]> On the scale of lengths

# On the scale of lengths

Lengths are intrinsically equivalent (via the speed of light) to times; here, I'll focus on lengths as spatial displacements between objects in mutual approximate-rest frames. (For an excellent treatment of them as heights above Earth's surface, see XKCD 482, with follow-up for small scales as XKCD 485; or, for an excellent zoom out from human scale to cosmic and then in to nuclear, see the classic short film Powers of Ten.) The Planck length, √(h.G/c)/c ≅ 40.507e−36 metres, is way off the scale's small end; that's 40 atto-attometres. That's as small compared to us as the attometre is compared to the exametre – see below !

attometre, am, 1e−18 m

Even atomic nuclei think this is small. A black hole with a mass of about a third of a million tons would have a radius of about half an attometre and would evaporate, by Hawking radiation, within a century (unless fed new matter to counteract the evaporation). Evaporation time for a black hole grows in proportion to the cube of its mass (aside from some abrupt changes in the constant of proportionality as the mass falls below various thresholds – each of which is the result of dividing the square of the Planck mass (possibly divided by 4.π, depending on your definition of it) by the mass of a species of fundamental particle).

femtometre, fm, 1e−15 m

If we interpret thermal neutron scattering cross-sections of nuclei (which really measure how readily the nuclei absorb neutrons) as areas; and their square roots as lengths (which we can think of as giving some indication of the scale of nuclear structures); we get a wide range of values – from zero (e.g. Be) to 16.4 pm (Xe); but many values fall in the femtometre range. The gamma rays produced when cosmic rays hit the atmosphere have wavelengths of order femtometres.

picometre, pm, 1e−12m

When an electron and positron annihilate one another, the two X-ray photons emitted have wavelengths of 2.43 pm. The division between gamma and X-ray radiation is orthodoxly around 10 pico metres. The Ångstrøm is 100 pico metres. The Bohr radius, ℏ/(me.α.c) ≅ 52.92 pico metres, is the length scale of electron orbits in the Hydrogen atom. The radii of atoms, as determined by considering the lengths of covalent bonds they form, range from 37 pico metres (Hydrogen) to 235 pico metres (Caesium). The lengths of covalent bonds in organic molecules range from around 95 pm (the O–H bond in CH3COOH) to 214 pm (the C–I bond in CH3I); non-covalent bonds can be longer, e.g. 392 pm for the K–K bond in K2.

nanometre, nm, 1e−9 m

Air molecules travel around 100 nm between collisions, under normal atmospheric conditions. That's roughly the diameter of a typical virus, 'though they come smaller; the tiny tobacco necrosis virus has a diameter of only 16 nm. The most common life-like species on Earth (numbering of order 1e27) are bacteriophages, usually between 20 and 200 nm long. Typical protein molecules are around 6 nm across. Carbon nanotubes have been made with diameters of two to three nm (and persuaded to emit infra-red light with a wavelength of one to two microns). Ultra-violet light has wavelengths down to about 10 nm; below that, electromagnetic radiation gets called X-ray. A photon with wavelength 91.19 nm has enough energy to fully ionize a Hydrogen atom from its ground state; divide this by any difference between inverses of squares of distinct positive integers to get the wavelength of a photon the electron can emit in the course of returning via suitable orbitals. A black hole with a radius of about 32 nano metres, mass 23 peta tonnes (about one three thousandth of the mass of the Moon), would be hot enough to boil Osmium (5300 K) at its surface; it would take about 2e24 times the (current) age of the universe to evaporate by Hawking radiation.

micron, micrometre, µm, 1e−6 m

Typical human body cells have diameters around 55 µm; the head of a sperm cell has diameter about 7.5 µm. Archaean cells range from about 0.1 µm to over 15 µm. Bacteria have diameters up to about 1.9 µm, with 1 µm being fairly typical; but the pleuro-pneumonia bacterium is only 0.15 µm across. The most plentiful type of life on Earth, the Prochlorococci (a type of cyanobacterium, living in vast numbers near the Ocean surface) have sizes of order 0.5 to 0.7 µm. The algal phytoplankton phycobiliphyta are of order 2 to 5 µm across. Ricketsiae cells (responsible for typhus and kindred diseases; but incapable of metabolising outside a host) are typically about 0.475 µm across. Visible light has wavelengths ranging from 0.38 (violet) to 0.78 (red) µm; the infra-red spectrum ranges from the latter up to around 325 µm. Interstellar dust clouds include dust grains up to about one µm in size.

