Thinking abouthow Starfleet command left Earth undefended by starships in TMP today, I decided, as I have before, that Starfleet Command was confident in the abiity of Earth's defense systems to defend Earth against any invading fleet that made it to Earth. Possibly Starfleet command hoped the klingons whould attack Earth so the defense system would destroy their battle fleet.
And as far as I know, Starfleet Comand was correct that the defense system could defeat any possible Klingon attack, ven though V'Ger was a different story.
So I began thinking about how near or far the closest planet ruled by the Klingons would be to Earth.
In the writer's guides and *The Making of Star Trek* (1968) it was said that the speed of a warp factor (in TOS) was *c*, the speedof light, multiplied by the warp factor cubed.
Thus warp factor 1 was 1 times *c*.
Warp factor 2 was 8 times *c*.
Warp factor 3 was 27 times *c*.
Warp factor 4 was 64 times *c*.
Warp factor 5 was 125 times *c*.
Warp factor 6 was 216 times *c*. That was the fastest speed for sustained travel.
Warp factor 7 was 343 times *c*. That was a dangeorus emergency speed.
Warp factor 8 was 512 times *c*. That was the fastests and most dangeorus emergency speed.
So if *Star Trek: The Motion Picture* used that warp factor, V'ger would be about 1,029 light days, or about 2.8172 light years from Earth when it was about 3 days away at warp 7. And the closest Starfleet ships to Earth would be at least 648 light days or 1.7741 light years away if they couldn't get to Earth in time at warp factor 6 (and maybe travelling for short periods at higher emergency speeds).
In our section of the galaxy the average separation between stars is about 5 light years, so V'ger was already closer to Earth than the nearest star, and if the nearest stars were at the strs nearest to Earth, they would still be several times as far away as V'Ger was.
So how does the TOS warp scale fit in with the speeds in various episodes?
Badly, since TOS and later *Star Trek* writers tended and tend to make starships travel at the speed of plot. I guess that tv and movie writers hated those high school problems where they had to calculate travel time from distance and speed, or speed from travel distance and tme, etc., because they hardly ever seemed to do it in science fiction movies and tv shows.
In many episodes the *Enterprise* and other ships seem to travel much faster than they would at the official war scale, in the extreme case of "That Which Surives" over 1,200 times faster.
So why not ditch the TOS warp scale since it isn't described in canon. Because in at least two TOS episodes, "This Side of Paradise" and "By Any Other Name" warp travel seems to be not much faster, if any, than the TOS warp scale. And the entire plots of *Star trek: Deep Space NIne* and *Star trek: Voyager* depend on TNG warp travel being both faster than TOS warp travel and at the same time much slower than the fastest speeds reached in TOS.
So one solution to the warp speed paradoxes is to imagine that there some type of wormholes or other space time tunnels connecting different star systems. A space ship can enter the mouth of such a tunnel and emerge from the mouth of the tunnel in another star system in much less time than it would take to travel the entire distance between the two systems, even at top warp speed.
So some starship voyages actually cross the spaces between stars using warp speed, while others bypass those vast distances by using those interstellar short cuts.
And so my theory is that on a space map the Federation would look like a collection of disconnected bubbles floating in space, consisting of stars systems connected by those shortcuts surrounded by spherical zones of space claimed by the Federation. And in some places two or more neighboring star systems would both be part of the Federation and the Federation would claim a larger zone of space around those systems.
And if other star traveling societies have discovered those space shortcuts their territories would also tend to look like disconnected soap bubbles in space maps.
Including the Klingon Empire. So my theory is that in the era of TMP the Klingon Empire rules a star system only a few light years from the Solar System, and claims a volume of space of space several light years around that system.
Thus when the Klingons detected V'Ger heading toward that system on a line toward Earth they would have sent three warships to that system to get ahead of V'Ger and then turn back and head toward V'Ger like they were coming from Earth, and attack V'Ger to make V'Ger think Earth was hostile to V'Ger.
