TNGTM pg 55 Warp Factor vs Power chart equation

[feld]

Folks,

I’m terribly sorry because I am certain tech fandom knows this but I cannot seem to find it. i would greatly appreciate it if anyone could point me to the math behind the well known “sawtooth” warp factor power consumption graph on pg 55 of the venerable Star Trek The Next Generation Technical Manual.

Note I’m not looking for the well established warp factor vs Cochranes (speed of light multiple) but the sawtooth shaped power graph

Apologies that my Google fu has failed me. I’ve found old posts on this board that duplicate the chart and have their own approximations to it but I wondered if Mr. Sternbach or Mr Okuda ever just posted the equations…

Thanks in advance for any help you can provide.

 

[Kamen Rider Blade]

Spoiler: The TNG:TM Power Consumption : Warp Factor Chart

Yes, I marked it up for my own purposes of measuring where things are.
Too my knowledge there was no formula for the power consumption side relative to Warp Factor.

Spoiler: TNG:TM Manual Power Consumption : Warp Factor in nicer format.

As you can tell, power consumption spikes dramatically as you approach each Warp Factor Threshold before it slides down a bit.
But the over-all power consumption increases is pretty drastic with each Warp Factor IMO.

Spoiler: TNG:TM Power Consumption relative to Warp Factor using my Warp Factor Scale 3.0

The only difference between my Warp Factor Scale & the TNG-era one was that I ignore the hand drawn curve to infinity after Warp Factor 9 and naturally used the TNG era formula and let it run on naturally to infinity.
I've discovered that almost all practical use cases in show could be easily explained with Warp Factor 1.0 - 1000.0, anything faster than that will be too fast.

No need for Wf 9.999… until infinite 9's needs to be mentioned, natural numbers can be easily uttered by actors instead of mentioning how many 9's after the decimal seperator.
It's a far simpler system with no BS hand-drawn curve to infinity after Warp Factor 9.0.

Spoiler: Warp Factor to Power consumption curve in ST:ENT

Here it is used in ST:ENT

 

[knightgrace]

AND what little that I can add, is that warp factors are treated in TNG, in particular as the gears in an automatic nine speed transmission in your car...

Which raises another question...

What would a one speed transmission mission for a starship be like? Remember that this is a metaphor. What it means is that as you approach the next gear shif, the amount of power to get to the next gear(warp factor), will drop...in a car or bus, by about six hundred rpm. In a semi, this occurs at about 1,650 rpm... in a bus, about 2,200 to 2,400 rpm.

Keep in mind that a semi today has 13 to 18 speed transmissions.


One of the possible issues is that a Shuttlecraft might have a two speed, covering from 1c to...let's say 3053c+.

A small actual ship, might have a a six speed transmission covering from 1c to 3053c+.

The Enterprise-D, being so massive requires a nine speed.

Keeping further in mind that the highest gear handles all speeds above a certain x light speed. My car, reaches sixth gear around forty miles per hour. The semi that I drove reached eight high at 55 miles per hour. But cars top out around forty...why?

Because of the lack of load. Which means that two speed is valid.

 

[feld]

Spoiler: The TNG:TM Power Consumption : Warp Factor Chart

Yes, I marked it up for my own purposes of measuring where things are.
Too my knowledge there was no formula for the power consumption side relative to Warp Factor.

As you can tell, power consumption spikes dramatically as you approach each Warp Factor Threshold before it slides down a bit.
But the over-all power consumption increases is pretty drastic with each Warp Factor IMO.

The only difference between my Warp Factor Scale & the TNG-era one was that I ignore the hand drawn curve to infinity after Warp Factor 9 and naturally used the TNG era formula and let it run on naturally to infinity.
I've discovered that almost all practical use cases in show could be easily explained with Warp Factor 1.0 - 1000.0, anything faster than that will be too fast.

No need for Wf 9.999… until infinite 9's needs to be mentioned, natural numbers can be easily uttered by actors instead of mentioning how many 9's after the decimal seperator.
It's a far simpler system with no BS hand-drawn curve to infinity after Warp Factor 9.0.

