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I did a time-lapse animation looking towards the sunrise from Sydney, Australia on 25th of April. You can see it here around 1m 25s in (that link should jump to just before it appears very near where the Sun is about to rise):

http://www.youtube.com/watch?v=lqr_KuLd2-c&t=1m25s

It then proceeds towards the right of screen over around the next 21 seconds (around 10½ minutes) before disappearing off the right. You might need to send the video to full screen to see the small white 'bullet of haze' that is moving against a very wide background.

One of my first thoughts was that it might be a meteor skimming the upper atmosphere, but I was less confident of that given it seemed to be coming from the general direction of the Sun, and I'd thought that was a direction unlikely to produce meteors.

That was until I read Eta Aquarid Meteor Shower in 2016 which states:

The best time to view the Eta Aquarids is in the early mornings, right before dawn.

Well, at least that pins down the same time of day, if not the direction from which it came.

What is the likelyhood that this feature is a piece of comet debris that skimmed Earth's atmosphere?

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Things entering the atmosphere (whether a meteor or an atmosphere-skimming object) travel too fast to be visible for 10 minutes. Normally a pass of a satellite (at above atmosphere height) takes 2-3 minutes and most meteors are over in seconds.

Even the famous daylight fireball from the 1970s was over in a very short time.

I would guess what you have there is a plane at high altitude, though it's moving a bit slower than I would expect. (I'm open to suggestions.)

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    $\begingroup$ I'd have expected vapor trails from an aircraft (that's what you are suggesting?) would have been less 'puffy' at the front of the moving object, but if there are no better argued suggestions, I'll go with this answer. The compelling point you made (that I did not realize) was that such debris reentries are typically very short in duration. $\endgroup$ – Andrew Thompson Apr 26 '16 at 15:01
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    $\begingroup$ The daylight fireball traveled from Utah to Alberta Canada in 100 seconds, a distance of approximately 1200 kilometers. That is around the same as the distance between Sydney (where I was) and Adelaide (half way across the continent). The object I saw did not go out of (visible to the eye) sight within around 10 minutes, so I think the answer to my question is: "The likelihood of it being debris from space is so close to 0 as to be effectively indistinguishable from 0". Thanks for the link. This is answered. $\endgroup$ – Andrew Thompson Apr 27 '16 at 3:44
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    $\begingroup$ certainly a plane. Contrails don't always form, it depends on the state of the atmosphere: temperature, humidity, etc, at the altitude of the plane. I'm told that as Australia is a destination, but not between other major continents, high altitude planes, and contrails are rare (lifeonperth.com/contrailsoverperth.htm ) I live under busy skies, and see a lot of high altitude aircraft. $\endgroup$ – James K Apr 28 '16 at 17:29
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As Andy mentioned, it seems far too slow to be any space debris, and also a bit slow to be a plane. I feel like it might have been a lense flare from the sun. Try setting the camera up in the same spot for another video and see what happens :)

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