Where did the Enterprise travel during its voyage in "Amok Time"?
Nobody knows, but "Amok Time" and two episodes of Star Trek: Enterprise, and Star Trek: The Motion Picture give some clues.
In the fourth season episode "Home", 22 October 204, when on leave on Earth, T'Pol invites Tucker to come to Vulcan with her and visit her mother.
The duration of the voyage is not specified during the episode but possibly can be deduced from data in the episode. On Vulcan, when Tucker learns that T'Pol plans to marry Koss, he says:
In "Daedalus", 14 January 2005, Emory describes his hopes for the new sub-quantum transporter technology:
So Vulcan's star system should be between sixteen point zero and seventeen point zero light years from Earth.
Wikipedia has a "List of star systems within 16-20 light years" that includes 12 star systems containing 16 individual stars and brown dwarfs, as well as 5 known exoplanets, that have distances between 16.0 and 17.0 light years.
I note that a light year is defined as the distance traveled by light in a vacuum in an average Julian calendar year of 365.25 days, and so is equal to 365.25 light days.
In Star Trek: The Motion Picture a voyage to Vulcan is suggested:
Obviously the Enterprise could reach Vulcan many times faster at its highest warp factor than at its lowest warp factor, and Scott didn't specify what warp factor would be necessary to reach Vulcan from Earth in approximately four days.
But if the trip would take 3.5 to 4.5 Earth days and the distance to Vulcan is 16.0 to 17.0 light years, and thus 5,844 to 6,209.25 light days, the average speed during the voyage would be about 1,298.6 to 1,773.8 times the speed of light. On the official TOS warp scale, which has never been established in canon, warp ten would be 1,000 times the speed of light, warp eleven would be 1,331 times the speed of light, and warp twelve would be 1,728 times the speed of light.
Of course when "Amok Time" was first broadcast on 15 September 1967, fans had no idea whether Vulcan was ten light years from Earth, or one hundred, or one thousand.
Sometime before "Amok Time" opens, the Enterprise is ordered to head for Altair VI. Perhaps they finish their mission in some star system and then head for Altair. Or maybe they are travelling between two star systems when they are ordered to turn and head for Altair instead.
Anyway, the Enterprise should be heading from a point S (for somewhere) that might be in a star system or in interstellar space, presumably in a straight line to the Altair (Alpha Aquilae) system, point A.
So you can imagine or draw a line from point S to point A.
Partway along the line, at point 1, the episode begins.
Spock has been behaving emotionally, and after an outburst he says:
So you can imagine, or draw, that the Vulcan system is at point V, partway between point 1 and point A (Altair), and off to one side of the line from 1 to A.
Did Spock mean that Vulcan was 1.4 light days from the line between point 1 and point A, so that they would turn at a right angle when they passed closet to Vulcan, travel 1.4 light days to Vulcan, leave Sock there, and travel another 1.4 light days back to the line between point 1 and point A, and continue on to point A (Altair), have traveled 2.8 more light days than in a straight line?.
Probably not.
Imagine or draw a map where point 1 and point A are 20 light years apart, and Vulcan is halfway between them, but off to the side. Thus there would be two equal right triangles, and the straightest course to Vulcan and then to Altair would be along the hypotenuses of the two triangles.
The second longest legs of the triangles would be 10 light years or 3652.5 light days long. The squares of one of those legs would be 13,340,756 light days. The hypotenuse of each triangle would 1.4 light days longer than that, or 3,653.9 light days, and the square of the hypotenuse would be 13,350,985. Subtracting the square of the other side from that leaves 10,229, and the square root of that is 101.13851 light days, or 0.27 light years.
So in this hypothetical situation, by travelling along the hypotenuses of the triangles they visit Vulcan at point V over a hundred light days off their original course, while still adding only 2.8 light days to the total distance to reach Altair. No doubt Spock had such a course correction in mind.
And i have to wonder whether plotting a course that probably took them less than one light year from Vulcan triggered Spock's Pon Farr to start a little sooner than it might otherwise have.
After talking with Spock, Kirk decides to leave him at Vulcan on the way to Altair.
Then a message from Starfleet Command is received.
So the time they have to get to Altair VI has been shortened by 7 days. According to the official TOS warp scale, a starshp can travel 4 cubed, or 64, times the speed of light, at warp factor 4. So at warp 4 a strarship can travel 64 light days or 0.175 light ears in one day. At warp 6, 6 cubed or 216 times the speed of light, a starship can travel 216 light days, or 0.591 light years, in one day.
if the voyage was going to take 8 days at warp 4, it would be 512 light days, which would take 2.370 days at warp 6.
