reevaluation of the quondam dust trend in the middle atmosphere

Reevaluation of the quondam dust trendin the middle atmosphere

Miroslav Kocifaj and Helmuth Horvath

Quondam lunar eclipse photometry data offered valuable information on the optical properties of themiddle atmosphere, including dust particles. However, in comparison with nonspherical grains, thesimple model of spherical particles has a different effect on solar radiation penetrating horizontallythrough the atmosphere. It is shown that the systems, in which the smallest size fraction of dust particlesdominates, reduce irradiation of the Earth’s shadow more efficiently if the grains are of irregular shape.In contrast, the populations contaminated by a certain amount of large particles cause an opposite effect.Depending on the actual form of the size distribution function of the irregular grains, the irradiancewithin the center of the Earth’s shadow may change by 2 orders of magnitude in the visible spectrum. Itis therefore evident that dust properties retrieved in the past are eligible candidates for reevaluation tocorrect a view on the dust trend in the middle atmosphere. Sample calculations are presented for thelunar eclipse observed on 19 January 1954. © 2005 Optical Society of America

OCIS codes: 010.1290, 290.2200, 010.1110.

1. Introduction

A global distribution of aerosol particles in theEarth’s atmosphere is monitored for a long time toevaluate their influence on climate, radiation budget,albedo of the system Earth’s atmosphere, or to getdetailed information on chemical, physical, and me-teorological processes in the individual atmosphericlayers. The aerosol system in the middle atmospherehas undergone significant changes in the past de-cades. For instance, the atmosphere is ever morecontaminated by space vehicle products. The devel-opment of such artificial structures in the upper at-mosphere is accompanied by rather unusual opticalphenomena caused by the solar light scattering bycombustion products and by their interaction withatmospheric constituents.1 The investigation of theoptical phenomena can offer information on the an-thropogenic pollution of the near-Earth space envi-ronment, on the processes of the interaction betweenpollutant emissions and the environment, and on dy-namic processes in the upper atmosphere.

Particles flying in the high atmosphere are usually ofextraterrestrial origin. Small meteoritic particles, whichaccompany larger ones in interplanetary space, do nothave sufficient energy to penetrate into the lower layersof the Earth’s atmosphere, which is accompanied by va-porization, but they are stopped in the atmosphere with-out being heated above their melting points. A sizablefraction of the annual influx of extraterrestrial materialsis completely vaporized during deceleration in the upperatmosphere between altitudes of 120 and 80 km.2 Thefinal behavior depends on particle composition. Longago it was assumed3 that interplanetary materialshould consist of small submicrometer (possibly) irongrains. Later, Rosinski and Pierrard4 suggested thatmeteoric dust particles observed in the mesosphereare mainly anhydrous oxides of silicon, magnesium,aluminum, iron, and calcium. Magnesium or iron sil-icates were usually also assumed to be representativeof dust particles of extraterrestrial origin.5 Jones6

showed that Fe1�xS could be an important element ininterplanetary dust material and such asteroidal7and cometary particles can be found in the Earth’satmosphere.8 In Ref. 9 small meteoric dust particleswere modeled, which act as condensation nuclei forthe ice particles of noctilucent clouds at about 85 kmaltitude. As for morphology, the ice crystals are def-initely nonspherical and may have a size of approxi-mately 0.1 �m. In addition, Rietmeijer2 has shownthat large submicrometer-sized particles have astrictly irregular shape. A great amount of informa-tion on such dust particles was gained by remote

The authors are with the Institute for Experimental Physics,University of Vienna, Boltzmangasse 5, 1090 Wien, Austria. M.Kocifaj ([email protected]) is on leave from the AstronomicalInstitute of the Slovak Academy of Sciences.

Received 2 November 2004; revised manuscript received 15April 2005; accepted 20 April 2005.

0003-6935/05/347378-16$15.00/0© 2005 Optical Society of America

7378 APPLIED OPTICS � Vol. 44, No. 34 � 1 December 2005

sensing methods, when the response of the particlepolydisperse system on an incident electromagneticradiation field was detected. As is known, the irreg-ular particles interact with radiation in a more com-plex manner when compared with the spheres.10

These facts were notoriously not taken into accountfor almost all optical data collected in the pastdecades—dust particles in the middle and upper at-mosphere were assumed to be simply spherical. Itraises therefore quite important questions on how thepicture of the quondam dust trend is affected, be-cause the optical data usually served as a basis forviewing the dust trend in the middle atmosphere. Aninventory of existent results is therefore highly de-sirable.

Previously, the information on dust particles in themiddle atmosphere was retrieved from lunar eclipsephotometry data. Eclipses and allied phenomenawere also of the utmost importance in the investiga-tions of planetary atmospheres. They have provided anumber of interesting facts on the structure and com-position thereof at relatively small costs. The theorydealing with optical effects during lunar eclipses wasdeveloped and frequently used more than half a cen-tury ago.11–15

According to Link’s theory, the optical density D ofthe Earth’s umbra is associated with optical proper-ties of the Earth’s troposphere (and lower strato-sphere), while irradiation of the Earth’s penumbra isaffected mostly by characteristics of the atmosphereat altitudes above 20–30 km.16 The passage of the raythrough the atmosphere entails losses of severalkinds. In an ideal atmosphere containing only mole-cules and atoms of the gases, losses of light arisepartly due to molecular scattering according toRayleigh’s law, and partly due to the real absorptionaccompanied by the appearance of absorption linesand bands. The particles floating in the atmospherecause a scattering of light that is governed by rela-tively complex laws. Finally, the losses of light causedby differential refraction in the atmosphere are due tothe change in the natural divergence of the rays is-suing from the source of light. In such a case, how-ever, gains in light may also occur due to theconcentration of the pencil of rays. The sunlightpasses tangentially through the atmosphere to reach(pen)umbra regions, so the path of rays in the atmo-sphere is 10 or more times longer than for the lightreaching the Earth’s surface (see Fig. 1 and alsoRef. 12). As a characteristic behavior, the absolutevalues of optical density of the umbra rapidly de-crease at its border. This fact correlates with a weakextinction in the stratosphere. However, the quantityD responds to meteoric falls that increase atmo-spheric pollution in the middle atmosphere. A depen-dency of D on the angular distance � from the centerof the Earth’s shadow can be interpreted as a gradualchange of the optical properties of the Earth’s atmo-sphere with altitude. Optical density measured in theEarth’s umbra reflects both the instantaneous con-sistency and the seasonal state of the backgroundaerosols in the lower stratosphere or in the tropo-

sphere. Occasionally, volcanic ash should also bementioned.17 A volcanic erruption may influence lu-nar eclipses for months or years after.18 Bouska andMayer18 have shown that experimental data for pen-umbra optical density was fitted well by the theoret-ically computed curve12: it indicated a relativelysmall turbidity of the middle atmosphere. Neverthe-less, the optical density in the central parts of theEarth’s shadow was quite large. Such a fact implied agreat pollution in the lower atmosphere, possiblyoriginating from volcanic events observed in March1963 (Mount Agung). On the other hand, the lunareclipse observed in 1969 already showed slight fluc-tuations of the optical density close to the border ofthe Earth’s shadow.19 Apparently the optical effectscorrelated with the activity of a less important me-teor shower (probably the Perseid meteor stream).The Perseids could be excluded in this case, becauseonly the penumbra was influenced, i.e., only highatmospheric layers were impacted due to recent dustinflux.