The Klingon high command no doubt figured that if three warships could defeat V'Ger no harm would be done, since V'Ger wouldn't have been powerful enough to defeat Earth's defense system anyway. And they probably also figured if the three warships couldn't defeat V'Ger, it might be powerful enough to defeat Earth's defenses, so sacrificing a mere three warhships to increase the probability that V'Ger would attack and destroy Earth would be worthwhile.
So what star is known to be close enought to Earth that V'Ger could be still be inside the bubble of space claimed by the Klingons around it (though brobably close to the border), while only about 2.8l72 ligth years from Earth?
The closest star system to the Solar System is Alpha centauri. Alpha Centauri A & B are about 4.37 light years from Earth. If the border is halfway between Earth and Alpha Centauri A & B it would be 2.185 light years from Earth.
But Alpha Centuari is often described as part of the Federation in non canan novels, etc., and has never been described as having been misruled by Klingons for even a short time.
On Earth, in the 20th Century, in "Tomorrow is Yesterday" Kirk said:
> KIRK: All right, Colonel. The truth is, I'm a little green man from Alpha Centauri. A beautiful place. You ought to see it.
And when Kirk met Zefram Cochrane in "Metamorphosis" he said:
> KIRK: Zefram Cochrane of Alpha Centuri, the discoverer of the space warp?
Kirk either ment that Zefram Cochrane was from Alpha Centauri, or that Cochrane gained fame connected with Alpha Centauri, perhaps by leading the first warp expedition to there.
And Memory Alpha lists many later mentions of Alpha Centauri inconsistent with the Klingons ever ruling it.
[https://memory-alpha.fandom.com/wiki/Alpha_Centauri][2]
Possibly at some time the Alpha Centauri system was split between the Federation and the Klingons, with The Federation or the Klingons ruling Alpha Centauri A & B, and the Klingons or the Federation ruling the third star, Alpha Centauri C or Proxima Centauri.
> Alpha Centauri C is about 13,000 AU (0.21 ly; 1.9×1012 km) from Alpha Centauri AB, equivalent to about 5% of the distance between Alpha Centauri AB and the Sun.[16][35][47] Until 2017, measurements of its small speed and its trajectory were of too little accuracy and duration in years to determine whether it is bound to Alpha Centauri AB or unrelated.
[https://en.wikipedia.org/wiki/Alpha_Centauri#Orbital_properties][3]
So possibly the border between the Federation and the Klingons includes a plane about 12,500 AU or 0.105 light years between Alpha centauri A & B and Proxima Centauri.
The next closest star to the Solar System is Barnard's Star, also known as BD + 04 degreees 3651a and by other catalog designations. Its distance is given as 5.9629 plus or minus 0.0004 light years, 0r 5.9625 to 5.9633 light years.
So if the border is halfway between Barnard's Star and the Solar System it would be 2.98125 to 2.98169 light years from the Sun. So how could V'ger engage the three Klingon ships on their side of the Border while only 2.8172 lgiht years from Earth.
Maybe the Klingons claim a border which is farther from their stars than the Federation claims from Federation stars, so that the border is at least 3.1453 lght years from Barnard's Star.
*STar Trek: The Motion Picture* opens when the Klingons engage V'Ger, and when a spy drone from the Epsilon IX station (which is near the Federation border), intercepts a signal from the Klingons:
> [Epsilon IX - interior]
>
> LIEUTENANT: Our sensor drone is intercepting this on Quad L fourteen.
>
> BRANCH: That's in Klingon boundaries. Who are they fighting?
And then they plot the course of V'Ger:
> 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!
Then the film cuts to Spock on Vulcan as he senses V'Ger, an unspecified amount of time later.
Then the film cuts to:
Starfleet Command (on Earth) an unspecified time later. Apparently Starfleet command is now informed of V'Ger, but hasn't yet decided to send the *Enterprise* to intercept it, but Kirk has already made up his mind about it. So the time could be hours or days after V'Ger fought the Klingons in Klingon space.
> 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.
So when Kirk talks to Scot later and says:
> 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.
