In Broadcast order the 18th episode of TOS is "Arena", broadcast on 1 9 January, 1967.
When the
Enterprise arrives at Cestus III, a message from "Commodore Travers" asks for Kirk to bring his tactical people.
SPOCK: Captain. I wonder why he's insistent that our tactical aides come down.
KIRK: This colony is isolated, exposed, out on the edge of now here. He probably wants additional advice.
Kirk, Spock , & McCoy beam down with Kelowitz, Lang, and O'Herlihy, and find the Cestus III outpost devastated..
SPOCK: Those messages we got, Captain, the one directing us here yesterday.
KIRK: Faked. All this happened several days ago. (to a man in a yellow shirt) Lang, over there. Look for survivors. Kelowitz, (blue shirt) that way. (to the red shirt) O'Herlihy, stick with me.
Sulu is in command of the
Enterprise. There is no clue where Scott is.
Captain's log, Stardate 3045.6 The Enterprise has responded to a call from Earth observation outpost on Cestus Three. On landing, we have discovered that the outpost has been destroyed.
Aliens attack on Cestus III while a ship attacks the
Enterprise.
KIRK: We're helpless down here. And the Enterprise
SPOCK: Sulu is an experienced combat officer.
As far as I remember, Sulu's only experience of ship to ship combat in TOS so far was in "Balance of terror" and the threat of combat in "The Corbomite Maneuver". Presumably Sulu has some other combat experience previous to TOS.
After the aliens are beamed back to their ship:
KIRK: I want a search party of thirty medical personnel beamed down immediately to search for survivors. Notify the transporter room. Lock onto us. We're beaming up.
Captain's log, supplement. We have beamed back to the Enterprise and immediately set out in pursuit of the alien vessel. It appears to be headed toward a largely unexplored section of the galaxy.
KIRK: The reason is crystal clear. The Enterprise is the only protection in this section of the Federation. Destroy the Enterprise, and everything is wide open.
If Kirk believes that section of the Federation would be unprotected and wide open for attack if the
Enterprise was destroyed, why h is Kirk risking the destruction of the
Enterprise by pursuing the alien vessel closer to other alien warships and their hypothetical powerful main bases?
KIRK: Is the alien still making warp five?
SULU [OC]: Affirmative, sir.
KIRK: Initiate warp six.
SULU [OC]: Affirmative, sir.
If warp factor five is is 125 times the speed of light, according to the official but not necessarily canon TOS warp formula, and if warp factor six is 216 times the speed of light, the
Enterprise would rapidly close the distance between it and the alien ship while travelling at warp factor six while the alien travels at warp factor five. The speed of light is 299,792,458 meters per second, or approximately 300,000 kilometers per second.
So warp factor six, about 64,800,000 kilometers per second, is about 27,300,000 kilometers per second faster than warp factor five, about 37,500,000 kilometers per second. While the
Enterprise is travelling at warp 6 and the alien ship at warp 5, the
Enterprise will get 27,300,000 kilometers closer to the alien ship every second, 1,638,000,000 kilometers closer ever minute, and 98,280,000,000 kilometers closer ever hour. In other terms, the
Enterprise would gain 91 light seconds every second, 91 light minutes every minute, 91 light hours every hours, and 91 light days every day, that it was travelling at warp six and the alien ship at warp five.
Captain's log, Stardate 3046.2. We are in hot pursuit of the alien vessel which destroyed the Earth outpost on Cestus Three.
There are 0.6 stardate units between 3045.6 and 3046.2. Unfortunately there is no indication of how many hours or days might have passed between those two stardates.
Assuming that it is at least one hour, there should be at least one and two thirds hours per stardate unit in "Arena", which is not very helpful.
KIRK: Mister DePaul.
DEPAUL: Yes, sir.
KIRK: Position.
DEPAUL: Twenty two point three parsecs beyond latest chart limit, sir.
A parsec is about 3.26156 (three point two six one five six) light years. So twenty two point three parsecs is about 72.73287 (seventy two point seven three two eight seven) light years.
Since Cestus III could not have been all the way to the latest chart limit, they must have traveled farther than that to reach a point seventy two point seven three two eight seven light years beyond the farthest charts.
A light year is the distance traveled by light in a Julian calendar year 365.25 days long. A light year thus contains 365.25 light days (the distance light travels in a day) and 8,766 light hours (the distance light travels in an hour), and 525,960 light minutes (the distance light travels in a minute), and 31,557,600 light seconds (the distance light travels in a second).
