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

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3. Dust production rate by interstellar dust impacts

3.1. From one EKO

For impacts by interstellar dust, the cross section of an EKO is assumed simply as [FORMULA]. The hard surface model leads to the production rate of dust escaping from one target as

[EQUATION]

Thus the dust production rate from an object with a hard surface is independent of the target size s, if we assume that the minimum velocity of the ejecta is independent of the target size (see Fig. 1). We note, however, that for small objects (i.e. [FORMULA]), [FORMULA] in Eq.(9) is replaced by [FORMULA]. As a result, the production rate of dust escaping from a small object becomes

[FIGURE] Fig. 1. The production rate of dust escaping from a target body with radii ranging from 10 m to 100 km, and covered by a hard icy surface (solid line), or by ice particles (dashed line).

[EQUATION]

In the gravity regime, the crater volume in particles targets [FORMULA] is given by Schmidt & Holsapple (1982) as

[EQUATION]

where [FORMULA] is defined in Eq.(14). From the definition of [FORMULA] in Eq.(12), and from Eqs.(10), (14), (15) & (23), we obtained

[EQUATION]

On the other hand, from Eqs.(13), (10), (14), (15) & (23),

[EQUATION]

If [FORMULA] cm, [FORMULA] in Eqs.(24) & (25). Therefore the escape velocity is always higher than [FORMULA]. This implies that the conditions which apply to Eq.(22) do not apply for a particles surface.

Therefore, the production rate of dust escaping from one target body is derived from Eqs.(19) and (20) as

[EQUATION]

for the upper estimate, and

[EQUATION]

for the lower estimate.

These results demonstrate why the dust production rate in the case of icy particles surfaces depends on the target radius s for all target bodies, in contrast with the two different s -dependence dust production rates derived for the hard surface case (see Fig. 1).

3.2. From all EKOs

Next, we estimate the total dust production rate [FORMULA] by summing the ejecta mass from one target body, estimated in Sect. 3.1, over the entire Edgeworth-Kuiper Belt region. The size distribution of EKOs is a key factor. Stern (1995, 1996) investigated the rate of mutual collisions of EKOs with radii from 0.1 km to 162 km, and predicted a total production rate of collisional debris. Since it is worthwhile to compare the production rate by impacts of interstellar dust with that produced by mutual collisions of EKOs, we shall adopt the size distribution model of EKOs used by Stern (1995, 1996).

Stern's model assumes that EKOs obey a power law size distribution,

[EQUATION]

where [FORMULA] is the number density of EKOs having radii between s and [FORMULA], and [FORMULA] is a constant. Using the two models of EKO size distribution used in Stern (1995, 1996), we estimate the total dust production rate due to impacts by interstellar dust over all EKOs.

We note that the minimum size limit of EKOs is an important parameter in estimating the total dust production rates, both for dust production by EKOs mutual collisions and for that by impacts of interstellar dust on EKOs. Observations by the Hubble Space Telescope found 29 objects with radii ranging from 5 to 10 km in the Edgeworth-Kuiper Belt region (Cochran et al. 1995). Much smaller objects that are too faint to be detected, however, may exist in the Edgeworth-Kuiper Belt. Recent works on collisional evolution among EKOs assume the minimum radius of an EKO to be [FORMULA] km (Stern 1995, 1996; Davis & Farinella 1997). In order to make a comparison with the dust production rate by mutual collisions of EKOs predicted by Stern (1996), we also assume that the minimum radius of the object is 0.1 km. In addition, we adopt a maximum radius of 162 km, which is also the same value used by Stern (1996).

3.2.1. Nominal model

According to Jewitt & Luu (1995), there are about [FORMULA] QB1 sized objects ([FORMULA] 50 km in radius) in the Edgeworth-Kuiper Belt region. As a simple power-law with [FORMULA], the first model connects this population with about [FORMULA] comets which Stern (1995, 1996) defined as bodies with radii between 1 and 6 km.

Note that Stern (1995, 1996) uses a power law with [FORMULA], and he defines the [FORMULA] QB1 sized objects as bodies with radii [FORMULA] 100 km, whereas Jewitt & Luu's (1995) estimate applies for radii [FORMULA] 50 km. In this work we define the QB1 sized objects as bodies with radii [FORMULA] 50 km.

The normalization constant [FORMULA] is calculated from the statistics of the [FORMULA] QB1 sized objects by

[EQUATION]

From Eq.(29), we obtained [FORMULA] = [FORMULA].

For the hard surface model, Eqs.(21) and (22) lead to

[EQUATION]

Substituting [FORMULA] = 10 and [FORMULA] cm s-1 into Eq.(30), we find that the total dust production rate [FORMULA] ranges from [FORMULA] g s-1 to [FORMULA] g s-1.

