Ldef And The Flux Model

As stated above, LDEF offers the most useful retrieved surface dataset. Foil penetrations and crater data span the fim to mm size regime, thus the data are self consistent over a very large particle mass range. Furthermore, the geometry of LDEF is also of great benefit for statistically resolving meteoroid fluxes from debris contaminants. LDEF was gravity gradient stabilised, such that it maintained a fixed orientation with respect to Earth. The satellite was an elongated cylinder with 12 side faces (including north, south, east and west faces), and two end faces; Earth and space. The space face is of particular importance, as it had no Earth shielding, and was essentially spinning with respect to interplanetary space (i.e. once per orbit). Additionally, as LDEF's orbit precessed, the exposure of the space face to interplanetary space was to a large extent randomised (see McBride et al. [39] for a more detailed explanation of LDEF's exposure over its entire lifetime) such that the exposure to meteoroids is also essentially random. This is in contrast to the orbital debris exposure which is highly directional. Additionally (and critically), relatively little orbital debris could strike the space face due to the highly oblique impact angles presented to co-orbiting particles (even debris in highly elliptical orbits with low altitude perigee would not contribute significantly due to the highly oblique impact angles). This then makes the space face of LDEF a very good meteoroid detector despite being in LEO.

It is therefore natural, that we would want to investigate in detail whether the LDEF data are consistent with our current understanding of the dust flux at 1 AU, i.e. the Grun et al. [16] flux. Indeed, as the space face of LDEF offers an excellent self-consistent dataset over a wide particle size regime, we would seek to test the Grun et al. flux, with the possibility of better defining the accepted interplanetary flux model over the sampled size regime. It is for this reason that considerable effort was put into consolidating the LDEF datasets, and a detailed geometric model of the LDEF exposure was constructed, paying particular attention to the minutiae of the implementation. This is described below.

For LDEF's 14 faces, all available impact data sources onto aluminium surfaces were collated with well defined selection criteria and specified search areas (see McDonnell et al. [40] for a full description). True crater diameters Dc, defined at the nominal material surface, were derived from crater lip diameters Dr where necessary (using Dc=0.75Dr [29]) and converted to ballistic limit Fmax values (i.e. maximum thickness of aluminium that could be penetrated) for direct comparison with the high reliability foil MAP penetration data, using the empirical relationship Fmax=0.87Dc, derived from impact studies [41], The consolidation of the data involved derivation of 'data-fits' (performed by S.F. Green; see [40]) which passed through the bulk of the flux points, with due regard for statistical significance of each dataset and incomplete sampling. Subjective upper and lower limits were defined to accompany each of the 14 data-fits. Consideration of all 14 data-fits with regard to meteoroids and debris, clearly offers a wealth of information, although it is the space face data-fit which is of the greatest importance for meteoroid investigations due to the relative lack of debris contamination.

4.2. The flux model

To assess the LDEF data, an isotropic meteoroid model was constructed following standard techniques used by a number of workers, but differing somewhat in detail, particularly with the use of a better defined velocity distribution (hence better defining velocity dependent effects). The model includes the following features: the geometry of LDEF's exposure; a meteoroid mass (or flux) distribution; a meteoroid velocity distribution 'at 1 AU'; gravitational flux and velocity enhancement to LEO; velocity dependent Earth shielding to LDEF faces; relative impact direction effects and spacecraft velocity (flux) enhancement; a conversion to impact damage output.

The meteoroid flux distribution can be user-defined, but initially uses the Grün et al. [16] distribution, which gives cumulative mean flux values for a spinning flat plate detector at 1 AU, i.e. outside the gravitational influence of the Earth (but moving in an Earth-like orbit). Note, if we define the isotropic flux as Fq then the flux intensity is given by Fafrr.

When using a meteoroid flux distribution in a model that will consider the threshold response of various detection techniques (which might be very velocity dependent) one should incorporate a meteoroid velocity distribution rather than just use a mean velocity value. The velocity distribution n(v^) of meteoroids at 1 AU has generally been derived from ground based observations of photographic meteors (corrected for the effect of the

Velocity (km s"1)

Figure 2. The Harvard Radio Meteor Project (HRMP) meteoroid velocity distribution, following Taylor [50,51], corrected for gravitational enhancement to take the distribution to 1 AU (i.e. as seen from a massless Earth). Also shown for comparison (dotted curve) is the distribution of meteoroids entering the top of the atmosphere.

