# On the scale of masses

One can formally equate mass with

• energy via a factor of c.c = 8.98755179e16, using the famous equation E = m.c.c – albeit this equates real-world amounts of energy to tiny masses and real-world masses to stupidly big energies. Consequently, I list some of the huger energies here under their mass equivalents and some of the tinier masses (notably that of the neutrino) under their energy equivalents.
• length and time via factors of c.c/G = 1.2471e27 kg/m and c.c.c/G = 403.8e33 kg/s, respectively; the former amounts to identifying a mass with the size of the black hole with that mass, which doesn't relate well to every-day experience. One can equally relate mass (via energy) to time inversely by dividing h/c/c = 7.3724958e-51 kg.s by the mass; this gives the time scale on which the uncertainty principle lets virtual particles of the given mass pop in and out of existence despite the law of conservation of mass-energy; but the result for even the tiniest things listed here is singularly brief.

The kilogram is the SI unit of mass; but (partly because it's name includes a quantifier) it's more usual to measure in terms of the gram (rather than milli kilogramme) for quantities below the kg and the tonne (rather than kilo kilogramme) for quantities much above a hundred kg; and to use quantifiers relative to these for (respectively) smaller and larger masses. Since the range of masses relevant in the universe is vast, I'll stretch to using some non-standard quantifiers (see the New Hackers' Dictionary for origins). Even so, there are things with masses off the top end of the scale, so I provide a final entry to cover everything from 1e33 tonne up to 1e49 tonne (the order of magnitude of the mass of the Universe as a whole).

yocto-gramme, 1e-24 g

The standard atomic mass unit, AMU, is about 1.66 yocto-gramme, slightly less than one GeV; a 12C carbon atom has a mass of almost 20 yocto-gramme. The time-scale on which the uncertainty principle allows the law of conservation of energy to ignore discrepancies of AMU order is 4 yocto-seconds, which is roughly the time scale on which a 4 gramme black hole would evaporate.

zepto-gramme, 1e-21 g

The LHC, CERN's Large Hadron Collider, is designed to accelerate protons up to speeds close enough to the speed of light that their mass increases by a factor of about 7460, to about 12 zepto-grammes (about 15 thousand AMU, the mass of about 63 Uranium atoms or a complex organic molecule with around a thousand non-hydrogen atoms in it – I suspect proteins are about this big). But see next entry …

atto-gramme

Individual genes have masses of order 40 atto gramme or 24 million AMU.

The LHC can accelerate more than just protons: it can get lead ions up to high speeds, making collisions possible at energies as high as 1148 TeV (82 times as high (lead's atomic number is 82), so 1.23 million AMU), which is just over 2 atto-grammes. If an LHC collision ever produces a black hole, this'll be its mass: it'll have a temperature on the order of 6e43 Kelvin and evaporate in under 1e-78 seconds – a time as tiny compared to the Planck time as the Planck time is tiny compared to 25 ns. Even if it somehow contrived to last longer than that, its radius is of order 75e-15 of the Planck length and it wouldn't be massive enough to have any appreciable gravitational attraction to anything, so it'd be unable to trap anything else in its hole.

femto-gramme

pg, pico-gramme

Bacteria typically have masses of pico gramme order.

ng, nano-gramme

µg, micro-gramme

The Planck mass, √(c.h/G) is about 54.6 µg; this is only slightly more than the 46.55 µg mass-equivalent of the chemical energy released by one tonne of TNT detonating. As little as 40 µg of a natural neurotoxin, secreted by certain South-American and Meso-American frogs (so-called poison dart frogs), is lethal. Amoeba have masses of order 4 µg.

A black hole with a mass of order a little over a micro-gramme (over five hundred milliard times what collisions at the LHC can produce) would evaporate in the Planck time, 0.14e-42 seconds; the furthest it could possibly travel in that time is the Planck length, 40.5e-36 metres, which is insanely tiny even on the scale of the internal structure of protons. All the same, this would duly explode with the same amount of destructive power as 25 kg of TNT – but concentrated in a much smaller space !

mg, milli-gramme

The difference in mass between one mole of 14N nitrogen atoms – most conveniently available as half a mole of molecular N2, which makes up roughly 80% of our atmosphere, with a tiny proportion of 15N – and a quarter mole of 56Fe (which contains the same number (14 moles) of nucleons; that is, protons and neutrons, albeit with more neutrons and fewer protons) is just over 19 mg. To put it another way, if all the 14N atoms in about 15 litres (roughly half a cubic foot) of our atmosphere were to suddenly decide to fuse into an equivalent amount of 56Fe, the amount of energy released would be about as much as 0.4 kilo-ton of TNT detonating.

g, gramme

When the USA demonstrated that it had nuclear weapons and was willing to use them on civilian populations, the first bomb they used turned approximately 0.7 grammes of matter into energy. When a magnitude 4 earthquake hits, it releases seismic stress equivalent, as energy, to about 14 grammes of mass.

kg, kilogramme

Humans are born weighing a few kg (taking a random example from a friend's child, 3.8 kg) and grow to several dozen kg; I spent a few years above 100kg and (in 2016) am happy to usually be back below that threshold. Many other domestic animals have weights in the same range; plenty more are not far outside it.

