Time, Distance, and Speed Problems in Star Trek

Many of us remember having to do time, distance and speed math problems in high school.

How long will it take a vehicle at speed X to travel a distance Y, for example.

Or if train A heads east as at speed C, and train B heads west at speed D, and if they are E miles apart when they both leave, when will the two trains pass each other?

And no doubt high school kids varied greatly in how well they did in such questions and how much they liked or hated that type of math problem.

And I kind of get the impression that many writers of Star Trek episodes, beginning with TOs and continuing with other series and movies, really hated those types of math problems and never bothered calculating the time it would take to travel from Earth to the star visited in the episode, if it was a real star, or to calculate how long it would take a starship to travel the distance mentioned in the episode.

And I get the impression that few members of the production staff ever bothered to calculate whether the writer of an episode got the time, distance, and speed correct in their script and correct any errors that they found.

And so, beginning with TOS, and continuing with other Star Trek productions, sometimes everything add ups and starships make voyages with travel times consistent with their stated speeds and either the TOS warp scale or the TNG warp scale.

And sometimes a starship visits a star that it should not possibly be able to reach, and sometimes a star closer to Earth than many which have been visited is considered to be on the frontier of exploration.

And some times a starship travels a distance tens or hundreds of times faster than it should be able to at the official warp scale for the era.

In TOS the extreme example of a ship traveling much faster than the official warp scale is in "That which survives", where the Enterprise is thrown far from the planet Kirk has just beamed down to:

Spock decides to return to the planet where Kirk's landing party is:

Later:

According to the official TOS warp formula found in writer's guides and The Making of Star Trek, warp factor 8.4 equals the speed of light multiplied by 8.4 cubed. Since 8.4 cubed is 592.704, warp 8.4 is 592.704 times the speed of light.

One of the few things which scientific authorities have done to make things easier for science fiction writers is to define a light year as the distance which light (in a vacuum) travels in one Julian year of 365.25 days (instead of a Gregorian calendar year of 365.2425 days).

A light day is the distance light travels in one day, a light hour is the distance light travels in one hour. Since there are 365.25 days in a year, and 24 hours in a day, there are exactly 8,766 light hours in a light year. So a distance of 990.7 light years equals 8,684,476.2 light hours.

Traveling 8,684,476.2 light hours in 11.337 hours requires a speed of 766,029.47 times the speed of light. But the official formula makes warp factor 8.4 equal to 592.704 times the speed of light. So apparently Lt. Rahda calculated that the Enterprise would reach its destination 1,292.4317 times as soon as it would if it traveled the entire distance at the official speed of warp 8.4.

And various fans have calculated that the Enterprise seems to exceed its official warp speed by about 1,292.4317 times in "That Which Survives" and publicized that in various publications since at least as early as the late 1970s forty years ago.

So forget the number 47. 1,292.4317 should be the number that Star Trek fans should be obsessed with, and variations on 1,291.4317 should be used as a stardate, serial number on a part, "phone number" equivalent, year in the calendar of an alien planet, or other number, in various fan fictions, authorized novels, and official episodes and movies to show that the writers are aware of various problems in Star Trek.

One common theory to explain how starships sometimes travel much farther and faster than they should according to the official warp speed formulas is that there is a factor which multiplies warp speed to many times greater than the speeds calculated by the official warp speed formula.

But if that is the case, why wasn't the official warp speed formula changed by scientific bodies and/or Starfleet to reflect the actual warp speeds. If, for example, warp factor four always equals 200 times the speed of slight, why wasn't the official warp formula changed to make warp factor four be 200 times the speed of light?

And one answer is possibly there are regions of "fast space" and "slow space" where warp speeds are multiplied by different amounts, so that the same warp factor will equal different speeds in different regions of space. For example, in one region of space warp factor four might equal 18,234.493 times the speed of light, in another 382.108 times the speed of light, in another 836.465 times the speed of light, etc.,etc., etc.

If explored space includes many different regions where warp factors are multiplied by different amounts, the official warp scale might use the basic speeds which a warp factor would produce if there wasn't any multiplication factor involved.