The time must be lesss than one a hour after Kirk Talked to Sonak. If less than 3 days means 2.5 to 3 days, Kirk talks to Sonak less than 2.541666 to 3.041666 days before V'Ger will reach Earth.
Later, in the Rec Room, Kirk tells the crew:
> 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.
53.4 hours is 2.225 days. Kirk ends the meeting by saying:
> KIRK: Viewer off. ...Pre-launch countdown will commence in forty minutes.
How long the pre-launch countdown will last is not specified. The countdown might as muchas 11 hours or as little as 5 minutes. So kirk might be speaking 0.25 hours after speaking to Sonak to 11.25 minutes after speaking to Sonak. So Kirk would have been speaking to Sonak about 53.65 to 64.65 hours, or 2.2354 to 2.69375 days, before V'Ger would reach Earth.
But Kirk could have speaking to Sonak hours or days after V'Ger destroyed the Klingon ships. Arbitarily assuming that Kirk spoke to Sonak 0.25 to 2 days after V'Ger dstroyed the Klingon ships, It would take V'ger 2.4854 to 4.69375 days at warp 7 to reach Earth, travelling a distance of 852.4922 to 1,609.9562 light days, or 2.334 to 4.4078198 light years.
I note that if warp seven was a rounded number, V'Ger ouldhave been travelling a little faster than warp factor seven. At warp 7.1 V'Ger culd travel 357.911 tiems the speed of light instead of 343 times the speed of light. At Warp 7.2 V'Ger could travel 373.248 times he speedof light, and so on. If V'ger was travelling as fast as warp 7.5 it might have been rounded upto warp 8 instead of down to warp 7. V'ger could travel about 421.875 times the speed of light.
Thus it is possible for V'Ger to have passed by Barnard's Star on its way to Earth and still be within Klingon space when it fought the three klingon ships, if one or more of those suggested factors applies to the situation.
Later, when they find the original Voyager 6 probe, they deduce what happened to it.
> DECKER: NASA. National Aeronautics and Space Administration. Jim, this was launched more than three hundred years ago.
>
> DECKER: Voyager VI ...disappeared into what they used to call a black hole.
>
> KIRK: It must have emerged sometime on the far side of the Galaxy and fell into the machine's planet's gravitational field.
> SPOCK: They built this entire vessel so that Voyager could fulfil it's programming.
So how long would it take V'Ger to return from the far side of the galaxy at warp factor 7?
At 343 times the speed of light V'Ger would travel 0.939 light yearsper Earth day, 343 light years per Earth year, 3,430 lightyearsper Earth decade, 34,300 lightyears per Earth century, and 343,000 light years per Earth millennium.
If Decker's "over three hundred years ago" means sometime between 300 and 350 years, then V'Ger could have traveled about 102,900 to120,050 light years. But the Machine planet could have taken a verylong time to build the vast machine which housed V'Ger - years, decades or centuries.
Earth is not at the edge of the galaxy, nor at its center.
> The Sun is near the inner rim of the Orion Arm, within the Local Fluff of the Local Bubble, between the Radcliffe Wave and Split linear structures (formerly Gould Belt).[92] Based upon studies of stellar orbits around Sgr A* by Gillessen et al. (2016), the Sun lies at an estimated distance of 27.14 ± 0.46 kly (8.32 ± 0.14 kpc)[34] from the Galactic Center. Boehle et al. (2016) found a smaller value of 25.64 ± 0.46 kly (7.86 ± 0.14 kpc), also using a star orbit analysis.[93] The Sun is currently 5–30 parsecs (16–98 ly) above, or north of, the central plane of the Galactic disk.[94] The distance between the local arm and the next arm out, the Perseus Arm, is about 2,000 parsecs (6,500 ly).[95] The Sun, and thus the Solar System, is located in the Milky Way's galactic habitable zone.[96][97]
[https://en.wikipedia.org/wiki/Milky_Way#Sun's_location_and_neighborhood][4]
Thus the latest esimates put the Sun about 25,180 to 27,600 light years from the center of the galaxy.
