Why Can't Elephants Jump?: And 101 Other Tantalising Science Questions - New Scientist (2010)
Chapter 5. Our planet, our solar system
Where on our planet is the furthest point from any sea? I’m hoping it is the middle of Asia somewhere because I’m travelling that way soon, and want to stand at that point in swimming trunks, snorkel and mask.
Slapton, Northamptonshire, UK
The furthest point from the sea or, to give its technical name, the continental pole of inaccessibility (CPI), does lie in Asia. It is located at 46° 17′ N, 86° 40′ E, in the Dzoosotoyn Elisen in Xinjiang, China, and is 2,648 kilometres from the nearest coastline, at Tianjin on the Yellow Sea. Although its location was calculated long ago, it wasn’t visited by surveyor-explorers until 27 June 1986, when it was reached by British cousins Nicholas Crane and Richard Crane.
The Cranes travelled there by bicycle, crossing the Hindu Kush and Gobi deserts, to raise funds for the Intermediate Technology Development Group (which has since been renamed Practical Action) – a charity that supports technological advances in developing countries.
Twenty years before that, however, the CPI attracted the attention of another group with very different interests. Its unique geographical status gave it considerable significance among western nuclear strategists debating the relative merits of weapons systems. For proponents of the submarine-launched Polaris missile, the ability to hit any point on the Earth – even if there is nothing there worth hitting – became a key point in the public relations battle with the sponsors of land-based and air-launched weapons.
When the A3 version of Polaris brought the CPI within range in the late 1960s, it was hailed as a technological triumph – particularly by the UK’s Ministry of Defence. They did not, however, trumpet the fact that to strike the pole a large nuclear-powered submarine would practically have to visit Tianjin docks.
Ironically, by the time the western navies acquired the capacity to bombard all of China with submarine-launched missiles, the region around the CPI was probably featuring more prominently on the targeting lists of generals in Moscow, rather than London or Washington DC, as Xinjiang acquired vital strategic significance in the Sino-Russian confrontation of the last quarter of the 20th century.
Finally, while appreciating the irony in your correspondent’s choice of attire, he should reflect that the CPI is subject to extreme climatic continentality: summers are hotter and winters are colder than many places of similar latitude because it is so far from the moderating influence of the ocean. ‘Elisen’ means ‘desert’ in the local Chinese Uighur dialect, and although the location is certainly sandy, it is no beach.
Indeed, this part of Xinjiang might be considered an extension of the Gobi, which is a decidedly cold desert where temperatures drop to -40 °C in winter. At the other extreme, during daytime in summer it can reach a blistering 50 °C, though temperatures can vary by as much as 32 °C within a 24-hour period. If he insists on wearing his trunks, I suggest that he keeps a warm pullover handy, just in case.
Norwich, Norfolk, UK
Fooled in Blackpool?
From the top of Blackpool Tower (approximately 150 metres) on the UK’s west coast, can you see the curvature of Earth along the Irish Sea horizon? I thought I could, but my friend disagreed. If I’m wrong, how high would we have needed to be?
Bolton, Lancashire, UK
While camped at 6,000 metres altitude in Peru in 1962, I and some colleagues asked ourselves this question about the Pacific horizon. We actually only saw the curvature by comparing the horizon (about 277 kilometres away) with a nylon thread stretched tight and level between two ice axes.
While standing atop Blackpool Tower, if you sight the seaward horizon over a level, 1-metre straight edge, which is held 1 metre in front of you, trigonometry shows that the ideal horizon would appear to be almost a millimetre higher at the centre of the straight edge than at the ends. This is a much smaller effect than typical atmospheric distortion which, in effect, means there is no visible curvature.
From our camp in Peru, the difference was almost 6 millimetres – easily visible when compared with a straight edge. Even so, the curvature was not apparent when simply looking at the horizon.
Byron Bay, New South Wales, Australia
As the radius of the Earth is 6,373 kilometres, a little trigonometry tells us that if you are at the top of a tower of height h metres, the horizon will be at a distance of approximately (2 × 6373 × h)1/2 kilometres.
