Diburnium Density vs. Phasers

[2046]

Summary: Phasers do have a problem penetrating dense materials. It's just that the density of the material in question is absolutely crazypants high.

----------------------------

In "That Which Survives" the Enterprise encounters a peculiar, unnamed world. Described as hollow later in the episode (and full of replicas of one of the actresses to play Catwoman in the 60's), we are told the Catwoman planet reads as only thousands of years old, is approximately the size of Earth's moon, yet has a mass and atmosphere comparable to Earth's.

SPOCK: I suggest no supernatural explanation, Captain. I merely point out that the facts do not fit any known categories of planets.
KIRK: Let's take the facts one by one. That should explain something.
SPOCK: Undoubtedly. The age of this planet would seem to be only a few thousand years. It would be impossible for vegetation to evolve in so short a period.
KIRK: Its size is approximately that of Earth's moon.
SPOCK: But its mass and atmosphere are similar to Earth.
KIRK: That would be difficult to explain.
SPOCK: It would be impossible, Captain. An atmosphere could not evolve in so short a period of time.
KIRK: And yet it has.
SPOCK: Evidently. But the inconsistencies are so compounded as to present a seemingly impossible phenomenon.
KIRK: But a fascinating one, Mister Spock.
SPOCK: Precisely, Captain.

The basic substance the world is made of is later identified as a diburnium-osmium alloy which couldn't have happened naturally (and indeed the world was an outpost of the Kalandans, led by Losira, that died out thousands of years prior). The material looks like pink rock on the rocky world and even exists just under a thin layer of topsoil, where there is any soil. However, when Kirk tries to dig a grave for an unfortunate crewman using his phaser, he discovers to his surprise that it doesn't work. With thanks as always to Chakoteya.Net:

SULU: That's the same red rock.
KIRK: My phaser didn't cut through it.
MCCOY: Whatever it is, it has a mighty high melting point.
KIRK: Eight thousand degrees centigrade. It looks like igneous rock, but infinitely denser.

In other words, Kirk seemed to believe the alloy had a melting point of 8000 degrees, presumably because it would be somewhat superior to what is phaser would be expected to generate. And further, while it looked like simple igneous rock (which has an average density of around 3000 kg/m^3), Kirk believed it was "infinitely" more dense ... an exaggeration, naturally, but not the sort of thing one would say in lieu of 'just a wee bit'.

(Kirk does use the term that way, after all. In "Errand of Mercy" he compares a possible fate to living under Klingon occupation and says they'd be infinitely better off. In "Whom Gods Destroy" Kirk claims to Garth, who is trying to get a passphrase from Kirk, that there are an infinite number of countermoves to a particular move in 3-D chess, a game with X pieces and Y possible squares to inhabit.)

Note also Kirk's surprise at not cutting through the 'rock'. He clearly expected his brief shot to either cut the grave or at least get it good and started, not merely pock-mark the surface (and in a rectangle, no less!). But instead, the lieutenant is buried beneath some other gray rocks scattered about the surface.

While the Catwoman planet is identified as being hollow, we are not told any details of the depth of the alloy layer. However, we can get some estimates for the planet as a whole. After all, if it is approximately the size of the moon then it is around 3500km in diameter, or around a quarter that of Earth. But the real story is in volume. The moon's volume is about 2.1958E10 km^3... a mere two percent of the volume of the Earth.

This means that for the planet to have the same mass as Earth, it must be about fifty times more dense than the average density of Earth. Earth's total density is about 5,500 kg/m^3, meaning the Catwoman planet density should be somewhere in the range of 275,000 kg/m^3!

Um, yeah. That's a lot. Like, crazypants a lot. That's 275 times denser than water, and almost 100 times denser than the igneous rock average from earlier (making Kirk's "infinitely" close enough). It's also 35 times denser than steel, and almost twice the density at the core of the sun!

Suffice it to say, you won't see an element of that density anywhere on the periodic table, or a material like that on modern Earth outside of a collider. And the scary thing is that it is alloyed with osmium, one of the densest elements known at a 'mere' 22,500 kg/m^3, twice the density of lead. If the ratio of diburnium and osmium was 50/50 by volume, diburnium would be almost twice as dense as our 275,000 kg/m^3 figure! And that doesn't even touch on the hollowness.

(Also note that osmium's melting point is about 3000 degrees, rather less than 8000. Indeed, the highest known melting points top out around 4200 degrees, which is the melting point of tantalum hafnium carbide. In other words, Kirk thinks his phaser ought to be able to melt (or worse) a grave-sized amount of anything we know of today.)

