﻿ Our planet, our universe - Abraham Lincoln - James M. McPherson ﻿

## Why Don't Penguins' Feet Freeze?: And 114 Other Questions - Mick O'Hare (2009)

### Chapter 6. Our planet, our universe

Pole poll

What time is it at the North Pole?

Nigel Goodwin

Nottingham, UK

There are two answers to this question. The first is that the time for a person is the time determined by her or his circadian rhythm. Initially, this physiological time will be close to the time for the longitude where the individual lived before visiting the pole.

Over a period of weeks at the pole, this time will drift as the individual settles into a rhythm with a period that is usually about 25 hours long.

Of course, there is also a local time, independent of people, unless you are a philosopher residing somewhere other than the pole.

So the second answer is that the time is either daytime (for the six months of summer) or night-time (for the six months of winter).

I have not been at the pole near the equinoxes, but I would imagine that there are also several continuous weeks of twilight when the sun is just below the horizon.

Will Hopkins

University of Otago, New Zealand

The crux of the question is this: how should a person born and bred at the North Pole, who has never heard of Greenwich or Tokyo or any other place on the Earth, start measuring time?

This can be done as follows. Suppose that it is the dark period of the North Pole when the sun is below the horizon all the time. Fix a horizontal board at the pole and draw a circle on it with two diameters perpendicular to each other. Label the ends of the diameters A, B, C, D round the circle.

At the North Pole, you can see the stars revolving in planes that are parallel to the horizon. The plane of the horizon coincides at the poles with that of the celestial equator.

Then choose some star on the horizon and define 0 hour as the instant when this star passes across the line of sight of point A when viewed from the circle’s centre (the pole). The crossings of the star across B, C and D will correspond to 6, 12 and 18 o’clock respectively.

It is then easy to draw other straight lines on the board representing intermediate hours.

If I were required to do this exercise at present (at the North Pole), for reference I would choose the faintest of the three stars in Orion’s belt because it lies almost exactly on the celestial equator, is the brightest of all the stars that lie that near, or on the celestial equator, and it is clearly visible to the naked eye.

The next problem at the Pole would be how to decide what time it was in the summer, when no stars can be seen because it is constantly daylight.

Having drawn the hour lines in the winter, you must wait for the sun to appear above the horizon. At the moment it is sighted at the approach of the Arctic spring, we make a note of its direction on the board. The hour line on which it falls can be called the time of sunrise in the 24 hours system that was devised during the winter.

The sun will revolve, like the stars in winter, in a plane parallel to the horizon, but unlike our reference star, which always revolves in the same plane, the sun’s plane will be higher up day by day, ultimately reaching a highest level of 23.5 degrees from the horizon.

Then it will become lower and lower again until, six months after our first sighting, the sun will disappear below the horizon.

D. S. Paransis

Luleå University of Technology Sweden

This isn’t a sensible question: time is independent of location. When it is 1800 GMT in London, it is also 1800 GMT at the North Pole, in Timbuktu or on the far side of the moon.

One could ask: ‘what time zone is it at the North Pole?’ but this also fails. Time zones are defined politically and administratively rather than by geography. Because the North Pole is floating on the high seas no time zone is defined for it.

Attempts to define a natural time astronomically also fail. Noon is when the sun is due south, but at the North Pole the sun is always due south. Noon is when the sun is at its highest, but the height of the sun is essentially constant at the North Pole. Noon is halfway through the period of daylight, but at the North Pole it is light for six months and then dark for six months.

Mike Guy

Cambridge, UK

Time, from a geophysical point of view, is related to the position of the sun over the Earth and to the position of the observer. Because any direction from the North Pole is south, the sun is always in the south and whatever the time is at the North Pole, it is always the same time.

What time is that? The International Date Line runs through the North Pole, leaving the pole sitting eternally between one date and the next. In other words, it is always midnight at the North Pole.

This, of course, explains how Father Christmas manages to deliver presents to every good little boy and girl throughout the world in the space of a single night.

He just heads out of his grotto due south (which from the North Pole is any direction), drops off as many presents as he can fit on his sleigh and then he heads back home where it is exactly the same time as when he left. So he can then drop off more prezzies, return home, and so forth.

Patrick Whittaker

Hounslow, Middlesex, UK

The North Pole is, of course, the true spiritual home of the politician because in answer to the question ‘what time is it?’, she or he can, with all honesty, say ‘what time do you want it to be?’