On the other hand, from Eqs.(26) and (27), the total production rate of dust from a surface of icy particles is

[EQUATION]

and

[EQUATION]

3.2.2. Constant mass model

The second model assumes a constant EKO mass distribution in every logarithmic size bin, and gives [FORMULA] (Stern 1995, 1996). Again, the normalization constant [FORMULA] is calculated from the statistics of the [FORMULA] QB1 sized objects (Jewitt & Luu 1995),

[EQUATION]

giving [FORMULA]. This model produces [FORMULA] comets with radii between 1 km and 6 km, and this result is consistent with the estimation by Duncan et al. (1995) that the total number of comets is roughly [FORMULA].

Following similar arguments as those for the nominal case described above, the total dust production rate for the hard surface model is,

[EQUATION]

Substituting [FORMULA] = 10 and [FORMULA] cm s-1 into Eq.(34), we find that the total dust production rate is [FORMULA] g s-1 and [FORMULA] g s-1 respectively.

For the case of icy particles surface model, we derive

[EQUATION]

and

[EQUATION]

3.3. Discussion

The total dust production rate [FORMULA] due to impacts by interstellar dust over the entire Edgeworth-Kuiper Belt is summarized in Table 1 for the cases considered.


[TABLE]

Table 1. Total dust production rate [FORMULA] due to impacts by interstellar dust over the entire Edgeworth-Kuiper Belt.


The production rate of dust escaping from a larger object ([FORMULA] 100 km) is about four orders of magnitude higher than that from a smaller object ([FORMULA] 0.1 km) (see Fig. 1). On the other hand, the smaller objects are at least 7 orders of magnitude more numerous than the larger target bodies from Eq.(28). Therefore, the major part of the dust produced in the Edgeworth-Kuiper Belt region originates from the smaller objects. In other words, the total dust production rate due to impacts by interstellar dust depends strongly on the number of small objects ([FORMULA] 1km) in the Edgeworth-Kuiper Belt.

The minimum radius of an EKO was set at 0.1 km for comparison with the dust production rate by mutual collisions of EKOs (Stern 1996). However, much smaller objects that are too faint to be detected may exist in this region. Since most of the dust produced by impacts of interstellar dust comes from small objects, a reduction in the minimum radius of an EKO from 0.1 km to 10 or 1 m would lead to dust production rates higher than those estimated in this paper.

The sensitivity of our results to the value of [FORMULA] in Eq.(5) is tested. By using [FORMULA], corresponding to the theoretical lower limit (Holsapple & Schmidt 1982), we re-derived the total dust production rate over all EKOs for the hard surface model. We find that the total dust production rates are [FORMULA] g s [FORMULA] g s-1 for the nominal EKO's size distribution model, and [FORMULA] g s [FORMULA] g s-1 for the constant mass model. These results are very similar to those derived for [FORMULA]. Therefore we conclude that the assumption of [FORMULA] for the ice targets does not have a significant influence on the total dust production rate, as long as [FORMULA].

Next, we estimate the optical depth of a dust cloud consisting of grains with the properties estimated in Table 1. The optical depth [FORMULA] is defined by

[EQUATION]

where [FORMULA] is the lifetime of the grains, [FORMULA] its cross section for extinction, l the width of the Edgeworth-Kuiper Belt, and [FORMULA] the volume of the Edgeworth-Kuiper Belt region. For simplicity, we assume that all the grains have a radius [FORMULA] cm and hence [FORMULA] cm2. Poynting-Robertson drag dominates the orbital evolution of grains with [FORMULA] cm in the Edgeworth-Kuiper Belt region (Liou et al. 1996) . Therefore, we take the timescale of the Poynting-Robertson drag [FORMULA] as [FORMULA], i.e.

[EQUATION]

where a and [FORMULA] are in CGS-units and [FORMULA] is in AU (Wyatt & Whipple 1950). We further assume that the dust grains are in circular orbits at [FORMULA] AU. Substituting [FORMULA] cm, [FORMULA] g cm-3 and [FORMULA] AU into Eq. (38), we obtain [FORMULA] s. Assuming the Edgeworth-Kuiper Belt region is a band with thickness of 16deg around the ecliptic, with a width [FORMULA] AU (Jewitt & Luu 1995), we calculated [FORMULA] cm3. Substituting these values into Eq. (37), [FORMULA] is calculated to be [FORMULA] for [FORMULA] g s-1, and [FORMULA] for [FORMULA] g s-1. Stern (1996) predicted the optical depth of debris produced by mutual collisions to be between [FORMULA] and [FORMULA]. Although our estimates are slightly higher than those predicted by Stern (1996), more detailed analysis of the optical properties of thin dust clouds is required to predict their detectability from the Earth.

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© European Southern Observatory (ESO) 1998

Online publication: December 8, 1997
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