If the journey was going to take 9 days at warp 4, it would be 576 light days, which would take 2.66 days at warp 6
If the journey was going to take 10 days at warp 4, it would be 640 light days, which would take 2.96 days at warp 6
If the journey was going to take 11 days at warp 4, it would be 704 light days, which would take 3.259 days at warp 6.
But 7 days plus 3.259 days equals 10.259 days, not 11 days.
So at the time that Chekov plotted the new course, they should have been only about 10 days from Altair via Vulcan at warp 4, or about 640 light days, or about t.75 light years. It would be highly unusual, though perfectly possible, for Vulcan's star to be less than 1.75 light years from Altair. But are any real stars known to be less than 1.75 light years from Altair?
Of course many fans say that the official TOS warp scale is not canon and suggest that starships are actually much faster than that.
Anyway, at point 2, they change course and turn directly toward Altair.
Later, in his cabin, Kirk has an idea and calls the bridge.
So at a previous time, when they were on course for Altair at a point 3, Spock ordered the course changed to head toward Vulcan. Presumably Kirk soon orders the course changed back to Altair and at point 4 on the way to Vulcan the ship changes course and heads for Altair.
Kirk talks privately to Spock and sends him to sickbay to be examined.
Meanwhile,on the bridge:
This implies that all these course changes happened during a single watch.
McCoy talks to Kirk:
According to my calculation based on the official TOS warp scale, they should be three days from Altair at warp six, which would give them four or possibly five days to get Spock to Vulcan alive. But of course McCoy may know how many hours or days the Enterprise is scheduled to spend at Altair for the ceremonies.
And if the TOS official warp scale is not correct, they might be four days or five days or something from Altair and not have enough time to get Spock to Vulcan after reaching Altair. Kirk talks to Spock and decides to get Spock to Vulcan:
In McCoy's office:
At this moment they should be at point 5, en route to Altair. The voyage to Altair will take three days, or possibly an unknown number of days.
So draw a line between point 5 (their present position) and point A (Altair) and make it three days travel long. Without worrying how many light days or light years a day's travel is. And draw concentric circles at half a day's travel, one day's travel, a day and a half travel', two day's travel, two and half days's travel, etc. around point A, and another set of such concentric circles around point 5 Vulcan should be at any position where its distance from point 5 and his distance from pint A should equal four days.
And you can do the same assuming that point 5 is X day's travel from point A. Chose any value of X, and select a point V for Vulcan which is a total of X plus 1 days from point 5 and from point A.
Kirk calls the bridge:
In Spock's cabin, Chapel says:
So if it takes eight days at warp factor eight, and Spock might possibly still be alive, at the official TOS warp scale the total distance traveled to Vulcan would be 512 light days times 8 days, or 4,096 light days, about 11.21 light years.
So if it takes seven days at warp factor eight, and Spock would more probably be alive, though probably at death's door, at the official TOS warp scale the total distance traveled to Vulcan would be 512 light days times 7 days, or 3,584 light days, about 9.8 light years.
So if it takes six days at warp factor eight, and Spock would probably be very weak, at the official TOS warp scale the total distance traveled to Vulcan would be 512 light days times 6 days, or 3,072 light days, about 8.41 light years.
So if it takes five days at warp factor eight, at the official TOS warp scale the total distance traveled to Vulcan would be 512 light days times 5 days, or 2,560 light days, about 7 light years.
So if it takes four days at warp factor eight, at the official TOS warp scale the total distance traveled to Vulcan would be 512 light days times 4 days, or 2,048 light days, about 5.6 light years.
So if it takes three days at warp factor eight, at the official TOS warp scale the total distance traveled to Vulcan would be 512 light days times 3 days, or 1,536 light days, about 4.2 light years.
So if it takes two days at warp factor eight, at the official TOS warp scale the total distance traveled to Vulcan would be 512 light days times 2 days, or 1,024 light days, about 2.8 light years.
But many fans believe that TOS warp factors must be many times faster than the official TOS warp scale.
As it happens, the 12 known star systems that are between 16.0 and 17.0 light years from Earth include both 40 Eridani, which many fans believe to be Vulcan's star system, and Altair. So leaving Altair out of consideration, that leaves 11 star systems that might possibly be Vulcan's.
