Nuclear weapon yield

Log–log plot comparing the yield (in kilotonnes) and mass (in kilograms) of various nuclear weapons developed by the United States.

The explosive yield of a nuclear weapon is the amount of energy released such as blast, thermal, and nuclear radiation, when that particular nuclear weapon is detonated, usually expressed as a TNT equivalent (the standardized equivalent mass of trinitrotoluene which, if detonated, would produce the same energy discharge), either in kilotonnes (kt—thousands of tonnes of TNT), in megatonnes (Mt—millions of tonnes of TNT), or sometimes in terajoules (TJ). An explosive yield of one terajoule is equal to 0.239 kilotonnes of TNT. Because the accuracy of any measurement of the energy released by TNT has always been problematic, the conventional definition is that one kilotonne of TNT is held simply to be equivalent to 10¹² calories.

The yield-to-weight ratio is the amount of weapon yield compared to the mass of the weapon. The practical maximum yield-to-weight ratio for fusion weapons (thermonuclear weapons) has been estimated to six megatonnes of TNT per tonne of bomb mass (25 TJ/kg). Yields of 5.2 megatonnes/tonne and higher have been reported for large weapons constructed for single-warhead use in the early 1960s.^[1] Since then, the smaller warheads needed to achieve the increased net damage efficiency (bomb damage/bomb mass) of multiple warhead systems have resulted in increases in the yield/mass ratio for single modern warheads.

Examples of nuclear weapon yields

In order of increasing yield (most yield figures are approximate):

Comparative fireball radii for a selection of nuclear weapons.^{[citation needed]} Contrary to the image, which may depict the initial fireball radius, the maximum average fireball radius of Castle Bravo, a 15-megatonne yield surface burst, is 3.3 to 3.7 km (2.1 to 2.3 mi),^[7][8] and not the 1.42 km displayed in the image. Similarly the maximum average fireball radius of a 21-kilotonne low altitude airburst, which is the modern estimate for the Fat Man, is .21 to .24 km (0.13 to 0.15 mi),^[8][9] and not the 0.1 km of the image.

In comparison, the blast yield of the GBU-43 Massive Ordnance Air Blast bomb is 0.011 kt, and that of the Oklahoma City bombing, using a truck-based fertilizer bomb, was 0.002 kt. The estimated strength of the explosion at the Port of Beirut is 0.3-0.5 kt.^[10] Most artificial non-nuclear explosions are considerably smaller than even what are considered to be very small nuclear weapons.

Yield limits

The yield-to-mass ratio is the amount of weapon yield compared to the mass of the weapon. According to nuclear-weapons designer Ted Taylor, the practical maximum yield-to-mass ratio for fusion weapons is about 6 megatonnes of TNT per tonne (25 TJ/kg).^{[11][self-published source?]} The "Taylor limit" is not derived from first principles, and weapons with yields as high as 9.5 megatonnes per tonne have been theorized.^[12] The highest achieved values are somewhat lower, and the value tends to be lower for smaller, lighter weapons, of the sort that are emphasized in today's arsenals, designed for efficient MIRV use or delivery by cruise missile systems.