Spoiler: Warp Factor to Power consumption curve in ST:ENT

Here it is used in ST:ENT

Ok thank you. Yours was the main post I found on this topic. Did you fit the power per cochrane curve using a least squares or something? Or just pick off enough points to reproduce it for your own use? Only reason I bring it up is that the out of universe note in the TM on the same page mentions Mr. Okuda using a Excel spreadsheet to get “speeds and times” and I wondered if they’d made some kind of crazy function for the power as well.

If you haven’t already (who am I kidding…everyone else in Trek tech fandom has probably done this before me) go back calculate how much antimatter it’d take to do warp six for three years. If I did my math right you’ll find that it’s pretty close to the TM’s 3000 m3 antimatter tankage for a Galaxy class…which they state lasts for a “standard mission duration” of three years. I assumed slush antideuterium had twice the density of slush hydrogen (I used 170 kg/m3). That implies the authors did that math.

I work in aerospace and am continually impressed by some of the details in this book. Have been increasingly so for 34 years.

Of course…given that this all holds together…with a few assumptions we can calculate time to fuel exhaustion for a Galaxy for any warp factor profile…always acknowledging that warp factors are less a measure of how fast you’re going and more a measure of how hard you’re pushing…like a ship’s turns per knot ratio on its propellers. It doesn’t mention how hard space pushes back…

Given a few MORE assumptions you could estimate hotel load, how much combat time the ship would have left, how many fully loaded photorps she could fuel…lotsa stuff.

And if you really wanted to go nuts you could do this for any ship by assuming some relationships between starship mass and or volume and the power per cochrane…

Yeah…this way madness lies…but I like this kind of madness…

 

[Kamen Rider Blade]

Ok thank you. Yours was the main post I found on this topic. Did you fit the power per cochrane curve using a least squares or something?

No, I didn't go that far.

Or just pick off enough points to reproduce it for your own use?

I just wanted to see what part of the Warp Factor Scale did the Power Consumption Curve start increasing drastically and it was close to the half way points between Whole Number Warp Factors.
After that, it seemed to fall off back to the original slope.

Only reason I bring it up is that the out of universe note in the TM on the same page mentions Mr. Okuda using a Excel spreadsheet to get “speeds and times” and I wondered if they’d made some kind of crazy function for the power as well.

I never found any formula within the TNG:TM for the power consumption, I'm assuming it's a "hand-drawn curve" just like the "hand-drawn curve" to Infinity after Warp Factor 9.0 in the TNG era formula which is silly & non-sensical IMO.

If you haven’t already (who am I kidding…everyone else in Trek tech fandom has probably done this before me) go back calculate how much antimatter it’d take to do warp six for three years.

Actually I haven't done that, so if you could do it, be my guest.

If I did my math right you’ll find that it’s pretty close to the TM’s 3000 m3 antimatter tankage for a Galaxy class…which they state lasts for a “standard mission duration” of three years. I assumed slush antideuterium had twice the density of slush hydrogen (I used 170 kg/m3). That implies the authors did that math.

Interesting. I'd love to see your calculations and work for that if you don't mind.
What Cruise Speed did you use for the Galaxy Class?
Memory Alpha only lists a intial Cruise Speed of Wf 6.0

I work in aerospace and am continually impressed by some of the details in this book. Have been increasingly so for 34 years.

What part of AeroSpace do you work in?

The TNG:TM along with all it's details was designed by those who had a similar mentality.
Design a vessel that's plausible given it's technological capabilities within the storyline universe.

 

[FalTorPan]

I’m terribly sorry because I am certain tech fandom knows this but I cannot seem to find it. i would greatly appreciate it if anyone could point me to the math behind the well known “sawtooth” warp factor power consumption graph on pg 55 of the venerable Star Trek The Next Generation Technical Manual.