Most of the past measurements were done only forone fixed area on the lunar surface (usually some-where within “Mare” regions). Nakamura et al.20 im-proved the technique for obtaining more detailedinformation on the 3D optical properties of theEarth’s atmosphere using the planar structure of theoptical density of the Earth’s shadow. They increasedthe computation accuracy of D, so the retrieval ofparticle optical and physical properties was improvedconsequently. In general, the better the qualityand�or quantity of data, the better the characteriza-tion of airborne particles. For instance, one can re-construct both the particle size distribution and theparticle modal radius rm when analyzing the spectralbehavior of D��. Therefore a dependency of opticaldensity D on wavelength � of an incident radiation

Fig. 1. Geometry of the lunar eclipse (I—solar plane, II—lunarplane). Point S corresponds to the center of the solar disk, E is thecenter of the Earth, and M lies in the center of the Earth’s shadowat the lunar plane. The optical density of the shadow D�� ismeasured at point N, whose angular distance from the center of theshadow is given by �. The projection of point N at the solar planeis given by point E�—its angular distance from the center of thesolar disk is also �. The other quantities are described inSection 2 or in Table 1.

1 December 2005 � Vol. 44, No. 34 � APPLIED OPTICS 7379

represents a subsidiary source of information on par-ticles.

In this paper we focus on differences in size distri-butions obtained from spectral profiles of D��when �i� using the conventional Mie theory (forspherical particles), as has been done traditionallyand �ii� using optical data for irregularly shaped par-ticles, which are both documented by high-altitude(NASA) sampling, and for which efficient algorithmsused to calculate the optical data have existed since1988.21

To analyze an effect of particle shape on D, thebasic theoretical relations need to be considered first.Therefore, in Section 2, the basic equations referringto the theory of lunar eclipse photometry are pre-sented. These equations can be used to calculate theoptical density of the Earth’s shadow under differentconditions and for various aerosol systems present inthe Earth’s atmosphere. A distinct influence of par-ticle shape on D is discussed in Section 3 (with de-tailed descriptions in Subsections 3.A and 3.B). Theeffect of the shape of particles suspended in the at-mosphere on information derived from D�� is dem-onstrated in Section 4 using published experimentaldata. Section 5 contains our concluding remarks. Tomake the nomenclature employed within the papermore comprehendable, we introduce a list of symbolsin Table 1, as we refer to standard terminology usedin atmospheric optics.22–23 In addition, the geometricaspects of the lunar eclipse are documented in Fig. 1to improve the understanding of the theory presentedin the following sections.

2. Origin of (Pen)umbra Optical Depth

The principal causes of changes in the light intensityas it passes through the atmosphere are refraction,which simply spreads the light over a wider area, andextinction, which actually removes a fraction of thelight from the incident beam. We use Link’s16 defini-tion for the optical density of the Earth’s shadow, butwith a negative sign,

D�� log�E��e�� , (1)

where � denotes angular distance of a certain pointwithin the Earth’s shadow from its center. Due tolight divergence in the atmosphere, the irradiance ofthe shadow at distance � is

e�� RS

��RS

T��0

�0

b��, ��d��d�, (2)

where RS is an angular radius of the solar disk (asseen from the Earth), and �0 characterizes the max-imum angle under which the elementary ring at thesolar disk is visible from point E= [radius and width ofthe ring are � and d� (see Fig. 1)]. The transmissioncoefficient T changes with the attenuation efficiencyof the solar radiation in the Earth’s atmosphere.24

Excluding the atmospheric effects (i.e., no eclipsephenomena are taken into account), the irradianceE�� of the same point in the space simply equals

E�� RS

��RS�0

�0

b��, ��d��d�. (3)

For further analysis, the ratio of E�� and e�� issufficient, and so we calculated both functions in (thesame) relative units to have them comparable witheach other. In light of this, the luminosity b� of acertain element of the solar disk is expressed in therelative units12

b��, �� 1 � � ��

RS�RS

2 � R2��, ��, (4)

where � is a wavelength-dependent coefficient25 and

R��, �� 2 � �2 � 2�� cos �, (5)

as is evident from Fig. 1. The integrations in Eqs. (2)and (3) cover all the contributions from the solar diskto the total irradiation of the examined point N at thelunar plane. For geometric reasons, the maximumangle �0 acquires a form

�0 � arccos��2 � �2 � RS2

2�� (6)

if one of the following two conditions is fulfilled:�i� � RS or �ii� � � RS along with � � RS��. Other-wise (i.e., � � RS along with � � RS��), the explicitassignment �0 is valid.

The quantity T can be expressed as a function ofminimum altitude h0 of rays in the Earth’s atmo-sphere, which contribute to the irradiation of themeasured point within the Earth’s shadow. Withrespect to the functional dependency between� and h0,16

� � � S � M��1 �h0

�� h0�, (7)

one can write T�� T��h0�, where S and M areparallaxes of the Sun and the Moon, � is the radiusof the Earth, and � is the astronomical refraction inthe spherically symmetric atmosphere.

