Tag Archives: North Korea

Disease, atom bombs, and R-naught

A key indicator of the speed and likelihood of a major disease outbreak is the number of people that each infected person is likely to infect. This infection number is called R-naught, or Ro; it is shown in the table below for several major plague diseases.

R-naught - communicability for several contagious diseases, CDC.

R-naught – infect-ability for several contagious diseases, CDC.

Of the diseases shown, measles is the most communicable, with an Ro of 12 to 18. In an unvaccinated population, one measles-infected person will infect 12- 18 others: his/her whole family and/ or most of his/her friends. After two weeks or so of incubation, each of the newly infected will infect another 12-18. Traveling this way, measles wiped out swaths of the American Indian population in just a few months. It was one of the major plagues that made America white.

While Measles is virtually gone today, Ebola, SARS, HIV, and Leprosy remain. They are far less communicable, and far less deadly, but there is no vaccine. Because they have a low Ro, outbreaks of these diseases move only slowly through a population with outbreaks that can last for years or decades.

To estimate of the total number of people infected, you can use R-naught and the incubation-transmission time as follows:

Ni = Row/wt

where Ni is the total number of people infected at any time after the initial outbreak, w is the number of weeks since the outbreak began, and wt is the average infection to transmission time in weeks.

For measles, wt is approximately 2 weeks. In the days before vaccine, Ro was about 15, as on the table, and

Ni = 15w/2.

In 2 weeks, there will be 15 measles infected people, in 4 weeks there will be 152, or 225, and in 6 generations, or 12 weeks, you’d expect to have 11.39 million. This is a real plague. The spread of measles would slow somewhat after a few weeks, as the infected more and more run into folks who are already infected or already immune. But even when the measles slowed, it still infected quite a lot faster than HIV, Leprosy, or SARS (SARS is a form of Influenza). Leprosy is particularly slow, having a low R-naught, and an infection-transmission time of about 20 years (10 years without symptoms!).

In America, more or less everyone is vaccinated for measles. Measles vaccine works, even if the benefits are oversold, mainly by reducing the effective value of Ro. The measles vaccine is claimed to be 93% effective, suggesting that only 7% of the people that an infected person meets are not immune. If the original value of Ro is 15, as above, the effect of immunization is to reduce the value Ro in the US today to effectively 15 x 0.07 = 1.05. We can still  have measles outbreaks, but only on a small-scale, with slow-moving outbreaks going through pockets of the less-immunized. The average measles-infected person will infect only one other person, if that. The expectation is that an outbreak will be captured by the CDC before it can do much harm.

Short of a vaccine, the best we can do to stop droplet-spread diseases, like SARS, Leprosy, or Ebola is by way of a face mask. Those are worn in Hong Kong and Singapore, but have yet to become acceptable in the USA. It is a low-tech way to reduce Ro to a value below 1.0, — if R-naught is below 1.0, the disease dies out on its own. With HIV, the main way the spread was stopped was by condoms — the same, low tech solution, applied to sexually transmitted disease.

Image from VCE Physics, https://sites.google.com/site/coyleysvcephysics/home/unit-2/optional-studies/26-how-do-fusion-and-fission-compare-as-viable-nuclear-energy-power-sources/fission-and-fusion---lesson-2/chain-reactions-with-dominoes

Progress of an Atom bomb going off. Image from VCE Physics, visit here

As it happens, the explosion of an atom bomb follows the same path as the spread of disease. One neutron appears out of somewhere, and splits a uranium or plutonium atom. Each atom produces two or three more neutrons, so that we might think that R-naught = 2.5, approximately. For a bomb, Ro is found to be a bit lower because we are only interested in fast-released neutrons, and because some neutrons are lost. For a well-designed bomb, it’s OK to say that Ro is about 2.

The progress of a bomb going off will follow the same math as above:

Nn = Rot/nt

where Nn is the total number of neutrons at any time, t is the average number of nanoseconds since the first neutron hit, and nt is the transmission time — the time it takes between when a neuron is given off and absorbed, in nanoseconds.

Assuming an average neutron speed of 13 million m/s, and an average travel distance for neutrons of about 0.1 m, the time between interactions comes out to about 8 billionths of a second — 8 ns. From this, we find the number of neutrons is:

Nn = 2t/8, where t is time measured in nanoseconds (billionths of a second). Since 1 kg of uranium contains about 2 x 1024 atoms, a well-designed A-bomb that contains 1 kg, should take about 83 generations (283 = 1024). If each generation is 8 ns, as above, the explosion should take about 0.664 milliseconds to consume 100% of the fuel. The fission power of each Uranium atom is about 210 MeV, suggesting that this 1 kg bomb could release 16 billion Kcal, or as much explosive energy as 16 kTons of TNT, about the explosive power of the Nagasaki bomb (There are about 38 x10-24 Kcal/eV).

As with disease, this calculation is a bit misleading about the ease of designing a working atomic bomb. Ro starts to get lower after a significant faction of the atoms are split. The atoms begin to move away from each other, and some of the atoms become immune. Once split, the daughter nuclei continue to absorb neutrons without giving off either neutrons or energy. The net result is that an increased fraction of neutrons that are lost to space, and the explosion dies off long before the full power is released.