Paul Birchall

Mickelover, Derbyshire, UK

Deep breath

Is it true that every time we take a breath of air or swallow a mouthful of water, we consume some of the atoms breathed or swallowed by Leonardo da Vinci (as I read in a children’s science book in 1960)?

Steve Moline

Wentworth Falls, New South Wales, Australia

We do indeed breathe in a considerable number of molecules that once passed through Leonardo’s lungs and, unfortunately, Adolf Hitler’s or anyone else’s for that matter. The calculation is not too difficult and is as follows:

The total mass of the Earth’s atmosphere is about 5 × 1021 grams. If we take air to be a mixture of about four molecules of nitrogen to one of oxygen, the mass of 1 mole of air will be about 28.8 grams. One mole of any substance contains about 6 × 1023 molecules. So there are about 1.04 × 1044 molecules in the Earth’s atmosphere.

A single mole of any gas at body temperature and atmospheric pressure has a volume of about 25.4 litres. The volume of air breathed in or out in the average human breath is about 1 litre. So we can assume that Leonardo da Vinci, in one breath, breathed out about 2.4 × 1022 molecules.

The average human takes, say, 25 breaths per minute, so during his 67-year lifetime (1452 to 1519) Leonardo would have breathed out about 2.1 × 1031 molecules. So, about 1 molecule in every 5 × 1012 molecules in the atmosphere was breathed out by Leonardo da Vinci.

However, because we breathe in about 2.4 × 1022 molecules with each breath, there is a pretty good chance we breathe in about 4.9 × 109 molecules that Leonardo breathed out. In fact, you can also show in a similar way that you probably breathe in about 5 molecules that he breathed out in his dying gasp.

Of course, there are some pretty crude assumptions involved here in order to arrive at the conclusion. We assume that there has been a good mixing of Leonardo’s molecules with the rest of the atmosphere (quite likely in 500 years), that he didn’t recycle some of his own molecules, and that there is no loss from the atmosphere due to later users, combustion, nitrogen, fixation and so on. But there is still scope for a considerable loss of molecules without it affecting the main point of the calculation.

By knowing that the number of molecules in the hydrosphere is 5.7 × 1046 molecules, similar calculations can be made for water. These show that a mouthful of liquid contains about 18 × 106 molecules that passed through Leonardo during his lifetime. So, in addition to breathing in his breath, there is also a pretty good chance of picking up some of Leonardo’s urine in every glass of water that you drink.

Peter Borrows

Epping, Essex, UK

The law of conservation of matter ensures that atoms are constantly being recycled in the Universe. Gravity ensures that most of those on the Earth stay there. Some of the atoms floating around were breathed by da Vinci, although the number of these atoms compared to all those in the Earth’s atmosphere would make them pretty few and far between.

However, considering the length of time in which, say, the dinosaurs inhabited the Earth, you can be pretty sure that every breath you take contains what was once part of one or more of these creatures, and that every apple you eat has many atoms that were once part of an animal, even a human. Of course all this could have some very worrying implications for vegetarians.

Glenn Alexander

Wollongong, New South Wales, Australia

This question provides some food for thought for homeopaths. There is a very high probability that a cup of water contains a few homeopathic molecules that are effective in countering every illness we may have, and at no cost.

Lassi Hyvarinen

Le Vésinet, France

There are more hours of daylight after noon than before it, particularly in summer. Does this mean midday is in the wrong place?

Dean Sherwin

Midday refers to the moment when the sun crosses the local meridian, which is one of the imaginary lines joining the North and South Poles at 90° to the equator. If you set your watch so that noon occurs as the sun crosses the meridian, the day will be of equal length either side of noon.

However, this system would mean that you would need to reset your watch if you travelled only a short distance east or west. To avoid the confusion this would cause we use time zones – areas throughout which we say that the time is the same, irrespective of the actual meridian. Time zones are nominally 15° wide but in practice they vary in size and shape because of political, geographical and practical considerations. The difference between the local meridian and the time setting of your time zone can be quite apparent if you live near the edge of an oddly shaped time zone.

David Eddy

Perth, Western Australia

The development of time zones is usually attributed to the development of the railway system in the US, which runs predominantly east to west. Until the advent of the railways, most towns followed local time and the clock midday was well aligned with the solar midday. Then trains began to travel from town to town so quickly that the constant compensation for different local times caused timetable difficulties, prompting the development of time zones.

Keith Anderson

Kingston, Tasmania, Australia

The standard time used in Britain is based upon the Greenwich meridian and the latitude of your correspondent in Reading is almost the same as that of Greenwich, but his longitude is 1° West. Sunrise, local noon and sunset therefore occur about four minutes later than at Greenwich, and Reading’s local time is actually four minutes later than the standard clock time used in Britain.