So at warp factor five, a starship would travel 125 light seconds every second, and so on. At warp factor six, a starship would travel 216 light minutes every minute, and so on.
So if they had traveled seventy two point seven three two eight seven light years at warp factor five, it would have taken them 212.525446 (two hundred twelve point five) days to travel that distance. And if they had traveled seventy two point seven three two eight seven light years at warp factor five, it would have taken them 122.9892628 (one hundred twenty two point nine) days to travel that distance.
And the audience gets the impression that only hours, instead of hundreds of days, have passed, and thus that the two warp factors mentioned must be at least hundreds or thousands of times as fast as the official TOS warp scale!
KIRK: Mister Sulu, status, alien vessel.
SULU: They must be aware we're after them, sir. They've gone to warp six also.
KIRK: Warp factor seven.
SULU: Aye, aye, sir.
KIRK: Something the matter, Mister Spock?
SPOCK: A sustained warp seven speed will be dangerous, Captain.
Warp factor seven on the official TOS era scale would be 343 times light speed, 127 light speeds faster than warp factor six. At warp factor seven, the
Enterprsie would gain 127 light seconds every second that the alien ship was travelling only at only warp six, and 127 light minutes every time the alien ship was travelling at only warp six, and 127 light hours every hour that the alien ship was travelling at only warp six.
About a minute later:
SULU: Alien ship maintaining interval, Captain. Now at warp seven.
KIRK: Warp factor eight.
(Everyone looks surprised. Scotty is aghast.)
Warp factor eight is 512 times light speed on the official TOS era warp scale, and thus 169 light speeds faster than warp factor seven. At warp factor eight, the
Enterprise would gain 169 light seconds every second that the alien ship was travelling only at only warp seven, and 169 light minutes every time the alien ship was travelling at only warp seven, and 169 light hours every hour that the alien ship was travelling at only warp seven.
Later they are finally closing on the alien ships. I don't know if the aliens went to warp factor eight for a while but couldn't keep it up, of if they have been at warp factor seven the whole unspecified time and the
Enterprise has been getting closer every second, minute, and hour in the interval.
SULU: Captain.
KIRK: Yes, Mister Sulu.
SULU: Closing on target, sir.
KIRK: Good. Mister DePaul.
DEPAUL: Yes, sir.
KIRK: Our position.
DEPAUL: Two two seven nine pl, sir. Uncharted solar system at two four six six pm.
KIRK: Is it on the alien's course?
DEPAUL: No, sir. He's headed away from it.
:
So the alien ship has already passed the uncharted solar system. The
Enterprise is scanned from that solar system.
UHURA: Captain, sensors report we're being scanned.
KIRK: By the alien ship?
UHURA: No, sir. It's from that solar system ahead.
So the alien ship has already passed that solar system and the
Enterprise is has not yet reached it and gone past. The gap between the two ships seems to be wider than that entire solar system. So how are the limits of a solar system defined in TOS era
Star Trek? How many millions, or billions, or trillions of kilometers would be between the two ships?
SULU: Captain!
KIRK: Yes, what is it?
SULU: The alien. It's slowing down. Warp five, four, two. It's going sublight, sir. Sir? It's stopped dead in space.
With the
Enterprise at warp eight and the aien ship stopped, the
Enterprise would close the distance by 512 light seconds (or .153,493,737.5, kilometers, more than one Astronomical Unit) every second, and 512 light minutes (more than eight and half light hours) every minute. The orbit of Neptune is aobut 4.2 light hours from the Sun, so if the Entprise was catching up to the alien ship at warp factor eight for a full minute it would travel the entire distance across Neptune's orbit from one side of the Sun to the other side!.
With the
Enterprise at warp eight and the alien ship stopped, the
Enterprise would close the distance by 512 light seconds (or .153,493,737.5, kilometers, more than one Astronomical Unit) every second, and 512 light minutes (more than eight and half light hours) every minute. The orbit of Neptune is about 4.2 light hours from the Sun, so if the
Enterprise was catching up to the alien ship at warp factor eight for a full minute it would travel the entire distance across Neptune's orbit from one side of the Sun to the other side!.
SULU: Range is one eight one zero. One seven six zero. Range is one seven zero zero. One six four zero. Range is one five nine zero. One five five zero and closing, sir.
