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Simulations suggest that a paper airplane launched from the International Space Station would maintain its orientation and gradually descend for 3.5 days before chaotically tumbling down and burning up.

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Additional insights we found via Universe Today
From an orbital height of about 400 kilometers (249 miles), the atmosphere is too thin to burn up the paper, even when traveling about 7.67 kilometers per second (17,200 miles per hour), the orbital speed of the ISS.
However, models show that at around 120 kilometers (75 miles) above the ground, drag forces would bring the paper plane into an uncontrollable descent, as happens for many paper airplanes flown near Earth's surface.
Tests at the Kashiwa Hypersonic and High Enthalpy Wind Tunnel at the University of Tokyo have shown that exposure to just seven seconds of these forces bends the nose of the paper airplane and begins to char the nose and wingtips.
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