Velocity (km s"1)

Figure 2. The Harvard Radio Meteor Project (HRMP) meteoroid velocity distribution, following Taylor [50,51], corrected for gravitational enhancement to take the distribution to 1 AU (i.e. as seen from a massless Earth). Also shown for comparison (dotted curve) is the distribution of meteoroids entering the top of the atmosphere.

Earth's gravitational acceleration). Dohnanyi [42] obtained a distribution from 286 observations taken from Hawkins and Southworth [43], and Erickson [44] used the same data but attempted a more rigorous reduction to meteor number with a constant mass threshold. Kessler [45] used 2090 sporadic meteors obtained by McCrosky and Posen [46] to give a more statistically reliable distribution (see [47] for a comparison between these distributions). However, probably the most statistically reliable dataset comes from the Harvard Radio Meteor Project (HRMP) where ~ 20000 meteor observations were obtained [48,49]. This data set is often used in various modelling work. Taylor [50] reappraised the data using an improved analysis of ionisation probability and mass distribution index. Taylor also identified a numerical error in the original code used to reduce the data which resulted in a significant under-estimation of numbers of fast meteors (particularly 50 to 70 km s-1 meteors where the under-estimation is by a factor of ~100). We use this Taylor [50,51] corrected velocity distribution of meteoroids encountering the Earth's atmosphere, and convert to the distribution n(vwhich would be seen at 1 AU. This is shown in Figure 2. This gives a useful form which can then be applied to any altitude (e.g. LEO or GEO) within the model accounting for gravitational enhancement.

We consider the flux of particles encountered by a moving flat plate detector (i.e. an LDEF face) by numerically integrating the particle intensity over all viewing angles, particle mass, and particle velocity. The total instantaneous flux contribution to the detector is then given by

F= f f f f — G n(v oc) cos A — sin 6 d6 d<j> dVoo dM (1)

JMJvooJiJe 7r ve where 0, <f> are spherical polar co-ordinates with respect to the spacecraft frame, ve is the gravitationally enhanced meteoroid velocity given by ve = y/vl + v'^ (2)

(where vis the meteoroid velocity at 1 AU, and uesc is the escape velocity at the spacecraft altitude), urei is the relative velocity of the incoming meteoroid with respect to the spacecraft (i.e. wrei/ve accounts for the spacecraft moving through the meteoroid environment), and the angle A is the instantaneous impact angle to the face (measured from the face normal). G is the gravitational flux enhancement given by [52]:

(using a realistic 'working range' for v^ of v^ > 1 km s-1).

In numerical evaluation of Equation 1, no instantaneous flux contribution is added if the meteoroid cannot impact the face (i.e. if A > n/2), or if the spacecraft is shielded by the Earth. A tangent to the Earth's circumference passing through the satellite subtends an angle 6C with the direction of the Earth's centre. Thus meteoroids cannot strike the spacecraft if approaching from within the cone defined by this angle 8C (i.e. within the solid angle subtended by the Earth from the point of view of the satellite). If one assumes that meteoroid trajectories can be represented by straight lines, this 'critical' 8C angle is simply given by sin#c = re/r where re is the radius of the Earth, and r is the distance of the satellite from the Earth's centre. In reality the trajectories will be curved due to gravitational influence, and hence the angle 0C is somewhat velocity dependent (see e.g. Kessler [53]). We are thus dealing with a function, 0c{voo), and the angle 8C is described by

Note that when considering a meteoroid approach direction (and whether it is shielded), one must obviously use the actual meteoroid velocity vector with respect to Earth, and not the relative impact velocity vector.

For an instantaneous d9, d<f>, dv^ and dM step, a damage equation may be used to obtain the ballistic limit Fmax of a spacecraft surface, i.e. each flux contribution is binned at the appropriate Fmax value for the 6} v^, M element. Clearly, different detection techniques would use an appropriate relationship. The penetration equation used here follows the empirically derived '1992C' equation of McDonnell and Sullivan [54]:

0.476 q 134

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