Celestial Mechanic

2009-Aug-05, 05:03 AM

Celestial Mechanic Explains the Earth's Rotation--Part One: A Blast From the Past

My old friends from grad school and my newer friends from UW are here for a coffee and a bit of physics. Our cups refilled and more of Tensor's doughnuts in hand, we return.

CM: "I want to say a few things about the Earth's rotation. Over the years I've thought a bit about the Earth's rotation since we use it as a time-piece and as part of the definition of two different coordinate systems. I've come up with a different idea for handling the motion of the ecliptic and treating the precession and nutation by a different method."

DB: "Shouldn't this be in Against the Mainstream?"

CM: "No, because it uses the tested methods of celestial mechanics to arrive at its results. It is not that far from the mainstream's presentation, indeed it could become the mainstream view if it explains the mainstream results more simply and with less computation. This thread is entirely mainstream."

BH: "Can you give a brief summary without giving too much away?"

CM: "It doesn't matter if I give it away. In brief, I propose to treat the ecliptic and the equator equally, with both precessing around the pole of the Laplace or invariant plane of the Solar System.

CM: "To explain why I have chosen this, I'd like to tell about my early years of study in celestial mechanics."

Jimmy K.: "Get ready for a blast from the past!"

CM: "Oh, come on, it won't be that bad. I won't tell any stories about wild times at OU. In fact these recollections might be boring, unless of course you are interested in celestial mechanics.

CM: "As a child I was interested in astronomy, and I recall some of my early struggles with trying to understand such things as sidereal time, the magnitude system, and why the harvest moon rose fewer minutes later than other full moons. This is not easy stuff for a 10-year old! I would eventually teach myself logarithms at 11, trig (plane and spherical!) and a bit of differential calculus when I was 12, but it would take me until I was 15 to really master integral calculus. That's one reason why I have some sympathy for 17 and 18-year olds who get differential and integral calculus in consecutive semesters."

Virginia: "I had trouble with it in advanced placement calculus."

JK: "So did I."

CM: "Oddly enough, although I read about differential and partial differential equations in high school, I never tried my hand at celestial mechanics. My orientation was more towards geometry, especially in higher dimensions, and number theory.

CM: "I finished high school and arrived at the University of Oklahoma the day before my 17th birthday, almost 40 years ago. OU had (and I hope still has) a fabulous library, indeed it was called the 'Yale of the Southwest'. It had just acquired its one-millionth volume before I got there; it would acquire its two-millionth volume shortly before I left 18 years later."

JK: "Eighteen years!"

CM: "It's a long story."

V: "I'll bet!"

CM: "But I won't be telling it anytime soon. Anyway, during my freshman year I made the acquaintance of book called Theory of Eclipses by Buchanan. I wanted to try calculating eclipses, in particular lunar eclipses since I also ran across Jean Meeus' Canon of Solar Eclipses that year. The first textbook of celestial mechanics that I studied from was Celestial Mechanics by Brouwer and Clemence.

CM: "My first attempt during spring break of 1970 at predicting the lunar eclipse of 1971 Febrary 10 was a disaster, I was off by two hours and the magnitude off by 0.2. Of course I was not handling perturbations properly. Ultimately I would calculate lunar eclipses. One of the things that impressed me then were the then-current analytical theories of lunar and planetary motion. I had access to the Tables of the Inner Planets in Volume VI of Papers of the Astronomical Ephemeris and Nautical Almanac, which were used (with some modifications) through 1984, and Improved Lunar Ephemeris. One of the things that struck me was the difference in the ways the planetary and lunar theories were expressed.

CM: "The perturbations of the planetary theories were expressed as cosine series of the form K*cos(j*g+j'*g'+C), with j and j' as integers (possibly zero) and g and g' the mean anomalies of the planet and the perturbing planet respectively. The lunar theory had sine series for the longitude and latitude perturbations, a cosine series for the sine parallax (the inverse of the radius vector), and all arguments were linear combinations of four arguments, D, M, M' and F, without any phases, for example sin(M-2*F+2*D). What I wanted to do was develop the planetary theories in the same form as the lunar theory."

BH: "What are D, M, M' and F?"

CM: "Sorry. D is the mean elongation, equal to the difference between the mean longitude of the Moon, L, and the mean longitude of the Sun, L'. M and M' are the mean anomalies of the Sun and Moon, respectively. And F is the argument of latitude, equal to the difference between the mean longitude L and the longitude of the Moon's orbit on the ecliptic, Omega."

JK: "Why is there no F'?"

V: "Because the Sun is in the ecliptic, it has no inclination and no node. Also, don't forget that the mean anomalies M and M' are the differences between the mean longitudes and longitudes of pericenters for the Moon and Sun respectively."

CM: "Thank you, Virginia. Anyway, I dropped out of school but continued to live in Norman, OK and work for the university. In 1973 I purchased the Dover reprint of Ernest W. Brown's An Introductory Treatise on the Lunar Theory, probably the best two dollars I ever spent."

JK: "Two dollars! You've got to be putting us on!"