The standard unit of destructive power of bombs is the amount of energy released, on detonation, by one tonne of trinitrotoluene (TNT); and people have actually made nuclear bombs with yields in the 10 to 100 mega-ton range. Such a weapon doesn't really contain millions of tonnes of TNT (that'd make delivery rather difficult); it actually gets its energy by rearranging some atomic nuclei into ones with higher binding energies: a ten mega-ton nuke effectively converts about 0.47 kg (roughly a pound) of mass into energy. A black hole with a mass of 0.47 kg would be roughly as destructive, in the few atto-seconds it would take to evaporate away.

tonne

A black hole with a mass of 100 tonne would have a temperature in the exa Kelvin range and rising fast as it explodes; within a fraction of a second its mass would be down to one tonne, with a temperature rapidly approaching zetta Kelvin, and it'd be less than a tenth of a µs from total annihilation. Everything near it would likely have already been annihilated, as converting the first 99 tonnes of mass into energy in under a second amounts to a bomb more than twenty thousand times as destructive as anything even the mighty superpowers have been rash enough to make (much less actually detonate).

kilo tonne

An empty Boeing 777 airplane has a mass of 136 tonnes (300,000 pounds); its full load of fuel, passengers and baggage has about the same mass, so its mass at take-off is over a quarter kilotonne. Black holes with masses of order a 100 kilo tonne evaporate in a matter of years, if my calculations are correct; their temperatures are of peta Kelvin order (and rise as they evaporate).

mega tonne

Black holes with masses less than about 100 mega tonne (1e11 kg, the mass of a small mountain) evaporate on time-scales less than the age of our Universe; their temperatures are of tera Kelvin order and above.

giga tonne
The portion of Earth's atsmosphere above a square of Earth's surface with 10 km sides (or any similar-sized patch of the surface, e.g. some cities) has a mass of roughly a giga tonne. Comet Tempel 1 is reckoned to have a mass in the 70s of giga tonnes; the masses of 19P/Borrelly and 81P/Wild are reckoned to be about 20 and 23 giga tonnes, respectively.
tera tonne

The asteroid Apollo has a mass around 90 tera tonnes, while Icarus, Adonis and Hermes have masses around 11 tera tonnes. Mars's two moons, Phobos and Deimos, have masses of 10.8 and 1.8 tera tonne, respectively. Halley's comet is reckoned to have a mass of about 0.22 tera tonnes.

peta tonne

Saturn's inner moons Prometheus, Pandora, Epimetheus, Janus, Mimas and Enceladus have masses ranging from about 0.1 to about 73 peta tonnes. The asteroids Albert, Eros and Amor have masses around 0.3, 14 and 11 peta tonnes, respectively. A black hole with a mass of 23 peta-tonnes would have a temperature above 5300 Kelvin (hot enough to boil Osmium, the element with highest boiling point) and a radius of 32 nm; but it would take 28e33 years to evaporate.

Lord Kelvin reckoned the Earth's atmosphere to contain 1020 millions of millions of tons (1.03 peta tonne) of Oxygen. This would put the mass of the atmosphere at around 5 peta tonne. Multiplying the surface area of the Earth times the atmospheric pressure at sea level gives the total weight of the atmosphere; dividing this by the gravitational field strength at the Earth's surface should give a reasonably accurate (if anything, slightly low, since the atmosphere is generally above the Earth's surface, so subject to slightly lower field strength) estimate of the atmosphere's mass: I get 5.28 peta tonne, roughly matching Lord Kelvin.

exa tonne

The Moon's mass is 73.5 exa tonnes; the Galilean satellites of Jupiter range in mass from just under 48 exa tonnes (Europa) to just over 148 exa tonnes (Ganymede); Saturn's Tethys, Dione, Rhea and Iapetus range from just over 0.62 exa tonnes (Tethys) to 2.31 exa tonnes (Rhea), and Titan's mass is almost 134 exa tonnes. Pluto's mass is a mere 15 exa tonnes. The largest asteroid, Ceres, has a mass of 0.87 exa tonnes. A black hole with a mass of 0.4 exa tonnes would have a diameter of almost 1.2 microns and have a temperature fairly close to human body temperature; a 45 exa-tonne black hole would have a diameter of 0.133 mm and the same temperature as the cosmic microwave background – so it would lose as much energy by Hawking radiation as it gains by absorbing that background; it would be close to equilibrium in deep intergalactic space, but the equilibrium is unstable, so it is unlikely that integalactic space is actually populated with such black holes.

zetta (1e21) tonne

The Earth's mass is almost six zetta tonnes; Mercury's is just under a third of a zetta tonne and the other two inner planets lie in between. Uranus's mass is 86.63 zetta tonnes.

yotta (1e24) tonne

Neptune, Saturn and Jupiter have masses 0.103, 0.567 and 1.90 yotta tonnes, respectively. A black hole with a mass of 0.1 yotta tonne would have a diameter of about a light nanosecond (a.k.a. foot) and a temperature of 1.2 mK.

harpi (1e27) tonne

The Sun's mass is just shy of 2 harpi tonnes. A black hole wih the same mass would have a radius just under 3 km (10 light microseconds) and a temperature of 61.7 nK.

grouchi (1e30) tonne

The most massive stars known are reckoned to be around 150 times as massive as The Sun, so about 0.3 grouchi-tonne. At least two of the stars making up Pismis 24-1 are individually over 100 times as massive as The Sun, for a total of at least 0.4 grouchi-tonne.

cosmic masses: 1e33 to 1e49 tonne

The biggest known black hole (with a radius 357 times that of Earth's orbit round the Sun – this is bigger than the whole of the heliosphere, out to the bow-shock where the solar wind disturbs the inter-stellar gas – and a temperature of about 3.4 atto-Kelvin, but surrounded by a super-heated accretion disk of in-falling matter) has a mass 18 milliards times that of the Sun; i.e. about 36e36 tonnes – that's as massive as a small galaxy. Our Milky Way galaxy has a mass of about 6e38 tonnes – not even twenty times as big ! Along with all the galaxies even vaguely close by, the Milky Way is falling towards The Great Attractor, whose mass is of order 1e43 tonne, enough to be felt even a fifth of a Gly away.

Written by Eddy.