Solar System Dynamics Group Impact orbits and close approaches of Potentially Hazardous Asteroids #### Method of computation

We apply the cracovian algorithm for the least squares correction of orbital elements. We used a special feature of the cracovian calculus for finding the impact orbits.
1. We can construct a set of randomly selected different orbits which represent the confidence region, e.g. the 6-dimensional region of osculating orbital elements where each set of orbital elements are compatible with the existing observations.
(Sitarski G. 1998 Motion of the minor planet 4179 Toutatis: Can we predict its collision with the Earth? )
2. We apply the method of the least squares solution with the "forced" equality constraints, performing all the computation in rectangular coordinates, to find an impact orbit if it exists.
(Sitarski G. 1999 How to find an impact orbit for the Earth-asteroid collision )
3. To obtain planetary coordinates necessary for numerical integration of equations of motion by the reccurent power series, we use our numerical ephemeris of the Solar system DE405/WAW compatible with the JPL DE405 ephemeris.
(Sitarski G. 2002 Warsaw ephemeris of the solar system DE405/WAW )

#### Algorithm for finding the impact orbits

1. After selection (and weighting) of observational material according to the statistical criteria, we improve the rectangular coordinates and velocity components of the asteroid.
2. We select randomly the sample of a few thousands of virtual orbits using Sitarski's method. It allows to generate any number of virtual orbits where its RMS's fulfil the 6-dimensional normal statistics ( example 1 ). These virtual orbits form the region of confidence in the 6-dimensional space of possible osculating elements where dispersion of each orbital element is given by its error estimated from the least squares method of orbit determination.
3. Equations of motion for each of these orbital clones were next numerically integrated forwards up to the year 2100 to find dates and values of minimum distances to the Earth.
4. We apply our method of least squares correction with "forced" equality constraints to find an impact orbit for each "dangerous" date.
5. We can "clone" the impact orbit, compute impact parameters, and can determine a path of the impact locations on the Earth surface ( example 2 ). The method based on the cracovian least squares correction with the "forced" equality constraints allows us to find a number of randomly selected orbits applying a random number generator with Gaussian distribution with dispersion σ=μ where μ is the mean residual of the nominal orbit.
If we apply the above method of the randomly selected orbits to the impact orbit, we can perform only one long-term integration to find quickly an arbitrary number of the randomly selected "clones"; using the random number generator we should put now σ=fsμ where fs<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 until T. 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 sclae-factor fs<1. (Sitarski G. 2006 Generating of "Clones" of an Impact Orbit for the Earth-Asteroid Collision )