And as more and more Star Trek productions are produced, a three dimensional map of our local region of the galaxy could include more and more volumes of space labeled with how much they seem to multiply warp speeds according to the evidence of various episodes.

And it seems to me quite possible that some regions of space might need to be much faster according to some episodes than they have to be according to other episodes. Thus there may be contradictions between various Star Trek productions in how "fast" or "slow" various regions of space should be.

And another problem with that theory is that sometimes a starship travels a distance in a time that is reasonably consistent with the warp speed scale in use in that era. According to the theory of "fast space" and "slow space" that would require some regions of space to not multiply warp speeds at all. Voyages that take days or weeks in other regions of space might take years, decades, or centuries in that region of space. But what if in another Star Trek production a starship travels though that same region of space many times faster than its official warp speed?

In my next post I will give an example of how a voyage at speeds close to the official warp scale messes up the theory of "fast" and "slow" regions of space where warp speeds are multiplied by different amounts.

 

I am shocked, shocked writers ignore warp calculations when they write scripts.

 

In the thread Last Star trek Episode you watched?

https://www.trekbbs.com/threads/last-star-trek-episode-you-watched.144775/page-699#post-13117396

Thee was a discussion of warp speeds in post numbers 13958, 13959, and 13960 onpage 698.

I wrote in post number 13961 on page number 699:

Spock Riding wrote in post number 13962 on page number 699:

And the reason why that explanation doesn't work for me is:

The fact that some long interstellar voyages in TOS take about as long as they would take according to the official TOS warp scale.

What is the planet colonized by the Sandoval group in "This side of Paradise":

The planet is Omicron Ceti III, the third planet out in increasing distance from the star Omicron Ceti, according to the standard science fiction planetary numbering system.

Where is Omicron Ceti?

According to Wikipedia:

Wikipedia also gives the distance as approximately 300 light years or approximately 99 parsecs.

And:

https://en.wikipedia.org/wiki/Mira

since "This Side of Paradise" was written decades before the Hipparcos measurements, the writers probably pictured Omicron Ceti as being about 220 light years from Earth if they thought about the distance at all.

According to the official warp speed formula for TOS:

Warp factor one is 1 times the speed of light, and would take 200 years to travel 200 light years, 220 years to travel 220 light years, 300 years to travel 300 light years, and 400 years to travel 400 light years.

Warp factor two is 8 times the speed of light, and would take 25 years to travel 200 light years, 27.5 years to travel 220 light years, 37.5 years to travel 300 light years, and 50 years to travel 400 light years. Warp factor two was the fastest speed of freighters in TOS according to "Friday's Child".

Warp factor three is 27 times the speed of light, and would take 7.407 years to travel 200 light years, 8.148 years to travel 220 light years, 11.111 years to travel 300 light years, and 14.814 years to travel 400 light years.

Warp factor four is 64 times the speed of light, and would take 3.125 years to travel 200 light years, 3.4375 years to travel 220 light years, 300 years to travel 300 light years, and 400 years to travel 400 light years.

Warp factor five is 125 times the speed of light, and would take 200 years to travel 200 light years, 220 years to travel 220 light years, 4.6875 years to travel 300 light years, and 6.25 years to travel 400 light years.

Warp factor six is 216 times the speed of light, and would take 0.9259 years to travel 200 light years, 1.0185 years to travel 220 light years, 1.388 years to travel 300 light years, and 1.8518 years to travel 400 light years. Warp factor Six was the fast long term speed of the Enterprise according to various sources and episodes.

Warp factor seven is 343 times the speed of light, and would take 0.583 years to travel 200 light years, 0.641 years to travel 220 light years, 0.8746 years to travel 300 light years, and 1.166 years to travel 400 light years. Warp a factor Seven was an emergency speed of the Enterprise.

Warp factor Eight is 512 times the speed of light, and would take 0.390 years to travel 200 light years, 0.429 years to travel 220 light years, 0.5885 years to travel 300 light years, and 0.78125 years to travel 400 light years. Warp factor eight was the highest emergency speed of the Enterprise in TOS.

