31 pixels: how to draw a logo on the Moon, and the law that already bans it

Part IThe screen and the eye

Chapter 1A favicon in the sky

The standard measure of sharp eyesight is 20/20 vision (6/6 in metric countries). It means you can read a letter 5 arcminutes tall whose strokes and gaps are 1 arcminute wide: the "minimum angle of resolution" is 1′.¹ At the Moon's mean distance of 384,400 km,² one arcminute spans 111.8 km. The Moon is 3,474 km across, so it spans about 31 of these "pixels": 29.4 when it is farthest and 33.5 when it is closest. Its disk holds about 760 of them (this page's arithmetic). That is the resolution of a small icon on a computer screen.

Three limits in the eye line up near this scale. Light passing through a pupil 2.5–3 mm wide is blurred by diffraction to about 0.8–0.9′. The cones at the centre of the retina, packed at an average peak of about 199,000 per square millimetre,³ sit about half an arcminute apart, enough to sample stripes up to about 60 cycles per degree. And the eye's lens and cornea add their own blur, which matters most when the pupil is wide. Looking at the Moon at night, your pupil is open to perhaps 5–6 mm (this page's estimate from a standard pupil formula), so it is the eye's optical imperfections, not diffraction, that set the limit.

Twenty-twenty is a pass mark, not an average. Healthy adults aged 18–29 average about 0.7′, and one study concluded that 6/6 is inadequate as a norm.⁴ For them the Moon is about 42–45 pixels across. The sharpest eyes in one study of 387 professional baseball players were 20/8.9,⁵ for which the Moon is about 70 pixels wide. A 2025 experiment that asked how many pixels the eye can actually see put the limit for black-and-white detail at the centre of vision at about 94 pixels per degree: a pixel about 0.64′ wide, or about 49 across the Moon.⁵ Figure 1 shows the Moon at several eyesights, drawn from a NASA map made with the Lunar Reconnaissance Orbiter.⁶

Figure 1👁️ Try it

The Moon at your eyesight

Drag the line · tap a pixel

Smallest detail you can see

Pixels across

One pixel is

Pixels on the Moon

About as sharp as

The full Moon divided into square pixels one "minimum angle of resolution" wide, for several eyesights: 20/40 (2′), 20/20 (1′), the average of healthy 18–29-year-olds (about 0.7′; Elliott et al. 1995), 20/10 (0.5′), the sharpest of 387 professional baseball players (20/8.9; Laby et al. 1996) and 10× binoculars. The Moon spans 31.1′ at its mean distance, 29.4′ when farthest and 33.5′ when closest. Left of the line: the near side from NASA's Scientific Visualization Studio (LRO data: a colour map adapted from the Lunar Reconnaissance Orbiter's Wide Angle Camera mosaic, NASA/GSFC/Arizona State University), shown in grey with the maria about half as bright as the highlands. Right of the line: each pixel shows that map's average brightness; "Blur" smooths the pixels, closer to how the eye really loses detail. "Southern sky" rotates the view by 180°. Tap a pixel to name what is in it; the six Apollo landing sites are plotted from their published coordinates.This is the Moon as your eye can resolve it: each square is the smallest detail you can make out. Drag the line to compare with a sharp Moon, pick different eyesights, tap a pixel to see what's in it, and flip to the Southern Hemisphere view (upside down!). Every footprint humans ever left on the Moon fits in a handful of these pixels. (The map is NASA's, made from Lunar Reconnaissance Orbiter photos.)

Two odd facts follow. First, the full Moon, at 31′, is smaller than the big "E" at the top of an eye chart, which is 50′ tall at the reading distance: it is only about six 20/20 letters wide. Second, it never turns away. The Moon is tidally locked, always showing Earth the same face (apart from a slight rocking called libration), so anything drawn on the near side stays on screen. But the screen is not held the same way up for everyone. Seen from the Southern Hemisphere, the Moon appears rotated by about 180° compared with the view from the north, and during the night its image turns in the sky. A logo that everyone can read needs to survive being turned upside down: "SOS" would work; most brand names would not.

Chapter 2Seeing is not resolving

If the eye's pixel is 112 km wide at the Moon, how can we see stars, whose disks are thousands to tens of thousands of times narrower than an arcminute? Because seeing something and resolving it are different. Resolution is about telling two things apart. Detection is about noticing that anything is there at all.

For small targets, the eye cannot tell a tiny bright spot from a larger, dimmer one: it responds to the total extra light. In 1939, Selig Hecht and Esther Mintz showed that a single long dark line on a bright background is visible when it is just half an arcsecond wide, one hundred-and-twentieth of the eye's pixel. What sets that limit is a dip of about 1% in the light falling on a row of cones, not the eye's resolution.⁷ At the Moon, half an arcsecond is about 1 km.

The key measurements were made in 1946 by H. Richard Blackwell, who collected about 450,000 judgements from 19 young observers of the faintest disks they could detect, from 0.6′ to 6° wide, on backgrounds from starlight to daylight.⁸ His brightest background, 1,000 foot-lamberts (about 3,400 candelas per square metre), is close to the full Moon's surface brightness, so his table is, in effect, a ready-made "logo on the Moon" table:

Disk diameter	Threshold contrast	Contrast × area (arcmin²)
0.6′	41%	0.12
3.6′	1.5%	0.16
9.7′	0.46%	0.34
18′	0.35%	0.90
55′	0.28%	6.8

Thresholds for 50% detection in a forced-choice test, on a 1,000-foot-lambert background (Blackwell 1946, Table VIII). The last column is this page's arithmetic.

