Were and when?
My Second post in this thread is much better organized and presents its conclusions at the top with the explanations for them below for those who are interested. Thus it is recommended that you read the second post instead of this one.
Space and time are connected.
Suppose that some of the creators wanted to make Star Trek: the Motion Picture the most scientifically accurate and realistic Star Trek production yet. One way they might have done so would have been to remember high school problems about calculating distance and time.
remember? Problems like: "If Train A leaves from City C at a speed of 50 miles per hour and Train B leaves from City D at a speed of 60 miles per hour, and the cities are 220 miles apart, when will the two trains pass each other - or crash if they're on the same track?"
So maybe the creators of Star Trek: the Motion Picture decided to make V'Ger's trip to Earth and the Enterprise's trip to intercept V'Ger scientifically plausible and do the necessary calculations.
Star Trek: the Motion Picture opens with three Klingon ships fighting V'Ger and then shows the Epsilon IX station intercepting the Klingon messages.
[Epsilon IX - interior]
LIEUTENANT: Our sensor drone is intercepting this on Quad L fourteen.
BRANCH: That's in Klingon boundaries. Who are they fighting?
LIEUTENANT: Unknown, sir.
TECHNICIAN: I have an exterior visual.
(the battle cruisers are destroyed by energy bolts from the Cloud)
LIEUTENANT: We've plotted a course on that Cloud, Commander. It will pass into Federation space fairly close to us.
BRANCH: Heading?
LIEUTENANT: Sir, it's on a precise heading for Earth!
So we can imagine a line and V'ger travelling on that line toward Earth. Somewhere between V'ger's current position and Earth there is the Federation Border. Epsilon IX is probably very close to the spot here V'Ger's path crosses the Federation border. Since V'Ger fought the Klingons inside Klingon boundaries, there is also a Klingon border between V'ger and Earth.
Possibly there is one border, between Federation and Klingon space, as there is in many regions of space, or possibly the Klingon and Federation borders are separated by unclaimed space between them in this region of space.
The next scene shows Spock's Kolinahr failure, probably but not certainly after the previous scene.
The Next scene is at starfleet Headquarters. Kirk has certainly heard about V'Ger heading toward Earth:
KIRK: Here? At Starfleet? The Enterprise is in final preparation to leave dock.
SONAK: Which will require twenty more hours at minimum.
KIRK: Twelve! I'm on my way to a meeting with Admiral Nogura which will not last more than three minutes. Report to me on the Enterprise in one hour.
SONAK: Report to you, sir?
KIRK: It is my intention to be on that ship following that meeting. Report to me in one hour.
The next scene is in an orbiting complex.
SCOTT:SCOTT: Those departure orders, twelve hours, Starfleet cannot be serious.
KIRK: Why aren't the Enterprise transporters operating, Mister Scott?
SCOTT: A wee problem, sir, really. Just temporary. Admiral, we have just spent eighteen months redesigning and refitting the Enterprise. How in the name of hell do they expect to have her ready in twelve hours?
KIRK: Take me over, please.
[Travel Pod]
SCOTT: She needs more work, a shakedown.
KIRK: Mister Scott, there's an alien object with unbelievable destructive power less than three days away from this planet. ...The only starship in interception range is the Enterprise. Ready, or not, she launches in twelve hours.
So now, an unknown time after V'ger was first detected in Klingon space by Epsilon IX, it is less than three days from Earth, though probably more than 2.5 days.
Kirk goes to the bridge and orders the crew to assemble on the rec deck at 0400 hours.
Soon after the bridge scene is the transporter accident to Commander Sonak, about an hour after Stardate 7410.2.
When the crew is on the rec deck Kirk plays the recordings of V'ger.
KIRK: That's all we know about it, except that it's now fifty-three point four hours away from Earth. Enterprise is the only Federation starship that stands in its way. Our orders are to intercept, investigate, and take whatever action is necessary, ...and possible.
Then they get the message of V'ger destroying Epsilon IX at about the time it crosses the Federation border.
KIRK: External view!
(Epsilon IX is destroyed in the same way as the Klingon battle cruisers)
KIRK: Viewer off. ...Pre-launch countdown will commence in forty minutes.
Since we don't know how long the Pre-launch countdown lasts, this moment is probably more than one hour and less than eleven hours and twenty minutes, after Kirk talked to Sonak and Spock. Thus at Stardate 7410.2 V'ger was about 54.4 to 64.7333 hours (2.266 to 2.69 days), or less from Earth.
Later Enterprise launches:
KIRK: Impulse power, Mister Sulu. Ahead, warp point five. ...Departure angle on viewer.
SULU: Departure angle.
KIRK: Viewer ahead.
