Power levels from TNG: "Masterpiece Society"

[359]

Another thread prompted me to actually get around to doing this instead of just thinking about doing this.

In the episode TNG: "Masterpiece Society" there is a simple quote giving the power output of the Enterprise's warp core, "We have a matter-antimatter warp reaction system, the most powerful in the Starfleet. Normally, it kicks plasma up into the terawatt range. Why?" But then we are privy to other certain events which are impossible when restricted to such 'low' power levels.

Namely the power requirements come from doing things to and around this ball of neutronium.

Its density is stated to be "one hundred billion kilograms per cubic centimeter" which is 1*10^17 kg/m^3, one-quarter that of theoretical neutronium, but close enough to say it is neutronium. From this image it should be close to 4.9 km across. This gives it a total mass estimate of 10^17 * (4/3)*π*(4900/2)^3 6.16*10^27 kg. While that is a lot of matter (many times more than a planet), it's probably not actually enough to maintain the pressure needed to support neutronium.

First we know that the Enterprise's engines are easily expected to move this thing, the limiting factor is the tractor beam and its system's ability to handle enough power to move the fragment. As seen in the image above, the fragment is not too far from the planet, probably somewhere between 600,000 and 1 million kilometers out. Let's assume the fragment will pass by the planet in only another 200,000 km thus causing significant tectonic stresses. We are told directly after that image that the "core fragment is going to pass by Moab Four in six days." Using our assumption of 200,000 km until minimum distance gives us a velocity of 385 m/s for the fragment's speed.

It is later stated that the Enterprise will need to alter the fragment's course by 1.2 degrees in order to save the colony. So the perpendicular velocity needed to achieve a 1.2 degree shift in an object traveling at 385 m/s is 385*tan(1.2°) approximately 8.06 m/s. For a mass of 6.16*10^27 kg this is a kinetic energy change of (1/2)(6.16*10^27)*8.06^2 2.00*10^29 J. The tractor beam was on for a total of two minutes of screen time for an average applied power of 1.66*10^27 watts (approximately 4*10^8 gigatons per second).

Also, at several times the Enterprise was actively maintaining its altitude over the fragment. At an altitude of 6-ish, let's say ten, kilometers and an estimated mass of the Enterprise of 6,510,000 Mg that's a gravitational force of (6.674*10^-11)(6510000000*6.16*10^27)/10000^2 2.68*10^19 newtons. For something with the mass of the Enterprise, this is an acceleration of 4.1 billion meters per second per second just to hold position.

Possible sources of error that could affect the results is the actual size of the giant ball of nutronium. However, in reality it would need to be actually bigger to not destabilize, so thats a moot angle. The estimation of the fragment's speed is questionable, but at only 385 m/s and 48 hours out before necessitating evacuation (66,000-ish km distant), 1.6 degrees isn't going to even one fraction of a wit of difference against that gravitational pull (actually that wit would be roughly 1,300 km but on the scale of greater than planet level gravity, that's not too relevant). So factoring in reality it either has to be further away than estimated (and thus must be traveling faster) or the degree of change must be greater than stated (and thus require more energy to move). Also, actually nutronium is four times denser, but that's not terribly significant on the order of 10^29. Most of the possible sources of error point upward by an order of magnitude or more, however the size estimation error on the fragment would lend itself to smaller values based on camera angles, but within the same order of magnitude.

So, if one were to claim the impossibility of what is shown and decide to go with more realistic estimates, this would only drive up the power requirements. Since there is only one downward source of error which is relatively insignificant, the estimations of power should be fairly accurate.

Tn the same episode we get terawatt level power and... something else level power; ridiculously low and ridiculously high. Not to mention engine accelerations that would push up to c in less than a tenth of a second (ignoring any and all mass gain, mass lightening, and mass whatnots).

Terawatt or petaterawatt, take your pick (but please leave all of the cherries).

[359]

Ah, I knew I had forgotten something.

Another source of error in the power estimate is assuming the scenes are continuous and that the real time was no longer than the camera time. With a greater beam duration the tractor beam's average power output would be decreased. But its still limited to 48 hours until disaster, so not too relevant even then. In fact to get down to the terawatt range the event would need to last on the order of a billion years. Somehow I doubt even the ship floating in space would last that long. :)

[2046]

A fantastic article!

I had always assumed that, realistically speaking, the core fragment should not have been so close to the planet as that image suggests. I am sure that was the intent, but, to put it mildly, having an object 1000 times more massive than Earth . . . six times the mass of Jupiter . . . swinging through our system within hundreds of thousands of kilometers of Earth and at such a pace seemed to me like it would jack up our orbit, plain and simple. We would, briefly or otherwise, become a moon of it.

But now, pondering the numbers, I'm not so sure.

Just to do some rough order-of-magnitude figuring, Jupiter is around six hundred million kilometers away. Your estimates put this thing a thousand times closer. We might not notice Jupiter's gravity from here . . . it's about a hundredth of the gravitational pull of the moon, I gather (10^20 vs. 10^18) . . . but we'd definitely notice it at that range. Move Jupiter a thousand times closer, and its gravitational pull would increase by the square of the distance, meaning, unless I'm mistaken, that it would be a million times greater than it is now (up to 10^24). That would make it have ten thousand times greater pull than the moon, which is also ten times more than that of the sun.

I suppose that maybe a few days of that might not really jack up the orbits too much, but it'd be a helluva trick. The orbital disruption might be minimized if the event occurred in just the right way . . . which is where the redirection might come in most handy, over and above the near passage . . . but it'd still be a neat trick. I'm thinking something on the order of the complication of the orbit-swap relationship of Epimetheus and Janus, here, though I have no idea how to make it actually work in principle.

Part of me still wants to say it would make more sense if the fragment were at a greater range and, by necessity, faster speed. Still, though, the fact is that you're generally correct . . . even if that part of me were right, the true range at passage couldn't be too much larger, otherwise the gravitational force would be too small. It needs to be somewhere north of the moon's pull to account for the tectonic stresses.

Perhaps the most amazing thing would be if they actually got all of this right. Sadly, I don't expect that sort of accuracy out of TNG-era Trek.

[2046]

Also, one more thing that would drive up the acceleration figure is that the ship was, per the side view with the tractor beam that you scaled the fragment from, more like two kilometers from the fragment, if that. And of course, thanks to the tractor beam action and the starfield beyond, it isn't like we can pretend the ship was in orbit or anything.

The only argument I can imagine at the moment is that we shouldn't see stars beyond due to the crazy-brightness we would expect . . . indeed, even the E-D's lit areas should appear dark . . . but that sort of thing happens in every space shot of sci-fi, just about. The mystery cameraman turns the contrast down to -11 or something.

Good show, sir.