The gaalctic disc of the MIlky Way Galaxy is about 75,000, or 84,700, or 100,000 light years in diameter. If those estimates are correct, the farthest part of the galactic disc would be somewhere between about 62,680 and about 77,600 light years from Earth, which V'Ger would travel in about 182.74 to 226.239 years. But as I wrote, it might have taken the Machine Planet years, decades, or centuries to rebuild V'Ger.
I also note there are some structures of stars beyond the rim of the galactic disc.
And the Galactic Disc is surrounded by a speroidal galactic halo of scattered stars and globular star clusters. Most are within about 100,000 light years of the galactic center, but some are out to distance of 200,000 light years. If the machine planet is that halo, it could take V'Ger up to 367 or even 658 years to reach Earth.
The Milky Way is also surrouneded by a large halo of very hot X-Ray emtting gases, and by a large harlo of Dark matter. If the Machine planet orbited a rare star in the Dark matter halo it might take V'Ger many centuries to Earth Earth at Warp factor seven.
And people might ask why Kirk speculated that the Machine Planet which found V'Ger was on the far side of the Galaxy.
One Kirk learned when Voyager VI disappeared, he would know how long V'Ger had to get to Earth at warp factor Seven. In just a few centuries, V'Ger wouldn't be able to travel much more than 100,000 light years, which would be very far away if the Machine Planet was in our galaxy, but not nearly far enough for the Machine Planet to be in another galaxy. So that pretty much leaves a very far part of the galactic disc or a very far Globular star cluster on the other side of the galaxy.
And of course if V'Ger was coming from the far side of the galaxy, what part of the galaxy would V'Ger pass though to get to Earth? The central region of the Galaxy, of course.
So if people on Earth detected V'Ger coming from the direction of the galacic center they would wonder how far away V'Ger cam from. Did V'Ger come from some place much closer than the galactic center, or from the galactic center, or from some space in the galactic disc beyond the galactic center, or from some distant galaxy far beyond the far side of the galaxy? And people who favored one or the other would probably calculate how long V'Ger would have to travel at warp factor seven to reach Earth from the distance they favored (and b probably bet on).
So I calculate that V'Ger was detected approaching Earth from a direction very close to the direction to the galactic center at warp factor Seven. And once kirk had a rough idea of how long V'Ger had been travelling, Kirk had good reason to believe that the Machine Planet whcih found V'Ger was on the far side of the galaxy.
And by an interesting coincidence the direciton from Earth to Barnard's Star is close to the direction from Earth to the center of the galaxy.
Since the distances and directios more or less add up if the Klingons had a base on a planet of barnard's Star and the Machine Planet was on hte opoosite side of thegalaxy, and VGer travelled between them at warp factor seven on theofficial (though notcanon) TOS warp scale, It seems quite probable that someone connected with the making of *Star Trek: The Motion Picture* may have worked it out for the producers and sriptwriters.
After all, the calculations I have made are simple travel distance, time, and speed calculations, and not exactly rocket science. They would be easy for someone like biochemist and science fiction writer Dr. Isaac Asimov, listed as a special science consultant, or the other special science consultant, Baron Jesko von Puttkamer whose day job in the US space program involved literal rocket science.
So since *Star Trek: The Motion Picturea* which is the production which probably has the highest score in the scale of science fiction hardness https://tvtropes.org/pmwiki/pmwiki.php/SlidingScale/MohsScaleOfScienceFictionHardnes has such a splendid example, deliberate or accidential, of distance, speed and travel time agreeing, it seems wasteful for *Star Trek* fans to throw away the TOS warp factor scale.
Instead they should accept that the TOS warp factor scale is bascially correct in TOS but there are other factors, such as the space short cuts I suggested, which enable stars to sometimes reach their destinations in a much shorter time that if they traveled the entire distance at the stated warp speeds.