So if colonization ships were faster than freighters but not faster than the safe cruising speed of the Enterprise, it should have taken the colonists 0.9259 to 14.814 years to reach Omicron Ceti from Earth.

How long did the colonists travel in "This Side of Paradise"?

So Kirk says it took the colonists a year to make the trip from Earth, presumably based in either the log of the ship that transported them or else his general knowledge of space travel speeds. And I would guess that Kirk would not have said "a year" if the travel time was less than 0.50 years or more than 2.00 years.

Which means that if the ship of the colonists used regions of space where the warp factors were multiplied by space conditions, at most the average multiplication amount would be 29.628. Which is pretty puny compared to the over 1,200 times which would be necessary for the voyage in "That Which Survives". And there is no need to imagine there was any multiplication of warp speed on the journey. If the colony ship traveled about 200 light years at a speed as fast as warp factor 6 it could make the voyage to Omicron Ceti in 0.9259 years, which is pretty close to Kirk's vague "a year".

So some people might imagine a cylinder of "slow" space where warp speeds are not multiplied much, if at all, between Earth and Omicron Ceti. But if that was the case, wouldn't the ship transporting the colonists simply go around that cylinder of "slow" space, travelling through regions of "fast" space to reach Omicron Ceti sooner by travelling a longer distance through space where it could go faster? If the colony ship traveled at warp factor 4, 64 times the speed of light, through regions of space where warp speed was multiplied by 100 times, it could travel 6,400 light years in one year, enabling it to reach Omicron Ceti by a roundabout route through "fast" regions of space much faster than by going straight through "slow" regions of space.

So in order for the only practical route to Omicron Ceti to be through regions of "slow space" where warp speeds are multiplied only a few times, if at all, Omicron Ceti would have to be surrounded by a roughly spherical region of "slow space" where warp travel would be only a few times faster than the TOS formula, if at all. And that sphere of "slow space" should be at least 200 light years in radius and extend close to Earth, and thus it might include many stars from various Star Trek productions that should be in regions of "fast space".

What about the Ceti Alpha system in "Space Seed" and WOK? that doesn't seem like it is a year's travel from Earth. It seems to take just a few days for Khan in the Reliant coming from Ceti Alpha and Kirk in the Enterprise coming from Earth to meet near the Regula One station and the Mutara Nebula. So where is Ceti Alpha?

There is no star system named Ceti Alpha, but Ceti Alpha is often assumed to be Alpha Ceti, or Menkar. The distance of Alpha Ceti is about 249 light years plus or minus 8 light years, or about 241 to 257 light years. Thus it should probably be less than 50 light years closer or farther from the Sun than Omicron Ceti.

The coordinates of the direction to Alpha Ceti in the equatorial system are given as Right ascension 3 hours, 2 minutes, 16.77307 seconds, and Declination plus 4 degrees, 5 minutes, 23.0596 seconds. The equatorial coordinates of Omicron Ceti (Mira) are Right ascension 2 hours, 19 minutes, 20.79210 seconds, Declination minus 2 degrees 58 minutes, 39.4956 seconds.

After calculations, I find that the Right ascensions of Alpha Ceti and Omicron Ceti differ by 11.509447 degrees and their declinations differ by 7.0621808 degrees. Thus they should be no more than about perhaps 15 degrees apart in the sky. If they are exactly the same distance from the Sun and Earth they should be only a few tens of light years from each other, and as I wrote above Alpha Ceti should probably be no more than 50 light years closer or farther from the Sun than Omicron Ceti.

So I estimate that the distance between Alpha Ceti and Omicron Ceti is probably a fraction of their distance from Earth, and thus the fact that Alpha Ceti was only a few days away from Earth in WOK indicates that it should't have taken the colony ship a year to take the Sandoval group to Omicron Ceti.

This shows that the theory that there are regions of fast space and slow space in Star Trek has some flaws.