Even if Altair and Vulcan's system were on opposite sides of our solar system, they would have be less than 34 light years apart.
So if Vulcan and Altair are 34 light years, or 12,418.5 light days, apart, and the voyage between them lasts only one day, the speed would be 12,418.5 times c, the speed of light.
If the voyage lasts 2 days, the speed would be 6,209.25 c. If the voyage lasts three days, the speed would be 4,139.5 c. If the voyage last 4 days,the speed would be 3,104.625 c. If the voyage lasts 5 days, the speed would be 2,483.7 c. If the voyage lasts 6 days, the speed would be 2,069.75 c. If the voyage lasts 7 days, the speed would be 1,774.0715 c. If the voyage lasts 8 days, the speed would be 1,552.3125 c.
And if Vulcan and Altair are less than 34 light years apart, the speed will be correspondingly slower.
There are several astronomical coordinate systems which picture the stars as being attached to an imaginary spherical shell centered on Earth and thus all at the same distance from Earth. After all, several of them were created before astronomers knew the distance to even a single star outside our solar system.
In the equatorial coordinate system, Altair is at right ascension 19 hours 50 minutes, and at declination plus 8 degrees 52 minutes. there are 60 minutes of arc in a degree of Arc, and 90 degrees of arc from the celestial equator to the celestial pole. In right ascension, a hour is 15 arc degrees, and a minute is 15 arc minutes. Altair is about 16.73 plus or minus 0.05 light years from Earth.
In the equatorial coordinate system, 40 Eridani is at right ascension 04 hours 15 minutes, and declination minus 7 degrees, 39 minutes minutes. 40 Eridania is about 16.35 plus or minus 0.03 light years from Earth.
So Altair and 40 Eridani are about 16.5 degrees apart in declination and about 130 degrees about in right ascension, and they shoud be spearated by at least 20 light years.
And the other 10 candidates for Vulcan's system can also be compared with Altair.
Nobody knows, but "Amok Time" and two episodes of Star Trek: Enterprise, and Star Trek: The Motion Picture give some clues.
In the fourth season episode "Home", 22 October 204, when on leave on Earth, T'Pol invites Tucker to come to Vulcan with her and visit her mother.
The duration of the voyage is not specified during the episode but possibly can be deduced from data in the episode. On Vulcan, when Tucker learns that T'Pol plans to marry Koss, he says:
In "Daedalus", 14 January 2005, Emory describes his hopes for the new sub-quantum transporter technology:
So Vulcan's star system should be between sixteen point zero and seventeen point zero light years from Earth.
Wikipedia has a "List of star systems within 16-20 light years" that includes 12 star systems containing 16 individual stars and brown dwarfs, as well as 5 known exoplanets, that have distances between 16.0 and 17.0 light years.
I note that a light year is defined as the distance traveled by light in a vacuum in an average Julian calendar year of 365.25 days, and so is equal to 365.25 light days.
In Star Trek: The Motion Picture a voyage to Vulcan is suggested:
Obviously the Enterprise could reach Vulcan many times faster at its highest warp factor than at its lowest warp factor, and Scott didn't specify what warp factor would be necessary to reach Vulcan from Earth in approximately four days.
But if the trip would take 3.5 to 4.5 Earth days and the distance to Vulcan is 16.0 to 17.0 light years, and thus 5,844 to 6,209.25 light days, the average speed during the voyage would be about 1,298.6 to 1,773.8 times the speed of light. On the official TOS warp scale, which has never been established in canon, warp ten would be 1,000 times the speed of light, warp eleven would be 1,331 times the speed of light, and warp twelve would be 1,728 times the speed of light.
Of course when "Amok Time" was first broadcast on 15 September 1967, fans had no idea whether Vulcan was ten light years from Earth, or one hundred, or one thousand.
Sometime before "Amok Time" opens, the Enterprise is ordered to head for Altair VI. Perhaps they finish their mission in some star system and then head for Altair. Or maybe they are travelling between two star systems when they are ordered to turn and head for Altair instead.
Anyway, the Enterprise should be heading from a point S (for somewhere) that might be in a star system or in interstellar space, presumably in a straight line to the Altair (Alpha Aquilae) system, point A.
So you can imagine or draw a line from point S to point A.
Partway along the line, at point 1, the episode begins.
Spock has been behaving emotionally, and after an outburst he says:
So you can imagine, or draw, that the Vulcan system is at point V, partway between point 1 and point A (Altair), and off to one side of the line from 1 to A.
