Residual mass from atmospheric ablation of small meteoroids

Planet. Space SCI., Vol. 33, No. 3. pp. 315-320.1985 Printed in Great Britam

RESIDUAL

MASS FROM ATMOSPHERIC SMALL METEOROIDS

ELISABETH

Physics Department,

6

J. NICOL,

Mount

JOHN MACFARLANE

Allison University,

ABLATION

0032-0633/8513.00+0.w 1985 Pergamon Press Ltd.

OF

and R. L. HAWKES

Sackville, New Brunswick,

Canada

EOA 3C0

(Received 20 Auyusf 1984) Abstract--Numerical solutions of the equations of meteor ablation in the Earth’s atmosphere have been obtained using a variable step size Runge-Kutta technique in order to determine the size of the residual mass resulting from atmospheric flight. The equations used include effects of meteoroid heat capacity and thermal radiation, and a realistic atmospheric density profile. Results were obtained for initial masses in the range 10-7-10~2 g, and for initial velocities less than 24 km s”’ (resultsindicated no appreciable residual mass for meteors with velocities above 24 km s-l m this mass range). The following function has been obtained to provide the logarithm of the ratio of the residual mass following atmospheric ablation to the original preatmospheric mass log I = 4.7-0.3311,

-0.01302,

+ 1.2 log m, +0.08 log’ M, -O.O83v,

log m,.

The pre-atmospheric mass and velocity are represented by mm and 0,. When the results are expressed in terms of the size of the residual mass following atmospheric ablation as a function of the initial mass and velocity, it is found that the final residual mass is almost independent of the original mass ofthe meteoroid, but very strongly dependent on the original velocity. For example, the residual mass is very nearly lo-’ g for a meteoroid with velocity 18 km s- ’ for initial masses from 10-’ to LO- 3 g. On the otherhand,aslight changein theinitialvelocityto 20 kms-’ wit1 shift the residuaimass to approx. lo- tr g. This strong velocity dependence coupled with the weak dependence on the original mass has important consequences for the sampling of ablation product micrometeorites.

INTRODUCTION

Most previous

studies

of the ablation

of meteoroids

have been concerned principally with the region in atmospheric flight when the major part of the mass is evaporated (e.g. Lebedinets and Suskova, 1968 ; Hawkes and Jones, 1975). This is because the major observational effects of the meteor trail (ionization and luminosity) are produced almost exclusively in this region. This paper presents numerical solutions to the equations of meteor flight in the atmosphere concentrating on obtaining results for the small residual masses which survive atmospheric flight. The justification for pursuing this line of inquiry is at least 2-fold. On the one hand such results are valuable for the interpretation of micrometeorite ablation products recovered in the atmosphere, or on terrestrial surfaces. Secondly, the multiple ablation products may produce important effects on the Earth’s atmosphere. We discuss each of these points below. In recent years a number of investigators have been able to sample directly micrometeorite material using spacecraft, rocket, balloon and high altitude aircraft sampling techniques (Brownlee, 1978). Furthermore, it has become increasingly possible to use chemical and physical properties which distinguish extraterrestrial dust from terrestrial dust, and hence to perform surface

collections ofmicrometeorites, in particular fromdeepsea sediments (Parkin and Tilles, 1968; Brownlee, 1978). The material collected from both the atmosphere below about 95 km and from terrestrial surfaces is obviously an aggregate of direct micrometeorite material (which never reaches the evaporation point because ofthe relatively large surface area to mass ratio for small meteoroids allowing reradiation of the incident energy), and products from larger meteoroids which are partially ablated upon atmospheric entry. By solving the equations of meteor ablation we can determine the relative importance of the contributions of each in different mass regions, as well as determine the sampling bias inherent in such studies. The latter point is particularly important and will be addressed at the end of the paper. Secondly, it will be possible to estimate the atmospheric flux of meteoroid products of different masses, and to assess the possible importance of this input for atmospheric phenomena. For example, the largely discounted theory linking meteoroid influx with global rainfall through injection of raindrop nuclei sized grains (e.g. Bowen, 1953 but see also Whipple and Hawkins, 1956) might be reconsidered should there be a peak in the residual masses at an appropriate size. With respect to the latter point the more recent work of

31.5

316

et al.

