Ice on the Moon


A review of the lunar polar ice controversy, how it will be resolved, and what it means to our return to the Moon


From The Space Review, November 6, 2006


Paul D. Spudis


We’ve known for many years that the Moon has no atmosphere and hence, no stable surface water.  Modern inquiries into polar ice started with a paper written in 1960.  Harrison Brown and associates noted that crater floors near the poles of the Moon would be both permanently dark and very cold, receiving heat only from space and the interior of the Moon (a cold, geologically dead object.)  Such a combination makes a “cold trap,” where ice remains stable for geological time spans.  As water-bearing comets and meteorites were known to strike the Moon, the slow addition of water – molecule by molecule – could result in large quantities of ice over the multi-billion year history of the Solar System.


A bonanza of geological information provided by the Apollo missions revolutionized our understanding of early Solar System history.  However, the samples dampened the enthusiasm of  scientists for the idea of lunar ice.  The rocks and soils were drier than any terrestrial sample, and even most meteorites.  Lunar samples showed the Moon has no indigenous water and that the tons of water from impacting comets had never altered or modified a returned lunar sample.  True enough, we had not visited the poles or even any high latitudes, but our improved understanding of the history and environment of the Moon deepened the ingrained skepticism among scientists that water would be found there.


Constrained to near-equatorial regions by operational and safety considerations, the poles weren’t investigated by the Apollo missions.  But the idea that water might exist in these forbidding, dark cavities was kept alive in a paper by Jim Arnold of the University of California.  Arnold was well aware that the Apollo samples were bone-dry, but noted that Brown’s original arguments were still valid and advocated flying a lunar polar orbiting satellite with a gamma-ray spectrometer.  Such instruments detect and measure polar ice by looking for the emission of a gamma-ray line caused by the presence of hydrogen.  Along with a variety of other measurements designed to follow-up on Apollo discoveries, the so-called Lunar Polar Orbiter (LPO) spacecraft would map the chemical composition of the whole Moon.


But after Apollo, NASA lost interest in the Moon and LPO was never selected for flight.  Still trying to unlock her secrets, scientists contented themselves with analyzing the returned samples and remotely studying the surface of the Moon.  In the early 1990’s, the Space Exploration Initiative of President George H. W. Bush caused the idea of polar water to surface again, though it was not considered in the “critical path” of lunar return.  The SEI plans called for a bigger, heavier, and slower lunar return – a re-cast Space Station Freedom.  Polar ice could be relevant, but incidental to lunar return, not an enabling asset.


In 1992, SEI was “zeroed out” by Congress after the election defeat of President Bush and lunar missions were once again farther away than ever.  Or were they?  A large part of the vigorous space activities of the 1980’s centered around research for defense against ballistic missiles, President Ronald Reagan’s Strategic Defense Initiative, a.k.a., “Star Wars.”  One concept for space-based defense was to launch a myriad of very small, yet capable satellites to both locate and intercept ballistic missiles in flight, the so-called “Brilliant Pebbles” system.  These small spacecraft would carry a variety of sensors that could be used for scientific purposes.  During the early 1990’s, the SDI Organization created the Clementine program that  built and launched a small, Brilliant Pebble-derived spacecraft to the Moon.


Clementine orbited the Moon in 1994 for 71 days, mapping the Moon globally in 11 wavelengths and measuring its topography by laser ranging.  Although the spacecraft didn’t carry instruments to specifically look for ice, the mission team improvised an experiment to do this.  We beamed radio waves into the polar darkness and listened for echoes with the distinctive characteristics of interaction with ice.  This bistatic radar experiment (so-called because the spacecraft transmitted while we listened to the echoes on Earth) found evidence in the dark areas near the south pole of the Moon for material with high circular polarization ratio (CPR; see below).


After Clementine demonstrated that significant scientific results could come from small missions, NASA sent Lunar Prospector in 1998 to orbit the Moon for 18 months.  This mission carried an instrument designed to measure neutrons.  By looking at the energy of neutrons coming from the lunar poles, the LP team found “excess” hydrogen at both poles.  If in the form of water ice, they estimated that hundreds of millions of metric tonnes could exist at the poles of the Moon.