 

Roddenberry once provided around the first or second pilots that maximum speed of the Enterprise was 0.73 ly/hr, be that either warp 6 or 8. This equates to 6400 c. If we assume GR meant that max. speed at warp 8, then the cubed formula becomes: speed = 12.5 x 8^3. I like to think that this is deep space or interstellar normal factor between star systems away from their gravity and solar wind effects.

Even higher speeds can be achieved under special space conditions such as the recently transported Enterprise in That Which Survives. The planet's transporter beam left something like a slowly, collapsing subspace tunnel or a path of ionization as seen in The Gamesters Of Triskelion, back to the planet (note that this is about 120 times the 6400 c max. speed under normal conditions as @MAGolding calculated above).

On the other extreme end, both in Operation: Annihilate and The Paradise Syndrome has the Enterprise diving into a star's gravity well and denser solar wind at high warp speeds of warp 8 and 9, respectively, while apparently moving slower than the cubed formula with no adjustment. Like others, I like the idea that the factor now is a fraction due to the space environment, something as bad as 0.2 x W^3.

My KISS warp scale factors for space environmental conditions:

1. High gravity/magnetic fields; High solar wind/matter (in close orbit of star, in a magnetic storm), x = ~0.2 or <1.0
2. Inside star system (between planets, diffuse solar wind/matter), x = 1.0
   
3. Outside star systems (between star systems, nearly no solar wind/matter), x = ~12.5 (or variable 10 to 50)
4. Special (cosmic strings, subspace paths/corridors, etc.), x = 100 to 1000 or more...

First, colony ships most likely move slower than top of the line Starships. Even the Enterprise doesn't travel faster than Warp 3 (Squire of Gothos) unless there is an emergency. If you assume the colony ship travelled 220 light years at Warp 3 (27 c) using the standard cubed formula, then it takes the ship 8.15 years. Nope. To make it in one year using that formula, then the colony ship needed to travel 220 c or about Warp 6. The Enterprise might be capable of sustained Warp 6 for a year, but a small colony ship? Nope. If average x = 8 over the trip for example, then 220 light years in one year can be made at Warp 3...sounds more plausible to me.

 

YWFMV
Your Warp Factors May Vary

If warp speeds are dependent on local stellar and subspace conditions, then it may be a case that Warp 4 here is many times faster than Warp 9 there. When a navigator/conn officer sets a course, he or she may take into account known "warp highways" to enable a ship to travel vast distances rather quickly. But if a ship is lost in an unknown quadrant of the Galaxy and doesn't know where such subspace shortcuts exist, then it could be a long trip home, IMO.

 

We reach.

 

Galaxy hopping is, I believe, a trope created for sci-fi thanks to Star Trek's fudging around the corners. STV-TFF was very retro-60s, as were a couple TOS episodes, in being able to traverse the entire galaxy - or at the very last its radius in a jiffy. I find it easier to suspend disbelief if individual episodes don't allude or play it up but when we're told Charon is at the southernmost edge of the galaxy and they get there in mid-episode and return back for the start of the next, it stops being easy to suspend said disbelief. Or being carried 999.7 light years and back when Losira has her way with the ship... or when they do dogfights and have to turn fast at warp speed such as in "Elaan of Troyius", all while in a show that - for the 1960s unlike today - nailed the time dilation factor in looking at a planet 900 light years away such as Trelane (who is not related to any Q since he's revealed to be a kid with his glowing mommy and daddy taking him home at the end (in one of many classic TOS FTW moments) and requires external technological help to do his wizardry as opposed to being organically able without the tinkering (the only way it begins to be convincing), and as VOY reminded us we don't need the Q explained or brought down to our level either, it just rots any sense of disbelief one was originally able to have (and at least Q was one of those incorporeal beings TOS was littered with, just taken to a more complex level than wearing a silly toga and saying how humans are oh-so-noble... sorry for the tangents...))

