Sunspots by Binocular Eyepiece Projection

The angular diameter of the sun and moon are both about 0.5° varying slightly with the distance the sun and moon are away from the earth (due to their elliptical orbits).
For a single lens/mirror system (the "objective" lens) with focal length FL, the diameter Id of the sun or moon image in the focal plane is approximately:
       Id = FL x θ^rad = FL x 1/2(π/180) ~ 0.009FL
where θ^rad is the angular diameter of the sun/moon image in radians. With a binocular or telescope system, the eyepiece provides magnification of the image formed by the main lens/mirror. The resultant image for a bright object like the sun can be projected and the focus adjusted to achieve large images of size suitable for photographing. The sun should NEVER be viewed directly or using any optical aid such as binoculars or even the smallest telescope unless equipped with proper solar filters over the main objective lens. Eyepiece projection discussed here allows for safe viewing. With eyepiece magnification and projection with a measured image diameter of Id, the EFFECTIVE focal length (EFL) is:
       EFL = Id / 0.009



The images below show binocular projection using Celestron 15x70 SkyMaster binoculars with the eyepiece adjusted to show a sun image diameter of ~ 2", a convenient size to show more prominent sunshots. This size image corresponds to an EFL ~ 560mm. The photo at bottom right is from the NASA SDO observatory for the same time frame (Oct 13, 2023 ~ 10:30 am EDT) showing more details of the sunspots (click for enlarged views).

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Sunspots with smartphone & solar filter

Oct 16, 2023 12:31 pm EDT
Partial cloud cover. S23 Ultra 10x f/4.9 periscope camera with zoom x42 ISO-400 1/30sec
Celestron ECLIPSMART Solar Safe Photo Filter.