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The Enterprise is near Earth and a Borg sphere travels back in time to a period before the Enterprise and the federation itself existed.

Almost immediately the entire earth changes to a Borg populated planet.

If the Borg went back and assimilated the Earth before the Enterprise existed and borgified (I know assimilated) all of mankind, the Enterprise should have disappeared immediately as it would have never been created.

Evidence of this in the STU would be the TNG episode "Yesterday's Enterprise." In the episode the Enterprise C comes through a rift bringing it into it's future to the 24th century (the time period of TNG). Because the Enterprise C's defense of a Klingon outpost was vital to forming an alliance between the Klingons and the Federation at the Battle of Narendra III, that alliance was never formed and the federation was now at war with the Klingons. In an instance the Enterprise and everything about the timeline has changed (heck Tasha's on the bridge). Picard sent the Enterprise C back into the rift and as soon as it went back to it's original time in the Enterprise D's past and fulfilled it's mission the timeline reverted back. It happened instantaneously, as soon as it cleared the rift the Enterprise D had their regular uniforms, compliment, etc. back.

In TNG Tapestry, I know it's Q time travel so don't just focus on this one, Picard goes back to his youth. When he doesn't get into a fist fight as a young cadet he ends up not being a risk taker. When Q brings him back into his time, he's a low level officer, not even command rank.

In TNG Firstborn, future Alexander (K'mtar) travels back from the future trying to change his young self into a warrior so that Worf won't die as soon as he does in the future. Eventually K'mtar wants to kill his younger self for being such a wuss. He fails as Worf catches K'mtar. Worf says that Alexander has already changed history by coming back in time. Things may not at all happen the way Alexander fears, and when he returns to his own time he may well find Worf alive. K'mtar says that he has failed, because the boy he was remains the same. Worf says that Alexander is the same, but Worf has changed, and now he understands that Alexander will have a noble future even if he is not a warrior. K'mtar embraces him and says, "I love you, father." Worf replies, "And I you, Alexander." So both K'mtar (future Alexander) and Worf acknowledge that changing the past affects the future, so much so that Worf states that when he gets back to the future (pun intended) Worf may be alive and well.

If a change is made in the past it affects the present and future. Why in the movie First Contact didn't the Enterprise disappear as soon as the Earth was assimilated in Earth's past?

(I awarded a correct answer as it is an in universe answer, however if anyone can further expound on that answer I'd appreciate it. I understand that Data was a bit puzzled so there probably isn't an answer, but it seems that if the Enterprise were never created, a wake, rift, temporal, or otherwise wouldn't protect them.)

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    They explain it almost instantly don't they? – Daft May 8 '15 at 8:47
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    @Daft Yeah, they do. – Shadur May 8 '15 at 10:40
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    Star Trek hasn't always prided itself on timeline continuity and logical consistency... – a CVn May 8 '15 at 12:43
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    Because there wouldn't be a story without a hack to say they were still there? It's ok with me and I don't even like ST:TNG. :) – Almo May 8 '15 at 15:13
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    "...the Enterprise should have disappeared immediately as it would have never been created." ...assuming, baselessly, that all time-travel that alters the course of history uses the same temporal physics. – Lexible May 8 '15 at 17:04
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From the transcript:

DATA: Population ...approximately nine billion. ...All Borg!

TROI: How?

PICARD: They must have done it in the past. ...They went back and assimilated Earth. ...Changed history.

CRUSHER: Then if they changed history why are we still here?

DATA: The temporal wake must somehow have protected us from the changes in the time-line.

I'd add that there is another circumstance of a ship being protected from changes to the timeline: Voyager once had temporal shielding protect it from changes in the timeline in the Year Of Hell double-episode. This is a little different because the wake is isolating the Enterprise from changes in its own past, but the idea is basically the same. Clearly, the wake is acting as a kind of temporal shield.

Addendum: The OP has asked for more information on how the wake would protect them. There's no in-universe explanation, as far as I'm aware, so we're left with little bits of information that we can cobble together with supposition. In First Contact, we're told the following:

DATA: Sensors show chronometric particles emanating from the sphere.

PICARD: They're creating a temporal vortex.

So we can assume safely that the wake is composed of "chronometric particles". I'm guessing this is another term for chronitons, which have been shown to have temporal properties, since Voyager's temporal shield, which as mentioned, is similar to the wake in how it protects them from changes to the timeline, uses chronitons. I'll use that term because it's shorter. Let's say, then, that the wake is composed of chronitons. We've seen in other ST canon that chronitons, when there are enough of them in one place, can essentially dislodge a ship from it's position in time, causing time travel (it happened to the Defiant at one point, IIRC, because they're generated by Romulan cloaking devices).

I would suggest that this is what happened- that the ship was essentially partially outside of time, able to interact with things but not influenced by changes to the timeline. This next bit is completely unsupported supposition. Perhaps there's some kind of trail in a higher dimension between the Enterprise and it's past, one that gets cut off because of some unknown property inherent to chronitons. An interference of sorts. This might explain why they can interact but aren't affected by temporal changes.

The thing about all this is that, as the OP has said, even Data doesn't really know. The fact is that Federation science knows little about temporal mechanics. They can only observe the effects of certain actions, but they're a far cry from explaining them in any meaningful way.

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    Precedent for a ship being protected from changes in the timeline. – PointlessSpike May 8 '15 at 11:01
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    First Contact was released in 1996; "Year of Hell" was 1997. "Year of Hell" can't serve as a precedent unless there's some sort of time tra— Never mind. – jwodder May 8 '15 at 12:40
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    There is also City on The Edge of Forever for precedent - shielded by The Guardian, the Enterprise landing party remained on the surface despite The Enterprise's absence in orbit and, presumably, history. – Politank-Z May 8 '15 at 13:37
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    The temporal wake was made up entirely of plotitonium energy. It's the kind of energy that can only be generated by writers from Star Trek and Doctor Who and is generated by writing storylines that contradict previously established canon. – BBlake May 8 '15 at 14:40
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    @bblake drafting or water wakes work like that in real life, allowing a car or ship or bird behind a leading vehicle to not be affected by wind or water resistance. In a physics sense, that's the most logical, science based thing that star trek has shown – user16696 May 8 '15 at 15:03

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