(external space, Enterprise passes Jupiter)
Captain's log, stardate 7412.6. one point eight hours from launch. In order to intercept the intruder at the earliest possible time, we must now risk engaging warp drive while still within the solar system.
DECKER: Captain, assuming we have full warp capability, accelerating to warp seven on leaving the solar system will bring us to IP with the intruder, twenty point one hours.
If the Enterprise did launch in 12 hours, stardate 7412.6 should be 2.4 stardate units and 13.8 hours after Kirk talked to Sonak and Scott, thus making about 5.75 hours per stardate unit in this era.
If the Enterprise will intercept V'ger in 20.1 hours at warp factor seven, and the official TOS warp scale is used and warp seven is 343 times the speed of light, Enterprise will intercept V'ger at a point 6,894.3 light hours from Earth. Since a light year is defined as the distance light travels in a Julian calendar year, 365.25 days long, there are 8,766 light hours in a light year. So the projected intercept point is about 0.786 light years from Earth.
But when they try to go to warp an engine imbalance creates a wormhole,. Then Kirk says:
KIRK: That object out there is less than two days from Earth. We need to intercept while it still is out there.
If it is less than 48 hours but more than 20.1 hours from Earth this should be about 5.4 to 33.1 hours after the rec deck scene.
Spock comes aboard, and by stardate 7413.4, about 18.4 hours after Kirk talked to Sonak and Scott on stardate 7410.2, has corrected the engine imbalance.
Captain's log, stardate 7413.4. Thanks to Mister Spock's timely arrival, and assistance, we have the engines rebalanced into full warp capacity. Repair time less than three hours, ...which means we will now be able to intercept intruder while still more than a day from Earth.
SULU: Warp point eight, ...point nine, ...warp two, ...warp five, ...warp six, ...warp seven, sir.
So now the projected intercept point is more than a day from Earth. Thus they will intercept V'Ger when V'Ger is between 24 and 48 hours, possibly between 24 and 36 hours, from Earth at V'gr's speed.
Later when they arrive at V'Ger and have managed to call off it's attack:
KIRK: That thing is twenty hours away from Earth. We know nothing about it yet.
I don't see how they spent at least 4 hours travelling alongside V'ger in the few onscreen minutes after intercepting it.
Later:
Captain's log, stardate 7414.1. Our best estimates place us some four hours from Earth. No significant progress thus far reviving Ilia memory patterns within the alien probe. This remains our only means of contact with our captor.
This should be about 3.9 stardate units, or about 22.425 hours after Kirk talked to Sonak and Scott on Stardate 7410.2. And that doesn't add up to at least 53.4 hours from Earth in the rec deck room scene. If the rec deck scene is 53.4 hours from Earth and is between stardates 7410.2 and 7412.6, and stardate 7414.1 is 4 hours from Earth, stardate 7414.1 is about 49.4 hours after the rec deck scene, and there should be about 12.6666 to 32.333 hours in a stardate unit.
And some time later:
[Enterprise bridge]
UHURA: A faint signal from Starfleet, sir. Intruder Cloud has been located on their outer monitors for past twenty-seven minutes. ...Cloud dissipating rapidly as it approaches.
SULU: Starfleet reports forward velocity has slowed to sub-warp speed. We are three minutes from Earth's orbit.
One other point:
KIRK: Gentlemen. At last report you were on Vulcan. Apparently to stay.
McCOY: Yes, you were undergoing the Kolineer discipline.
KIRK: Sit down.
SPOCK: If you are referring to the Kolinahr, Doctor, you are correct.
McCOY: Well, however it's pronounced, Mister Spock, it's the Vulcan ritual supposed to purge all remaining emotions.
KIRK: The Kolinahr is also the discipline you broke ...to join us. Will you, please, ...sit down!
SPOCK: On Vulcan I began sensing a consciousness of a force more powerful than I have ever encountered. Thought patterns of exactingly perfect order. I believe they emanate from the intruder. I believe it may hold my answers.
This seems to prove that Spock was on Vulcan or in the Vulcan Solar system when he sensed V'ger.
KIRK: Mister Scott! Shall we give the Enterprise a proper shakedown?
SCOTT: I would say it's time for that, sir, aye. We can have you back on Vulcan in for days, Mister Spock.
SPOCK: Unnecessary, Mister Scott. My task on Vulcan is completed.
So the rebuilt Enterprise can reach Vulcan in "four days", or about 3.0 to 5.0 days.
At warp 6 a ship could travel 1.77 light years in 3 days, 2.36 light years in 4 days, and 2.956 light years in 5 days.
At warp 7 a ship could travel 2.817 light years in 3 days, 3.756 light years in 4 days, and 4.695 light years in 5 days.