Meanwhile, astronomers on Earth began publishing results questioning the Clementine and Lunar Prospector results.  With the giant Arecibo radiotelescope, radar images were taken from the Earth.  They found radar reflections with high CPR lying in both permanent darkness and in sunlit areas.  Ice is not stable in sunlight so they postulated that all high CPR is caused by surface roughness; if any ice is at the lunar poles, it must be in a finely disseminated form, invisible to radar mapping.


Thus we arrive at the present.  Is there water ice on the Moon?  If so, does it exist in accessible quantities making a permanent human presence on the Moon easier?  What additional information is needed to resolve this controversy?


What’s the story?


In contrast to some recent claims, this debate is still open and nothing has occurred in the last few years to cause participants in the debate to abandon their positions. In a nutshell, poor or incomplete coverage by a variety of marginal data has led to much heat, while casting little light on the issue of lunar polar water.  Here, I  present the evidence to the reader, noting the strengths and weaknesses of each data set, and attempt to identify the remaining unanswered questions.


Clementine bistatic radar.  As the Clementine spacecraft orbited the Moon, it transmitted radio waves toward the poles and we listened to the reflected radio waves bounced back to Earth.  This experiment was bistatic, i.e., the transmitter and receiver were in different places.  Bistatic radar has the advantage of observing reflections through the phase angle, the angle between transmitted and received radio rays (Figure 1).  This phase dependence is important.  It’s similar to the effect one gets from looking at a bicycle reflector at just the right angle – at certain angles, the internal planes in the transparent plastic align and a very bright reflection is seen.  Similarly, in both radio and visible wavelengths on the Moon, we see an “opposition surge,” an apparent increase in brightness looking directly down sun (zero phase).  Clementine orbited the Moon such that we could observe its phase dependence (Fig. 1) and we specifically looked for this “opposition surge,” called the Coherent Backscatter Opposition Effect (CBOE).  CBOE is particularly valuable to identify ice on planetary surfaces.


Clementine transmitted right circular polarized (RCP) radio and we listened on Earth in both right- and left-circular polarized (LCP) channels.  The ratio of power received in these two channels is called the circular polarization ratio (CPR.)  The dry, equatorial Moon has CPR less than one, but the icy satellites of Jupiter all have CPR greater than one.  We know these objects have surfaces of water ice; in this case, the ice acts as a radio-transparent media in which waves penetrate the ice, are scattered and reflected multiple times, and returned such that some of the waves are received in the same polarization sense as they are sent – they have CPR greater than unity (Figure 2.)



Figure 1.  The Clementine bistatic radar experiment.  Two orbits of radar data were taken; orbit 234 passed directly over south polar darkness; the control orbit 235 was over similar terrain that receives sunlight.  An enhancement of CPR was seen in orbit 234 data, centered around zero-phase (b=0), over the dark region, but not over the sunlight region (orbit 235), even though both are highlands terrains.  This relation was interpreted to be caused by the presence of ice in the dark areas near the south pole.  From Nozette et al. (1996) Science 274, 1495-1498.


The problem with CPR alone is that we can also get high values from very rough surfaces, such as a rough, blocky lava flow which has angles that form many small corner reflectors.  In this case, a radio wave could hit a rock face (changing RCP into LCP) and then bounce over to another rock face (changing the LCP back into RCP) and hence to the receiver (Figure 2.)  This “double-bounce” effect also creates high CPR in that “same sense” reflections could mimic the enhanced CPR one gets from ice targets.

Figure 2.  High CPR can be caused by both surface roughness scattering or by ice volume scattering.


Bistatic geometry can help in the interpretation of radar scattering.  Both monostatic and bistatic radar measure CPR but bistatic radar also measures the angular dependence on reflection, which is distinctly narrower for volume (ice) scattering.  In the case of the Clementine experiment, we measured two orbits of the lunar south pole, one over an area of polar darkness and the other over a nominally sunlit zone near the pole.  Results (Figure 1) are intriguing; we see evidence of a CPR enhancement (symmetric about the zero phase angle; see peak in orbit 234 curve) over the dark region, where ice would be stable, but not over the control (orbit 235) sunlit area.  The Clementine team interpreted this response as CBOE, caused by ice in dark areas near the south pole.  From the strength of the enhancement and its angular width, they reasoned that ice was mixed with regolith dirt and present in a deposit about 2 m thick with an average concentration of about 1.5 wt.%.  It should be noted that this doesn’t require an intimate mixture of ice and dirt, but is the average over hundreds of square kilometers.  Thus, areas could exist of nearly pure ice in some places, and virtually none elsewhere.