In general, the flux density of the solar radiation isattenuated because of extinction and differential re-fraction. The effect of extinction is obvious, so let usillustrate the contribution of refraction. One can as-sume the flux of light is emanating from the Sun tothe Moon and irradiates the surface dS (Fig. 1) incase no refraction is taken into account. Because ofthe refraction in the atmosphere, the pencil will de-viate and spread to the area dS�. The attenuation byrefraction will then be given by the ratio dS�dS�. As


Table 1. Symbols, Units, and Definitions

Symbol Quantity Unit Definition

� Wavelength �m Wavelength of the incident solar radiation� Angular distance arc min Angular distance from the center of the Earth’s shadowRS Angular radius arc min Angular radius of the SunR Angular distance arc min Angular distance from the center of the solar diskε0 Angle arc min Maximum angle under which the elementary ring at

the solar disk is visible from point E= (Fig. 1)ε Angle arc min Angle that defines the position of a certain point at the

elementary ring (Fig. 1)� Angular distance arc min Angular distance of the elementary ring from point E=

(Fig. 1)�S Parallax of the Sun arc min Angle subtended by the Earth’s equatorial radius at

the center of the Sun at the mean distance betweenEarth and the Sun

�M Parallax of the Moon arc min Angle subtended by the Earth’s equatorial radius atthe center of the Moon at the mean distance betweenthe Earth and the Moon

dS Area km2 Surface at the lunar plane illuminated by flux of lightwhen no refraction takes place in the Earth’satmosphere

dS= Area km2 Surface at the lunar plane illuminated by flux of lightwhen refraction takes place in the Earth’satmosphere

�� Coefficient dependent on thewavelength

Dimensionless Coefficient that characterizes the luminositydistribution at the solar disk

c� Light dispersion Dimensionless Coefficient that refers to the light dispersion in theEarth’s atmosphere

h0 Altitude km Minimum altitude of the ray in the Earth’s atmosphereh Altitude km Altitude of the general point on the sunbeam trajectory� Radius of the Earth km Radius of the EarthH Scale height km Altitude up to which a homogeneous molecular

atmosphere would extend�G Air density kg m�3 Air density at the groundr Particle radius �m Radius of the particlerm Particle modal radius �m Modal radius of the particle size distribution function

f(r)a Parameter of the particle

size distribution functionDimensionless Refers to modified gamma size distribution function f�r�

� rae�br

b Parameter of the particlesize distribution function

�m�1 Refers to modified gamma size distribution function f�r�� rae�br

B Euler angle rad Specify the particle’s orientation with respect to thelaboratory reference frame

� Euler angle rad Specify the particle’s orientation with respect to thelaboratory reference frame

Euler angle rad Specify the particle’s orientation with respect to thelaboratory reference frame

m Refractive index Dimensionless Complex refractive index of the particlex Size parameter Dimensionless Size parameter defined as 2�r/�e�(�) Relative irradiance Dimensionless Irradiance of a certain point at distance � from the

center of the Earth’s shadow (including atmosphericeffects)

E�(�) Relative irradiance Dimensionless Irradiance of a certain point at distance � from thecenter of the Earth’s shadow (excluding atmosphericeffects)

D�(�) Optical density Dimensionless Negative common logarithm of the ratio of irradiancesE�(�) and e�(�) (Fig. 1)

b�(�, ε) Relative luminosity Dimensionless Relative amount of energy the element at the solarsurface radiates in unit time

(h0) Angle of refraction arc min Refraction of the rays of light passing through theatmosphere horizontally

i�(h) Apparent zenith angle rad Angle subtended at a general point on the trajectory byits radius vector with the ray (Fig. 1)

Tref(h0) Transmission coefficient Dimensionless Factor by which the radiation flux is reduced (due toatmospheric refraction) when reaching altitude h0

Continued


for refraction and extinction, we can split the T func-tion into two parts,13,26,27

T��h0� � T�ref�h0�T�

ext�h0�, (8)

where Tref � dS�dS� corresponds to the refractioneffect and Text is referred to the extinction. The func-tion Tref is almost independent of aerosol properties,while Text contains valuable information on the aero-sol system. Except for atmospheric molecular absorp-tion bands, the function Text will employ the form

T�ext�h0� � exp��2�

h0

�

��h� � ��h� � ��h��cos i��h�

dh�,

(9)

where ��, ��, and �� are the coefficients for molecularscattering, ozone absorption, and aerosol extinction,respectively, and angle i� reflects Snell’s law for lightrefraction in the optically inhomogeneous atmo-sphere (the spherical shell model of the atmosphere isadopted). In other words, the i� is the angle betweenthe light ray and the radius vector—measured fromthe center of the Earth to the general point on thebeam trajectory (the altitude of the beam is h). Mul-tiplying by a factor of 2 follows from the geometry ofthe analyzed problem—the sunbeams reach the alti-tude h0 at only one half of their trajectory in the

Earth’s atmosphere. Relating the index of refractionin the Earth’s atmosphere to the air density by theGladstone–Dale relation, one can write the functioni� with sufficient accuracy as follows:

sin i��h� � 1 �h � h0

�� c�

�G

�0�e�h0�H � e�h�H�, (10)

where c� is light dispersion ��0.000293 in the visiblespectrum28), �G is air density at the ground, �0

� 1.29 kg�m3 is the air density for the standardtemperature and pressure conditions of 273.15 Kand 760 mm of Hg, and H is the scale height for themolecular atmosphere. The exponential law [inEq. (10)] for air density is an appropriate simplifica-tion when an analytical expression for i� is wanted;otherwise, the apparent zenith angle of beam i� canbe calculated numerically by using some more real-istic atmospheric model (e.g., Ref. 29). Although theEarth’s atmosphere is not isothermal, it is useful tocharacterize it by representative values of �Gand H,which are understood as the best-fit parameters to ameasured density variation.30 We used H �8 km and �G � �0.

The aerosol extinction coefficient �� is associatedwith aerosol optical thickness �� by

��h0��h0

�

��h�dh. (11)

Table 1. (continued)

Symbol Quantity Unit Definition

Text(h0) Transmission coefficient Dimensionless Factor by which the radiation flux is reduced (due toatmospheric extinction) when reaching altitude h0

Tair Transmission coefficient Dimensionless Factor by which the radiation flux is reduced (due tomolecular scattering) when reaching altitude h0

Tozone Transmission coefficient Dimensionless Factor by which the radiation flux is reduced (due toozone absorption) when reaching altitude h0

Taerosol Transmission coefficient Dimensionless Factor by which the radiation flux is reduced (due toaerosol extinction) when reaching altitude h0

��(h) Aerosol coefficient Extinction km�1 Fraction of radiant flux lost from a collimated beam perunit thickness of aerosol

��(h) Ozone coefficient Extinction km�1 Fraction of radiant flux lost from a collimated beam perunit thickness of ozone due to absorption

�(h) Rayleigh coefficient Extinction km�1 Fraction of radiant flux lost from a collimated beam perunit thickness of air molecules due to scattering

