Nuclear weapon yield

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

The explosive yield of a nuclear weapon is the amount of energy released 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 kilotons (kt—thousands of tons of TNT), in megatons (Mt—millions of tons of TNT), or sometimes in terajoules (TJ). An explosive yield of one terajoule is 0.239 kt of TNT. Because the accuracy of any measurement of the energy released by TNT has always been problematic, the conventional definition accepted since the dawn of the Atomic Age is that one kiloton of TNT is simply to be 10¹² calories equivalent, which is only approximately equal to the energy yield of 1,000 tons of TNT.

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 megatons of TNT per metric ton of bomb mass (25 TJ/kg). Yields of 5.2 megatons/ton and higher have been reported for large weapons constructed for single-warhead use in the early 1960s. Since this time, the smaller warheads needed to achieve the increased net damage efficiency (bomb damage/bomb weight) of multiple warhead systems, has resulted in decreases in the yield/weight ratio for single modern warheads.

Examples of nuclear weapon yields

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

Bomb	Yield		Notes
	kt TNT	TJ
Davy Crockett	0.01	0.042	Variable yield tactical nuclear weapon—mass only 23 kg (51 lb), lightest ever deployed by the United States (same warhead as Special Atomic Demolition Munition and GAR-11 Nuclear Falcon missile).
Hiroshima's "Little Boy" gravity bomb	13–18	54–75	Gun type uranium-235 fission bomb (the first of the two nuclear weapons that have been used in warfare).
Nagasaki's "Fat Man" gravity bomb	20–22	84–92	Implosion type plutonium-239 fission bomb (the second of the two nuclear weapons used in warfare).
W76 warhead	100	420	Twelve of these may be in a MIRVed Trident II missile; treaty limited to eight.
W87 warhead	300	1,300	Ten of these were in a MIRVed LGM-118A Peacekeeper.
W88 warhead	475	1,990	Twelve of these may be in a Trident II missile (treaty limited to eight).
Ivy King device	500	2,100	Most powerful pure fission bomb, 60 kg uranium, implosion type.
B83 nuclear bomb	variable		Up to 1.2 megatonnes of TNT (5.0 PJ); most powerful US weapon in active service.
B53 nuclear bomb	9,000	38,000	Was the most powerful US bomb until 2010; it was not in active service for many years before 2010, but during that time, 50 were retained as part of the "Hedge" portion of the Enduring Stockpile until completely dismantled in 2011; a variant of the two-stage B61 is the B53 replacement in the bunker-busting role; the B53 was similar to the W-53 warhead that has been used in the Titan II Missile; decommissioned in 1987.
Castle Bravo device	15,000	63,000	Most powerful US test.
EC17/Mk-17, the EC24/Mk-24, and the B41 (Mk-41)	various		Most powerful US weapons ever: 25 megatonnes of TNT (100 PJ); the Mk-17 was also the largest by size and mass: about 20 short tons (18,000 kg); The Mk-41 or B41 had a mass of 4800 kg and yield of 25 Mt, this equates to being the highest yield-to-weight weapon ever produced; all were gravity bombs carried by the B-36 bomber (retired by 1957).
The entire Operation Castle nuclear test series	48,200	202,000	The highest-yielding test series conducted by the US.
Tsar Bomba device	50,000	210,000	USSR, most powerful nuclear weapon ever detonated, yield of 50 megatons, (50 million tons of tnt). In its "final" form (i.e. with a depleted uranium tamper instead of one made of lead) it would have been 100 megatons.
All nuclear testing as of 1996	510,300	2,135,000	Total energy expended during all nuclear testing.[1]

Comparative fireball radii for a selection of nuclear weapons. Contrary to the image, which may depict the initial fireball radius, the maximum average fireball radius of Castle Bravo, a 15 megaton yield surface burst, is 3.3 to 3.7 km (2.1 to 2.3 mi), and not the 1.42 km displayed in the image. Similarly the maximum average fireball radius of a 21 kiloton low altitude airburst, which is the modern estimate for the fat man, is .21 to .24 km (0.13 to 0.15 mi), and not the 0.1 km of the image.

As a 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. Most artificial non-nuclear explosions are considerably smaller than even what are considered to be very small nuclear weapons.

