Methodology

I designed this data visualisation largely because I was fed up of news agencies in 2016-17 reporting things like: "North Korea has launched an inter-continental ballistic missile. It reached an altitude of 4,500 km and flew a distance of 965 km, according to South Korea's military."

Now, if you're a layman, or even an engineer or physicist with not much knowledge of orbital mechanics or flight path geography, you probably have very little idea what those numbers mean - realistically speaking. You may be thinking:

"How high is that?"

"Could that fly far enough to reach [city/country]?"

"So did it go high enough to get into space, or what?"

Newspaper editors seem to have a habit of failing to qualify their reported numbers with basic relative comparisons, and it's even rarer to see a comprehensive diagram or visualisation presented to give readers an actual picture of why it's big news that a little East-Asian country is firing little rockets at distances you can't picture in your head unless you have facts memorised such as the radius and circumference of the Earth, the altitude at which the atmosphere stops and space starts, or how far up man-made satellites usually fly.

I had the idea to develop a diagram that showed, as simply as possible, the extent of these missile launch trajectories relative to the size and curvature of the Earth, its atmosphere and the immediate vicinity of low earth orbit, as well as distances and ranges relating to notable cities, nations and landmasses - all to scale.

I discovered that the Nuclear Threat Initiative has produced an excellent database and some great diagrams of their own to to document the ballistic missile launches and nuclear tests that North Korea has conducted since 1984. Link: http://nti.org/6928A

Using the data from the CNS North Korea Missile Test Database, I produced the above and below figures accounting for missile launch counts and trajectories, split up by projectile type.

(note: only successful launches are recorded, with the exception of satellite launch attempts.

 

For the main animated diagram presented at the top of the page, a mathematical framework was developed to project 2nd order parabolic trajectories about a circular representation of Earth, augmenting each parabola array (generated from missile path apogee and ground distance data) by a transformation matrix to effectively rotate the arc about the centre of the 'Earth' in my diagram. Important factors were accounted for such as the angle between launching and landing locations, and the sagitta (distance between apex of the arc length along the Earth's surface, and the midpoint of the chord created by launch and landing locations) adding to the apogee after developing from the basic flat ground assumption. Also note that atmospheric effects, vehicle mechanics such as staging and throttling, the velocity (and thus range) bias given by the angular momentum of the Earth, and other similar constraints are assumed to make negligible difference to the shape of the parabolas. This was a necessary assumption given that extremely little precise data is publicly known about these missile launches, other than the altitudes reached, ground distances travelled, and the type of launch vehicle. I also assumed that all missiles were fired from the same point in North Korea - I picked a point near Hamhūng which is a reasonable estimate near the approximate latitudinal and longitudinal midpoint of the country, and around the right latitude for most launch sites recorded in the database.

The MatLab code for generating and transforming the parabolas goes something like this, iterating through every missile launch datapoint from 1984-2017:

Given that there is (to my knowledge) limited public information about the precise landing locations of the missiles in either the Sea of Japan or the Pacific Ocean, and that I preferred to produce a 2D infographic for simplicity, making a precise model of a 3D Earth wasn't very feasible. The next best thing was to produce what I have called a 'circle of equivalent ranges' - which can be thought of as squashing the earth into a 2D disc and representing physical locations around its edge.

 

That said, there is more complexity in that the physical landmarks are not arranged by longitude, as this would defeat the point of depicting how far the missiles could reach realistically. It is not realistic to simply plot a straight line over a map in 2D projection. This would be a 'Rhumb Line' distance, intersecting all meridians at the same angle. This is often a feasible path, but mathematically it is not the shortest way of getting from A to B on a spherical Earth. In this case, it was necessary to use Great Circle distances, which are the true shortest paths between locations, taking into account the curvature of a sphere's surface - therefore, the most reasonable option for planning a missile guidance path or any efficient flight path in most scenarios. Represented below are rhumb lines in red and great circle distances in green, showing the difference on a Mercator projection (left) and on a spherical Earth model (right), for two journeys: London (UK) to Beijing (China), and Barcelona (Spain) to Wellington (New Zealand).

Rhumb lines appear to be the shortest routes on 2D maps due to their inherent projection distortions, but are bested by great circle distances in reality, over our near-spherical Earth: which is why all commercial flights take routes that, on a flat map, appear to travel greater distances: curving North and over in the Northern Hemisphere, or curving South and under in the Southern Hemisphere - like this (neat project btw). Great circle distances are categorically the shortest, most fuel-efficient routes around the Earth, and there is no doubt that North Korean missile launches would follow the same pattern.

So, to convey the appropriate information in my diagram, I arranged landmarks around the 'circle of equivalent ranges' in order of their great circle range from my designated virtual launch location near Hamhūng, as mentioned earlier. I used the ruler tool on Google Earth to obtain the values for distances to cities (points) and between land mass extremities (arc-swept), in great circle terms. Then I converted these distances into arcs (of Earth's radius) with centre angles that I could plot directly onto my circular diagram.

Finally, I added extra circles to represent the edge of earth's atmosphere (the Karman Line, about 100km altitude by consensus) and the average orbital altitude of the International Space Station (around 408km), to convey how far the missile launches have extended into space - which I think is not often realised!

atmosphere,

100km

© Michael Middleton 2018

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