Type-I Phaser Firepower

[359]

With phasers it is often difficult to prove with complete certainty the levels of firepower shown due to their odd effects. Most often when set to full power they cause effects such as this:
http://ds9.trekcore.com/gallery/albums/ ... re_265.jpg
or this:
http://tng.trekcore.com/hd/albums/1x10/ ... hd_208.jpg

But in TNG: "Hide and Q" we see the effects of a phaser on a rock without any of the odd effects normally associated with phaser hits. Here we can see the rock as the phaser beam hits it:
http://tng.trekcore.com/hd/albums/1x10/ ... hd_187.jpg
And here the result:
http://tng.trekcore.com/hd/albums/1x10/ ... hd_191.jpg
http://tng.trekcore.com/hd/albums/1x10/ ... hd_190.jpg

I calculated this to be about 0.8008 m^3 of material removed by the shot.
The scaling reference is attached to this post (1.8 meters used for Worf's height.)
V = (4/3)πabc
V = (4/3)π(1.2995)(1.2995)(.3019)
V = 2.1355
2.1355/2 = 1.0678
1.0678*.75 = .8008

Given that an energy weapon caused these effects the explosion would be a result of rapidly expanding vapor that was once solid rock. And because the rock is sticking out of sandy ground we know that it is composed of fairly tough material, due to relative erosion rates etc... For this analysis I will use granite.

Granite requires 15.89 GJ/m^3 to vaporize. Because of the amount of debris etc... I will only calculate 40% of the removed material as vaporized, this puts the event at 5.1 GJ.
.5(15.89*.8008) = 5.0899

The weapon discharged for a duration of 0.6 seconds, so its power output for this event would be 8.5 GW.
5.0899/.6 = 8.4831


This is on the same order of magnitude as DS9: "Return to Grace" (1.57 GW). Incidentally, it is also on the same order of magnitude as other phaser events if you calculate for thermal effects:

TNG: "Chain of Command" (9.7 GW)
DS9: "Rapture" (3.97 GW)
Whenever Garak meet an old enemy :) (1.3 GW)
Various disintegrations of various people (1.3 GW)

[Mike DiCenso]

I know this sounds a bit macabre, but maybe you should do a firepower calculation using the explosion of poor Commander Remmick's head from phaser fire in TNG's "Conspiracy". ;-)
-Mike

[359]

That would be somewhat more difficult, humans are composed of a variety of materials. And it is hard to determine the cause and energy in the explosion. Plus it had appeared that the bugs had altered his interior which would alter the results. Also, the phasers were set to kill, or something near that, not full, so there would be little utility in such an analysis.

Additionally, looking over the episode I noticed these lines:

Riker: "Increase to warp six."
LaForge: "Aye sir. Full impulse."

[2046]

I eyeballed less material and underestimated the effects intentionally, so I certainly appreciate your more thorough and direct analysis. When time permits I will quote you, if I may.

http://st-v-sw.net/STSWground-newphas1.html

[Mike DiCenso]

That would be somewhat more difficult, humans are composed of a variety of materials. And it is hard to determine the cause and energy in the explosion. Plus it had appeared that the bugs had altered his interior which would alter the results. Also, the phasers were set to kill, or something near that, not full, so there would be little utility in such an analysis.

Not really. You have to show that the bugs were not literally inside Remmick's head, but a mother and a bunch of the smaller ones inside his chest cavity:







By all accounts, his head is still perfectly normal looking... well as normal as having two high-energy weapon beams on full hitting it. Furthermore, most of the human body by weight is water, so calculating the heating and rapid thermal expansion based off of that should give a good ballpark number.
-Mike

[359]

OK, so here is goes:

The average mass of a human head is about 5 kg

The human body is roughly 60% water, with brain/muscles being 75% water, and bone being 22%. Given the head is a large amount of both brain and bone, brain making up 1.4 kg, and skull weighing about 1 kg, it comes back to about 60% water.

Given that the head has a mass of 5 kg, and that it is about 60% water, that gives a mass of 3 kg of water in the average human head.

To cause a pressure build up necessary for such an explosion would require vaporization of a portion of that water. Water boils at 100°C, the human body maintains a temperature of 37°C, so the water needs to be heated by 63°C and then changed into a gas.