I doubt that any actual equation was used in the creation of the sawtooth curve. It was probably just a curve drawn to provide a rationalization for why Star Trek ships seem to travel at sustained speeds that are almost exclusively integer warp factors (Example: Warp Factor 4), and almost never at non-integer warp factors (Example: Warp 3.5) when the speed difference between consecutive integer warp factors is so vast.

 

[feld]

I work in space nuclear technology at NASA. My employer, as I hope is obvious, doesn’t endorse Star Trek or my musings thereon. Just so my fellow taxpayers don’t think I’m wasting their time. But like many in aerospace Star Trek changed my life. I got a physics degree, became a U.S. Navy nuclear submarine officer, and then joined NASA in no small part due to the influence of TMP on an impressionable young boy.

I will certainly show my math here but I need to write it out fully and check my work. First time I did it it was from memory on my phone late at night holding a sick kid. I probably made some dumb math error.

I’m pretty sure an equation COULD be written to duplicate the power per cochrane curve. My whole purpose for this thread was to see if Mr Okuda or Sternbach had done that. Seems like the curve was hand drawn. Next step is to digitize the curve and pull points off for future use. Eugene’s Law doesn’t bother me so I don’t mind the infinite power at the end. I know why they did it.

 

[Kamen Rider Blade]

Eugene’s Law doesn’t bother me so I don’t mind the infinite power at the end. I know why they did it.

The TOS Warp Factor formula doesn't have a hand-drawn curve to infinity.
They let the formula naturally run along the numbers to infinity.

The TNG Warp Factor formula is fine IMO except for the hand drawn curve to infinity after Warp Factor 9.0.
Which is silly IMO.

Combine the best of both worlds (No Hand Drawn curves to infinity & TNG era Warp Factor formula), and you have a perfect formula to measure things.

So that's my personal take on it.

 

[feld]

Ok so I’ve owed a calculation of fuel consumption for a Galaxy class starship to this thread for a little while. It is stated that the Galaxy’s 3000 cubic meter (m3) is enough for a normal mission period of three years. It is elsewhere in the book stated that the ship’s propulsion system is most efficient at warp 6. From the famous warp table on pg 55 that’s 392 cochranes.

Looking at the “sawtooth” part of the curve that is supposed to indicate power per cochrane I’m going to silently correct its units to megawatts per cochrane and pick off the power per cochrane figure for Warp 6 (call it 3.1x10^6 MW/cochrane = 3.1x10^12 W/cochrane).
Multiply that by 392 cochranes to get power consumption for warp drive alone:

392 cochranes x 3.1x10^12 W/cochrane = 1.22x10^15 W

The Galaxy’s design requirements earlier in the book state that the ship’s warp coil efficient needs to meet or exceed 88% up to Warp 7. I’m going to assume they can do 90% efficiency at Warp 6.

1.22*10^15 W / 0.9 =1.35×10¹⁵ W

So matter/antimatter is good for 9 x 10^16 J/kg but it’s important to recall that this is for a 1:1 intermix ratio and that only half of that is antimatter. Now the Galaxy only goes to 1:1 intermix at Warp 8 but this need not concern us because we’re just figuring out how long the thing can keep up Warp 6 and the antimatter tank is the limiting factor. Recalling that a watt is a joule per second. Either way the amount of antimatter you need per second to reach that power figure above is:

(1.35 x 10^15 W) / (9 x 10^16 J/kg) / 2 = 0.0075 kg/s

The Galaxy’s total antimatter supply is 3000 m3. The antimatter used is slush antideuterium. Now antideuterium is expected to have the same density as deuterium. Deuterium’s density is about twice hydrogen’s for our purposes but the slush part complicates things a bit. I poked about a bit on the internet at settled on 170 kg/m3 for anti deuterium density. That means the Galaxy has:

3000 m3 * 170 kg/m3 = 510,000 kg of antideuterium

At the burn rate calculated above we’ll go thru that in:

510,000 kg / 0.0075 kg/sec =
68,000,000 secs / 3600 sec/hr / 24 hrs/day / 365 days/year = 2.16 years

So if this antimatter load is supposed to last for 3 years then Galaxy could be going warp 6 for 2 of those three years. This doesn’t seem like a crazy duty cycle to me. I can totally see the ship averaging taking two weeks to get somewhere, then spending a week in orbit basically burning very little fuel, and then repeating the whole process. Worth noting that a couple of weeks at warp six on the TNG scale is about 15 light years.