[2]: https://memory-alpha.fandom.com/wiki/Alpha_Centauri
[3]: https://en.wikipedia.org/wiki/Alpha_Centauri#Orbital_properties
[4]: https://en.wikipedia.org/wiki/Milky_Way#Sun's_location_and_neighb
And as far as I know, Starfleet Comand was correct that the defense system could defeat any possible Klingon attack, ven though V'Ger was a different story.
So I began thinking about how near or far the closest planet ruled by the Klingons would be to Earth.
In the writer's guides and *The Making of Star Trek* (1968) it was said that the speed of a warp factor (in TOS) was *c*, the speedof light, multiplied by the warp factor cubed.
Thus warp factor 1 was 1 times *c*.
Warp factor 2 was 8 times *c*.
Warp factor 3 was 27 times *c*.
Warp factor 4 was 64 times *c*.
Warp factor 5 was 125 times *c*.
Warp factor 6 was 216 times *c*. That was the fastest speed for sustained travel.
Warp factor 7 was 343 times *c*. That was a dangeorus emergency speed.
Warp factor 8 was 512 times *c*. That was the fastests and most dangeorus emergency speed.
So if *Star Trek: The Motion Picture* used that warp factor, V'ger would be about 1,029 light days, or about 2.8172 light years from Earth when it was about 3 days away at warp 7. And the closest Starfleet ships to Earth would be at least 648 light days or 1.7741 light years away if they couldn't get to Earth in time at warp factor 6 (and maybe travelling for short periods at higher emergency speeds).
In our section of the galaxy the average separation between stars is about 5 light years, so V'ger was already closer to Earth than the nearest star, and if the nearest stars were at the strs nearest to Earth, they would still be several times as far away as V'Ger was.
So how does the TOS warp scale fit in with the speeds in various episodes?
Badly, since TOS and later *Star Trek* writers tended and tend to make starships travel at the speed of plot. I guess that tv and movie writers hated those high school problems where they had to calculate travel time from distance and speed, or speed from travel distance and tme, etc., because they hardly ever seemed to do it in science fiction movies and tv shows.
In many episodes the *Enterprise* and other ships seem to travel much faster than they would at the official war scale, in the extreme case of "That Which Surives" over 1,200 times faster.
So why not ditch the TOS warp scale since it isn't described in canon. Because in at least two TOS episodes, "This Side of Paradise" and "By Any Other Name" warp travel seems to be not much faster, if any, than the TOS warp scale. And the entire plots of *Star trek: Deep Space NIne* and *Star trek: Voyager* depend on TNG warp travel being both faster than TOS warp travel and at the same time much slower than the fastest speeds reached in TOS.
So one solution to the warp speed paradoxes is to imagine that there some type of wormholes or other space time tunnels connecting different star systems. A space ship can enter the mouth of such a tunnel and emerge from the mouth of the tunnel in another star system in much less time than it would take to travel the entire distance between the two systems, even at top warp speed.
So some starship voyages actually cross the spaces between stars using warp speed, while others bypass those vast distances by using those interstellar short cuts.
And so my theory is that on a space map the Federation would look like a collection of disconnected bubbles floating in space, consisting of stars systems connected by those shortcuts surrounded by spherical zones of space claimed by the Federation. And in some places two or more neighboring star systems would both be part of the Federation and the Federation would claim a larger zone of space around those systems.
And if other star traveling societies have discovered those space shortcuts their territories would also tend to look like disconnected soap bubbles in space maps.
Including the Klingon Empire. So my theory is that in the era of TMP the Klingon Empire rules a star system only a few light years from the Solar System, and claims a volume of space of space several light years around that system.
Thus when the Klingons detected V'Ger heading toward that system on a line toward Earth they would have sent three warships to that system to get ahead of V'Ger and then turn back and head toward V'Ger like they were coming from Earth, and attack V'Ger to make V'Ger think Earth was hostile to V'Ger.
The Klingon high command no doubt figured that if three warships could defeat V'Ger no harm would be done, since V'Ger wouldn't have been powerful enough to defeat Earth's defense system anyway. And they probably also figured if the three warships couldn't defeat V'Ger, it might be powerful enough to defeat Earth's defenses, so sacrificing a mere three warhships to increase the probability that V'Ger would attack and destroy Earth would be worthwhile.