Up to a few arcminutes, the product of contrast and area at threshold is almost constant, about 0.12–0.16 square arcminutes. In that regime, painting a spot twice as dark is as good as painting twice the area. Beyond a few arcminutes the needed contrast falls ever more slowly, levelling off at about 0.3% for large targets,⁹ so making a mark bigger buys less and less. Real observers are less sensitive than a laboratory forced-choice test suggests: Blackwell's later work, discussed by Andrew Crumey, multiplies laboratory thresholds by about 2.4 for confident everyday seeing, and older eyes need more.⁹

This gives a single currency for any mark on the Moon: contrast × area. A mark that you could confidently notice needs about 0.3 arcmin², roughly a third of a pixel painted at full contrast, or six pixels painted at 5%. Remarkably, this calculation was done for the Moon as early as 1959. A US Air Force study of what a lunar mission could show from Earth (more about it in Chapter 10) concluded that a sunlit white sphere would need to be 13 miles across to be seen against the bright Moon, 0.4 miles against the dark part of the Moon, and 700 feet against a dark night sky.¹⁰ A white ball 13 miles wide covers 0.03 arcmin² with a contrast of about 500–800% against lunar soil: 0.13–0.23 arcmin² in all, right on Blackwell's threshold (this page's arithmetic).

Chapter 3The cheapest logo

Detecting a blob is not reading a logo. To recognise a shape, its parts must be resolved, which brings resolution back in. The eye's sensitivity to patterns of different fineness is described by the contrast sensitivity function, measured with striped gratings, classically by Fergus Campbell and John Robson in 1968: it peaks at about 4 cycles per degree (stripes about 7′ wide) and falls to nothing near 50 cycles per degree (stripes a little over half an arcminute wide), as read from their graphs.¹¹ A standard engineering model, Peter Barten's formula, captures how sensitivity depends on stripe width, brightness and the size of the pattern.¹²

That allows a simple optimisation (this page's arithmetic). Take a logo drawn on a 5 × 5 grid, like an eye-chart letter, with strokes of width w. Its main stripe frequency is about 30/w cycles per degree. Use Barten's formula at the full Moon's brightness (about 4,000 cd/m²), with the optical blur of the 5–6 mm pupil of Chapter 1, for the threshold contrast, and multiply it by 2, an empirical factor (this page's calibration) that turns a sine-wave threshold into the contrast of a letter-like pattern and roughly reproduces both large-letter (about 1.6%) and 20/20-letter thresholds on eye charts.¹³ Then suppose the cost of a "dilute" pen, such as dark powder, is proportional to painted area times contrast. The result:

Stroke width	Logo width	Contrast needed	Relative cost
0.8′	450 km	over 100%	impossible
1.0′	560 km	44–62%	2.3–3.0×
1.8–2.1′	1,000–1,180 km	4.5–7%	1.0×
3.0′	1,700 km	2.6–2.7%	1.2×
6.2′ (fills the Moon)	3,500 km	1.6%	2.9–3.2×

Barten's formula with the optical blur of a pupil 5–6 mm wide (Barten's allowance σ = √(0.5² + (0.08d)²) arcminutes for a pupil d millimetres wide). With the formula as published, which assumes the small pupil of an eye adapted to a bright field, the cheapest row is 1.7′, about 950 km and 5.7%, and a 1,000-km logo needs about 5%. Either way, a 1,000-km logo costs within 4% of the minimum (this page's arithmetic).

So the cheapest recognisable logo is neither "big and faint" nor "small and bold", but medium and faint: about 1,000 km across, a little over a quarter of the Moon's width, with strokes of about 1.8–2.1′, at a contrast of about 5–7%. Below that size, the eye's blur forces the contrast up faster than the area falls. Above it, the eye stops trading contrast for area, so cost grows with area. Blackwell's disk data agree on the large-size side: their cost is flat within a factor of 1.4 from 0.6′ to 3.6′, then roughly doubles by about 10′. In this budget, a whole recognisable logo needs a contrast × area of about 2.5 pixels' worth: for example, 40 pixels painted at 6.4%. Like Blackwell's laboratory numbers, this is a threshold, for young observers who know where to look; held to the "confident" standard behind Chapter 2's 0.3 (Blackwell's factor of about 2.4), it would be about 6 pixels' worth, or roughly 11–17% at this size (this page's arithmetic).

Figure 2🎯 Try it

Big and faint, or small and bold?

Guess, then drag the logoDrag the logo or the curve

Which logo costs least to paint?

