6. Secular resonances

 

Figure: Map of the Near Earth Asteroids affected by one or more of the main secular resonances in the phase space (minimum $e$, $a$): the asteroids with $g\simeq g_6$ have been plotted with a circle, the ones with $s\simeq s_6$ with a square and the ones with $g\simeq g_5$ with a cross.

From the computation of the proper frequencies it has been possible to draw a map of the dynamical structure of the Near Earth Asteroids, in particular identifying the location of the most important secular resonances.

In the plane (minimum $e$, $a$) we have plotted all the 987 NEA; we have marked with a circle the ones which we expect would be affected by the resonance $\nu_6$ (that is $g-g_6\simeq 0$), with a square the ones by the resonance $\nu_{16}$, (that is $s-s_6\simeq 0$), and with a cross the ones by the resonance $\nu_5$ (that is $g-g_5\simeq 0$). We have used a simple criterion, marking the ones with the secular divisor associated to the resonance smaller than $2$ arcsec per year. A better map should be made by using actual estimates of the resonance widths, but these values are reasonable, given the results of the numerical integrations such as [Gronchi and Michel 2000].

The well known transport mechanism towards high eccentricities due to the $\nu_6$ resonance is clearly visible in figure 6. It had already been studied by a secular theory and by numerical integrations in the main belt [Morbidelli and Henrard 1991,Froeschle and Scholl 1986]. Semianalytical theories confirmed its existence also near the Earth crossing region [Michel 1997,Michel and Froeschlé 1997], which could not be studied by the same theory for the singularities appearing in the perturbing function.

Also the region of the Hungaria can be noticed: it is characterized by the presence of the $\nu_{16}$ resonance.

In addition to these already known dynamical structures determined by secular resonance, a new one appears from this plot: looking at the crosses also the $\nu_5$ resonance seems to play a role in the increase of the eccentricity, mostly inside the Earth crossing region. This is consistent with the identification of the same resonance for orbits near-Earth but not crossing, as obtained by a semianalytic theory [Michel and Froeschlé 1997]. We are not aware of a numerical confirmation of this resonant effect; this confirmation would be useful, and even more useful would be an assessment of the possible role of the $\nu_5$ resonance in injecting already Earth-crossing objects in orbits impacting the Sun.