At warp 8 a ship could travel 4.205 light years in 3 days, 5.607 light years in 4 days, and 7.01 light years in 5 days.
It seems to me that they would travel faster when trying to intercept V'ger than on a shakedown cruise, so it doesn't seem likely Scott would calculate the travel time to Vulcan using warp 8 instead of warp 7.
It seems to be generally agreed that Vulcan is 40 Eridani:
According to Gene Roddenberry, James Blish and multiple other background sources, the Vulcan system was the star 40 Eridani A. One official source that suggested this proposal was the 1980 reference work Star Trek Maps (pp. 25-26). An alternative possibility, included in the equally official Star Trek Spaceflight Chronology (first published in 1979), was that Vulcan's sun was Epsilon Eridani. Roddenberry favored 40 Eridani, due to the comparative ages of the two systems. In a letter printed in Sky and Telescope magazine in July 1991, Roddenberry wrote, "Based on the history of life on Earth, life on any planet around Epsilon Eridani would not have had time to evolve beyond the level of bacteria. On the other hand, an intelligent civilization could have evolved over the aeons on a planet circling 40 Eridani. So the latter is the more likely Vulcan sun." He also made the presumption that the planet orbits the primary star. [12] This association was continued in the books The Worlds of the Federation (p. 18) and Star Trek: Star Charts (p. 58). Both sources cite Vulcan as the second planet in the system, which Star Charts (pp. 19 & 45) places in Sector 005 in the Beta Quadrant.
MiscellaneousEdit
But the most certain thing Memory Alpha could say was"
Location Edit
Vulcan was located "a little over" sixteen light years from Earth. Vulcan was located near Andoria. (ENT: "The Andorian Incident", "Home")
In the 2270s, a Constitution-class heavy cruiser traveling at warp speed from Earth could reach Vulcan in four days. (Star Trek: The Motion Picture)
http://memory-alpha.wikia.com/wiki/Vulcan_(planet)
40 Eridani is about 16.45 plus or minus 0.07 light years, or 16.38 to 16.52 light years. To travel that distance in "four days" - 3.0 to 5.0 days - requires a speed of 1,196.559 to 2,011.31 times the speed of light. That is about 2.337 to 9.311 times as fast as the official TOS warp factors 6, 7, and 8.
Possibly TMP uses a new warp scale in which warp seven is about 1,196.559 to 2,011.31 times the sped of light. Maybe Scotty calculated the trip time to Vulcan knowing that certain unspecified factors would enable the Enterprise in much less time that according to the warp formula.
Presumably Spock's Vulcan shuttle used the same warp factors and/or unknown additional factors to enable it to make the trip from Vulcan to near Earth in the same time as Scott calculated. Thus Spock would have left Vulcan three to five days before meeting the Enterprise. And we might suppose that Spock left Vulcan two to five days before stardate 7410.2 when Kirk talked to Sonak and Scott.
Thus at V'ger's speed it was probably roughly approximately (53.4 plus 48=101.4) to (72 + 120 = 192) hours, or 4.225 to 8 days, from Earth When Spock left Vulcan.
When the Enterprise launched, V'ger was less than 3 days minus half a day from Earth, or 1.5 days to 2.5 days from Earth. It was also at least 40 minutes (0.666 hours) less than 53.4 hours from Earth, or less than 52.734 hours or 2.197 days from Earth. When the Enterprise tried to enter warp 1.8 hours (or 0.075 days) after launch, V'ger was between 1.425 and 2.197 days from Earth. If the Enterprise would intercept V'ger after travelling 20.1 hours (or 0.8375 days) at warp seven, V'ger would have traveled 20.1 hours (or 0.8375 days) toward Earth at it's speed.
20.1 hours (or 0.8375 days) would be 0.3812 to 0.5875 of the total time V'Ger needed to reach Earth. V'ger would have traveled 0.3812 to 0.5875 of its remaining distance to Earth while the Enterprise traveled 20.1 hours at warp seven.
If the Enterprise is using the official TOS warp formula, it should expect to intercept. V'ger about 0.786 light years from Earth. But possibly they plan to travel at warp factor seven and somehow travel 1,196.559 to 2,011.31 times the speed of light, as in Scott's calculated travel time from Earth to Vulcan. in that case they would expect to intercept V'ger at a distance of about 2.743 to 4.611 light years from Earth.
Then they have the wormhole problem and turn off the warp engines. After a few minutes Kirk says that V'Ger is less than 2 days (48 hours) from Earth. But obviously more than the 0.8375 days (20.1 hours) they expected to intercept V'ger in. Thus when the Enterprise tried to enter warp V'ger should have been between 1.425 and 2.000 days from Earth. Thus intercepting V'ger in 0.8375 days would mean traveling somewhere between 0.441875 and 0.5877 of the distance V'ger has to travel to Earth.