��(h0) Aerosol optical thickness Dimensionless Negative natural logarithm of ratio of incident radiantflux and the flux transmitted through aerosolenvironment

Cext(�, r) Extinction cross section �m2 This quantity equals the total monochromatic powerremoved by the particle from the incident beamdivided by the incident monochromatic energy flux

Qext(�, r) Efficiency coefficient forextinction

Dimensionless Dimensionless quantity defined as a ratio of Cext��r2

(for spherical particles of radii r)fh0(r) Particle size distribution �m�3 Columnar aerosol size distribution function at base

altitude h0

Sh0(r) Particle size distribution �m�1 Columnar surface aerosol size distribution function atbase altitude h0

s(r, h) Particle size distribution m�2 Altitude-dependent particle size distribution function


While �� is usually measured in km�1, the function ��

is dimensionless. For spherical homogeneous parti-cles, the aerosol optical thickness can be computedusing conventional Mie theory,31

��h0� � �0

�

Qext��, r�r2fh0�r�dr, (12)

where Qext � Cext� r2 is the efficiency factor for ex-tinction, Cext is the extinction cross section, and fh0�r�is a columnar size distribution of airborne particles(at the base altitude h0) computed for various particleradii r.

3. Extinction by Dust Particles

The spherical geometry is rare in both terrestrial andextraterrestrial particulates. An interaction of thesolar radiation with such particulates is quite com-plex. The particle characteristics, such as physical,optical, and chemical properties, and especially mor-phology, are important parameters determining thewavefront of scattered radiation. For instance, thenonsphericity of dustlike particles eliminates the in-terference structure and ripple typical for monodis-perse scattering patterns (Fig. 2). Such matter mayresult in the disappearance of the subsidiary mode,which usually occurs with Mie particles.

Extinction is observed when the light is scatteredat the scattering angle 0° (forward transmitted ra-diation). The computational results obtained byBorghese et al.32 indicate that efficiency factors for

extinction for small nonspherical particles (or hollowspheres) can also differ from that for compactspheres. At the same time a change of Qext in a rangeof some tens of percent has a measurable effect on theretrieval of particle characteristics.33 This is quitesufficient motivation to analyze how the particleshape influences the processing of photometry datacollected during lunar eclipses. To guarantee repre-sentativeness of numerical simulations we avoid per-forming calculation for model particles, which neverrender the natural irregularities of airborne parti-cles. No simple analytical model exists to fit real par-ticle morphology as any geometrically well-definedmodel is still too regular to represent a real roughgrain.34

Because of these facts, for the considerations givenbelow we used several real interplanetary dust par-ticles analyzed by means of electron microscopy.35

The particles were collected by NASA during a seriesof flights in the stratosphere.36 Two grains, L2008pand U2015 B10, were selected for further processingas their aspect ratios (which bring together the larg-est and smallest characteristic length of the particle)fit well the value known for atmospheric parti-cles.37,38 For these two grains we also reconstructedand digitized the 3D structure (see, e.g., Fig. 3), tomake the scattering computations possible.

Several powerful methods are available to calcu-late the light scattering and absorption by non-spherical dust particles.10 We are using the discretedipole approximation21 (DDA) and its numericalimplementation called DDSCAT (Ref. 39) becausethe method is easily applicable to various geome-tries and material configurations.40 Although theDDA is reasonably fast, it is quite memory consum-ing. In general, the DDA works effectively when thedust radius does not exceed �2.5�. This condition isfulfilled for optically active particles with sizes lessthan approximately 1 �m (if measurements cover thevisible spectrum). This size range is also representa-tive for interplanetary dust populations (see, e.g.,Ref. 41), which may contribute significantly to opticaldensity inside the Earth’s shadow. We assume thatsmall-scale inhomogeneities are evenly distributedover the particle volume. In such a case it is possibleto define an effective refractive index m of the parti-cle, for which the scattering calculations have to beperformed.42

The atmospheric environment contains a mixtureof particles with different sizes, shapes, or orienta-tions, so the efficiency factor for extinction must beaveraged over the particle ensemble. Such computa-tion is rather straightforward, and all orientations ofthe particles are equiprobable. The extinction crosssection Cext for the set of randomly oriented particlesis then a triple integral,

�Cext� �1

8 2�0

2

dB��1

1

d cos ��0

2

Cext�B, �, ��d�,

(13)

Fig. 2. Extinction cross section divided by projection area a2,where a is the radius of the sphere volume equivalent to thenonspherical particle. The long-dashed curve corresponds to non-spherical iron-rich silicate particles with an aspect ratio of 1.4. Thesolid curve is drawn for volume equivalent spheres. The position ofthe first mode of both curves is almost identical and indicated bythe dashed vertical line.


where Euler angles B, �, and � specify the particleorientations with respect to the laboratory referenceframe and Qext � Cext� r2 (for nonspherical particleswe define r as the radius of a volume equivalentsphere). The randomly oriented irregular targets thatare very large compared to the wavelength shouldtherefore have Cext� r2 � 2 (Fig. 2), depending on theshape, because they will have larger projected areasthan the volume equivalent sphere. For example,Cext� r2 for a randomly oriented cube should con-verge to �3� ��4 �3�2�3 � 2.48. However, it maybe difficult to approach asymptotic behavior withDDSCAT because of computational limitations tox � 20 or 30. We used a module to select the appro-priate number of the elementary dipoles to fulfill con-ditions required by DDA. The shape of an examinedparticle is fixed. We can replace one cube (dipole) byfour, nine, etc. subcubes. On the other hand, the back-tracking procedure is also possible when the accuracyrequirements are fulfilled and one would like to speedup the numerical simulations.

