radiation data and the procedure * This work was done while the author was a National presented in this paper can be used for the calculation Research Council-NASA Research Associate; on sabbatical leave from Tel-Aviv University. of any desired solar radiation quantity in engineering 1. INTRODU~ION
353
354
J. APPELBAUM and D. J. FLOOD
design. New information about Mars may be forthcoming in the future from new analysis of previously collected data, from new Earth-based observation, or from future flight missions. The Mars solar radiation data will thus be updated accordingly.
as long as on the Earth, corresponding to the greater length of the Martian year (Table 1). Furthermore, they are distinctly unequal in duration as a result of the appreciable eccentricity of the Martian orbit. For that reason, the Martian year is not divided into months. Table 1 gives the duration of the Martian seasons in terrestrial and Martian days (a Martian day 2. OFHCAL DEPTH a sol, 1 sol = 24.65 hr). Areocentric longitudes Ls The most direct and probably most reliable esti= 0” and 180” correspond to the spring and fall equimates of opacity are those derived from Viking lander nox for the northern hemisphere, respectively, and L, imaging of the Sun. Figures 1 and 2 show the seasonal = 90” and 270’ correspond to northern and southern variation of the normal incidence of the optical depth summer solstices,respectively. at the Viking lander locations VLl and VL2, respectively. The season is indicated by the value of L,, ar3. GLOBAL AND LOCAL DUST STORMS eocentric longitude of the Sun, measured in the orbital The intensity of Martian global and local dust plane of the planet from its vernal equinox, L, = 0”. Figures 1 and 2 were derived from references by Pol- storms [ 8,9] is defined in terms of opacity of the dust lack [ 6,7] and Zurek [ 8 ] and were discretized for each it raises. Global dust storms are those which obscure 5” of L, value. As mentioned before, the optical depth planetary-scale sections of the Martian surface for is assumed to remain constant throughout the day. many Martian days (~01s))whereas local dust storms Opacities are minimum during the northern spring (L, are lessintense, and form and dissipate in a few days = O”-90”) and summer (L, = 90”-180”), and max- or less. From a photovoltaic system design point of imum during southern spring (L, = 180”-270”) and view, the intensity, frequency, and duration of these summer (L, = 270°-360”), the seasonsduring which storms may be viewed as partially cloudy and cloudy most local and major dust storms occur. When dust days for which additional energy storage in the phostorms are not present, the optical depth is typically tovoltaic system must be taken into account. The about 0.5. Two global dust storms occurred during the characteristicsofglobal and local dust storms are listed periods of each observation as indicated by the high below. values of the optical depth (they are lower bound values). 3.1 Global dust storms Mars has seasonscomparable to those of Earth. 1. One, or occasionally two global dust storms of planetary scale may occur each Martian year. The However, the seasonsare on the average about twice 3.5
2.5
0
60
120 180 240 AREOCENTRJCLONGITUDE. Ls, DEG
300
360
Fig. 1. Opticaldepth asmeasuredfor Viking Lander VLl as function of areocentric longitude.