The difference in range decreased by 50, 60, 60, 50, and 40 of the unspecified units. Assuming that it decreased by about 40 to 60 units per second, and the
Enterprise was travelling at warp 8, it would be decreasing the range by 512 light seconds or 8.53 light minutes every second. So those units would equal approximately 8.53 to 12.8 light seconds. Thus the total distance at the beginning of Sulu's count down would be approximately 72.400 to 108,600 light seconds, and the total distance remaining at the end of Sulu's count down would be about 62,000 to 93,000 light seconds, or about 17.2 to 25.83 light hours.
(Suddenly, the ship decelerates, and everyone hangs on to something as the lights dim.) SULU: Warp six, warp five, four, warp three, warp one. Sublight, Captain. We're stuck, Captain. It's impossible, but. It's impossible.
KIRK: From warp eight? Have you lost your mind?
SPOCK: Same as the alien, Captain.
Assuming that the average speed of the
Enterprise during that deceleration was 256 times the speed of light, for every second that it took the
Enterprise to decelerate it would have traveled 256 light seconds or 4.26 light minutes, a tiny fraction of the remaining distance to the alien ship.
If the alien ship left Cestus III aft warp factor five, and if that was the official TOS scale 125 times the speed of light, each second it was at warp 5 and the
Enterprise wasn't it would get 125 light seconds or 2 light minutes farther way. Each minute it would get 125 light minutes or t.2 light hours farther away. Each hour it would get 125 light hours or 5.2 light days or 0.0142 of a light year farther away.
And it may have taken Kirk, Spock, & McCoy minutes or hours to get the 30 medical personnel and their necessary equipment and supplies down to Cestus III before leaving to pursue the alien ship. Of course, if the ships had to travel at hundreds or thousands of times the speeds of the official TOS wrp scale, the distances traveled per unit of time and the separation between the two ships would have been hundreds or thousands of times as far., and thus possibly the
Enterprise was one or more light years behind the alien ship during the chase.
The Metrons transport Captain Kirk to the surface of what they call a planet in their system.
KIRK [OC]: The Enterprise is dead in space, stopped cold during her pursuit of an alien raider by mysterious forces, and I have been somehow whisked off the bridge and placed on the surface of an asteroid, facing the Captain of the alien ship. Weaponless, I face the creature the Metrons called a Gorn. Large, reptilian. Like most humans, I seem to have an instinctive revulsion to reptiles. I must fight to remember that this is an intelligent, highly advanced individual, the Captain of a starship, like myself, undoubtedly a dangerously clever opponent.
Some people might have seen the Disney channel show
Jessie (2011-2015), where a ten-year-old boy keeps a pet Asian water monitor (
Varanus salvator).. in real life, my sister had a pet snake when she was about that age. So I don't think that all humans always have an instinctive revulsion to all reptiles.
Aboard the
Enterprise.
SPOCK: Lieutenant Uhura, have sensors learned anything about the nature of the force which holds us here?
UHURA: No, sir. They report they definitely emanate from that solar system ahead.
:
So the Metron solar system is still ahead of the
Enterprise and behind the Gorn ship, and they are still separated by at least the unspecified size of the Metron solar system.
KIRK: This is Captain James Kirk of the Starship Enterprise. Who ever finds this please get it to Starfleet Command. I'm engaged in personal combat with a creature apparently called a Gorn.
(We see that the recording is also being transmitted and translated by the Gorn's device too)
KIRK [OC]: He's immensely strong. Already he has withstood attacks from me that would have killed a human being. Fortunately, though strong, he is not agile. The agility and, I hope, the cleverness, is mine.
(Back on the rock)
KIRK: The Metrons, the creatures that sent us both here, said that the surface of the planet provides the raw material to construct weapons. There's very little here. Scrub brush, rocks, an abundance of mineral deposits, but no weapons in the conventional sense. Still, I need to find one. Bare-handed against the Gorn, I have no chance.
MCCOY: Now, you're the one that's always talking about logic. What about some logic now? Where's the Captain, Mister Spock?
SPOCK: He's out there, Doctor. Out there somewhere in a thousand cubic parsecs of space, and there's absolutely nothing we can do to help him.
A thousand cubic parsecs could be a volume of space that was 10 parsecs by 10 parsecs by 10 parsecs. Since a parsec is 3.261 light years, a thousand cubic parsecs would be about 34,677 cubic light years and have about 148 solar systems in it. Is it possible that the Gorn ship is more than 32 light years ahead of the
Enterprise, or does Spock have some other reason for saying Kirk could be within a thousand cubic parsecs?
