Rise: have we underestimated torpedoes?
Posted: Wed Jan 06, 2016 1:55 am
This little calculation came from fairly early on when I was into Star Wars vs Star Trek stuff. I initially rejected it because, well, it seemed kind of wanky. Like, "The-Die-is-Cast-had-a-baby-with-the-antimatter-bomb-from-Obsession-level" wanky. Then I discovered this board.
I discovered that the Enterprise-D was capable of dragging around neutron stars (TNG: The Masterpiece Society). That's at least 1e27 J right there. (assumptions: the neutron star was ~5 km in diameter, had the stated density of 1e17 kg/m^3, and its delta-V was changed by less than 1 m/s)
I discovered that a "technologically primative" species was capable of turning a significant chunk of its own planet into an asteroid field (TNG: Booby Trap). 1e30 J or more. (assumptions: 1% of the planet's mass was launched into space, forming the asteroid field we saw)
I discovered that the Defiant would have been capable of destroying the ocean-like great link while being escorted by multiple dominion vessels (DS9: Broken Link). Harder to gauge, but could be as high as 1e24 J. (assumptions: the great link takes up a volume comparable to that of the Mediterranean sea, it takes half as much energy to sterilize the great link as it would take to heat the Mediterranean sea from 4 degrees Celsius to boiling).
Suddenly, the numbers I was getting seemed to have plenty of company in Star Trek canon. Since I now know they won't find themselves alone, I will release these numbers to the world!
This whole thing is based on what Kim said about what should have happened when they shot the asteroid: there shouldn't have been any fragments larger than a centimeter in diameter.
Now, contrary to most of what passes for "scientific analysis" in this debate, it is nearly impossible to "vaporize" an asteroid. A nuke or a weaponized laser will deliver all of its energy to a relatively thin layer on the surface of the asteroid. If the layer is heated faster than it can exchange heat with the rest of the asteroid, said layer vaporizes. If it vaporizes fast enough, it can blow the rest of the asteroid apart. Adding more power doesn't necessarily make the layer any thicker, it just makes it explode more violently. The only way I can picture large-scale vaporization occurring is if the necessary energy is imparted at a very low power (when compared to the power of a weaponized laser), allowing the heated region to exchange heat with its surroundings. Such a method would be impossible if said energy were being delivered via a nuke - or an antimatter explosive for that matter.
Incidentally, similar considerations apply to particle beams. Only instead of a layer of vaporized material heated by photons, you get a conical region heated by a cascade of collision products. But I digress.
What is required to gauge the effectiveness of a photon torpedo is not an analysis in terms of the heat capacity of the asteroid, but in terms of the energy it would take to reduce the 100+ meter asteroid down to 1 cm fragments.
Which is where this study comes in. There's a lot of fun stuff in there, but on page 703, there's an interesting equation:
log(Ml/Mt) = a - B*log(Ei)
where Ml is the mass of the largest fragment, Mt is the mass of the target asteroid, Ei is the energy density in J/kg, and "a" and "B" are empirically determined constants.
In short, if we know the mass of an asteroid before it goes boom, and we know the mass of the largest piece left after it goes boom, we can figure out how big the boom was in the first place.
It's actually fairly easy to manipulate logarithms, but with all the [sup] and [sub] tags*, it's easier for me just to skip to the final product:
Ei = ((Mt/Ml)^(1/B)) * 10^(a/B)
(*in the first version of this post, I had attempted to use superscript and subscript tags to get rid of all those messy carats and parentheses. Unfortunately, those tags don't appear to work on this forum. So all superscript and subscript have been edited out as of 1/11/2016.)
Fortunately for us, they provide values for a and B on page 702 (I just now realize how quickly I'm moving, but this post is monstrously long already, so I'm counting on being able to clarify in the comments section).
For basalt, a = 1.29, and B = 0.77. We're going to assume that the asteroid is basalt-like, as assuming that it's tuff-like would inflate our result by more than two orders of magnitude.
Throwing in those numbers, our new equation is:
Ei = 9(Mt/Ml)^1.30) * 10^1.68
Darkstar has a calculation on his site indicating that the asteroid is more than 300 meters long, and some 200 meters wide, but we'll assume it's a 100 meter sphere for our purposes. Assuming the largest fragment that should have been left would have been a 0.01 meter sphere, we get a value of 1e12 for Mt/Ml.
Plugging that in, Ei = 1.91e17 J/kg. A 100 m wide spherical asteroid would have a volume of some 5.24e5 m^3, and basalt has a density of 2900 kg/m^3, giving us a total of 2.9e26 J. That's about the same amount of energy as was released by the Kirk-bomb from Obsession.
