3 Limitation on the heliocentric dynamic parameters of TCB

3.1 The heliocentric position

As discussed in Sect. 2.1, in this paper we use the time given by Ben-Menahem (see Table 4) with an uncertainty prudently estimated equal to $\pm 1$ min. In two minutes the Earth moves along its orbit approximately by  $\pm2.4\times 10^{-5}~{\rm AU}$. We therefore assumed the same uncertainty for the components of the heliocentric vector of the position of the TCB at the time of its collision with the Earth. For our purposes the accuracy of the position is very high, and if we knew the velocity vector with similar accuracy, the problem of the origin of TCB would be simpler. However, we have very poor estimation of the velocity, both its magnitude and its direction.

In the discussion that follows, we consider the Earth orbiting on a circular orbit (with the radius $1~{\rm AU}$) in the ecliptic plane. Also we assume, that the TCB moved on a heliocentric elliptical orbit.

3.2 The ascending node

Knowing the exact moment in time at which the explosion took place and that it happened very close to the Earth's surface enables us to estimate the direction of the line of nodes of the orbit. From the theory of the Earth's orbital motion we found the ecliptic longitudes of two points at which this line crosses the ecliptic:

$$\lambda= 279.1\hbox{$^\circ$ }\;\;\; {\rm or} \;\;\; \lambda=99.1\hbox{$^\circ$ }\;\;\;( 2000.0).$$ (6)

To decide which of these longitudes corresponds to the ascending node we need additional information. For this purpose we used parameters of the geocentric trajectory from Table 4 illustrated in Fig. 3. As we see, both radiant rectangles are plotted below the line of the ecliptic, hence all the TCB geocentric velocity vectors, as well as the corresponding heliocentric vectors which resulted, point to the Northern ecliptic hemisphere. Therefore the TCB must have collided with the Earth at the ascending node of the orbit, and consequently the longitude of this node is equal to:

$$\Omega=279.1\hbox{$^\circ$ }\;\; (2000.0).$$ (7)

3.3 The eccentricity, the perihelion distance and the argument of perihelion

The collision with the Earth occurred at the heliocentric distance $1~{\rm AU}$ and at the ascending node of the orbit, so we have the following crossing condition:

$$\frac{q(1+e)}{1+e\cos(\omega)}=1 \; {\rm AU}$$ (8)

where q is the perihelion distance, e the eccentricity, $\omega$the argument of the perihelion of TCB orbit.

Condition (8) is not fulfilled for all trios $(q,e,\omega)$, in particular, the orbits for which

  

(q>1) (9)

$$\Big(e<1,q<\frac{1-e}{1+e}\Big)$$ (10)

are ruled out, in the sense that they are either too large or too narrow to cross the Earth's orbit. The limiting cases are the orbits tangent to the Earth's circle at the perihelion or the aphelion points (see Fig. 4).

Figure 4: The limiting cases for the TCB heliocentric orbits. All orbits with perihelion distances q>1 or aphelion distances Q<1 cannot cross the Earth's orbit (dashed circle).

The fact that the collision took place at the ascending node and was observed on the daytime side of the Earth limits the argument of perihelion, i.e.:

$$180\hbox{$^\circ$ }\leq\omega\leq 360\hbox{$^\circ$ }.$$ (11)

Condition (10) can be used to find the smallest value of the aphelion distance $Q=1 \; {\rm AU}$ of the possible TCB orbits, and the smallest semi-major axis $a=0.5\;{\rm AU}$.

Figure 5: a) The left panel illustrates q-e distribution of 666 NEO (Apollo open circles, Aten open squares, Amor x-es, comets +es), for which the minimum distance between them and the Earth's orbit is smaller than $0.1~{\rm AU}$. The sloping solid lines trace the limits for the orbits with $a\leq 0.6, a\leq 1,a\leq 3$ AU. The dashed lines correspond to the orbits with (from bottom to top) $\omega =180\hbox {$^\circ $ }$ and $\omega =270\hbox {$^\circ $ }$. We only found nine objects (two comets and seven NEA) with the perihelion distance $q \leq 0.2~{\rm AU}$. b) The right panel shows the same distribution of elements of the TCB orbits calculated for two sets of parameters $a_{{\rm G}},h_{{\rm G}}, V_{{\rm G}}$ taken from Table 4. At the top, we have the points corresponding to the asteroidal solutions for the TCB, while at the bottom, on the right, the cometary ones.

However, as we see in Fig. 5, the perihelion distances are smaller than $0.2~{\rm AU}$ for a few of the small bodies observed. If we agree that this is true in case of the TCB, then the following lower limit for the semi-major axis is:

$$a>0.6~{\rm AU}.$$ (12)

The upper limit for q is given by condition (9), and there is no explicit upper limit for the semi-major axis.

Using condition (12) and assuming that the TCB's orbit was elliptical enable us to obtain the explicit lower and upper limits of the heliocentric orbital speed of the TCB at the moment of the collision, in [kms^-1]:

$$V_{{\rm Hmin}}=17.2, V_{{\rm Hmax}}=42.1.$$ (13)

In the right-hand panel of Fig. 5 we plotted the distribution of the values of e and q corresponding to the geocentric intervals of the TCB parameters given in Table 4. We see that for a part of the cometary solutions the eccentricities of the orbits are beyond the assumed upper limit. In Table 5 we summarized the assumed and the deduced limits on the heliocentric dynamic parameters of the TCB.

 

 

Table 5: The lower and upper limits of the TCB's heliocentric dynamic parameters. Note that except of being inside given ranges some elements have to fulfil the condition (8).

	minimum	maximum
$\Omega$	$279.1\hbox{$^\circ$ }$	-
$\omega$	$180\hbox{$^\circ$ }$	$360\hbox{$^\circ$ }$
i	0	$180\hbox{$^\circ$ }$
e	0	1
q [AU]	0.2	1
Q [AU]	1	$\infty$
a [AU]	0.6	$\infty$
$V_{\rm H}$ [kms-1]	17.2	42.1