OrbFit Impact Solutions for Asteroids (99942) Apophis and (144898) 2004 VD17

The all-encompassing term Space Science was coined to describe all of the various fields of research in science: Physics and astronomy, aerospace engineering and spacecraft technologies, advanced computing and radio communication systems, that are concerned with the study of the Universe, and generally means either excluding the Earth or outside of the Earth's atmosphere. This special volume on Space Science was built throughout a scientifically rigorous selection process of each contributed chapter. Its structure drives the reader into a fascinating journey starting from the surface of our planet to reach a boundary where something lurks at the edge of the observable, light-emitting Universe, presenting four Sections running over a timely review on space exploration and the role being played by newcomer nations, an overview on Earth's early evolution during its long ancient ice age, a reanalysis of some aspects of satellites and planetary dynamics, to end up with intriguing discussions on recent advances in physics of cosmic microwave background radiation and cosmology.


Introduction
The best systems with the exact impact solutions for dangerous asteroids are presented by the JPL Sentry System: http://neo.jpl.nasa.gov/risk/ and by the NEODyS CLOMON2: http://newton.dm.unipi.it/neodys/index.php?pc=4.1 From many years on the top of these lists were two asteroids: (99942) Apophis (is still up now, October 2011) and (144898) 2004 VD17 -now is removed from the list of the dangerous asteroids. Thanks to the courtesy of those who made free available OrbFit software and its source code at: http://adams.dm.unipi.it/~orbmaint/orbfit/ It is now possible to compute individually dates of possible impacts of selected dangerous asteroids or the energy of impact and others impact factors. In this respect we investigated the motion of these recently discovered minor planets: (99942) Apophis and (144898) 2004 VD17 -the most dangerous for the Earth, according to the Impact Risk Page of NASA: http://neo.jpl.nasa.gov/risk/.
To compute exact impact solutions of asteroids it is necessary to include some additional small effects on the asteroid's motion. The inluence of relativistic effects, the perturbing massive asteroids, the Yarkovsky/YORP effects, solar radiation pressure, different ephemeris of the Solar System were investigated. To compute gravitational forces perturbing the motion of (99942) Apophis and (144898) 2004 VD17 from different massive asteroids, the free software Solex from A. Vitagliano was used: http://chemistry.unina.it/~alvitagl/solex/. SOLEX computes positions of the Solar System bodies by a method which is entirely based on the numerical integration of the Newton equation of motions (Vitagliano, A. 1997). With the use of Solex it was possible to compute all close approaches between (99942) Apophis and (144898) 2004 VD17 with all nearly 140000 numbering asteroids. Similar work with (15) Eunomia using Solex was done by Vitagliano and Stoss (2006). Selected orbit solutions for (99942) Apophis and (144898) 2004 VD17 were presented during Meeting on Asteroids and Comets in Europe -May 12-14, 2006 in Vienna, Austria. At that time the new version of OrbFit (3.3.2) was released and gave better results of computations of impact probability mainly with the use of non linear monitoring and multi ple solutions method (Milani et al., 2002, Milani et al., 2005a and www.intechopen.com Space Science 60 2005b). The main goal of our work was to compare our results generated by OrbFit with the results presented by CLOMON2 system which uses the same OrbFit software and with the results of JPL NASA SENTRY. The second purpose was to prove how differently small effects in motion of asteroid change impact solutions. It was possible thanks to public available source code of the OrbFit software. The orbital uncertainty of an asteroid is viewed as a cloud of possible orbits centered on the nominal solution, where density is greatest. This is represented by the multivariate Gaussian probability density and the use of this probability density relies on the assumption that the observational errors are Gaussian (Milani et al., 2002)

