- Email: [email protected]

HYPERVELOCITY

DUST IMPULSES HALLEY PROBES

ON THE COMET

MAX K. WALLIS

Department

of Applied

Mathematics (Rewired

and Astronomy. in ,jnal

University

College, Cardiff,

U.K.

form 23 June 1986)

Abstract-Energy and momentum arguments lead to useful bounds on the effective impulse of dust grains impacting the high velocity space probes passing through comet Halley’s dust coma. Impulse enhancement by a factor of X-10 above the impacting momentum is favoured. while the marginal mass for dust grain penetration of the Giotro bumper shield is 445ng. The reduced impulse to the bumper from grains that penetrate it is assessed. in relation to interpretation of data from the DIDSY momentum sensors.

I. INTRODUCTION

Modelling impacts of dust grains at several tens of kilometres per second is relevant for interpreting microcraters on lunar and minor body surfaces, for calibrating thin-film dust detectors as carried on the ffelios spacecraft (Pailer and Grim. 1980) and for designing shielding and calculating deviation of the VEG.4 and Giorro spacecraft when transversing comet Halley’s dusty atmosphere (Reinhard, 1979: Wallis, 1984). Because of the back-thrust of sputtered material, a hypervelocity dust grain impacting a solid target exerts a substantially higher impulse than the grain’s momentum. The enhancement factor is independent of mass if the grain is smaller than that which can penetrate the target, but depends on impact speed and composition insofar as some energy goes into liquifying and vaporizing the material of the grain and target. Stanyukovich (1959) predicted that at high enough speeds (> 10 kms-‘) the impulse becomes proportional to the impact energy. The material structure is fully disrupted at large energy density (pressures of some IOMb or more) so there is little dependence on density and intermolecular bonding. Then the impact is analogous to a high explosive pulse of energy to a tiny volume. This argument leads to a penetration depth scaling as p r m”3 Vz’3. used for fitting experimental data by Charters and Summers (1958). Later authors have, however, used higher indices as VI’.~’ or rt~‘.~ (CourPalais, 1979: Pailer and Griin, 1980) and V”.76 or V”-’ (McDonnell, 1979; Pailer and Grim, 1980). Since the experimental data are generally at < 10 km s- 1 impact speeds, use of such formulas at 68.3 km s- ’ (Giotto) and 78 km s- ’ (VEGA) gives substantial uncertainties. In particular, the estimated mass of a dust grain that just penetrates Giotto’s I mm Al

bumper shield (setting penetration depth p = 1 mm) ranges from M,ri, = 0.9 to 5pm (Cour-Palais and Pailer and Griin formulae). Naumann (1965) reports formulae for p that give still more widely dispersed values. A numerical simulation commissioned for the Giotto project (Arnaudeau et al., 1984) found that a 5pm grain almost penetrated the 1mm Al plate. However, this simulation had some limitations in neglecting ionization plus subsequent recombination in the impact plasma (over-high pressure leads to a lower total impulse) and in ignoring flexural wave losses into the plate (for the “long” impact time > 700 ns with shear wave transmission at 3 km s ‘). The Giotto dust-impact experiments (McDonnell et al., 1986) adopted a marginal penetration mass of 1 pg, based on extrapolation from thin foil measurements at < 16 km s- ’ (McDonnell, 1979). Part of the difference is linguistic, as the 1 /lg gives the first sign of a lighttransmitting hole.

2. IMPULSE

ENHANCEMENT

Let us introduce a factor k expressing the enhancement of the impulse on a thick target compared with the impacting grain’s momentum. Experimentally for the impulse is enhanced by a few speeds VL lOkms-‘, times; in the absence of higher speed experiments, k was estimated by McDonnell (1982) to lie in the range k = 22-70

(1)

at V= 68 km s- ‘. These high values could be queried because Bjork et al. (1967) reported k = 18 at 72 km s ’ (k = 6 at 20 km s ‘), while the numerical simulation (Arnaudeau et al., 1984) resulted in a still lower value k = 7.9 at 69 km s- ’ for the 5 pg impact that only marginally penetrated the target. Subsequently, McDonnell et al. (1984) gave k = 12 & 3,

1087

1088

M. K.

using a semi-empirical extrapolation from the lower limit of k = 3 observed for a lZkms_’ impact. However, the VEGA project team (Sagdeev et al., 1983) combine momentum-energy arguments with empirical efficiencies to derive values that appear to be around those of (1). According to Sagdeev et al., the mass of dust plus aluminium target material melting on impact is M = imVZ/~Emeltr

.&,,,[Al]

= 4.0 x lO’ergg_

‘, (2)

where x is an efficiency factor taken as 0.2-0.25 from laboratory experiments. However, most of the material ejected from the impact craters at I/> 20 kms- ’ consists of vapour, and little is solid, so (2) is to be replaced by M = fmV=/(+ < u2 > + XE, + Emelt)

(3)

where E, is the specific energy of sublimation (1.05 x 10” ergg- ’ for Al), the new efficiency factor is larger, i > x, and the specific energy going to expansion velocities is < u2> = qE,.

q = 0.333

(4)

within the second uncertain factor ‘I. The expanding vapour taken to have speeds u at angle 0 to the normal has momentum M < u cos ff >, whose ratio to initial m V is k = < ucos0

> v(

+ 2;iE,

neglecting the momentum contribution tion 1 - 3 of liquid ejecta. Let us parameterize the expressions the velocity distribution function f(u,fl) 3z co&

2E,,,,),

(5)

from the fracby assuming

u3 [email protected]*’

(6)

Since A, = 0.66 to 2% accuracy for various iy between 0 and 2, expression (5) becomes k = E.V(qE,)t,‘(p5s + 2iE, + 2E,,,,), A = 0.93;s. This result means that k I fiv(2;iE,

= - $ln[

Evaluating

%uze-~“‘n2du]

(7)

= %.

the integrals: fx cosj+‘ff sin#dB s = A*= co& sin0 dO s0 cc u3+ae-&‘dU

X

JO

s

=

cc

u2 + ze-

;$

(2~3~4

&“&

0

where A, = 2/(3r$,

Al = 3n*/g, A, = 8/3(5x)*.