For a tower 150 metres high, the horizon will be 44 kilometres away and displaced downwards from a true horizontal line by about 0.39 degrees. If you hold a 1-metre stick horizontally 1 metre in front of you, seemingly touching the horizon at the midpoint of the stick, the ends will appear to be 0.8 millimetres above the horizon. That’s pretty hard to see with the naked eye.
La Courneuve, France
When out in the mid-ocean, up at the top of the main mast, the horizon is a horizontal line right round the field of view. The higher the mast, the lower the horizon appears to be, but it is still a horizontal line.
Wilmslow, Cheshire, UK
The short answer to this question is that the curvature is not obviously visible from anywhere on the Earth’s surface. Pilots of Lockheed U-2 and SR-71 Blackbird aircraft suggest that the Earth’s curvature only becomes clear at an altitude of about 18 kilometres. Indeed, it has been photographed from Concorde cruising at this altitude. The curvature can be inferred at sea level, though. For example, ships disappear over the horizon from the bottom upwards, as if sinking into the sea.
Willenhall, West Midlands, UK
Seeing the curvature of the Earth can mean either seeing the surface of the Earth in front of you fall away towards the horizon, in the same way that you see the ground fall away when standing on a rounded mountain top, or seeing the horizon as a curved rather than horizontal line.
It is actually possible to see the curvature of the Earth, in the second sense outlined above, at any height: for example, sitting on the beach, standing on the deck of a ship or looking out of a plane window. This is to be expected, because a view from any point on a sphere such as the Earth will give the horizon as a disc. The height of the viewpoint will simply determine its size.
The visual cues employed to see the curvature of the Earth are many, but judging the line of the horizon relative to the horizontal is generally not one of them. Instead, two more obvious cues are noting that the distance of the horizon is the same in any direction, and seeing that the texture gradient – the way a view changes in appearance and perspective with distance – of the sea or land is constant within that distance.
I agree that increased viewpoint height will yield a richer set of cues, especially those associated with seeing the horizon in the second sense, delivering a more obvious curvature. Nonetheless, this curvature can still be noted at sea level.
Psychologist and vision scientist
Liphook, Hampshire, UK
Every time I read yet another theoretical contribution to your debate over whether the curvature of the Earth is apparent from the top of Blackpool Tower, I turn to the front cover of the magazine just to check that the word ‘scientist’ is really still there.
I would have thought that by now someone would have followed the scientific method: just do the experiment and report the result. I would certainly have had a look if I lived a little closer.
Ross-on-Wye, Herefordshire, UK
Put that light out
If the sun was extinguished or there was a permanent worldwide eclipse, how long would it take for us all to freeze to death, and what could we do to try to avoid it?
A back-of-the-envelope calculation suggests the whole Earth might freeze solid within 45 days, radiating away its thermal energy according to the Stefan-Boltzmann law, which relates energy loss of a body to its temperature. My calculations assume the vast bulk of the Earth’s captured solar energy is stored in the oceans, which have an average temperature of 15 °C down to a depth of 35 metres. Energy carried by water at greater depths doesn’t count because it would rapidly become isolated from the surface by ice floes.
With its smaller heat capacity, the land would freeze much more quickly than the oceans. Air over relatively warm oceans would rise, pulling in cold air from the continents. This would chill the surface waters and might increase the circulation of water, exposing it to the chilling, perpetual night.
Interestingly, the volcanic dust thrown up by the eruption of the Tambora volcano in 1815 acted both for and against cooling. The dust blocked out the sun, but it also reduced the escape of thermal radiation from the Earth by dint of a greenhouse effect. Sunlight dimmed by 25 per cent for a while, leading to a dip in global temperatures of 0.7 °C in 1816. But the fall in temperature was small despite a big reduction in sunlight, suggesting that the Earth might take longer to freeze than 45 days. Indeed, freezing may well be delayed further by the natural greenhouse effect that comes with our atmosphere and the thermal inertia of our oceans.