Stroke width of the logo

Logo width

Contrast needed

Cost vs cheapest

Pens strong enough (Chapter 7):

A logo on a 5 × 5 grid with strokes of width w has a main spatial frequency of about 30/w cycles per degree. The (Weber) contrast needed to recognise it is taken as twice Barten's (2003) sine-wave threshold for that frequency, field size and luminance (about 4,000 cd/m² for the full Moon; about 0.3 cd/m² for earthshine), a factor calibrated on eye-chart letters. The cost of a dilute pen is taken as painted area (half the logo's square) times contrast, relative to the cheapest size. "Moon's own pattern" adds contrast masking (threshold ∝ mask contrast^0.6, Legge and Foley 1980) with an assumed lunar background contrast of 25% at coarse scales, falling to 5% near 30 cycles per degree. The eye's optical blur is Barten's allowance for the 5–6 mm pupil of Chapter 1 (σ ≈ 0.67′ at 5.5 mm). "Daytime pupil" uses Barten's formula as published, which builds in the small pupil of an eye adapted to a bright field (σ ≈ 0.54′) and moves the optimum to strokes of about 1.7′ at about 5.7%. The small Moon shows the logo darkened by the needed contrast, either sharp or through a 20/20 eye (averaged into 1′ pixels and smoothed); drag on it or on the curve to change the size. The pen contrasts are those of Chapter 7. All this page's arithmetic; Barten's formula is extrapolated beyond 1,000 cd/m² and below half-degree fields, so read it as a factor-of-two guide.Should a Moon logo be huge and faint, or small and bold? Make a guess, then drag the logo's size. Too small and your eye blurs it, so it needs to be very dark. Too big and you're paying to cover half the Moon. The cheapest size is in between. The little Moon shows the logo as your eye would see it.

Two complications make the job harder. The Moon is not a blank screen: its own dark and light patches act as clutter, and patterned backgrounds raise the contrast needed to see a target on them ("masking").¹⁴ With a rough allowance for the Moon's own pattern, the optimum moves to strokes of about 1.4′ at about 13% contrast. And these are thresholds for young, attentive observers; older eyes need several times more contrast. The model is also stretched: Barten's formula is calibrated up to 1,000 cd/m² and for patterns at least half a degree across, and the full Moon is brighter and the logo smaller. The numbers are best read as a factor-of-two guide.

Part IIThe surface

Chapter 4A screen made of asphalt

The full Moon looks dazzling white. It is actually dark. Its geometric albedo is about 0.12,² about the same as old, worn asphalt; new asphalt is about 0.05.¹⁵ The dark lava plains called maria reflect only 7–10% of sunlight, and the brightest highlands and young craters about 20–24%.^16,17 It looks white only because it is the brightest thing in a black sky. Its surface brightness, about 4,000–5,000 cd/m², is about that of a sunlit road (this page's arithmetic). And the light it sends to Earth is feeble: full moonlight on the ground is typically 0.05–0.1 lux, and at most about 0.3 lux; often-quoted values of 1–2 lux are wrong.¹⁸

The Moon has another trick. You might expect the half Moon to be half as bright as the full Moon, since half as much is lit. It is about eleven times fainter (this page's arithmetic from the standard lunar phase law used by Krisciunas and Schaefer).¹⁹ Even a perfect matte sphere would only give a factor of about three. The extra brightening comes mostly from shadows: at half Moon the rough, porous soil is full of them, and they vanish as the Sun moves behind us (the grains themselves also scatter light preferentially back the way it came). In the last few degrees before full this becomes a sharp spike, the opposition surge, which that law leaves out. The Clementine spacecraft looked at the Moon with the Sun directly behind it and found its brightness rose by more than 40% between 4° from opposition and exact opposition.²⁰ Two effects contribute. At full Moon, every grain hides its own shadow from us ("shadow hiding"). And light waves that take a path through the soil and the same path in reverse arrive in step, reinforcing the light sent straight back ("coherent backscatter").²¹ In effect, moondust is a weak natural retroreflector.

Figure 3🌗 Try it

Half the Moon, a tenth of the light

Drag the Moon round its orbit

From above (not to scale)

From Earth, same camera settings

Days from full Moon

Lit fraction

Lit part, on average

Whole Moon vs full

A logo would show

Total brightness of the Moon relative to full, against phase angle α (the Sun–Moon–observer angle, about 12.2° per day from full), from the lunar phase law used by Krisciunas and Schaefer (1991), Δm = 0.026|α| + 4 × 10⁻⁹ α⁴ (α in degrees), compared with the lit fraction (1 + cos α)/2 and with a matte (Lambertian) sphere. "Lit part, on average" is the whole-Moon brightness divided by the lit fraction. The Moon "from Earth" is drawn pixel by pixel from Figure 1's NASA map with the Lommel–Seeliger law, which has no limb darkening, scaled so that its total matches that law; the exposure is fixed, and the earthlit night side is drawn far brighter than it really is. The close-up is a cartoon of shadow hiding at the spot where the Sun and the Earth both stand α/2 from overhead (so it is always lit and in view). Below about 1.5° the Moon would be at least partly in Earth's penumbra, so an uneclipsed full Moon never reaches 0°; the law also leaves out the narrow opposition surge of the last few degrees. The rule of thumb for logos: brightness logos show best near full, relief logos near the terminator; one study found the rays of a small, fresh crater vanishing into the background in images taken at 60° solar incidence (Elliott et al. 2018).You'd think half a Moon is half as bright as a full one. Drag the Moon around the Earth, or press Play, and watch the real brightness: the full Moon is about 11 times brighter than the half Moon. The close-up shows why: at full Moon, with the Sun behind you, every grain of dust hides its own shadow.

