4 Limitation on the geocentric dynamic parameters of TCB

For convenience, we discuss this problem using Öpik's coordinates (described in Öpik 1976; Carusi et al. 1990; Valsecchi 1992), namely the geocentric coordinate system ( $U,\theta ,\phi$) - where U is the geocentric speed of the body, $\theta$is the elongation of the vector $\vec{U}$ from the vector $\vec{V}_{{\rm E}}$ (the Earth's orbital velocity), whereas $\phi$ is the angle between the plane perpendicular to the ecliptic and including the vector $\vec{V}_{{\rm E}}$, and the plane defined by the vectors $\vec{U}$ and  $\vec{V}_{{\rm E}}$ (see Fig. 6).

Figure 6: Öpik's ( $U,\theta ,\phi$) coordinate system. The origin of the frame is placed in the Earth's centre. The x-axis points in the opposite direction to the Sun; the direction of the y-axis is the same as the direction of the vector  $\vec{V}_{{\rm E}}$, the Earth's heliocentric circular velocity; the z-axis is perpendicular to the plane of the Earth's orbit.

The coordinates of the velocity $\vec{U}$ are given by:

 

$$U\sin\theta\sin\phi$$

$$U\cos\theta$$ (14)

$$U\sin\theta\cos\phi.$$

To calculate the components of the vector $\vec{U}$ we transformed the alt-azimuth components of $\vec{V}_{{\rm G}}$ into the equatorial ones, and then we used the transformation described in Valsecchi et al. (1999).

In Fig. 7, we show the ($U,\theta$) distributions for the 610 observed NEO approaching the Earth's orbit closer than $0.1~{\rm AU}$, and on the right, the same plot for the points corresponding to the TCB solutions listed in Table 4.

As shown by Carusi et al. (1990), the variables U and $\theta$ have quasi secular invariance properties, so they conserve the information about the original dynamic parameters for longer periods of time than the Keplerian osculating elements. In Fig. 7a the two populations of NEO are seen clearly: the comets are concentrated along the line of the parabolic orbits, while the asteroids lay below and occupy a greater area of the  $(U,\theta)$ plane. In Fig. 7b, the bottom region was calculated using the set of data (I) given in Table 4 and as we see it is placed entirely inside the asteroidal region of the $(U,\theta)$ plane. It does not contain any single comet. The upper region, obtained using the set of data (II), lays both in the asteroidal and the cometary region of the ($U,\theta$) plane. We also see that part of it lays in the hyperbolic range excluded from our considerations.

If we assume that the relative size of the surfaces occupied on the $(U,\theta)$ plane by the asteroidal (set I of Table 4) and cometary (set II of Table 4) solutions of the TCB orbits represent a measure of the probability of the TCB's origin, according to the results of Fig. 7b an asteroidal origin of the TCB seems more probable than a cometary one.