The Influence of sigma_LOV and radar observation
The orbital elements of (99942) Apophis in Tab. 1 were computed by the author using all 1007 observations up to this date (Sep. 14th, 2006) and software OrbFit where M -mean anomaly, a -semimajor axis, e -eccentricity,   -argument of perihelion,   -longitude of the ascending node, i 2000 -inclination of the orbit. These orbital elements are referred to the J2000 equator and equinox. [RE] -minimum distance, the lateral distance from LOV (line of variation, which represent the central axis of the asteroid's elongated uncertainty region); impact probability -computed with a Gaussian bidimensional probability density; IW -computed solutions by author of this paper; nr denotes solution without radar observations and equal tosigma_LOV -approximate location along the LOV in sigma space; values of sigma are usually in the interval [-3,3] which represent 99.7 % probability of occurrence of real asteroid in this confidence region (Milani et al. 2002). The impact probability is not reported if the computed value is less than 1E-11. The presented  are only the input data in OrbFit software, not the real  -positive or negative, along the LOV. For example = 3 denotes that the real  is between -3 and +3. For different setting of value we observe slightly different impact solutions mainly in the date of possible impact. The differences between the results from the NEODyS (CLOMON2) and the JPL NASA (SENTRY) are evident because they are independent systems as state at: http://neo.jpl.nasa.gov/risk/doc/sentry_faq.html. For example impact probabilities different by a factor of ten or so are not extraordinary.  presented e.g. in the paper of Yeomans et. al. (1987). No impact solutions for =1 were found. Time of computations of single solution was about 3 hrs with 1.7 MHz processor.

(99942) Apophis: Approaching asteroids
To compute exactly impact solutions for (99942) Apophis it is necessary to include gravitational perturbations of approaching massive asteroids. Usually SENTRY include 3 massive asteroids: (1) Ceres, (2) Pallas and (4) Vesta, CLOMMON2 -as SENTRY or 4 asteroids: (1) Ceres, (2) Pallas, (4) Vesta and (10) (1866) Sisyphus. These selected asteroids together with the 4 massive ones (Ceres, Pallas, Vesta and Hygiea) were included to equations of motion of (99942) Apophis. The computations of influence of gravitational perturbations of these asteroids for the motion of (99942) Apophis were performed using software OrbFit ver. 3.3.1. The masses of asteroids were taken from Michalak (2001) and from Solex as computed by A. Vitagliano. First of all we must include Ceres in our gravitational model which has about 30 % of the mass of the main belt asteroids and the asteroids which have the closest approaches with (99942)  From Table 3 we can see that there is significant role of massive asteroids in motion of Apophis, specially after 2042. Some impact solutions does not exist in given year. For example, in April, 2069 there are only impact solutions with additional perturbing effect from together: Ceres, Eros, Alinda, Toro, Sisyphus and the second solution with perturbations from Ceres, Pallas, Vesta and Hygiea. Fig. 2 shows the changes of differences in mean anomaly between asteroid (99942) Apophis on nominal orbits for different cases. In Fig. 2(a) there are differences in mean anomaly between (99942) Apophis with and no relativistic effects included. Fig. 2(b) presents differences in mean anomaly of (99942) Apophis between orbits computed without perturbing asteroids and with perturbation from: 1 -Ceres, Pallas and Vesta, 2 -Ceres, Pallas, Vesta, and Hygiea and 3 -Ceres, Pallas, Vesta, Hygiea and Eros. It is clear from Fig.  2(a) that a relativistic effects play a great role in motion of asteroid -over 30 degs difference in mean anomaly between asteroids with and no these effects in the next 100 years. However in Fig. 2(b) the infuence of close approaching asteroids is evident but these effects are several times smaller than the relativistic effects. The rapidly changes in differences in mean anomaly in Fig. 2 are connected with the close approaches of (99942) Apophis with the Earth in the years: 2029 (0.00025 AU) and 2057 (0.022 AU) for the nominal orbits. Hence chaoticity of the motion of the asteroid appears (Wlodarczyk, 2001). The infuence of number of perturbing asteroids on impact solutions for (99942)

(99942) Apophis: The JPL Ephemerides
The question was appeared how the model of the Solar System used influences for the impact solutions of (99942)