After some time Spock comes aboard, and after repair time of less than three hours, or about 2.5 to 3.0 hours, Enterprise is ready to go to warp. This could be about 2.5 to 4.0 hours after the first attempt to go to warp. Thus V'ger should be about 1.2584 to 1.896 days from Earth when they go to warp the second time. And Kirk says they will intercept V'Ger while it is still more than a day from Earth (about 1.0 to 2.0 days).
If they originally planned to intercept V'ger between 0.441875 and O.5877 of the distance V'ger had still to travel to Earth, V'ger would travel between 0.558125 and 0.4123 of the total distance to Earth during that period. Thus the Enterprise would travel between about 0.7917 and 1.4254 times as far, and thus between about 0.7917 and 1.4254 times as fast as V'ger. Similarly V'Ger would travel between about 0.701 and 1.263 times as fast as the Enterprise.
As written above, V'ger should be about 1.2584 to 1.896 days from Earth when they go to warp the second time. And Kirk says they will intercept V'Ger while it is still more than a day from Earth (about 1.0 to 2.0 days). Thus the Enterprise should travel for less than 0.2584 to 0.896 days to intercept V'ger more than 1.000 days from Earth.
Half of 1.2584 to 1.896 days is 0.6292 to 0.948 days. If the Enterprise and V'ger traveled at the same speed they would meet halfway between them after 0.6292 to 0.948 days. And V'ger would be less than a day from Earth, not more than a day.
If V'ger has 1.2584 to 1.896 days to travel and the Enterprise meets V'ger when V'ger still has more than 1.001 days to travel, then the Enterprise travels at least 1.118 to 3.387 times as fast as V'ger.
Thus it seems that the Enterprise should be travelling about 1.118 to 1.4254 times as fast as V'ger, and thus that V'ger should be travelling 0.701 to 0.894 times as fast as the Enterprise.
If the Enterprise travels at the official TOS warp factor Seven, or 343 times the speed of light, V'Ger's speed should be about 240.4 to 306.6 times the speed of light.
The Enterprise should intercept V'Ger about 1.0 to 2.0 days from Earth at a point about 0.658 to 1.679 light years from Earth.
V'ger should cross the Federation border and destroy Epsilon IX 53.4 hours (2.225 days) from Earth about 1.464 to 1.867 light years from Earth. And farther away than the Enterprise reaches it, of course.
V'ger should be less than three days from Earth (about 2.0 to 3.0 days) on stardate 7410.2, and thus about 1.316 to 2.518 light years from Earth. But this is before V'Ger crosses the Border, and so 1.464 to 2.518 light years from Earth.
And Spock Should have left Vulcan about 2.0 to 5.0 days before stardate 7410.2. This should be when V'ger is about 4.225 to 8.000 days from Earth, and thus about 2.364 to 6.715 light years from Earth.
But if Spock's Vulcan Shuttle and the Enterprise could travel at speeds of about 1,196.559 to 2,011.31 times the speed of light between Earth and Vulcan, maybe the Enterprise travels at 1,196.559 to 2,011.31 times the speed of light when travelling at warp seven to intercept V'Ger.
So if V'ger travels at 0.701 to 0.894 times the Enterprise's speed of 1,196.559 to 2,011.31 times the speed of light, V'ger would travel 838.787 to 1,789.111 times the speed of light.
The Enterprise should intercept V'Ger about 1.0 to 2.0 days from Earth at a point about 2.296 to 9.796 light years from Earth.
V'ger should cross the Federation border and destroy Epsilon IX 53.4 hours (2.225 days) from Earth about 5.292 to 10.898 light years from Earth. And farther away than the Enterprise reaches it, of course.
V'ger should be less than three days from Earth (about 2.0 to 3.0 days) on stardate 7410.2, and thus about 4.592 to 14.694 light years from Earth. But this is before V'Ger crosses the Border, and so 5.292 to 14.694 light years from Earth.
And Spock Should have left Vulcan about 2.0 to 5.0 days before stardate 7410.2. This should be when V'ger is about 4.225 to 8.000 days from Earth, and thus about 9.702 to 39.186 years from Earth.
And presumably V'Ger destroyed the Klingon ships within Klingon boundaries shortly before, or at the same time that Spock sensed Vger, and thus when V'ger was either about 2.364 to 6.064 light years from Earth, or about 9.702 to 39.186 light years from Earth.
If V'Ger traveled at about 240.4 to 306.6 times the speed of light, it could travel about 72,120 to 122,640 light years in 300 to 400 years, the approximate time limits for its journey if there was no time travel involved.
If V'Ger tra share