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The Solar Wind creates auroras due to interactions with Earth's Magnetic Field. Do we witness any similar effects from the Galactic Wind interaction with the Sun's Magnetic Field?

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The auroras are not interaction of the solar wind and the Earth's magnetic field. The light is produced, because the high energy particles from the sun excite atoms in the earth's atmosphere. The earth's magnetic field "guides" the particles towards the magnetic poles of the earth, that's why you don't see the aurora often anywhere else.

Similarly high energy particles from the galactic wind could excite atom's near the magnetic poles of the sun. But I don't know if the effect is observed.

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We have observed aurora-like emissions on Jupiter, Saturn, Uranus, and Neptune. The emissions on these larger gas giants (well, ice giants really) are often in the UV wavelength range, so special telescopes are needed to see them. You should note that the emissions are incredibly weak. So to see anything near the sun is probably a hopeless endeavor since it is the "loudest" radio, microwave, visible light, UV-light, X-ray, and $\gamma$-ray (well, I am not so sure about $\gamma$-rays) source in the sky.

The aurora are actually created through an interaction between the solar wind (its dynamic pressure and magnetic fields) and the Earth's magnetic field. The particles that ultimately lead to the emission are high energy particles (10's of keV electrons) hitting the atmosphere and exciting nitrogen (green light) and oxygen (red and green light). These particles come from the Earth's geomagnetic tail (i.e., the region opposite the sunward direction), not the solar wind directly. They are accelerated during their journey from the tail to the auroral regions (i.e., magnetic poles) through multiple processes. Some even have enough energy to produce UV and X-ray emissions, though these are not the dominant emissions on Earth.

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