Nevertheless, Earth would still be able to support a colony of humans. There would still be plenty of energy in the form of fossil and nuclear fuels, and geothermal heat mines. But without plants to replenish our oxygen supply, it would quickly run out, so we would need to build biospheres with artificial light for plant photosynthesis.
Thankfully, switching off the sun is an experiment the Earth will not undergo for another 5 billion years.
Willenhall, West Midlands, UK
It would be interesting to discover what effect on global cooling or warming the 2010 Icelandic volcanic eruptions have had – Ed.
You can estimate how long it would take us to freeze by extrapolating from the rate of cooling that happens overnight. In areas with clear skies, the temperature can drop to freezing in less than 12 hours. In places with heavy cloud cover, water vapour traps infrared radiation before slowing down the rate at which it radiates away into space, so cooling takes much longer. Here, the temperature drops by perhaps 5 °C in 12 hours. However, without the thermal energy of the sun constantly evaporating water, this insulating cloud cover would quickly disappear. It is likely that most parts of the Earth’s surface would be frozen within a few days. The only exception would be near the coastline, where it might take a few weeks because of the amount of heat stored in the oceans.
Could we stop this? Perhaps we could quickly burn the world’s forests to release large amounts of carbon dioxide to help trap infrared thermal radiation. However, with only fossil fuels, nuclear and geothermal energy left to rely on, we’d still freeze quickly. And if we didn’t, we’d soon run out of food and oxygen.
Department of Chemical Engineering
University of Newcastle
New South Wales, Australia
The sun can’t be turned off like a light bulb. It glows because its surface is about 5,500 °C and is heated by the nuclear fusion inside its core which is even hotter – about 15,000,000 °C. Even if fusion in the core could be switched off suddenly, the sun would continue to radiate light just as the heating element on an electric stove gives off heat for a time after you switch it off.
Obviously, the sun is bigger and hotter than a stove, so would continue to radiate heat and light for a long time. In addition, the energy produced in the core of the sun takes time to work its way out – millions of years if you track the energy by following the paths of individual photons. The sun would cool a tiny bit each year, but as the sun cools it would contract, releasing gravitational energy that would heat it and offset some of the cooling. That’s how white dwarf stars continue shining. Suffice it to say that it would take many millions of years before our descendants even noticed.
However, if the sun suddenly vanished, the Earth would cool quickly. Unprotected people would start freezing in days, but they could survive much longer if they went down into deep mines, warmed by the Earth’s heat.
New Scientist contributing editor
Auburndale, Massachusetts, US
Even if the sun went out, ecosystems around hydrothermal vents along the Earth’s mid-ocean ridges would continue to chemosynthesise using geothermal energy for a few thousand million years. So, business as usual for tube worms.
Alnwick, Northumberland, UK
The previous correspondent was wrong to claim that it would be business as usual for tube worms if the sun went out. It is a modern myth that communities of chemosynthesising organisms at hydrothermal vents live in self-sufficient isolation. They don’t. Most use the oxygen found in seawater for their metabolism. That comes, of course, from photosynthesis powered by sunlight in the near-surface waters and on land.
Freshwater, Isle of Wight, UK
From reading your books I now know what percentage of the UK’s surface area is roads, but having just returned from the Netherlands I would like to know what percentage of the surface area of that country is water.
Official statistics say that 19.3 per cent of the Netherlands’ surface area is water. But any area of water less than 6 metres across is counted as land. So the total area of water will be well over 20 per cent.
K. A. H. W. Leenders
The Hague, Netherlands
Ah – the Netherlands. The land of dykes, canals and windmills – and 18.41 per cent water. The Netherlands is the world’s fourth most watery nation, behind the Bahamas (27.76 per cent), Guinea-Bissau (22.48 per cent) and Malawi (20.49 per cent).
The total area of the Netherlands is 41,526 square kilometres, of which 33,883 square kilometres is land – 27 per cent of it below sea level – and 7,643 square kilometres water. Despite the best efforts of nature, the amount of land is increasing, thanks to modern versions of the dykes and windmills for which the country is famous: since the 13th century, 10 per cent of the total land area of the country has been reclaimed from the sea as polders. These are beds of artificial lakes that are bounded by dykes, pumped dry by windmills and drained by canals.