E. J. NIOL

Rosinski and co-workers in which global influxes of 520 firn particles of apparent extraterrestrial origin were related to rainfall should be mentioned (e.g. Rosinski et al., 1975). Also it should be stressed that a problem still persists in terms of explaining the rapid production of raindrop sized particles by condensation-accretion processes in warm clouds (see Hawkes, 1975 for a review), and hence any possible mechanisms should be thoroughly investigated. A number of other atmospheric effects of meteoroids have been proposed which are dependent on the mass flux and size spectrum of the incident meteoroids including ablation products. For example noctilucent clouds may well have a meteoric origin. See Hughes (1978) for a review of other atmospheric effects of meteors. While previous studies concerning the ablation of small meteoroids have considered the question ofmassvelocity limits for amicrometeorite to pass through the atmosphere without significant ablation (Whipple 1950, 1951), the treatment in this paper will include investigation of the residual masses of partially ablated meteoroid grains. Furthermore, the use of numerical solution techniques will permit accurate inclusion of all important factors. THEORETICAL

FORMULATION

TABLE 1. EQUATIONSDESCRIBING TO

OF THE PROBLEM

The ablation of a solid macroscopic meteoroid in a constant scale height atmosphere can be solved

THE

METEOROID ABLATION PRIOR

ONSET OF INTENSIVE EVAPORATION

dT -

dt

= ___

A

AP‘JJ3 ~ - 4GE(T4 - T;)

Cm1/3 213 Pm

2

Meaning Meteoroid shape factor Bulk meteoroid specific heat Acceleration due to gravity Height Sum of latent heats of fusion plus vaporisation Mass Preatmospheric mass Temperature Atmospheric temperature Meteoroid boiling point Time Velocity Preatmospheric velocity Zenith angle Drag coefficient Emissivity Heat transfer coefficient Atmospheric density Meteoroid bulk density Stefan-Boltzmann constant

Tb

and

(1)

- rlzp,o2 dv z=p+g m1/3P,z/3 dh --YCOSZ z=

dm --=O dt

(3) (4)

analyticalIy if one assumes that the incident energy is used only for evaporation of the meteoritic material, and such solutions are found in numerous works (e.g. McKinley, 1961). Clearly such an approximation leads to zero residual mass. A more exact formulation allows for expenditure of the incident energy for heating of the meteoroid mass prior to reaching the boiling point, and reradiation as well. In essence the problem can be specified by four coupled differential equations. In the time prior to the onset of intensive evaporation (temperature T less than the boiling temperature 7’J we have the equations listed in Table 1. The variables and constants used are outlined in Table 2.

TABLEZ.SYMBOLSUSEDINABLATIONBQUATIONS

Symbol

(T<

m=m,)

Value assumed 1.21 1.0x iO’ergg-’ K-r 96Ocms-’ Varies: start solutions = 140 km 6.0 x 10”’ erg g-”

at k

Varies Various values used Varies 280 K 21OOK Varies Varies Various values cos z = 0.7 1.0 1.0 1.0 Varies with height according to U.S. Standard Atmosphere (1962) 3.5 g cm-3 5.67 x 10e5 erg s-t crnm2 Km4

