Saturday, October 30, 2010

Triumph of Newtonian Mechanics – the Story of Neptune


As a teacher, I very much enjoyed teaching a one-semester course on the History of Science to undergraduate students in my college long ago and during just the few years when the subject was part of the curriculum. Later it was axed from the ‘syllabus’ mainly because no one else volunteered to teach it. Also, the curriculum framers did not appreciate the value of the subject as part of the repertoire of teacher competencies.  Thereafter I always tried to bring in a historical perspective to my teaching of Physics and the methodology of teaching science.
One reason for my love for the History of Science was the opportunity it gave me to highlight the main characteristics of the ‘Scientific Method’ underlying the evolutionary, and often revolutionary, discoveries in Science dating back to the times of Aristotle and Archimedes.  I have made a reference to these broad characteristics in my earlier post tilted “Is science education promoting scientific temper?”  I have also tried to illustrate some aspects of the scientific method embedded in an unfinished ‘detective’ work of my own, titled “An Enduring Enigma’, in a later post.

Somewhat simplistically, the scientific method consists in:  (1) observing something that draws our attention or interest, (2) coming up with a tentative explanation (hypothesis) that is consistent with the observations, (3) using the hypothesis to make plausible predictions, (4) testing these predictions by experiments or further observations and modifying the hypothesis in the light of the findings, and (5) repeating steps 3 and 4 until there are no discrepancies between theory and experiment and/or observations.  Of course there are numerous examples of scientific discoveries where these processes were circumvented or short-circuited and where serendipity, chance and even luck, have also played a major role.  The end result is often the formulation of a law or set of laws (such as Newton’s Laws of Motion) or a definitive theory (like the Theory of Relativity) or the establishment or confirmation of what may be termed a scientific fact. 
One of the finest examples of the scientific method is the long sequence of events leading up to the formulation of Newton’s Laws of Motion and their subsequent application to study the motion of heavenly bodies, culminating in the discovery of planet Neptune. This thrilling triumph of Newtonian Mechanics is the main theme of the present post.

Aristotelian Legacy

Apart from the Sun and the Moon, the planets (Mercury, Venus, Mars, Jupiter and Saturn) were recognized as a distinct class of celestial objects, setting them apart from a huge number of stars, since the dawn of civilization.  It was all too obvious for them to be seen as moving around the Earth.  This geocentric view of the universe, based on common sense, is a legacy from the times of the great Aristotle, vigorously upheld and defended by the Roman Catholic Church.  However,  the observation that the planets departed significantly from circular paths was a major irritant, particularly since the similar looking stars were all ‘well behaved’, holding their places firmly on a ‘celestial sphere’ rotating around the Earth like a well oiled machine.  The discrepancy with the misbehaving planets was sought to be overcome by a very intricate mechanism of ‘cycles and epicycles’ of planetary paths devised by Ptolemy and others to preserve geocentricity at all costs.  The Church was happy that, in the process, the human egocentricity was also being preserved and protected.  This feeling of snug satisfaction and status quo lasted well over a millennium during much of which, human civilization was in hibernation and passing through an utterly dark age.

The Copernican Revolution

Though a heliocentric view of the planetary system had been advocated by others, it was left to Nicolaus Copernicus (1473 – 1543), a Polish clergyman and astronomer, to project it with such powerfully convincing arguments that it began to attract serious attention in spite of strong resistance from the Church. Copernicus wrote a treatise called De Revolutionibus Orbium Coelestium setting out his case in considerable detail, but withheld its publication during his own lifetime for fear of adverse reaction from the Church.  It saw the light of day on his death bed and the Copernican system gained widespread acceptance in due course.  It marked a major revolution in the history of human thought.   It also signaled the beginning of an end to the long Dark Age and a renaissance in human civilization.