Like I said, ludicrously high. Though there may be some mitigating factors. First and foremost, the study got its numbers by shooting steel bullets at rock targets. The velocities involved were substantially less than one kilometer per second, and the targets were (with only one exception) never heavier than a kilogram. The highest energy densities were in the tens of thousands of Joules per kilogram. So we may have a simple issue of pushing an otherwise accurate model further than it was meant to go.
But there are lesser factors to consider. Note that the asteroid in Rise was apparently nickel-iron, not basalt. If the comparison of basalt to tuff is anything to go by, a harder asteroid actually reduces the amount of energy required. In the case of basalt and tuff, the difference is some two orders of magnitude. A tuff-like asteroid would have required an energy density of 5.25e19 J/kg, compared to the 1.91e17 kg required by the basalt-like asteroid. If iron is to basalt as basalt is to tuff, then the energy density required to fragment an iron-nickel asteroid could be as low as 10e15 J/kg. That still gives us a good 1e24 J, but that's actually pretty much in line with Garak's expectations (reducing the planet to a "smoking cinder" and obliterating the ocean-like great link) of what the Defiant could do with only a minute's worth of pulse phasers and maybe two volleys of torpedoes.
For that matter, putting the maximum yield of a photon torpedo in the hundreds of teratons (1 TT = 4e21 J) fits in well with the "primative" planet destroyers from Booby Trap, the planetary bombardment from TDiC, and the Kirk-bombs in Obsession and The Immunity Syndrome. Plus there's the fact that Voyager's shields were capable of surviving explosions that demolished planets in episodes like Think Tank. Why have shields that can tank teratons if you don't expect to face weapons that can deliver teratons?
High yield torpedoes also explain why ships that can fight at ranges in the light-seconds often choose to fight at ranges in the tens of kilometers at most: if you get that close, the enemy can't one-shot you without one-shotting himself as collateral damage.
That still might not be enough to convince you. Heck, I'm not 100% convinced myself. But the more I read this forum's archives, things like TDiC look less and less like outliers and more and more like expected upper limits. Before discovering StarFleetJedi, I thought I had misapplied the math. Now? My weird numbers seem to fit into a surprising pattern.
EDIT I: Is it possible to do subscripts and superscripts on this forum? If so, how?
EDIT II: Evidently it is not possible. All superscript and subscript tags have been removed.
I discovered that the Enterprise-D was capable of dragging around neutron stars (TNG: The Masterpiece Society). That's at least 1e27 J right there. (assumptions: the neutron star was ~5 km in diameter, had the stated density of 1e17 kg/m^3, and its delta-V was changed by less than 1 m/s)
I discovered that a "technologically primative" species was capable of turning a significant chunk of its own planet into an asteroid field (TNG: Booby Trap). 1e30 J or more. (assumptions: 1% of the planet's mass was launched into space, forming the asteroid field we saw)
I discovered that the Defiant would have been capable of destroying the ocean-like great link while being escorted by multiple dominion vessels (DS9: Broken Link). Harder to gauge, but could be as high as 1e24 J. (assumptions: the great link takes up a volume comparable to that of the Mediterranean sea, it takes half as much energy to sterilize the great link as it would take to heat the Mediterranean sea from 4 degrees Celsius to boiling).
Suddenly, the numbers I was getting seemed to have plenty of company in Star Trek canon. Since I now know they won't find themselves alone, I will release these numbers to the world!
This whole thing is based on what Kim said about what should have happened when they shot the asteroid: there shouldn't have been any fragments larger than a centimeter in diameter.
Now, contrary to most of what passes for "scientific analysis" in this debate, it is nearly impossible to "vaporize" an asteroid. A nuke or a weaponized laser will deliver all of its energy to a relatively thin layer on the surface of the asteroid. If the layer is heated faster than it can exchange heat with the rest of the asteroid, said layer vaporizes. If it vaporizes fast enough, it can blow the rest of the asteroid apart. Adding more power doesn't necessarily make the layer any thicker, it just makes it explode more violently. The only way I can picture large-scale vaporization occurring is if the necessary energy is imparted at a very low power (when compared to the power of a weaponized laser), allowing the heated region to exchange heat with its surroundings. Such a method would be impossible if said energy were being delivered via a nuke - or an antimatter explosive for that matter.
Incidentally, similar considerations apply to particle beams. Only instead of a layer of vaporized material heated by photons, you get a conical region heated by a cascade of collision products. But I digress.
