There is no evidence in the episode that the Enterprise was coming from Earth or anywhere near Earth, only that SB12 was the closest Starbase.
There is a theory to explain many time/speed/distance problems in
Star Trek that starships can use a system of space warps or stargates to travel instantly between one star system and another, thus cuttong down on transit time greatly.
Starbase 12 was the command starbase in "Space Seed" when the Enterprise's journey apparently would take it near Alpha Ceti (if Alpha Ceti was Ceti Alpha) , and was also the command starbase or nearest starbase in "Who Mourns for Adonias?" set at Pollux (Beta Geminorum).
Beta Germinorum or Pollux is about 33.8 light years from Earth. Alpha Ceti or Menkar is about 240 to 260 light years from Earth.
In the equatorial coordinate system, one of three or four astronomical coordinate systems, the direction to Pollux is 7 hours and 45 minutes right ascension, declination plus 28 degrees. In the equatorial coordinate system, the direction to Alpha Ceti is 3 hours and 2 minutes right ascension, and declination plus 4 degrees. So the declinations of Pollux and Alpha Ceti differ by 24 degrees of arc (out of 360 degrees). And the right ascension of Pollux and Alpha Ceti differ by 71 degrees of arc, almost a right angle, since each hour is 15 degrees.
So Starbase 12 certainly seems to have authority over a large region of space.
But if there is a system of stargates leading from star system to star system, a starship might jump 45 light years though one stargate, 11 light years through the next, 318 light years through the third, and so on during its journey. The star systems that are close to a starbase in the system of stargates could be scattered all over space.
So possibly the
Enterprise was travelling from Star A to Star B to get to the system of stargates connected to Star B and use that system of stargates to reach its destination, when it crossed the path of the
Botany Bay from Star C to Star D close enough to detect
Botany Bay. Stars A, B, C, and D would be orbiting the center of the galaxy independently, and the directions between them would slowly shift slightly over time while the
Botany Bay made its centuries long voyage. So the
Enterprise could be the first ship to pass close enough to the
Botany Bay to detect it.
Another possibility is that the
Botany Bay was headed for Tau Ceti, Tau Ceti is about 11.9 light years from Earth, at the direction of right ascension 1 hour, 44 minutes, declination - 15 degrees 56 minutes, and one of hte nearest more or less Sun like stars. Tau Ceti is now believed to have five planets, two of them in or near its habitable zone..
Suppose that, after visiting Earth in "Tomorrow is Yesterday", or on some other occasion, the
Enterprise was sent to Tau Ceti to patrol the Romulan Neutral Zone, and traveled through the 11 light years of space between Earth and Tau Ceti instead of using the hypothetical system of stargates to reach Tau Ceti, and so found the
Botany Bay on its way to Tau Ceti.
Is there a Romulan Neutral Zone near Tau Ceti? Maybe.
In "Whom Gods Destroy" Spock sees two Captain Kirks, one of whom is the shape changing Garth of Izar:
SPOCK: Fascinating. What maneuver did we use to defeat the Romulan vessel near Tau Ceti?
KIRK 1: Very good, Spock. The Cochrane deceleration.
KIRK 2: Spock, you know the Cochrane deceleration's a classic battle maneuver. Every Starship Captain knows that.
So there may be a section of the Romulan Neutral Zone near Tau Ceti. There is also a section of the Romulan Neutral Zone between Gamma Hydra and Starbase 10 in "The Deadly Years". Gamma Hydra is presumably Gamma Hydrae, about 133 to 135 light years from Earth, and with directional coordinates right ascension 13 hours 18 minutes, declination -23 degrees 10 minutes, almost in the opposite direction from Tau Ceti.
In "Balance of Terror", set in a section of the Romulan Neutral Zone, Kirk sends messages to the nearest command base, on Earth or on a starbase somewhere. At the end of the episode:
KIRK: How many men did we lose, Bones?
MCCOY: Only one. Tomlinson. The boy who was getting married this morning. His fiancée is at the chapel.
(Kirk turns to go, and Rand enters)
RAND: We finally received an answer from Command base, sir. They say they'll support whatever decision you have to make.
So it took less than 24 hours to send a message and get a reply. In my study of the episode I think that it took about ten hours to send a message and get a reply.
In "The Enterprise Incident", near the Romulan Neutral Zone:
KIRK: You understand that Starfleet Command has been advised of the situation?
TAL [on viewscreen]: The subspace message will take three weeks to reach Starfleet. The decision is yours, Captain. One hour.
Assuming that a subspace radio message to the nearest command base took 5 to 12 hours in "Balance of Terror", and that Tal's "three weeks" was 17 to 25 days, or 408 to 600 hours, "the Enterprise Incident" could be about 34 to 120 times aw far away from Starfleet Command as "Balance of Terror" was.
So if "Balance of Terror" was at Tau Ceti, "The Enterprise Incident" could be between 375 and 1,320 light years from Earth, while if "Balance of Terror" was at Gamma Hydrae, "The Enterprise Incident" could be between 4,556 and 16,080 light years from Earth.
The minimum distance between Earth and "Balance of Terror" should be farther than the distance to the nearest command base, otherwise Kirk would have sent his messages to Starfleet headquarters on Earth. And of course sometimes there are instant subspace communications with Starfleet headquarters, which sets a certain minimum speed for subspace communications and thus the distance in "Balance of terror".
The Pioneer space probe was slow too, and yet it made it into Klingon space in only 300 years. And the Cryosatellite (a contemporary to the Botany Bay) was slow too, and it made it almost to the Neutral Zone in only 400 years.
Real science does not work in Star Trek.
Giving up on real science doesn't seem like a good idea to me.
the cryosatellite wasn't "slow", since it wasn't going anywhere. It would have been parked in a convenient orbit, around Earth or around the Sun, where it would be easy to get to when it would be possible to bring one or more of the occupants back to life.
The propulsion system of the cryosatellite doesn't matter since it wasn't intended to go anywhere, just to keep its station. Possibly some zany aliens thought it was fun to take primitive Earth probes and spaceships from our solar system to distant stars.
Mytran in post number 116 I think quoted a post from Alchemist:
Here's an interesting excerpt from an earlier draft of the script (December 8, 1966). Scott is referring to the controls of the Botany Bay, of course.
SCOTT
Completely automated. Hasn't been a human hand at
those controls for over a hundred years.
KIRK
What power, Scotty?
SCOTT
Atomic, for the most part. But they've added an ion particle
drive which is pretty advanced for her time. One of the first ones, I'd say.
KIRK
Then it could approach light speed?
SCOTT
Close to it.
The December 8 script would not be the final draft 9 December 1966,or the revised final draft 12 December 1966, or the second revised final draft 13 December 1966 and of course there may have been changes made while the episode was filmed before the scenes in the Botany Bay were filmed on 22 December 1966.
It is possible that the December 8 script has other historical references that may be inconsistent with the episode that was filmed and is canon.