It seems little wonder, then, that Spock marveled at the material used by the Kelvans in "By Any Other Name":

SPOCK: Fascinating material. Similar to diburnium, but considerably more dense. I doubt even phaser fire could disturb its molecular structure.


SCOTT: This is it. Now all we have to do
SPOCK: Whatever it is we must do, it is impossible.
SCOTT: Why?
SPOCK: This material surrounding the projector is the same as I discovered on the planet. Readings indicate it is impervious to any of our weapons. We cannot penetrate the casing to get to the machine.

If the material is considerably more dense than 275,000kg/m^3 (or even higher depending on the osmium content on the Catwoman planet), then it is no wonder the phasers couldn't begin to touch it! If it was a cool million kilograms per cubic meter, it would only be 1000 times less dense than a white dwarf!

There is, however, one problem with the information we are provided.

The Moon has one-sixth of Earth's gravity ... 16% or so ... despite only having 1.2% of its mass. The reason the gravity is so high is that the radius is so small. At Earth's radius ... that is to say, if you could make a massless surface to stand on at a distance from the core equal to the distance of Earth's surface to Earth's core ... the gravity would be 1.2% of Earth's, or around 1/83rd instead of 1/6th.

Catwoman's planet, meanwhile, is said to have Earth's mass and the moon's size. That means the surface gravity should be over 13g. (And lest one be tempted to think that things will be different because the world may be hollow, as I have seen some commenters online succumb to, bear in mind the shell theorem.)

One easy solution would be to have the world be in a tidal lock with a nearby massive but dim sun, perhaps something like a brown dwarf, but unfortunately this won't work . . . the planet has at least a partial day-night cycle of greater and lesser studio lighting and sky color. We might then suppose instead a massive, dense moon in a geosynchronous orbit ... this works rather better, inasmuch as it keeps the light levels as high as they were observed in the supposedly night-time scenes, though of course there is no mention of a moon nor is it spotted in Enterprise orbit scenes. However, the moon would have to be even more interesting than the planet itself to have enough gravity at a distance to subtract up to 12g.

In any case, the presence of an Earth-normal atmosphere as Spock also mentioned does constrain us. Higher gravity equals a higher atmospheric density, but if the atmosphere is Earth-like then the gravity should be, as well. We can fudge a little here and assume he simply meant the atmosphere was the same at the surface, meaning it was several times thinner under normal gravitational circumstances, but then we're still faced with the gravity problem of Kirk and company running around effortlessly while weighing something like 2000 pounds, a feat Shatner wouldn't perform again until Generations.

Even fudging the numbers a bit ... 1.33 Moon radii and 0.66 Earth masses, for instance ... we still come out just shy of 5g. And to match the visuals I'd want no more than 1.5g on the highest end. Even pretending the densest part of the planet is on the opposing surface (thus faking 2 Moon radii out of it, despite whatever odd atmospheric effects this might cause), we still only get down to 2.2g.

The gravity problem is significant, and we are constrained to the idea that whatever causes the beamdown site to have 1g should be a permanent phenomenon. After all, when the Enterprise was thought lost, Kirk's concerns were food and water, not the notion of having something happen that would subject them to 13+ times Earth's gravity at some point.

It seems to me the best solution is to add a little extra complexity to the stellar dwarf idea, such as the dwarf itself having a dim side that allows for the brighter/darker cycle. This has one delightful side benefit . . . it also would mean that we could say the Catwoman Planet orbited the Riddler Star . . . here referring to the fact that the guy who played the Riddler in the 60's also played one of the last two folks from Cheron from "Let That Be Your Last Battlefield", a world where the people's skin was white on one side and black on the other.

The benefits here are tremendous from an explanatory perspective. For instance, Spock specifically noted a mass and atmosphere similar to Earth's. Earlier we tried to fudge this last point, but if we don't fudge it then we ought to have a planet with an atmosphere several times denser than ours at the surface. But if a significant area of the planet shows much less gravity due to the presence of a nearby star to which it is tidally locked, then a large part of the atmosphere would be subject to Earth-ocean-tidal-esque forces and perhaps not seem so dense. The only possible glitch I see is that I have no idea what this would do insofar as wind patterns.

