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In the film Empire of the Sun (1987), the Nagasaki atom bomb explosion is depicted as being visible (albeit faintly) from Shanghai. Would that really have been possible? According to Google, the two cities are about 810 km apart.

Update: I've uploaded the clip from the movie to YouTube.

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This is not possible. And this has nothing to do with the actual brightness of explosion. A similar explosion on the Moon will probably be visible from the earth if the weather conditions are good and if you know when and where to look.

According to the Wikipedia, the Nagasaki bomb was exploded at the height of 503 m, about 1/2 km. To be directly visible from 810 km distance it had to be exploded at the height more than 60 km.

(Calculation: 810 km=437 nautical miles=7.289 degrees. The radius of Earth is 6370 km; 6370/cos 7.289=6423. Subtract 6370, we get 63 km. This is the height at which the bomb had to be exploded to be visible at 810 km. I neglected refraction but the difference between 60 km and 0.5 km is so large that refraction can be neglected.)

Of course there are described cases of strong anomalous refraction, but they usually occur in polar regions http://en.wikipedia.org/wiki/Novaya_Zemlya_effect, and one does not expect it on the latitude of Japan.

EDIT. A suggestion in a comment was that a reflection from a cloud could be seen. Again, the information found in Wikipedia shows that clouds do not reach such heights, except in polar regions (stratospheric clouds). In addition to this, a reflection would be much weaker than the original flash, and perhaps cannot be seen at 800 km.

EDIT2. I am surprised by the number of comments to this answer. Nobody would discuss the question whether the Great Fire of London could be visible from Switzerland (about the same distance), because of some super-unusual refraction or some super-high clouds, or whatever. The only difference is that a nuclear explosion involves a very bright (and very short) flash. But this difference is irrelevant.

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    My first reaction on seeing this question was to suggest it be moved to a math or physics stack to raise the odds of someone being able to provide this exact calculation. Good job. – T.E.D. Mar 28 '15 at 14:40
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    When you fly an airplane at 10 km altotude, do you ever see clouds ABOVE you? – Alex Mar 28 '15 at 18:26
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    @Alex: Actually, yes; wispy cirrus clouds can exist at much higher altitudes than even planes commercial transports fly, up to 60,000 feet or higher. – Pieter Geerkens Mar 28 '15 at 21:21
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    @Alex: Unfortunately, I can't fly my plane at anywhere near that altitude (its ceiling is about 15K ft). However, per Wikipedia cumulonimbus clouds (thunderheads) can reach altitudes of 75K ft/23K m: en.wikipedia.org/wiki/Cumulonimbus_cloud#Appearance Also consider the case of high-altitude clouds midway between the explosion and viewing point. – jamesqf Mar 28 '15 at 22:13
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    Why don't you simply suggest that the light was reflected from a UFO which was passing by? – Alex Mar 29 '15 at 1:19
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The link provided by @T.E.D. combined with the observations of Major Charles Sweeney that the height of the flash, off the cirrus clouds over Nagasaki, was "at least 6 miles (or 31,000 feet) [above the explosion]" giving a refraction-free visibility for such a reflected flash as about 230 miles or 370 km.

The underside of the great clouds over Nagasaki was amber-tinted, as though reflecting the conflagaration at least 6 miles below. Beneath the top cloud mass, white in colour, there gradually climbed a turbulent pillar of black smoke and dust which emitted a second fireball less vivid than the first.

distance in miles =~ sqrt( 7 * height-in-feet / 4) = sqrt(7 * 31,000 / 4)  
                  =~ 230 miles ~ 380 km

However, as noted further down in the link provided by T.E.D.:

Unfortunately, the refraction varies considerably from day to day, and from one place to another. It is particularly variable over water: because of the high heat capacity of water, the air is nearly always at a different temperature from that of the water, so there is a thermal boundary layer, in which the temperature gradient is far from uniform.

...

In conditions that produce superior mirages, there are inversion layers in which the ray curvature exceeds that of the Earth. Then, in principle, you can see infinitely far — there really is no horizon.

...

However, there is one situation in which objects can be made out at great distances: when they are silhouetted against a bright background, such as the setting Sun, or (just after sunset) a bright twilit sky.

...

What's the record for visibility without help from the silhouetting effect? I think that might belong to the report of the expedition led by Korzenewsky (1923), who reported seeing snow-capped peaks of a mountain range 750 km away. Conditions were perfect: the lower atmosphere was in shadow at sunset; the peaks were quite high (4650 meters, or over 15,000 feet); they were covered with white snow, increasing their visibility; and there must also have been considerable looming to bring these distant features above the observers' horizon.

This observation by Korzenewsky is very similar in circumstances to the reported sighting from Shanghai, so I would not rule it out completely without further investigation of the meteorological conditions that day. It was late summer, with clear calm conditions (breeze < 5 mph at ground zero), which is typical of mirage conditions.

This link goes into the detailed meteorological conditions necessary for a long-distance superior mirage, such as the reported observation must have been. The required temperature inversion could easily have existed in late morning in early August over the cooler waters of the Sea of Japan.

Update:
Sweeney can accurately say that "the underside of the great clouds over Nagasaki" were at least 6 miles high, because he himself was at 30,000 feet for the bombing run and aftermath.

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Note that the atomic cloud after the explosion is said to have reached a height of 70,000 feet (about 21 km). Although this still appears to be too low, with refraction and other effects, it might have been possible to see it from Shanghai.

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With Earth curvature in mind, 810km gives a 50km high curve. So the earth is flat or the clip in the movie is full of crap. On the other hand, after doing some research, they state nuke explosions can be seen from 600 miles distance, but 600 miles gives a curve of 71km, I don't think we can see a plume, unless the earth is flat.

Curve calculator here http://dizzib.github.io/earth/curve-calc/index.html

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    You have ignored the mirage and refraction effects that have commonly been noted in these examples. – Pieter Geerkens Jul 29 '15 at 21:10

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