CM: "No, I'm not. Let's refill our cups and I'll get my copy of the book and show you."

To be continued...

My old friends from grad school and my newer friends from UW are here for a coffee and a bit of physics. Our cups refilled and more of Tensor's doughnuts in hand, we return.

CM: "I want to say a few things about the Earth's rotation. Over the years I've thought a bit about the Earth's rotation since we use it as a time-piece and as part of the definition of two different coordinate systems. I've come up with a different idea for handling the motion of the ecliptic and treating the precession and nutation by a different method."

DB: "Shouldn't this be in Against the Mainstream?"

CM: "No, because it uses the tested methods of celestial mechanics to arrive at its results. It is not that far from the mainstream's presentation, indeed it could become the mainstream view if it explains the mainstream results more simply and with less computation. This thread is entirely mainstream."

BH: "Can you give a brief summary without giving too much away?"

CM: "It doesn't matter if I give it away. In brief, I propose to treat the ecliptic and the equator equally, with both precessing around the pole of the Laplace or invariant plane of the Solar System.

CM: "To explain why I have chosen this, I'd like to tell about my early years of study in celestial mechanics."

Jimmy K.: "Get ready for a blast from the past!"

CM: "Oh, come on, it won't be that bad. I won't tell any stories about wild times at OU. In fact these recollections might be boring, unless of course you are interested in celestial mechanics.

CM: "As a child I was interested in astronomy, and I recall some of my early struggles with trying to understand such things as sidereal time, the magnitude system, and why the harvest moon rose fewer minutes later than other full moons. This is not easy stuff for a 10-year old! I would eventually teach myself logarithms at 11, trig (plane and spherical!) and a bit of differential calculus when I was 12, but it would take me until I was 15 to really master integral calculus. That's one reason why I have some sympathy for 17 and 18-year olds who get differential and integral calculus in consecutive semesters."

Virginia: "I had trouble with it in advanced placement calculus."

JK: "So did I."

CM: "Oddly enough, although I read about differential and partial differential equations in high school, I never tried my hand at celestial mechanics. My orientation was more towards geometry, especially in higher dimensions, and number theory.

CM: "I finished high school and arrived at the University of Oklahoma the day before my 17th birthday, almost 40 years ago. OU had (and I hope still has) a fabulous library, indeed it was called the 'Yale of the Southwest'. It had just acquired its one-millionth volume before I got there; it would acquire its two-millionth volume shortly before I left 18 years later."

JK: "Eighteen years!"

CM: "It's a long story."

V: "I'll bet!"

CM: "But I won't be telling it anytime soon. Anyway, during my freshman year I made the acquaintance of book called Theory of Eclipses by Buchanan. I wanted to try calculating eclipses, in particular lunar eclipses since I also ran across Jean Meeus' Canon of Solar Eclipses that year. The first textbook of celestial mechanics that I studied from was Celestial Mechanics by Brouwer and Clemence.

CM: "My first attempt during spring break of 1970 at predicting the lunar eclipse of 1971 Febrary 10 was a disaster, I was off by two hours and the magnitude off by 0.2. Of course I was not handling perturbations properly. Ultimately I would calculate lunar eclipses. One of the things that impressed me then were the then-current analytical theories of lunar and planetary motion. I had access to the Tables of the Inner Planets in Volume VI of Papers of the Astronomical Ephemeris and Nautical Almanac, which were used (with some modifications) through 1984, and Improved Lunar Ephemeris. One of the things that struck me was the difference in the ways the planetary and lunar theories were expressed.

CM: "The perturbations of the planetary theories were expressed as cosine series of the form K*cos(j*g+j'*g'+C), with j and j' as integers (possibly zero) and g and g' the mean anomalies of the planet and the perturbing planet respectively. The lunar theory had sine series for the longitude and latitude perturbations, a cosine series for the sine parallax (the inverse of the radius vector), and all arguments were linear combinations of four arguments, D, M, M' and F, without any phases, for example sin(M-2*F+2*D). What I wanted to do was develop the planetary theories in the same form as the lunar theory."

BH: "What are D, M, M' and F?"

CM: "Sorry. D is the mean elongation, equal to the difference between the mean longitude of the Moon, L, and the mean longitude of the Sun, L'. M and M' are the mean anomalies of the Sun and Moon, respectively. And F is the argument of latitude, equal to the difference between the mean longitude L and the longitude of the Moon's orbit on the ecliptic, Omega."

JK: "Why is there no F'?"

V: "Because the Sun is in the ecliptic, it has no inclination and no node. Also, don't forget that the mean anomalies M and M' are the differences between the mean longitudes and longitudes of pericenters for the Moon and Sun respectively."

CM: "Thank you, Virginia. Anyway, I dropped out of school but continued to live in Norman, OK and work for the university. In 1973 I purchased the Dover reprint of Ernest W. Brown's An Introductory Treatise on the Lunar Theory, probably the best two dollars I ever spent."

JK: "Two dollars! You've got to be putting us on!"

CM: "No, I'm not. Let's refill our cups and I'll get my copy of the book and show you."

To be continued...