• The 25 Mt yield option reported for the B41 would give it a yield-to-mass ratio of 5.1 megatonnes of TNT per tonne. While this would require a far greater efficiency than any other current U.S. weapon (at least 40% efficiency in a fusion fuel of lithium deuteride), this was apparently attainable, probably by the use of higher than normal lithium-6 enrichment in the lithium deuteride fusion fuel. This results in the B41 still retaining the record for the highest yield-to-mass weapon ever designed.^[12]
• The W56 demonstrated a yield-to-mass ratio of 4.96 kt per kilogram of device mass, and very close to the predicted 5.1 kt/kg achievable in the highest yield-to-mass weapon ever built, the 25-megatonne B41. Unlike the B41, which was never proof-tested at full yield, the W56 demonstrated its efficiency in the XW-56X2 Bluestone shot of Operation Dominic in 1962,^[13] thus, from information available in the public domain, the W56 may hold the distinction of demonstrating the highest efficiency in a nuclear weapon to date.
• In 1963 DOE declassified statements that the U.S. had the technological capability of deploying a 35 Mt warhead on the Titan II, or a 50–60 Mt gravity bomb on B-52s. Neither weapon was pursued, but either would require yield-to-mass ratios superior to a 25 Mt Mk-41. This may have been achievable by utilizing the same design as the B41 but with the addition of a HEU tamper,^{[citation needed]} in place of the cheaper but lower energy density U-238 tamper, which is the most commonly used tamper material in Teller–Ulam thermonuclear weapons.
• For current smaller US weapons, yield is 600 to 2200 kilotonnes of TNT per tonne. By comparison, for the very small tactical devices such as the Davy Crockett it was 0.4 to 40 kilotonnes of TNT per tonne. For historical comparison, for Little Boy the yield was only 4 kilotonnes of TNT per tonne, and for the largest Tsar Bomba, the yield was 2 megatonnes of TNT per tonne (deliberately reduced from about twice as much yield for the same weapon, so there is little doubt that this bomb as designed was capable of 4 megatonnes per tonne yield).
• The largest pure-fission bomb ever constructed, Ivy King, had a 500 kilotonne yield,^[2] which is probably in the range of the upper limit on such designs.^{[citation needed]} Fusion boosting could likely raise the efficiency of such a weapon significantly, but eventually all fission-based weapons have an upper yield limit due to the difficulties of dealing with large critical masses. (The UK's Orange Herald was a very large boosted fission bomb, with a yield of 800 kilotonnes.) However, there is no known upper yield limit for a fusion bomb.

Large single warheads are seldom a part of today's arsenals, since smaller MIRV warheads, spread out over a pancake-shaped destructive area, are far more destructive for a given total yield, or unit of payload mass. This effect results from the fact that destructive power of a single warhead on land scales approximately only as the cube root of its yield, due to blast "wasted" over a roughly hemispherical blast volume, while the strategic target is distributed over a circular land area with limited height and depth. This effect more than makes up for the lessened yield/mass efficiency encountered if ballistic missile warheads are individually scaled down from the maximal size that could be carried by a single-warhead missile.

Yield Efficiency

The efficiency of an atomic bomb is the ratio of the actual yield to the theoretical maximum yield of the atomic bomb. Not all atomic bombs possess the same yield efficiency as each individual bombs design plays a large role in how efficient it can be. In order to maximize yield efficiency one must make sure to assemble the critical mass correctly, as well as implementing instruments such as tampers or initiators in the design. A tamper is typically made of uranium and it holds the core together using its inertia. It is used to prevent the core from separating too soon to generate maximum fission, so as not to cause a "fizzle". The initiator is a source of neutrons either inside of the core, or on the outside of the bomb, and in this case it shoots neutrons at the core at the moment of detonation. It is essentially kick starting the reaction so the maximum fission reactions can occur to maximize yield. ^[14]

Milestone nuclear explosions

The following list is of milestone nuclear explosions. In addition to the atomic bombings of Hiroshima and Nagasaki, the first nuclear test of a given weapon type for a country is included, as well as tests that were otherwise notable (such as the largest test ever). All yields (explosive power) are given in their estimated energy equivalents in kilotons of TNT (see TNT equivalent). Putative tests (like Vela incident) have not been included.

Note

• "Staged" refers to whether it was a "true" thermonuclear weapon of the so-called Teller–Ulam configuration or simply a form of a boosted fission weapon. For a more complete list of nuclear test series, see List of nuclear tests. Some exact yield estimates, such as that of the Tsar Bomba and the tests by India and Pakistan in 1998, are somewhat contested among specialists.

Calculating yields and controversy

Yields of nuclear explosions can be very hard to calculate, even using numbers as rough as in the kilotonne or megatonne range (much less down to the resolution of individual terajoules). Even under very controlled conditions, precise yields can be very hard to determine, and for less controlled conditions the margins of error can be quite large. For fission devices, the most precise yield value is found from "radiochemical/Fallout analysis"; that is, measuring the quantity of fission products generated, in much the same way as the chemical yield in chemical reaction products can be measured after a chemical reaction. The radiochemical analysis method was pioneered by Herbert L. Anderson.