Solarradiation on Mars
355
2.5
0
60
120 180 AREOCENTRICLONtITlJDE,
L,,
240 DEG
300
360
Fig. 2. Opticaldepth asmeasuredfor Viking Lander VL2 asfunction of areocentriclongitude. duration may vary from 35 to 70 days or more. 3. Local dust storms last a few days. Although global dust storms do not occur every 4. The opacity of local dust storms may be assumed year, their occurrence is fairly frequent. greater than 1. 2. Global dust storms begin near perihelion, when insolation is maximum (southern spring and sum4. SOLAR RADIATION AT THE TOP OF mer) in the southern mid-latitude. MARS ATMOSPHERE 3. The first global dust storm observed by VL ( 1977) spread from a latitude of 40”s to a latitude 48”N The variation of the solar radiation at the top of the Mars atmosphere is governed by the location of in about 5 to 6 days. 4. The opacity during the global dust storm is greater Mars in its orbit and by the solar zenith angle, and than 1. is of direct beam radiation. The beam irradiance, in W/m’, is given by: 3.2 Local dust storms 1. Local dust storms occur at almost all latitudes and throughout the year. However, they have been observedto occur most frequently in the approximate latitude belt 10” to 20”N and 20” to 4O”S, with where S is the solar constant at the mean Sun-Earth more dust clouds seenin the south than in the north, distance of 1 AU, i.e., S = 1371 W/m’; r is the inthe maioritv - _ of which occurred during southern stantaneous Sun-Mars distance in AU (heliocentric distance) given by ref. 10: spring. 2. Basedon Viking orbiter observations,it is estimated r= 41 -e2) that approximately 100 local storms occur in a given (2) 1 + e cos 0 Martian year. Table 1. Mars seasonalduration Areocentric longitude of the sun, Ls
Duration
to to to to
90” 180" 270" 360"
or
the
season
Mars
Martian days 0 90 180 270
of
0"
Spring Summer Autumn Winter
Autumn Winter Spring Summer
194 178 143 154 669
Terrestrial days 199 183 147 158 687
J. APPELBAUM and D. J. FLOOD
356 800
600
0
60
120 AREOCENTRIC
180 LONGITUDE.
240 L,.
300
360
DEG
Fig. 3. Beam irradiance at the top of Mars atmosphere as function of areocentric longitude.
where a is the Mars semimajor axis in AU, and e is the Mars eccentricity, i.e., e = 0.093377; and 0 is the true anomaly given by: 0 = L, - 248”
(3)
where L, is the areocentric longitude and 248” is the areocentric longitude of Mars perihelion. The SunMars mean distance in astronomical units (AU) is 1.52369 15; therefore, the mean beam irradiance at the top of the Martian atmosphere is 137 I / 1.52369 15* = 590 W/m*. The instantaneous beam irradiance is givenbyeqns(l)to(3): SPRING SUmER
AUTUMN WINTER
G = 590 [l + ecos(L,
:
G &,, =
G&OS
2
(5)
where z is the zenith angle of the incident solar radiation given by: cos z = sin C#J sin 6 + cos i$ cos 6 cos 0
sin 6 = sin &sin L,
(7)
15
5
% 0 F: z -5 3 -10 z 2 -15 d - -20 -25
0
90
180
270
360
AREOCENTRICLONGITUDE, Ls, DEG
Fig. 4. Variationof solardeclinationangle6,with areocentric longitude, LS.
(6)
where C#J = latitude; 6 = declination angle; w = hour angle measured from the true noon westward. The solar declination angle is given by
10 Y
kg
(4)
and is shown in Fig. 3. The beam irradiance on a horizontal surface is
20 H
- 248”)12 (1 - e2)*
ob
0 = 900
0 = -900
T = 18:00
T = 06:OO
+ 0 = 00 T = 12:OO
Fig. 5. Solar time and hour angle relation.
357
Solar radiation on Mars 600
500
0
12
14
16 SOLAR TINEE.H
18
12
14
16 SOLAR TIR,
Fig. 6. Diurnal variation of beam irradiance on a horizontal surface at top of Mars atmosphere.
where 6, = 24.936” is the Mars obliquity of rotation axis. The variation of the solar declination angle is shown in Fig. 4. The four seasonspertain here to the northern hemisphere, the reverseis true in the southern hemisphere. The ratio of Mars to Earth length of day is 24.65 / 24. It is convenient, for calculation purposes, to define a Mars hour, H, by dividing the Martian day (sol) into 24 hours. Using the same relationship between the Mars solar time T and the hour angle as for the Earth, we write: w = 15T-
0
20
180.
(8)
This is shown in Fig. 5. The final solar radiation results can then be adjusted by the above ratio to correspond to actual (terrestrial) time. Examples of the solar radiation calculation procedure and results in this paper pertain to L, = 69",
18
20
H
Fig. 7. Diurnal variation of hourly beam insolation on a horizontal surface at top of Mars atmosphere in Mars watt hours WH).
