Calc The Dark Forest - If I destroy you, what business is it of yours?

The Droplet makes itself known, and terror ensures.

1. The Droplet destroys the combined SFJCs space fleet
The first Trisolaran probe, the Droplet, reaches the Solar System, and humanity is not close to prepared for the horrors this can unleash, with it wiping out the entire space fleet of 2015 ships (with only two ships escaping, and only around 55000 survivors out of 1.3 million).

“What is its mass?” Ding Yi asked.
“We don’t have an exact value right now. A rough value, obtained through high-precision gravitational instruments, is less than ten tons.”

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There were various hypotheses about the manner in which the droplet might self-destruct, but the maximum release of energy would come from self-destruction by antimatter annihilation. Since the droplet had a mass of less than ten tons, the largest energy burst that needed to be considered was that produced by the annihilation of five tons each of matter and antimatter.

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The Droplet weighs less than 10 tons, with people estimating how powerful an antimatter explosion would be from it with 5 tons of matter and 5 tons of antimatter. I've a feeling it's going to be pretty close to 10 tons, but to play it safe, let's say it weighs 9 tons for our low end, and have 10 tons for our high end.

By three hundred kilometers it had already passed the third cosmic velocity and was continuing to gain speed. The alert was upgraded to level two, but was still ignored.
By the time the fragment had flown roughly 1,500 kilometers from the explosion site toward the corner of the formation, only fifty-one seconds had elapsed. By the time it reached the corner, it was traveling at a speed of 31.7 kilometers per second.

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After passing through Infinite Frontier, the droplet continued onward at a speed of thirty kilometers per second. In the space of three seconds it had crossed ninety kilometers, passing first through Yuanfang, Infinite Frontier’s neighbor in the first row, and then through Foghorn, Antarctica, and Ultimate, leaving their hulls red-hot, as if the warships were giant lamps lined up.

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This time the angle it described was even smaller: just fifteen degrees off a total reversal, executed nearly instantaneously even as it maintained a constant speed. Then, after a small heading adjustment that brought it in line with the second row of warships in the fleet’s array—or what was now the first row, in light of the recent destruction—it sped toward the first ship in that row, Ganges, at thirty kilometers per second.

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Following this broken path, the droplet drilled through the ships in the third row one after the other, never dropping its speed below thirty kilometers per second.

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LI WEI: I’ve noticed something about the speed.
ZHAO XIN: What speed?
LI WEI: The speed of the small projectile. Its speed when it’s launched from each warship is slightly slower. Then it accelerates to thirty kilometers per second during flight. Then it strikes the next warship, and when it launches from that warship prior to the explosion, its speed is a little slower. Then it accelerates....
ZHAO XIN: That doesn’t mean anything....
LI WEI: What I mean is ... it’s a little like drag.
ZHAO XIN: Drag? How so?
LI WEI: Every time this projectile passes through a target, the drag slows it down.

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The initial attack speed of the droplet is 30km a second, or really, about 31.7km a second.

T = 30km/1s
= 30000/340.29
= Mach 88.1600987

(Low end)

KE = (0.5)mv^2
= (0.5) X 9000 X 30000^2
= 4050000000000 joules
= 967.9732313575525 tons of TNT

(High end)

KE = (0.5)mv^2
= (0.5) X 10000 X 30000^2
= 4.5e12 joules
= 1.075525812619503 kilotons

That's not the end of it though; it then increases in speed.to sixty kilometers a second.

The merciless slaughter in space continued. As the distance between ships grew, the droplet accelerated and had soon doubled its speed to sixty kilometers per second. Exhibiting a cool and precise intelligence inits continuous attacks, it solved the traveling salesman problem in localregions with perfect accuracy, hardly ever retracing its path.

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T = 60km/1s
= 60000/340.29
= Mach 176.320197

(Low end)

KE = (0.5)mv^2
= (0.5) X 9000 X 60000^2
= 1.62e13 joules
= 3.8718929254302101 kilotons

(High end)

KE = (0.5)mv^2
= (0.5) X 10000 X 60000^2
= 1.8e13 joules
= 4.3021032504780115 kilotons

Finally, the droplet increases in speed one final time to wipe out the rest of the fleet, reaching 170km a second.

The droplet’s speed was now ten times the third cosmic velocity, or roughly 170 kilometers per second. Its course took it smashing through debris that liquefied under impact, splashing away at high speed to collide with other debris and giving the droplet a brilliant tail. First it resembled a comet bristling with rage, but as the tail lengthened, it turned into a huge silver dragon that stretched ten thousand kilometers.

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T = 170km/1s
= 170000/340.29
= Mach 499.573893

(Low end)

KE = (0.5)mv^2
= (0.5) X 9000 X 170000^2
= 1.3005e14 joules
= 31.0826959847036335 kilotons

(High end)

KE = (0.5)mv^2
= (0.5) X 10000 X 170000^2
= 1.44500e14 joules
= 34.5363288718929269 kilotons

That's not all we can get from this terrifying little thing though.