So what star is known to be close enought to Earth that V'Ger could be still be inside the bubble of space claimed by the Klingons around it (though brobably close to the border), while only about 2.8l72 ligth years from Earth?
The closest star system to the Solar System is Alpha centauri. Alpha Centauri A & B are about 4.37 light years from Earth. If the border is halfway between Earth and Alpha Centauri A & B it would be 2.185 light years from Earth.
But Alpha Centuari is often described as part of the Federation in non canan novels, etc., and has never been described as having been misruled by Klingons for even a short time.
On Earth, in the 20th Century, in "Tomorrow is Yesterday" Kirk said:
> KIRK: All right, Colonel. The truth is, I'm a little green man from Alpha Centauri. A beautiful place. You ought to see it.
And when Kirk met Zefram Cochrane in "Metamorphosis" he said:
> KIRK: Zefram Cochrane of Alpha Centuri, the discoverer of the space warp?
Kirk either ment that Zefram Cochrane was from Alpha Centauri, or that Cochrane gained fame connected with Alpha Centauri, perhaps by leading the first warp expedition to there.
And Memory Alpha lists many later mentions of Alpha Centauri inconsistent with the Klingons ever ruling it.
[https://memory-alpha.fandom.com/wiki/Alpha_Centauri][2]
Possibly at some time the Alpha Centauri system was split between the Federation and the Klingons, with The Federation or the Klingons ruling Alpha Centauri A & B, and the Klingons or the Federation ruling the third star, Alpha Centauri C or Proxima Centauri.
> Alpha Centauri C is about 13,000 AU (0.21 ly; 1.9×1012 km) from Alpha Centauri AB, equivalent to about 5% of the distance between Alpha Centauri AB and the Sun.[16][35][47] Until 2017, measurements of its small speed and its trajectory were of too little accuracy and duration in years to determine whether it is bound to Alpha Centauri AB or unrelated.
[https://en.wikipedia.org/wiki/Alpha_Centauri#Orbital_properties][3]
So possibly the border between the Federation and the Klingons includes a plane about 12,500 AU or 0.105 light years between Alpha centauri A & B and Proxima Centauri.
The next closest star to the Solar System is Barnard's Star, also known as BD + 04 degreees 3651a and by other catalog designations. Its distance is given as 5.9629 plus or minus 0.0004 light years, 0r 5.9625 to 5.9633 light years.
So if the border is halfway between Barnard's Star and the Solar System it would be 2.98125 to 2.98169 light years from the Sun. So how could V'ger engage the three Klingon ships on their side of the Border while only 2.8172 lgiht years from Earth.
Maybe the Klingons claim a border which is farther from their stars than the Federation claims from Federation stars, so that the border is at least 3.1453 lght years from Barnard's Star.
*STar Trek: The Motion Picture* opens when the Klingons engage V'Ger, and when a spy drone from the Epsilon IX station (which is near the Federation border), intercepts a signal from the Klingons:
> [Epsilon IX - interior]
>
> LIEUTENANT: Our sensor drone is intercepting this on Quad L fourteen.
>
> BRANCH: That's in Klingon boundaries. Who are they fighting?
And then they plot the course of V'Ger:
> 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!
Then the film cuts to Spock on Vulcan as he senses V'Ger, an unspecified amount of time later.
Then the film cuts to:
Starfleet Command (on Earth) an unspecified time later. Apparently Starfleet command is now informed of V'Ger, but hasn't yet decided to send the *Enterprise* to intercept it, but Kirk has already made up his mind about it. So the time could be hours or days after V'Ger fought the Klingons in Klingon space.
> 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.
So when Kirk talks to Scot later and says:
> 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.
The time must be lesss than one a hour after Kirk Talked to Sonak. If less than 3 days means 2.5 to 3 days, Kirk talks to Sonak less than 2.541666 to 3.041666 days before V'Ger will reach Earth.
