A Tale of Two Worlds : Part 2

After a weather and political delay, we resume our tale of other worlds.

Before continuing our tale, I'm going to bluntly advertise for the next session of my Introduction to Astronomy course which starts in January. 

This course is for school children ages 8 and up, and surveys the science of astronomy from the contents of our solar system, to how stars form age and die, to the farthest regions of our universe, and the variety of strange objects that we have discovered within it. 

I touch on the fundamental physics involved in our knowledge of the universe surrounding us - how we use the light from distant objects to understand their distance and composition, and even introduce thermonuclear physics and general relativity at a level that even the 8 year olds can grasp. 

Weekly presentations are followed by observing sessions in my front yard (in Southbury) using a telescope capable of showing the major planets, star clusters, nebulae and galaxies, along with the occasional comet, meteors, and plenty of artificial satellites.  We will meet for 19 weeks (Thursdays at 7 p.m.) from January through May.  The cost of this course is $150 per student; parents are encouraged to stay and learn as well (for free).  Full details and contact information are at http://www.turnerclasses.com

Ah, but where were we?  Oh yes, the second of our two worlds.

After the discovery of Uranus, its orbit about the Sun was determined with increasing accuracy over the next several years, allowing exact predictions of where it should appear in the sky month after month into the indefinite future.  As time went on however, it was noticed that where Uranus was predicted to be differed from where it was actually observed by a steadily increasing amount. 

By the 1840s, 60 years after the discovery of Uranus, the error in the predicted and observed position of Uranus became large enough to suggest that something was fundamentally wrong with the assumptions that were made in calculating the orbit of the planet.

The mathematics involved in computing the orbit of a planet are actually quite simple if the only objects to be considered are the planet and the Sun.  A single force of gravity pulls on the planet in the direction of the Sun, and the momentum of the planet's motion keeps it in an elliptical path around the Sun.  However, this simple condition is not what occurs in the solar system.

Gravitational attraction occurs between all objects in the universe.  The force increases as the masses of the attracted objects increase, and it increases as the distance between the objects decreases.  Here on Earth, the largest source of gravity is the Earth itself, as it is both enormous compared to the objects on its surface, and we are close to its center compared to our distance to other massive objects such as the Sun.  But nonetheless, all of the objects on Earth and elsewhere are attracted to one another.  The person sitting across the room from you is very slightly pulling you toward them, and (as creepy as it may sound) even you and I, separated by many miles, are ever so slightly pulled toward each other.

Going back to the orbit of a planet, there are several other planets also orbiting the Sun at varying distances from the planet being studied, and each of these exerts a small gravitational pull of its own upon the planet. Because each of these other planets is itself in motion, and all planets are pulling on all other planets as they move, the simple problem of calculating an orbit suddenly becomes exceedingly complex.  In fact, with as few as three objects involved, an exact general solution for the motions has been impossible to achieve, and only very high precision approximations are possible.

In the case of the orbits of planets in our solar system, the smaller planets can be ignored, and for Uranus, only the effects of the Sun, Jupiter and Saturn were considered in the predicted positions determined in the 1820s.  However, the steady disagreement between computed and observed positions of Uranus suggested that either Newton's law of gravity did not apply to objects as distant as Uranus, or there was another large body other than the Sun and the known planets affecting the orbit of Uranus.

In the early 1840s, the British mathematician John Couch Adams realized that it would be theoretically possible to determine the mass and the orbit of a hypothetical planet that would explain the observed deviations in the position of Uranus using only the observations of Uranus itself in the calculation.  Adams began attacking the problem in 1844, working on it part-time while tutoring other students to earn his livelihood.  Meanwhile, in France, another mathematician, Urbain Le Verrier, began to work the same problem in 1845.  Partly because of the intense rivalry between England and France in the mid-19th century, these men knew nothing of each other's work.

We should take a moment here to consider what these men were attempting to accomplish, and the conditions under which they were working.  The calculations involved take the form of lengthy differential equation calculations which can only be approximately solved, and require many repeated iterations to achieve a level of accuracy sufficient to be better than the observed errors in the position of Uranus.  Each iteration required weeks of effort - recalling that in the 1840s there were no mechanical devices to assist in calculations, and every line of computation was written out manually on paper with a quill pen, using candle or gas lights to work into the early hours of the morning.

By 1846, both Adams in England and Le Verrier in France had arrived at solutions for the orbit of an unknown body which would explain the changes observed in the observed positions of Uranus.  From these calculations they could begin to predict where in the sky one should point a telescope to see the predicted object. But both men were surprised by the lack of interest their predictions generated in the communities of astronomers in their countries.  Being primarily mathematicians, their work was neither well understood nor appreciated by the astronomers of their day.

Finally, upon seeing Le Verrier's work published, and understanding the importance to national pride, if not science, the Royal Astronomer of England (Sir George Airy) urged British astronomers to begin a search, using the predictions of Adams as a guide.  At about the same time, generating no interest in his own country, Le Verrier reached outside of France to the Berlin Observatory, sending a letter with coordinates to be used in searching for the unknown object.  When this letter arrived on September 23, 1846, the director of the observatory (Johann Encke) had just left on a vacation to the Alps.  His understudy, Johann Galle, decided to begin a search that very night. 

Unlike the British astronomers who had attempted a search, Galle had very recent star maps of the search area available to him, allowing him to easily separate stars with known and constant positions from the object he was seeking.  Amazingly, within a few hours of starting the quest, Galle located a bluish-green object showing a very miniscule disk quite definitely not on his star map within a very small distance from where Le Verrier had predicted it would be found. Subsequent observations confirmed an excellent match to the orbit and size predicted by both Adams and Le Verrier.  The planet Neptune had been discovered.

What is unique about the discovery of Neptune is the clear triumph of mathematics snd physics in being able to discover an object purely through computation, only directly observed after the discovery had been completed on paper.  The mighty intellectual efforts of Galileo, Kepler, and most directly Isaac Newton, were forever confirmed and validated on that September night in 1846.

Speaking of Galileo, a pair of observations he made in December, 1612 and January, 1613 add a footnote to this tale.  During this time, Galileo was intensely studying the motions of the moons of Jupiter.  In his notebook from December 28th, we find a diagram of Jupiter and its moons, with a "fixed star" also marked.  In his later notebook entry from January 27th, he again notes this "star", but also indicates that it has moved relative to other stars near it in the sky. 

Using modern computer models of the orbit of Neptune, we can predict the positions of the planets at any date in the future or past (what took Adams and Le Verrier years to compute can now be accomplished in a matter of minutes). What we find for the dates of Galileo's observations is that Neptune and Jupiter where very close to one another in the sky on those dates, with the position of Neptune almost exactly as Galileo had drawn it! 

Had he realized what he was observing, Galileo would have discovered Neptune some 220 years prior to Adams and Le Verrier.  But then the story of this great discovery would have been significantly less interesting, at least in my opinion.

This post is contributed by a community member. The views expressed in this blog are those of the author and do not necessarily reflect those of Patch Media Corporation. Everyone is welcome to submit a post to Patch. If you'd like to post a blog, go here to get started.

Will Wilkin November 13, 2012 at 05:20 PM
Thanks for a great read on the role of math in the discovery of Neptune. I bet you could write additional articles on the role of math in so many aspects of astronomy. I think good math teachers are hard to find, so much of the so little math I studied seemed to have no explanation or meaning, maybe that's why the math textbooks today are full of words and the only numbers seem to be for listing ideas.
Jimmie Pursey November 13, 2012 at 09:13 PM
you have some interesting ideas, Bob.


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