Yield limits

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 is about 6 megatons of TNT per metric ton (25 TJ/kg). 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-weight ratio of 5.1 megatons of TNT per metric ton. 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-weight weapon ever designed.
• While the W56 demonstrated a yield-to-weight ratio of 4.96kt per kg of device weight, and very close to the predicted 5.1kt/kg achievable in the highest yield to weight weapon ever built, the 25 megaton 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, 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-weight 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, 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 kilotons of TNT per metric ton. By comparison, for the very small tactical devices such as the Davy Crockett it was 0.4 to 40 kilotons of TNT per metric ton. For historical comparison, for Little Boy the yield was only 4 kilotons of TNT per metric ton, and for the largest Tsar Bomba, the yield was 2 megatons of TNT per metric ton (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 megatons per ton yield).
• The largest pure-fission bomb ever constructed Ivy King had a 500 kiloton yield, which is probably in the range of the upper limit on such designs. 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 750 kilotons.) However, there is no known upper yield limit for a fusion bomb.
• Because the maximum theoretical yield-to-weight ratio is about 6 megatons of TNT per metric ton, and the maximum achieved ratio was 5.2 megatons of TNT per metric ton, there is a practical limit on the total yield for an air-delivered weapon. Most later generation weapons have eliminated the very heavy casing once thought needed for the nuclear reactions to occur efficiently, and this has greatly increased the achievable yield-to-weight ratio. For example, the Mk-36 bomb as built had a yield-to-weight ratio of 1.25 megatons of TNT per metric ton. If the 12,000 pound casing of the Mk-36 were reduced by 2/3s, the yield-to-weight ratio would have been 2.3 megatons of TNT per metric ton, which is about the same as the later generation, much lighter 9 megaton Mk/B-53 bomb.
• Delivery size limits can be estimated to ascertain limits to delivery of extremely high yield weapons. If the full 250 metric ton payload of the Antonov An-225 aircraft could be used, a 1.3 gigaton bomb could be delivered. Likewise, the maximum limit of a missile-delivered weapon is determined by the missile gross payload capacity. The large Russian SS-18 ICBM has a payload capacity of 7,200 kg, so the calculated maximum delivered yield would be 37.4 megatons of TNT. A Saturn V-scale missile could deliver over 120 tons, giving a calculated maximum yield of about 700 megatons.

Again, it is helpful for understanding to emphasize that 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 2/3 power of its yield, due to blast "wasted" over a spherical 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/weight efficiency encountered if ballistic missile warheads are individually scaled-down from the maximal size that could be carried by a single-warhead missile.

Calculating yields and controversy

See also: Fermi problem

Yields of nuclear explosions can be very hard to calculate, even using numbers as rough as in the kiloton or megaton 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.

While 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 and thermonuclear Ivy Mike's 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), and the strength of the shock wave.

Standard bomb's energy distribution, in the "moderate" kiloton range, near sea level

Blast	50%
Thermal energy	35%
Initial ionizing radiation	5%
Residual fallout radiation	10%

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 kiloton of blast energy.

Fermi later recalled that:

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 tons of T.N.T.

The surface area(A) and volume(V) of a sphere are: $A=4\pi r^{2}$ $\!V={\frac {4}{3}}\pi r^{3}$

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(14 km)². Multiply by 1 atm to get energy of 3×10¹⁴ J ~ 80 kT TN.

Picture of the blast, captured by Berlyn Brixner were used by G.I. Taylor to estimate the yield of the device during the Trinity test

A good approximation of the yield of the Trinity test device was obtained in 1950 from simple dimensional analysis as well as an estimation of the heat capacity for very hot air, by the British physicist G. I. Taylor. Taylor had initially done this highly classified work in mid-1941, and published a paper which included 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 number having dimensions of length that can be constructed from these quantities is:

$R=S\left({\frac {{E{t}}^{2}}{\rho }}\right)^{\frac {1}{5}}$

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

$Y={\biggl (}{Cp \over \Lsh Cv}{\biggr )}$ ,

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 was 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 kilotons 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 γ below about 2 is:

$S={\Biggl (}{75(y-1) \over \ 8\pi }{\Biggr )}^{\frac {1}{5}}$

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 S.T.P. (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.

A derivation of the Taylor formula is also said to be capable of determining the arrival time of the blast wave as a function of yield.

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

$E=[{M}.{L^{2}}.{T^{-2}}]$

(think of the expression for kinetic energy, $E={\frac {mv^{2}}{2}}$

$\rho =[{M}\cdot {L^{-3}}]$

$t=[T]$

$r=[L]$

and then derive an expression for, say, E, in terms of the other variables, by finding values of $\alpha$ $\beta$ and $\gamma$ in the general relation

$E={\rho ^{\alpha }}\cdot {t^{\beta }}\cdot {r^{\gamma }}$

such that the left- and right-hand 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 this data is 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

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