For logos, this matters. A logo drawn in brightness (dark or bright material) shows best near full Moon, when shadows vanish; one study found the rays of a small, fresh crater vanishing into the background in images taken with the Sun 60° from overhead.²² A logo drawn in relief, as walls or trenches, would show only near the terminator, where shadows are long: to cast a shadow one pixel long with the Sun 1° above the horizon, a wall would need to be about 2 km high (this page's arithmetic). And the night side is not black: it is lit by earthshine, roughly 10,000–50,000 times fainter than the full Moon (this page's conversion of measured surface brightnesses).²³ At that brightness the eye needs far more contrast, so the cheapest daylight logo would vanish on the crescent Moon. In 1787 William Herschel saw bright spots on the earthlit Moon and reported them as volcanoes;²⁴ the brightest was probably the young crater Aristarchus, the Moon's own logo in earthshine.

Chapter 5Fresh is bright, and brightness is a clock

Why is the Moon so dark? Not because its rocks are black. Crushed lunar rock is fairly bright. What darkens it is space weathering. Over time, the solar wind and the constant rain of micrometeorites coat soil grains with vapour that condenses into tiny particles of metallic iron. Particles a few nanometres across both redden and darken the soil; larger ones, over about 40 nm, darken it without reddening.²⁵ How much of the work is done by the solar wind and how much by micrometeorites is still debated. Either way, the result is that fresh lunar soil is bright and old soil is dark. That is why young craters like Tycho are surrounded by bright rays of freshly excavated material.

Brightness is therefore a clock. A spectral measure called optical maturity, OMAT, tracks the weathering.²⁶ Tycho, about 109 million years old, has an OMAT of 0.278 in its ejecta; Copernicus, about 800 million years old, has faded to 0.178, close to the background.²⁷ Soil becomes fully mature after something like 0.8–1 billion years. Some rays do not fade this way, because they are bright by composition rather than youth: Lichtenberg's rays are still visible after more than 1.6 billion years.²⁸

The clock has not been reset while anyone was watching. Giordano Bruno, a 22-km crater just beyond the Moon's north-eastern limb, is one of the youngest large craters: counts of the small craters on its ejecta give an age of about 1–10 million years.²⁹ In 1976 it was suggested that it formed on 18 June 1178, when, by a Canterbury monk's account, five or more men saw "the upper horn" of the new Moon "split in two".³⁰ Almost certainly not: an impact that size would have rained about 10 million tonnes of debris into Earth's atmosphere, causing a week-long meteor storm that no known record mentions.³¹

Figure 4⏳ Try it

Brightness is a clock

Press play: a billion years

A Tycho-sized crater, from orbit

A painted layer 1 mm thick, up close

Age of the mark

Thick crater ejecta

A 1-mm painted layer

At this age

Top: optical maturity (OMAT) of bright crater ejecta against age, an illustrative two-point fit through Tycho (109 Myr, 0.278) and Copernicus (about 800 Myr, 0.178) with an assumed mature background of 0.13: OMAT = 0.13 + 0.177 e^{−t/609 Myr} (this page's arithmetic; no fitted fade curve has been published). Higher OMAT means fresher and brighter. Bottom: the fraction of a thin (about 1 mm) surface layer not yet churned by impact gardening, e^{−t/17,600 yr}, from the overturn rate of Speyerer et al. (2016) (this page's arithmetic; the top millimetre may be churned faster). The pictures are illustrations driven by the two curves: the rays' extra brightness follows OMAT, and the close-up shows the edge of a painted stroke (paint on the left, bare soil on the right), where randomly placed small impacts remove the layer at the rate that gives the bottom curve. Rays bright because of their composition, like Lichtenberg's, do not fade this way. Drag along the plot to change the age.Fresh moondust is bright, and it darkens with age. Press play, or drag along the plot. A thick crater splash takes hundreds of millions of years to fade. A thin coat of paint (left in the close-up; bare soil on the right) is punched through by tiny impacts and stirred into the dust within tens of thousands of years.

The weathered layer is deep. In August 2026 a spent Falcon 9 upper stage struck the Moon and left an 18-metre crater. The Lunar Reconnaissance Orbiter team judged that its dark, streaky ejecta likely came from the mature top half-metre, and the bright material near the rim from deeper down.³² At a site like that, raking the surface would not expose bright material: you would have to dig.

The surface is also constantly stirred. By comparing images taken years apart, the Lunar Reconnaissance Orbiter team found that small impacts and their secondary debris churn the top 2 cm of 99% of the surface in about 81,000 years, more than a hundred times faster than earlier estimates.³³ They concluded that astronaut footprints and rover tracks will be erased in a few tens of thousands of years, not the millions often quoted. Dust is not the eraser. The LADEE spacecraft found no electrostatically lofted grains at altitudes of 3–250 km,³⁴ and laboratory experiments show charged grains hopping only centimetres.³⁵ That can blur the edge of a footprint, not a 100-km stroke.

Chapter 6Every mark we have made

Humans have been marking the Moon since 1959, mostly by accident. The Lunar Reconnaissance Orbiter has photographed most of these marks (a few crash sites, including that of Luna 2, have never been found). None of it comes close to being visible from Earth.