Did Spock mean that Vulcan was 1.4 light days from the line between point 1 and point A, so that they would turn at a right angle when they passed closet to Vulcan, travel 1.4 light days to Vulcan, leave Sock there, and travel another 1.4 light days back to the line between point 1 and point A, and continue on to point A (Altair), have traveled 2.8 more light days than in a straight line?.
Probably not.
Imagine or draw a map where point 1 and point A are 20 light years apart, and Vulcan is halfway between them, but off to the side. Thus there would be two equal right triangles, and the straightest course to Vulcan and then to Altair would be along the hypotenuses of the two triangles.
The second longest legs of the triangles would be 10 light years or 3652.5 light days long. The squares of one of those legs would be 13,340,756 light days. The hypotenuse of each triangle would 1.4 light days longer than that, or 3,653.9 light days, and the square of the hypotenuse would be 13,350,985. Subtracting the square of the other side from that leaves 10,229, and the square root of that is 101.13851 light days, or 0.27 light years.
So in this hypothetical situation, by travelling along the hypotenuses of the triangles they visit Vulcan at point V over a hundred light days off their original course, while still adding only 2.8 light days to the total distance to reach Altair. No doubt Spock had such a course correction in mind.
And i have to wonder whether plotting a course that probably took them less than one light year from Vulcan triggered Spock's Pon Farr to start a little sooner than it might otherwise have.
After talking with Spock, Kirk decides to leave him at Vulcan on the way to Altair.
Then a message from Starfleet Command is received.
So the time they have to get to Altair VI has been shortened by 7 days. According to the official TOS warp scale, a starshp can travel 4 cubed, or 64, times the speed of light, at warp factor 4. So at warp 4 a strarship can travel 64 light days or 0.175 light ears in one day. At warp 6, 6 cubed or 216 times the speed of light, a starship can travel 216 light days, or 0.591 light years, in one day.
if the voyage was going to take 8 days at warp 4, it would be 512 light days, which would take 2.370 days at warp 6.
If the journey was going to take 9 days at warp 4, it would be 576 light days, which would take 2.66 days at warp 6
If the journey was going to take 10 days at warp 4, it would be 640 light days, which would take 2.96 days at warp 6
If the journey was going to take 11 days at warp 4, it would be 704 light days, which would take 3.259 days at warp 6.
But 7 days plus 3.259 days equals 10.259 days, not 11 days.
So at the time that Chekov plotted the new course, they should have been only about 10 days from Altair via Vulcan at warp 4, or about 640 light days, or about t.75 light years. It would be highly unusual, though perfectly possible, for Vulcan's star to be less than 1.75 light years from Altair. But are any real stars known to be less than 1.75 light years from Altair?
Of course many fans say that the official TOS warp scale is not canon and suggest that starships are actually much faster than that.
Anyway, at point 2, they change course and turn directly toward Altair.
Later, in his cabin, Kirk has an idea and calls the bridge.
So at a previous time, when they were on course for Altair at a point 3, Spock ordered the course changed to head toward Vulcan. Presumably Kirk soon orders the course changed back to Altair and at point 4 on the way to Vulcan the ship changes course and heads for Altair.
Kirk talks privately to Spock and sends him to sickbay to be examined.
Meanwhile,on the bridge:
This implies that all these course changes happened during a single watch.
McCoy talks to Kirk:
According to my calculation based on the official TOS warp scale, they should be three days from Altair at warp six, which would give them four or possibly five days to get Spock to Vulcan alive. But of course McCoy may know how many hours or days the Enterprise is scheduled to spend at Altair for the ceremonies.
And if the TOS official warp scale is not correct, they might be four days or five days or something from Altair and not have enough time to get Spock to Vulcan after reaching Altair. Kirk talks to Spock and decides to get Spock to Vulcan:
In McCoy's office:
At this moment they should be at point 5, en route to Altair. The voyage to Altair will take three days, or possibly an unknown number of days.
So draw a line between point 5 (their present position) and point A (Altair) and make it three days travel long. Without worrying how many light days or light years a day's travel is. And draw concentric circles at half a day's travel, one day's travel, a day and a half travel', two day's travel, two and half days's travel, etc. around point A, and another set of such concentric circles around point 5 Vulcan should be at any position where its distance from point 5 and his distance from pint A should equal four days.
And you can do the same assuming that point 5 is X day's travel from point A. Chose any value of X, and select a point V for Vulcan which is a total of X plus 1 days from point 5 and from point A.