When I first did this math in my head I got a little over three years so I might have dropped a factor of two somewhere but I’ve checked this figure twice and I cannot find an error.

Anyway…there you go. Someone had asked to see my math and there it is. Sorry for the wall of text.

If I get a minute I can digitize that warp power per cochrane graph and should be able to pretty quickly write an excel spreadsheet to give you fuel consumption rates for any warp factor for the Galaxy class Enterprise. I think this would make a great fan calculator on Star Trek Minutae or Daystrom Institute…but I just enjoy seeing that they appear to have done at least some of this math.

 

[Kamen Rider Blade]

Ok so I’ve owed a calculation of fuel consumption for a Galaxy class starship to this thread for a little while. It is stated that the Galaxy’s 3000 cubic meter (m3) is enough for a normal mission period of three years. It is elsewhere in the book stated that the ship’s propulsion system is most efficient at warp 6. From the famous warp table on pg 55 that’s 392 cochranes.

Looking at the “sawtooth” part of the curve that is supposed to indicate power per cochrane I’m going to silently correct its units to megawatts per cochrane and pick off the power per cochrane figure for Warp 6 (call it 3.1x10^6 MW/cochrane = 3.1x10^12 W/cochrane).
Multiply that by 392 cochranes to get power consumption for warp drive alone:

392 cochranes x 3.1x10^12 W/cochrane = 1.22x10^15 W

The Galaxy’s design requirements earlier in the book state that the ship’s warp coil efficient needs to meet or exceed 88% up to Warp 7. I’m going to assume they can do 90% efficiency at Warp 6.

1.22*10^15 W / 0.9 =1.35×10¹⁵ W

So matter/antimatter is good for 9 x 10^16 J/kg but it’s important to recall that this is for a 1:1 intermix ratio and that only half of that is antimatter. Now the Galaxy only goes to 1:1 intermix at Warp 8 but this need not concern us because we’re just figuring out how long the thing can keep up Warp 6 and the antimatter tank is the limiting factor. Recalling that a watt is a joule per second. Either way the amount of antimatter you need per second to reach that power figure above is:

(1.35 x 10^15 W) / (9 x 10^16 J/kg) / 2 = 0.0075 kg/s

The Galaxy’s total antimatter supply is 3000 m3. The antimatter used is slush antideuterium. Now antideuterium is expected to have the same density as deuterium. Deuterium’s density is about twice hydrogen’s for our purposes but the slush part complicates things a bit. I poked about a bit on the internet at settled on 170 kg/m3 for anti deuterium density. That means the Galaxy has:

3000 m3 * 170 kg/m3 = 510,000 kg of antideuterium

At the burn rate calculated above we’ll go thru that in:

510,000 kg / 0.0075 kg/sec =
68,000,000 secs / 3600 sec/hr / 24 hrs/day / 365 days/year = 2.16 years

So if this antimatter load is supposed to last for 3 years then Galaxy could be going warp 6 for 2 of those three years. This doesn’t seem like a crazy duty cycle to me. I can totally see the ship averaging taking two weeks to get somewhere, then spending a week in orbit basically burning very little fuel, and then repeating the whole process. Worth noting that a couple of weeks at warp six on the TNG scale is about 15 light years.

When I first did this math in my head I got a little over three years so I might have dropped a factor of two somewhere but I’ve checked this figure twice and I cannot find an error.

Anyway…there you go. Someone had asked to see my math and there it is. Sorry for the wall of text.

If I get a minute I can digitize that warp power per cochrane graph and should be able to pretty quickly write an excel spreadsheet to give you fuel consumption rates for any warp factor for the Galaxy class Enterprise. I think this would make a great fan calculator on Star Trek Minutae or Daystrom Institute…but I just enjoy seeing that they appear to have done at least some of this math.