2. The Droplet deflects a laser beam

The speed of the droplet’s strikes and its invisibility to radar meant that defensive systems on the first ships hit never responded. But as the distance between the warships grew and the droplet’s striking distance increased, defensive systems on all warships were recalibrated based on the droplet’s target characteristics. This meant that Nelson was the first ship to attempt to intercept, using laser weapons to increase the accuracy of firing on the small, high-speed target. When struck by the multiple beams, the droplet emitted a powerful visible light, even though the Nelson had fired gamma-ray lasers that were invisible to the naked eye. The droplet’s imperceptibility to radar had never been understood, since it had a completely reflective surface and a shape that was perfectly diffuse, but perhaps the ability to alter the frequency of reflected electromagnetic waves was the secret to this invisibility. The light emitted from the droplet when it was struck was so bright it drowned out the nuclear fireballs going off all around it, forced the monitoring systems to dim their images to avoid damage to their optical components, and caused sustained blindness to anyone who looked directly at it. In other words, this superpowerful light was indistinguishable from darkness in its effect. The droplet, wrapped in this all-engulfing light, entered Nelson and was extinguished, plunging the battlefield into pitch darkness. Moments later, the nuclear fireballs reestablished their dominance and the droplet emerged from Nelson unscathed and sped straight for Green, eighty-odd kilometers away.

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The Droplet is hit by laser beams from the Spaceship Nelson, which causes a burst of light so bright that it dims out the nuclear fireballs going off around it and forces the monitoring systems to dim their images to avoid damaging them, and causes sustained blindness in anyone who looked directly at it. The brightness eclipses even the nearby nuclear explosions. Nuclear explosions being as bright as the Sun.

The Light of the Atom Bomb: In brightness, a nuclear detonation is comparable to the sun​

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Thus, the brightness of the explosion would be even greater than that of the Sun (that being 100,000 lux). The conversion factor from lux to apparent magnitude is -2.5 log I - 14.2.

L = -2.5 log I - 14.2
= -2.5 log 100,000 - 14.2
= -26.7

Now, for the distance, it talks about the burst of light outshining the nuclear fireballs going off all around it. The nearest ship is 80km (80000m) away, so that's a safe bet for a low end distance.

Our formula is as follows...

-26.7 = -26.73 - 2.5log((L/3.846*10^26)(146000000000/(80000))^2)

Let's go through this step by step.

(146000000000/(80000))^2) = 3330625000000
-26.7 - -26.73 = 3330625000000/((L/3.846*10^26)
0.03 = -2.5Log(3330625000000/((L/3.846*10^26))
0.03/-2.5 = (3330625000000/((L/3.846*10^26))
10^(-0.012) = (3330625000000/((L/3.846*10^26))
0.972747224 = (3330625000000/((L/3.846*10^26))
0.972747224 X (3.864*10e26) = 3330625000000
3.75869527e27/3330625000000 = 1.12852551e15 joules = 269.724070267686443 kilotons

However, looking over the quotes again, and I found something of interest...

Then Infinite Frontier exploded. It and the four warships after it were hit in the fusion fuel tanks. But unlike Mantis’s conventional, high-temperature explosion, this explosion was a fusion reaction triggered in Infinite Frontier’s fuel. No one ever figured out whether the fusion reaction had been sparked by the droplet’s ultra-high-temperature propulsive halo or some other factor. The fireball of the thermonuclear explosion appeared at the point of impact with the fuel tank and swiftly expanded until it illuminated the entire fleet against the velvety background of space, outshining the Milky Way.

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Each fireball is able to illuminate the entire fleet, and the droplet deflecting the gamma laser beam exceeds that still. At the very least, each row is 2000km long (or 2000000m long).

In one minute and eighteen seconds, the droplet had completed a two-thousand-kilometer course, passing through each of the hundred ships in the first row of the combined fleet’s rectangular formation.

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Thus we can also add our high end.

-26.7 = -26.73 - 2.5log((L/3.846*10^26)(146000000000/(2000000))^2)

Once again, going through this step by step...

(146000000000/(2000000))^2) = 5329000000
-26.7 - -26.73 = 5329000000/((L/3.846*10^26)
0.03 = -2.5Log(5329000000/((L/3.846*10^26))
0.03/-2.5 = (5329000000/((L/3.846*10^26))
10^(-0.012) = (5329000000/((L/3.846*10^26))
0.972747224 = (5329000000/((L/3.846*10^26))
0.972747224 X (3.864*10e26) = 5329000000
3.75869527e27/5329000000 = 7.05328442e17 joules = 168.57754349904396918 megatons

Final Results
Droplets initial attack speed = Mach 88.160
Droplets initial attack speed kinetic energy (low end) = 967.973 tons of TNT
Droplets initial attack speed kinetic energy (high end) = 1.076 kilotons
Droplets secondary attack speed = Mach 176.320
Droplets secondary attack speed kinetic energy (low end) = 3.872 kilotons
Droplets secondary attack speed kinetic energy (high end) = 4.302 kilotons
Droplets highest observed attack speed = Mach 499.574
Droplets highest observed attack speed kinetic energy (low end) = 31.083 kilotons
Droplets highest observed attack speed kinetic energy (high end) = 34.536 kilotons
The Droplet withstands a laser attack (low end) = >269.724 kilotons
The Droplet withstands a laser attack (high end) = >168.578 megatons