Later, in the Rec Room, Kirk tells the crew:
> 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.
53.4 hours is 2.225 days. Kirk ends the meeting by saying:
> KIRK: Viewer off. ...Pre-launch countdown will commence in forty minutes.
How long the pre-launch countdown will last is not specified. The countdown might as muchas 11 hours or as little as 5 minutes. So kirk might be speaking 0.25 hours after speaking to Sonak to 11.25 minutes after speaking to Sonak. So Kirk would have been speaking to Sonak about 53.65 to 64.65 hours, or 2.2354 to 2.69375 days, before V'Ger would reach Earth.
But Kirk could have speaking to Sonak hours or days after V'Ger destroyed the Klingon ships. Arbitarily assuming that Kirk spoke to Sonak 0.25 to 2 days after V'Ger dstroyed the Klingon ships, It would take V'ger 2.4854 to 4.69375 days at warp 7 to reach Earth, travelling a distance of 852.4922 to 1,609.9562 light days, or 2.334 to 4.4078198 light years.
I note that if warp seven was a rounded number, V'Ger ouldhave been travelling a little faster than warp factor seven. At warp 7.1 V'Ger culd travel 357.911 tiems the speed of light instead of 343 times the speed of light. At Warp 7.2 V'Ger could travel 373.248 times he speedof light, and so on. If V'ger was travelling as fast as warp 7.5 it might have been rounded upto warp 8 instead of down to warp 7. V'ger could travel about 421.875 times the speed of light.
Thus it is possible for V'Ger to have passed by Barnard's Star on its way to Earth and still be within Klingon space when it fought the three klingon ships, if one or more of those suggested factors applies to the situation.
Later, when they find the original Voyager 6 probe, they deduce what happened to it.
> DECKER: NASA. National Aeronautics and Space Administration. Jim, this was launched more than three hundred years ago.
>
> DECKER: Voyager VI ...disappeared into what they used to call a black hole.
>
> KIRK: It must have emerged sometime on the far side of the Galaxy and fell into the machine's planet's gravitational field.
> SPOCK: They built this entire vessel so that Voyager could fulfil it's programming.
So how long would it take V'Ger to return from the far side of the galaxy at warp factor 7?
At 343 times the speed of light V'Ger would travel 0.939 light yearsper Earth day, 343 light years per Earth year, 3,430 lightyearsper Earth decade, 34,300 lightyears per Earth century, and 343,000 light years per Earth millennium.
If Decker's "over three hundred years ago" means sometime between 300 and 350 years, then V'Ger could have traveled about 102,900 to120,050 light years. But the Machine planet could have taken a verylong time to build the vast machine which housed V'Ger - years, decades or centuries.
Earth is not at the edge of the galaxy, nor at its center.
> The Sun is near the inner rim of the Orion Arm, within the Local Fluff of the Local Bubble, between the Radcliffe Wave and Split linear structures (formerly Gould Belt).[92] Based upon studies of stellar orbits around Sgr A* by Gillessen et al. (2016), the Sun lies at an estimated distance of 27.14 ± 0.46 kly (8.32 ± 0.14 kpc)[34] from the Galactic Center. Boehle et al. (2016) found a smaller value of 25.64 ± 0.46 kly (7.86 ± 0.14 kpc), also using a star orbit analysis.[93] The Sun is currently 5–30 parsecs (16–98 ly) above, or north of, the central plane of the Galactic disk.[94] The distance between the local arm and the next arm out, the Perseus Arm, is about 2,000 parsecs (6,500 ly).[95] The Sun, and thus the Solar System, is located in the Milky Way's galactic habitable zone.[96][97]
[https://en.wikipedia.org/wiki/Milky_Way#Sun's_location_and_neighborhood][4]
Thus the latest esimates put the Sun about 25,180 to 27,600 light years from the center of the galaxy.
