Comets do not actually circle around the Earth specifically. Instead, comets follow elliptical orbits around the Sun, just like planets and other celestial objects in our solar system. The reason comets appear to move in such a way that they approach the Earth and then head out beyond the Sun is due to their orbital paths and the gravitational interactions within the solar system.
Comets originate from distant regions of the solar system, such as the Kuiper Belt and the Oort Cloud. These regions are located much farther from the Sun compared to the Earth's orbit. When a comet's orbit brings it closer to the Sun, it experiences the Sun's gravitational pull, which causes it to accelerate and move faster as it approaches the inner solar system.
As the comet gets closer to the Sun, its icy nucleus heats up and begins to vaporize, releasing gas and dust into space. This process forms a glowing coma (a hazy envelope) around the nucleus and often develops a tail that points away from the Sun due to the solar wind. It is this coma and tail that become visible from Earth, creating the characteristic appearance of a comet.
After reaching its closest point to the Sun, known as perihelion, the comet's trajectory starts to take it away from the Sun once again. This happens because the comet's orbital momentum carries it outward, and the Sun's gravity gradually slows it down as it moves away. The shape of the comet's orbit determines how close it approaches the Sun and how far it travels into the outer regions of the solar system.
While some comets have highly elongated and eccentric orbits, others have orbits that are less extreme. Comets with shorter orbital periods, such as Halley's Comet, have orbits that bring them back into the inner solar system relatively frequently, allowing us to observe them during their periodic returns. Other comets have much longer orbital periods, spanning hundreds or even thousands of years, which means they spend most of their time in the distant reaches of the solar system and are rarely visible from Earth.
In summary, comets follow elliptical orbits around the Sun, and their appearances from Earth result from their journeys through the solar system. It is not the Earth's orbit specifically that determines their trajectory, but rather the interplay between the Sun's gravity, the comet's initial velocity, and the shape of its orbit.