A Grand Synthesis

Tycho Brahe (1546 – 1601), an eccentric Danish nobleman and brilliant astronomer, realized the need for much greater accuracy and reliability in the measurement of planetary positions and built a gigantic astronomical observatory equipped with some highly sophisticated measuring instruments much before Galileo’s invention of the telescope.  He compiled a huge database of the shifting positions of planets and other celestial objects over a period of twenty years and hoped to use the observational data to actually support the old geocentric system and negate the Copernican arguments.  In this he was sadly mistaken.  His data was used by his German assistant, Johannes Kepler (1571 - 1630), an outstanding mathematician, to strike a death blow to the Ptolemaic system and provide conclusive evidence in favour of the Copernican system.

After an extensive mathematical analysis of Tycho Brahe’s data on planetary positions, Kepler extracted three empirical ‘laws’ governing the motion of the planets in the solar system and brought about a grand synthesis.  He showed that (i) the planetary orbits were actually elliptical, though the departure from circularity was very marginal except for Mars, (ii) an imaginary line joining any planet to the Sun at the focal point of the ellipse swept out equal areas in equal intervals of time, and (iii) there was a mathematical relationship between the periods of motion and the distances of different planets from the Sun.  The Copernican picture was provided a quantitative canvas by Kepler’s monumental work.

Though Kepler’s discovery showed how the planets moved around the Sun, it provided no answer to the central question of why they did so.  The answer had to wait a considerable time.  Yet, Kepler’s findings marked a major milestone in the history of Astronomy.  They meant that the motions of celestial bodies were highly predictable; if the position of a planet was known at one point of time it could be worked out for any other point of time, past or future, with considerable precision.  Astronomy was no longer an inexact science.

Galileo paves the way

Galileo Galiei (1564 – 1642) is credited with starting a scientific revolution of an unprecedented nature and regarded as the founder of the modern scientific method.  He underscored the crucial importance of observation and experimentation in deciding between conflicting ideas and practiced both throughout his illustrious life.  His invention of the telescope and its use in exploring celestial objects – the satellites of Jupiter, phases of Venus, the shape of Saturn, earth-type lunar features, etc., – marked the true beginning of Astronomy and its divorce from the clutches of blind authority and religious dogma.  His observations and discoveries relating to the solar family drove the final nail in the coffin of the geocentric, and egocentric, world view.  Though he incurred the wrath of the mighty Roman Catholic Church for his views and underwent insufferable ignominy and mental torture, he left posterity with the unmistakable message that truth is always the final victor in any intellectual conflict.

Galileo’s numerous experiments on the motion of objects on earth paved the way for an understanding of the root causes of such motion in the hands of his successor, Isaac Newton (1642 – 1727), who was born in England the same year that Galileo died in Italy.
Newton, not particularly known for his modesty, is reported to have said; “If I have seen further than others, it is by standing upon the shoulders of giants”.  The biggest of these giants was undoubtedly Galileo.

Let Newton be…

Newton’s impact on Science was so enormous it is best summed up by Alexander Pope’s famous epitaph:
Nature and nature's laws lay hid in night;
God said "Let Newton be" and all was light.

[Ironically, Newton’s (corpuscular) theory of light was a total flop]

Not since the advent of Einstein in 1905 was so much achieved in science by one individual.  Newton’s greatest contribution was the formulation of the laws of motion and the universal law of gravitation that laid a firm foundation for all of physical science and astronomy of future centuries.  For the first time in human history, fundamental questions relating to the why of nature’s behavior could be faced and answered through the application of Newton’s Laws at both macroscopic and microscopic levels.   They were shown to be truly universal until the early twentieth century when their applicability at the sub-atomic level was shown to be wanting.

Newton’s Laws are indeed laws in the fundamental sense of the term and not empirical ones like those of Kepler cited earlier.  According to these laws: (i) Any object preserves its state of rest or of uniform motion in a straight line unless acted upon by an external force, (ii) An external force applied to a body produces an acceleration (of the body) which is directly proportional to the applied force (the constant of proportionality is called the ‘inertial mass’ of the system), (iii) For every force acting on a body, there is always an equal and opposite force on the agency exerting such force (Law of Action and Reaction), and (iv) Every object in the universe attracts every other body in the universe with a force that is directly proportional to the product of the masses of the two bodies and inversely proportional to the square of the distance between the two (Universal Law of Gravitation).   These statements may sound rather obtuse to readers without a background of science. On the other hand, people with a background of Physics may find them lacking in rigour.  A great deal of discussion and illustration are necessary to understand these fully and it is really not necessary for the average reader to do this to appreciate the essential content of this post.  What matters most is that these laws hold the key to a complete understanding of the motion of planets in the solar system and indeed the motion of all heavenly bodies anywhere in the universe.