On a slightly different but related tangent:

The Ironic thing is that Geordi stated in New Ground stated that the 'Power Efficiency' of the energy of the Soliton Wave converted to the Soliton Wave Rider to go to warp was 98%

La Forge reports that the ship's speed is warp 2.35, slightly faster than they expected. Picard asks if the wave is affecting the Enterprise's warp drive, but Data replies that it is not. La Forge reports that the power efficiency of the wave is 98 percent. Data remarks that is 450% more efficient than their own warp engines.

That statement to me implies that however the Enterprise is converting the energy from the EPS to the Warp Coils, they have about 21.7…% energy conversion efficiency for the Galaxy Class Enterprise-D.

BTW, I sent you a PM, it's not related to this topic.

 

[feld]

The EPS is a separate plasma circuit energized by magnetoplasmadynamic taps across the PTC’s. I *think* the efficiencies mentioned above are revenant ones from the TM for the warp coils. They’re found in the ship’s requirements in the first section.

Contrary to most fan practice I am happy to ignore stuff on screen and all I’m trying to do is see how the ship described in the TM might work. It’s all I can do because the on screen data was written by caffeine inspired writers on tight deadlines who had better things to do than calculate warp efficiency.

 

[Kamen Rider Blade]

The EPS is a separate plasma circuit energized by magnetoplasmadynamic taps across the PTC’s. I *think* the efficiencies mentioned above are revenant ones from the TM for the warp coils. They’re found in the ship’s requirements in the first section.

Contrary to most fan practice I am happy to ignore stuff on screen and all I’m trying to do is see how the ship described in the TM might work. It’s all I can do because the on screen data was written by caffeine inspired writers on tight deadlines who had better things to do than calculate warp efficiency.

Fair Enough.

 

[DSG2k]

As an aside, a fellow on Twitter brought up "Demon"[VOY] and the TNG TM chatter about the Bussard collectors. A point of whether the Bussards should totally refuel the ship's deuterium tanks versus merely aiding in fuel mileage.

I ended up doing some back-of-the-envelope stuff, which I'm paraphrasing here:

Even casting a big net and trawling hella-fast, I can't imagine the ramscoops perfectly replenish the tank. Feld's calculation necessitates 7.5 grams/second antimatter at warp six per the TNGTM ... the deuterium input would be more, still, since warp plasma is the output. For ease, let's just say that the deuterium needs would be 10 grams/second for standard cruising speed.

We can figure out how wide a net we'd need to cast assuming replenishment, or we can estimate a net. Let's try the latter.

Some estimates of the interstellar medium are as low as 1 atom per cubic centimeter. That's a million atoms of hydrogen per cubic meter. (Note, of course, that this is just hydrogen and not deuterium, but we're rolling with it.)

Let's say the Bussards cast a net of 100 square kilometers ( which seems hella-large, since it would involve casting a magnetic field out to 10 kilometers laterally from the ship, but maybe that's not abnormal). At a velocity of 1 kilometer/second, then, we'd have 100 cubic kilometers of space scooped per second. That's 100,000,000,000,000 atoms per second.

How many grams is that? Well, Avogadro is underwhelmed at that point . . . the gram figure is gonna need a really small SI prefix. However, all we need to do to fix that is go faster. At 6,022,140,760 km/s, you should get to a gram per second, assuming perfect collection.

See, no problem, it works, right? Ah, but that velocity is 20,000c, or almost warp 9.9 per Tom Paris in "The 37's"[VOY2]. At TNG TM speeds, warp six is low-hundreds of lightspeed.

 

[Sgt_G]

In my not-so-humble opinion, the idea of Warp 10 == Infinite Speed is wrong. The chart clearly states that you can't reach Warp 10 because the energy required is near-infinite. Even that defies logic. I accept that it requires more energy than current technology can deliver, not that it's "infinite".

 

[publiusr]

“There be dragons” ….of the gaps