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Letcher's Lunars Math
From: Frank Reed CT
Date: 2004 May 20, 15:20 EDT
From: Frank Reed CT
Date: 2004 May 20, 15:20 EDT
In Part 3 of George H's compendium on lunar distances, he talked a little about Letcher's method for clearing lunar distances as described in his book "Self-Contained Celestial Navigation with H.O. 208"...
George H wrote:
"The way he proposes for its use would be suitable only for lunar distances up to 90 degrees, whereas any lunar observer would certainly require its application to angles up to 120 degrees. I will therefore give a modified calculation method that will cover the full range of useful sextant angles."
This isn't right. I can't figure out what George was thinking here, but the equation in Letcher's book is fine just the way it is and can be used without modification.
George H wrote:
"I am not sure where this method comes from, as Letcher does not quote any reference, nor does he show how it is derived from first-principles. "
I strongly suspect he derived it himself since it is not all that difficult to do so. It is similar to the equation that I posted in "Easy Lunars" except that he prefers a (rather neat) graphical method for calculating the refraction part of the linear terms and he uses specific mathematical forms for both refraction and the Moon's parallax instead of using the altitude correction tables in the Nautical Almanac.
The clearing equation that I posted in "Easy Lunars" is
d = d0 + dh1*A + dh2*B + Q
where d0 is the measured lunar (converted to a "center-to-center" distance by adding/subtracting semidiameters of the two objects), dh1 is the altitude correction for the Moon, dh2 is the altitude correction for the Sun or other body, Q is the small "quadratic correction", and A and B are the "corner cosines" which tell us how much of each altitude correction acts along the arc connecting the two bodies. A is given by
A = (sin(h2)-cos(d)*sin(h1))/(sin(d)*cos(h1))
and B is the same with h1 and h2 swapped. Q is given by
Q = 0.5*(dh1)^2*cot(d)*(1-A^2).
Note that if you use 0.55 instead of 0.5 above, the equation covers a wider range of cases as I've mentioned before. I'm leaving it in the original form here since that's closer to Letcher's version. Also note that if dh1 is measured in degrees, then you have to divide Q by 57.3. If dh1 is measured in minutes of arc, then Q is divided by 3438.
This equation can be derived from straight-forward calculus. You take the equation for the standard spherical triangle for the lunars problem and then do a two-dimensional Taylor series expansion. The resulting series has an infinite number of terms, but by knowing upper limits on dh1 and dh2, only the first few terms have to be kept to achieve the required accuracy for lunar distance sights. Historically, these series expansions were called "approximate" or "approximative" solutions of the lunar distance problem. This terminology is misleading. They are as accurate as they need to be when done right.
Letcher, I suspect, derived the same equation as above and then he chose to insert specific mathematical forms for dh1 and dh2, which was also a common practice in 18th and 19th century clearing methods. For dh1, the altitude correction of the Moon, Letcher used
dh1 = HP*cos(h1)-0.95*cot(h1)
for the linear term, and for the quadratic term he used
dh1 = HP*cos(h1).
Dropping second part, the refraction, for the Moon in the quadratic term is not unreasonable since the quadratic term is very small but it does lead to an error of a little more than 0.1 minutes of arc in some cases where the distance is above 90 degrees or so and the altitude of the other body is low. For the other body's altitude correction, Letcher uses
dh2 = -0.95*cot(h1).
This terms gives the altitude change due to refraction only and, for both the Moon and the other body, the refraction is accurate only at standard temperatures and pressures. It does not include an option for parallax of the other body. Also this simple expression for refraction is valid only down to 15 degrees of altitude. It's not a good idea to shoot lunars when objects are at lower altitudes anyway, so this isn't really a problem. As Letcher notes in his book, his choices for the functional form of dh1 and dh2 mean that his equation will have errors amounting to as much as 0.3 minutes of arc. In fact for Mars or Venus (when close to the Earth) and under unusual atmospheric conditions, the error could be even larger than this.
If you decide to crunch through the algebra to verify that you can get Letcher's equation by following the steps above, you'll find that his term "y" is the same as the quantity "A" in my version (A is the corner cosine for the Moon) multiplied by cos(h1). This makes little difference to the calculational steps involved, but it also removes the simple geometric interpretation of this intermediate quantity (which is very useful for teaching purposes and error-detection to a lesser extent) as the "percentage" of the altitude correction that acts along the arc of the lunar.
