Loren Pechtel wrote:Jonathan_S wrote:The closer your deflection vector is to one of those alternate intercept courses the less the rocks point of impact will shift for a given acceleration. So you'd want to apply acceleration in an efficient direction. I suspect that any direction direction perpendicular to the current vector would be pretty close to optimal. (But fortunately you don't need to actually work one of those optimal vectors out by eye, or longhand, that's what computers are for

-- or just apply enough brute force that it doesn't matter if you're inefficient about it

)

I don't think any of those alternate intercepts are close enough to the target trajectory to matter.

Well there

are and infinite number of them

And so many of those wouldn't be so very different from the target trajectory.

The Earth's mean orbital speed around the sun is about 30 km/s. (And it's 12,742 km wide). So let's assume, simplest case, that the rock is crossing perpendicular through Earth's orbit -- that means if you make the rock arrive 5 minutes earlier, and nothing else changes, and it'll pass about (300s * 30 km/s - 6,400 km) 2,600 km ahead of the edge of the Earth* -- hitting the same point in Earth's orbit just 86 seconds before Earth starts crossing it. So only a slight nudge could cause it to still impact. (FWIW it takes the Earth about 7 minutes to fully cross any given point in its orbit -- 12,742 km / 30 km/s ~= 425 seconds)

Make it cross Earth's orbit 1 hour earlier and it'd pass about 101,000 km ahead of the Earth; but apply a bit of the correct side vector and now it'll hit the Earth about an hour earlier (and, due to Earth's rotation, likely about 1,600 km further east).

Similarly if you decelerated to cross the orbit an hour later you'd need to shove it the other way. And for every second of those two hours (or any other length of time) there's a corresponding deflection along the Earth's orbital track that would cause it to still impact the Earth.

(Okay, technically, I think that gets trickier if you need to fly past the sun to hit it; because now you need to worry about staying far enough away it doesn't tear your rock apart and adjust for how much its gravity will deflect your trajectory. So on some approaches being 4-8 months earlier or later might be an issue)

Fortunately for Earth there's a far larger infinity of courses that

don't lead to an impact -- so even picking your deflection by pure chance you'd be far more likely to cause it to miss than to cause it to impact at a different time.

(Adjust numbers as necessary for the size and orbital velocity of the planet in question)

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* I'm neglecting the diameter of the rock itself; but hopefully it's small enough not to be a major factor.