Specific Heat of water: 4,184 J/kg*°C
Enthalpy of Vaporization of water: 2,260 kJ/kg
q = mc∆T
q = 3*4184*63
q = 790776 J = 790.776 kJ

q = mHv
q = 3*2260
q = 6780 kJ

6780+790.776 = 7570.776 kJ

Vaporizing the water in one's head would take 7.6 MJ, but all that probably is not required.

In the episode, Riker and Picard's phaser beams initially strikes Remmick's torso (00:42:35) forcing him back into a sitting position. After 4 seconds of sustained fire, Riker's beam sweeps up to Remmick's head (00:42:39). The beam terminates and his head explodes a little under a second later (00:42:40). During the explosion, Picard's beam, still on Remmick's torso, terminates as well.

The beam duration was about one second in length, which puts the output of the phaser in this event at 7.6 MW.

________

When time permits I will quote you, if I may.

Sure.

[Mike DiCenso]

Nice job with the "Conspiracy" calcs, 359. It dawns on me that you have more examples worth looking at, even if they don't involve vaporization:


* TOS "Where No Man Has Gone Before"': Kirk in desperation while fighting the mutated Gary Mitchell fires his phaser rifle at a large cliff face, causing it to explode violently and dislodge a large rock that falls down and crushes Mitchell, killing him.

* TOS "The Enemy Within": Sulu and several others are stranded on the surface of a planet where the local temperature is rapidly dropping. Sulu uses his Type-II phaser in an unusual wide beam setting to heat a large boulder to provide warmth.

* TOS "A Private Little War": Dr. McCoy uses his little Type-I hand phaser to heat several modest-sized boulders to the glowing point to provide warmth for Kirk who is sick from a creature's poisoned bite.

* TNG's "The Enemy": Geordi is trapped in a pit, in order to escape he digs out some crystals that he can see with his visor, then digs a small, narrow trench, placing the crystals inside it, and then uses his phaser to heat the crystals until they melt.

*DS9 "Rocks and Shoals": Sisko, Dax and several other crew are stranded on a planet. They make camp in a cave and use a Type-II phaser to heat a large boulder to the glowing point to provide heat.

All of these events should easily provide megajoule level energy for hand phasers.
-Mike

[359]

* TNG's "The Enemy": Geordi is trapped in a pit, in order to escape he digs out some crystals that he can see with his visor, then digs a small, narrow trench, placing the crystals inside it, and then uses his phaser to heat the crystals until they melt.

The material looks metallic, like native nickel.

Nickel:
Liquid density: 7.8 g/cm^3
Melting point: 1455°C
Specific heat: .44 kJ/kg°C
Heat of fusion: 297 kJ/kg

http://tng.trekcore.com/gallery/albums/ ... emy043.jpg

Volume of material melted
~10cm^3

Mass of material melted
(10*7.8)/1000 = 0.078 kg

q = mc∆T
q = (.078)(.44)(1435)
q = 49.2492

q = mHv
q = (.078)(297)
q = 23.166

23.166+49.2492 = 74.2192 kJ

74.2192/9 = 8.0461 kW

So this would put the phaser's output in this instance at about 8 kilowatts.

[359]

DS9: "Rocks and Shoals"

Beam duration: 00:10:17 to 00:10:19
Melting point of Granite: 1250°C
Specific heat of Granite: 790 J/kg°C
Density of Granite: 2.7 g/cm^3 or 2700 kg/m^3

I estimate the rock was about 1.5 meters wide based on the height of the people in this image:
http://ds9.trekcore.com/gallery/albums/ ... ls_167.jpg
The volume of a sphere with a 1.5 meter radius is 14.1372 m^3, which gives a mass of

V = (4/3)πr^3
V = (4/3)π(1.5)^3
V = 14.1372

M = dV
M = 2700*14.1372
M = 38170.44

Granite has a melting point of about 1250°C, guessing at the temperature increase based color difference and that their cloths are drying quickly, but without burning, so I would say that they raised it's temperature by 100°C with that two second discharge.
http://ds9.trekcore.com/gallery/albums/ ... ls_165.jpg
http://ds9.trekcore.com/gallery/albums/ ... ls_166.jpg

q = mc∆T
q = (38170.44)(790)(100)
q = 3015464760

3015464760/2 = 1507732380

That puts this instance at a surprising 1.5 GW.

