How long does it take an asteroid to get from the asteroid belt, to Earth? I know the asteroids don't go directly from the asteroid belt to Earth, but I know it's not quick, either. The time probably varies, but what's the average time?
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$\begingroup$ A year or two... very roughly $\endgroup$– uhohJun 26, 2019 at 10:16
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$\begingroup$ You can get a rough estimate for things like that by computing a Hohmann transfer orbit. $\endgroup$– PM 2RingJun 26, 2019 at 10:46
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1 Answer
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The full orbital period of an object orbiting the sun is $$T=2\pi \sqrt{\frac{a^3}{GM_\odot}},$$ where $a$ is the semi-major axis of the orbit and $M_\odot$ the mass of the sun.
If the asteroid has apoapsis in the asteroid belt ($r_{max}= 3$ AU) and periapsis near Earth ($r_{min}=1$ AU) then the semimajor axis will be $a=(r_{max}+r_{min})/2=2$ AU and going in from the belt takes half a period, so the answer is $\pi\sqrt{(2 AU)^3/GM_\odot}=1.4140$ years. An asteroid coming from further out, say 4 AU, would take 1.9762 years. From 2 AU, 0.9184 years.
Of course, were one to just give an asteroid a big push it could move nearly arbitrarily fast (up to the lightspeed limit if we want to be absurd) along an Earth-intersecting orbit, but I suspect the question was about normal asteroids.
However, it is worth noting that long-periodic comets can move fast (tens of km/s). Such a comet (or an interstellar asteroid) would have a velocity on the order of $\sqrt{2GM_\odot/(1AU)}\approx 42.1285$ km/s if it came in from the outer edge of the solar system. That would cross the 2 AU from the asteroid belt to Earth in about 82 days.
answered Jun 26, 2019 at 15:59
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4$\begingroup$ If you're thinking of asteroids which hit Earth "naturally", it is very unlikely that they get displaced from a stable orbit in the asteroid belt onto a collision course with Earth. It's much more likely that they get displaced onto an "Earth-crossining" orbit and then remain in it for millions of years, before they happen to actually collide with Earth. $\endgroup$ Jun 26, 2019 at 16:03