Rocket blasts. Landing engines scour the soil around a lander, leaving a "blast zone". The peer-reviewed measurement gives Apollo blast zones a mean area of about 29,000 m², 3–12% brighter than their surroundings at the angle of illumination measured; a conference paper gives 23,800 m² and 15–20%.³⁶ Luna landers made zones of about 2,400 m² and Surveyor about 300 m²;³⁶ Chang'e-3 made about 2,500 m², 10% brighter.³⁷ The brightening seems to come mostly from smoothing the soil's fluffy micro-texture, which removes tiny shadows, rather than from exposing fresh material. A 2026 modelling study finds that a 100-tonne Starship-class lander would scour about 26,000 m² on landing, five times its estimate for an Apollo landing alone, and about 45,000 m² with an ascent, about twice the zone Apollo 11 left.³⁸
Footprints. Where astronauts walked, the soil is 15–30% darker.³⁶ Trampling compacts it.
Crashes. The spent third stages of Apollo rockets made craters 35–40 m wide, with bright rays up to more than a kilometre long; a rocket body in 2022 left a double crater of 18 and 16 m; the 2026 Falcon 9 stage made 18 m.^39,32 A crash can darken the ground as well as brighten it: ispace's Resilience lander, which crashed in Mare Frigoris on 5 June 2025, left a dark smudge with a faint bright halo.⁴⁰

Adding these up gives roughly 0.2 km² of blast zones, or up to about 3–12 km² including the rays of the Apollo stage impacts (this page's estimate; no total has been published). That is between 1/60,000 and 1/1,000 of a single pixel, at contrasts of 10–20%. The Moon does better on its own: a natural crater 222 m wide, named McGetchin, formed in 2024 and changed the surface more than 100 km away.⁴¹

Figure 5👣 Try it

Every mark we have made, at true scale

Zoom in · tap a mark

Field of view

One screen pixel here

Biggest landing mark

All our marks

One eye pixel at the Moon (1′: a square 111.8 km on a side, 12,500 km²) with every kind of mark people have made drawn at true size, side by side rather than where they really are: a typical Apollo blast zone (about 29,000 m², drawn 10% brighter; Clegg et al. 2014) with its descent stage (about 4 m across, 9 m with legs) and footprints, Luna, Surveyor and Chang'e-3 blast zones (about 2,400, 300 and 2,500 m²), an Apollo third-stage crater (35–40 m, rays up to more than a kilometre), the 2022 double crater (18 and 16 m), the August 2026 Falcon 9 crater (18 m, with its dark ejecta) and, for comparison, the natural McGetchin crater (222 m, 2024) with a dashed circle 100 km out: changes to the surface were found even farther away. Zoom out to 250 km to see that circle. "Stack them all" gathers the blast zones of every landing into one square of about 0.2 km², and the dashed squares add the rays of the Apollo stage impacts (3–12 km²): this page's estimate, as in the text. The shapes of the halos, rays and ejecta, McGetchin's distant splotches and the background craters are illustrations; the sizes of the marks are measured. Drag to move around, double-tap to zoom in.One pixel of your eye, 112 km wide, with all our marks drawn to scale. Zoom in to find them, or stack them all into one square.

Telescopes on Earth cannot see any of the hardware either. The Hubble Space Telescope resolves about 100 m at the Moon's distance; an Apollo descent stage is 4–9 m across and would need a mirror tens of metres wide. Nor could the closest human eyes since 1972: on 6 April 2026 the Artemis II crew passed about 6,500 km above the Moon's far side, where a 1′ eye pixel spans about 1.9 km. An Apollo-sized blast zone, about 29,000 m² and at most about 12% brighter than its surroundings, would fill under 1% of that pixel: hundreds of times below Chapter 2's threshold for noticing (this page's arithmetic).⁴²

Brands have already made the trip. When Intuitive Machines' Odysseus touched down in February 2024, one of its panels carried Columbia Sportswear's logo and heat-reflecting fabric.⁴³ Like every other mark here, it is far too small to see from Earth.

The Moon does show changes bright enough for the naked eye, but only for a moment. Telescopes that watch its earthlit side record flashes from meteoroids striking it. One of the brightest recorded, on 11 September 2013 in Mare Nubium, peaked at magnitude 2.9, as bright as a star easily seen without a telescope, faded below naked-eye brightness within about a quarter of a second, and glowed faintly in telescopes for more than 8 seconds; its discoverers put the energy at about 16 tonnes of TNT, from a rock roughly a metre across.⁴⁴ It was far smaller than an eye pixel, yet bright enough to notice: Chapter 2's rule, on the side of the Moon where, as Chapter 9 shows, light is the cheapest pen.

Part IIISeven pens

Chapter 7Pens that push dirt

Chapter 3 gives the target: a recognisable logo needs a contrast × area of about 2.5 pixels' worth, for example 25 pixels changed by 10%, or 50 by 5%. A stronger pen does best covering the same strokes sparsely, like a halftone; bolder, narrower strokes cost more (Chapter 3). Every pen has to deliver that budget. The pens differ in how much contrast they buy per kilogram, per joule or per year. All numbers below are this page's arithmetic, good to a factor of a few.

1. Rocket blasts. Each Apollo-class landing brightened about 24,000–29,000 m² by about 10%.³⁶ That was measured away from full-Moon lighting (Chang'e-3's 10% at a phase angle of 30°), and phase-ratio images show that a blast zone sends less light straight back than the soil around it.³⁷ Near full Moon its brightening should therefore be smaller, so the counts below are lower limits (this page's inference). One pixel needs about 430,000–525,000 landings; a whole logo, around 11–13 million. A modelled 100-tonne Starship-class lander would blast about 26,000–45,000 m² (the larger figure if an ascent stage also lifts off), so it would cut those counts by about a factor of two if its modelled zone is set against Apollo's measured ones, or by about five if the model's own Starship-to-Apollo ratio is the better guide.³⁸ The blasts also fling the finest dust at up to 2–3 km/s, some of it faster than the Moon's escape speed of 2.4 km/s, so a little may leave the Moon altogether.