120”, 153”, 249’, and 299” at Viking lander VLl location 4 = 22.3”N. Areocentric longitude L, = 69" corresponds to aphelion; L, = 249’ to perihelion; L, = 153” to mean radiation of 590 W/m*; L, = 120” corresponds to the lowest atmosphere opacity of 0.4; and L, = 299” to the highest opacity of 3.25 (Fig. 1). For a given L, and latitude 4, one can calculate the zenith angle z as function of solar time T using eqns (6) to (8). The beam irradiance on a horizontal surface is then determined using eqns (4) and (5). The diurnal variation of the beam it-radianceon a horizontal surface at the top of the Mars atmosphere for the above-mentioned values of L, is shown in Fig. 6. Becauseof symmetry, only the afternoon values are shown in the figure. The sunset hour angle is given by ref. 11: w, = cos-‘( -tan 4 tan 6)
(9)
Table 2. Hourly and daily beam insolation on a horizontal surface at top of Mars atmosphere [VL: $J= 22.3”N]
r
Hourly
Iobh
(Hhr/m2)
L
for
Mars hours
16:OO
14:oo
472 510 551 467 451
461 449
480 386 374
337 363 378
270 264
H ending 17:oo
240 256 254 129 130
at:
la:00 131 138 116 a 10
Daily H,bh,
(1 sol) Whr/m2 I
19:oo
26 24 7 ---
4248
4562 4745 3542 3441
J. APPELBAUM and D. J. FLOOD
358
Table 3. Normalized net flux function f(z, 7) at the Martian surface Optical depth
Zenith
T
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1
0.885 .866 .847 .828 .810 .793 .776 .760 .745 .732 .713 .697 .682 .666 .651 .637 .622 .609 .596 .582 .552 .518 .486 .460 .434 .411 .370 .294 .228
1:: 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.25 2.50 2.75 3.00 3.25 3.50 4.00 5.00 6.00
20
30
40
50
60
70
80
0.883 .865 .846 .827 .810 .791 .773 .756 .740 .725 .709 .692 .677 .661 .646 .630 .615 .600 .587 .573 .542 .509 .478 .450 .424 .400 .360 .286 .223
0.882 .860 .841 .821 .802 .785 .766 .750 .733 .717 .700 .683 .667 .650 .633 .618 .601 .586 ,571 .558 .522 .492 .462 ,434 .410 .387 .347 .275 .215
0.880 .858 .836 .815 .796 .775 .755 .736 .717 .700 .682 .662 ,646 .629 .612 .597 .581 .568 .551 .537 .501 .469 .440 .414 .390 .367 .330 .258 .200
0.876 .851 .826 .802 .778 .755 .733 .710 .690 .670 .651 .632 .613 .596 .580 .563 .546 .531 .514 .500 .462 .430 .401 .376 .354 .333 .296 .230 .178
0.870 .836 .806 ,778 .752 .725 .700 .675 .650 .628 .604 .585 .567 .546 .530 .512 .494 .480 ,464 .448 .410 ,378 .353 .330 .308 .290 .258 .203 .153
0.857 .813 .774 .740 .708 .677 .646 .616 .587 .560 .539 .518 .498 .478 .460 .441 .424 .408 .393 .378 .343 .316 .293 .273 .254 .240 .212 .166 .130
0.830 .758 .708 .667 .628 .593 .555 .520 .487 .455 .433 .413 .394 .379 .362 .348 .332 .318 .304 .293 .265 .242 .224 .206 .193 ,180 .160 ,.130 ,103
0.755 .640 .562 .502 .452 .414 .383 .360 .336 .317 .300 .288 .273 .262 .251 .240 .232 .224 .217 .208 .190 .174 .158 .150 .140 .132 118 :094 .080
Td = & cos-'( -tan + tan 6).
(10)
It is of interest to calculate the solar beam insolation on a horizontal surface in watt hours per square meter ( Whr/m 2), for a desired period of time between hour angles wr and 02. This is obtained by integrating eqn (5), i.e., =
E
Gob P -2
X s
WI
Z,
10
and the number of Mars daylight hours is
Zobh
angle dw
(sin~sin6+cos~cos6cosw)dw
(11)
85 0.635 .470 .412 .373 .342 .318 .298 .280 .264 .252 .239 .230 .220 ,210 .202 ,195 .188 181 :176 .170 156 :145 .136 128 :120 110 :100 .080 .068
define an hour. The daily solar insolation, Hobhron a horizontal surface, in watt hours per square meter, is often needed. This is obtained by integrating eqn ( 11) over the period from sunrise to sunset. One gets 24**
Hobh = -G a
1
+ cos 4 cos 6 sin w, .