This means that, at Reading, the duration of daylight after noon, as shown on the clock, is on average longer than the duration of daylight before noon. East of the Greenwich meridian, afternoon daylight is, on average, shorter than morning daylight. At Greenwich the difference between the duration of morning and afternoon daylight, averaged over a year, is zero.

On any particular day, the difference between the duration of morning and afternoon daylight depends not only upon the latitude and longitude of a place but also upon the equation of time. This is the difference, in time, between the mean sun, which gives us clock time, and the true sun. It is caused by the eccentricity of the Earth’s orbit around the sun, and the tilt of the Earth’s axis in relation to its orbital plane. The equation of time varies during the year from minus 14 minutes to plus 16 minutes, and it is the main reason for the difference between the time that you will calculate by looking at a sundial and that shown on a clock. There is also a slight difference between morning and afternoon caused by that day’s portion of the sun’s annual movement around the ecliptic.

The combination of the above effects can create a difference between morning and afternoon daylight of more than half an hour at Reading.

None of this means, however, that midday is in the wrong place, merely that the standard time system, whose simplicity and uniformity are essential for communication, is necessarily an approximation to the sun’s complex apparent motion.

The further lengthening of the afternoon daylight, and shortening of before-noon daylight, during the months of British Summer Time are of course, the intended result of the forward movement of clocks by one hour.

David Le Conte

The Astronomical Society of Guernsey

Midday Greenwich Mean Time is only the middle of the day at the Greenwich meridian. If you are west of Greenwich, as Reading is, the sun rises later and sets later, so 1200 GMT will be earlier than the midpoint between sunrise and sunset. The sun travels 360° in 24 hours, or 15° in one hour. Hence, as I write this in north London (0° 10’ West), 1200 GMT is 24 seconds before midday, but if I lived in Swansea (3° 56’ West), 1200 GMT would be nearly 16 minutes before midday.

Using Central European Winter Time, 1200 is 6 minutes before midday in Berlin (13° 30’ East), but it is more than 50 minutes before midday in Paris (2° 15’ East).

The most extreme example is Lisbon in Portugal (9° West), which has recently adopted Central European Time – during summer, 1200 is two and a half hours before midday.

Nigel Wheatley

London, UK

Clear day blues

Why (on a clear day) is the sky blue?

Jaspar Graham-Jones

Southampton, Hampshire, UK

The sky is blue because of a process called Rayleigh scattering. Light arriving from the sun hits the molecules in the air and is scattered in all directions. The amount of scattering depends dramatically on the frequency, that is, the colour of the light. Blue light, which has a high frequency, is scattered ten times more than red light, which has a lower frequency. So the ‘background’ scattered light we see in the sky is blue.

This same process also explains the beautiful red colours at sunset. When the sun is low on the horizon, its light has to pass through a large amount of atmosphere on its way to us. During the trip, blue light is scattered away, but red light, which is less susceptible to scattering, can continue on its direct path to our eyes.

Rick Eraho

Cleckheaton, West Yorkshire, UK

The sky is blue because of a process known as Rayleigh scattering. According to classical physics, an accelerated charge emits electromagnetic radiation. Conversely, electromagnetic radiation may interact with charged particles causing them to oscillate. An oscillating charge is continually being accelerated and hence will re-emit radiation. We say that it becomes a secondary source of radiation. This effect is known as the scattering of the incident radiation.

The atmosphere is, of course, composed of various gases that together form air. We may treat each air molecule as an electron oscillator. The electron charge distribution of each molecule presents a scattering cross-section to the incident radiation. This is essentially an area upon which the incident radiation must fall for scattering to occur. The amount of scattered radiation will depend upon the magnitude of this cross-section. In Rayleigh scattering the cross-section is proportional to the fourth power of the frequency of the incident radiation. Sunlight is composed of various visible frequencies ranging from low frequency (red) to higher frequency (blue) light. Because it is of a higher frequency than other visible components, the blue part of the sun’s spectrum will be scattered more strongly. It is this scattered light that we see and so the sky appears to be blue.

Incidentally we are also able to explain why sunsets are red. When the sun is close to the horizon its light must travel through more atmosphere. The blue light will be scattered strongly whereas red light, because it is of lower frequency is less prone to scattering and so is able to travel straight to the observer.