However, it’s not just the sea that the besieged Dutch battle. The Netherlands lies at the mouth of three major rivers: the Rhine, the Meuse and the Scheldt. Apart from the famous dykes protecting half the land from sleeping with the fishes, other dams and levees along these rivers protect against freshwater flooding.
Woodville South, South Australia
On a high
A number of athletics and cycling world records have been set at high-altitude venues, for example during the 1968 Olympic Games in Mexico City. Presumably the air is thinner so there is less resistance, enabling them to run or cycle faster. But surely oxygen uptake at altitude is more difficult, so there must be a point at which altitude no longer favours athletes. What is this point and why? And which tracks or velodromes come nearest to it?
Mexico City is situated at about 2,250 metres above sea level so, as the questioner correctly points out, air is less dense here because of the reduced atmospheric pressure – 580 millimetres of mercury (mmHg) compared with 760 mmHg at sea level – so you can’t make a decent cup of tea because the water boils at 92 °C as opposed to 100 °C.
This reduced density of the air undoubtedly reduces the work that needs to be done by a cyclist, who must cut a path through it, but it poses problems in terms of oxygen availability. Although oxygen always makes up 21 per cent of the atmosphere, the fractional pressure it exerts in Mexico City is only 120 mmHg, compared with 160 mmHg at sea level. This means there is a noticeable reduction in oxygen pressure at the interface between air and blood in the lungs’ alveoli, leading to mild hypoxia and a reduction in the amount of oxygen delivered to body tissue.
In the short term, athletes will breathe harder, leading to respiratory alkalaemia (increased blood pH), and heart output will increase to circulate the blood faster in an attempt to compensate. After several weeks’ acclimatisation, the density of blood vessels in muscle and the number of red blood cells both increase, carrying more oxygen to the working muscle. In addition, the kidneys excrete extra bicarbonate to compensate for the alkalaemia caused by rapid respiration and consequent reduction in carbon dioxide levels. These effects will not confer any advantages to the exerciser while they remain at altitude, though they would briefly be of benefit if the athlete were to return to sea level.
At yet higher altitudes, above about 2,500 metres, heart rate and oxygen delivery are stretched to the utmost, beyond the ability of the body to compensate for them, and work output and athletic performance decrease.
It would appear that reduced air density is the only explanation for the performances at the 1968 Olympics, though doubtless athletes undertook extensive altitude training.
Department of Veterinary Cell Sciences
University of Glasgow, UK
Altitude affects both running and cycling in two opposing ways. The power needed to overcome air resistance varies approximately with the velocity cubed and in direct proportion to the density of the air. The important consequences of this are that air resistance is far more significant at high speeds, and that it can be reduced by going to altitude, where the air is less dense.
The other effect of thin air is that the athlete receives less oxygen. In a race lasting less than 20 seconds, most of the energy comes from oxygen-independent glycolysis, in which the muscles break down carbohydrates without requiring large amounts of oxygen. This, combined with the sprinters’ high speed, means that sprint race times will be quicker at high altitude.
Competitors in longer events depend more on aerobic respiration, so for any running race taking more than about a minute the benefits of altitude are lost. In cycling, the equation differs, because the higher velocity means that up to 90 per cent of the energy expended by a cyclist is used to counter air resistance, so almost all world record times would be faster at altitude.
The 400 metres athletics event sits somewhere between a flat-out sprint and an aerobic distance race. A 1991 study in the Journal of Applied Physiology suggests that the ideal altitude for this event would be between 2,400 and 2,500 metres, close to that of Mexico City. Indeed, one of the more enduring records set at the 1968 Olympics was achieved by Lee Evans in the 400 metres.