KIRK: A large deposit of diamonds on the surface. Perhaps the hardest substance known in the universe. Beautifully crystallized and pointed, but too small to be useful as a weapon. An incredible fortune in stones yet I would trade them all for a hand phaser, or a good solid club. Yet the Metrons said there would be weapons, if I could find them. Where? What kind?
Like Mudd's Women", this seems to contradict "Catspaw" about the value of diamonds.
KIRK: This may be my last entry. I am almost exhausted. Unless I find the weapon the Metron mentioned I have very little time left. Native sulphur, diamonds. This place is a mineralogist's dream. Yet there is something about sulphur. Something very old. Something? If only I could remember.
As I remember, when the Metrons show what is happening on the viewscreen, Uhura seems frightened by the sight of the Gorn. Even though Saurian brandy was a well known drink, and There were Saurians serving in starfleet according to TMP and DISC, and there was Sord, a member of another reptilian species in "The Jihad". And maybe Uhura is reptile phobic and a lot less accustomed to highly nonhuman aliens than most Starfleet members.
When the Metron appears to KIrk, in their real or assumed form, Kirk says that they look like a boy, but the Metron says they are about 1,500 Earth years old, without saying whether they are an adult by Metron standards.
When Kirk is transported back to the bridge:
SULU: Captain.
KIRK: Mister Sulu.
SULU: It's impossible, but there's Sirius over there when it should be here. And Canopus. And Arcanis. We're. All of a sudden, we're clear across the galaxy, five hundred parsecs from where we are I mean, were. I mean
KIRK: Don't try and figure it out, Mister Sulu. Just plot a course for us back to Cestus Three.
SULU: Aye, aye, sir.
I don't know what star is meant by Arcanis, but Sirius and Canopus are real stars.
A distance of 500 Parsecs is about 1,630.5 light years. Since the galactic disc of the Milky Way galaxy is about 100,000 light years in diameter and about 1,000 light years thick, a distance of 1,600 light years is less than two percent of the diameter of he galactic disc. that doesn't seem b very close to "clear across the " the galaxy. Maybe Sulu meant that they were clear across the thickness of the galactic disc.
KIRK: I don't. Not anymore. And maybe in a thousand years or so, we'll be able to prove it. Never mind, Mister Spock. It doesn't make much sense to me either. Take us back to where we're supposed to be, Mister Sulu. Warp factor one.
SULU: Warp factor one.
So apparently they are going back to Cestus III, to pick up the medical team and resume patrolling their region of space, which could be about one or two hundred light years closer or farther away than the Metron system was. So they might be travelling 1,400 to 1,800 light years at a speed of warp factor one, supposed to be the speed of light. So it should take them only about 1,400 to 1,8000 years to pick up the medical team they left on cestus III. I'm sure the medical team brought enough supplies to last, and I'm sure nobody will invade that section of the Federation while the
Enterprise is away.
At warp factor six it would take only about 8 years to get back to Cestus III.
In order to get back to Cestus III within one year a speed of 1,400 to 1,800 times the speed of light would be necessary. A speed of 16,800 to 21,600 times the speed of light would be necessary to get back to Cestus III in about one month. A speed of 73,050 to 93,921 times the speed of light would be necessary to get back to Cestus III in about one week. A speed of 511.350 to 657,450 times the speed of light would be necessary to get back to Cestus III in about one day.
If the
Enterprise traveled 22.3 parsecs or 72.73287 (seventy two point seven three two eight seven) light years, or 637,576.32 light hours.in a time period between one hour and one week (168 hours) that would require a speed of about 3,795.1 to 637,576.32 times the speed of light.
At speeds of 3,795.1 to 657.450 times the speed of light, it would take about 0.15 to 26 years to cross the entire galactic disc 100,000 light years in diameter. That would make the basic plots of two entire series, DS9 and VOY, nonsense.
I also note that the faster and the farther the Gorn ship flees from Cestus III without meeting another Gorn warship or a main Gorn base, the more dubious their claim to sincerely believe that Cestus III is within their territory becomes. So the ships travelling as slow and as small a distance as possible is good for the idea that the Gorns were defending their territory.
So to me that suggests that the solution to similar problems suggested in my post about "The Galileo 7" could also be used to explain the time, speed, and distance problems in "Arena".