millimetre, mm, 1e−3 m

An amoeba's diameter is about 0.2 mm; this is also roughly the length of the tiny wasp Megaphragma mymaripenne; a human ovum is about 0.14 mm across; male wasps of the species Dicopomorpha echmepterygis are even smaller than that. A black hole with a diameter of 0.133 mm would have a mass of 45 exa tonnes (almost as much as Jupiter's moon Europa) and the same temperature as the cosmic microwave background. Large parts of a jet engine (of order a metre across) have to be round to within a tenth of a mm or so. A spherical droplet of water whose mass is the Planck mass, √(h.c/G), has diameter 0.47 mm. So-called microwave radiation has wavelengths ranging from 1 to 100 mm (so could better be described as milliwave radiation); all electromagnetic radiation with greater wavelengths is classified as radio.

metre, m

A pendulum of length 0.248 metres at the Earth's surface swings with a period of one second. In a nanosecond, light travels 0.299792458 metres (exactly, thanks to the modern definition of the metre); this is quite respectably approximated by various cultures' archaic units based on the length of an adult man's foot; it is also roughly the amount by which the Earth's crust moves up and down, twice per day, in tidal response to the moon's gravitation. Most human adults who were adequately nourished when growing up are between 1.5 and 2 metres tall; at birth, we start out around half a metre long. UHF radio wavelengths are up to a metre; VHF radio ranges from one to ten metres; the range from ten to a hundred metres is called short wave. The atmospheric pressure at Earth's surface is roughly equal to the added pressure each 10 m of water adds as a diver descends. I live three floors above ground level: my window is about 9 m above the street outside. Norway's national average for length of road per capita is 22.87 metres per person (of which 9.49 m is municipal, 6.66 m is rural and the remaining 6.72 m of major road splits as 2.21 m trunk plus 4.51 m primary.

kilometre, km, 1e3 m

The Eiffel tower is over 0.3 km tall. That's about the furthest anything, including light, can have travelled, after the big bang, in the roughly ten microseconds before matter went through a phase transition from quark-gluon liquid to a hadron-lepton plasma. Long wave radio has wavelengths of order a kilometre and more.

Earth's solid surface's radius varies by as much as 31.7 km between the Arctic depths and the top of Mount Kilimanjaro. The Earth's spin causes a centrifugal bulge so that the equatorial ocean surface is 21.48 km further from the Earth's centre than the notional sea level at the poles. The oceans of planet Earth are as much as 10.9 km deep (Mariannas trench); the solid surface is as much as 8.88 km above the ocean surface (Everest). Earth's tallest active volcano, Ojos del Salado, reaches to over 6.7 km above ocean level. The magma responsible for Yellowstone park's geothermal activity is at depths around 5 to 15 km below the surface. The atmosphere stretches out to around 100 km (that's the altitude at which meteors start burning up); an altitude of 60 miles (96 km) is the internationally agreed-upon beginning of space.

In my youth I walked, on separate occasions, 10 km in one hour and 48 km in six hours; and was once in a scratch crew which rowed something like 100 km (Cambridge to Denver sluice and back) between dawn and midnight (with some breaks to eat and rest) – so I am inclined to suppose that a day's travel in ancient times fell, roughly, in the 40 km to 100 km range.

mega metre, Mm, 1e6 m

An aurora over the Earth's poles seldom reaches below 0.06 Mm and can reach up to 1 Mm. I live (in Oslo, Norway) roughly 1 Mm from where I was born (near Leicester, England). Planet Earth has diameter 12.75 Mm and circumference 40.1 Mm. The Moon has diameter 3.47 Mm and circumference 10.92 Mm. A 50 Hz AC electricity supply (such as the UK's national power grid) emits radio waves with a wavelength of 6 Mm – roughly The Earth's radius. Arctic terns fly 40 to 100 Mm each year (Earth's circumference plus detours).

