Why is there a warp barrier at warp 9.1 and not at warp 9.0?

Question

This question is about this plot from the Star Trek: The Next Generation Technical Manual:

Reference site worth reading

This plot shows energy discontinuities at the integer warp factors warp 1 through warp 8.

Presumably (out-of-universe) this is to justify why Picard often picks round numbers for the warp factor -- it's more efficient and safer. (Related: How does Picard choose a warp speed?)

It also explains why they can have notions (in First Contact and Enterprise) of "breaking the warp barrier", "breaking the warp 2 barrier", "breaking the warp 5 barrier", etc. They are not arbitrary numbers; they represent some real warp physics effect.

But if you look at the plot carefully, you can see that the last discontinuity is not at warp 9.0. Actually, it's around warp 9.1, easily read since the plot already has clear vertical lines at warps 9.0 and 9.2.

Why is there a warp barrier at warp 9.1 and not warp 9.0 ?

Assuming there is no in-universe explanation, perhaps there is an out-of-universe explanation for why someone decided to plot it like this.

Answer 1

Likely a plotting error...

First of all, the black curve that we see is mostly given by the following formula: the warp factor w is related to the velocity v by w=(v/c)^(3/10), where c is the speed of light.

However, Michael Okuda (co-author of the TNG Technical Manual ) altered the relation after warp 9. Here is an email from Okuda, in which he discusses the plot that we see in the question:

Date: Fri, Jan 27, 1995 02:09 AM EST

From: MOkuda

Subject: Re: Star Trek Warp

To: Yar of Spit

The warp factors we've used in ST:TNG were computed in an arbitrary way to fit some specific characteristics we needed.

First, the speed for any given warp factor had to be greater than it was in the original Star Trek series. This was primarily to satisfy fan expectations.

Second, the new warp speeds couldn't be TOO much faster, or it would be possible for the ship to cross the galaxy in a fairly brief time. (In a way, maintaining this restriction made Voyager's story situation possible. If we hadn't done this, Voyager could have gotten home too quickly.)

We used an exponent of (I think) 3.33 or 3.33333... for warp factors less than 9.

Between 9 and 10, I gradually increased the exponent so that it approached infinity as the warp factor approached 10. Lacking knowledge of calculus, I just drew what looked to me to be a credible curve on graph paper, then pulled the points from there. I think I re-created the curve fairly accurately in the Star Trek: The Next Generation Technical Manual.

Hope this helps.

-Mike

(Source)

I believe the slightly off behaviour of the blue curve near warp 9 might be due to Okuda's alteration of the exponent for the black curve. The defining function for the black curve changes at warp 9, which in turn changes the shape of the black curve at the far right of the plot. He probably also changed the shape of the blue curve there too — the black curve is steeper after warp 9 and so there is likely a corresponding change in the blue curve. So just as the black curve is piecewise, I suspect the blue curve is piecewise too, and there is probably something amiss with how the two blue curves were joined near warp 9.

The statement "I think I re-created the curve fairly accurately" suggests that the plot may have been open to some mild error, and the cusp at warp 9.1 may be such an error.

Note: This email was not sent to me (I am not "Yar of Spit"), but the email is publicly available on several, older web sites.