This conclusion was tempered by the recognition that Clementine found enhanced CPR only during one observation; the limited time of the mission at the Moon (71 days) precluded repeating that measurement.  In addition, the Clementine spacecraft was not optimized for this experiment, so the data have very low resolution – basically a spot about 300 km across.  Nevertheless, the results of this experiment have not been refuted.  The most recent Earth-based radar studies confirm that high CPR does indeed exist within the dark area near the south pole.  Given the size of the Clementine resolution cell, the observed CPR enhancement could be explained by the same area of high CPR observed in ground based radar images of the crater Shackleton.  The controversy is not whether an area of high CPR exists in the permanently shadowed interior of Shackleton crater but over what is causing the high CPR signature.


Lunar Prospector Neutron Spectrometer.  NASA’s Lunar Prospector spacecraft carried an instrument that measured neutrons emitted from the Moon as a function of their energy.  Medium energy neutrons are strongly absorbed by hydrogen.  Thus, by measuring the flux of neutrons in this energy range, we can estimate how much hydrogen is present in the lunar soil.  The LP neutron experiment sampled only the upper 40 cm or so of the Moon.  As the  spacecraft was a spinner, its instruments looked simultaneously in all directions and the effect of such a view is to limit surface resolution to roughly the altitude of the spacecraft.  The best resolution of the LP neutron data is 30-40 km.  Unlike both the Clementine radar experiment and Earth-based radar, the LP instrument looked directly into the entire polar dark area of the Moon.


The Lunar Prospector observed strong absorption of medium energy neutrons at both poles.  Initially it was thought that there was more hydrogen at the north pole, but later analysis showed roughly equal amounts at both poles.  The actual enrichment (up to 200 parts per million; Figure 3) is only about a factor of two greater than the highest concentrations of solar wind hydrogen seen in the  Apollo soil samples.  But the LP team suggested that if this hydrogen was present as water ice (which is stable only in polar dark areas), the average concentration of ice was around 1.5 wt. %, a significant value.  Moreover, with the low resolution of the LP neutron data, significantly higher concentrations within the shadow cannot be ruled out; a uniform, low average ice concentration of about 1-2 wt.% or a very heterogeneous distribution with very high concentrations (in some places up to over 40 wt.%) are equally consistent with the data.



Figure 3.  Lunar Prospector neutron spectrometer maps of the lunar poles.  These low resolution data indicate elevated concentrations of hydrogen at both poles; it does not tell us the form of the hydrogen. Map courtesy of D. Lawrence, Los Alamos National Laboratory.


Curiously, data from fast neutrons detected by Lunar Prospector suggest that the uppermost surface is depleted in hydrogen, down to about 10 cm below the surface.  Such a depletion suggests a non-solar wind origin for the polar hydrogen, as hydrogen implanted by solar wind would be expected to be high in the uppermost lunar surface.


As you might expect, the LP neutron results have been questioned.  Some have suggested that the reduction in neutrons is caused by the presence of another light element, with sulfur  suggested as an alternative.  However, cometary ice is very abundant and known to constantly hit the Moon.  Lunar sulfur is not rare, but is relatively low in cosmic abundance and any process that would concentrate sulfur in the polar dark areas would also concentrate the more abundant extra-lunar hydrogen.  Recent claims that the LP neutron data indicate a low, uniform concentration are not correct; we know nothing about the distribution of the hydrogen below the resolution of the neutron spectrometer (i.e., scales smaller than 30 km.)


Earth-based radar data.  Radar has been used to study the Moon for decades with many observations made in preparation for the Apollo missions.  This work largely concentrated on the equatorial regions (target sites for Apollo), but later work has focused on the lunar poles.  Although some of their early work supported the concept, the most strident  objections to the presence of lunar polar ice has come from planetary radar astronomers.


Nick Stacy mapped the south pole of the Moon using the Arecibo telescope in 1992 for his Ph.D. dissertation.  The Arecibo group found several zones of high CPR, although its distribution is patchy and discontinuous.  They noted that some areas of high CPR occur within craters that might be permanently shadowed (at that time, lighting maps of the poles did not exist).  Although couched in appropriately cautious terms, Stacy noted that one high CPR zone occurs  within the crater Shackleton and that it appears to continue down into the portion of the crater floor in Earth shadow, out of view of the Arecibo dish.  Attributing most of the high CPR to blocky, rough surfaces associated with craters, Stacy reserved the possibility that some high CPR spots could be ice if they occurred deep within permanently dark crater floors.


