What direction is my asteroid coming from?
I have an asteroid. I want it to hit Earth. The best way to hit Earth is from behind the Sun, which makes it harder to detect you if you're an asteroid. Now, I have a basic understanding of things like aphelion/perihelion, semi-major and minor axes, and orbital eccentricities, but not enough that I feel comfortable just making something up to describe the path of my asteroid.
So here's my question for you. How do I describe the path an asteroid takes from behind the sun to impacting with the Earth off the coast of Rio de Janeiro at a rough 45 degree angle (give or take a few degrees)?
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1 answer
To properly determine the orbital path of the asteroid, you'd need to do a numerical simulation (the three-body problem here likely not having an analytical solution). There are certainly $n$-body tools in place, although you would have to modify them a little bit to use them (you could also build your own using a symplectic integrator. You would have to specify the position and velocity vectors of the asteroid and Earth at the time of impact (both masses should have a negligible impact on the Sun, pun intended). After that, you simply reverse all of the velocity vectors and the sign of acceleration due to gravity (essentially, treat it as a repulsive force); the latter could involve modifying the source code to an $n$-body integrator. Then, simply watch as the asteroid - and all three bodies, in fact - trace their paths back in time. Add the other planets for a more computationally expensive but far more accurate asteroid trajectory.
If you don't want to do all that, you could just pick and choose a random Near Earth Object (NEO) that might do the trick and look at its orbit. As of this month, we've identified over 15,000 of them, so chances are good that there's one with detailed enough information that satisfies your criteria. In general, many NEOs have elliptical orbits, as the following graphic of Potentially Hazardous Asteroids shows:
Image in the public domain.
It's nearly impossible to look at every orbit in that image, but it's safe to say that most have high eccentricities, likely due to constant perturbations by the four inner planets, and possibly Jupiter.
More specifically, given that your asteroid crosses earth's path, it's either an Aten asteroid (semi-major axis less than 1 AU) or an Apollo asteroid (semi-major axis greater than 1 AU). You can search through the JPL Small Body Database with parameters that include one or both of these groups, and pick one that seems suitable. The Minor Planet Center also has excellent data; here, for instance, are the Aten asteroids. Note that many have relatively large eccentricities.
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