Artificial satellites orbit the Earth at altitudes ranging from 0.16 to 36 Mm; ham radio satellites orbit at altitudes of between 100 and 300 miles (0.16 to 0.48 Mm); the ISS orbits at a variable altitude of around 0.4 Mm; communication, weather and global-positioning satellites orbit at altitudes ranging from 250 miles (0.4 Mm) to 12,000 miles (about 20 Mm); and synchronous orbit, 42.2 Mm from Earth's centre, is at an altitude of 35.8 Mm.

The radius range of Earth's inner van Allen belt is from about 1.2 to 3 times the solid surface's radius (although it sometimes comes as close as 200 km above the surface), roughly 8 to 19 Mm. The outer van Allen belt extends in radius between four and eleven times that of the surface, around 25 to 70 Mm. (There are also sometimes transient radiation belts in addition to these two.) The radius to Earth's magnetopause's sun-ward (closest) point is six to fifteen times the solid radius, so roughly 40 to 90 Mm.

giga metre, Gm, 1e9 m

The Sun's apparent surface has diameter 1.392 Gm and circumference 4.372 Gm. The Moon's surface is only 0.376 Gm from Earth's (but their centres are 0.3845 Gm apart) and its orbit's circumference is 2.416 Gm. Mercury orbits The Sun at a distance of 57.9 Gm. One light second is just under 0.3 Gm; in a minute, light travels 18 Gm. There are about 0.16 Gm (i.e. just over half a light second) of nerve fibres in an adult human body. It's not beyond the bounds of credibility that this may have something to do with our reaction times being of order a fifth of a second – the time it takes light to travel 60 Mm. An arctic tern flies around 2.4 Gm in the course of a life-time.

tera metre, Tm, 1e12 m

The standard Astromonical Unit, a good approximation to the mean radius of Earth's orbit, is a fraction under 0.15 Tm. Venus's orbit's radius is 0.108 Tm. Neptune's orbit's radius is slightly over 4.5 Tm. The Kuiper belt stretches from about there to about 7.5 Tm from The Sun. The bow-shock between the solar wind and the inter-stellar gas is about 34 Tm from The Sun; and some man-made objects were about that far away in the early 21st century, having left Earth only three decades (or so) earlier. In an hour, light travels 1.08 Tm; in a day, it travels 25.9 Tm. One estimate of the diameter of VY Canis Majoris places it at 2.8 Tm (2.6 light hours); this is almost the diameter of Saturn's orbit.

peta metre, Pm, 1e15 m

In a week, light travels 0.18 peta metres; in a year it travels 9.46 peta metres. The Oort cloud extends to about 12 peta metres (1.29 light years) from The Sun. The nearest visible star (aside from The Sun) is Proxima Centauri; it is 39.92 peta metres (1.294 parsecs or 4.22 light years) away.

One parsec – or paralax second – is 30.86 peta metres (3.26 light years); the apparent position of a star x parsecs away, as seen by an Earth-bound observer, changes by 1/x seconds of arc as Earth moves from one end to the other of a diameter of its orbit at right angles to the line from our Sun to that star. Such a change in apparent position of a star is known as paralax, which gives its name to this method of determining distances. One second of arc is one sixtieth of one minute of arc; one minute of arc is one sixtieth of one degree of arc; 360 degrees of arc constitute one full turn; at a distance of 1 parsec, then, the diameter of Earth's orbit subtends one second of arc; that is, two astronomical units equal 2.π.parsec/360/60/60 ≈ 4.85 µparsec. The distances to other solar systems range upwards from just above one parsec.

exa metre, Em, 1e18 m

There are 6 giga people on Earth, more or less; and each has about a sixth of a giga metre of nerve fibres (actually many aren't adults, so the average is less than this); that makes about one exa metre of human nervous fibre in this universe. That's about 100 light years – a bit over one three hundredth of the distance to our galaxy's centre – or about 30 parsecs; the method of paralax suffices to determine the distance to stars up to this far away – their apparent positions, seen from Earth, wobble by a thirtieth of an arc second in six months.