Should you find the Riddler Star idea distasteful, bear in mind that most alternatives are unpleasant, either from Occam's perspective or in other ways. We could suppose all sorts of off-the-wall ideas, such as a localized or near-surface 'natural' anti-grav effect (one would think that would have been worth a mention), or one projected from the ship (a temporary condition, one would think, after the ship's abrupt departure, and an incredible can of worms to open ... not to mention the fact that the tremor felt at beamdown would've had an obvious cause, rather than being a mystery). There could've been an also-beamed-down gravity modifier (same notation), or a conveniently invisible bio-augmentation-handwavium apparatus worn by the crew (which would presumably be a doodad that could run out of gas, making it another of Kirk's concerns ... see also "Melora"[DS9]). Another idea is of a subspace field of some kind, with Spock having reported the unadjusted mass . . . but the planet is specified to not even have a magnetic field, so one would think a high-energy subspace field would have been worth a note.

The only other alternative I can see is to simply dismiss the statements of Spock, Kirk, Sulu (who identified the planet's composition), and others, treating them all as dim-witted fools who didn't know what the devil they were talking about.

I'm generally no fan of this analysis method. It may be the standard approach by Star Wars tech inflationists, but that doesn't mean it's any good, and indeed the opposite is indicated. Their approach is to disregard Trek character statements at the first sign of an excuse to do so, so that they may then insert their own claptrap unimpeded. After all, they became SW tech inflationists in order to twist the fusion and steel of Star Wars canon into something that would "make it a better comparison to Trek", so it only follows that they would be Trek deflationists at the same expense of canon.

Literally, you'd have to have Spock getting the mass wrong, Kirk getting the reason his phaser had an issue wrong, Sulu being wrong about the planet, and even McCoy and D'Amato saying some dumb things in there. And, of course, they'd all be working from balky sensor equipment on ship or in hand. That just doesn't mesh. If the entire planet's gravity was 1g, then the mass of the world would be 7.5% Earth's mass, makng the density fairly close to osmium. So Kirk's line about "infinitely denser" would actually only reflect a difference of six or seven times, and Spock, who spends the entire episode badgering people about precision, would be off by 92.5%.

In other words, you have to throw out all sense of any reliability in Trek, making the entire Trek tech analysis field a pointless enterprise (even moreso than just because we should all go outside and soak in some sunshine).

That's a bad idea anyway, but also for another reason . . . the things you'd be tossing out are all consistent with each other. Kirk's analysis, Spock's report, and so on all point toward the same sort of densities and physical properties. Only the observed gravity at the beamdown site suggests anything different.

And the place obviously has a sun of some sort, so it becomes the most likely explanatory culprit.

[Mr. Oragahn]

I think that TOS does suffer from more absurd levels of bad science. We do have the super-decibel ground-to-orbit sound gun after all, the one with a power worth so many Death Stars that I can't care to remember the multiplier exactly, and the Enterprise sort of tanking that ... in space.

There's also another problem with the tidal lock idea and the massive net gravitational differential.
If on one side (and more like in a limited region at the center of the proper side), it would be Earth like because of the pull of some massive star, then on the other, what compensates for the insane gravity would suddenly add.
Geology issues aside, it is true that we could imagine the works on fluids (air, water - there's no magfield so perhaps we can forget magma here) would be insane. Tempest, I think, would probably not even begin to describe the mess you'd have on a daily basis.
The total pull (planet + star) on one side would be so massive that air alone might be crush beyond the speed of sound.

Perhaps there's a solution. A tidal lock, with a side always facing the star, and such a difference of final gravity on the surface, might have moved most of the atmosphere on one side in some kind of asymetrical and perhaps very stable state after all.
We'd still have a "belt" of air (somewhere at the terminator of that moon, where the ground would be perpendicular to the star's direction) that would still be compressed by the moon's own gravity and pulled sideways starwards (what-a-term), but even this circular region encircling the light/dark rim of the planet may very well be maintained in place.

This stability would be even more necessary since scientists have recently found that the mere friction of flux of fluids such as an atmosphere circling a planet does have an influence on the planet's spin!

The other point I'm not so sure of is of how close to a nearly dead star the moon-world would need to be for this already insane surfacic gravity to be almost nullified down to ~1g.
I'd suspect very close... too close in fact. And the moon would need to circle the star very, very fast, to count on centrifugal force to keep it orbitally locked.

And then there's the idea of the star having a darker side... ?
Plus if the star had such a feature, the moon would probably experience insane frequency shifts between "night" and day.

Eventually, I'd have suggested a binary star, with one being rather dim but large enough to mask the sister's light.

Or is the moon-sized planet a planet's moon? Assuming the moon would be miraculously locked facing the local star(s), the only source of real night would be the planet's shadow.

