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Why are the compositions of comets and asteroids different?

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Why are the compositions of comets and asteroids different?

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I am a math teacher and astrophysics is my hobby. I have read a lot about comets and asteroids but I still can not understand this: If both were formed under the same conditions, during the early days of the solar system, then why are not similar in composition? While asteroids and comets did form at the same time, they did not form under quite the same conditions. The solar system formed from the solar nebula, a cloud of gas and dust. At the center of the nebula, the Sun was being born through gravitational collapse. Because of this collapse, which releases heat, the centeral regions of the nebula were hotter and denser, while the outer regions were cooler. In the inner part of the nebula, only rock or metal could remain solid, and this is where the asteroids formed. In the outer part of the nebula, beyond the so-called frost line at 3 AU (between Mars and Jupiter) it was cool enough for water to freeze, and this is where the comets formed.

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