## 3. Dust production rate by interstellar dust impacts## 3.1. From one EKOFor impacts by interstellar dust, the cross section of an EKO is assumed simply as . The hard surface model leads to the production rate of dust escaping from one target as Thus the dust production rate from an object with a hard surface is
independent of the target size
In the gravity regime, the crater volume in particles targets is given by Schmidt & Holsapple (1982) as where is defined in Eq.(14). From the definition of in Eq.(12), and from Eqs.(10), (14), (15) & (23), we obtained On the other hand, from Eqs.(13), (10), (14), (15) & (23), If cm, in Eqs.(24) & (25). Therefore the escape velocity is always higher than . 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 for the lower estimate. These results demonstrate why the dust production rate in the case
of icy particles surfaces depends on the target radius ## 3.2. From all EKOsNext, we estimate the total dust production rate 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, where is the number density of EKOs having
radii between 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 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 modelAccording to Jewitt & Luu (1995), there are about
QB Note that Stern (1995, 1996) uses a power law with
, and he defines the
QB The normalization constant is calculated
from the statistics of the QB From Eq.(29), we obtained = . For the hard surface model, Eqs.(21) and (22) lead to Substituting = 10 and
cm s On the other hand, from Eqs.(26) and (27), the total production rate of dust from a surface of icy particles is and ## 3.2.2. Constant mass modelThe second model assumes a constant EKO mass distribution in every
logarithmic size bin, and gives (Stern 1995,
1996). Again, the normalization constant is
calculated from the statistics of the
QB giving . This model produces 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 . Following similar arguments as those for the nominal case described above, the total dust production rate for the hard surface model is, Substituting = 10 and
cm s For the case of icy particles surface model, we derive and ## 3.3. DiscussionThe total dust production rate due to impacts by interstellar dust over the entire Edgeworth-Kuiper Belt is summarized in Table 1 for the cases considered.
The production rate of dust escaping from a larger object ( 100 km) is about four orders of magnitude higher than that from a smaller object ( 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 ( 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
in Eq.(5) is tested. By using
, 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 g s
g s Next, we estimate the optical depth of a dust cloud consisting of grains with the properties estimated in Table 1. The optical depth is defined by where is the lifetime of the grains,
its cross section for extinction, where © European Southern Observatory (ESO) 1998 Online publication: December 8, 1997 |