Frankly, all these theories require excuses and really far fetched excuses, perhaps half of them not even working.

I'd rather go with the small planet being an artificial construct, or having something in its core that generates an artificial gravity. Perhaps some initially "diluted" red-matter generated core (which would probably be a singularity now), either natural or artificial, something that would constantly generate a pull towards the crust, but somehow not destructive enough to have destroyed said crust.

A red-matter produced singularity, which is already a super piece of non-science there, generates such a pull that even a starship can get free of it!
Here, we don't need that, hence the diluted "additive".
Plus the planet being a sphere, any pull corewards would have a hard time breaking the crust, especially if artificial, and would have in fact the surprising effect of strengthening said crust.

Also, all the theories are nice and fancy, but such insane setups would have probably be worth a remark or something.

[Darth Spock]

Ohhhhhh yeah, thank you 2046 for reminding me why I allowed myself to even consider such a frivolous hobby. A fun, goofy motivator to move my arse and work my brain, do some research and relearn stuff I skimped on in my youth, heh. It's a sad thought, but I wonder how many people stay stupid because they're afraid to admit they need to learn. Getting to the point, reading over those points it does seem to present a real mucky situation, yes? As much as I like the fun coincidences of a Catwoman Planet orbiting a Riddler star, if the visuals retain a significant degree of canon reliability, the opening views of the planet seem to indicate that the planet is actually rotating, and that the clearly defined terminator moves over the surface perpendicular to, and in direct relation to this rotation. Ouch. I think that kills the idea that the planet is tidally locked with its star, and the possibility of the "ghost planet" actually being a satellite with night caused by its planet's shadow as well, an idea I liked for a bit too, Mr. Oragahn.

One other idea I considered, was that the gravity could be offset by extreme centrifugal force, the planet does seem to be rotating excessively fast in the opening scenes, though I haven't yet actually tried screen-capping it to estimate its rotation. I'm pretty sure most planets from the TOS special effects rotated too fast anyway.... Of course, such insane rotational speed is going to make its own problems too. One of which, in my preliminary estimates, would make night and day come so fast so as to require the entire episode take place in less than 20 minutes to match the sky we saw on the surface. Still, I wonder if a combination of factors couldn't whittle things down a bit. Wiggle the size and mass a bit, bump up the centrifugal force, maybe there is another orbiting body present, etc.

Mental gymnastics aside, remembering this is a fictional universe with loads of preexisting magical technology to cure all ills, I can't help but think there is a way to take Occam's razor and shave this hairy beast down a bit. Reviewing the dialogue, we learn that the mystery planet is roughly the size of Earth's moon, but it's mass and atmosphere are similar to Earth. No mention is made of gravity, but under the circumstances, that the atmosphere is already similar to Earth, they may be taking this fact to indicate that surface gravity is likewise Earth-like, hence their willingness to beam down. The fact that the surface should not exhibit these attributes could be included among the factors that make this planet so compelling, despite not having been explicitly mentioned in the dialogue. Nevertheless, it wouldn't stop them from taking advantage of the situation and beaming down for a closer look.

Clearing the Enterprise crew of any mistakes, what is there to bar the existence of anti gravity technology on the planet itself? We've seen from references elsewhere that artificial gravity, and I would in turn assume anti-gravity, generally has a localized effect, thus it would not necessarily interfere with their estimate of its mass from orbit. While such a field would be worth a mention, it is entirely possible they missed it during preliminary scans. Note that the existence of the super computer and its power source within this "hollow" planet was originally missed.

Which leads me somewhere else.... While the statement about the planet being hollow struck me as more of an off the cuff remark, its odd inclusion, coupled with the massive power of this "station" gets me wondering. Dyson spheres, Romulan Warbirds, Hirogen communication modules.... You see where I'm headed here. Yeah, you're gonna hate me for this, but may as well get it out of the way. An artificial singularity sounds like a likely power source for a several thousand year old artificial planet capable of teleporting the Enterprise almost a thousand light years away. And they're really dense.

This might be a good time to point out that I am not in the "phasers suck" club. Also, D'deridex-class ships show that functional power generator singularities aren't always that dense, and the exact depth of the planets thin candy shell all leave plenty of room crazypants material density. But, in light of the title of this thread, I'm inclined to say that this is not the slam dunk you're looking for.

[2046]

The Remastered version's purplish planet does not visibly rotate when contrasted against the moving starfield behind it, so near as I can tell.