In the end, if the episode has other and actual strengths, the real science stuff are just nitpicks are more easily overlooked or forgiven entirely. After all, low budget aside some ideas behind "That Which Survives" were fairly cool, STV had some great Big Three character moments and Sybok, Elaan had Dr Shrinker and some hawt guards wearing miniskirts and plastic placemats used for shoulder pads that the 80s were very envious of, "The Squire of Gothos" is simply a fun romp with Trelane wearing Michael Nesmith's fancy outfit, and - naturally - there's this:

Spoiler

Holy campfest, Batman!

 

On the flip side of the thread title, an episode of Discovery had a shuttle go to Warp 1 to reach a destination, and the scene inside the shuttle lasted as long as it would have taken for the shuttle to reach the destination at 1c.

 

I believe warp travel is actually measured by how many commercial breaks it takes to get to your destination.

"Good God, Jim, that planet is a full two acts away. We'll never get there in time!"

 

Jesus. If anything deserved the label "TLDR"....

 

You're overthinking this, the ship has always travelled at the speed of plot. Some writers might have done the math using the warp scale, others didn't give a f*** and that's okay because it really doesn't matter, the reason warp factors were used in the first place was to be vague and stop the audience from saying "Well, if they travel at 2 trillion miles per second, how did their 200 trillion mile trip only take a minute?"

 

Short Version: Warp speeds are too slow. "Pathfinder" (VOY) states that Voyager was travelling at an average speed of Warp 6.2. Can't go at maximum all the time. That would wreck the engine. Let's round that down to Warp 6. According to the TNG Scale, Warp 6 is 392 times the speed of light. Let's round that up to 400.

If Voyager started off at 75,000 light years away from home in "Caretaker", it would take them 187.5 years to get back. Not 75. So warp speed in TNG, DS9, and VOY needs to be more than twice as fast as what the scale has.

Finito.

 

Umm, basically every time we get a "it takes ages to get back home" estimate, it is given with the accompanying greeting "even at maximum speed", this meaning the speed that is not available to the ship. Janeway was never going to get home in 75 years, and this is exactly what she said.

But the rest of Trek works by Roddenberry's rough assumption: the ship can move at thousands of times the speed of light, and regularly does, even if not for long stretches. In "Where No One Has Gone Before", a very reasonable 9,000 c is given as the utterly unsustainable maximum speed of the E-D (2.7 million ly in 300 years). We can call that warp 9.8 or thereabouts, as per the pilot where Picard finds out firsthand how fast his ship can really go. That is the true TNG scale. And it pretty much covers all the TNG bases.

If Kirk's ship could do 6,400 c, that, too, is quite reasonable, and covers most of the TOS bases. After all, the prime criterion is that the ship can get from star A to star B in plot time, without A and B being next-door neighbors. At Roddenberry speed, Kirk could cover fifty lightyears in about three days, which is perfectly fine in meeting the criterion.

We can then give Archer something like 4,000 or 5,000 c to play with, and again it works out pretty well.

It is only with TOS outliers that we have to get creative. Two stand out: the 1000 lightyears in what looks like less than a day (rather than a thousand hours) at warp 8.4, from "That Which Survives", and the 22.3 parsecs or 72.7 lightyears in what looks like less than a day (rather than a hundred hours) at just warp 6, from "Arena". All the rest are vague, missing key parameters, and generally fudgeable.

I have no problem with fudging "Arena", too: the heroes may be 22.3 parsecs "beyond latest chart limit", but perhaps they already were 19 parsecs beyond that limit when visiting Cestus, the cartographers having not completed their work there quite yet. Or whatever. Only "That Which Survives" is a true outlier, with all the parameters available and undeniable.

At the other end, warping near stars is slow, sometimes making maximum warp significantly slower than light (as we see with our own eyes whenever Kirk slingshots around the Sun). Which is a fun way to eliminate all the lower outliers at one stroke. No worries there. And every long distance trip that appears suspiciously slow is merely calculated by factoring in the pit stops...

Really, "she flies at plot speed" is something that yields consistent speeds without any writer effort, since the plots are not particularly dissimilar!

Timo Saloniemi

 

Let me dig out the old chart:

It needs updating, but I think it's fair to say the current Trek producers put spectacle and emotion far ahead of stuff like accurate travel times.