(13)
Table 2 givesthe hourly beam insolation z&h, and the daily beam insolation &bh, on a horizontal surface at the top of the Mars atmosphere for actual terrestrial time. The diurnal variation of the hourly beam insolation in Mars watt hours (WH) is shown in Fig. 7.
or 5. SOLAR
Zobh
RADIATION
ON THE
SURFACE
OF MARS
12* = n
The variation of the solar radiation on the Martian surface is governed by three factors: (i) the Mars-Sun + cos $ cos G(sin w2 - sin wi) . (12) distance, (ii) solar zenith angle, and (iii) the opacity I of the Martian atmosphere. The global solar radiation A commonly used quantity is the hourly insolation, is composed of the direct beam and diffuse compoin watt hours per square meter. In that case(Joand w2 nents. The direct beam irradiance, Gb, on the Martian * Replace 12 by 12.325 in eqns ( 11) and ( 12) to get the insolation with reference to actual (terrestrial) time.
* * Replace 24 by 24.65 in eqn ( 13) to get the insolation with reference to actual (terrestrial) time.
359
Solar radiation on Mars 600
net flux functionf( Z,T) where the parameters are the zenith angle z and the optical depth T. This table pertains to an albedo of 0.1 but can be used for higher albedo values to a first approximation. Using this data we calculated the global solar irradiance. We assumed that the diffuse irradiance is obtained by subtracting the beam from the global irradiance. The solar irradiance components, on a horizontal Martian surface, are related by
500
400
300
Gh = Gbh+ Gdh
(16)
where Gh global irradiance on a horizontal surface; Gbh direct beam irradiance on a horizontal surface; Gd,, diffuse irradiance on a horizontal surface. The diffuse irradiance of the Martian atmosphere may be a result of a different mechanism than that for the Earth atmosphere, nevertheless, to a first approximation, we will apply eqn ( 16) as for Earth-terrestrial calculations.
600 500
400
300
200 100
0
10
20
30 40 50 60 SUN ZENITH ANGLE. Z. DEG
70
80
(B) EFFECT OF SUN ZENITH ANGLEWITH OPTICAL DEPTHAS A PARMTER.
Fig. 8. Variation of global irradiance with optical depth and sun zenith angle on a horizontal surface. (A) Effect of optical depth with sun zenith angle as a parameter. (B) Effect of sun zenith angle with optical depth as a parameter.
surface normal to the solar rays is related by Beer’s law to the optical depth, 7, of the intervening atmospheric haze: Gb = G,bexp[-Tm(z)]
OPTICAL DEPTH, T
6
2 cc
(A) EFFECTOF OPTICAL DEPTHWITH SUN ZENITH ANGLEAS A PARANETER.
5 Y
600
(14)
where m(z) is the air mass determined by the zenith angle z, and can be approximated, for zenith angles up to about 80”, by 1 m(z) g cos z .
(15)
0 SUN ZENITH ANGLE. Z. DEG
The net solar flux integrated over the solar spectrum on the Martian surface was calculated by Pollack [ 121 based on the multiple wavelength and multiple scattering of the solar radiation. Derived data from this calculation are shown in Table 3 by the normalized
(B) EFFECT OF SUN ZENITH ANGLEWITH OPTICAL DEPTHAS A PARAKTER.
Fig. 9. Variation of beam irradiance with optical depth and sun zenith angle on a horizontal surface. (A) Effect of optical depth with sun zenith angle as a parameter. (B) Effect of sun zenith angle with optical depth as a parameter.
360
J. APPELBAUM and D. J. FLOOD
200
0 i = . E 5 9 s w Y k E
4 0
1
2
3 4 OPTICAL DEPTH. T
5
6
8
10 12 14 SOLARTIME. H
6
(A) EFFECT OF OPTICAL DEPTH WITH SUN ZENITH ANGLE AS A PARAMETER.
