What the dinosaurs did not know!!
The deflection of an asteroid is a daunting task that requires top level technologies and design methodologies, and most of all it requires a lot of time. Many deflection concepts have been proposed in recent years, many lacking of an appopriate theoretical analysis. Within the European Space Agency, the Advanced Concepts Team provides tools to perform trade-offs among deflection options.
The Deflection FormulaDeflection strategies act on the asteroid by means of some force interaction (gravitational, propulsion, impact) the goal of a deflection effort is, of course, to increase the miss-distance between the Earth and the asteroid to some predefined security value (10000 Km is a typical value). Regardless of the deflection strategy choosen, the resulting Earth miss distance is governed by a simple equation [Izzo 2004]:
An explanation of the various symbols can be found in the related literature (see the studies and publication page). This formula is valid for long term deflections, that is in those cases in which a good planning is possible of a deflection countermeasure. It tells us a number of things that improve our understanding on the asteroid deflection mechanism.
A simple Example
A back of the envelope calculation might clarify the challenges of this problem. Say we have an asteroid of 200m of diameter, assuming an average density of 2.6 Kg/m^3, its mass will roughly be 1e10 Kg. The orbit of the asteroid is assumed to have a .7 eccentricity and a 3.7 AU semi-major axis. Say we want to assess the deflection concept of attaching some engines to the asteroid.
where U is the relative velocity between the asteroid and the impactor, m is the impactor mass, M the asteroid mass and h is a coefficient that accounts for the increased momentum exchange due to expelled material (this can be as high as 5 if the impactor is carefully designed). Again some fast calculation shows that in the case previously discussed a value of 1.5e7 is needed for the product (eta)mtsvU where ts is expressed in years. This, given that we manage to properly design the impactor trajectory and shape, may require an impactor mass as high as 10000 Kg.
These back of the envelope calculations shows how difficult it can be to deflect an asteroid and what technology challenges it could pose. Things, fortunately are not always this bad. They can be much better, but also much worse. As an example, repeating the calculations for asteroids that are of a larger size, say 1 km, not much can be done (essentially they cannot be deflected, only destroyed....).
The case of ApophisWe cannot escape but shortly speak about Apophis. This asteroid is named after the greek Good of Chaos and owes its name to his orbit that will bring it to come very close to the Earth in 2029 to then modify its orbit and return to our planet in 2036 at a distance that is hardly predictable and whose estimate has been subject of quite some discussion (see here). The Advanced Concepts Team studied the possible delfection options for this asteroid already when it was called 2004 MN4, that is from the very early days of the warning. Fortunately, Apophis (with a very small chance) would need to be deflected of a very small amount (1km roughly) and launch windows exist to perform such a deflection as late as 2026 (three years before its 2029 close encounter) with a 700kg spacecraft. After this date the task of deflecting the asteroid would become much larger and numbers would look a bit like those in the example above. Apophis is a good study case for the international community as it represents an as-real-as-it-gets scenario to challenge us in developing our technologies in case a real threat one day would come.
References Izzo, D., Bourdoux, A., Walker, R., and Ongaro, F., Optimal Trajectories for the Impulsive Deflection of NEOs, Acta Astronautica, 59(1-5), pp.294-300, 2006. (link)
 Carnelli, I., Gálvez, A., and Izzo, D., Don Quijote: A NEO deflection precursor mission, submitted to NASA Workshop: Near-Earth Object Detection, Characterization, and Threat Mitigation, 2006. (link)
 Izzo, D. and Rathke, A., The asteroid deflection formula for Earth quasi co-orbiting asteroids, European Space Agency, the Advanced Concepts Team, ACT technical report (ADFECA06), 2006. (link)
 Rathke, A. and Izzo, D., Keplerian Consequences of an impact on an asteroid and their relevance for a deflection demonstartion mission, Near Earth Objects, our Celestial neighbours: Opportunity and Risk, proceedings IAU symposium No.236, 2006. (link)
 Izzo, D., Optimization of Interplanetary Trajectories for Impulsive and Continuous Asteroid Deflection, Journal of Guidance Control and Dynamics, 30(2), pp.401-408, 2007. (link)
 Murdoch, N., Izzo, D., Bombardelli, C., Carnelli, I., Hilgers, A., and Rodgers, D., Electrostatic tractor for near Earth object deflection, Proc 59th Int Astronautics Conf, 2008. (link)
 Sanjurjo, M., Peláez, J., Izzo, D., Purcell, O., Yam, C.H., and Scheeres, D.J., Asteroid Deflection by Means Of Electromagnetic Forces During an Earth Flyby, 1st IAA Planetary Defense Conference, 2009. (link)
 Izzo, D., Olympio, J., and Yam, C.H., Asteroid Deflection Theory: fundamentals of orbital mechanics and optimal control, 1st IAA Planetary Defense Conference, 2009. (link)