Subsequent work by the Arecibo group has moved away from this cautiously positive interpretation to a definitive assertion that none of the high CPR zones seen around the pole are caused by the presence of ice.  In at least four papers published between 1997 and 2006, they have presented increasingly more detailed image data, each showing the same relations: patchy, high CPR found in both sunlight areas and in permanent darkness.  The latest paper from the Arecibo group, published in October 2006 to a barrage of publicity (including an overwrought press release in which one investigator called the “door on the debate” on lunar polar ice detected by radar “closed”) shows the south pole of the Moon in unprecedented surface resolution, about 20 meters per pixel.  Yet again, we see the high CPR patch in Shackleton (Figure 4), but this time, it is accompanied by an image and analysis of another crater, Schomberger G, which is alleged to have the same distribution of high CPR within it.  As Schomberger G is in sunlight (and has high CPR in portions of its interior), the authors conclude that the high CPR in Shackleton is similarly caused by surface roughness and not by the presence of ice within the permanently dark area of the crater.


Figure 4.  New CPR maps of two polar craters (right); yellow-red colors indicate high CPR values.  The interior of Shackleton (89.9° S, 0°; 19 km diameter) is permanent darkness, but the interior of Schomberger G (77° S, 8° E; 17 km diameter) is at least partly sunlit during the lunar day.  High CPR is found in both craters, but do they both have the same cause?  After Campbell et al. (2006) Nature 443| 19 October 2006| doi:10.1038/nature05167


As all parties agree that high CPR is found in the polar regions of the Moon, the debate is over what this relation means.  The Arecibo group claim that the distribution patterns in Schomberger G and Shackleton are the same (Figure 4); hence, the high CPR patches represent rocky outcrops on and within these craters, not ice.  However, high CPR can be caused by either roughness or ice; in itself, high CPR is not uniquely diagnostic of either (Figure 2).  I contend that because of its non-unique nature, high CPR within Shackleton could be ice; as near as can be determined, the high CPR patch occurs within a zone of permanent darkness.


Why even entertain this notion?  After all, if ice is unstable on any part of the Moon that sees sunlight, doesn’t that mean that high CPR here indicates roughness, not ice?  In fact, similar relations are seen on the planet Mercury (Figure 5).  The polar features of Mercury were initially discovered by Dewey Muhleman and colleagues at Caltech using very low resolution, global disk images.  Although these images show a prominent high CPR zone near the north pole of Mercury (Fig. 5), they also show high CPR zones in mid-latitudes and equatorial regions.  The interpretation of the authors of this work was that two mechanisms produce high CPR on Mercury; near the equator, surface roughness must be the cause of high CPR, but at the poles, water ice in permanent shadow could not ruled out (like the Moon, Mercury’s pole is normal to the plane of its orbit around the sun).  Thus, two scattering mechanisms were invoked.  In principle, there is no reason why such a relation would not also occur on our Moon.  In such a case, high CPR can be caused by both roughness and ice.  If a spot is in sunlight, it must be surface roughness, but if it’s in the permanent darkness, ice cannot be ruled out.



Figure 5.  High CPR areas on the planet Mercury.  The discovery of polar ice on Mercury (Muhleman and co-workers, 1992) stunned most planetary scientists.  The discovery image (left) shows the north polar deposits (red) at top, but also show high CPR zones near the equator and in mid-latitudes.  Such a relation indicates at least two different high CPR-creating mechanisms.  Left image from NRAO, Muhleman, 1992; right north polar image from Harmon, NAIC, 2000.


It is claimed by the Arecibo group that the distribution of high CPR within the two craters Shackleton and Schomberger G are identical.  As Schomberger G is in partial sunlight, high CPR seen within it cannot be caused by ice.  As a planetary geologist, I see significant differences in the distribution of CPR in the two craters (Figure 4.)  In Schomberger G, high CPR is found as a quasi-continuous upper “layer,” with CPR values decreasing deeper into the crater.  At Shackleton, the upper crater wall is complex and high CPR is discontinuous; the large zone of high CPR within the crater at about 8 o’clock (Figure 4) starts below the rim, but continues down into the crater, disappearing into the shadow caused by the Earth-Moon geometry.  I leave it to the reader to decide for themselves whether the distribution of high CPR is identical in these two craters.  All of the interior of Shackleton is in permanent darkness, shielded from sunlight and has been continuously for at least the last 2 billion years.  So in theory, ice may have accumulated within it.


