
For the tl;dr crowd: The best time to go out is Thursday night Aug. 11/Friday morning Aug. 12 after your local midnight. Find a place with wide-open skies, lie back, and look up. You don't need a telescope or anything like that. The meteors should zip across the sky about once per minute or so on average, appearing to come from near the constellations of Cassiopeia and Perseus in the northeast.
Meteor showers occur when, as the Earth orbits the Sun, it plows through the debris left by a comet (or, in one case, a weird asteroid) that is also orbiting the Sun. Comets are basically dust and gravel held together by water ice, and as the Sun warms the comet, the ice sublimates (turns directly to gas) and the rocky bits are sloughed off. They orbit the Sun in more or less the same path as the comet, and when the Earth rams through them, they enter our atmosphere at high speed, heat up, and glow.
And we get a meteor shower.
Due to perspective (the same effect that makes raindrops always seem to be coming from ahead of you when you drive a car through a rain shower) the meteors appear to come from a point in the sky in the constellation of Perseus, near the W-shape of Cassiopeia. They radiate away from there, so that point in the sky is called the radiant, and the shower gets its name from the constellation.
By the way, I did a whole episode of Crash Course Astronomy on meteors. It's not specific to the Perseids, but it's chock full o' fun science.
Now, on to the FAQ!
When is the best time to watch?
Technically, the meteor shower starts around July 17 and lasts until late August. Realistically, though, it peaks over the week around Aug. 11-13. This year, the best time to watch is Thursday night/Friday morning after local midnight (that's when your part of the Earth is facing into the oncoming meteoroids and you see more). However, a night before or after will be fine, too. The later you wait in the evening the better, but even a couple of hours after dark will be fine.
How many will I see, and how often?
The Perseids average about 60-100 meteors per hour. I usually see a lot fewer than that because my skies aren't particularly dark. Meteors are random, in that you may see three in a row in a few seconds, then nothing for five more minutes.
A lot of people have the misconception that you'll see meteors zipping across the sky everywhere like fireworks. It's not like that; expect to see one per minute or so. Trust me, it's still wonderful!
Speaking of which, don't expect to go outside and see tons of meteors right away. Your eyes take a few minutes to adjust to the darkness (it takes about a half hour to get fully adapted) so give it a few after going out. Patience!
And remember, if there's an outburst, you might see many more! It's worth going out to see.
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