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Solar storms have a direct effect on Earth, creating charged particles. What are the direct and indirect effects of large solar storms on Earth's atmosphere? Do solar storms affect Earth's weather?

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Those charged particles you mention are not created by any kind of interaction with the Earth, but rather the term solar storm is used to refer to the temporary disturbance of Earth's magnetic field caused by bombardment of the charged particles which originate from the Sun. The stream of particles flow outward from the sun through the solar system(The particles escape from the sun because its outer atmosphere is very hot, and the atoms there move too rapidly for the sun's gravity to hold onto them).

However, these charged particles are different to the usual solar radiation (visible and ultraviolet light) that penetrates Earth’s atmosphere and warms the surface. Most of the charged particles that are aimed at the Earth are deflected back into space by Earth's magnetic field, although a small amount may enter the thermosphere (upper atmosphere). This may cause some warming in the thermosphere, however, carbon dioxide and nitrogen oxide, coolants in the thermosphere, absorb the energy and then re-radiate heat back into space. A small fraction of the extra heat from the solar flare radiates to layers of the atmosphere below the thermosphere, but it is insignificant compared to the normal amount of heating the lower layers of the atmosphere already experience from normal solar radiation.

References:

Bates, D.R., 1969. PHYSICS OF THE UPPER ATMOSPHERE. QUEEN'S UNIV BELFAST (NORTHERN IRELAND) DEPT OF APPLIED MATHEMATICS AND THEORETICAL PHYSICS.

Mlynczak, M.G., Martin‐Torres, F.J., Crowley, G., Kratz, D.P., Funke, B., Lu, G., Lopez‐Puertas, M., Russell, J.M., Kozyra, J., Mertens, C. and Sharma, R., 2005. Energy transport in the thermosphere during the solar storms of April 2002. Journal of Geophysical Research: Space Physics, 110(A12).

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