Sam Baylis, Malvern
Why does the Moon appear as bright as a cloud in the midday sky, when it is a very dark body with an albedo of 0.07? Albedo is the ratio of the intensity of light reflected from an object to that of the light it receives from the Sun, and the albedo of clouds is around 0.6 to 0.8. This does not seem to apply to the difference between the brightness of the Moon compared with the brightness of clouds.
Altrincham, Cheshire, UK
The albedo of the darker areas on the Moon is indeed around 0.07, but that of the mountains, rayed craters and cratered highlands is considerably higher (between 0.10 and 0.15). As viewed in a bright blue sky through binoculars or a small telescope, the dark areas appear almost indistinguishable from the surrounding sky.
Department of Physics and Astronomy
University College London, UK
The light we see reflected from the Moon is just that, reflected light. But we see the clouds by transmitted light because we are below the clouds and sunlight is passing through the clouds, not reflecting from them. Only high-flying jet travellers see light reflected from the tops of clouds and, even through the tinted aircraft windows, the clouds are 10 times as bright as the Moon, as the albedo figures suggest.
Gatton, Queensland, Australia
Clouds have to be quite thick to reflect most of the sunlight. Unfortunately, when they are sufficiently thick, they usually cover the whole sky and their bright tops cannot be seen. The converse is also true: when the sky is only partially covered with clouds, they are usually thin and no brighter than the Moon. However, if it happens that there are well-developed but sparse cumuli in the sky, and the Sun and Moon are on the opposite sides of the sky, then it can be observed that the top of a large cumulus is about ten times as bright as the Moon, as expected from their albedo ratio.
J. J. Thomson Physical Laboratory
University of Reading
In the hills south of Rome there is an area which, in the distant past, was volcanic. On the drive from Rocca di Papa to Albano there is a well-known gentle slope in the road which has an extraordinary property. If you stop your car, put it into neutral and then slowly release the brakes, the car will gradually, but perceptibly roll up the hill. I have observed this phenomenon twice, once as a passenger and once as the driver, both in broad daylight. Is there any physical explanation?
The same effect occurs on the A719 road in Ayrshire, UK at the Electric Brae. It has special warning signs because of the likelihood of meeting cars coasting uphill backwards, as baffled drivers are confused by their senses. Coming round the shoulder of a hill to enter a small, steep valley, the driver sees the road apparently falling towards the stream. But the road is traversing the side of the valley, rather than crossing the stream at right angles and it is actually slightly uphill.
Department of Engineeering
University of Aberdeen, UK
Near Neepawa in Manitoba, Canada, there is a road called Magnetic Hill. You drive down a long gentle slope, stop, then release the brake and your car moves backwards, apparently up the hill. The local residents have finally had to admit that the hill is not really magnetic, but is actually a very convincing optical illusion produced by the local topography. No doubt the same applies on the road from Rocca di Papa to Albano.
The simplest explanation is an optical illusion, similar to one experienced by a friend and myself while we were on a cycling tour of northern Portugal. We were completely flummoxed to find ourselves having to pedal hard to make progress along a gently sloping, yet clearly downhill, stretch of road. The situation became positively surreal when a local came towards us also on a bike and apparently uphill, but with his feet resting on the handlebars, freewheeling.
Potters Bar, Hertfordshire, UK
I have had the same experience twice in the past. I was driving across western Germany a number of years ago in an old Volkswagen Beetle. The autobahn along which I was travelling was, for many kilometres, a series of hills like a shallow sawtooth. My car could only manage about 100 kilometres per hour on the level and the prospect of driving uphill on this autobahn for 5 to 10 minutes at a time clearly exposed the power limitations of the engine. The uphill drive was foot-tothe- floor stuff the whole way.
At a certain point I became aware that I had lost the sense of whether I was travelling uphill or downhill, because while my eyes told me I was going uphill, the engine revs and the car’s speed (which had reached 110 km/h) told me I was definitely going downhill.
I later had the same experience in Cornwall. The roads there are almost exclusively up and down, and at times it is impossible to tell whether your car is level or not.