Frank R
[ ] Mystic, Connecticut
[X] Chicago, Illinois
George H wrote:
"The way he proposes for its use would be suitable only for lunar distances up to 90 degrees, whereas any lunar observer would certainly require its application to angles up to 120 degrees. I will therefore give a modified calculation method that will cover the full range of useful sextant angles."
This isn't right. I can't figure out what George was thinking here, but the equation in Letcher's book is fine just the way it is and can be used without modification.
George H wrote:
"I am not sure where this method comes from, as Letcher does not quote any reference, nor does he show how it is derived from first-principles. "
I strongly suspect he derived it himself since it is not all that difficult to do so. It is similar to the equation that I posted in "Easy Lunars" except that he prefers a (rather neat) graphical method for calculating the refraction part of the linear terms and he uses specific mathematical forms for both refraction and the Moon's parallax instead of using the altitude correction tables in the Nautical Almanac.
The clearing equation that I posted in "Easy Lunars" is
d = d0 + dh1*A + dh2*B + Q
where d0 is the measured lunar (converted to a "center-to-center" distance by adding/subtracting semidiameters of the two objects), dh1 is the altitude correction for the Moon, dh2 is the altitude correction for the Sun or other body, Q is the small "quadratic correction", and A and B are the "corner cosines" which tell us how much of each altitude correction acts along the arc connecting the two bodies. A is given by
A = (sin(h2)-cos(d)*sin(h1))/(sin(d)*cos(h1))
and B is the same with h1 and h2 swapped. Q is given by
Q = 0.5*(dh1)^2*cot(d)*(1-A^2).
Note that if you use 0.55 instead of 0.5 above, the equation covers a wider range of cases as I've mentioned before. I'm leaving it in the original form here since that's closer to Letcher's version. Also note that if dh1 is measured in degrees, then you have to divide Q by 57.3. If dh1 is measured in minutes of arc, then Q is divided by 3438.
This equation can be derived from straight-forward calculus. You take the equation for the standard spherical triangle for the lunars problem and then do a two-dimensional Taylor series expansion. The resulting series has an infinite number of terms, but by knowing upper limits on dh1 and dh2, only the first few terms have to be kept to achieve the required accuracy for lunar distance sights. Historically, these series expansions were called "approximate" or "approximative" solutions of the lunar distance problem. This terminology is misleading. They are as accurate as they need to be when done right.
Letcher, I suspect, derived the same equation as above and then he chose to insert specific mathematical forms for dh1 and dh2, which was also a common practice in 18th and 19th century clearing methods. For dh1, the altitude correction of the Moon, Letcher used
dh1 = HP*cos(h1)-0.95*cot(h1)
for the linear term, and for the quadratic term he used
dh1 = HP*cos(h1).
Dropping second part, the refraction, for the Moon in the quadratic term is not unreasonable since the quadratic term is very small but it does lead to an error of a little more than 0.1 minutes of arc in some cases where the distance is above 90 degrees or so and the altitude of the other body is low. For the other body's altitude correction, Letcher uses
dh2 = -0.95*cot(h1).
This terms gives the altitude change due to refraction only and, for both the Moon and the other body, the refraction is accurate only at standard temperatures and pressures. It does not include an option for parallax of the other body. Also this simple expression for refraction is valid only down to 15 degrees of altitude. It's not a good idea to shoot lunars when objects are at lower altitudes anyway, so this isn't really a problem. As Letcher notes in his book, his choices for the functional form of dh1 and dh2 mean that his equation will have errors amounting to as much as 0.3 minutes of arc. In fact for Mars or Venus (when close to the Earth) and under unusual atmospheric conditions, the error could be even larger than this.
If you decide to crunch through the algebra to verify that you can get Letcher's equation by following the steps above, you'll find that his term "y" is the same as the quantity "A" in my version (A is the corner cosine for the Moon) multiplied by cos(h1). This makes little difference to the calculational steps involved, but it also removes the simple geometric interpretation of this intermediate quantity (which is very useful for teaching purposes and error-detection to a lesser extent) as the "percentage" of the altitude correction that acts along the arc of the lunar.
Frank R
[ ] Mystic, Connecticut
[X] Chicago, Illinois