For nuclear explosive devices where the fallout is not attainable or would be misleading, neutron activation analysis is often employed as the second most accurate method, with it having been used to determine the yield of both Little Boy^[16][17] and thermonuclear Ivy Mike's^[18] respective yields.

Yields can also be inferred in a number of other remote sensing ways, including scaling law calculations based on blast size, infrasound, fireball brightness (Bhangmeter), seismographic data (CTBTO),^[19] and the strength of the shock wave.

Enrico Fermi famously made a (very) rough calculation of the yield of the Trinity test by dropping small pieces of paper in the air and measuring how far they were moved by the blast wave of the explosion; that is, he found the blast pressure at his distance from the detonation in pounds per square inch, using the deviation of the papers' fall away from the vertical as a crude blast gauge/barograph, and then with pressure X in psi, at distance Y, in miles figures, he extrapolated backwards to estimate the yield of the Trinity device, which he found was about 10 kilotonnes of blast energy.^[23][24]

Fermi later recalled:

I was stationed at the Base Camp at Trinity in a position about ten miles [16 km] from the site of the explosion... About 40 seconds after the explosion the air blast reached me. I tried to estimate its strength by dropping from about six feet small pieces of paper before, during, and after the passage of the blast wave. Since, at the time, there was no wind[,] I could observe very distinctly and actually measure the displacement of the pieces of paper that were in the process of falling while the blast was passing. The shift was about 2 1/2 meters, which, at the time, I estimated to correspond to the blast that would be produced by ten thousand tonnes of TNT.^[25][26][27]

The surface area (A) and volume (V) of a sphere are [math]\displaystyle{ A = 4\pi r^2 }[/math] and [math]\displaystyle{ V = \frac{4}{3} \pi r^3 }[/math] respectively.

The blast wave, however, was likely assumed to grow out as the surface area of the approximately hemispheric near surface burst blast wave of the Trinity gadget. The paper is moved 2.5 meters by the wave, so the effect of the Trinity device is to displace a hemispherical shell of air of volume 2.5 m × 2π(16 km)². Multiply by 1 atm to get an energy of 4.1×10¹⁴ J ~ 100 kT TNT.^[quantify]

This photograph of the Trinity blast, captured by Berlyn Brixner, was used by G. I. Taylor to estimate its yield.

A good approximation of the yield of the Trinity test device was obtained in 1950 by the British physicist G. I. Taylor from simple dimensional analysis and an estimation of the heat capacity for very hot air. Taylor had initially done this highly classified work in mid-1941 and published an article with an analysis of the Trinity data fireball when the Trinity photograph data was declassified in 1950 (after the USSR had exploded its own version of this bomb).

Taylor noted that the radius R of the blast should initially depend only on the energy E of the explosion, the time t after the detonation, and the density ρ of the air. The only equation having compatible dimensions that can be constructed from these quantities is

[math]\displaystyle{ R = S\left(\frac{E t^2}{\rho}\right)^{\frac{1}{5}}. }[/math]

Here S is a dimensionless constant having a value approximately equal to 1, since it is low-order function of the heat capacity ratio or adiabatic index

[math]\displaystyle{ \gamma = \frac{C_P}{C_V}, }[/math]

which is approximately 1 for all conditions.

Using the picture of the Trinity test shown here (which had been publicly released by the U.S. government and published in Life magazine), using successive frames of the explosion, Taylor found that R⁵/t² is a constant in a given nuclear blast (especially between 0.38 ms, after the shock wave has formed, and 1.93 ms, before significant energy is lost by thermal radiation). Furthermore, he estimated a value for S numerically at 1.

Thus, with t = 0.025 s and the blast radius being 140 metres, and taking ρ to be 1 kg/m³ (the measured value at Trinity on the day of the test, as opposed to sea-level values of approximately 1.3 kg/m³) and solving for E, Taylor obtained that the yield was about 22 kilotonnes of TNT (90 TJ). This does not take into account the fact that the energy should only be about half this value for a hemispherical blast, but this very simple argument did agree to within 10% with the official value of the bomb's yield in 1950, which was 20 kilotons of TNT (84 TJ) (see G. I. Taylor, Proc. Roy. Soc. London A 200, pp. 235–247 (1950)).

