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In the beginning of the episode "Silent Enemy" in Star Trek: Enterprise, they are dropping communication relays. They seem to have another one, Echo Three, on board at the end of the episode.

A central plot point of the episode is that the other two relays are destroyed so they can't contact anyone for help. If they had a third one on board, what prevented them from just using it to call for help?

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    It's been a while since I saw the episode, but generally, you position relays within the range of a communication source, and the relay receives and re-sends the communications. If S is source, R is relay and E is Enterprise, and --- is the distance any given source or relay can reliably broadcast, I'd expect that the situation prior to the two relays being destroyed was S---R---R---E. If the relays are destroyed, the situation is S---x---x---E. The Enterprise would have to move toward the source by ------ to drop the relay for it to do any good, by which point they'd be in range anyway. Commented Apr 4 at 13:01

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Note: This answer sticks to information only from the Star Trek shows that had aired up until this point in time (TOS, TNG, DS9, VOY, ENT) and the semi-canonical Star Trek: The Next Generation Technical Manual, with only the most mild extrapolation from on-screen events.

This is the only Enterprise episode that features the Echo subspace relays in any meaningful capacity. They get name-checked in a handful of other episodes (such as Fusion and Shuttlepod One) but little else.

That being said, it's clear from the context that the Echo series satellites are the early precursors to the subspace communication relay stations discussed in both the Star Trek: The Next Generation Technical Manual and shown extensively on-screen in the TNG episode Aquiel.

As described in both sources -- and mentioned in CaptAlgorithm's comment on the original question -- the relay stations are designed to receive and rebroadcast subspace messages that would otherwise slow down/degrade/etc. during travel.

Much like nodes in a computer network, if a ship is using relay stations to get the message to a distant location, and if the relay station is itself not in range of the destination, then the relay stations need another station to which they can relay messages in order to get the messages through.

Because humans have comparatively low-power subspace communications gear at this point in time (in this episode, without the stations, they're unable to even contact a Vulcan ship that is a couple of days away), and from the context in the episode, it's plain that somewhat distant relay stations are required to contact distant targets.

As such, the third relay on board won't help them, because it needs to be closer to the target than they are in order to retransmit the signals.

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    That's exactly it. There's no point having two walkie talkies if you're holding both of them in your hand.
    – Valorum
    Commented Apr 4 at 14:03
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    I really like that analogy. :-D Commented Apr 4 at 14:06

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