 

What do you mean by that? TOS was the one with the screwy travel times; all the newer shows are doing splendidly in comparison.

Timo Saloniemi

 

Merely that it's not a current priority and expecting consistency will only lead to disappointment. For example, Spock covers a good third of the galaxy in a shuttlecraft in Discovery season 2.

 

Or then that's a zoom-in graphic. Actual plot-relevant, dialogue-established travel time screwiness was a TOS thing; after this, the writers just learned better and stopped making dialogue about this thing, unless there was a plot need. In which case they generally got it right.

Timo Saloniemi

 

PART ONE (of five):

According to the TOS warp scale in the writers' guides and The Making of Star Trek, the speed of a warp factor equals the factor number cubed times the speed of light. So warp factor one is the sped of light, warp factor two is 8 times the speed of light, warp factor three is 27 times the speed of light, warp factor four is 64 times the speed of light, warp factor five is 125 times the speed of light, warp factor six is 216 times the speed of light, warp factor seven is 343 times the speed of light, warp factor eight is 512 times the speed of light, warp factor nine is 729 times the speed of light, warp factor ten is 1,000 times the speed of light, and so on and so on.

In many TOS episodes the Enterprise traveled distances that were specified more or less accurately, at various stated warp factors, or at warp factors presumed to be less than warp eight, the maximum rated emergency speed, and the trips lasted times that were stated or could be estimated.

And such travel times were often much less than could be calculated from the TOS warp scale, which tends to imply that the actual speeds must have been much greater than the TOS warp scale.. But how much faster the speeds were thn the TOS warp scaled varied greatly from episode to episode.

A very common fan theory is that local space conditions vary in different three dimensional regions or volumes of space, and that those variations affect how many times faster than the TOS warp scale various warp factors are in those different volumes of space. In some volumes of space a specific warp factor might be five, or fifty, or five hundred times as fast as the TOS warp scale indicates.

PART TWO:

One problem with the local space conditions changing how fast a ship travels at a certain warp factor is the fact that local space conditions can be, and are, studied from Earth with contemporary astronomical techniques.

For example, the Introduction to Navigation booklet in Star Trek Maps (1980) claims that the Cochrane factor multiplies or divides the speed of a warp factor based on the density of matter within a volume of space. According to relativity mass bends space, creating gravity, and a starship in normal space has to travel along the curve of bent space to its destination. But a stars hip in warp drive can travel straight, avoiding the curvature of space, and so has a shorter distance to travel than a starship traveling in curved space would have to travel. Thus the curvature of space multiplies the speed of a warp factor by varying amounts in different volumes of space.

The more interstellar matter there is within a specific volume of space, the more that matter will bend space, and so the more the speed of a warp w factor will be multiplied for a starship travelling at that warp factor within that volume of space.

https://en.wikipedia.org/wiki/Interstellar_medium

The Sun is within a region of low density called the Local Bubble:

https://en.wikipedia.org/wiki/Local_Bubble

Here is a link to a space map of local interstellar clouds:

https://en.wikipedia.org/wiki/Inter...lar_Cloud_and_neighboring_G-cloud_complex.gif

Here is a link to a map on a different scale:

https://www.centauri-dreams.org/2010/02/10/mapping-the-interstellar-medium/

Here is a link to another map:

https://en.wikipedia.org/wiki/Local_Bubble#/media/File:Local_bubble.jpg

And another:

https://en.wikipedia.org/wiki/Local_Bubble#/media/File:LocalBubble.png

As astronomical technology advances, astronomers will be able to make better and better three dimensional maps of the density of the interstellar medium. And it seems to me that there will be a low probability that those maps will agree with the maps of the volumes of space that need to have higher or lower density in order for the voyages in TOS to make sense.

An opposite theory that a starship can travel faster at a specific warp speed in regions of space with lower density of the interstellar medium is likely to have the same problem with the actual density of the interstellar medium in various volumes of space being different from what is required by the theory.

Since the Introduction to Navigation booklet in Star Trek Maps (1980) was published, astronomers have learned a lot about the distribution of "dark matter" in the universe, which is unseen and interacts with normal matter only through gravity.