Residual mass from atmospheric ablation of small meteoroids Following the onset of intensive evaporation the temperature stays essentially constant, while the mass decreases. This means that equations (2) and (3) remain as above, while (1) and (4) change. The new set of equations is listed in Table 3. Lebedinets and Suskova (1968) have solved similar equations subject to the constraint that the atmosphere has a constant scale height. They also used a different evaporation law, a radiation term which ignored absorption from the surrounding atmosphere, and neglected the gravity term in equation (2). Since the scale height varies extensively in the atmospheric region concerned, a better approximation is needed if one is concerned with the relatively small residual masses involved (Lebedinets and Suskova did not continue their solutions to the point of estimating residual masses, hence this point does not seriously affect their work). We have chosen to model the atmosphere by a lo-term Chebyshev polynomial fit to the data of the U.S. Standard Atmosphere (1962), which is certainly valid within the scope of the other approximations inherent in the study. Hawkes and Jones (1975) [see also Hawkes (1979) for more detailed and refined calculations] have used the same technique. Their calculations, however, covered only one representative velocity, and did not extend the solutions to the point where residual masses could be calculated. The acceleration due to gravity term is small compared with the other terms in equation (2) and its inclusion produces little change. The solutions were arrived at by a quartic coupled Runge-Kutta technique. Since the time derivatives of the variables change markedly during the meteor’s atmospheric flight, it is essential for purposes of efficiency and accuracy to use a variable step size. The solutions were implemented with FORTRAN programs, equations (l)-(4) being solved initially until the temperature reached the boiling point. The program then shifted control to a solution with equations (la)

TABLE 3. EQUATIONS DESCRIBING METEOROID FOLLOWlNGTHEONSETOFINTENSIVEEVAPORATION(~=

ABLATION '&)

dT (14

dt dv

TAp,d ----+g &/3 2/3 Pm

iii=

(2)

dh z=

(3)

-“cosz ~uE(T:-TT,~)-~

AP‘$

(44

317

and (4a), with control being shifted back when the rate of energy radiated from the meteoroid’s surface was greater than the rate of incident energy from the intercepted air mass. For the values of the constants appropriate to the physical properties of the meteoroid, we carefully considered the comments of a number of authors including Opik (1958) McKinley (1961) Lebedinets and Suskova (1968), Verniani (1969), and Hawkes and Jones (1975). Based on those considerations we have adopted the values given in Table 2 as representative of the meteoroids in the mass range considered here. We assumed no change in the constants during atmospheric flight. Clearly this is an approximation, as is the modelling of all meteoroids with a single set of constants since the work of Ceplecha and co-workers (Ceplecha, 1958; Hawkes et al., 1984) suggests a number of distinct groups with different physical parameters. These subgroups are confirmed in some recent sensitive television observations offaint meteors (Sarma and Jones, 1984). However, we feel that the above approximations do not invalidate the main conclusions of our study, and furthermore feel that it is impossible at this time to define more precisely the physical characteristics of the meteoroid population.

RESULTS

Computer runs were conducted for geocentric velocities of 15, 18,20 and 24 km s- ‘. At each velocity value numerical solutions were obtained for initial masses of 1O-7-1O-2 g inclusive (at intervals of a factor of 10). A number of additional runs were completed at velocities above those quoted above, but we consider the resulting residual masses to be so small to be of little interest (further the macroscopic particle equations used above would probably fail for these small residual masses). The upper end of the mass range used is the approximate point where one can no longer assume isothermal heating of the meteoroid (see expressions in Lebedinets and Suskova, 1968). The computer solutions were conducted from a starting point of 140 km height and 280 K initial temperature. This starting height is sufficient to result in errors much less than those inherent in the other approximations of the method. The starting temperature is representative of the equilibrium temperature of particles with typical albedo values at 1 a.u. Rather large variations in this value would have only minor effects on the results. TO provide an analytical tool for interpreting the results we fitted the results to a function of the form specified in Table 4. We define the dependent variable r as the ratio of the residual mass following atmospheric

318

E. J.

NICOL

et al.

TABLE 4. LEAST ~QUARFS FIT OF RATIO OF RESIDUAL MASS TO INITIALMASSASAFUNCTIONOFINITIALMASSANDVELOCITY log r = a+b,u,+b,uZ,+c,, +c, log2 In,+&,

log m, log m,

(5)

a = 4.7kO.7.

b, = -0.33+0.05. c0 = 1.2*0.1.

d = -0.083+0.005. b, = -0.013~0.001. c1 = 0.08+0.01.

