Solar System Dynamics Group

Example 2 Trajectory of impact at the Earth surface

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2008 ST₇

The small asteroid 2008 ST₇ was discovered on September 24, 2008 by the Mount Lemmon Survey, seventeen days after it had passed within 880 000 km of Earth. It has a diameter of about 50 meters.

The impact calculations given below were based on

Number of original observations: 108
Arc : 2008 09 24 - 2008 10 20
Nominal rms: 0².32 (205 residuals)


From this set of data we derived that collisions with Earth are possible in years:
2074, 2082, 2089, 2094A, 2094B, 2094C

We focus here on three different posibilities of impact in 2094, namely:

Table 1A. Solution 2094A. Orbital elements of impact orbit for two dates (equ. J2000)

Impact date: 2094 09 09.992 (UT)   V_impact= 15.268 km/sec   Ψ_impact = 55°.31  

V_impact

The velocity of the asteroid relative to the Earth at the moment of impact.

Ψ_impact

The impact angle is measured from the vertical axis that goes through the impact point. Thus, the impact angle is 0 degrees for a vertical impact.

rms

Root mean square for the derived impact orbit.

Table 1B. Solution 2094B. Orbital elements of impact orbit for two dates (equ. J2000)

Impact date: 2094 09 10.064 (UT)   V_impact= 15.306 km/sec   Ψ_impact = 30°.80  

Table 1C. Solution 2094C. Orbital elements of impact orbit for two dates (equ. J2000)

Impact date: 2094 09 9.903 (UT)   V_impact= 15.307 km/sec   Ψ_impact = 48°.43  

If we apply the method of the randomly selected orbits to the one of the derived impact orbit, we have to perform only one long-term integration to find quickly the arbitrary number of randomly selected VAs. Using a random number generator we put the dispersion of the assumed Gaussian distribution: σ=f_sμ where μ is the mean residual of the nominal orbit and f_s<1 is a scale factor. Let us choose an epoch T, say, a week before the impact orbit, then starting from the initial date t₀ we integrate equations of motion (and also a set of special differential equations) from the epoch t₀ (in our example from the epoch 2008 11 30, see first rows in tables 1A, 1B and 1C) until T (until 20940903, see second rows in tables 1A, 1B and 1C). Thus, we get the starting data for an effective integration of the equations of motion from the epoch T to the impact date. The number of 'clones' of the impact orbit will depend on the accepted value of the scale factor f_s<1.
(for details see Sitarski G. 2006 Generating of "Clones" of an Impact Orbit for the Earth-Asteroid Collision)


Table 2A lists the impact parameters of 129 impact clones derived according to the procedure described above for the solution 2094A. The impact clones are sorted according to the increasing value of the angle Ψ_impact (fourth columns). The geographical longitude, λ, and latitude, φ, of the impact place at the Earth surface are given in the second and third column, respectively. The derived impact points form at the Earth surface the trajectory shown by the most northern green path in Figure 1, while the red point placed on this path represents the impact orbit given in Table 1A and the first row given in Table 2A. The other two green trajectories were derived from the solutions 2094C and 2094B, respectively.

Fig.1 Three paths of the hypothetical impacts at the Earth surface for small asteroid 2008 ST7. From the top to bottom green trajectories correspond to the sample of clonned impact orbits 2094A, 2094C, 2094B, respectively. Red points represent virtual impact orbits with the smallest angle Ψ ;— these impact orbits are given in the tables 1A, 1B, 1C.

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