Motion in the Heavens

It is relatively easy to show that Kepler’s empirical laws of planetary motion follow as a logical consequence of the application of Newton’s laws of mechanics to celestial bodies.  All the hard work that Kepler put in would have been unnecessary if the laws of motion were known during his time!  This is just one illustration of the extraordinary power of Newtonian mechanics.
Newton’s second law of motion, together with the universal law of gravitation, can be used to calculate the orbital parameters of any planet in motion around the Sun.  The position of the planet can be calculated at any point in time and verified against observational evidence.  In operational terms, one has to solve a two-body problem (such as the Sun-Earth or the Earth-Moon) with appropriate input data and boundary conditions.  It is possible to obtain an exact solution to such a two-body problem.  In actual fact we have to handle a multi-body problem since the other members of the solar system are also involved in the gravitational interactions to varying, though small, extents. These are called perturbations. Exact solutions to such complex problems are impossible and one has to employ a variety of approximation techniques to obtain the desired results.
Mathematicians have perfected techniques for obtaining numerical solutions to such complex perturbation problems to any prescribed degree of precision.  The higher the precision required the more the number of steps in the calculation process.  In the olden days such calculations could only be carried out manually and extremely laboriously.  Months and even years of work were involved.  The advent of the computer in the second half of the last century changed all that.  Today, high speed computers can carry out millions of such calculations in the wink of an eye.  Even modestly priced home computers can be put to such uses if one has the necessary software.  One notable application of such techniques is the guidance and navigation of spacecrafts in real time, the bread and butter of the space age.

A Perturbed Planet

Mercury, Venus, Mars, Jupiter and Saturn are planets known to us since antiquity.  In 1781, William Herschel, an amateur astronomer and musician in England used his own telescope to discover Uranus, a new planet outside the orbit of Saturn. Visible even to the naked eye, it was just waiting to be discovered.  Herschel’s discovery catapulted him to fame and he went on to become a full-fledged astronomer of great repute.

The first accurate mathematical calculations of Uranus' orbital parameters were published in 1792. Within a few years, it was obvious that there was something wrong with the motion of the planet, as there was a small but distinct departure of the observed positions from the calculated ones.  As the accuracy of the observations was not in doubt, questions began to be raised about the calculations themselves, but these were found to be consistently reproducible.  So, there was a real problem on hand and a solution had to be found.  The ingenuity of the human mind began to work, especially in the minds of two very competent mathematicians – John Couch Adams in England and Urbain Le Verrier in France.  They both postulated the existence of an undiscovered planet outside the orbit of Uranus, producing a small residual perturbation in the path of the latter.  But, calculating the location and the orbital parameters of such a hypothetical planet was a formidable challenge and both of them, independently of each other, set out to solve the knotty problem. 

Predicting a Planet

What followed was one of the most enthralling and contentious episodes in the history of science.  In 1843, Adams was apparently the first to predict the expected location of the new planet and sent his calculations to the Astronomer Royal, George Airy, who appears to have just sat on the communication and made no attempt to discover the planet with the telescopes at his command.  Unknown to both, Le Verrier in 1846 completed his calculations, which turned out to be more precise that of the Englishman, and had the good sense to send the findings to Johann Galle of the Berlin Observatory when he found difficulty in arousing the interest of any of his own countrymen.


Galle and his assistant responded by successfully locating the predicted planet (later to be called Neptune) within hours of receipt of Le Verrier's communication on September 23, 1846. Neptune was discovered within 1° of where Le Verrier had predicted it to be, and about 12° from Adams' prediction.  The news created a sensation.  A bitter Anglo-French war of words followed on the question of who should get the bragging rights for the discovery.  Eventually, sanity prevailed and this amazing hare and tortoise race had a happy ending.  Both won! And the referee too! Historians of Astronomy decided to give equal credit to both Le Verrier and Adams.  Adams appears to have been distinctly lucky to share the spoils, but his positive role in the episode was never in question.