And I would think the presence of any sort of anti-grav field would've borne mention, especially with the confusion about the earthquake on beam-down. I mean, we heard them mention the place had no magnetic field ... surely something far less mundane (and deadly if it disappeared) would warrant a note of concern regarding its permanence. I still think only permanent solutions fit the logic of the episode.

I think the hollowness part was perhaps somewhat overstated in the episode, coming as it did after "For the World Is Hollow...". I mean, if I built a planet I'd certainly want a kickass zero-gee central party-room, but there's really no evidence of it here.

I say that because the central computer was in a cave into which Spock and Mr. Quick-On-The-Trigger could beam . . . if it was kilometers deep I would not expect that could've occurred, especially through that stuff. I'm kinda surprised they could even beam into it at all.

[Trinoya]

Whoa, I haven't examined a phaser figure like this in a long time...

Nice catch!

Also: Since he quotes the melting point at 8,000 and we can therefore assume TOS hand phasers put out less than that figure... we end up somewhere at double the melting point of Tungsten.

This could go a long way to explain why the hull of a Galaxy class was able to withstand to survive 12,000 degrees Celsius.

... still whoa... that's um... This makes the whole 'phasers not setting everyone on fire nearby' thing that much more odd.

[Mr. Oragahn]

Just how hollow is that planet?
And what is that computer? Did we see it? Was it big? Or did we only see an interface, a console of some sort, with the "cpu" unseen?

[Who is like God arbour]

• • • • • • [...]
        
        
        • • SULU:
        
        That's the same red rock.
        
        • • KIRK:
        
        My phaser didn't cut through it.
        
        • • MCCOY:
        
        Whatever it is, it has a mighty high melting point.
        
        • • KIRK:
        
        Eight thousand degrees centigrade. It looks like igneous rock, but infinitely denser.

In other words, Kirk seemed to believe the alloy had a melting point of 8000 degrees, presumably because it would be somewhat superior to what is phaser would be expected to generate.

• • • • • • [...]

I do not understand this as though as Kirk means that the "red rock" has a melting point of 8.000 degrees centigrade. He didn't know the properties of the "red rock" to ascertain its melting point.

On the other side, he knows the capacity of his phaser.

[2046]

Either we are saying the same thing there or else you are suggesting the 'red rock' alloy should be higher than 8000 ... my version is more conservative.

[Who is like God arbour]

That's what I'm suggesting.

Kirk - knowing the capacity of his phaser but not the melting point of the "red rock" - expected the phaser to be able to heat the "red rock" to 8.000 degrees centigrade.

But - as the "red rock" did not melted - the 8.000 degrees centigrade was not enough to reach the melting point of the "red rock".

Conclusion: The "red rock" has a melting point above the 8.000 degrees centigrade that can be reached with the phaser.

[2046]

Had he said "above 8000" I would go that way, too.

I figure Kirk was ballparking 8000 based on his inability to affect it with brief shots. That is, if a normal material would at least glow or change reflectivity on Setting X he might know to go to Setting Y based on a rough rule of phaser thumb against rock, but here he's on max and it laughed at him and questioned his parentage. Max might represent anything, temperature-wise, and glow may not necessarily be his criterion, but given he estimates it to 8000 I would presume he expects the phaser to get near it. I wouldn't go lower than 5000 without a fine argument, and prefer 6 or 7000.

True, you can have a range of temperature for glow based on constituent elements ... some glowy rock is oven temp, some sloshing magma four times higher, but his estimate would seem most likely based on simpler principle.

I suppose he could be way off base if the stuff is more thermally conductive than he realizes, but by the same token that may very well play in to his assertion of absurd density.

[Darth Spock]

The Remastered version's purplish planet does not visibly rotate when contrasted against the moving starfield behind it, so near as I can tell.

HAH! You're absolutely right. I saw the starfield moving, but I could've sworn I saw the terminator moving over the landscape. When I took a closer look though, I could see the shadows cast over the mountains and landscape hadn't shifted after all. So the planet does not seem to have an excessively high rotation speed contributing to it's synodic period. My bad.

And I would think the presence of any sort of anti-grav field would've borne mention, especially with the confusion about the earthquake on beam-down. I mean, we heard them mention the place had no magnetic field ... surely something far less mundane (and deadly if it disappeared) would warrant a note of concern regarding its permanence. I still think only permanent solutions fit the logic of the episode.

