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Approximately 4.6 billion years ago, the solar system was a cloud of dust and gas. Eventually, gravity collapsed the cloud in on itself, forming the sun, of what is now our solar system. The remaining material began to clump up, larger particles, bound by gravity, attracted small particles. These "clumps" slowly became larger and larger, forming planetary bodies, such as Earth.
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When the first rocky particles of Earth coalesced, the center of the planetary body, or the core, was formed. For much of Earth's first billion years, the planet was molten, as well as its core. Overtime, the Earth's crust formed as the planet cooled down, and a few billion years later, the core cooled down to what it is now today, (mostly) solid iron.
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There are many theories surrounding the formation of the moon. It is most likely believed that a Mars-sized planet collided with Earth. The collision launched debris out, and the debris eventually clumped together (due to gravity) and began orbiting Earth. It is believed that the Moon formed around the same time as Earth did.
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When the Earth first formed, the entire planet was molten due to the high temperatures of the material making up the planet. Overtime, the planet gradually lost its heat and began to cool down. The surface of the Earth cooled down enough to form what is today known as the Earth's crust.
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The end of the heavy bombardment was a tumultuous time period in which many asteroids impacted the terrestrial planets in our solar system, including Earth. This completed Earth's accretion, as an atmosphere formed shortly after, protecting the planet from further collisions.