A good approximation to Taylor's constant S for [math]\displaystyle{ \gamma }[/math] below about 2 is^[28]

[math]\displaystyle{ S = \left(\frac{75(\gamma - 1)}{8\pi}\right)^{\frac{1}{5}}. }[/math]

The value of the heat capacity ratio here is between the 1.67 of fully dissociated air molecules and the lower value for very hot diatomic air (1.2), and under conditions of an atomic fireball is (coincidentally) close to the STP (standard) gamma for room-temperature air, which is 1.4. This gives the value of Taylor's S constant to be 1.036 for the adiabatic hypershock region where the constant R⁵/t² condition holds.

As it relates to fundamental dimensional analysis, if one expresses all the variables in terms of mass M, length L, and time T:^[29]

[math]\displaystyle{ E = [M \cdot L^2 \cdot T^{-2}] }[/math]

(think of the expression for kinetic energy, [math]\displaystyle{ E = mv^2/2 }[/math]),

[math]\displaystyle{ \rho = [M \cdot L^{-3}], }[/math]

[math]\displaystyle{ t = [T], }[/math]

[math]\displaystyle{ r = [L], }[/math]

and then derive an expression for, say, E, in terms of the other variables, by finding values of [math]\displaystyle{ \alpha }[/math], [math]\displaystyle{ \beta }[/math], and [math]\displaystyle{ \gamma }[/math] in the general relation

[math]\displaystyle{ E = \rho^\alpha \cdot t^\beta \cdot r^\gamma }[/math]

such that the left and right sides are dimensionally balanced in terms of M, L, and T (i.e., each dimension has the same exponent on both sides).

Other methods and controversy

Where these data are not available, as in a number of cases, precise yields have been in dispute, especially when they are tied to questions of politics. The weapons used in the atomic bombings of Hiroshima and Nagasaki, for example, were highly individual and very idiosyncratic designs, and gauging their yield retrospectively has been quite difficult. The Hiroshima bomb, "Little Boy", is estimated to have been between 12 and 18 kilotonnes of TNT (50 and 75 TJ) (a 20% margin of error), while the Nagasaki bomb, "Fat Man", is estimated to be between 18 and 23 kilotonnes of TNT (75 and 96 TJ) (a 10% margin of error).

Such apparently small changes in values can be important when trying to use the data from these bombings as reflective of how other bombs would behave in combat, and also result in differing assessments of how many "Hiroshima bombs" other weapons are equivalent to (for example, the Ivy Mike hydrogen bomb was equivalent to either 867 or 578 Hiroshima weapons — a rhetorically quite substantial difference — depending on whether one uses the high or low figure for the calculation).

Other disputed yields have included the massive Tsar Bomba, whose yield was claimed between being "only" 50 megatonnes of TNT (210 PJ) or at a maximum of 57 megatonnes of TNT (240 PJ) by differing political figures, either as a way for hyping the power of the bomb or as an attempt to undercut it.

See also

• Effects of nuclear explosions — goes into detail about different effects at different yields
• List of nuclear weapons

References

External links

• "What was the yield of the Hiroshima bomb?" (excerpt from official report)
• "General Principles of Nuclear Explosions", Chapter 1 in Samuel Glasstone and Phillip Dolan, eds., The Effects of Nuclear Weapons, 3rd edn. (Washington D.C.: U.S. Department of Defense/U.S. Energy Research and Development Administration, 1977); provides information about the relationship of nuclear yields to other effects (radiation, damage, etc.).
• "THE MAY 1998 POKHRAN TESTS: Scientific Aspects", discusses different methods used to determine the yields of the Indian 1998 tests.
• Discusses some of the controversy over the Indian test yields
• "What are the real yields of India's nuclear tests?" from Carey Sublette's NuclearWeaponArchive.org
• High-Yield Nuclear Detonation Effects Simulator

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