Abu Dhabi, United Arab Emirates
I have experienced the same phenomenon at a similar site in Israel, near Jerusalem, exactly as described. There, the road is cut into the side of the hill, so that there is a drop on one side and a steep, rocky embankment on the other. The fact that the car appears to roll uphill is due to an optical illusion caused by the relative alignment of the embankment and the road. It is, of course, rolling downhill.
I recall seeing a report of a similar occurrence in Australia many years ago. The road was near the landmark Hanging Rock, which was the setting for a mystery novel and a more famous film. Given its location, people were, I think, quite content to enjoy this apparent aberration and assume, or hope for, some unexplained, attractive force at work. The explanation provided was more mundane, but still curious. After careful measurement, the road was shown to be sloped, slightly, in the opposite direction to which it seemed.
St Lucia, Queensland, Australia
A similar phenomenon can be experienced when driving across the centre of the island of Cheju Do off the southern coast of South Korea. This is a volcanic island with the remains of craters and eroded plugs and the experience is just as the questioner describes.
East Molesey, Surrey, UK
This is almost certainly an optical illusion, like a similar road at Spook Hill, Lake Wales, Florida. This road seems to slope downwards for a short distance before climbing a hill, yet cars left in neutral roll backwards, apparently up and out of the depression. The mystery disappeared when an investigator used a spirit level to show that the dip is illusory, and the road is in fact uphill all the way. These observations were published in the autumn 1991 issue of Skeptical Inquirer.
The optical illusion that makes a downhill slope appear to be uphill is quite common in mountainous districts. It occurs when the true horizon is obscured by the surrounding hills and more distant hills in the direction of travel are rather lower than the nearer ones, giving one the impression of approaching the summit of a pass. This effect is enhanced when the valley floor approximates to a plane surface but is slightly tilted downwards in the direction of travel.
The fact that the questioner only observed this phenomenon in daylight tends to confirm the above explanation, as the surrounding hills would not be so obvious at night, even under a full moon. A further visit accompanied by surveying equipment would convince even the sceptical.
Tunbridge Wells, Kent, UK
I remember from a university surveying course the case of a water main in the Republic of Ireland where water was supposed to travel by means of gravity over a long distance. The drop was carefully calculated but very small. When water was introduced to the completed main, the engineers were horrified to see it flow uphill, back towards its source.
The reason turned out to be a local gravity anomaly which meant that gravity in the region acted not perpendicular to the surface of the geoid but at a slight angle backwards towards the water source. The very small anomaly was enough to overcome the even smaller gradient on the water main. Therefore the water really did flow uphill. I assume a small pump solved the problem.
Kendal, Cumbria, UK
I have experienced an occasion where a major river appears to run uphill. In the USA, Route 128 leaves Moab, Utah, to the east and runs alongside the Colorado river for several miles. At a certain stretch, the river appears distinctly to be flowing uphill – a highly disorienting impression. The rocks of the canyon in which the river flows are stratified and I suspect that it is the line of their bedding which plays the trick on the eye.
Like David Cope, I once saw a river ‘running uphill’. It was on a flooded piece of unasphalted track near my school, in Bradford, West Yorkshire. A beck had burst its banks in heavy rain and was running along the track. My class was out on the dreaded weekly cross-country run and we all stopped to comment on it. Although we were wet and tired at the time, our teacher brought us back the following week in warmer weather with string and spirit levels to prove that the track really did run downhill. Proof, perhaps, that innovative teaching really does have an effect? I remember his demonstration 30 years later.
I remember seeing a related phenomenon while on holiday in Cornwall in 1978. But rather than a hill you could roll up, this was, apparently, a sloping lake. The gradient appeared quite noticeable, although I did not try unpowered waterskiing on it.
From what I remember, the lake was somewhere near Cape Cornwall, although I have failed to find it again on subsequent visits to the area. Can any locals let me know where it is?
Widnes, Cheshire, UK
It would be impossible to prove or disprove the antigravity phenomenon using a spirit level as described earlier. If the antigravity effect can act on a motor car, it can probably also act on a few millilitres of fluid containing a bubble, causing the bubble to float downhill instead of uphill.