Computers are very helpful in the analysis of bombs and plagues, as are smart people. The Manhattan project scientists got it right on the first try. They had only rudimentary computers but lots of smart people. Even so, they seem to have gotten an efficiency of about 15%. The North Koreans, with better computers and fewer smart people took 5 tries to reach this level of competence (analyzed here). They are now in the process of developing germ-warfare — directed plagues. As a warning to them, just as it’s very hard to get things right with A-bombs, it’s very hard to get it right with disease; people might start wearing masks, or drinking bottled water, or the CDC could develop a vaccine. The danger, if you get it wrong is the same as with atom bombs: the US will not take this sort of attack lying down.

Robert Buxbaum, January 18, 2019. One of my favorite authors, Issac Asimov, died of AIDS; a slow-moving plague that he contacted from a transfusion. I benefitted vastly from Isaac Asimov’s science and science fiction, but he wrote on virtually every topic. My aim is essays that are sort-of like his, but more mathematical.

Estimating the strength of an atom bomb

As warfare is a foundation of engineering, I thought I’d use engineering to evaluate the death-dealing power of North Korea’s atomic/hydrogen bomb, tested September 3, 2017. The key data in evaluating a big bomb is its seismic output. They shake the earth like earthquakes do, and we measure the power like earthquakes, using seismometers. I’ve seen two seismographs comparing the recent bomb to the previous. One of these, below, is from CTBTO, the Center for Test Ban Treaty Oversight, via a seismometer in western Kazakhstan (see original data and report).

Seismic output of all North Korean nuclear tests.

Seismic output, to scale, of all declared DPNK nuclear tests as observed from IMS station AS-59 in Western Kazakhstan

North Korea’s previous bomb, exploded 9 September 2016, was reported to be slightly more powerful than the ones we dropped on Hiroshima and Nagasaki, suggesting it was about 20 kilotons. According to CTBTO, it registered 5.3 on the Richter scale. The two tests before that appear somewhat less powerful, perhaps 7-10 kilotons, and the two before that appear as dismal failures — fizzles, in atomic bomb parlance. The MOAB bomb, by comparison, was 9 Tons, or 0.009 kiloTons, a virtual non-entity.

To measure the output of the current bomb, I place a ruler on my screen and measure the maximum distance between the top to bottom wiggles. I find that this bomb’s wiggles measure 5 cm, while the previous measures 5 mm. This bomb’s wiggles are ten times bigger, and from this I determine that this explosion registered 6.3 on the Richter scale, 1.0 more than the previous — the Richter scale is the logarithmic measure of the wiggle amplitude, so ten times the shake magnitude  is an addition of 1.0 on the scale. My calculation of 6.3 exactly matches that of the US geological survey. The ratio of wiggle heights was less on the, NORSAR seismometer, Norway, see suggesting 5.8 to 5.9 on the Richter scale. The European agencies have taken to reporting 6.1, an average value, though they originally reported only the 5.8 from NORSAR, and a bomb power commensurate with that.

We calculate the bomb power from the Richter-scale measure, or the ratio of the wiggles. Bomb power is proportional to wiggle height to the 3/2 power. Using the data above, ten times the wiggle, this bomb appears to be 10^3/2 = 31.6 times as powerful as the last, or 31.6 x 20kTon = 630kTon (630,000 tons of TNT). If we used the European value of 6.1, the calculated power would be about half this, 315 kTons, and if we used the NORSAR’s original value, it would suggest the bomb had less than half this power. Each difference of 0.2 on the Richter scale is a factor of two in power. For no obvious reason we keep reporting 120 to 160 kTons.

NORSAR comparison of North Korean bomb blasts

NORSAR comparison of North Korean blasts — suggests the current bomb is smaller; still looks like hydrogen.

As it happens, death power is proportional to the kiloton power, other things being equal. The bombs we dropped on Hiroshima and Nagasaki were in the 15 to 20 kTon range and killed 90,000 each. Based on my best estimate of the bomb, 315 kTons, I estimate that it would kill 1.6 million people if used on an industrial city, like Seoul, Yokohama, or Los Angeles. In my opinion, this is about as big a bomb as any rational person has reason to make (Stalin made bigger, as did Eisenhower).

We now ask if this is an atom bomb or a hydrogen-fusion bomb. Though I don’t see any war-making difference, if it’s a hydrogen bomb that would make our recent treaty with Iran look bad, as it gave Iran nuclear fusion technology — I opposed the treaty based on that. Sorry to say, from the seismic signature it looks very much like a hydrogen bomb. The only other way to get to this sort of high-power explosion is via a double-acting fission bomb where small atom bomb sets off a second, bigger fission bomb. When looking at movies of Eisenhower-era double-acting explosions, you’ll notice that the second, bigger explosion follows the first by a second or so. I see no evidence of this secondary-delay in the seismic signature of this explosion, suggesting this was a hydrogen bomb, not a double. I expect Iran to follow the same path in 3-4 years.

As a political thought, it seems to me that the obvious way to stop North Korea would be to put pressure on China by making a military pact with Russia. Until that is done, China has little to fear from a North Korean attack to the south. Of course, to do that we’d likely have to cut our support of NATO, something that the Germans fear. This is a balance-of-power solution, the sort that works, short of total annihilation. It was achieved at the congress of Vienna, at the treaty of Ghent, and by Henry Kissinger through détente. It would work again. Without it, I see the Korean conflict turning hot again, soon.

Robert Buxbaum, September 11, 2017.