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Swift–Tuttle Comet has orbital period of 133.28 years. Yet, people say that debris and dust from this comet causes this meteor shower. This does not make any sense. Or do they want to say that from 1992, every year, for 28 years Earth picked up debris left behind so many years ago while passing the August part of orbit? It's logical to assume that all this debris would be exhausted by now...

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The comet sheds material each time it get close to the sun. The sun heats this dirty snow ball, the ice evaporates and tears dust and smaller rocks with it.
These dust particles then follow a similar orbit around the sun as the comet. But as they are ejected with some velocity and react differently to the solar wind & radiation, their orbit is slightly different from their parent comet. Different orbit means generally different orbital velocity and, over time, the particles spread out along the comets orbital path. They form a steam of material that constantly dissipates and is replenished each time the comets approaches perihelion.
And each time the earth passes through the part of the solar system this stream occupies, we see meteor.

Here is a nice interactive animation: https://www.meteorshowers.org/view/Perseids enter image description here

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    $\begingroup$ WOW, this animation really makes everything clear from just one look at it! $\endgroup$ – ScienceDiscoverer Aug 9 at 9:48
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The comet releases lots of dust particles. The sunlight pushes these particles into orbits not quite the same as the comet Some particles will be pushed into faster orbits, some will be pushed into slower orbits. Over time the dust particles get spread out in a band that goes all the way around the sun. So we don't just see meteors in the years that the comet was in the inner solar system, we see meteors every year.

There is no "August part" of the orbit. The dust particles are in orbits of about 133 years (some more, some less) Each year we pass through a different part of the 133 year long meteor stream.

The meteors you see on the 12 and 13th may have been released by the comet thousands of years ago.

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