Astron. Astrophys. 321, 652-659 (1997)

## 3. Dynamics

We are interested in estimating the probability that fragments from (7) Iris, produced by collisions with smaller bodies, may be injected into the chaotic region associated with the 3:1 mean-motion resonance with Jupiter. Of course, the probability of this kind of event depends critically on the minimum velocity needed to reach the resonance, and this is a function of the distance in the phase space between Iris and the edges of the 3/1.

In principle, the conditions of closest approach between Iris and the 3/1 resonance could be derived by computing the resonant proper elements of Iris. Resonant proper elements have been used first by Morbidelli et al. (1995) in order to determine the conditions of close approach between family members and edges of neighbouring mean-motion resonances. These authors have shown that, roughly speaking, each asteroid "sees its own resonances" due to the interplay between oscillations of the asteroid orbital elements and pulsations of resonance edges. Resonant proper elements are defined as the orbital elements of an asteroid in the conditions of closest approach to a given resonance.

However, we are not interested here in deriving simply the conditions of closest approach, but rather we want to have an idea of the typical dynamical behaviour of Iris, in order to check what is the typical range of semi-major axis and eccentricity values, in order to evaluate the limits in the critical ejection velocity values required to inject fragments into the 3/1 resonance at 2.501 AU.

Therefore we have integrated the orbit of Iris using a RADAU15 integrator (Everhart 1985) for 1 million years backwards and forwards on a Sun-SPARC workstation at the Observatory of Milan. We used the orbital elements supplied by Ephemerides of Minor Planet 1996 (see Table 1). The data were stored every 500 years. We included all the planets with the exception of Pluto and Mercury, the mass of the latter being added to the Sun.

As could be expected due to the supposedly stable position of Iris in the main belt, the semimajor axis is very stable (Fig. 1 lower left), whereas the inclination has a variation of about 4 degrees (Fig. 1 upper left). The eccentricity ranges from 0.16 (present is 0.23) to 0.27 (Fig. 1 upper right). The values of perihelion and aphelion never reach those of any planets on the timescale investigated (Fig. 1 lower right). The integrations confirm that Iris is in a stable orbit as could be expected also on the basis of the proper elements quality codes given by Milani et al. (1994).

 Fig. 1. Orbital parameters of (7) Iris. In the upper left corner the inclination versus the longitude of ascending node is shown. In the upper right eccentricity is shown whereas in the lower left the semimajor axis is plotted. The lower right corner shows the value of Iris' perihelion and aphelion versus the epoch, dashed-line is at the position of Mars' orbit.

The above results can be applied to derive the minimum ejection velocity that a fragment from Iris must achieve in order to reach the border of the 3:1 mean motion resonance. As for a possible role played by the secular resonance, we decided to disregard it, since this resonance is fairly distant from Iris in the phase space, and at most 0.2% of fragments, according to Farinella et al. 1993, could reach it.

A good approach to estimate the minimum ejection velocity needed to reach the 3/1 for different values of the eccentricity of Iris is to apply Gauss's formulæ which give the variations in orbital elements experienced by an orbiting body suffering an abrupt velocity change (Brouwer & Clemence 1961):

where n is the mean motion, while a, e and i are the semi-major axis, eccentricity and inclination of the body, respectively. f and are the true anomaly and the argument of perihelion of the body at the instant of the velocity change. are the components of the velocity change vector. For sake of simplicity, we will assume a distribution of with all the directions equally likely, i.e. . Taking into account that:

where P is the period of revolution, and are respectively the mass of the Sun and the asteroid (the latter being fully negligible, of course) we obtain from (1) the following expression for the velocity value needed to give a variation in semi-major axis:

The first two terms of (5) are, once the parameters are fixed, constants, while the latter term is a function of the true anomaly of the parent at the instant of collision.

However, we should also take into account that the above expression gives a velocity at infinity, whereas we have to consider also the fact that the real fragments must overcome also the gravitational field of the parent body. In other words, the real ejection velocity is given by:

where is the escape velocity from Iris. This can be easily estimated on the basis of the nominal size of Iris derived by IRAS: from the diameter value of 199.83 km (Tedesco & Veeder 1992), and assuming an average density of 2.5 we obtain for a value of about 120 m/sec to be quadratically added to .

The final result concerning the ejection velocity needed to reach the 3/1 resonance is shown in Fig. 2a as a function of true anomaly f and eccentricity of Iris.

 Fig. 2a. Required velocity to reach 3:1 resonance versus true anomaly (f) is shown. Solid line is referred to the current value of eccentricity, whereas short-dashed line is for the minimum value and the long-dashed line is for the highest.

In deriving the results shown in the above figure, the adopted value of (see Eq. (5)) corresponds to the closer edge of the 3/1 mean-motion resonance as given by Farinella et al. (1993). The computation has been performed taking into account also the V-shaped profile of the resonance edge, when different possible values of Iris' eccentricity are considered.

It is interesting to note that the range of variation is quite big, up to factor of 2, depending on position along the orbit. Also the dependence on eccentricity is not negligible, for most values of f. The long-dashed line corresponds to the highest value of the eccentricity range of Iris, as found in the numerical integrations shown above, while the solid line corresponds to the present eccentricity value and the short-dashed refers to the lowest eccentricity limit.

The results indicate that the resulting velocities are fairly high, but not totally unrealistic. Even assuming that the maxima of the curves shown in Fig. 2a are too high to allow an injection of fragments into the 3/1, it is still true that over a wide range of true anomalies (covering at least one half of the orbit) the velocity values are still acceptable. In particular, we should note that the computed velocity values have the meaning of average modules of velocity for fragments ejected in an isotropic velocity field, and refer to fragments having three equal velocity components. In reality, the fragments actually injected into the resonance should be those ejected mostly toward the resonance edges. These fragments, obviously do not have three equal velocity components, and the module of velocity needed to reach the resonance edge is obviously lower. As an example, if we consider fragments ejected exactly along the direction ( ; ) and we recompute the velocity value needed to reach the 3/1 border, we obtain the results shown in Fig. 2b. It is easy to see that the resulting values are sensibly lower than in the case shown in Fig. 2a. As a consequence, the values shown in Fig. 2a may be quite pessimistic, although it is better to be on the safe side in this kind of computation, since the relative fraction of fragments ejected in favourable directions should not be very high in any case. As a conclusion, we can say that over long timescales the normal collisional history of an object like (7) Iris should provide many events in which at least a small fraction (the high-velocity teil of the distribution, and/or the objects ejected in favourable directions. According to Farinella et al., 1993, this fraction could be between about 1 and 4 %) of collisional fragments might reach the 3/1 resonance, yielding a fairly weak, but continuous contribution to the process of injection of meteoritic material in the inner zone of the solar system. It is known, in fact, that the 3/1 resonance should be one of the major dynamical routes from the asteroid main belt to the zone of terrestrial planets (Wisdom, 1983, 1985; Yoshikawa, 1990; Farinella et al. 1994).

 Fig. 2b. Required velocity to reach 3:1 resonance versus true anomaly (f) in the case of and . Solid line is referred to the current value of eccentricity, whereas short-dashed line is for the minimum value and the long-dashed line is for the highest.

As for other possible factors to be taken into account in the computation of the minimum velocity, any influence of the rotation of the target on the escape velocity (Zappalà et al. 1984) is negligible in this case. For a sphere, the equatorial velocity on the surface is:

where r is Iris' radius and p is its rotational period. We simply obtain about 22.6 m/sec which is negligible compared to that required to reach 3:1 (of the order of 1 km/sec).

© European Southern Observatory (ESO) 1997

Online publication: June 30, 1998