The change in color would suggest higher temperatures, but the fact that Sisko could touch the cloth afterwards indicates otherwise. Something to note about how the phaser heats the rock is that is does so evenly, if that were just any 1.5 GW energy beam there would have been explosions, as the granite taking a direct hit would vaporize. Instead we see a rock that is evenly heated over a large restricted area (the hot rock quickly transitions into cool rock at the base of the glow). This appears similar to how phasers 'vaporize' targets; controlled, even, and over a restricted area. This suggests that phasers are thermal effects weapons, but with odd secondary effects, such as even heating and restricted effect areas.


EDIT: Should be 750 MW for the above, I forgot to half the sphere's volume.

[Lucky]

That would be somewhat more difficult, humans are composed of a variety of materials. And it is hard to determine the cause and energy in the explosion. Plus it had appeared that the bugs had altered his interior which would alter the results. Also, the phasers were set to kill, or something near that, not full, so there would be little utility in such an analysis.

Not really. You have to show that the bugs were not literally inside Remmick's head, but a mother and a bunch of the smaller ones inside his chest cavity:







By all accounts, his head is still perfectly normal looking... well as normal as having two high-energy weapon beams on full hitting it. Furthermore, most of the human body by weight is water, so calculating the heating and rapid thermal expansion based off of that should give a good ballpark number.
-Mike

The problem with that scene is that it looks like there is a shield over the guy being shot.

[2046]

No shield.

1. On the headshot, the outer layers vaporized first, which is strange. At worst, one could suggest some sort of alteration of him by the lifeform, but that would only apparently drive up the energy . . . we can still use a human head. After all, if the skin was weakened and altered, it would be rather more obvious.

2. The body shot has little rings around it but that's because the early TNG phasers usually put out an area effect on impact, like little waves or ripples coming off of it. I was rather fond of the effect myself, and missed it when it was gone.

[359]

I just stumbled across a PDF about geothermal energy from MIT. In it are various thermal properties of granite.
http://www.psfc.mit.edu/library1/catalo ... 1_full.pdf

So using this data it takes 25,900 J to vaporize 1 cm^3 of granite or 25.9 GJ/m^3.

Granite:
Density: 2.7 g/cm^3
Specific Heat (solid): 1.05 J/g/°C
Specific Heat (liquid): 1.57 J/g/°C
Heat of Fusion: 335 J/g
Heat of Vaporization: 5050 J/g
Melting Point: 1240°C
Vaporization Point: 3100°C
Room Temperature: 20°C

m = dv
m = 2.7 g/cm^3*1 cm^3
m = 2.7 g

q = mc∆T
q = 2.7 g*1.05 J/g/°C(1240°C-20°C)
q = 3458.7 J

q = mHv
q = 2.7 g*335 J/g
q = 904.5 J

q = mc∆T
q = 2.7 g*1.57 J/g/°C(3100°C-1240°C)
q = 7884.54 J

q = mHv
q = 2.7g*5050 J/g
q = 13635 J

3458.7 J+904.5 J+7884.54 J+13635 J = 25882.74 J

Before I found this I have been using 15.89 GJ/m^3 from st-v-sw.net's Ground Combat page, but in light of this new information I will be using this instead. That means that my calcs for TNG: "Hide and Q" should increase by 63% to 13.8 GW using this new value.

[Mike DiCenso]

Will your new information change other calculations as well?
-Mike

[359]

It would change the DS9:"Rapture" and TNG:"Chain of Command" calculations I mentioned in my initial post (+63%). Aside from that, nothing else I have done comes to mind.

[Mike DiCenso]

This is on the same order of magnitude as DS9: "Return to Grace" (1.57 GW). Incidentally, it is also on the same order of magnitude as other phaser events if you calculate for thermal effects:

Where and how do you get 1.57 GW from "Return to Grace"? Are you talking about the millisecond recharge on the Cardassian rifle Kira showed Ziyal, or the effects of the freighter's phaser on the target practice asteroid?
-Mike