I commend (or maybe condemn

) 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
Altair Vulcan, a few days can be 3 days at the least and 5 days at most (at least that's my definition of "few"). At the given speed of Warp 8 (v=0.73 lys/hour), if 3 days out, then the Enterprise is 52.5 lys out. If 5 days out, then the Enterprise is 87.6 lys out. Sound plausible and logical.
The problem with warp drive is that it has to as fast as it is in some episode and also as slow as it is in other episodes.
A light year is the distance that light travels in a Julian calendar year of 365.25 days, so it contains 365.25 light days, and 8,766 light hours. 0.73 light years equals 6,399.18 light hours, so 0.73 light years per hour is 6.399.18 c. That is 12.49 times the official TOS warp 8 at 512 c, 18.65 times warp 7, 29.62 times warp 6, 51.19 times warp 5, 99.98 times warp 4, 237 times warp 3, 799.89 times warp 2, and 6,399.18 times warp 1.
The
Enterprise is at Mira or Omicron Ceti in "This Side of Paradise" . Beaming down to the colony, Kirk looks around and says:
KIRK: Another dream that failed. There's nothing sadder. It took these people a year to make the trip from Earth. They came all that way and died.
Later is is said the the colonists were on Omicron Ceti for three years and that they left Earth four years earlier.
So the voyage took approximately one year. Assume that the voyage took between 0.5 and 2.0 Earth years. How far is Omicron Ceti from Earth?
The distance to Mira is uncertain; pre-
Hipparcos estimates centered on 220
light-years;
[17] while Hipparcos data from the 2007 reduction suggest a distance of 299 light-years, with a
margin of error of 11%.
[2]
Assuming that Omicron Ceti is between 200 and 340 light years from Earth, the average speed of the voyage from Earth to Omicron Ceti would be between 100 c - which is less than warp 5, 1.56 times warp 4, 3.7 times warp 3, 12.5 times warp 2, and 100 times warp 1 - and 680 c, which is 1.32 times warp 8, 1.82 times warp 7, 3.14 times warp 6, 5.44 times warp 5, 10.62 times warp 4, 25.18 times warp 3, 85 times warp 2, and 680 times warp 1.
In "By Any Other name" Kelvins capture the
Enterprise for a voyage to the Andromeda Galaxy.
IRK: What's the point of capturing my ship? Even at maximum warp, the Enterprise couldn't get to Andromeda galaxy for thousands of years.
ROJAN: Captain, we will modify its engines, in order to produce velocities far beyond the reach of your science. The journey between galaxies will take less than three hundred of your years.
Spock Fascinating. Intergalactic travel requiring only three hundred years. That is a leap far beyond anything man has yet accomplished.
I deduce from Kirk's words that it would take the
Enterprise somewhere between 1,000 years and 10,000 years to make the voyage to Andromeda at maximum warp. The Andromeda Galaxy is about 2,540,000 light years from Earth, so maximum warp speed should be between 254 c - which is 1.17 times warp 5, 0.74 times warp 7, and 0.49 times warp 8 - and 2,540 C - which is 11.75 times warp 6, 7.40 times warp 7, and 4.96 times warp 8.
The Kelvins plan to make the trip in less than 300 years, so probably between 250 and 300 years., giving an average speed of 8,466.66 c and 10,160 c.
Those two episodes indicate that warp factors shouldn't be more than a few times as fast as the official TOS warp scale.
ROJAN: Drea has computed and laid a course for Kelva, Captain.
CHEKOV: We've jumped to warp eight.
ROJAN: And we'll go faster yet. Increase speed to warp eleven.
KIRK: Increase speed to warp eleven, Mister Chekov.
DREA: On course and proceeding as planned. We will approach the barrier within an hour.
Rojan may plan to increase the warp factor even more once past the barrier. If so, warp 11 might be 1,331 c as per the TOS warp scale. At that speed the barrier should be less than 1,331 light hours, or 55.45 light days, or 0.15 light years, from the planet.
If Rojan plans to make the entire trip to the Andromeda galaxy at warp factor 11, warp factor 11 would equal 8,466.66 c to 10,160 c, 6.36 to 7.633 times warp factor 11 on the TOS scale. In that case the barrier would be less than 8,466.66 to 10,160 light hours, 352.77 to 423.33 light days, or 0.96 to 1.159 light years, from the planet, which still seems too close.
Retcon canon has chosen 40 Eridani for Vulcan, so be it. 40 Eridani is a triple star system where 40 Eridani A is a main-sequence dwarf of spectral type K1 (about 0.84 solar mass and 0.457 solar luminosity), 40 Eridani B is a 9th magnitude white dwarf of spectral type DA4, and 40 Eridani C is an 11th magnitude red dwarf flare star of spectral type M4.5e which orbit each other and both orbit 40 Eridani A at about 400 AU distance every 8000 years. The habitable zone of 40 Eridani A, where a planet could exist with liquid water, is near 0.68 AU from A (probably a little closer since Vulcan is on the hot side). At this distance Vulcan would complete a revolution in 223 Earth days (according to the third of Kepler's laws) and 40 Eridani A would appear nearly 20% wider than the Sun does on Earth. Since Vulcan has no moon, the celestial bodies seen in the movies are probably close passing planet(s) a little further out from the sun. (One of which the ice planet we see Spock and Kirk marooned in the JJ film.) I guess on these close approaches, Vulcan undergoes high gravitational stresses causing periodic high volcanic activity.
Is 40 Eridani absolutely canon?
Set decoration from Star Trek: Enterprise and Star Trek: Discovery confirm that 40 Eridani A is Vulcan's star. No need to investigate further than that, I'm afraid.
Is one or both of those set decoration" items clearly visible and readable in the episode?
A picture of a
star chart supposedly used for
Star Trek: Discovery was tweeted by
Ted Sullivan on November 28, 2017. According to this map, the star Vulcan was also known as
40 Eridani A.
[1]
https://memory-alpha.fandom.com/wiki/Vulcan_system
And if "Vulcan 40 Eridani A" or words to that effect can be read on any chart in any episode of
Star Trek: Discovery that would be canon proof that 40 Eridani is the Vulcan system.
Tim Thomason says there is also a scene in an
Enterprise episode with a visual identification of 40 Eridani as Vulcan's system. If that can be read clearly in that episode it would be canon proof that Vulcan is a planet of 40 Eridani.
FWIW, the old 3D star program
Celestia puts the distance between Altair and 40 Eridani at 29.45 ly.
So that seems to be a better distance from Vulcan to Altair than my estimate of at least 20 light years.