Astron. Astrophys. 329, 785-791 (1998)

2. Model construction

We shall estimate the dust production rate of hypervelocity impacts on EKOs by interstellar dust. The surface condition of the target is an important parameter in the cratering process. Since the escape velocity of EKOs is small (less than about cm s^-1), for hard icy surfaces the effect of material strength dominates over the effect of gravity in the cratering process. If the surface of an EKO is covered by a layer of icy particles (Luu & Jewitt 1996), however, the effect of gravity dominates the cratering process. Therefore, we shall estimate the dust production rate separately for both a hard surface and for a surface covered by icy particles.

2.1. Model for a hard surface of ice material

The first model assumes that EKOs have a hard surface of ice material. In previous works, impact experiments onto water ice targets were performed to investigate the crater volume (e.g., Lange & Ahrens 1987; Frisch 1992; Eichhorn & Grün 1993). Frisch (1992) used particles with masses g to g for the projectile, while Lange & Ahrens (1987) applied a particle mass of 8 g for the projectile. On the other hand, Eichhorn & Grün (1993) used smaller particles with masses between g and g as the projectiles. The data of Eichhorn & Grün (1993) is appropriate for the study of craters produced by the impacts of interstellar dust grains, which have an average mass of about g (Grün et al. 1993 ). Eichhorn & Grün (1993) compared their results with those obtained by Frisch (1992) and Lange & Ahrens (1987), and gave an expression for the crater volume [cm³ ], as a function of the kinetic energy [eV] of the projectile, which holds over 10 orders of magnitude in kinetic energy:

where [g cm^-3 ] is the density of the ice, and [cm s^-1 ] and [g] are the impact velocity and mean mass of interstellar dust grains respectively. We assume that the density of ice targets is 1.0 g cm^-3, the mean particle mass of interstellar dust is g, and the impact velocity is 26 km s^-1 (Grün et al. 1993 ). Substituting these values into Eqs.(1)-(2), we obtain,

Since the impact velocity is sufficiently high (26 km s^-1), the total mass of the ejecta is about four orders of magnitude higher than the mass of the impacting particle.

It should be noted that some of the excavated material may melt or vaporize. According to Melosh (1989), the ratio of the mass of melted material to the mass of the projectile is given by

where is the specific internal energy for melting the target material. Substituting erg g^-1 for an ice target (Melosh 1989) and g, we found g. Therefore, the mass of the melted material is about 0.2% of the total mass of excavated material. In addition, the former is always larger than the vapor mass by a factor of nearly 10 (Melosh 1989). Consequently, both the masses of the melted and vaporized material are negligibly small compared to the total mass of excavated material.

Impact ejecta with velocity smaller than the escape velocity of the target body would eventually fall back and deposit on the surface. The amount of escaping ejecta depends on the velocity distribution of excavated material and on the gravity of the target body. Unfortunately, the velocity distribution of icy ejecta has not yet been investigated in previous impact experiments onto icy targets. Therefore, the amount of ejecta escaping from the icy target bodies is estimated in the following way.

When the target is composed of hard materials, the effect of material strength dominates the cratering process; this is referred to as the strength regime (e.g., Housen et al. 1983). According to Housen et al. (1983), the volume of ejecta with velocity higher than can be expressed in the strength regime by:

where R is the crater radius, Y the target strength, a constant, and is a parameter related to energy and momentum coupling in cratering events. By presenting the physical arguments for , Holsapple & Schmidt (1982) restricted to the range . Physically, when , the projectile energy is important for the crater dimensions. On the other hand, when , the projectile momentum is important (Holsapple & Schmidt 1982). From the results of impact experiments, Housen et al. (1983) reported for a basalt target, and 0.51 for a sand target. The cratering process in an ice target is expected to be similar to that for a basalt target rather than that for a loose sand target. Since the value of for a basalt target seems to be the theoretical upper limit of , we assume that the value of for an ice target is equal to or smaller than 3/4. When the target body has a larger mass (e.g., the Moon), the amount of escaping ejecta is very sensitive to the velocity distribution of the ejecta. In that case, the value of is a key factor. But since the majority of EKOs have low escape velocities, the amount of escaping ejecta is not so sensitive to the value of . Therefore in the following we adopt for impacts on hard icy surfaces.