Thus, three data sets exist, each unique, on the possibility of lunar polar ice.  But what are they telling us?


Synthesis:  Best guess on polar volatiles


No single piece of evidence for lunar ice is decisive, but I think the preponderance of evidence indicates that water ice exists in permanently dark areas near the poles.  However, its origin and the processes associated with its deposition are unclear.  The ice could be of cometary, meteoritic, or solar wind origin; each mode would have interesting implications for its  composition.  If largely of cometary origin, other volatile species of great utility may also be present, such as ammonia (NH3), methane (CH4), and various organic substances.  Nitrogen is particularly useful in  supporting human life, both for breathing air and for agriculture.  Whatever the source, polar ice is a useful resource for future lunar inhabitants.


Much remains unclear about the nature of ice is on the Moon.  Rates of deposition of polar ice and implications for its physical nature are unknown.  We can, however, make some inferences from the data in hand.  Ice deposits cover a minority of the polar terrain and concentrations of it could vary widely over a small area, leading to a very heterogeneous deposit.  This supposition is suggested by the patchy distribution of high CPR spots in the Earth-based radar data (not all of which are caused by ice; see section above.)  The concentration and distribution of the ice is unknown, but if very heterogeneous as suggested, deposits could locally cover between 10-50 % of a given patch of dark area.  Individual bodies of trapped ice could be on the order of meters to tens of meters in size, as suggested by the patchy extent of high CPR areas seen in the polar darkness.


From the fast neutron data of Lunar Prospector, the uppermost 10 cm or so of the polar dark regions are depleted in hydrogen.  Radar data suggest volume scattering at depths on the order of several tens of wavelengths of the S-band radar (~13 cm.)  Thus, ice occurs between depths of 10 cm and 2-3 m.  From our current understanding of the creation, turnover, and evolution of the lunar soil, the ice is probably not “pure” but contains contaminants and solid inclusions of varying concentrations.  Although water ice is expected to dominate the deposit, other minor species of cometary origin could be present in useful quantities.  The terrain of a lunar highlands region (found at both lunar poles) can be very rugged, with local slopes exceeding 30 degrees.  However, as shown by the Apollo 16 highland landing site, such areas can be negotiated reasonably well, if the correct paths are chosen.



So what?


Water ice on the Moon makes living there easier, cheaper, and thus, more likely.  Solar wind hydrogen is found everywhere on the Moon, but in vanishingly small quantities.  Ice at the poles is a concentrated source of both hydrogen and oxygen – two substances vital to supporting human life and creating a space transportation infrastructure.  We can extract what we need out of the equatorial regolith, but it’s much harder and more energy intensive than at the poles.  Extracting solar wind hydrogen requires heating soil to about 700° C, at which point 90% of the adsorbed gas is driven off.  In contrast, icy regolith heated to about 100° C gives off water as an easily collected and stored gas.  Per unit mass, it takes roughly two orders of magnitude less energy to extract hydrogen from icy polar regolith than it does by roasting soil at the equator.


Although polar ice is important, it is not a requirement to successfully live and work on the Moon.  The poles of Moon are primarily attractive due to the near-permanent sunlight found in several areas.  Such lighting is significant from two perspectives.  First, it provides a constant source of clean power and allows humans to live on the Moon without having to survive the two-week-long (14 day) lunar night experienced on the equator and at mid-latitudes.  Second, because these areas are illuminated by the sun at grazing angles of incidence, the surface never gets very hot or very cold.  Sunlit areas near the poles are a benign thermal environment, with an estimated temperature of about -50° ± 10° C.  Having water near these locales would be a huge bonus.  The most compelling reason to go to the poles is to solve the problem of surviving the extended lunar night – a task that, at most other places on the Moon, would probably require spending billions of dollars for a nuclear reactor.