Well, I would agree normally. That the Kalandans bothered making a planet in the first place, I would expect as many natural phenomenon be used as possible, and a properly balanced planet is certainly more elegant than an artificial gravity "fix" after the fact. It's the fact that a tidally locked planet would kill one whole half of the world that bothers me, it's quite a waste of surface area, especially if they had to pick a special star to facilitate the effect of day and night and compensate for excessive surface gravity on the planet. Especially one that is going to have an odd, lopsided atmosphere as Mr. O already pointed out. These are all things that I think would normally be worth noting too.

At any rate, I like it, as far as finding an all natural astrometric solution goes, I'm just not completely sold on the idea in favor of an unnatural solution. Part of the reason they wanted to go down was because the planet didn't make sense, and it wouldn't be the only time Starfleet crew members beamed into an unnatural, unexplained safe haven, even though it shouldn't be there, so as to get a closer look.

Also: Since he quotes the melting point at 8,000 and we can therefore assume TOS hand phasers put out less than that figure... we end up somewhere at double the melting point of Tungsten.

This could go a long way to explain why the hull of a Galaxy class was able to withstand to survive 12,000 degrees Celsius.

... still whoa... that's um... This makes the whole 'phasers not setting everyone on fire nearby' thing that much more odd.

That's what I'm suggesting.

Kirk - knowing the capacity of his phaser but not the melting point of the "red rock" - expected the phaser to be able to heat the "red rock" to 8.000 degrees centigrade.

But - as the "red rock" did not melted - the 8.000 degrees centigrade was not enough to reach the melting point of the "red rock".

Conclusion: The "red rock" has a melting point above the 8.000 degrees centigrade that can be reached with the phaser.

That's more the way I took the scene too, unless phasers have a thermographic sight on them, I wouldn't think he'd know what temperature he had heated the rock to. Especially considering that he adjusted his phaser and tried again, it sounds more like 8000 degrees is what the "red rock" was subjected to, and more in the second try.

I figure Kirk was ballparking 8000 based on his inability to affect it with brief shots. That is, if a normal material would at least glow or change reflectivity on Setting X he might know to go to Setting Y based on a rough rule of phaser thumb against rock, but here he's on max and it laughed at him and questioned his parentage. Max might represent anything, temperature-wise, and glow may not necessarily be his criterion, but given he estimates it to 8000 I would presume he expects the phaser to get near it. I wouldn't go lower than 5000 without a fine argument, and prefer 6 or 7000.

I would assume the phaser beam on that setting is usually associated with that temperature range. It wouldn't be the last time, TNG Vengeance Factor put significant importance on temperature regarding phaser settings as well. Like you say, if the material laughed at him, not even leaving a blemish on the surface, I'd take that to mean the material was subjected to 8000 degrees and didn't even budge. I would take that as a side point even, since I doubt he planned on "melting" a grave.

[2046]

So, for the record, the consensus is that I am a Star Trek deflationist insofar as Kirk's phaser is concerned. I can live with that.

[Darth Spock]

So, for the record, the consensus is that I am a Star Trek deflationist insofar as Kirk's phaser is concerned. I can live with that.

Henceforth, you shall be known as Darth Deflater!

[Mr. Oragahn]

So, for the record, the consensus is that I am a Star Trek deflationist insofar as Kirk's phaser is concerned. I can live with that.

Henceforth, you shall be known as Darth Deflater!

Dunno. Sounds like he might have something to do with flat tires, price cuts or bowels movements.

[Mr. Oragahn]

• • • • • • [...]
        
        
        • • SULU:
        
        That's the same red rock.
        
        • • KIRK:
        
        My phaser didn't cut through it.
        
        • • MCCOY:
        
        Whatever it is, it has a mighty high melting point.
        
        • • KIRK:
        
        Eight thousand degrees centigrade. It looks like igneous rock, but infinitely denser.

In other words, Kirk seemed to believe the alloy had a melting point of 8000 degrees, presumably because it would be somewhat superior to what is phaser would be expected to generate.

• • • • • • [...]

I do not understand this as though as Kirk means that the "red rock" has a melting point of 8.000 degrees centigrade. He didn't know the properties of the "red rock" to ascertain its melting point.

On the other side, he knows the capacity of his phaser.

I think the simple road is that Kirk knowing the limit of the phaser, he threw a figure that would give a conservative melting point to this rock. Perhaps if he had been made to conclude his sentence with "at least"...

Nevertheless, it is straight forward: Kirk fires his weapon at rock, sees that the rock tanks the shot, and most likely based on some other experience, knows then this rock would be pegged at a 8000°C melting point rock material, without necessarily cutting off the possibility of a higher melting point.

But to me this gives a good idea of what a phaser is expected to do against a "known" rock of such thermal aptitudes.