D. Roberts

Physics Department
University of Sheffield, South Yorkshire, UK

Chinese puzzle

The Great Wall of China is commonly cited as being the only non-natural object visible from space. For an object to be visible when viewed from above, the eye must be able to resolve it in two dimensions. The Great Wall of China is immensely long, but very narrow. If the eye is able to resolve its width from space, many other objects such as the Great Pyramid of Cheops should be large enough to be seen in two dimensions, despite having a much smaller total area. Is the ability of the eye to resolve objects in the smaller of the two dimensions affected by the magnitude of the larger (and if so why) or is the claim made for the Great Wall of China incorrect?

A. R. MacDiarmaid-Gordon

Sale, Cheshire, UK

The claim is incorrect. It is well known as one of the most widely believed urban legends, perhaps second only to the famous one about mass suicide by lemmings.

A person with perfect eyesight is able to resolve up to about one minute of arc without binoculars or a telescope. The Great Wall of China is, very approximately, 6 metres wide. This means that it is not directly visible above an altitude of about 20 kilometres, or just over twice the height of Mount Everest. Even if its shadow is taken into account, this would only make it visible, in places, up to perhaps about 60 kilometres at the most. Because of atmospheric drag, this is still below the height necessary for a stable spacecraft orbit.

There are, however, many man-made objects which are visible from outer space, the largest being the Dutch polders or reclaimed land. Cities too can be seen at night because of the bright streetlights.

D. Fisk

Ipswich, Suffolk, UK

It is well known that the human eye can pick out long objects much more easily than short ones, so the Great Wall of China is certainly a candidate for being visible from the moon. However, the wall is, in places, a broken-down edifice and is often scarcely visible on the ground, never mind from space. H. J. P. Arnold, photographic expert and skilled astronomer, has studied this problem and concludes that seeing the wall from the moon is a physical impossibility.

Neil Armstrong of Apollo 11 has stated that the wall is definitely not visible from the moon. Fellow astronaut Jim Lovell of Apollo 8 and 13 made very careful observations and says that the claim is absurd. Jim Irwin of Apollo 15 has said that seeing the wall is out of the question.

Photographs from uncrewed probes do show that the route of the wall is sometimes shown by sand that is blown on to the windward side, but that the wall itself is not visible. The end, perhaps, of yet another legend.

Robert Brown

Ashby-de-la-Zouch, Leicestershire, UK

Can anyone explain in simple and common-sense terms why there is simultaneously a high tide on both sides of the Earth?

Pat Sheil

Sydney, New South Wales, Australia

In considering the origin of tides we must disregard the Earth’s daily rotation around its axis and concentrate only on the revolution of the Earth-Moon system.

This revolution takes place around the system’s common centre of gravity, which is about halfway from the surface to the centre of the Earth, and causes every point in the Earth’s interior or on its surface to describe a circle of radius equal to the distance of the common centre of gravity from the Earth’s centre.

Therefore, at every point there is a centrifugal force of the same magnitude and in the same direction: away from the moon, parallel to the line joining the Earth-Moon centres. This centrifugal force is distinct from the one caused by the Earth’s rotation, which we are disregarding.

Every point of the Earth also experiences a gravitational force as it is pulled towards the moon, the direction of this force being different for different points of the Earth.

The resultant of these two forces creates the tide-generating force. If we now consider two points on the Earth’s surface, one directly below the moon and the other on the far side, it turns out that the moon’s gravitational force at the near point is greater than the centrifugal force which, as we have seen, is away from the moon.

The far point is farther away from the moon by one Earth diameter and the moon’s gravitational force there happens to be smaller than the centrifugal force, so the net force on water at the far point is away from the moon.

In most popular accounts, the simultaneous occurrence of tides at the two opposite points is explained by asserting that while the moon pulls the water at the near point some distance, it pulls the Earth’s body a little less.

But this explanation does not clarify why a system like that will not simply collapse under the mutual gravitational attraction between the Earth and the moon.

D. S. Parasnis

Department of Geophysics

Luleå University of Technology, Sweden

Ignoring the effects of other bodies, the centre of mass of the Earth and the centre of mass of the moon are both in free fall, following orbits around the common centre of mass of the Earth-Moon system, which gravity and centrifugal acceleration precisely cancel out.

Over most of the Earth’s surface, though, this cancellation is not precise, because you’re either nearer to, or farther away from, the moon, but still forced to orbit at the same rate as the Earth’s centre of mass.

For the ocean on the side of the Earth facing the moon, lunar gravity dominates centrifugal force, so water bulges towards the moon.