16
18
20
Fig. 12. Diurnal variation of global Gh, beam Gbh, and diffuse Cd,,k-radiance on a horizontal Mars surface at Viking Lander VLI.
400
Gbh =
300
G&OS
10
0
20
30 40 50 60 SUN ZENITH ANGLE. Z. DEG
70
80
(B) EFFECT Cf SUN ZENITH ANGLE WITH OPTICAL DEPTH AS A PARAMETER.
Fig. 10. Variation ofdiffuse irradiance with optical depth and sun zenith angle on a horizontal surface. (A) Effect of optical depth with sun zenith angle as a parameter. (B) Effect of sun zenith angle with optical depth as a parameter. The global irradiance by
Gh on a horizontal
surface is given
Gh = GobcosZ F The beam irradiance obtained by 600
"(.I:
Q= 22.3'
(17)
Gbh on a horizontal
N, L, = 153'e T = 0.5.
surface is
G,h = 590 Id/"'
Z exp
(18)
The diffuse irradiance on a horizontal surface is obtained from eqns ( 16 ) to ( 18 ) . Figures 8 to 10 describe the variation of the global, beam and diffuse irradiantes, respectively, on a horizontal Martian surface: and are given in pairs as functions of the optical depth r and zenith angle z. The irradiances were calculated based on Table 3 data and the mean irradiance of 590 W/m’. The variation of the global irradiance Gh, eqn ( 17), is shown in Fig. 8. The beam irradiance Gbh is obtained using eqn ( 18) and is shown in Fig. 9. The beam irradiance shows a sharp decrease with increasing optical depth, and a relatively moderate decrease with increasing zenith angle. The diffuse irradiance Gdh is shown in Fig. 10. The diffuse irradiance shows a sliding maximum with the variation of the zenith angle. The solar radiation (global, beam and diffuse) variation (diurnal, hourly and daily) can be calculated based on the preceeding equations and the f (2,~) data of Table 3. The following examples pertain again to the Viking lander VLl location and areocentric longitudes L, = 69”, 120”, 153”, 249’, and 299”. Daily solar insolation are also given for L, = O", 30", 60", 90”, 150”, 180”, 210”, 240”, 300”, and 330”. For a
500
100
0
4
6
8
10 12 14 SOLARTIME. H
16
18
20
Fig. 11. Diurnal variation of global G, , beam Gbh,and diffuse Gdhirradiance on a horizontal Mars surface at Viking Lender VLI. * The factor 0.9 comes from the expression ( 1 - albedo) in the denominator. For an albedo of 0.1, the denominator is 0.9.
4
6
8
10 12 14 SOLARTIME, H
16
18
20
Fig. 13. Diurnal variation ofglobal Gh, beam G,,, and diffuse Gdhirradiance on a horizontal Mars surface at Viking Lander VLI.