As a historian of science put it; “This was the high-water mark of Newtonian physics -- to be able, given the laws of physics and the peculiar motion of one object, to reach out into the depths of space and uncover a previously hidden object -- and caused an even greater sensation than the discovery of Uranus”.

Here is a rough sketch of the orbits of the outer planets of the Solar System.  Pluto, the outermost object shown here, was discovered only in the twentieth century.

Here is a recent photograph of planet Neptune taken with a spacecraft passing close by.  Through a small earth based telescope it will look just like any ordinary star.

Galileo and Neptune

This tale has an extraordinary twist to its tail.  The last word had not been said about the discovery of planet Neptune.  In 1980, Charles T Kowal and Stillman Drake reported a sensational discovery about a discovery attributable to the great Galileo.  They presented compelling evidence to show that Galileo had indeed observed planet Neptune accidentally as far back as 28 Dec 1612 when he was exploring the neighborhood of Jupiter in the night sky – a whopping 234 years before its officially credit discovery and 169 years before the discovery of Uranus itself!  He had marked its position very clearly and accurately in his notebook, but seems to have mistaken it for a star.  He had observed it on two more occasions the following month and during one of them he even seems to have suspected a barely perceptible shift in its position compared to a nearby star.  If only he had followed up on any hunch he may have had, planetary astronomy would have been at least two centuries older and who knows how far it would have progressed in subsequent centuries.  What an opportunity lost!

Last year (celebrated worldwide as the International Year of Astronomy, appropriately marking the four hundredth anniversary of Galileo’s invention and first astronomical use of the telescope), a University of Melbourne physicist, David Jamieson, unearthed some inconclusive evidence that Galileo’s suspicions about this ‘star’ which was actually planet Neptune went deeper than presupposed.  He has proposed a chemical analysis of some of the pages in Galileo’s diary which might point to an afterthought on his part.  It looks as if the last word is yet to be said on the discovery of Neptune.

History Re-enacted

This enthralling story of the discovery of Neptune was re-enacted when tiny unaccounted discrepancies in the calculated and observed positions of Neptune itself began to accumulate over a period of time, leading to the prediction of a trans-Neptunian planet.  After a long period of futile search lead by American astronomer Percival Lowell, Clyde Tombaugh of the Lowell Observatory in USA discovered this planet, called Pluto, in 1930.  This was so faint, so small and so far away from the Earth that special observational techniques had to be developed to find it.  Its discovery did not fully account for the residual perturbation of Neptune.

Pluto differs from the other planets of the solar system so much that it is no longer regarded as a planet.  It is now placed under a distinct class of objects, called the Kuiper Belt, lying beyond the orbit of Neptune.

A Wrap-up

I have narrated the story of Neptune’s discovery as a splendid example of the method of science inherent in most scientific discoveries and the predictive power of a well developed theory or law of nature.  The observations related mostly to planets and other celestial objects that displayed an unexpected behavior and drew special attention.  An initial hypothesis was the geocentric theory, but this did not survive the test of detailed observational evidence.  So it had to be given up in favour of a workable hypothesis in the form of the heliocentric system.  When the accuracy of observed data came into question, a whole set of measuring devices were designed and built to increase the accuracy many fold.  When the picture of circular planetary orbits failed to fit in with the more accurate and improved observations that became available, it was modified in favour of the more realistic elliptical orbits, a circular orbit being just a special and unusual case of an elliptical orbit.  Systematic analyses of the huge amounts of accumulated data lead to the formulation of a synthesized model of the motion of planets in the whole of the solar system.  The invention of the telescope opened up a whole new theatre of operations.  The resulting discoveries not only strengthened the heliocentric theory but also laid the firm foundation for a set of fundamental laws of nature.  These laws were successful not only in explaining a whole lot of observed phenomena but could also make testable predictions. These predictions were not only verified to place the laws on an even firmer pedestal, but they also led to many unforeseen and further verifiable consequences.  So the process has gone on, ensuring ever expanding horizons and an accelerated progress of science and technology.