The gaalctic disc of the MIlky Way Galaxy is about 75,000, or 84,700, or 100,000 light years in diameter. If those estimates are correct, the farthest part of the galactic disc would be somewhere between about 62,680 and about 77,600 light years from Earth, which V'Ger would travel in about 182.74 to 226.239 years. But as I wrote, it might have taken the Machine Planet years, decades, or centuries to rebuild V'Ger.
I also note there are some structures of stars beyond the rim of the galactic disc.
And the Galactic Disc is surrounded by a speroidal galactic halo of scattered stars and globular star clusters. Most are within about 100,000 light years of the galactic center, but some are out to distance of 200,000 light years. If the machine planet is that halo, it could take V'Ger up to 367 or even 658 years to reach Earth.
The Milky Way is also surrouneded by a large halo of very hot X-Ray emtting gases, and by a large harlo of Dark matter. If the Machine planet orbited a rare star in the Dark matter halo it might take V'Ger many centuries to Earth Earth at Warp factor seven.
And people might ask why Kirk speculated that the Machine Planet which found V'Ger was on the far side of the Galaxy.
One Kirk learned when Voyager VI disappeared, he would know how long V'Ger had to get to Earth at warp factor Seven. In just a few centuries, V'Ger wouldn't be able to travel much more than 100,000 light years, which would be very far away if the Machine Planet was in our galaxy, but not nearly far enough for the Machine Planet to be in another galaxy. So that pretty much leaves a very far part of the galactic disc or a very far Globular star cluster on the other side of the galaxy.
And of course if V'Ger was coming from the far side of the galaxy, what part of the galaxy would V'Ger pass though to get to Earth? The central region of the Galaxy, of course.
So if people on Earth detected V'Ger coming from the direction of the galacic center they would wonder how far away V'Ger cam from. Did V'Ger come from some place much closer than the galactic center, or from the galactic center, or from some space in the galactic disc beyond the galactic center, or from some distant galaxy far beyond the far side of the galaxy? And people who favored one or the other would probably calculate how long V'Ger would have to travel at warp factor seven to reach Earth from the distance they favored (and b probably bet on).
So I calculate that V'Ger was detected approaching Earth from a direction very close to the direction to the galactic center at warp factor Seven. And once kirk had a rough idea of how long V'Ger had been travelling, Kirk had good reason to believe that the Machine Planet whcih found V'Ger was on the far side of the galaxy.
And by an interesting coincidence the direciton from Earth to Barnard's Star is close to the direction from Earth to the center of the galaxy.
Since the distances and directios more or less add up if the Klingons had a base on a planet of barnard's Star and the Machine Planet was on hte opoosite side of thegalaxy, and VGer travelled between them at warp factor seven on theofficial (though notcanon) TOS warp scale, It seems quite probable that someone connected with the making of *Star Trek: The Motion Picture* may have worked it out for the producers and sriptwriters.
After all, the calculations I have made are simple travel distance, time, and speed calculations, and not exactly rocket science. They would be easy for someone like biochemist and science fiction writer Dr. Isaac Asimov, listed as a special science consultant, or the other special science consultant, Baron Jesko von Puttkamer whose day job in the US space program involved literal rocket science.
So since *Star Trek: The Motion Picturea* which is the production which probably has the highest score in the scale of science fiction hardness https://tvtropes.org/pmwiki/pmwiki.php/SlidingScale/MohsScaleOfScienceFictionHardnes has such a splendid example, deliberate or accidential, of distance, speed and travel time agreeing, it seems wasteful for *Star Trek* fans to throw away the TOS warp factor scale.
Instead they should accept that the TOS warp factor scale is bascially correct in TOS but there are other factors, such as the space short cuts I suggested, which enable stars to sometimes reach their destinations in a much shorter time that if they traveled the entire distance at the stated warp speeds.
[2]: https://memory-alpha.fandom.com/wiki/Alpha_Centauri
[3]: https://en.wikipedia.org/wiki/Alpha_Centauri#Orbital_properties
[4]: https://en.wikipedia.org/wiki/Milky_Way#Sun's_location_and_neighb
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