L

\

-7

$ ablation to the initial meteoroid mass. The values of the least squares fit coefficients, with standard deviations of the coefficients, are provided in Table 4 as well. We present in Fig. 1 our results for r as a function of preatmospheric velocity and mass. As expected, meteoroids with masses of less than 10m6 g pass through the atmosphere without ablation providing their original velocity is less than about 14 km s - ‘. If the original velocity is increased to 20 km s-i the initial mass must be about lo- 7 g in order for no ablation to occur. The well behaved form of the functional fit suggested in Fig. 1 means that some extrapolation to other mass or velocity values would be reasonable. While the format of Fig. 1 is useful in order to establish bounds of the ablation/no ablation regimes, for many purposes it is more informative to plot the residual mass, not the ratio of the residual mass to the

-4

-8

-7

-6 log Initial

-5 rmss

-

-6

-

-8

initial mass, as the dependent variable. This is done in Fig. 2. The most surprising result is the very weak dependence of the final residual mass on the initial mass. The strong dependence on the initial velocity is to be expected. For example, a 10e7-g residual mass results from initial masses of 1O-7-1O-2 g provided that the initial velocity is about 18 km s-l. A slight increase of the velocity to 20 km s-l will result in a change of a factor of 10 in the final residual mass to 10m8 g. We discuss the importance of these results in the following section.

16

-7

-4

-5

-6 log

initlat

-3

lg)

FIG. 1. PLOT OF LOGARITHM OF RATIO OF THE RESIDUAL ABLATION MASS TO THE INITIALMASS AS A FUNCTION OF THE LOGARITHMOFTHEINITIALMASSANDTHEVELOCITY.

-

-5

-4

mass

-3

-2

(g)

FIG.~.PLOTOFLOGARITHMOFTHERESIDUALABLATIONMASSASAFUNCTIONOFTHELOGARITHMOFTHEINIT~AL MASSANDTHEVELOCITY.

Residual

mass from atmospheric

DISCUSSION AND CONCLUSIONS The first point which is obvious from Figs. 1 and 2 is the very strong velocity dependence of ablation products. Large ablation products (10-s g or greater) clearly only result from low initial velocity meteors (26 km s-l or less). This means that micrometeorites larger than this size sampled in the atmosphere or on terrestrial surfaces are almost exclusively the result of a very selected sample (in terms of orbital characteristics and hence origin) of the total meteroid complex. The fact that distinct groupings, similar in overall features but different in detail from those found for brighter photographic meteors, have recently been found in studies of faint television meteors (Hawkes et al., 1984; Sarma and Jones, 1984), emphasizes the importance of this selection criterion. Furthermore, the contribution from shower meteors (hence more recent cometary material) should be very small in this mass range (since no major showers have geocentric velocities less than 24 km s-l). A second important point is obvious from Fig. 2. It can be seen that a certain residuai mass tends to result from a characteristic velocity, even with a rather wide range of initial masses. This is an important and rather unsuspected finding since it means that if one can determine (from physical and chemical arguments) that a recovered micrometeorite is an ablation product, then one can argue statistically from its residual mass what its approximate initial velocity must have been. This might provide an alternate strategy for studying orbital grouping of the meteoroid complex. Considering the differences in the orbital groupings discovered in the faint television meteor results of Hawkes et al. (1984) and Sarma and Jones (1984), the possibility of extending investigations to much fainter meteors is significant. Furthermore, this provides a mechanism for the injection of large numbers of ablation products of a nearly uniform size if there is an influx of meteors at a certain velocity. Although there are no major showers witb velocities low enough to produce significant ablation products, a minor low velocity shower could produce a rather large increase in meteoroid products in a limited mass range by this mechanism. This mechanism could well be enhanced when products of ablating dustball meteors are included (see Jacchia, 1955; Fiocco and Colombo, 1964; Simonenko, 1968; Verniani, 1969 and Hawkes and Jones, 1975 for descriptions of dustball disintegration). We are currently carrying out calculations to investigate this possibility. Obviously the injection of particles of a uniform size could have a number of important atmospheric effects.