High Wycombe, Buckinghamshire, UK
One small footprint?
Why can’t one of our space telescopes, capable of seeing galaxies many light years away, be pointed at the site of the moon landings where one can assume there are some remnants from the visits? Would this definitively prove to any sceptics that humans landed on the moon?
Shrivenham, Wiltshire, UK
The resolving power of a telescope – the size of the smallest object it can see at a given distance – is inversely proportional to the diameter of its lens. In other words, to see something small a long way off you need a very big telescope.
Apollo 11’s Eagle lunar module measures about 4.3 metres across, and to see it from Earth, when we are at our closest to the moon, would require a telescope with an angular resolution of 670 billionths of a degree. If we take the wavelength of the reflected light from the moon as being 550 nanometres, the middle of the visible range, then to see the lunar module would require a telescope with a diameter of nearly 60 metres. The largest telescope now in existence, the Gran Telescopio Canarias on the Spanish island of La Palma, has a diameter of 10.4 metres.
Larger telescopes would be very expensive. The cost of building the European Southern Observatory’s proposed Overwhelmingly Large Telescope, with a diameter of 60 to 100 metres, is estimated at €1.2 billion.
Redhill, Surrey, UK
We can see distant galaxies but cannot see the much closer footprints left on the moon because galaxies and galaxy clusters represent a bigger target: they take up, or ‘subtend’, a much larger angle in the sky. Galaxies are also bright, making them stand out against the blackness of space. Footprints are simply impressions left on the lunar surface, offering no contrast at all. We would be reduced to looking for shadows cast by the tread.
Imagine two walkers about half a metre apart. In your mind’s eye, draw lines from the focal point of your eye to the two figures. The angle between the two lines gets smaller as the figures walk away. The smallest angle at which the two figures can still be resolved is a measure of the resolving power of an optical instrument, in this case your eye. We use telescopes because they have a greater resolving power, so they can distinguish between objects that subtend a smaller angle.
With the naked eye – whose pupil has an aperture of about 2 millimetres – the two ramblers would blur into one object at a distance of about 2 kilometres, assuming perfect eyesight in which the ability to resolve two objects is limited only by diffraction. The best terrestrial optical telescope, the Gran Telescopio Canarias on La Palma, has an aperture of about 10 metres, giving it about 5,000 times the resolving power of the naked eye. A telescope of this power would be able to resolve our two ramblers even if they were 10,000 km away. However, the moon is 380,000 km away, and at this distance the telescope has no chance of separating the walkers, let alone their footprints.
To determine the resolving power of a telescope we use the Raleigh criterion. This tells us that the angle subtended by the smallest object an optical telescope can detect is roughly the wavelength of visible light divided by the aperture of the telescope. Multiply that by the distance to the object and we get the minimum size that can be resolved.
Taking visible light to have a wavelength of 555 nanometres, the aperture of our terrestrial telescope to be 10 metres and the moon to be 380,000 km away, the smallest object that can be resolved on the moon would be about 20 metres across, assuming no atmospheric aberration. The ability to resolve footprints would require a telescope with an aperture of about 20 km.
If the Hubble Space Telescope were brought to within 40 km of the lunar surface it could achieve a resolution of 1 centimetre and make out footprints. Lunar-orbiting telescopes have come this close but their optics are not as good. Another option is to use an array of terrestrial telescopes to simulate a large effective aperture.
In any case I suspect that there are more interesting things to study, given that the sceptics would not be convinced anyway.
Willenhall, West Midlands, UK
On NASA’s website (bit.ly/PdSU) you can see the trail of footprints left on the moon by the Apollo 14 astronauts, photographed from the Lunar Reconnaissance Orbiter between 11 and 15 July 2009.
Department of Mathematics and Computer Science
St John’s University
Jamaica, New York, US
If the sceptics who doubt the moon landings don’t believe the photos taken on the moon by the people who were standing there, why on earth (no pun intended) would they believe pictures beamed down from a telescope operated by the very organisation they suspect of lying to them?
Boston, Massachusetts, US