PART THREE:

Can dark matter rescue the space density explanation for the varying speed of a warp factor?

It is possible that there is dark matter in the disc of the Milky Way Galaxy and that there are small local clumps and voids of dark matter in local interstellar space affecting the curvature of local interstellar space.

If the luminous matter of the galaxy is concentrated in a disc with a radius of 50,000 light years and a
thickness of 2,000 light years, it will occupy a volume of 15,707,956,326,794.9 cubic light years.

If the dark matter halo of the galaxy is spherical and has a radius of 100,000 light years, it would have a volume of 4,188,790,204,786,400 cubic light years, which is roughly 266.683 times the volume of the visible disc of the galaxy.

If about 95 percent of the mass of the galaxy is dark matter, it would be about 19 times as common as visible matter, so if it occupies about 266.683 times as much space as the visible matter it should be about 0.0712456 times as densely distributed in space. So unless dark matter has lots of small scale clumps and voids it is not likely to be a major factor in the density of matter within a volume of space.

And eventually astronomers will be find ways to map the density of dark matter in our local region of the galaxy, and thus it will be possible to compare the density of both visible and dark matter with that necessary for the matter density variation theory to work. And that will probably disprove the the matter density variation theory.

PART FOUR:

The Eddington Experiment.

An arc second is one sixtieth of an arc minute which is one sixtieth of a degree which is one three hundred a sixtieth of a full circle. So an arc second is about 0.0000007 of a full circle. That is a very small angle.

Try drawing a straight line from point A to point C, and try drawing an otherwise straight line from point A to point C that bends by about 1.75 degrees of arc at point B between points A & C. Now compare those two lines. It will be very hard to see the difference between those two lines with the naked eye. And try measuring the distance light travels along the two different paths between points A and C. They will be just about identical distances.

Now try drawing a line that bends at point B by only 1.75 arc minutes, one sixtieth as much as 1.75 degrees of arc. That line will be even harder to tell from a perfectly straight line, and the distance traveled will be even closer to a perfectly straight line.

Now try drawing a line that bends at point B by only 1.75 arc seconds, one sixtieth as much as 1.75 arc minutes. That line will be even harder to tell from a perfectly straight line, and the distance traveled will be even closer to a perfectly straight line.

And that is the deflection experienced by a light ray "grazing the surface of the Sun", where the Sun's gravity is many times stronger and bends space many times more, than the gravitational force and bending of space in interstellar space.

Even if a starshp passed right above the photospheres of countless thousands and millions of stars lined up surface to surface, the shortcut provided by the warping of space by those stars would be only a minute fraction of the total distance the starship needed to travel. It would multiply the speed of the starship's warp factor by only a tiny fraction of a percent, not by tens, hundreds, or thousands of times.

What if the starship at warp passed along the surface of a line of countless millions and billions of black holes almost close enough to touch each other? I haven't done the math, but in that case space right above the event horizons might be bent enough that going straight using warp drive might multiply the speed of the starship many times.

It is theoretically possible that a cosmic string could have similar gravitational and space bending effects as a long line of black holes.

https://en.wikipedia.org/wiki/Cosmic_string

So possibly by the time of TOS a number of cosmic strings have been mapped within the local region of the galaxy and starships often travel to the closest cosmic string and zoom along its surface at many time the normal speed of a warp factor until they come to the point in the string closest to their destination. Or perhaps they may use several strings in
succession in order to reach their destination by a rather zig zag course much faster than if they headed straight toward the destination.

Of course cosmic strings would have to be rather plentiful in this part of the Milky Way galaxy, and I don't know how common they are theorized to be.

https://en.wikipedia.org/wiki/Cosmic_string

Considering how much vaster a Hubble volume is than a tiny little galaxy, cosmic strings could be many times more common than the minimum of one per Hubble volume, without any cosmic strings being within millions or billions
of light years of our Milky Way Galaxy.

Continued:

 

Because those calculations had nothing to do with whether or not they would stay on the air.