The largest globular cluster in our galaxy, Omega Centauri, is 150 light years (1.42 exa metres) in diameter and 15 k ly (142 exa metres) away; it comprises about 10 million stars, implying a typical separation of about .7 ly, 6.6 peta metres.

zetta metre, Zm, 1e21 m

The Sun is about 0.28 zetta metres (30 kilo light years) from the centre of the Milky Way galaxy; objects up to ten times as far from the centre (and inwards to about a third as far) orbit at the same speed as The Sun (so with orbital periods proportional to radius). That implies that mass density, per unit area of the Milky Way's equatorial plane, falls off in inverse proportion to radius; consequently, there is no crisp edge to the Milky Way, though a diameter of 100 k ly is commonly quoted; that's about one zetta metre.

Our Milky Way galaxy has two stellar halos orbiting in opposite directions; an inner halo out to about 0.47 zetta metres and an outer halo out to over 6 zetta metres. It also has a central bar about 27 k ly (0.26 zetta metres) long; and its disc's average thickness is about 20 k ly (0.19 zetta metres).

The Small Magellanic cloud orbits our galaxy at a distance of about two zetta metres. Andromeda has a halo of stars out to a radius of about 0.5 M ly, i.e. about 5 zetta metres, from its centre (in all directions ? or just in the plane of its disk ?). Galaxies are commonly of order ten zetta metres (1 mega light year) apart; the nearest spiral galaxy to ours, Andromeda, is about 19 to 23 zetta metres away; the satellite galaxies in the local group we share with it are up to 100 zetta metres (10 M ly) away.

The nearest cluster (as opposed to mere group) of galaxies, the Virgo Cluster, is about 40 zetta metres across; cluster Abel 1185 (home of The Guitar, a colliding pair of galaxies, among hundreds of other galaxies) spans about 1 M ly, which is about 9.5 zeta metre.

yotta metre, Ym, 1e24 m

Our local group of galaxies has a speed of around 600 km/s (one five hundredth of the speed of light) relative to the cosmic microwave background: this is roughly the relative velocity the universe's expansion would give objects 28 mega light years apart; that distance is 0.26 yotta metres, or about 26 times the typical galactic separation (i.e. on the scale of gravitationally bound groups of galaxies). The Virgo Cluster of galaxies, which is close enough to drag our local group towards it, is 48 million light years away; that's 0.45 yotta metres. The Great Attractor, towards which the Virgo Cluster and much else around it is moving at 600 km/s, is about 2 yotta metres (APOD says about a quarter Gly) away, in the direction of Centaurus. Abel 1185 is about 3.8 yotta metres (400 M ly) away; the Perseus Cluster (Abel 426, dominated by NGC 1275) is about 2.4 yotta metres away and the Pisces-Perseus supercluster (comprising over 1000 galaxies) spans about 0.62 yotta metres. Interaction with the cosmic microwave background tends to slow really-high-energy cosmic rays, leading to the Greisen–Zatsepin–Kuzmin limit on how far they can travel; this is about 50 mega parsec or 1.54 Ym.

harpi metre, 1e27 m

The universe is 13.7 giga years old; this means there are photons reaching us that have travelled 13.7 G ly, 0.130e27 m. However, as the universe has expanded hugely since such photons were emitted, the space each traversed in its first light year is now of order a thousand light years wide, and similar ('though less so) for the rest of its journey; so the matter that emitted it is in fact significantly more than 13.7 G ly away. Current estimates place this matter 46 G ly = 0.435e27 m away. Of course, that's measured along the path such a photon has travelled: if the universe were topologically non-trivial – just as someone who's followed a geodesic on the Earth's surface for 40 Mm is in fact only about 53 km from where they started – that needn't mean the universe is actually 46 G ly wide. However, data from WMAP constrains the possibility for such wrap-around so that we can be confident that the cosmic topology scale is at least 78 G ly = 0.738e27 m. Dividing the speed of light by Hubble's constant yields 0.132e27 m or 14 G ly: if two objects in the universe are that far apart, and each is slow-moving with respect to nearby galaxies (formally: roughly co-moving, so that the cosmic microwave background has no over-all red or blue shift in any direction), then the space between them is growing enough that the distance between them grows at the speed of light.

Written by Eddy.