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Solar Flares obviously release extreme amounts of energy and extend thousands of miles out into space.

Because they are so big I would like to be able to observe some of these events through a telescope but I am wary of damaging both my eyes or the sensitive telescopic equipment.

How might I safely observe a solar flare through an amateur telescope? Might this be solved by using light filters to dim the amount of light entering the telescope or is a more specialist piece of equipment required?

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One truly safe way of solar flare observations is to not directly observe them (with your eyes or imaging technology) at all, such is the case with radio-astronomy. There are a number of examples of how this could be done, for example, the Sudden Ionosphere Disturbance Monitor, this device measures disturbances in the ionosphere that includes the effects of solar flares. This operates in the VLF (3-30kHz) range.

From the Society of Amateur Radio Astronomers, the advice is

For VLF solar flare observations you will need a strip chart recorder and a radio receiver capable of operating in the noisy 20 to 100 kilohertz radio band. These receivers are quite simple and may be home constructed.

(Plans for these can be obtained from that group)

A repository of further links (far too many to list here) can be found on the website Amateur Radio Astronomy on the Internet.

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Yes, there are filters which do block out the vast majority of light from the sun. I think it's actually only a very small (~1 angstrom) wavelength band of light which gets through. You can see some pretty amazing features, including sunspots, and solar flares. Here's a composite image as an example (taken through a Hydrogen alpha filter):


Those smallish looking edge features are solar prominences which are large enough to swallow up multiple Earth's. This is not too far off from what you could observe with a solar scope.

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Solar flares are explosive events that last not too long, so actually, your bigger problem will be to be able to see them when they are happening.

Normally they are classified by their X-ray flux, and they can be either X, M, C, B or A class which corresponds to the logarithmic scale of $Watts/m^2$ ($10^{-4}, 10^{-5}, 10^{-6}, 10^{-7}, 10^{-8}$ respectively). You can understand better what I mean by looking at the real time lightcurves obtained from GOES satellite

These peaks have different duration depending of the flare, there are some records of flares lasting few hours where the shortest can be shorter than a minute.

All the above applies to X-ray flares, which we cannot seen from Earth, we need to put some instrument on board spacecraft or rocket so we go above the part of the atmosphere that absorbs them.

Now, if you want to see them by using an optical telescope you have two options, either using the whole visible range by projection or through an Hydrogen alpha filter. In this case the flare classification is different than in the X-ray case. Where the flare class in X-ray is measured by the enhanced of flux produced, the optical class is based by the area it covers. The Solar Influence Data Analysis Center offers a table with their properties and the corresponding X-ray flare type:

|      Area | Area            | Class | Typical corresponding |
|  (sq deg) | (10^-6 solar A) |       | SXR Class             |
|    <= 2.0 | <= 200          |     S | C2                    |
|   2.1-5.1 | 200-500         |     1 | M3                    |
|  5.2-12.4 | 500-1200        |     2 | X1                    |
| 12.5-24.7 | 1200-2400       |     3 | X5                    |
|     >24.7 | >  2400         |     4 | X9                    |

For example, the famous Carrington's event (Sept. 1, 1859) was observed and hand-drawn from Carrington by projection. But, as this article says: "He was lucky enough to be in the right place at the right time..." you need too to be quite lucky to observe one of these.

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