Assuming that Galileo had observed and accurately recorded the position of Neptune as far back as 28 Dec 1612,  one can work backward and compare the calculated position on that day with the observed position as recorded by Galileo.  This has been done. Lo and behold, what is the outcome!  There is a small residual discrepancy which cannot be explained away as errors in calculation in this age of super computers.  Could it be an error in Galileo’s observations and recording?  It is a measure of the supreme confidence placed in Galileo’s extraordinary observational skills that, despite the poor quality of the telescopes used by him, astronomers simply dismiss this notion. In other words, the discrepancy is very real.  Part of it appears to have been accounted for by the discovery of Pluto in 1930, but the rest of it appears to be due to the recently discovered swarm of hundreds of thousands of small trans-Neptunian objects forming what is called the Kuiper Belt (Pluto is now regarded as the most prominent member of this belt).  The final verdict is yet to be pronounced.

Thursday, October 14, 2010

Jaipur, the Pink City – Personal Photo Album Part 3

Delhi, Agra and Jaipur constitute a ‘Golden Triangle’ of cities forming a major tourist attraction in India.  My official duties used to take me to Delhi quite frequently and I was able to explore its tourist attractions in fair detail several times.  On a few such occasions I also managed to visit Agra which hosts one of the seven man-made wonders of the world.  This was the subject of my earlier blog post dated 27th August this year.  I did not get a chance to visit Jaipur, capital of Rajasthan state and popularly known as the Pink City, until 3rd August 2005 on my way back to Delhi from an official visit to Ajmer.  I had waited far too long for such an opportunity and needlessly so.  Since I didn’t have adequate time for a leisurely walking tour of the city, my preferred style of sightseeing, I opted for a whole day conducted tour offered by the state tourism board.  I have never really liked conducted tours for various reasons, one of them to do with the poor professional quality of the guide.  The Jaipur tour was a major exception since I and my fellow tourists ended up in the hands of an exceptionally competent and knowledgeable guide.  He was very young, dynamic, friendly and could speak English quite fluently, something rather hard to come by among official departmental guides. 
The Pink City was founded in 1727 by Maharaja Sawai Jai Singh II, an extraordinarily enlightened ruler by any standards. He was also a famous astronomer and founded the great medieval astronomical observatory, called Jantar Mantar, at both Jaipur and Delhi, the one in Jaipur being larger and better known.  What he accomplished in India has some modest similarities to the achievements of the great Danish nobleman and astronomer Tycho Brahe in 16th century Europe. The city itself is steeped in Rajasthani history and culture as well as some distinctive architecture.

[All pictures in my posts can be blown up to their full size by clicking on a picture and opening it in a separate window]

Amber Palace

The main attraction in Jaipur is the palace located in the sprawling Amber Fort on a rugged hillock about 10 km from the city.  Here is a picture of the fort across a water way as we approached it:

Here is a view of the quadrangle inside the fort complex with a flock of pigeons enjoying their freedom:

The following is a view of the palace from the interior.  The architecture is similar to the ones seen in Agra and Delhi forts.

Pictured below is Ganesh Pol, the ornate entrance gate to the Amber palace, a stunning mix of Mughal and Rajasthani architectures.

