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Re: Chauvenet
From: Frank Reed CT
Date: 2003 Dec 19, 02:32 EST
From: Frank Reed CT
Date: 2003 Dec 19, 02:32 EST
H.Prinz wrote:
"Now for the math. The method is a good counter-example to proof an often
stated opinion wrong that methods for clearing the distance can be
divided into rigorous and approximative methods according to whether the
MZS triangle is solved from the mZs triangle via the common angle Z
(using the cosine formula twice), or whether MS is derived from ms by
applying corrections on either end that are estimated by means of small
plane rectangular triangles with hypotenuses Mm and Ss."
Yes, I've also been a bit annoyed by this distinction, too, since it creates the subtle impression that the so-called "rigorous" versions are better. A modern physicist would probably distinguish these situations with the expressions "closed form" and "series expansion". Neither is necessarily superior for practical calculation. The linear terms of the series expansion *can* be interpreted in terms of those "small plane triangles" at the corners of the big spherical triange, but they don't have to be. And that geometrical picture doesn't limit the series expansion in any fundamental sense.
The correction to the lunar distance expressed as a series expansion to quadratic order is:
distance correction = dh1*A + dh2*B + dh1^2*C + dh2^2*D + dh1*dh2*E + (etc.)
where dh1 is the altitude correction for the Moon, dh2 is the correction for the other body, and A, B, C, D, etc. are geometric factors that may or may not involve some significant calculation. Given what we know about the altitude corrections for the Moon and other body (that they are always less than a fiftieth of a radian), we can drop almost everything except the linear terms and one quadratic term and leave the correction as
dist. correction = dh1*A + dh2*B + dh1^2*C.
Within certain limits of the altitudes and the observed lunar distance, which are not that tough to obey, this truncated series will cover all the spherical trig. It's the method Letcher uses, for example. Chauvenet's "approximate" method is not far away from this series expansion.
You wrote of his method, that he
"consider for a sea-method the oblateness of the earth, the effect of which can change the final result for the distance by typically 6" or 0.1' of arc in mid-latitudes."
And often more than that. I've mentioned that I can get very good lunars with a high-quality sextant. If you omit the oblateness correction, you can't get that close.
And you wrote:
"This approximative method is in fact capable of yielding results that are more accurate
than those obtained by some procedures based on "rigorous" methods."
Right. I couldn't agree more.
"Now I have a practical question (if anything regarding lunars can be considered practical)."
Well, no, nothing about them can be considered 'practical', but I know what you mean!! ;-)
You wrote:
" Chauvenet's method relies also on his own table
of "reduced refraction". This in turn is based on Bessel's refraction
table. But Bessel's table is no longer used in its original form. The
N.A. incorporates slightly different tables, and the formula given there
(and in the Supplement) is again different. Choosing the right
refraction table is more of an art than a science (Jan, are you
listening?); at any rate, it is an empirical process. My question to
Frank therefore, is: Have you looked into this aspect?"
Very perceptive. I've wrestled with this business on several occasions. At some point, you have to make a call and drop all observations below some altitude cutoff. Should it be 15 degrees? Or 10? Or 2.5??
You asked:
"Do you actually use the method for practical exercises? It would seem to be a
contradiction to go all the way correcting for atmospheric conditions
and at the same time use a table that is officially considered to be
outdated."
I use the "method", but NOT the tables. I consider Chauvenet's method very clever but I don't fuss much over the calculational details which were, naturally, tailored to an era when calculational details were expensive. We can be more flexible today.
And:
"Having been produced by the abstract process I described above, the final tables for log A,
B, C, D are so arcane that even Chauvenet himself could probably not have reverse engineered them, had he forgotten his original procedure of devising them."
If he had been away from them for five years, you're probably right. But I bet if he had been sailing in the barren ocean of the eastern Pacific for 46 long, long days, he could have re-derived all of modern science and STILL had time to kill!
Frank E. Reed
[X] Mystic, Connecticut
[ ] Chicago, Illinois
"Now for the math. The method is a good counter-example to proof an often
stated opinion wrong that methods for clearing the distance can be
divided into rigorous and approximative methods according to whether the
MZS triangle is solved from the mZs triangle via the common angle Z
(using the cosine formula twice), or whether MS is derived from ms by
applying corrections on either end that are estimated by means of small
plane rectangular triangles with hypotenuses Mm and Ss."
Yes, I've also been a bit annoyed by this distinction, too, since it creates the subtle impression that the so-called "rigorous" versions are better. A modern physicist would probably distinguish these situations with the expressions "closed form" and "series expansion". Neither is necessarily superior for practical calculation. The linear terms of the series expansion *can* be interpreted in terms of those "small plane triangles" at the corners of the big spherical triange, but they don't have to be. And that geometrical picture doesn't limit the series expansion in any fundamental sense.
The correction to the lunar distance expressed as a series expansion to quadratic order is:
distance correction = dh1*A + dh2*B + dh1^2*C + dh2^2*D + dh1*dh2*E + (etc.)
where dh1 is the altitude correction for the Moon, dh2 is the correction for the other body, and A, B, C, D, etc. are geometric factors that may or may not involve some significant calculation. Given what we know about the altitude corrections for the Moon and other body (that they are always less than a fiftieth of a radian), we can drop almost everything except the linear terms and one quadratic term and leave the correction as
dist. correction = dh1*A + dh2*B + dh1^2*C.
Within certain limits of the altitudes and the observed lunar distance, which are not that tough to obey, this truncated series will cover all the spherical trig. It's the method Letcher uses, for example. Chauvenet's "approximate" method is not far away from this series expansion.
You wrote of his method, that he
"consider for a sea-method the oblateness of the earth, the effect of which can change the final result for the distance by typically 6" or 0.1' of arc in mid-latitudes."
And often more than that. I've mentioned that I can get very good lunars with a high-quality sextant. If you omit the oblateness correction, you can't get that close.
And you wrote:
"This approximative method is in fact capable of yielding results that are more accurate
than those obtained by some procedures based on "rigorous" methods."
Right. I couldn't agree more.
"Now I have a practical question (if anything regarding lunars can be considered practical)."
Well, no, nothing about them can be considered 'practical', but I know what you mean!! ;-)
You wrote:
" Chauvenet's method relies also on his own table
of "reduced refraction". This in turn is based on Bessel's refraction
table. But Bessel's table is no longer used in its original form. The
N.A. incorporates slightly different tables, and the formula given there
(and in the Supplement) is again different. Choosing the right
refraction table is more of an art than a science (Jan, are you
listening?); at any rate, it is an empirical process. My question to
Frank therefore, is: Have you looked into this aspect?"
Very perceptive. I've wrestled with this business on several occasions. At some point, you have to make a call and drop all observations below some altitude cutoff. Should it be 15 degrees? Or 10? Or 2.5??
You asked:
"Do you actually use the method for practical exercises? It would seem to be a
contradiction to go all the way correcting for atmospheric conditions
and at the same time use a table that is officially considered to be
outdated."
I use the "method", but NOT the tables. I consider Chauvenet's method very clever but I don't fuss much over the calculational details which were, naturally, tailored to an era when calculational details were expensive. We can be more flexible today.
And:
"Having been produced by the abstract process I described above, the final tables for log A,
B, C, D are so arcane that even Chauvenet himself could probably not have reverse engineered them, had he forgotten his original procedure of devising them."
If he had been away from them for five years, you're probably right. But I bet if he had been sailing in the barren ocean of the eastern Pacific for 46 long, long days, he could have re-derived all of modern science and STILL had time to kill!
Frank E. Reed
[X] Mystic, Connecticut
[ ] Chicago, Illinois
