Why do the Perseids only happen once per year, not twice or four times?

Question

The Perseid meteor shower is caused by debris left behind Comet Swift-Tutte. My understanding is that the meteor shower occurs once per year because that's when the Earth passes near the orbital path of the comet.

My question is why this shower only occurs once per year. I'm imagining that the Earth has an elliptical orbit around the sun, as does the comet. If the comet's orbit intersects the Earth's orbit at some point, since they're both ellipses, I would have thought that there would be (at least) two intersection points with the Earth's orbit, which would mean that we'd see this meteor shower twice or four times in the year as the Earth passes through these intersection points.

I'm assuming there's something wrong with my reasoning - can anyone clarify this for me?

Answer 1

Comets do orbit in ellipses, but there is no requirement for their orbital planes to match the Earth's exactly. Short-period comets have planes that are relatively parallel to the Earth's (with a good degree of lee-way), and long-period comets can come in from any direction and with any orbital inclination.

Here is a good representation of the orbit of comet 109P/Swift-Tuttle:

When the comet's orbit crosses the orbital plane of the Earth again, it is way out past Jupiter, and this means that there is just the single meteor shower from this debris track.

Also, it's worthwhile noting that while ellipses in general can intersect four times, keplerian orbits share a focus, which means that they can intersect at most twice (which they do when they're coplanar and the sizes are right).

Answer 2

Swift-Tuttle is orbiting the Sun in a different plane than the Earth: its orbital path has an inclination of 113° with respect to the mean ecliptic, i.e. its orbit is almost perpendicular to the Earth's orbit. Therefore their orbits only intersect in one place, as this picture makes clear (the cyan/blue curve is the comet's path):