ablation

of small meteoroids

319

The mass distribution index of the meteoroid population, if sampled below the ablation height (approx. 95 km),could be significantly altered in certain mass regions by this injection process. Furthermore, the overall index will be different for observations made below the ablation height. These effects, which we are currently investigating, may result in some of the variance observed in values of the mass distribution index for small meteoroids observed using rocket, balloon and surface collection techniques (see Hughes, 1972, 1975 for reviews). Acknowledgements-The research program of which work is a part is supported by the Natural Sciences

this and

Engineering Research Council of Canada. We would like to also acknowledge the helpful comments of Dr. J. Jones,

REFERENCES Bowen, E. G. (1953) The influence of meteoric dust on rainfall. Amt. J. Pkys. 6,490. Brownlee, D. E. (1978) Microparticle siudies by sampling techniques, in Cosmic Dust(Edited by McDonnell, J. A.), pp. 295-336. John Wiley & Sons, New York. Ceplecha, Z. (1958) On the composition of meteors. Bull. u.&-. Insts. Csl. 9, 154. Fiocco, G. and Colombo, G. (1964) Optical radar results and meteoric fragmentation. J. geophys. Res. 69, 1795. Hawkes, R. L. (1975) The 15-25 urn barrier to drop growth in warm rain. Atmosphere 13,62. Hawkes, R, L. (1979) Structure and Ablation ofDustball Meteors. Ph.D. thesis, University of Western Ontario, London, Ontario. Hawkes, R.L. and Jones, J.(1975)A quantitativemodelforthe ablation of dustball meteors. Mon. Not. R. astr. Sot. 173. 339. Hawkes, R. L., Jones, J. and Ceplecha, Z. (1984) The populations and orbits of double-station TV meteors. Bu/l. astr. Insts. Csl. 3546. Hughes, D. W. (1972) The meteoroid influx and the maintenance of the solar system dust cloud. Planet. Space sci.20, 1949. Hughes, D. W. (1975) Cosmic dust influx to the earth. Space Research XV, pp. 531-539. Akademie. Berlin. Hughes, D. W. (i978) Meteors, in Cosmic Dust (Edited by McDonnell, J. A.), pp. 123-185. John Wiley & Sons, New York. Jacchia, L. G. (1955) The physical theory of meteors. VIII : Fragmentation as cause of the faint-meteor anomaly. Astrophys. J. 121, 521. Lebedinets, V. N. and Suskova, V. B. (1968) Evaporation and deceleration ofsmall meteoroids, in Physics and Dynamics of Meteors(Edited by Kresak, L.and Millman, P. M.),pp. 193204. D. Reidel, Dordrecht. ,McKinley, D. W. R. (1961) Meteor Science and Engineering. McGraw-Hill, New York. Opik, E. (1958) Pfzysics of Meteor Flight in the Atmosphere. Interscience, New York. Parkin, D. W. and Tilles, D. (1968) Influx measurements of extraterrestrial material. Science 159,936. Rosinski, J., Nagamoto, G. T. and Bayard, M. 119751 Extraterrestrial particles and precipitation. J. atmos. terr. Ph?ls. 37, 1231.

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Sarma, T. and Jones, J. (1984) Double station observations of 465 TV meteors. Bull. as@. Znsts. Csl. (in press). Simonenko, A. N. (1968) The separation of small particles from meteor bodies, and its influence on some parameters of meteors. Physics and Dynamics of Meteors (Edited by Kresak and Millman), pp. 207-216. D. Reidel, Dordrecht. Verniani, F. (1969). Structure and fragmentation of meteors. Space Sci. Rev. 10, 230.

Whipple, F. L. (1950) The theory of micro-meteorites. Part IIn an isothermal atmosphere. Proc. Nat. Acad. Sci. 36,687. Whipple, F. L. (1951) The theory of micro-meteorites. Part II-In heterothermal atmospheres. Proc. Nat. Acad. Sci. 37, 19. Whipple, F. L. and Hawkins, G. S. (1956) On meteors and rainfall. J. Met. 13, 236.