On the opposite side, centrifugal force dominates, so water bulges away from the moon. Both bulges produce high tides.

In effect, sea level – which would otherwise be spherical – is stretched along the Earth-Moon axis into an ellipsoid, and as any point on the Earth rotates into and out of either bulge, the local tides flow, then ebb.

Greg Egan

Perth, Western Australia

The simultaneous high tides on opposite sides of the Earth are a result of an imbalance between gravitational forces and centrifugal forces. Tides are caused by the gravitational interaction of Earth and moon, and to a lesser extent the Earth-Sun interaction.

Although we think of the moon as orbiting the Earth, in fact the moon and the Earth both orbit their common centre of mass, which is close to, but not exactly at, the centre of the Earth. The centrifugal forces generated by the orbital motions of each body just balance the gravitational pull of the other body.

However, the balance is only exact at the centre of each body. On the side of the Earth nearest the moon, the moon’s gravitational pull is slightly greater and the centrifugal force slightly less than at the Earth’s centre, so water here is pulled out towards the moon. On the opposite side of the Earth, the gravitational pull is slightly less and the centrifugal force slightly greater, so here water is thrown outwards away from the moon.

Mark Bertinat

Chester, UK

Glenbrook Infants School went to the seaside for our summer outing. We had a nice time, but please can you tell me why the sea is salty. My mum doesn’t know.

John Connolly

London, UK

The sea is salty because the rivers that flow into it wash salts and other minerals out of the ground. The salts dissolve in the rivers and the rivers flow into the sea. As the sun evaporates the water from the sea to make clouds, it leaves the salts and minerals behind, so the sea is saltier than rivers and lakes.

Jack Cave-Lynch (aged 9)

Wellington, New Zealand

John Connolly is a brainy guy

The salty sea which is such fun

When splashing in the waves and sun

Is not freshwater from the tap

Or from a bottle with a cap;

So he will learn that salt and sea

Mix just like sugar into tea

And that many other kinds of salt

Dissolve into this briny malt,

Sodium chloride, the salt of table

Has other friends within its stable

Potassium Ch, magnesium Ch, and iodide

All flow solvent with the tide.

So now, dear John, you clever lad

Off you go – tell mum and dad!

Ray Heaton

Solihull, West Midlands, UK

Energy loss

What is the so-called ‘slingshot effect’ used to accelerate interplanetary spacecraft? It obviously makes use of the gravitational attraction of a planet, but my naïve understanding of physics tells me that any kinetic energy gained on approaching a body would be lost as potential energy on leaving. How does the spacecraft extract energy from the planet?

David Bates

Ely, Cambridgeshire, UK

I had the same problem as your questioner when I first heard about Voyager using the ‘slingshot effect’. Clearly a probe will not make any net gain in energy by simply falling through a stationary gravity field.

However, Jupiter and its gravity field are moving around the sun at a speed of about 1300 metres per second and any probe passing behind the planet will be accelerated by this moving gravity field much as a surfer is pushed forward by a wave. The energy does not come from the gravitational field but from the kinetic energy of the moving planet which is slowed by the tiniest amount in its orbit, causing it to drop ever so slightly closer to the sun.

The planet speeds up as it falls towards the sun, and paradoxically it ends up moving more quickly than it did before. Moving Jupiter closer to the sun by 10–15 metres (about the diameter of a proton) would yield more than 416 megajoules.

Mike Brown

Knutsford, Cheshire, UK

In one of her songs, the American artist Laurie Anderson uses the refrain ‘Now that the living outnumber the dead . . .’ Is this true? If so, when did it happen? If not, when might it happen, if ever? Do we have good estimates of population numbers before recorded history?

John Woodley

Toulouse, France

If the world population had always been increasing at its present rate, doubling within an average human lifespan, then the living would indeed outnumber the dead.

However, this is not what has happened. There have been very long periods in the past when the population hardly grew at all, but when deaths continued to accumulate. For historical periods, there is a surprising amount of information on population figures, including censuses conducted by both the Romans and the Chinese.

Before then, there are estimates based on the area of the world which was under cultivation or used for hunting, and of the numbers of people who could be supported per acre using these methods of food production. According to estimates assembled by J.-N. Biraben, the world population was about 500,000 in 40000 BC. It grew – but not at a steady rate – to between 200 and 300 million in the first millennium AD, and reached 1 billion early in the 19th century.

On multiplying the population numbers by the estimated death rates, you discover that the total number of deaths between 40000 BC and the present comes to something in the order of 60 billion. The present world population is still only about 6 billion.

