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Astron. Astrophys. 341, 296-303 (1999)

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3. Dust orbits and detection probabilities

3.1. Dust orbits

In general interplanetary particles are moving on Keplerian orbits around the Sun under the influence of the solar gravitational attraction. The radiation pressure force [FORMULA] counterbalances the gravitational force [FORMULA] to an effective force: [FORMULA]. The [FORMULA]-value is defined as the ratio of radiation pressure [FORMULA] to gravity [FORMULA] and leads to [FORMULA]. The velocity of a particle in a circular orbit in this photo-gravitional field (cf. Baguhl, 1993) is [FORMULA], where v is the Keplerian velocity, µ is the reduced mass and r is the solar distance.

The velocity which is needed to keep a particle in a circular orbit, or more generally in a bound elliptic orbit, decreases with increasing [FORMULA]-value. Compared to the case when only gravity applies, the velocity is reduced by the factor [FORMULA]. Since this is also the case for the escape velocity, particles with high [FORMULA]-values will be in unbound orbits. When the reduction in size of a given particle leads to a higher [FORMULA]-value, it will, with the same velocity, consequently be in a more elliptic, respectively hyperbolic orbit.

The [FORMULA]-value is independent of the solar distance r and depends on the physical properties, i.e. on the mass, material, composition and structure, as well as the size of dust particles. Fig. 1 shows the mass dependency of the [FORMULA]-value for different particles (see Wilck & Mann, 1996). For particles of given optical properties the [FORMULA]-value has a maximum at about [FORMULA]-[FORMULA] g so that it is likely that particles in this mass range are more easily launched into unbound orbits than particles of a different mass. This mass range corresponds to particle radii between 0.1 and 0.2 [FORMULA] assuming a bulk density of about 3 [FORMULA] for the dust (cf. Leinert & Grün, 1990).

[FIGURE] Fig. 1. The ratio, [FORMULA] of radiation pressure force and gravity force as a function of particle mass for different dust models. Solid line: young cometary; dotted line: old cometary; dashed line: asteroidal; dash-dotted line: interstellar particles (Wilck & Mann, 1996)

If [FORMULA]-meteoroids are detected at large distances from their perihelion, they move nearly radially outward and can easily be identified from their flux direction. The radial flux direction together with the velocity would also allow us to derive the probability that the particles are detected with the Ulysses dust experiment. We make an estimate as to which extent the flux of [FORMULA]-meteoroids can be described as radial at the position of the spacecraft. We are assuming parent bodies ([FORMULA]) moving in Keplerian orbits of a given eccentricity, e and perihelion distance, q which release a smaller particle due to either fragmentation or sublimation. For the Keplerian motion of a particle in the photo-gravitational field the velocity in the heliocentric system is given as

[EQUATION]

We assume that these particles are produced from bigger particles in the perihelia of their Keplerian orbits. We do not take into account the mechanism, how the particles are formed, i.e. whether sublimation or collisions of parent bodies are responsible for the origin of these [FORMULA]-meteoroids. The particles maintain their perihelion velocities, which, due to the reduced potential (or reduced attracting force), leads to a less bound, or even hyperbolic orbit.

Assuming the perihelion velocity of the fragment to be the same as the velocity of it's parent body the semi-major axis, [FORMULA] of the newly formed particle can be written as

[EQUATION]

using the indices 0 for the orbital elements of the parent body (see for instance Kr[FORMULA]sak, 1976).

We derive a set of possible orbits of [FORMULA]-meteoroids by assuming different solar distances of their formation, different [FORMULA]-values, and different eccentricities for the orbits of the parent body. We then determine the velocity and the radial velocity component as a function of the solar distance, r.

Table 1 gives the eccentricity and the perihelion distance of the orbit of the parent body and the assumed [FORMULA]-value of the fragment. This [FORMULA]-value is the lowest possible value, so that the new particle will be in an unbound orbit. From that we derive the parameters of the orbit at a distance of 2.5 AU from the Sun in the two right columns of the table. Assuming, in the first line, a circular orbit of the parent body at 0.3 AU and a [FORMULA]-value of 0.5 the velocity of the new particle is 19 km/s at 2.5 AU and the direction differs by [FORMULA] from the radial direction. Comparing this [FORMULA] to the detection cone of the experiment, which amounts to [FORMULA], such a flux of particles can be assumed as directed radially outward. The same is the case for particles with higher [FORMULA]-values or for particles with higher perihelion velocities, when already the parent bodies have non circular orbits. From that we can conclude that [FORMULA]-meteoroids which are formed inside 0.3 AU around the Sun with [FORMULA]-values [FORMULA] would produce a nearly radial flux of particles at the position of the spacecraft. Assuming the model calculations by Wilck & Mann (1996) for [FORMULA]-values of typical interplanetary dust, [FORMULA]-values [FORMULA] 0.5 would occur in the mass range from [FORMULA] to [FORMULA] g based on their model assumptions for asteroidal dust particles and in the mass range from [FORMULA] to [FORMULA] g based on their assumptions for young cometary dust particles. At this point we should note that our estimate does not include assumptions about the relative velocities between the parent body and the fragment. However, this can be expected to be small compared to the orbital velocities (see Ishimoto & Mann, 1996). The estimate for a parent body in a non-circular orbits would also change slightly, if the fragmentation of the parent body is not in the perihelion of the orbit.


[TABLE]

Table 1.


3.2. Detection probabilities

To answer the question, "when is it possible to detect a radial flux of particles coming with different velocities from the Sun?", we determine the relative velocities between the particle and the spacecraft. We derive the angle between the velocity vector of the dust particle relative to the spacecraft and the spin axis of the spacecraft along its orbit. From that we derive the effective detection area of the dust detector for a radial flux of particles and finally the detection probability as a function of time.

The effective detection area of the dust detector, which is averaged over a spin period (approximately 5 revolutions per minute), is a function of the angle between the impact direction and the spin axis. For a directed dust flux the maximum sensor area for particles with an impact direction parallel to the symmetry axis of the detector amounts to [FORMULA] (Grün et al., 1992a). It is lower for other impact directions. We derive the effective detection area and identify the time spans, in which [FORMULA]-meteoroids could be detected. This "effective area" was first mentioned by Wilck (1994) and used to estimate detection probabilities for dust fluxes of different orbital parameters. We calculate the effective area of the detector for particles that stream radially from the Sun. This calculation is made along the orbit of the spacecraft. The effective area is shown in Fig. 2 as a function of time. The described effective area was high in 1991 directly after launch, i.e. there was a high detection probability for [FORMULA]-meteoroids. A high detection probability for [FORMULA]-meteoroids occured again in 1994 and 1995 shortly before and after the passage through the ecliptic plane. The effective area, however is always below 20% of the geometric area of the detector, i.e. [FORMULA]-meteoroids always have a low detection probability. This is a result of the geometry of the dust experiment which, due to experimental reasons, never directly faces the Sun.

[FIGURE] Fig. 2. The effective detection area of the dust experiment for particles incoming from the radial solar direction is shown as a function of time. This area is high in 1991 directly after launch and again shortly before and after the flyby through the ecliptic plane.

[FIGURE] Fig. 3. Geometry of the dust detector: we assume that particles can be identified as [FORMULA]-meteoroids, if the angle between the Sun and the sensor axis of the detector at the time of the impact event is less than about [FORMULA]. This value results on the one hand from the model calculation ([FORMULA]) mentioned in the text and on the other hand from the geometry of the detector ([FORMULA]).

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© European Southern Observatory (ESO) 1999

Online publication: November 26, 1998
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