A. Spherical Particles: Semianalytical Approach

The dust particles in the middle atmosphere aresmall in size [predominantly smaller than 1 �m(Ref. 2)]. Such particles either belong to the smallestfraction of meteor showers, or they represent frag-ments or residuals produced during a first phase ofparticle burning in the upper atmosphere.43,44 In gen-eral, the particles, which are of equal mass, can flareat different altitudes when they are different in com-position. For instance, even though the density of ironis approximately two times greater than the densityof stone, stony particles flare in atmospheric layerssituated only �4 km above iron’s burning layer.45

Baggaley46 showed that the mean height of sporadicmeteors is about 82 km. This altitude coincides wellwith the region where icy grains of noctilucent cloudsare formed47—the meteoric dust here plays a role ofcondensation nuclei. The upper size limit of the ice

particles is about 0.05–0.3 �m.48 However, recent re-sults collide rather with the lower limit.49

When assuming that the particles are weak ab-sorbers [like ices or interplanetary particles withabundant Fe or Ni (Ref. 50)] but have a sphericalshape, the Qext can be supplied by an efficiency factorfor scattering Qsca. This factor employs the followingform in van de Hulst’s approximation,

Qsca�z� �2m � 1

m �2 �4z sin z �

4

z2 1 � cos z��,

(14)

where z � 2x�m � 1� and x is the size parameter x� 2 r��. One can see that the model size distribution

f�r� � r�n�1

N

cn exp��bnr� (15)

satisfies the condition f�0� � f�� 0 and simulta-neously enables it to fit a large scale of real distribu-tions. Employing Eq. (15) for the simplest case N� 1, the aerosol optical thickness �� at certain fixedaltitude will be proportional to

2m � 1m c1rmL� �

rm

14 �m � 1��, (16)

where the modal radius of the particle size distribu-tion equals rm � 1�b1 and

L�� 15�4 � 10�2 � 3�

��2 � 1�3 . (17)

The transmission coefficient Text [Eq. (9)] is a productof three coefficients,13

Fig. 3. Cosmic dust particle L2008p (collected in the Earth’s stratosphere) and its digitized model.


T�ext�h0� � T�

air�h0�T�ozone�h0�T�

aerosol�h0�, (18)

which depend on light extinction in the pure molec-ular atmosphere, on the ozone absorption, and on theaerosol attenuation, respectively. At h0 � 40 km thequantity of Tozone has an unimportant effect onT�

ext�h0�, and the fluctuations of Tair are negligible.Therefore in the middle atmosphere Text changes pre-dominantly with the aerosol component of the trans-mission coefficient. One can write

T�aerosol�h0� � exp �2M��h0��h0��. (19)

Here M��h0� is the normalized horizontal optical pathlength referred to aerosols (the so-called aerosol op-tical mass) computed for the minimum altitude h0.51

Combining Eqs. (16), (17), and (19), one can illus-trate how the refractive index m and the modal ra-dius rm of the particle ensemble [with a given sizedistribution, Eq. (15)] will influence the aerosol com-ponent of the transmission coefficient. As resultsfrom the above formulas, the logarithm of T�

aerosol isproportional to expression (16) at a fixed altitude. It isplausible to examine ln�T�

aerosol� as an exact functionof L�� when a stable aerosol system is present in theatmosphere (i.e., when the modal radius rm is con-stant). This is shown in Fig. 4. For several materials,the refractive index is almost independent of wave-length in the visible. Due to this fact, the depictedprofile of ln�T�

aerosol� can also be analyzed as a func-tion of m and � when rm is assumed to be constant. AsFig. 4 manifests, for decreasing size parameters, theattenuation of the sunlight decreases. This corre-sponds to observation at large wavelengths. In a lim-ited range of wavelengths the attenuation increaseswith decreasing wavelengths. However, the functionL�� has a mode at a certain value of �mod. After �modis exceeded, L�� decreases again. One can find theasymptotic formulas and limits of the function L��for nonabsorbing, homogeneous spherical particles

lim�→�

L � 15�rm

4 �m � 1�� 2

� ��2 � 0,

lim�→0

L � 3 � 10� �

rm

14 �m � 1��

2

� 3 � const �2 � 3.

(20)

It is evident that particles, which are small comparedto the wavelength, attenuate the radiation consider-ably when they consist of optically hard materials,i.e., materials whose refractive index is essentiallydifferent from the surrounding medium.52 On theother hand, the attenuation is dominant at large val-ues of x, when the refractive index of aerosol particlesis small.

When the experimental data of D�� are available,one can solve integral in Eq. (1) to retrieve the trans-mission coefficient T�� and consequently its altitude-dependent form T��h0� [using Eq. (7)]. Since therefraction part of the transmission coefficient Tref de-pends on the geometry of the lunar eclipse, we mayisolate T�

ext�h0� from Eq. (8). Using the extinction partof the transmission coefficient obtained this way, wemay finally retrieve the quantity of T�

aerosol, becauseTozone and Tair can be calculated numerically employ-ing a model of the standard atmosphere for gaseouscomponents.

Let us present a primitive method to evaluate themean effective refractive index and the modal radiusof the columnar gamma size distribution function forthe system of spherical particles: The gamma func-tion is chosen frequently in atmospheric models be-cause of its convenience in radiation transfercalculations.53 Our approach requires that the func-tion D�� is measured at three different wavelengths.Such spectral coverage is available with photometrydata obtained during lunar eclipses in past decades.16

Due to expression (16) and Eq. (19), the ratio ofln�T�1

aerosol��ln�T�2aerosol� depends only on m and rm at

the fixed altitude (wavelengths �1 and �2 are known);the ratio for ln�T�1

aerosol��ln�T�3aerosol� is similar. Since

we have two expressions, ln�T�1aerosol��ln�T�2

aerosol� andln�T�1

aerosol��ln�T�3aerosol� depending only on m and rm,

these unknown values can be determined. An exam-ple of numerical simulation for a simple size distri-bution [Eq. (15) with N � 1] is depicted in Fig. 5.While the modal radius rm can be evaluated ratherwell from Fig. 5, the profile of the refractive index mshows quite complex behavior. The symmetry fea-tures within the m-related diagram (the left graphicin Fig. 5) correspond to the specific form of the func-tional dependency of �� on m [Eq. (16)]. A complexbehavior of the m-related diagram is therefore quiteevident. On the other hand, the rm -related diagram(the right graphic in Fig. 5) has some characteristicfeatures. The larger the rm, the smaller the ratioln�T�1

aerosol��ln�T�2aerosol� in all evaluated spectral

bands. An estimation of the modal radius is thereforemore reasonable than recognition of the particle re-fractive index.

Fig. 4. Dependency of the logarithm of Taerosol on particle refrac-tive index m and size parameter x � 2 rm��. Modal radius rm of theparticle ensemble with a given size distribution [Eq. (16), N � 1]equal to 1�b1.