The influence of sigma_LOV and weighting
The orbital elements of (144898)   Mainly it have an effect on value of impact probability. Similar the problem of scaling of LOV (Milani et al., 2002) is neglecting in this case. Otherwise everywhere weighing is as CLOMON2, further settings are: multiple solution, use scaling (fn denotes impact solution without scaling), LOV with the largest eigenvalue; w=1 denotes without weighing of observations. On the MPML (Minor Planet Mailing List) forum the problem was connected with 4 first observations of (144898) 2004 VD17 recovered from 2002 year. It was appear that adding these observations does not affect on impact solutions considerably. In Tab. 6 fn denotes impact solutions without first four observations from 2002.
The computations of infuence of gravitational perturbations of these asteroids for the motion of (144898) 2004 VD17 were performed using software OrbFit 3.3.1. The masses of asteroids were taken from Michalak (2001) and from Solex90 as computed by A. Vitagliano (2006 All results in Tab. 7 are computed with DE405 ephemeris and using relativistic effects (without case IW-bnrel). We can see that the impact solutions for asteroid (144898) 2004 VD17 does not differ so much using different number of perturbing asteroids as in the case of (99942) Apophis. Fig. 4 shows the changes of differences in mean anomaly between asteroid (144898) 2004 VD17 on nominal orbits for different cases. In Fig. 4 (a) there are differences in mean anomaly between (144898) 2004 VD17 with and no relativistic effects included. Fig. 4 (b) presents differences in mean anomaly of (144898)   As in the case of (99942) Apophis the greatest infuence for motion (144898) 2004 VD17 have relativistic effects, about 10 times greater than the perturbing effects of additional massive asteroids. Even so we must use perturbing massive asteroids for computed precise impact solutions as Tab. 7 states. The rapidly changes in differences in mean anomaly in Fig. 4 are connected with the close approaches of (144898) 2004 VD17 with the Earth in the years: 2041 (0.01 AU), 2067 (0.03 AU) and 2102 (0.03 AU) for the nominal orbits. Hence chaoticity of the motion of the asteroid appears similar to this of (99942) Apophis but in the case of (144898) 2004 VD17 motion is less influenced.

(144898) 2004 VD17: The JPL Ephemerides
As in the case of (99942) Apophis using JPL Ephemerides DE405 and DE406 does not affect on the computed impact solutions in this short about 100 years time span.
Both they are based on the new error model (Chesley, Baer and Monet, 2010). The orbits are computed using star catalog debiasing and an error model with assumed astrometric errors RMS deduced from the tests of the paper cited above.
Also additional observations of (99942) Apophis and (144898) 2004 VD17 were added. Actually, both the Yarkovsky/YORP effect, which are part of a set of other astrodynamical effects that were taken summary only into account to prepare the present analysis, but that seems to be of significant influence in the orbital evolution of such objects. The preliminary results are in Table 9. The Yarkovsky and YORP (Yarkovsky-O'Keefe-Radzievskii-Paddack) effects are thermal radiation forces and torques that cause a drift of semimajor axes (computed value of da/dt in present work) of small asteroids and meteoroids and a change their spin vectors (obliquities). Because the Yarkovsky force depends on the obliquity, we can expect a complicated interplay between the Yarkovsky and YORP effects . Therefore it is difficult to estimate the influence of the Yarkovsky and YORP effects on the motion of asteroids separately. The result of the Yarkovsky effect is removal of small asteroids from the main belt to chaotic mean motion and secular apsidal or nodal resonance zones. Then they can be gradually transported to Earth-crossing orbits. Therefore the Yarkovsky and YORP effects are now considered in relation to objects crossing the Earth orbit, particularly they are important in the motion of potentially dangerous asteroids for the Earth. Using all 1490 observations of Apophis and the OrbFit software I computed value of the semimajor axis drift of Apophis equal to da/dt=+180 10 -4 AU/Myr connected with the Yarkovsky/YORP effects and got following impact solutions as are presented in Table 9. Additional perturbations from (1) Ceres, (2) Pallas and (4) Table 9. (99942) Apophis. Impact solutions with the Earth using semimajor axis drif, da/dt= =+180 10 -4 AU/Myr, computed by the author. Similar value of da/dt= (235+/-50) 10 -4 AU/Myr has computed Grzegorz Sitarski (private information).

(99942) Apophis
To compute impact solutions of Apophis we must know exact uncertainty from the Yarkovsky effects and physical parameter uncertainties of Apophis together with the astrometric biases and radiation pressure (Giorgini et al. 2007). No impact solutions were found for (144898) VD17 in the next 100 year in the future. Similar solution, no possible impact, were detected by the JPL NASA and by the NEODyS.

Conclusion
To compute precisely the impact solutions of (99942) Apophis and (144898) 2004 VD17 it is necessary to include small effects like relativistic effects, close approaching asteroids, the Yarkovsky/YORP effect. The use of the software OrbFit is helpful in computing exact possible impacts of asteroids with the Earth. Thanks for the OrbFit Consortium. Also the free software Solex was useful in this work.

Acknowledgment
Thank you to the researches from OrbFit Consortium working in four research laboratories: http://adams.dm.unipi.it/~orbmaint/orbfit/OrbFit/doc/help.html#authors for theirs free software and source code.
Thank you very much for Andrea Milani and Geny Sansaturio for discussions during MACE 2006 and for the help through e-mails.