Although no great claim can be made for the accuracy of the historical estimates, the errors can hardly be so large as to effect the conclusion that the living are far outnumbered by the dead. This has always been the case, and will continue to be so into the indefinite future.

Roger Thatcher

New Malden, Surrey, UK

In the Garden of Eden, the living (2) outnumbered the dead (0).

G. L. Papageorgiou

Leicester, UK

In the Indian epic poem Mahabharata the eldest Pandava, Yudisthira, was asked many questions, including the one posed above, by the god Yama, who was the keeper of the Underworld and all that is righteous, to test Yudisthira’s knowledge, power of reasoning and truthfulness.

Yama was disguised as a stork guarding a pond from which Yudisthira’s four brothers drank without being able to answer a single question and were all struck dead. The stork Yama asked ‘Who are the more numerous, the living or the dead?’ Yudisthira answered: ‘The living, because the dead are no more!’

Yama accepted this and all the other answers given by Yudisthira and, with great pleasure because Yudisthira was actually the son of Yama, blessed him and revived all of his dead brothers.

Shafi Ahmed

London, UK

Snow laughing matter

Would it be possible to reduce the impact of the greenhouse effect by painting roofs of buildings white to reflect sunlight in the same way the polar icecaps do? Does a paint exist that would mimic the reflective properties of snow?

Paul Nolan

Warrington, Cheshire, UK

Painting roofs white would reflect more sunlight and it might also compensate for global warming. The Global Rural Urban Mapping Project (GRUMP), undertaken by the Earth Institute at Columbia University in New York, shows that roughly 3 per cent of the Earth’s land surface is covered with buildings.

The Earth has an albedo of 0.29, meaning that it reflects 29 per cent of the sunlight that falls upon it. With an albedo of 0.1, towns absorb more sunlight than the global average. Painting all roofs white could nudge the Earth’s albedo from 0.29 towards 0.30. According to a very simple ‘zero-dimensional’ model of the Earth, this would lead to a drop in global temperature of up to 1 °C, almost exactly cancelling out the global warming that has taken place since the start of the industrial revolution. A zero-dimensional model, however, excludes the atmosphere and, crucially, the role of clouds. It would be interesting to see if more sophisticated models predict a similar magnitude of cooling.

Mike Follows

Willenhall, West Midlands, UK

A better use of roofs would be to use them as mini power stations by installing photovoltaic tiles. This would displace a significant proportion of the fossil carbon that we emit without relying on perturbing the Earth’s delicate and complex climate system. Sure prevention is much better than uncertain cure.

Mike Hulme

Norwich, UK

As the sun produces energy, it presumably loses mass and its gravity weakens. Are the planets slowly spiralling outwards? If so, by how much, and by the time the sun becomes a red giant, how far out would the Earth be?

Mike Ganley

Ferntree, Tasmania, Australia

The sun loses about 4 million tonnes per second, which is the mass equivalent of the energy it produces through ther monuclear reactions. Another few million tonnes are lost in solar wind and other particle emissions. However, even after 2 billion years, this loss constitutes only one 10-thousandth of the sun’s mass. So the change in the Earth’s distance from the sun will be of the same fractional order.

The situation will be more drastic when the sun eventually becomes a red giant, 6 billion years or so from now. Then, the solar radius will be 100 times its present value. Some of the latest estimates suggest that in its giant stage the sun may well engulf Mercury, Venus and Earth, while planets more distant than Mars will survive and continue to orbit the sun when it later becomes a white dwarf.

If we assume the final mass of the sun at the white-dwarf stage to be 0.6 of its present value, the dimensions of the planetary orbits in the very far future will be about 80 per cent greater than they are now for the reasons your questioner suggests.

C. Sivaram

Indian Institute of Astrophysics

Koramangala, Bangalore, India

Amazingly, although the sun converts 4 million tonnes of its mass into pure energy every second and will continue to burn hydrogen until it becomes a red giant several billion years from now, it will even then have lost only a tiny proportion of its present mass. In order for the Earth to conserve its angular momentum, the radius of its orbit will have to increase at the rate of only about 1 centimetre a year.

However, this will not be sufficient to compensate for the steadily increasing luminosity of the sun. So Earth is destined to follow in the footsteps of its celestial companion, Venus, and undergo a natural runaway greenhouse effect – unless human activity contrives to accomplish it rather sooner.

Mike Follows

Willenhall, West Midlands, UK

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