B. Nonspherical Particles

Traditionally the particles have been assumed to beisotropic, homogeneous spheres—for instance, min-eral aerosols, although it is mostly misleading. Manyalgorithms for the retrieval of the microphysical char-acteristics of the particle polydisperse system aretherefore still based on this simple approximation,among others, because the Lorenz–Mie theory is fastand can handle the whole size distribution of parti-

cles. Nevertheless it is a well-established fact thatscattering by nonspherical particles generally differsfrom that of spherical particles, so an assumption ofspherical, homogeneous particles may result in er-rors. However, independent of whether the particleshave spherical or irregular shapes, the solution of theinverse problem for fh0�r� remains unmodified in thebasic logic: solving Eq. (1), the vertical profile ofT�

ext�h0� is retrieved, and consequently one can con-struct the integral equation (9) to obtain the stratifi-cation of aerosol extinction coefficient �� using gaineddata of T�

ext�h0�. Employing Eq. (11) we can solve theFredholm integral in Eq. (12) for fh0�r�.

Stratospheric particles are often of extraterrestrialorigin, so there exists a correlation between the par-ticles and the currently active meteor showers. Par-ticles that belong to the same meteor stream shouldbe similar in optical properties. This fact can signif-icantly simplify the solution of the inverse problemsbecause of the scarcely variable particle refractiveindex.

To model the effect of various dust size distributions onD and to render the basic features with spectral andangular behavior of D, we fixed the total volume–mass ofsmall particles in the atmosphere, �0

� r3fh0�r�dr �const, i.e., a continuous influx of extraterrestrial ma-terial is assumed.54 We simulated columnar sizedistributions by modified gamma functions fh0�r�� Arae�br, which fulfill the above given condition, butdiffer in the modal radius [Fig. 6(a)]. The values of rm

varied from 0 up to 0.5 �m. Special attention waspaid to size distributions with diverse shapes havingthe same rm. The variation of D under these condi-tions [Fig. 6(b)] is important because of the fact thatavailable measurements of lunar eclipses offered atbest the modal radius, so it was not possible to dis-tinguish such distributions in the past. To performthe numerical computations we needed to define any

Fig. 5. Under conditions discussed in Subsection 3.B, the presentgraphic schema may serve for concurrent evaluation of both theeffective refractive index and modal radius of the size distributionfunction [Eq. (16)] of spherical homogeneous particles at certainfixed altitudes in the Earth’s atmosphere.

Fig. 6. Columnar surface size distributions sh0�r� � r2fh0�r�, which represent the constant mass of dust material in the Earth’satmosphere [normalization is applied to every function; fh0�r� � rae�br]. (a) fh0(r) with variable modal radii; (b) modal radius of the functionfh0�r� is fixed at 0.1 �m.


staring conditions. Since fh0�r� contains three param-eters, we proceeded the following way. The opticalthickness was set to 0.5 at wavelength 0.55 �m(a typical value for a clean atmosphere). As opticalproperties we used the ones of magnesium-rich sili-cates (which is an appropriate material for the dustparticles55). At the beginning we assumed sphericalparticles with a � 2, b � 20 �m�1. A scaling factor Awas needed to obtain an optical thickness of 0.5. Forall size distributions the total mass of particles in theatmosphere must be the same, since the source isconstant. For any other combination of parameters aand b, one can always get an adequate parameter A[employing �0

� r3fh0�r�dr � const] and construct thedesired size distribution function. In such a way wecomputed the optical density within the Earth’s

shadow for both randomly oriented nonspherical par-ticles and volume equivalent spheres. Sample numer-ical results for nonspherical iron-rich silicates56,57 aredepicted in Figs. 7 and 8. The morphology of theparticles is assumed to be identical to the cosmic dustsample L2008p, i.e., aspect ratio �1.4.

The course of D�� depends on the shape of the fh0�r�function even if the modal radius is close to 0 (mean-ing that the smallest fraction of particles dominatesin the middle atmosphere). An influence of the profileof fh0�r�|rm�0 on D is apparent over the whole visiblespectrum [Figs. 7(a)– 7(c)]. Optical densities D differmostly in the umbra, while the changes of fh0�r� in-fluence D only slightly within the penumbra. Underthe specific condition f�r�|rm�0, the weak attenuationof the sunbeams is observed when small particles

Fig. 7. Optical density D at distance � from the center of the Earth’s shadow: � � �a� 0.45 �m, �b� 0.55 �m, �c� 0.65 �m. The atmosphereconsists of randomly oriented nonspherical iron-rich silicate particles with an aspect ratio of 1.4. Modal radius rm � a�b of all the mentionedcolumnar size distributions fh0�r� � rae�br is fixed at 0 �m.


dominate in the atmosphere. It is documented inFigs. 7(a)–7(c) where aerosol systems are evidentlydepleted of large particles when b � 50 �m�1 (solidcurves). An influence of large particles is more im-portant when b � 10 �m�1. As shown, the opticaldensity can considerably vary within the center of theEarth’s shadow—it depended on the actual value ofthe parameter b. A similar range of the variability ofD was also found for distributions with a modal ra-dius larger than 0, e.g., for rm � 0.1 �m [Fig. 8(a)]. Itwas shown that the irradiance inside the shadow isminimal when the particle size distribution has amode close to rm � 0.1 �m [Fig. 8(b)]. The profile ofD�� is more complicated now: The attenuation ofsolar radiation increases when the modal radius offh0�r� changes from 0.06 to 0.1 �m—here the maxi-mum extinction of the solar radiation is observed. Afurther increase in particle size (or size parameter x)causes a reduction in attenuation.

The courses of D�� for irregular grains and volumeequivalent spheres are drawn in Fig. 9. The meanaverage radius equals 0.1 �m, and parameters a andb of the modified gamma function were chosen inaccordance with the limits of model simulations. Wemay observe an evident incompatibility between theprofiles of D�� for nonspherical particles and thevolume equivalent spheres. Whereas D is the loga-rithm of irradiances [Eq. (1)], the differences found inFig. 9 are not negligible. Variation of �D by about 0.5inside the shadow may manifest quite large changesof the irradiance. As the total irradiance is propor-tional to 10�D, its value can be three times smaller (orlarger, depending on the sign of �D) if spherical par-ticles are replaced by irregular grains. Consequentlythe past measurements of the optical density of theEarth’s shadow need to be reprocessed. In Fig. 9 onecan see that the systems in which the smallest frac-

tion of dust particles dominates attenuate the solarradiation more efficiently if the grains are of irregularshape (solid curves). In contrast, the populations con-taminated by a certain amount of large particlescause an opposite effect (the dashed curves in Fig. 9).