2. Boots and wheels. Trampling darkens the soil by 15–30%. Driving a rover with a 1-metre-wide track at 1 metre per second over one pixel takes 396 years; a logo needs about 5,000 rover-years. The energy is trivial, roughly 10¹¹ joules per pixel if compacting the soil costs about 10 J per square metre. Time is the cost.

3. Soot. Fresh black carbon absorbs light extremely well: about 7.5 m² of absorbing cross-section per gram.⁴⁵ An ideal haze of it, spread evenly, dims the soil by 5% with just 3.4 milligrams per square metre: 43 tonnes per pixel. For faint logos, the total mass hardly depends on the design, because it is set by the contrast × area budget: about 2,100 tonnes for a whole logo. Real powders clump and sink into the pores of the soil, so a practical coating might need 10–100 times more (an assumption). NASA's Inspector General estimates that deliveries under its commercial lander contracts will cost about $1.2 million per kilogram;⁴⁶ SpaceX advertises lunar cargo at $100 million per tonne ($100,000 per kilogram), no earlier than 2028, a price no flight has yet tested.⁴⁷ The ideal soot logo would then cost about $2.5 trillion at today's price, or about $210 billion at the advertised one, and practical coatings 10–100 times more. Because the soil is already dark, soot can never reach more than about 70–80% contrast.

4. Fresh rock. At sites like the one in Chapter 5, bright material lies below the top half-metre. Spreading a 1-mm layer of it over one pixel takes about 19 million tonnes of rock, and a logo about 50–160 million tonnes. It needs no launch, only a lunar mining industry. Gardening would mix the thin layer back into dark soil in something like 10⁴–10⁵ years, about as fast as space weathering would darken it. Real crater rays last far longer because they are thick, blocky deposits.

5. Impacts. Nature's pen. A naked-eye bright spot is a crater with fresh ejecta and rays, tens of kilometres across. Standard crater-scaling formulas⁴⁸ put the energy for a crater the size of Kepler (29.5 km) at about 1.3 × 10²¹ joules, and for Tycho (85 km) about 5 × 10²² J: 240,000 times the largest bomb ever tested, and roughly 3,000–6,000 times the energy of all of today's roughly 12,000 nuclear warheads⁴⁹ (this page's estimate of their total yield: 2,000–4,000 megatons). Delivered as a single impact, the whole arsenal would dig a crater of about 9 km at most. But impacts are the only pen that lasts 10⁸–10⁹ years: Tycho, 109 million years old, is still a naked-eye "logo", and its longest ray runs 2,203 km.²²

Figure 6🖌️ Try it

Paint the Moon

Paint on the globe, then look from Earth

Your design

Write a wordA–Z, 0–9, up to 7 (4 with the Bold brush)

Seen from Earth, by a 20/20 eye

Ground painted

What it takes

Lasts about

Visibility

The globe is the design view, on Figure 1's NASA map: paint anywhere on it, including the far side, which never faces Earth. Mirrors and lasers are explained in Chapters 8 and 9. It is lit from the Sun's direction at the chosen phase, with its night side drawn at half brightness so it can still be painted. Each eye pixel is 1′ (about 112 km, 12,500 km² seen face-on) at the Moon's mean distance; near the limb a pixel covers more ground, so the costs use the true painted area (surface pens) or the area seen from Earth (mirrors, lasers). The "seen from Earth" view shows each pen's real contrast at the chosen phase, averaged into 31 × 31 pixels and smoothed (about a 1′ blur); the earthlit night side is drawn far brighter than it really is. The visibility meter adds up contrast × area in units of one pixel at 100% contrast (marks made with soil pens count less where slanting light shows the terrain's shadows: fully where the Sun is within 45° of overhead or the phase angle is below 45°, and not at all where both exceed 75°, this page's rule of thumb from Figure 3); about 0.3 makes a mark you could notice and about 2.5 a recognisable logo, if its strokes are wide enough to resolve (Chapters 2–3). Soil-pen strokes 1 pixel wide are marked too thin to read below about 44% contrast (Chapter 3's table: 44–62% at 1′). Laser strokes 1 pixel wide are marked too thin as well: at earthshine brightness Figure 2's model needs about +730% at 1′, and +100% only from about 1.7′. Soil pens below about 11% are marked readable only on plain ground: with Figure 2's allowance for the Moon's own pattern, a logo needs about 11–13% at any stroke width from 1.4′ to 6′. "Southern sky" counts a design as reading the same if at least 85% of its pixels match the design turned by 180°, allowing a shift of one pixel. On the earthlit side thresholds are about 10 times higher. Per pixel (this page's arithmetic): rocket blasts +10% at 430,000–525,000 Apollo-class landings; boots and wheels −20% at 396 rover-years; soot darkening the soil by 5%, 10%, 30% or 70% at about 43, 88, 300 or 1,000 t (an ideal haze of the black carbon of Chapter 7, light crossing it twice: 833 t × −ln(1 − contrast)); fresh rock +50% at 19 Mt dug from below half a metre; flat mirrors +100% within each beam at about 0.56 km² of film for 15 beams (about 4.4 t at 8 g/m²), dark where the Sun has set; green lasers on the earthlit night side +100% of earthshine at about 75–260 MW. Launch prices: about $1.2 million per kg (the NASA Inspector General's 2024 estimate for its commercial landers) or $100,000 per kg (SpaceX's advertised price, no earlier than 2028).Pick a pen and paint on the globe (you can turn it, and even paint the far side, where nobody on Earth will ever see it). The second Moon shows what people on Earth would actually see, at full Moon, half Moon or crescent. The meter shows whether it's enough to notice (first mark) or read (second mark). Try the same heart with different pens, make the soot darker (30% shows, but costs more than 3 times as much), and try a laser on the crescent.