Kirk calls the bridge:
In Spock's cabin, Chapel says:
So if it takes eight days at warp factor eight, and Spock might possibly still be alive, at the official TOS warp scale the total distance traveled to Vulcan would be 512 light days times 8 days, or 4,096 light days, about 11.21 light years.
So if it takes seven days at warp factor eight, and Spock would more probably be alive, though probably at death's door, at the official TOS warp scale the total distance traveled to Vulcan would be 512 light days times 7 days, or 3,584 light days, about 9.8 light years.
So if it takes six days at warp factor eight, and Spock would probably be very weak, at the official TOS warp scale the total distance traveled to Vulcan would be 512 light days times 6 days, or 3,072 light days, about 8.41 light years.
So if it takes five days at warp factor eight, at the official TOS warp scale the total distance traveled to Vulcan would be 512 light days times 5 days, or 2,560 light days, about 7 light years.
So if it takes four days at warp factor eight, at the official TOS warp scale the total distance traveled to Vulcan would be 512 light days times 4 days, or 2,048 light days, about 5.6 light years.
So if it takes three days at warp factor eight, at the official TOS warp scale the total distance traveled to Vulcan would be 512 light days times 3 days, or 1,536 light days, about 4.2 light years.
So if it takes two days at warp factor eight, at the official TOS warp scale the total distance traveled to Vulcan would be 512 light days times 2 days, or 1,024 light days, about 2.8 light years.
But many fans believe that TOS warp factors must be many times faster than the official TOS warp scale.
As it happens, the 12 known star systems that are between 16.0 and 17.0 light years from Earth include both 40 Eridani, which many fans believe to be Vulcan's star system, and Altair. So leaving Altair out of consideration, that leaves 11 star systems that might possibly be Vulcan's.
Even if Altair and Vulcan's system were on opposite sides of our solar system, they would have be less than 34 light years apart.
So if Vulcan and Altair are 34 light years, or 12,418.5 light days, apart, and the voyage between them lasts only one day, the speed would be 12,418.5 times c, the speed of light.
If the voyage lasts 2 days, the speed would be 6,209.25 c. If the voyage lasts three days, the speed would be 4,139.5 c. If the voyage last 4 days,the speed would be 3,104.625 c. If the voyage lasts 5 days, the speed would be 2,483.7 c. If the voyage lasts 6 days, the speed would be 2,069.75 c. If the voyage lasts 7 days, the speed would be 1,774.0715 c. If the voyage lasts 8 days, the speed would be 1,552.3125 c.
And if Vulcan and Altair are less than 34 light years apart, the speed will be correspondingly slower.
There are several astronomical coordinate systems which picture the stars as being attached to an imaginary spherical shell centered on Earth and thus all at the same distance from Earth. After all, several of them were created before astronomers knew the distance to even a single star outside our solar system.
In the equatorial coordinate system, Altair is at right ascension 19 hours 50 minutes, and at declination plus 8 degrees 52 minutes. there are 60 minutes of arc in a degree of Arc, and 90 degrees of arc from the celestial equator to the celestial pole. In right ascension, a hour is 15 arc degrees, and a minute is 15 arc minutes. Altair is about 16.73 plus or minus 0.05 light years from Earth.
In the equatorial coordinate system, 40 Eridani is at right ascension 04 hours 15 minutes, and declination minus 7 degrees, 39 minutes minutes. 40 Eridania is about 16.35 plus or minus 0.03 light years from Earth.
So Altair and 40 Eridani are about 16.5 degrees apart in declination and about 130 degrees about in right ascension, and they shoud be spearated by at least 20 light years.
And the other 10 candidates for Vulcan's system can also be compared with Altair.
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) your analysis. I assume that the Enterprise is warping in from its patrol sector, perhaps 100-1000 lightyears out. The episode starts with them already headed for Altair Six, so, we don't know (or care) how long they they are en route, only that they are at some unknown distance from Altair Six. I'm in the camp that the cubed-root, warp factor scale is baloney, and that we should use Roddenberry's original max. speed of v=0.73 lys/hour or 6400 c. At this max. warp speed (probably Warp 8), it would take the Enterprise 27.4 hours to travel 20 lightyears. We probably need to pad the time up a little for inter solar system maneuvering speeds, but this would be a few hours at most; but the duration and timing of the ceremonies could delay departure by another day, or so. As to how far they are out from 