Substituting into Eq.(5), we obtained,

From the definition of ,

where is the minimum velocity of the ejecta. From Eqs.(6) and (7), we obtained,

Substituting Eq.(3) into Eq.(8), we obtained the total mass of ejecta with velocities higher than the escape velocity of target body, as

If we assume that the EKO with g cm^-3 has a spherical shape with radius s [cm], is presented by

From the observations by the Ulysses spacecraft, the flux of interstellar grains f is estimated to be cm^-2 s^-1 (Grün et al. 1993). Using this value and Eqs.(9) & (10), we can calculate the mass flux of escaping ejecta produced by the impact of an interstellar dust particle,

We set minimum velocities ranging from 10 cm s^-1 to cm s^-1 in Eq.(11). From impact experiments onto water ice targets, Frisch (1992) measured velocities of ice ejecta ranging from cm s^-1 to cm s^-1. Due to the detection limit of the experimental setup, the real minimum velocity could be lower than cm s^-1. Based on her laboratory measurements, Onose (1996) reported that the minimum velocity of ice ejecta was around tens of cm s^-1. Thus the minimum velocities ranging from 10 cm s^-1 to cm s^-1 employed here seem to be reasonable for icy ejecta.

2.2. Model for a layer of icy particles

The second model assumes that the surfaces of EKOs are covered by a layer of icy particles. We note that if the size of the particles is sufficiently larger than that of the interstellar dust grain, the impact by the latter produces a crater on the surface of an individual particle. This process can then be treated as the hard surface case presented in Sect. 2.1. On the other hand, if the layer of particles is composed of fine grains which are smaller than the interstellar dust grains, an impact crater will be produced in the layer of the particles. This case shall be examined in the following.

Gravity dominates over material strength for cratering in a layer of particles; this is generally referred to as the gravity regime (Housen et al. 1983). Therefore the cratering process is not sufficiently affected by the properties of the target material. Hence we assume that the cratering process in icy particles is similar to that of sand targets.

Housen et al. (1983) formulated the distribution of velocity in the lower velocity ( m s^-1) region of powdery ejecta, based on experiments of impact cratering on sand targets. Their result is expressed as,

where g is the surface gravity and R is the carter radius. On the other hand, the velocity distribution of powdery ejecta with velocities higher than several hundred m s^-1 has been detected recently by Yamamoto & Nakamura (1997). Their data are fitted by the same scaling formula to give the following relation:

In this study we estimate the impact ejecta from target bodies with radii ranging from hundreds of m to hundreds of km, with corresponding escape velocities from about tens of cm s^-1 to hundreds of m s^-1. The reason why derived from Eq.(12) is about one order of magnitude higher than that derived from Eq.(13) is unclear. This difference affects the total mass estimation of ejecta escaping from the target body. Therefore, we use Eq.(12) and Eq.(13) separately to obtain the upper and lower estimates of the total dust production rate from the surface of icy particles.

According to Schmidt & Holsapple (1982), the crater radius R in particles targets is given by the following:

where r is a radius of projectile. By using Eq.(10), the surface gravity is expressed as

where G is the gravitational constant. Substituting Eq.(15), , and an impactor density of g cm^-3 (Grün et al. 1985) into Eq.(14), the crater radius in the particles target is:

We have assumed that the porosity of particles was 0.5 (Yen & Chaki 1992) and the bulk density was 0.5 g cm^-3. Substituting Eq.(16) into Eqs.(12) and (13), the total volume of ejecta escaping from the target body is derived. The upper estimate is

whilst the lower estimate is

Substituting Eq.(10) into Eqs.(17) and (18), and using the value of f for the flux of interstellar dust grains used in Eq.(11), we are able to calculate the mass flux of escaping ejecta by the impact of an interstellar dust grain. The upper estimate is

whilst the lower estimate is

In both cases we quote the power index to 3 significant figures.

© European Southern Observatory (ESO) 1998

Online publication: December 8, 1997
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