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A couple days ago, I zoomed in with my 30x optical zoom camera, and after some exposure adjustments, a bright star in the night sky turned into this:Bright star with (planets)?

Are those other planets or other stars? Or is that a lense effect?

EDIT: The bright object in question was ~60 degrees above the horizon, and ESE of me (East-south-east). I took the picture on 3-25-16 from Madison, Wisconsin.

EDIT: Question answered, more clear picture added FYI :) Enjoy & sorry for double posting to Stack Exchange :( More clear pic - Enjoy!

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That should be Jupiter and his 4 Galilean moons. However on your picture 2 of them seem missing, maybe they were cut-off by the field-of-view. – AtmosphericPrisonEscape Mar 27 at 13:42
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Sometimes they hang out behind the planet. – Wayfaring Stranger Mar 27 at 14:36
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Cross-posted to Astronomy and Photography. Please don't do that. It's against site policy because it fragments answers and wastes people's time when they write an answer to something that's already been answered elsewhere. – David Richerby Mar 27 at 17:53
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Very nice picture, by the way. – Russell Borogove Mar 28 at 2:21
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@WayfaringStranger: Sometimes they also hang out in front of the planet, at which point they're equally invisible to this type of equipment. – Michael Seifert Mar 28 at 14:57
up vote 35 down vote accepted

That should be Jupiter and his 4 Galilean moons.

They are usually very well visible even with very cheap equipment and a nice experience for amateur astronomy. On your picture 2 of them seem missing, maybe they were cut-off by the field-of-view, or possibly as a commenter pointed out, they might be behind the planet.
You can test that notion actually, as Io runs around Jupiter pretty fast and can show significant movement relative to the planet on a timespan of ~2-3 hours (a typical observing session). So if after 2 hours another 'diamond' appears behind Jupiter, you can be pretty sure it to be the volcanic moon Io.

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The moon in front of Jupiter is strangely sharp. Do you think that has to do with the software in the camera, or is it a real optical effect? – LocalFluff Mar 27 at 16:07
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@LocalFluff: Wouldn't bet my eye on it if the moons differ in sharpness... But I'd say it could be related to the contrast, which decreases for the moon closer in. That should wash out dimmer features and thus make it appear sharper. So in that sense I'd speculate on a real, measurable effect. – AtmosphericPrisonEscape Mar 27 at 16:19
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@LocalFluff Jupiter looks overexposed. That causes blurriness. The dimmer moons are likely better exposed, so fewer photons in the wrong image sensor cell to blur things up: en.wikipedia.org/wiki/Image_sensor – Wayfaring Stranger Mar 27 at 17:44

You don't say what time you were looking. Here is a screenshot from Stellarium at 10pm Wisconsin time on 25th March 2016. Jupiter is in the ESE, but the altitude is a bit lower than 60 degrees. Seems fairly conclusive. You were seeing Ganymede and a Europa/Io combination.

Jupiter

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9:20 was the timestamp on the picture – Bobdabiulder Mar 27 at 20:46
    
Another picture shows this clearly, check the latest edit to the question – Bobdabiulder Mar 27 at 20:53

This appears to be Jupiter and two of its four "Galilean" moons, being the four discovered by Galileo with his telescope in 1610. I searched with Wolfram Alpha (http://www.wolframalpha.com/input/?i=jupiter+moon+configuration+march+25th+2016+9pm+US+central+time) to try and determine which moons you were looking at, and the answer was quite interesting.

Callisto was too close to Jupiter (as viewed from Earth) to be seen in your image. The outermost object is definitely Ganymede, and the other object appears to be a combination of Europa and Io, as these were very close (again, as viewed from Earth), so much so that they appear as one object in the image. Hope this helps.

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Wolfram never ceases to surprise me. +1'd – Mindwin Mar 28 at 19:13
    
@Mindwin Same lol – Bobdabiulder Mar 29 at 2:15

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