What direction is my asteroid coming from?

Question

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)?

Answer 1

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.

Answer 2

First off, this question is probably better off on Physics.SE or Astronomy.SE; as they'll be able to provide a better, more complete answer on asteroid trajectories than we ever could; but say this is for your story (Which it probably is, seeing as you've asked in WB).

Let's take your question from easiest to hardest, shall we?

Item 1: Asteroid hitting Earth? (Check!; this happens every year, it's just most of them are small enough that we don't notice)

Item 2: Asteroid impacting Earth (This one's more difficult, as an impactor is usually called a meteorite at this point, but according to this link, an impactor would need to be at least 100m in diameter (would make a 1.2km crater and impacts approximately once every 5,000 years or so)

Item 3: Asteroid Impactor from Behind the Sun - Okay, so the reason I'm putting this here is that, although fairly common for an asteroid to swing around the Sun and hit the Earth, it'd probably have been picked up by Satellites tracking it as it approached the Sun - Remember that orbital mechanics would indicate that for an object to hit us, it needs to target where we're going to be not where we've been.

Item 4: Hitting at a $45^o$ Angle: Err. Not quite sure how you'd calculate that one; The Earth's a sphere (even if it is a little elongated); you'd need to provide a frame of reference. In what co-ordinate do you mean by $45^o$ angle? (Conventional wisdom says bisecting the orbital plane, but in which direction, through the ascending node or descending node?)

Item 5: Hitting off the coast of Rio de Janeiro: Not likely. There's no guarantee asteroids will hit or even come close to any major cities or urban centers as there's no steering them. (This one got somewhere relatively populated) but statistically; your asteroid is more likely to strike the Pacific Ocean; then followed by the Atlantic Ocean, as these two bodies of water alone, make up most a lot of the Earth's surface.

Answer 3

Sounds like you're thinking of an apohele or a near apohele with it's aphelion just over 1 A.U.

I'm anxious about these guys because they're hard to see. They spend most of their time in the day sky. If they appear at night, they're close to the horizon where light pollution and atmospheric distortion hide them. When they're closest to earth, they are presenting us their night side which also makes them difficult to detect.

There are a multitude of possible apohele impact scenarios. I'll pick one out of a hat. Say asteroid's perihelion is .3 A.U. and aphelion is 1 A.U.

When this asteroid comes into earth's sphere of influence, it's path is better modeled as a hyperbola with a Vinf of 9.55 km/s. Set perigee of this hyperbola at 2578 km from earth's center (in other words, 3800 km below earth's surface). This rock would strike the earth at a speed of about 14.7 km/s and flight path angle at time of impact would be close to 45º

If you set up this rock's orbit so aphelion coincides with an ascending or descending node, you could pick the earth latitude of impact by adjusting inclination of the orbit.

Don't have time at the moment to show the math. Some terms to Google are Vis Viva Equation and Flight Path Angle.