Or [THREAD=573725]this more recent thread[/THREAD] which discusses methods of diverting a potentially hazardous object (PHO), and why simply fragmenting a PHO doesn’t measurably improve the situation, and indeed, may make it dramatically worse.
While simply detonating explosive on the side of the object may not provide a significant amount of side impulse, using a nuclear explosive to generate x-rays that are absorbed by a sacrificial polystyrene foam puck, which becomes a cloud of fast-moving gas that can apply a distributed impulse to the entire object, whether it is whole or fragments as descried in the thread above. It may be necessary to do a two or three stage system (using the heated plasma to heat successive strata) to reduce velocity and distribute the impulse along a larger time interval. By doing this repeatedly (similar to the nuclear pulse propulsion concept of Project Orion) it would be possible to move a mass of significant size–say, 1 km nickel-iron asteroid–to a trajectory that is not hazardous given even just a few months warning , provided the launch infrastructure and basic technology is already in place.
Your proposal to “strap enough of the space shuttle’s rocket boosters [assuming you to mean the Solid Rocket Boosters (SRB)] onto the asteroid to change its orbit,” suffers from a few problems. One is that the Shuttle SRBs produce 3,100,000 lbf of peak thrust for an action time of about ~130 seconds, which obviously requires some pretty strong structure to mount to. Mounting this to a rocky or icy surface would be an extremely challenging engineering problem. (The stand used for static test firing such motors rest the motor on its side, and hold the forward and rear attach points to enormous concrete foundations to make certain that it doesn’t come free and fly around like a buzzbomb firework.) Another is that the SRBs aren’t designed or qualified to be operated in deep space, and would have to be thermobarically protected and conditioned for the duration of their exposure. The nozzle design of the SRB is design to function more-or-less optimally at low altitude; nozzles to operate optimally in vacuum for the SRBs would be gigantic. (The nozzles for small solid rocket “kick motors” intended to be operated in vacuum are typically very large in comparison to the motor. They are often nearly the outer diameter of the motor and may be as long in axial length as the motor chamber, and even at this they don’t provide nearly optimal thermodynamic performance for fully expanded plume; they are a tradeoff between improving thrust performance and the additional weight an optimal nozzle bell would offer.) On top of all of this, there is no way to get a loaded SRB into orbit, much less into interplanetary space. A Shuttle SRB is so large that it has to be poured in four segments in order to be transported by barge. The assembled booster masses 1,300,000 lbm (590 metric tons), which far in excess of the payload capacity of the largest super-heavy space launch, never mind the insane size of payload fairing you’d need to enclose and protect an SR, or the fantastical L/D of the resulting vehicle. There is just no feasible way to transport such a vehicle, nor would it be possible to pour the segment grains or assemble a booster in freefall vacuum.
Given that we have a couple of centuries to look at this, the wisest course of action would be measured procrastination. Keep monitoring the object to refine the trajectory estimate and perturbations, make an iCal notice to start worrying about it around Q3 of FY2175, and see if someone doesn’t come up with some kind of genius propulsion technology that can drag this thing permanently out of harm’s way without the crude business of explosive propulsion or other schemes using the existing primitive technology.
Stranger