Let's do a comparison.
Mizar and Alcor are separated by about 12 minutes of arc (').
Jupiter and Saturn today at 6:37 PM Universal Coordinated Time will be about 6 arc-minutes apart.
The Moon, as previously discussed, is around 30 minutes.
From Observing and Measuring Visual Double Stars by R.W. Argyle (second edition), it is stated:
A normal pair of eyes should be able to see... 5.5 minutes without difficult and some may be able to make out... 3.6'.
This matches up with a very old note that I had.
The closest separation the eye can distinguish about 4 minutes of arc.
So, to conclude, people with good vision will be able to "split" the two planets. In other words, they will not merge.
A new vision test...
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Updated from Dr Ralph Chou.
The "normal" minimum angle of resolution of the eye is 1 arc-minute - that is what is referred to as 20/20 vision. Many young adults have 20/15 vision, a MAR of 0.75 arc-min. [That] is for daylight-adapted (photopic) vision. With dark adaptation, rod vision comes to the fore, and the MAR will become larger. A limit of 4 arc-min seems reasonable for mixed rod and cone vision in mesopic conditions that are more typical of a city nightscape.
There you have it with 4' being reasonable.
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