0.65 .40 -50 1.40 3.25
69”
120" 153" 249" 299"
T
Day-L,
diffuse
0.65 .40 .50 1.40 3.25
69" 120" 153" 249" 299"
Hourly
T
beam
global
Day-L,
Hourly
Hourly
173 128 167 244 172
(Whr/m*)
for
270 314 310 125 63
Mars
175 206 190 46 25
hours
H
80 101 75 2 1
18:00
ending
11 15 3 ---
19:oo
at: 1 Daily
3430 3965 3987 1951 1052
(1 sol) insolation, "h? Whr/m*
global
236 331 318 51 2
14:oo
Ibh
191 272 251 27 1
15:oo
(Whr/m2)
for
131 195 167 10 ---
16:OO
Mars
69 106 79 2 --
17:oo
hours
H
21 34 15 ---
18:00
ending
----
19:oo
at:
3 2
1 Daily
1816 2603 2371 323 12
(1 sol) insolation, "bhl Whr/m*
beam
164 127 165 226 151
14:oo
Idh
156 125 159 183 109
15:oo
(Whr/m2)
139 119 143 115 63
16:00
for
Mars
106 101 111 44 25
17:oo
hours
ending
60 67 60 2 1
18:00
H
10 13 3 ---
19:oo
at:
1 Daily
1615 1362 1617 1629 1039
(1 sol) insolation, Hdh* Whr/m*
diffuse
13.70 13.60 12.96 10.95 11.04
1 Daylight
13.70 13.60 12.96 10.95 11.04
1 Daylight hours Td, hr
13.70 13.60 12.96 10.95 11.04
1 Daylight hours -fd$ hr
Table 6. Hourly and daily diffuse insolation on horizontal surface at Mars surface [ VL 1: q5= 22.3ON]
I-
13:oo
insolation
Ih
Table 5. Hourly and daily beam insolation on a horizontal surface at Mars surface [VLI: 6 = 22.3”N]
259 362 354 71 3
13:oo
insolation
431 490 522 315 175
insolation
Table 4. Hourly and daily global insolation on a horizontal surface at Mars surface [ VLI: $J= 22.3”N]
r
118 roe 125 149 94
I Daily mean diffuse irradiance, W/m*
133 191 183 29 1
mean beam irradiance, W/m*
Daily
250 292 308 178 95
Daily mean global irradiance, Wtm*
E 9
0 3
362
J. APPELBAUM
and
D. J. FLOOD
0
60
120 180 240 300 AREOCENTRICLONGITUDE. L,, DEG
360
Fig. 16. Daily global insolation on a horizontal Mars surface at Viking Lander VLI. “12
14
16 SOLAR
TIE,
18
20
H
Fig. 14. Diurnal variation of hourly global insolation on a horizontal surface on Martian surface.
given L, and $, one can calculate the variation of the zenith angle z as function of the Mars solar time T using eqns ( 5 ) to ( 8). Referring to Fig. 1 for the given L,, the optical depth r is determined; with Table 3 and eqns ( 16) to ( 18) one can calculate the solar radiation variation for the given day. The results are shown in Figs. 11 to 13. Becauseof symmetry around 12:00, the graphs in Figs. 12 and 13 are the forenoon or afternoon variation. The figures show clearly that for higher opacities, the diffuse component dominates the solar radiation. The hourly insolation (global, beam and diffuse) on a horizontal surfacecan be calculated based on Figs.
11 to 13 by integrating hourly areas. The daily insolation Hh on a horizontal surface is the summation of the hourly values. The beam insolation, for a desired period of time, can be also calculated by: 12* Ibh =
7
Gob
s
O2(sin $I sin 6 a,
+ cos q5cos 6 cos w)exp[ -r/ (sin C$sin 6 + cos C#J cos 6 cos w)]dw.
(19)
Tables 4 to 6 give the hourly global IJ,, beam I&, and diffuse Idh insolation as well asthe daily global Hh, beam Hbh and diffuse Hdh insolation. Included in the tables are also the number of Martian daylight hours and the daily mean irradiance. For a day (L, = 299) with a relative high opacity, the daily mean global irradiance is still appreciable and is about 30% of that in a clearday. The diurnal variation of the hourly global insolation on a horizontal surface in Mars watt hours (WH) is shown in Fig. 14. The percentage of diffuse and beam insolation for the five analyzed L, days is shown in Fig. 15. The daily global insolation on a horizontal surface on Mars is shown in Fig. 16 for twelve areocentric longitudes covering a Martian year. Using the procedure outlined, one can calculate the variation of the solar radiation for any desired day to use for any engineering system design. 6. CONCLUSIONS
69
120 153 249 AREOCENTRICLONGITUDE. L,, DEG
299
Fig. 15. Percent of diffuse and direct beam insolation on a horizontal Mars surface.
Effective design and utilization of solar energy depend to a large extent on adequate knowledge of solar radiation characteristics in the region of solar energy system operation. In this paper we presented a procedure and solar radiation related data from which the diurnally, hourly and daily variation of insolation on Mars were calculated. This includes the global, beam and diffuse insolation on a horizontal surface, from * Replace the 12 by 12.325 in eqn (19) toget the insolation with reference to actual (terrestrial) time.