Chapter 8Borrow the Sun

Every pen so far changes how much sunlight the soil scatters. Scattering is wasteful: a matte surface throws light into a whole hemisphere, and only a sliver reaches Earth. A mirror does something different. It shows Earth a piece of the Sun's own surface.

The numbers make the point. The full Moon shines at magnitude −12.74 and the Sun at −26.74, a factor of 398,000 in total light. They look almost the same size, so the Sun's surface is about 376,000 times brighter per square degree than the Moon's (this page's arithmetic). A flat mirror reflecting 90% of sunlight is therefore worth 340,000 times its own area in sunlit moondust. About 37,000 m², a square of film 192 m on a side, would shine as brightly as a whole 12,500-km² pixel of full Moon: a point of magnitude −5.5, brighter than Venus, fixed on the Moon's face. A 15-metre square would shine like one of the brightest stars, though, set against the full Moon around it, it would add under 1% to its pixel's light: tens of times too little to notice (Chapter 2).

The catch is aim. The reflected beam is only as wide as the Sun looks, about 0.53°, so at the Earth it lights a patch about 3,600 km wide. Covering the whole moonward face of the Earth would take roughly 15–20 separately aimed beams (15 leave thin gaps near the rim; about 19 leave none), and every mirror would have to turn steadily, at about a quarter of a degree per hour, as the Sun crosses the lunar sky. A whole logo for the whole Earth would need about 1.4 km² of film: 5–17 tonnes for films a few microns thick, roughly 120 to 400 times less mass than ideal soot, without counting the steering hardware (this page's arithmetic).

Moondust is the enemy. The retroreflectors left by Apollo now return about ten times less light than when new, and ten times less again near full Moon, probably because of a thin coat of dust.⁵⁰ And retroreflectors themselves would be useless as a pen: they send each ray of sunlight back the way it came, to within a few arcseconds, so it returns only towards the Sun's disk,⁵¹ and seen from the Moon, Earth is never that close to the Sun's direction except during a lunar eclipse, when there is no sunlight to return.

None of this is fanciful. In 1993 a Russian experiment, Znamya-2, unfurled a 20-metre mirror from a cargo ship and swept a spot of reflected sunlight a few kilometres wide across Europe. In July 2026 the Space Bureau of the US Federal Communications Commission licensed a single test satellite carrying an 18 × 18 m mirror for the company Reflect Orbital, noting that the light it reflects lies outside the Commission's authority; in August, dark-sky and conservation groups asked the full Commission to reverse the decision.⁵²

Chapter 9Paint with light

Why send anything at all? A laser on Earth could project a logo. Lasers already reach the Moon: lunar laser ranging stations fire pulses at the Apollo reflectors, and the APOLLO station's beam, about an arcsecond wide, makes a spot about 2 km across on the surface.⁵¹ Their echoes also show that the strength of gravity has not drifted, to about one part in 100 trillion per year (Nº 1). Widening the beam to a 112-km pixel is easy.

Power is not. On the sunlit Moon, a laser has to compete with the Sun. Matching the light of one sunlit pixel takes about 3.3 terawatts of green light, allowing for the eye's high sensitivity to green and for the atmosphere. A whole logo needs about 8 TW of light, or about 27 TW of electricity at a realistic efficiency: more than humanity's entire energy supply, about 19 TW (this page's arithmetic).⁵³

On the night side, it is a different story. There the laser competes only with earthshine, roughly 10,000–50,000 times fainter. Matching the earthlit brightness of one pixel takes about 75–260 megawatts of light, and a readable logo, which in earthshine needs roughly ten times Chapter 3's contrast × area (some 25–45 pixels, each lit to twice its earthlit brightness), about 2–12 GW of laser light, or roughly 10–40 GW of electricity: tens of large power stations, switched on for as long as the logo is wanted (this page's arithmetic). So the cheapest logo drawn in light appears on the dark part of a crescent Moon, the opposite of the cheapest logo drawn in dust. It also involves no launch at all, which matters for the law (Chapter 11). Chapter 7's painting figure shows it: pick the laser pen and the crescent phase.

Part IVPeople and law

Chapter 10They did the sums first

Science fiction got there first. In Robert Heinlein's story "The Man Who Sold the Moon" (1950), a promoter hints to a soft-drink company that its rival might cover the Moon with its logo in black dust (a threat he has made up), and takes money to make sure nobody does. Arthur C. Clarke's "Watch This Space" (1956) has an experiment on the Moon hijacked to paint a soft-drink advertisement.