Science is an imperfect process.  At any given point in time, we have limited data of less than optimum quality and nearly always imperfectly or incompletely understood.  Our information on lunar polar ice is limited in both quality and quantity.  No question in modern science is “solved” and the presence of “consensus,” while a useful concept in marketing and politics, has no real value to the truth or falsity of scientific questions.  The way the universe is put together and works is quite independent of the collective opinions of the experts.


To answer the question of lunar polar ice, we need more and better data.  We must first    thoroughly map the polar deposits from lunar orbit.  India is preparing to launch their first mission to the Moon, Chandrayaan-1, in early 2008.  I am on a team that will build and fly the radar mapping instrument on that mission.  This radar will map both poles using a revolutionary new processing architecture that allows us to distinguish areas of high CPR caused by roughness and those caused by the presence of ice.  An even more advanced radar instrument will be on the U.S. Lunar Reconnaissance Orbiter (LRO) mission in late 2008, mapping in two frequency bands (potentially distinguishing roughness from ice) and in high resolution, showing patches of ice as small as 20 meters across.  Chandrayaan will systematically map the polar regions at moderate resolution (75 m/pixel.)  On the subsequent LRO mission, we will get high resolution coverage (10 m/pixel) at multiple wavelengths of promising targets seen in that data.  Since these two missions overlap and will orbit the Moon at the same time, we can use both instruments on the two spacecraft to make bistatic images of the polar deposits; such a mode of operation can observe scattering through the phase angle (looking for the CBOE effect, a good discriminator between ice and roughness.)  Together, these missions will map the extent and distribution of anomalous material in the polar regions of the Moon.


A key advantage of orbital mapping is the ability to look into all of the areas of permanent darkness.  In a recent article in Scientific American (“Radar Images Fail to Detect Ice at Lunar Poles”, October 2006), Don Campbell of Cornell University, part of the Arecibo team, notes that the lunar orbiters LRO and Chandrayaan “will get a better view of the polar terrain than we can from Earth.”


The LRO spacecraft will carry other instruments, including a thermal mapper to determine temperatures of the dark areas, a laser altimeter to measure the topography of the poles (needed to make definitive maps of sunlight and darkness) and other instruments designed to characterize the environment and deposits of the polar regions.  In addition, other nations (including China and Japan) are flying lunar orbiters carrying a variety of mapping instruments.  The Moon, once the most poorly mapped body in the Solar System, appears ready to become the most thoroughly charted and remotely studied object in the history of mankind.


The next step is critical.  After polar deposits have been mapped from orbit, we must land at a promising target and measure volatile substances in the soil.  As described above, no matter what high quality remote sensing data we obtain, there is always ambiguity in interpreting remote sensing data.  We must have ground truth.  Going into the polar darkness, digging up the soil, and measuring what’s there will finally answer, in a definitive manner, the nagging question we started with:  Is there ice on the Moon?


After the first lander, we should survey potential mining prospects, map the distribution of ice on small, local scales (hundreds of meters), and experiment with different extraction methods, water separation technologies, and resource processing and storage techniques.  The goals of lunar resource utilization are challenging, but significant experience can be gathered from small robotic landers prior to the arrival of people.  A program of robotic missions can provide critical strategic information as well as gaining operational experience and providing milestones for a human lunar return.


The Moon is not a hostile, barren rock in space – it is humanity’s stepping stone into the Solar System. The poles of the Moon are inviting “oases” for humans in the desert of near-Earth space.

To live there and at destinations beyond, we must identify resources that will support human life and enable the creation of a new spacefaring infrastructure.



Paul D. Spudis is a planetary scientist at the Applied Physics Laboratory, Laurel MD.  He was a member of the Clementine Science Team in 1994.  In 2004, was a member of the President’s Commission on the Implementation of U. S. Space Exploration Policy and was presented with the NASA Distinguished Public Service Medal for that work.  He is the recipient of the 2006 Von Karman Lectureship in Astronautics, awarded by the American Institute for Aeronautics and Astronautics.  He is the author or co-author of over 150 scientific papers and four books, including The Once and Future Moon, a book for the general public in the Smithsonian Library of the Solar System series, The Clementine Atlas of the Moon (with Ben Bussey), published in 2005 by Cambridge University Press, and Moonwake: The Lunar Frontier (with Anne Spudis), an adventure novel for young adults about life at a base on the Moon.  His web site can be found at: and his blog is at:



The views and opinions expressed in this paper are the author’s and are not necessarily those of the institution for which he works.


Belorussian translation