What is required to gauge the effectiveness of a photon torpedo is not an analysis in terms of the heat capacity of the asteroid, but in terms of the energy it would take to reduce the 100+ meter asteroid down to 1 cm fragments.
Which is where this study comes in. There's a lot of fun stuff in there, but on page 703, there's an interesting equation:
log(Ml/Mt) = a - B*log(Ei)
where Ml is the mass of the largest fragment, Mt is the mass of the target asteroid, Ei is the energy density in J/kg, and "a" and "B" are empirically determined constants.
In short, if we know the mass of an asteroid before it goes boom, and we know the mass of the largest piece left after it goes boom, we can figure out how big the boom was in the first place.
It's actually fairly easy to manipulate logarithms, but with all the [sup] and [sub] tags*, it's easier for me just to skip to the final product:
Ei = ((Mt/Ml)^(1/B)) * 10^(a/B)
(*in the first version of this post, I had attempted to use superscript and subscript tags to get rid of all those messy carats and parentheses. Unfortunately, those tags don't appear to work on this forum. So all superscript and subscript have been edited out as of 1/11/2016.)
Fortunately for us, they provide values for a and B on page 702 (I just now realize how quickly I'm moving, but this post is monstrously long already, so I'm counting on being able to clarify in the comments section).
For basalt, a = 1.29, and B = 0.77. We're going to assume that the asteroid is basalt-like, as assuming that it's tuff-like would inflate our result by more than two orders of magnitude.
Throwing in those numbers, our new equation is:
Ei = 9(Mt/Ml)^1.30) * 10^1.68
Darkstar has a calculation on his site indicating that the asteroid is more than 300 meters long, and some 200 meters wide, but we'll assume it's a 100 meter sphere for our purposes. Assuming the largest fragment that should have been left would have been a 0.01 meter sphere, we get a value of 1e12 for Mt/Ml.
Plugging that in, Ei = 1.91e17 J/kg. A 100 m wide spherical asteroid would have a volume of some 5.24e5 m^3, and basalt has a density of 2900 kg/m^3, giving us a total of 2.9e26 J. That's about the same amount of energy as was released by the Kirk-bomb from Obsession.
Like I said, ludicrously high. Though there may be some mitigating factors. First and foremost, the study got its numbers by shooting steel bullets at rock targets. The velocities involved were substantially less than one kilometer per second, and the targets were (with only one exception) never heavier than a kilogram. The highest energy densities were in the tens of thousands of Joules per kilogram. So we may have a simple issue of pushing an otherwise accurate model further than it was meant to go.
But there are lesser factors to consider. Note that the asteroid in Rise was apparently nickel-iron, not basalt. If the comparison of basalt to tuff is anything to go by, a harder asteroid actually reduces the amount of energy required. In the case of basalt and tuff, the difference is some two orders of magnitude. A tuff-like asteroid would have required an energy density of 5.25e19 J/kg, compared to the 1.91e17 kg required by the basalt-like asteroid. If iron is to basalt as basalt is to tuff, then the energy density required to fragment an iron-nickel asteroid could be as low as 10e15 J/kg. That still gives us a good 1e24 J, but that's actually pretty much in line with Garak's expectations (reducing the planet to a "smoking cinder" and obliterating the ocean-like great link) of what the Defiant could do with only a minute's worth of pulse phasers and maybe two volleys of torpedoes.
For that matter, putting the maximum yield of a photon torpedo in the hundreds of teratons (1 TT = 4e21 J) fits in well with the "primative" planet destroyers from Booby Trap, the planetary bombardment from TDiC, and the Kirk-bombs in Obsession and The Immunity Syndrome. Plus there's the fact that Voyager's shields were capable of surviving explosions that demolished planets in episodes like Think Tank. Why have shields that can tank teratons if you don't expect to face weapons that can deliver teratons?
High yield torpedoes also explain why ships that can fight at ranges in the light-seconds often choose to fight at ranges in the tens of kilometers at most: if you get that close, the enemy can't one-shot you without one-shotting himself as collateral damage.
That still might not be enough to convince you. Heck, I'm not 100% convinced myself. But the more I read this forum's archives, things like TDiC look less and less like outliers and more and more like expected upper limits. Before discovering StarFleetJedi, I thought I had misapplied the math. Now? My weird numbers seem to fit into a surprising pattern.
EDIT I: Is it possible to do subscripts and superscripts on this forum? If so, how?
EDIT II: Evidently it is not possible. All superscript and subscript tags have been removed.