Then came the space race, and with it a real wish to show the Moon being reached. In 1958 Soviet engineers considered exploding a nuclear device on impact as "irrefutable proof" of a hit; by Boris Chertok's account, the idea was dropped when calculations showed the flash, in vacuum, could not be reliably photographed from Earth.⁵⁴ Instead, in January 1959, Luna 1 released a kilogram of sodium 113,000 km from Earth, an artificial comet meant to be photographed from observatories.⁵⁴

In the United States, the Armour Research Foundation studied lunar missions for the Air Force in 1958–59, in work later known as Project A119; the astronomer Carl Sagan was among the contributors. Its report of 19 June 1959 gave the motive as "scientific, military and political".¹⁰ It worked out much of what this page asks: the 13-mile white sphere of Chapter 2; that about 100 tonnes of sodium would be needed to be seen against the bright Moon; that lines would beat dots against the Moon's patchy background; and that, for a 1-megaton device, the debris cloud would be visible through a small (16×, one-inch) telescope, though not with the naked eye. The study became public only in 1999–2000: Keay Davidson's 1999 biography of Sagan revealed it, and its director, Leonard Reiffel, confirmed it.⁵⁵

The modern law grew out of a billboard. In 1993 a company called Space Marketing proposed to put a Mylar sheet reported as half a mile to a mile long and about a quarter of a mile wide into orbit, about 150 miles up, where, its promoters said, it would look about half the size of the full Moon (by this page's arithmetic, between about a third and three-quarters of its width, depending on which reported length is right), for about $25 million.⁵⁶ That same spring, Columbia Pictures announced it would advertise the film Last Action Hero on the side of a rocket; the launch slipped and the advertisement never reached space. Representative Ed Markey introduced the Space Advertising Prohibition Act, with fines of up to $30 million, calling such advertising an inappropriate use of outer space.⁵⁷ It died in committee, but its idea did not.

Chapter 11Forbidden, almost word for word

In 2000, Congress banned the licensing of launches carrying "obtrusive space advertising". It is now 51 U.S.C. §50911. The definition, §50902(12), is advertising in outer space that can be recognised from the Earth's surface "without the aid of a telescope or other technological device".⁵⁸ That is precisely the design target of this page. Non-obtrusive advertising on rockets, payloads and launch facilities is expressly exempted. The Federal Aviation Administration proposed matching regulations in 2005 and withdrew them in 2006 as unnecessary;⁵⁹ the statute stands on its own.

Internationally, the Outer Space Treaty of 1967, with 118 parties at the start of 2026, says the Moon cannot be claimed by any nation "by means of use or occupation, or by any other means" (Article II), makes states responsible for the activities of their companies (Article VI), and requires "due regard" for others and the avoidance of "harmful contamination" (Article IX).⁶⁰ The 1979 Moon Agreement goes further, asking states to avoid disrupting the lunar environment, but only 17 countries have joined it, not including the main spacefaring ones.⁶¹ NASA's guidelines ask landers to keep 2 km away from Apollo landing sites, and US law now builds that into NASA's contracts.⁶² A newer, non-binding layer sits on top: in the Artemis Accords, signed by 77 countries as of October 2026, signatories say they intend to preserve "outer space heritage", which they take to include historically significant landing sites, artefacts and spacecraft.⁶³

There are gaps, which this page notes rather than resolves (this is not legal advice). The US statute governs launch licences, so a laser on the ground seems to fall outside it, though aviation and laser-safety rules would still apply. And when the FCC's Space Bureau licensed an orbital mirror in July 2026, it said the reflected light was outside its authority; that decision is now being challenged.⁵² Other countries' rules differ: a Russian law in force since 1 January 2026 allows Roscosmos to sell advertising on spacecraft and space infrastructure.⁶⁴

The deeper reason for restraint is in the numbers of this page. The Moon is the only screen that every human being shares. A mark big enough to read is at least hundreds of kilometres wide. It would outlast any company, any country and any language: a thin coat for tens of thousands of years, an impact for a billion. Tycho, 109 million years old, is a reminder of how long a mark on the Moon can last. Nobody has yet made a mark on the Moon that anyone on Earth could see. Zero is probably the right number.

Chapter 12Glossary and sources

Arcminute (′): A sixtieth of a degree. At the Moon's distance, 111.8 km.
Visual acuity, 20/20: The finest detail you can read, here as the "minimum angle of resolution": 1′ for 20/20.
Contrast: The change in brightness divided by the background brightness. A mark 10% darker than its surroundings has a contrast of 10%.
Contrast sensitivity function: How much contrast the eye needs to see stripes of each fineness.
Albedo: The fraction of light a surface reflects. Seen at full phase, with the Sun directly behind the observer, the Moon's is about 0.12 (its geometric albedo); counting light reflected in all directions, about 0.11 (its Bond albedo).
Opposition surge: The steep brightening of a dusty surface as the Sun moves directly behind the observer.
Space weathering: The slow darkening and reddening of lunar soil by the solar wind and micrometeorites.
OMAT: "Optical maturity", a measure of how weathered lunar soil is, from its colours.
Gardening: The stirring of the top layer of lunar soil by small impacts.
Earthshine: Sunlight reflected from the Earth onto the Moon's night side.
Tidal locking: The Moon's rotation matches its orbit, so it always shows Earth the same face.

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31 pixels

A 31-pixel screen

Seeing isn't resolving

The cheapest logo

A screen made of asphalt

Fresh is bright

Every mark we've made

Pick your pen

Borrow the Sun

Paint with light

They did the sums first

Forbidden, almost word for word

Thirty-one pixels