Both BLSSDWLF and Timo make great point for this episode. I look forward to your "math"; maybe NASA can step up to the plate and do this analysis as a training exercise on their super computer (hint, hint). Or you may need to use a Spirograph kit. My take on the orbit mechanics (no math): The asteroid was knocked out of its orbit toward the sun. It started making repeated passes by the planet. This happened three times since the harvest (I missed that line), it must be in a close and highly elliptical orbit for its entire orbit. If moving fast, and it must be to pass by the planet three times in a short period of time (less than one year or ~6 months), its closest approach to the sun must be very close, and its farthest point must be just past the planet. Also, I imagine that the asteroid orbit was changed each time it came near the planet. Maybe the previous close passes were in-bound, out-bound, in-bound, and the next would be out-bound directly into the planet? Or, the close approaches were all from one direction and most likely in the out-bound direction. Initially if the E wants to have the best chance to deflect the dense asteroid (it looked a little metallic and lumpy in the original special effect shot and almost the size of our moon), I would think E would try to intercept it at its farther away point from the sun were its speed is slowest and gravitational attraction force would be weakest. And maybe, they already missed the chance to do that; the rock is already past the planet in-bound to the sun for its final pass and then out again directly into the planet. Maybe then, E would want to get at it very close to the sun. I can see the E warping from one side of the sun where the planet happened to be, and zoom past the sun to get directly to the other side where the asteroid may have been at its closest point at the time. (Moving slow at warp speed!) The real reason that Spock would want to intercept the asteroid close to the sun because the internal stresses on the asteroid would be highest and gives E the best chance to fracture the rock into two pieces. Remember when comet Hale-Bopp broke up in Jupiter's gravity! My "eureka" moment. That was Spock's plan A. Spock did the math and concluded that the chance to affect its orbit with the ship's deflector beam was nil (or meteoroid beam in "The Cage", I wish they would have used that term...); this was only Spock's plan B. He calculated that fracturing the rock was his best chance, but it depended on too many unknown factors about the internal composition and stresses inside the irregular asteroid. Spock knew he could melt down the M/AM reactors in the ship's warp engines for this Plan A. He was committed to go all the way; which would break first: the rock or the ship? I love techno-drama.