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At what distance (in angular degrees) from the Sun, could we see Venus, in mid daylight, with a pair of binoculars (7x50)?

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It's an interesting question but a bit difficult to judge as you need a measure for the brightness of the sky as function of angular distance from the Sun - a figure seemingly hard to find as many papers are behind paywall I have no access to.

There is this poster by Jim at al or this ancient article by Abbot et al. Converting the former to magnitudes as used usually in astronomy I get an area brightness of the daytime sky of about -12 magnitudes, the latter gives the sky brightness as about $10^{-10}$ of the solar disk, so rather -16mag. The brightness of the sky also does vary quite a bit according to this study (I don't have access to the original data they quote by Lang 2010). However this indicates nicely also the brightness levels of full moon an eclipse sky and the solar corona:

Brightness of the sky, corona etc adopted from Lang 2010

On the other hand, there are also accounts that Venus can be spotted during daytime - if you know exactly where to look like here and here. A set of binoculars will make easier to spot.

As to the angular distance to the Sun when it is visible: The brightness of Venus does not vary that dramatically: In green is the magnitude of Venus and in yellow the phase of Venus which I plotted here for roughly one cycle with Stellarium.

Venus brightness as and phase

Thus in summary: going by the from Lang and the brightness of Venus, and given the two observational accounts above and my own observational experience, it seems safe to deduce that one can see Venus at least as close as a few solar disk radii away from the Sun - if you know exactly where to look.

A word of caution: don't try this close-to-Sun observation with a hand-held binoculars and exercise VERY GREAT CARE even with automated guiding. Looking at the Sun is dangerous to your eyes, and it will permanently damage your eye sight if you do so with a pair of binoculars, even done so very briefly. Safest and best viewed before sun rise or after sunset - or at large elongations where it has large angular separation from the Sun.

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    $\begingroup$ +10 for don't try this handheld, near the sun $\endgroup$
    – Dr Chuck
    Feb 12, 2023 at 14:05
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    $\begingroup$ Thank you from your answers. When I was 2nd mate, on merchant ships, I took frequently day sights of Venus. With a sun sight, we’d have a fix. Today, from my house I haven’t been able to spot it yet . I will keep working on it. Many thanks again $\endgroup$ Feb 12, 2023 at 22:37
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    $\begingroup$ I've seen Venus in daylight naked-eye many times, w/o the aid of binoculars. The brightness of the background sky makes a tremendous difference-- high-altitude locations usually feature darker daytime skies. Polarized sunglasses can help tremendously. A few times when Jupiter is very near the moon I've been able to use the moon as a reference point to spot Jupiter in 10x binoculars in broad daylight. (That last bit of information may be a useful data point to help answer the actual question here.) $\endgroup$ Feb 15, 2023 at 14:08

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