The extinction properties of the middle atmosphereare expressively manifest at the border of theshadow, i.e., between the umbra and the penumbra,where the slope of D�� is very steep, and irradiation

Fig. 8. Profile of D affected by the attenuation of sunbeams �� 0.55 �m� in the atmosphere consisting of nonspherical iron-rich silicateand particles (the aspect ratio is 1.4): (a) modal radius rm � a�b of all mentioned columnar size distributions fh0�r� � rae�br is fixed at 0.1 �mand (b) various modal radii are assumed.

Fig. 9. Optical density of the Earth’s shadow as it refers to anatmosphere consisting of irregular particles (thin curves) or vol-ume equivalent spheres (thick curves) with columnar size distri-bution given in the form fh0�r� � rae�br.


of the Moon’s surface increases with growing of � byas much as several orders of magnitude. As followsfrom Eq. (7), only a few parts of the atmosphere con-tribute to the irradiation of a given point in theEarth’s shadow because of a small integration rangeover � in Eqs. (2) and (3). For instance, the outer partsof the shadow at a distance of �10� from the edge areaffected mostly by the atmospheric layers above�10 km, only where there is no disturbance causedby cloudiness or atmospheric pollution. Due to theo-retical conditions discussed in Section 2, one can findthe mapping � � RS, � � RS� → h0

min, h0max� at de-

fined �. The interval of h0 corresponds to the mini-mum altitudes of the atmospheric layers, whichcontribute efficiently to the irradiation of the moni-tored area inside the shadow. When measuring theoptical density in the outer parts of the shadow, bothD and h0 will change rapidly with a small shift of theangular distance �. An excess of D in comparison tothe standard background can then be interpreted asa fall of new extraterrestrial material. For example,one of the most active meteor showers, the Perseids,brings a considerable amount of dust periodicallyto the Earth’s atmosphere. It has been shown byHansa58 that the effective radius of a large amount ofthose micrometeoroids is approximately 0.35 �m andthat the observed optical thickness of turbid atmo-spheric layers corresponded well to the mass contri-bution of this shower 3 � 10�11 g cm�2 s�1. At thesame time, the submicrometer-sized particles arealso optically active in the visible, so they will essen-tially influence solar radiation penetrating throughthe atmosphere. Particles of this size range were inthe focus of quondam observations of lunar eclipses,but their scattering properties were modeled usingthe conventional Mie theory that implies the spheri-cal particle shape. However, as discussed above, theproblem with stratospheric aerosols is that mineraland dust particles have nonspherical shapes, thusmaking questionable the applicability of Mie theoryin retrieval algorithms. Employing DDA and the Mietheory, we calculated the optical densities of theEarth’s shadow Dnonsph for irregular grains and Dsphfor volume equivalent spheres simultaneously.

The differences of �D � Dnonsph � Dsph were obtainedfor various parameters of the columnar size distribu-tion function. The results indicate that size distribu-tions with modal radii about 0.3 �m produce fairlylarge values of �D. Although �D�Dsph in the center ofthe Earth’s shadow can vary in the range of about15%, the fluctuations of averaged values �D�Dsph

�,

�D�Dsph��

1��

0

�

�Dnonsph��Dsph��

� 1�d�, (21)

are approximately 5% as displayed in Figs. 10(a)–10(c). To guarantee that only in-umbra data will beanalyzed, the top limit � in Eq. (21) was chosen suchthat D � �1 for all �. It is evident from Figs. 10(a)–10(c) that the modal radius rm, at which �D�Dsph

�

reaches the maximum, increases with the observa-tional wavelength—it corresponds most likely to theshift of the first mode of the function Qext�x�. Evensmall deviations of �D (within some percent) can bequite important. For instance, if the optical density inthe center of the shadow is D � �10 (in the red partof the visible spectrum), then the decrease of �D ofabout 10% characterizes the change of the irradianceto within 1 magnitude.

4. Lunar Eclipse on 19 January 1954

The density of the shadow depends on the structure ofthe atmosphere, which itself depends on geographiclatitude. In other words, the optical density will notonly depend on the distance from the center of theshadow, but also on the position angle within theshadow. This induces us to examine the geographiccircumstances on the eclipse, i.e., to consider placeson the Earth above which the rays of light that forma shadow on the Moon are passing. The measuredoptical density D within the Earth’s shadow can, insuch a way, provide detailed information on proper-ties of the atmosphere within a certain terminatorrange. According to the current astronomical ephe-merides, the terminator of the shadow represents allthe places on the Earth at which the upper limb of theSun rises or sets at the same time. The whole termi-nator does not always take part in the formation ofthe shadow, but only the part dependent on the po-sition of the considered point of the lunar plane,where the optical density is measured.

The lunar eclipse on 19 January 1954 contains awell-documented set of data.59 Optical measurementswere performed at four wavelengths59 covering thevisible spectrum. We analyzed the first phase of theeclipse in more detail, i.e., when the Moon enteredthe shadow. The active part of the terminator waslocated west from equatorial South America. Com-paring the measured optical densities DM with theo-retical values DTR (for Rayleigh scattering in the puremolecular atmosphere), Link found roughly a con-stant excess of DM � DTR � 1.0 at � � 0.53 �mand � � 0.62 �m in the region of � � �30�, 38�� (i.e.,close to the border of the Earth’s shadow). On theother hand, DM � DTR was about 0.7 in blue (� � 0.47�m) at the border of the shadow and decreased con-tinuously toward the center of the shadow. Such spec-tral behavior of DM reflects an actual turbiditysituation in the middle atmosphere and contains ba-sic information about physical properties of smallparticles in that environment.

Using Link’s data we calculated the optimized valuesof the parameters of the altitude-dependent modifiedgamma distribution function s�r, h� � A�h�ra�h�e�b�h�r atthree altitudes h � h0: 12, 16, and 20 km above sealevel. Here h0 corresponds to the atmospheric layers,which actively influenced the optical densities mea-sured inside the Earth’s shadow. We assume the pa-rameters [a, b] are constant in every layer, sowe have a12, b12� in h � �12, 16 km�, a16, b16� in h��16, 20 km�, and a20, b20� in h � �20, h��. However,


the contribution of high layers to the extinction partof the transmission coefficient [Eq. (19)] is negligible,because the sunbeams pass along horizontal trajec-tories through the atmosphere, so the longest opticalpath is in the layer located closely above h0. Forinstance, the M��h0 � 20 km� will be affected by onlyapproximately 5%–10% when fixing h� 5 kmabove h0. In such a way, the retrieved pair a20, b20�will be representative for altitudes h � � 20,�25 km�. The function A(h) is proportional to theprofile published in Ref. 29 for the standard atmo-sphere. In accordance with Eqs. (11) and (12) wecalculated the aerosol optical thickness as follows:

��h0� ��h0

�

��h�dh ��h0

��0

�

Qext��, r�s�r, h�drdh,

(22)

where values of h0 change continuously fromh0

min to h0max according to the actual value of �. We

have precalculated a large database of optical prop-erties for a set of particle models, which differ inshape, size, and chemical composition. The databasecontained the optical characteristics of both, non-spherical particles and volume equivalent spheres.For all these models the vertical profile of Text wascalculated employing the surface size distribution s(r,h) with any combination of parameters a � � 0, 5�and b � �10, 50� �m�1. The total aerosol opticalthickness was adjusted [i.e., A(h) was multiplied by acertain scaling factor] to fit the measured values ofD�� at all three wavelengths as best as possible. Weidentified the dustlike particles with optical proper-ties similar to magnesium-rich silicates to be repre-sentative to fit the measured values of DM. Asdiscussed previously, the particles were assumed to

Fig. 10. Relative differences between optical densities D as computed for nonspherical (the aspect ratio is 1.4) and spherical iron-richsilicate particles: � � �a� 0.45 �m, �b� 0.55 �m, �c� 0.65 �m. Columnar size distribution employed the form fh0�r� � rae�br.


be nonspherical with an aspect ratio of approximately1.4. The average deviation of our theoretical profileDT from measured values of DM was approximately0.05 in the region of � � �30�, 38��, i.e., the theoret-ical and measured values differ by less than 2%. Anexample of one individual minimization is presentedin Fig. 11, where optimum parameters of a modifiedgamma distribution were found to be a20 � 2and b20 � 10 �m�1. Averaging was made over� and �, where the maximum inaccuracy (up to 5%)corresponded to the blue part of the spectrum �� 0.47 �m�. The resulting altitude-dependent sizedistribution function s(r, h) is depicted in Fig. 12. Onecan see the evident differences between optimized

solutions for nonspherical particles (dashed curves)and volume equivalent spheres (solid curves) at alti-tudes of 12–20 km. The concentration of the sphericalparticles was found to be greater than the concentra-tion of nonspherical grains to fit the measured opticaldensities within the Earth’s shadow. Also the posi-tion of the modal radii of the size distributions fornonspherical particles and their spherical equiva-lents differ by approximately 0.1 �m. Whereas theMie theory was almost exclusively applied in the pastto simulate an interaction of aerosol particles withthe incident light, the scheme of quondam evolutionof the aerosol environment in the middle atmosphereis definitely affected. Even more striking results areobtained when no ozone absorption was taken intoaccount (as done by Link16) (Fig. 13). Such ap-proaches usually occurred in the past, because onlythe Rayleigh component of the optical density wasincluded to calculate the theoretical values of D. Anabsorption in the ozone layer was only deducted fromindividual measurements.16

5. Concluding Remarks

It is now well recognized that the scattering proper-ties of nonspherical particles can differ dramaticallyfrom those of volume equivalent spheres. For in-stance, nonspherical particles tend to scatter more inthe side angles than spherical particles. The exis-tence of complex morphologies with dry dust particlespresent in the middle atmosphere is manifest also byscattering patterns. In the past the characteristics ofaerosol polydisperse systems were obtained by con-tactless optical methods; under a simple assumptionabout the spherical shape of the particles. In such away the lunar eclipse photometry data were ana-

Fig. 11. Error of minimized solution for altitude-dependent sizedistribution �rae�br at h0 � 20 km. The dustlike particles are as-sumed to be nonspherical with an aspect ratio of 1.4. The minimumerror is about 0.02 of D (i.e., less than 1% of the measured opticaldensity).

Fig. 12. Altitude-dependent size distribution functions s(r, h) fornonspherical particles (dashed curves) and their volume equiva-lent spheres (solid curves). Ozone absorption is included.

Fig. 13. Altitude-dependent size distribution functions s(r, h) fornonspherical particles (dashed curves) and their volume equiva-lent spheres (solid curves). Ozone absorption is excluded.


lyzed, so the results are certainly affected and have tobe reevaluated.

Before reaching the (pen)umbra regions within theEarth’s shadow, the sunlight rays pass tangentiallythrough the atmosphere and their path is 10 or moretimes longer than for the light reaching the Earth’ssurface. A contrast between extinction properties ofboth nonspherical particles and volume equivalentspheres is then significantly increased.

When the mass content of atmospheric dust (having afixed size distribution) occurs in the form of sphericalparticles, the detected irradiance within the Earth’sshadow can be on average three times smaller or larger(depending on the actual situation) than the correspond-ing irradiance referred to in polydisperse systems of non-spherical particles. The systems in which the smallestsize fraction of dust particles dominates attenuate thesolar radiation more efficiently if the grains are of irreg-ular shape. In contrast, the populations contaminated bya certain amount of large particles cause an oppositeeffect. At the same time, an influence of the shape of theparticle size distribution function on optical density Dinside the Earth’s shadow is apparent over the wholevisible spectrum. Depending on the profile of the particlesize distribution function of randomly oriented non-spherical particles, the irradiance within the center of theEarth’s shadow may change by 2 orders of magnitude.The optical density of the Earth’s shadow also changesefficiently with the modal radius of the particles size dis-tribution. For nonspherical particles with the aspect ratio�1.4, it was found that the minimum values of D (themaximum attenuation of sunbeams in the Earth’satmosphere) refer to distributions with modal radii ofapproximately 0.1 �m.

Sample calculations have been presented for thelunar eclipse observed on 19 January 1954. Compar-ing the measured and theoretically computed opticaldensities within the Earth’s shadow, we gained in-formation on particle polydisperse systems in thestratosphere during the eclipse. It was shown thatthe total concentration of dust particles at altitudesbetween 12 and 20 km is overestimated when as-suming the particles to be spherical. When analyzingthe profile of the altitude-dependent surface size dis-tribution function s(r, h)—as obtained for nonspheri-cal and spherical particles—one can find that thepositions of their modes are shifted by approximately0.1 �m (Fig. 12). All such results have of course im-portant consequences with respect to the view of thequondam dust trend in the middle atmosphere (asevaluated in the past for spherical particles).

This work was supported by Project M772-N02 bythe Fonds FWF and by Scientific Grant AgencyVEGA (grant 2/3024/23). We thank Ch. Dellago forcomputer resources at the Institute of ExperimentalPhysics.

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reevaluation of the quondam dust trend in the middle atmosphere

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