Here is a blown-up view of the lower right portion (on the ground floor) of the previous picture showing its strikingly beautiful painting and inlay work:

Albert Hall Museum

The Albert Hall Museum, conceived on the lines of the famous Victoria and Albert Museum in London, is one of the oldest museums in the country.  Surprisingly, the tour itinerary did not include a stop here and I had no chance to get a good picture of the excellent heritage building housing it.  However, I managed to shoot a zoomed-in picture of its uppermost part capturing its stunning style and architecture.  Here it is:

The Hawa Mahal

The Hawa Mahal is the most popular landmark of Jaipur and in some ways symbolic of the city itself.  Abutting a busy street in the heart of the city, its uniquely rich exterior is in sharp contrast to a drab and dull interior.  The tourist bus slowed down sufficiently for me to capture this image of the famous structure without having to get down:

Jaigarh Fort

The Jaigarh Fort, also known as the ‘Victory Fort’ is located on top of a hill about 15 km away from the city and quite close to the Amber Fort.  Here is a view of this sprawling fort complex:

The Jaigharh Fort houses what is claimed to be the world’s largest cannon.  To see this, one has to pay an additional entrance fee!  Here is a picture of this fabled cannon:

Nahargarh Fort

Located north of Jaipur city, the Nahargarh Fort lies atop a rugged hillock and commands a great view of the city all around.  Here is a picture of the older and rather dilapidated portion of the fort in the foreground against the backdrop of part of the city:

The later additions to the fort are much better preserved and house a number of beautiful structures. Here is a spectacular view of the intricate and colourful ceiling of one of them:

Here is another view of the pink city from a vantage point on the fort. The triangular structure prominently visible at the central right portion of the picture is part of the famous Jantar Mantar, the astronomical observatory built by Sawai Jai Singh. 

Jantar Mantar

The Jantar Mantar observatory complex houses a number of large geometric mechanical devices mostly made up of stone for measuring time, tracking stars and planets, predicting eclipses and measuring celestial coordinates to a high degree of accuracy without employing any optical aid as in modern astronomical observatories.  Preserved in very good condition, Jantar Mantar has been declared a world heritage site.  Here is a picture of the largest structure, the majestic Samrat Yantra (easily recognizable in the previous picture), whose steep edge is at an angle of just under 27 degrees to the horizontal plane, corresponding to the latitude of the place.  It points north along the earth’s axis and the device is part of what is considered to be the world’s largest sundial.   

Our guide spent a good bit of time at Jantar Mantar explaining quite competently how the various instruments in the observatory were designed to be used.  Most of what he said fell on deaf ears and I appeared to be the only tourist showing any interest in his narrations. The invisible bond between us grew stronger.

Jal Mahal

The Jal Mahal is a beautiful palace surrounded on all sides by the picturesque Mansagar lake against the backdrop of the Nahargarh Fort.  Here is a picture of this taken from the roadside:

City Palace 

One of the major attractions of the pink city is the City Palace Complex which has a number of spectacular and palatial buildings, with a great museum as its centerpiece.  Here are two pictures from inside this complex:

The Birla Connection

The influence of the Birla group in the Pink City is underscored by two of the most famous modern buildings in the city – the Birla Mandir and the Birla Science and Technology Centre. 

The Birla Mandir (Laxminarayan Temple) built in 1988 out of pure white marble like the one in Hyderabad, is a stunning sight during both day and night.  At the foot of the Moti Dungri Fort housing former royalty, it is surrounded by lush greenery and large open spaces.  The following picture of the temple is particularly pleasing against the background of a blue sky with white clouds.  The carvings on the outside can be best seen when the picture is blown up to its full size. 

Here is a picture of the main building forming part of the famous Birla Science and Technology Centre which also houses a modern planetarium, in some respects complimenting the ancient Jantar Mantar.  It is a stunningly modern building with a breathtakingly beautiful architecture. 

Au Revoir Jaipur

The daylong tour of the Pink City and surroundings was really long and a warm weather added considerably to the discomfort of the tourists.  The trip should have terminated at the Birla Planetarium with a celestial show, but most tourists were in no mood for it and dropped out along the way.  I and the indefatigable guide were the only ones who reached the destination, but I opted to spend the rest of the day outdoors rather than inside a planetarium which was no stranger to me.  Before leaving, I paid a sincere and richly deserved compliment to my guide for all that he had done to make the day so memorable.  On his part, he paid me an equally rich compliment by saying that at twice his age I was hardly the kind of tourist he was used to meeting.  Even as I compose this post I greatly regret my thoughtlessness in not having written a formal letter of appreciation to the state tourism board for providing such an exceptionally able guide.