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When we look at the sun with our eyes it seems much larger than it actually is. When we use a solar filter we are seeing what the sun actually is.

So, what is the name of the thing the solar filter is eliminating? The larger glare part.

And, why does this happen? Why do we see this light that makes the sun appear bigger and not the smaller circle that is actually the sun?

photo of the sun where it looks like an orange ball, darker near the edges photo of the sun where it looks "glowing", with visible rays

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I think you're talking about the effect of a "fluffy glowing ball" around the solar disk, shown on the right in this photo:

This is called solar aureole, and it's caused by the aerosols in the air, which scatter light with a well-pronounced forward peak in the phase function:

(image source)

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    $\begingroup$ +1 this is the right answer! cf. Forest M. Mims III in Applied Optics: Solar Aureoles Caused by Dust, Smoke, and Haze see also Sun and Sky Photography. $\endgroup$
    – uhoh
    Jul 19 at 9:49
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    $\begingroup$ @uhoh In fact, after some thought, the description looks to me more like that of glare: although the aureole is brilliant so that one has to wince to see its details (otherwise it's plain white), it's not as fiery as the solar disk - and this fieriness does make a large glare that masks the aureole completely and makes the sun look enormous. $\endgroup$
    – Ruslan
    Jul 20 at 15:33
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What it the outer part of the sun, that we see with our eyes, called?

I am not sure there is a single word for this, since the effect is a little complicated. We might call it "the glare of the Sun".

But there are (at least) two things that will contribute to this.

  1. Optical and perceptual artifacts created by our visual system and by cameras when looking at very bright astronomical objects, especially those we should never look at with our eyes! :-) These are called glare for vision and lens flare for cameras. These are not real, in the sense that a perfect eye or perfect camera would not see them. They are artifacts generated by imperfect systems.
  2. What is real is small-angle scattering by both dust and tiny water droplets in the atmosphere (I notice that you've added the atmospheric-effects tag) which can be considered as Mie scattering for particles larger than a few wavelengths of light, and this will generally not add any color the way Rayleigh scattering the source of the blue diffuse sky radiation does, specifically because the particles are substantially larger than a wavelength.

Lens flare and glare (but in a photograph, so it's really flare):

Lens flare example Lens flare diagram

Sources: left: CCTV Lens flare (annotated) right: Lens flare scheme en below: Flashlight effect Sumo Jan08

Lens flare used to illustrate what glare looks like

Mie scattering:

While Mie scattering is generally strongest in the forward direction and for particles tens or hundreds of wavelengths in diameter is almost completely seen within a few degrees of the source, it does have some intensity in other directions. Clouds are the white (i.e. they reflect as the same color as the light that hits them) due to large angle Mie scattering.

Rayleigh and Mie scattering in the sky

Source: Cumulus Cloud (annotated)

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None of the other responses seem to answer the question "What is the name of the thing the solar filter is eliminating?"

In fact, the solar filter doesn't eliminate anything. It just makes everything a lot less bright.

The reason the white part of the sun looks so big is due to saturation of the film, CCD, or retina that you're using to look at it. There's a limit to the brightness that the film, CCD, or retina can register, and anything brighter than that is going to look white.

You see a lighted region around the sun due to scattering effects that others have mentioned, but it really is not that bright compared to the sun itself. When you use a solar filter, you can see that clearly. But with an unfiltered view, it is still bright enough to saturate the viewing medium, just like the sun does, and so it looks just as bright as the sun, even though it is nowhere close.

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The part of the sun you see (but you shouldn't look at the sun except through a filter) is the photosphere: https://en.wikipedia.org/wiki/Photosphere The scattering effect in your second photo is due to the atmosphere: it can be anywhere from almost nonexistent (say in clear mountain air) to obscuring the solar disk entirely, as when you have a heavy cloud layer.

You can see the same scattering effect with the moon, and sometimes with stars, though it's much less noticable because they're so much dimmer.

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