Energy Output (Incoming Asteroid)
At the end of the film, almost as a poetic encircling of the plot, the Cosmos twins (seemingly the last surviving pair of a prehistoric fairy humanoid race) relay to the humans of a unfathomable threat to their existence: a massive asteroid is directly en route to a collision with Earth sometime in the late 1990s, with very ominous consequences for those 'in the way'.
But never fear! The Cosmos reassure the farewell party that Imago Mothra is capable of physically diverting the meteor from its path and redirect it out of harm's way.
But how much energy would it require to accomplish such a task?
The precise size of the incoming asteroid is not specified, with the most information we have to go on still being somewhat vague in nature: that it is (presumably) large enough to trigger a sufficiently great cataclysm for humans to recognise its impact as "destroying Earth".
Now, this could very well mean that the bolide is capable of generating a great enough energy output upon impact to overcome Earth's GBE (2.24e+32 joules or ~53 zettatons of TNT equivalent) and irrepairably destroy the entire planet as its fragments disperse infinitely. The dialogue certainly does not exclude such rationale as a possible conclusion.
However, it's a sufficiently wishy-washy phrasing of the question (at least IMHO) that the Cosmos affirmation could just as easily qualify for the asteroid being big enough to cause sufficiently major damage to the Earth's biosphere and trigger a mass-extinction event that would wipe out humanity (the main species concerned with the incident). Which would still be tremendously powerful, but orders of magnitude weaker than total planetary destruction.
So let's approach the asteroid size with two avenues: (1) minimum bolide size capable of triggering a mass-extinction event and (2) minimum bolide size capable of overcoming Earth's GBE.
(1) Mass-Extinction
This should be a simple demonstration, given the extensive amount of research having been dedicated to a universally verified meteor impact in our prehistory, currently cited as the most probable cause for a mass-extinction which wiped out approx. 75% of contemporary species on the planet: the K-Pg/K-T extinction event, and more precisely, the Chicxulub impactor/crater.
Studies indicate that the impactor responsible for the Chicxulub crater would have to be a minimum size of 10 km in diameter. The only other two impact zones known to science that are 100% verifiable asteroid impacts and larger than Chicxulub, Vredefort in South Africa and Sudbury Basin in Ontario, Canada, are also believed to have been caused by impactors measuring in the 10 - 15 km diameter range, thus demonstrating a fair consistency in qualifying a "mass extinction" asteroid at the 10 km mark.
Assuming a decently symmetrical/ovular ellipsoid shape for the Chicxulub impactor (axis A = radius; axis B = 1/2 of axis A length; axis C = axis B):
Volume of an ellipsoid =
V = 1.31e+11 m^3
As for the density: the Chicxulub impactor is currently believed to have originated as a member of the Flora family of S-type asteroids within the Asteroid Belt; Florian asteroids are highly plausible candidates to being the 'parent' family to L chronditemeteorites, which in turn have an average density of 3.35 g/cm^3.
3.35 g/cm^3 = 3350 kg/m^3
Mass = 1.31e+11*3350 = 4.389e+14 kg
Average orbital velocity of an asteroid around our Sun is 25 km/s, which will probably best match up to our giant rock buddy here, particularly given that there's no figures for deep space-traversing asteroids.
KE = (0.5)*4.389e+14*25,000*25,000 = 1.37140625e+23 joules or 32.78 teratons of TNT equivalent.
Imago Mothra would, at minimum, have to capable of exerting an equivalent output of kinetic energy in order to offset the trajectory of an asteroid of that size and velocity.
(2) Planetary Destruction
But the "mass-extinction" avenue is, ultimately, the lower rung of the cosmic wazoo ladder.
How big would the impactor need to be in order to overcome Earth's GBE?
In short: a rogue celestial body approximately 3/5 (or 60%) the mass of Earth itself.
Earth's mass is 5.96e+24 kg
5.96e+24/5 = 1.192e+24 kg
1.192e+24*3 = 3.576e+24 kg
Assuming the same cosmic velocity...
KE = (0.5)*3.576e+24*25,000*25,000 = 1.1175e+33 joules or 267.09 zettatons of TNT equivalent.
Fucking. Yikes.
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An important note: the Cosmos twins state that prior to Battra's premature awakening and subsequent death at the hands of Godzilla, it was to emerge and transform into its Imago state around the deadlineof the meteor's appearance, then outrightdestroy the bolide prior to its impact when "in range" (presumably, "in range" = far enough that Battra doesn't risk collateral damage to Earth via asteroid fragmentation, but near enough so that Imago Battra can fly back to the planet in a relatively short span of time).
Much earlier in the film, when describing the artifical birth of Battra by the Cosmos civilisation 12,000 years prior to the modern day, Battra is explicitly stated to have equivalent energy output to Mothra.
So powerscaling either the low-end or high-end (lololol) result to Imago Battra and Second Mutation Heisei Godzilla is pretty much a dead cert.
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9# - Flight Speed (Imago Mothra)
It is nearly over, I swear
Mothra in Imago state departs the Earth, along with the Cosmos twins, to traverse into deep space and re-direct the giant asteroid. It manages to fly past the planet's atmosphere in a remarkably short timeframe as the film concludes.
Spoiler: http://i.imgur.com/W8Cb6Cb.png (Warning: image may appear huge as shit. If so, apologies~)
Angsize formula:
object degree size = 2*atan(Object_Size/(Panel_Height/tan(70/2)))
PoV-Mothra Angsize: 49.214099009724 degrees
PoV-Earth Angsize: 89.615848642702 degrees
Inputting those values into the angsizecalculator:
PoV-Mothra Distance: 191.05 metres PoV-Earth Distance: 3,558,700 metres
3,558,700 - 191.05 = 3,558,508.95 metres is the 'true' distance.
3,558,508.95/1000 = 3558.51 km
For the timeframe:
1:38:14 - Mothra last seen in atmosphere. 1:39:01 - First view of Mothra with Earth in background.
Thus, the sequence duration is 47 seconds.
3558.51/47 = 75.71 km/s or Mach 222.49
Correlates quite well with the previously calculated speed of Imago Battra, though obviously quite a bit higher at the same time. Again, outside of Imago Battra, no beneficial powerscaling to other daikaiju, IMO.
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Godzilla vs Mothra (1992) - Part 3 Energy Output (Huge Asteroid, Kinetic)* = 32.78 teratons [low-end] - 267.09 zettatons [high-end] of TNT equivalent. Speed (Imago Mothra, Flight)** = Mach 222.49
* = My immediate and overwhelming rationale for this would be to stick to the low-end unless one can irrefutably prove to the contrary; implication otherwise is that even Burning/Meltdown Godzilla is practically a flea in energy to the casual performance of Mothra and/or Battra.
** = Should also apply to Imago Battra's flight/combat speed.