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I've been presented with this problem:

Say that Jupiter, with its diameter of 142,000 km, was located where Mars now orbits. What would be the angular size of (the newly-relocated) Jupiter during a close approach, when its distance would be 79,300,000 km? Would we be able to see Jupiter as a round object with our unaided eye, or only as a point of light?

The angular size is easy enough to calculate given $ \frac{Angular \space Size}{206,000} = \frac{Linear \space Size}{Distance} $.
So $\frac{Angular \space Size}{206,000} = \frac{142000}{79300000}$ meaning Angular Size = 369 seconds of arc.

However, would the unaided eye be able to see this? Knowing that the eye can detect 1 minute of arc across, surely it can detect 6.15? Any help appreciated, thanks!

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    $\begingroup$ One way to look at it is, moon is about 30, so about 1/5 the diameter of the moon. Yes, certainly visible as a "round" object. $\endgroup$
    – Jeff Y
    Jan 26, 2016 at 20:12

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Yes, you could definitely see Jupiter as a round object if it's angular diameter was 369 arcseconds. The limit for seeing something as round appears to be a diameter of around 1 arcminute. Notes:

The M16 complex is visible to the naked eye on clear nights as a hazy patch

(and thus not a single point), and http://messier.seds.org/m/m016.html notes that M16 has an angular diameter of about 7 arcminutes.

In the real world, folks with excellent eyesight can resolve down to about 2 arcminutes. Folks with average eyesight can resolve about 3 arcminutes.

Of course, resolving a double star as two separate points of light isn't the same as seeing something as a circle, but provides a good upper limit.

The extreme crescent phase of Venus can be seen without a telescope by those with exceptionally acute eyesight, at the limit of human perception. The angular resolution of the naked eye is about 1 minute of arc. The apparent disk of Venus' extreme crescent measures between 60.2 and 66 seconds of arc,[4] depending on the distance from Earth. Nevertheless it is possible for observers with extremely acute eyesight to see a crescent Venus under ideal atmospheric circumstances.

Note also that Venus appears circle-like (not point-like) when it transits the Sun, but the same is true of Mercury, so this may be a special situation that doesn't count.

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