+ 2E,,r$,

(9)

giving under 30 even for -2+ 0 and strongly-peaked backward gas ejection with j = 3-5. With more realistic values, 1 = 0.25-0.4 and q as (4), expression (9) gives the peak values and, despite the large uncertainty in q, (8) leads to the resticted range k = [8.8 - 12.8](j + I),& + 2).

(10)

The efficiency 2 could be larger than the range taken, but if the simulation value 7.9 is to be encompassed with ,j = 3, (9) gives the limit 2 < 0.49.

3. IMPtISE

FROM PENETRATING GRAlNS

Grains larger than the marginal mass M,,,, have lower values of k. From the simulation of 5mg and lOOmg impacts (Arnaudeau et al.. 1983). the time for penetration is T= 100 ns for M = IO3 MCri,, T= 301~ jbr m = 2 x lo4 Mcri,, (11) T- 15n.s form = cc. For such large grains, the impulse to the shield may be proportional to the product of contact circumference and penetration time: circumference x m1 ‘3 leads to k - K(m/M,,J

so that qE,=

WALLIS

2,3

form > > > MC,;,.

(12)

The parameter K CCTmay be taken as Tr I i am- B with /S = 0.4 to satisfy (1 I), so a somewhat steeper dependence for intermediate size grains is implied. Moreover, grains of (l-lO)M,,,, only penetrate because of the back-ejection; they suffer significant deceleration so have longer transit times through the target plate and a contact area dependence closer to cross-section than to circumference. In the absence of detailed experimental and theoretical analysis, there is no functional form bridging the (l-l~)M~~i~ gap. The effective value of Tfor the 5,~g impact, given by the gas expansion time, is a few 100 ns, so the 1 + am-8 form cannot be extrapolated that far. The - 2/3 dependence in (12) is too shallow beyond 103M,,i,, but the “contact times” (11) are themselves too uncertain to be conclusive. For interpreting the Giotto DIDSY experimental data (McDonnell et a/.,

Hypervelocity

dust impulses

1986), formula (12) was used down to MCri,, but with a smaller marginal mass M,,, = 1 pg. Taking K constant, their comparable value K z 40 (using their k = 11) is compatible but rather low by the present estimates. 4. CONCLUSION

Energy-momentum arguments, together with the relation (3) assumed between expansion speed and sublimation energy, set rather strong limits on the enhancement factor k. The high values (1) appear to be excluded and values k i 10 (taking j = 3) for Giotto encounter speeds of 68.3 km s-’ are probable, in agreement with the Arnaudeau et al. (1984) simulation. This confirms the value 5pg for 1!4,,~,,the marginally penetrating mass (or 4 pg if k = IO), appropriate to the onset of lower values of k. For the VEGA encounters at 78 km s ‘, values k d I2 are appropriate. Grains larger than MC,;, impart impulses to the target plate with lower effective k, decreasing perhaps as mm “’ for very massive grains. Use of this formula may be accurate to a factor 5 for the largest ( - 1OOmg) grains detected by the DIDSYexperiment.

on the comet Halley probes

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REFERENCES

Arnaudeau, F., Winkelmuller, G. and de Rouvray, A. (1984) Eng. System Int., report EA/83-823 to ESA-ESTEC. Bjork, R. J., Kreyenhagen, K. M. and Wagner, M. J. (1967) NASA CR-757. Charters, A. C. and Summers, J. L. (1958) Proc. 3rd Hyperuelociry Impacf Symp. Armor Res., Foundation Chicago. Cour-Palais, B. G. (1979) The comet Halley micrometeoroid hazard, ESA SP-153, 85-92. McDonnell, J. A. M. (1979) The comet Halley micrometeoroid hazard, ESA SP-153, 93-97. McDonnell, J. A. M. (1982) 4th Giotto SWT Minutes, Appendix 5. McDonnell, J. A. M., Alexander, W. M., Lyons, D., Tanner, W., Anz, P., Hyde, T., Chen, A.-L., Stevenson, T. J. and Evans, S. T. (1984) Adu. Space Res. 4, 297. McDonnell, J. A. M. and 27 co-authors (1986) Nature 321, 338. Naumann, R. J. (1965) Proc. 7th Hyperuelocity Impact Symp. 4, l-33.

Pailer, N. and Griin, E. (1980) Planet. Space Sci. 28, 321. Reinhard, R. (1979) The comet Halley micrometeoroid hazard, ESA SP-153, 7-15. Sagdeev, R. Z., Anisimov, S. I., Galeev, A. A., Shapiro, V. D., and Shevchenko, V. I. (1983) Adrr. Space Res. 2, 133. Stanyukovich, K. P. (1959) ZhETF 36, 1605 [ = Sov. Phys. JETP 9, 11417.