4. The proper elements catalog

This generalized averaging principle has been used to compute, by a model including five planets (from Venus to Saturn) in circular, coplanar orbits, the secular evolution of 987 asteroids from the database of NEA orbits of NEODyS, which is available at the WWW address:

http://newton.dm.unipi.it/neodys/

From the output of this secular propagation we have computed proper elements and proper frequencies: the results of this computation are available through NEODyS, and also in catalog form at the following ftp address:

ftp://copernico.dm.unipi.it/pub/propneo/

There is a catalog file containing the following data:

1) the asteroid name,

2) the initial condition date (in Julian days),

3) the prime integrals of the theory, that is the semimajor axis $a$, the averaged perturbing function $\overline {R}$ and $Z$, which is the third component of the angular momentum. The qualitative behavior of $\omega$ (circulation or libration) is also specified. 4) the number of crossings with each planet (Venus to Saturn, numbered from 2 to 6) per period of $\omega$.

5) the proper frequencies $g-s$ of $\omega$, $s$ of the longitude of the node $\Omega$ and $g$ of the longitude of perihelion $\varpi$. In case $\omega$ is librating $g-s=0$ by definition, and we give the frequency $lf$ of the libration argument.

6) the proper elements $e_{min}, e_{max}, I_{min}, I_{max}$ (respectively minimum and maximum of eccentricity and inclination over one period of $\omega$) and, in the libration case, also $\omega_{min}, \omega_{max}$ (the minimum and maximum of the value of $\omega$) and the amplitude of the libration.

7) for each Earth crossing we have also computed the encounter conditions, namely the four quantities $(U, \theta, \phi, \lambda)$: they are a combination of the orbital elements computed at node crossings [Valsecchi et al. 1999]. These conditions have the property of being directly observable during a close approach. $U$ represents the normalized object velocity in a geocentric reference frame; $\theta$ and $\phi$ are two angles defined by the geocentric velocity vector, related to the radians in the case of a meteor stream; $\lambda$ is the longitude of the crossing point along the Earth orbit, that is the date of the crossing.

The proper elements, especially the semimajor axis $a$, the range of values of the eccentricity and the inclination are the best way to describe the medium term ($1\,000$ to $100\,000$ years) evolution of the orbit. The proper frequencies are used to detect secular resonances: when the frequencies $g$ and $s$ are close to some of the fundamental frequencies of the planetary orbits ($g_i$ and $s_i$ respectively), then the long term behavior can be dominated by secular resonance effects and the theory, as presented here, cannot provide stable proper elements. The encounter conditions are the same quantities which are directly observable for meteors; their computation for asteroids whose orbit is not crossing the Earth at the present time, but can cross at some time in the future and in the past, can be used in the search for parent bodies of meteor streams. The same variables contain the information on the average probability of impact over a long time span. These data are to be used also as input to study the fine structure of the sequences of close approaches taking place near node crossings [Valsecchi et al. 2000].