363
Solar radiation on Mars
which any desired solar radiation quantity can be derived for an engineering design. The global insolation on the surface of Mars was derived based on the normalized net solar flux functionf(z,r); the beam insolation was determined by Beer’s law relating the insolation to the optical depth of the Martian atmosphere; and the diffuse insolation was calculated as the difference between the global and the beam insolation. The optical depths were measured at the two Viking lander locations, but can also be used, to the first approximation, for other locations. One of the most important results of this study is that there is a large diffuse component of the insolation, even at high optical depth, so that solar energy system operation is still possible. In the absence of long-term insolation data on Mars, the data presented in this paper can be used until updated data are available from new analysis or future flight missions. Acknowledgment-We are very grateful to James B. Pollack from the Space Science Division, NASA Ames Research Center for supplying us with the data from which thef(z,T) table was derived. NOMENCLATURE
Radiation
values
Gb beam irradiance on Mars surface
Gh,
Hh
global, beam and diffuse irradiance on Mars horizontal surface beam irradiance at the top of Mars atmosphere Gob G obh beam irradiance on a horizontal surface at the top of Mars atmosphere H Mars hour ( l/24 sol) global, beam and diffuse daily insolation on > Hbh > Hdh Mars horizontal surface H obh daily beam insolation on a horizontal surface at the top of Mars atmosphere hr actual (terrestrial) hour global, beam and diffuse hourly insolation on I,, Ibh, Idh Mars horizontal surface I obh hourly beam insolation on a horizontal surface at the top of Mars atmosphere Gbh,
Gdh
Subscripts b beam values d diffuse values h horizontal values 0 values on top of Mars atmosphere Other values e eccentricity = 0.093377 L, areocentric longitude
airmass Sun-Mars distance solar constant = 1371 W/m2 at the mean Sun-
Earth distanceof 1 astronomicalunit (AU) Mars solar time Mars daylight hours zenith angle declination angle obliquity
true anomaly optical depth latitude hour angle sunset hour angle REFERENCES
1. J. Appelbaum and D. J. Flood, The Mars climate for photovoltaic system operation, Space Power 8 ( 3 ) , 307317 (1989). 2. J. Appelbaum and D. J. Flood, Photovohaic power system operation in the Mars environment, The 24th Intersociety Energy Conversion Engineering Conference, IECEC-89, Washington, D.C., pp. 841-848, Aug 6-l 1 (1989). 3. J. S. Levine, D. R. Kramer, and W. R. Kuhn, Solar radiation incident on Mars and the outer planets: Latitudinal, seasonal,and atmosphericeffects,ICARUS 31, 136-145 (1977). 4. E. Van Hemehijk, The influence of global dust storms on the mean seasonal daily insolation at the Martian surface, Earth, Moon, and Planets 33, 157-162 (1985). 5. E. Van Hemelrijk, The oblateness effect on the mean seasonal daily insulation at the Martian surface during global dust storms, Earth, Moon, and Planets 38, 209-216 (1987). 6. J. B. Pollack, et al., Properties of aerosols in the Martian atmosphere, as inferred from Viking Lander imaging data, Journal of Geophysical Research 82( 28), 4419-4496 (1977). I. J. B. Pollack, et al., Properties and effectsof dust particles suspended in the Martian atmosphere, Journal of Geophysical Research 84(B6), 2929-2945 (1979). 8. R. W. Zurek,Martian greatduststorms:An update,ICARUS 50,288-310 ( 1982). 9. A. R. Peterfreund and H. H. Kieffer, Thermal infrared properties of the Martian atmosphere, 3. Local dust clouds, Journal of Geophysical Research 84(B6), 28532863 (1979). 10. E. V. P. Smith and K. C. Jacobs, Introductory astronomy and astrophysics, W. B. Saunders, Philadelphia ( 1973). 11. J. A. Duffie and W. A. Beckman,Solar engineering of thermal processes, Wiley,New York ( 1980). 12. J. B. Pollack,R. M. Harberle,J. Schaeffer,and H. Lee, Simulation of the general circulation of Martian atmosphere I: Polar processes,Journal ofGeophysical Research 95, B2, 1447-1473 (1990).