Excel Astronomy for the Surveyor Pt 2.

#1 · July 12, 2010, 9:29 pm

Quote from scott-zelenak on July 12, 2010, 9:29 pm
The mathematical models for the use of astronomy in surveying require us to determine the apparent (or true) coordinates for a stellar object. The coordinates we will find in star catalogues are heliocentric mean places, referred to some specific mean equator and equinox. In order for surveyors to utilize these coordinates corrections must be applied in a certain order to coincide with the epoch of observation.
Precession and Nutation are the motions of the coordinate system itself.
Proper Motion or Motion in Space is the movement of the star and our sun.
Refraction, Aberration, and Parallax are the apparent displacements of the stars due to physical phenomena.
And Polar Motion is the movement of the coordinate system with respect to the solid model Earth.
In the following pictures I show the completed steps and the corrected coordinates in rose and the expected coordinates in yellow. The difference between my coordinates and those derived classically, is that I have not followed the classical methods is some cases. My main aim has been to provide an instantaneous computation which requires neither expensive Almanacs, complicated tables or reams of paper recording step by step operations.

Here, you can see in blue the data required to be entered in the spreadsheet to begin the computations required. The blue data is provided in the star catalogue and we add the date and time of observation. The most precise catalogues have generally been the Fundamental Catalogues, such as the FK4 and FK5. However, since the launch of the Hipparcos satellite, the best resource is a website known as SIMBAD.
The corrections to be applied to a stellar coordinate, given in Right Ascension (?) and Declination (?) begin with Proper Motion. Proper Motion is the movement of the star against the background and the movement of the sun in its course through the Milky Way and our galaxies movement. This correction, given the components of motion in ? (?) and ? (?’), of milliarcseconds, can be simply multiplied by the number of years and added to the original coordinates. Unfortunately, Proper Motion has some faults. The math tends to break down with circum-polar stars especially. Hence, we have the newer theory of Motion in Space.
Imagine that you have a clear globe (the surface of which represents the spherical coordinates) and you let a pencil represent the true motion of a star from the start to the end of the pencil, and you place it somewhere near the pole of your globe, then it is not hard to see that it is a complicated matter to describe the spherical coordinates of the end-point of the pencil given only the length of the pencil and its starting point in spherical coordinates. If the pencil was very short relative to the globe and close to equator you may just "add the spherical coordinates", but not when it is near the pole.
Motion in Space converts the spherical coordinates to a set of Cartesian coordinates and corrections over time. It then converts these back to spherical coordinates, all very accurately. For example, if we compute the motion of a star called Sirius, over a thousand years, the deviations ignoring unforeseen perturbations of the Earth approach three tenths of a second in ? and four seconds in ?.

Precession is the resultant of attractive forces of the sun, moon (luni-solar precession) and planets (planetary precession) acting upon our model Earth. In combination, these forces attempt to drag the equatorial plane in to the ecliptic plane, disturbing the motion of the Earths axis of rotation about the ecliptic pole.

Within the long period Precession is the shorter period component of Astronomic Nutation. Generally, this is the result of the Earth orbiting the Sun, the Moon orbiting the Earth and the fact that the Moons orbit does not lie in the ecliptic plane.

Annual Aberration is an apparent displacement of a stellar body due to the limitations of the speed of light and the relative motion of the observer and the observed object.
Annual Stellar Parallax is due to the separation of the Earth and the Sun and the Earths movement in its orbit about the Sun.

The mathematical models for the use of astronomy in surveying require us to determine the apparent (or true) coordinates for a stellar object. The coordinates we will find in star catalogues are heliocentric mean places, referred to some specific mean equator and equinox. In order for surveyors to utilize these coordinates corrections must be applied in a certain order to coincide with the epoch of observation.
Precession and Nutation are the motions of the coordinate system itself.
Proper Motion or Motion in Space is the movement of the star and our sun.
Refraction, Aberration, and Parallax are the apparent displacements of the stars due to physical phenomena.
And Polar Motion is the movement of the coordinate system with respect to the solid model Earth.
In the following pictures I show the completed steps and the corrected coordinates in rose and the expected coordinates in yellow. The difference between my coordinates and those derived classically, is that I have not followed the classical methods is some cases. My main aim has been to provide an instantaneous computation which requires neither expensive Almanacs, complicated tables or reams of paper recording step by step operations.

Here, you can see in blue the data required to be entered in the spreadsheet to begin the computations required. The blue data is provided in the star catalogue and we add the date and time of observation. The most precise catalogues have generally been the Fundamental Catalogues, such as the FK4 and FK5. However, since the launch of the Hipparcos satellite, the best resource is a website known as SIMBAD.
The corrections to be applied to a stellar coordinate, given in Right Ascension (?) and Declination (?) begin with Proper Motion. Proper Motion is the movement of the star against the background and the movement of the sun in its course through the Milky Way and our galaxies movement. This correction, given the components of motion in ? (?) and ? (?’), of milliarcseconds, can be simply multiplied by the number of years and added to the original coordinates. Unfortunately, Proper Motion has some faults. The math tends to break down with circum-polar stars especially. Hence, we have the newer theory of Motion in Space.
Imagine that you have a clear globe (the surface of which represents the spherical coordinates) and you let a pencil represent the true motion of a star from the start to the end of the pencil, and you place it somewhere near the pole of your globe, then it is not hard to see that it is a complicated matter to describe the spherical coordinates of the end-point of the pencil given only the length of the pencil and its starting point in spherical coordinates. If the pencil was very short relative to the globe and close to equator you may just "add the spherical coordinates", but not when it is near the pole.
Motion in Space converts the spherical coordinates to a set of Cartesian coordinates and corrections over time. It then converts these back to spherical coordinates, all very accurately. For example, if we compute the motion of a star called Sirius, over a thousand years, the deviations ignoring unforeseen perturbations of the Earth approach three tenths of a second in ? and four seconds in ?.

Precession is the resultant of attractive forces of the sun, moon (luni-solar precession) and planets (planetary precession) acting upon our model Earth. In combination, these forces attempt to drag the equatorial plane in to the ecliptic plane, disturbing the motion of the Earths axis of rotation about the ecliptic pole.

Within the long period Precession is the shorter period component of Astronomic Nutation. Generally, this is the result of the Earth orbiting the Sun, the Moon orbiting the Earth and the fact that the Moons orbit does not lie in the ecliptic plane.

Annual Aberration is an apparent displacement of a stellar body due to the limitations of the speed of light and the relative motion of the observer and the observed object.
Annual Stellar Parallax is due to the separation of the Earth and the Sun and the Earths movement in its orbit about the Sun.

#2 · July 12, 2010, 9:30 pm

Sorry, I went over the 5000 character limit for a post.

I have not considered Einsteins’ General Relativity, which causes a deflection of the path of light from a stellar object as it passes close to the Sun. In any case, for any elongation greater than 15 degrees the effect is less than 0.03”. And what surveyor would observe stars next to the sun anyway?
I also haven’t considered the geocentric parallax as it’s effects are inconsequential.
My total deviation appears to be 1.30 seconds in time and 0.85 seconds in arc for the computation of the position of Theta Persei over twenty eight years. I think that's pretty good for a spreadsheet. Be interesting to hear from Mr. Nolton on this.

Cheers, I need a beer, Scott.

#3 · July 12, 2010, 9:56 pm

Is this a job for TeX?

> The corrections to be applied to a stellar coordinate, given in Right Ascension (?) and Declination (?) begin with Proper Motion. Proper Motion is the movement of the star against the background and the movement of the sun in its course through the Milky Way and our galaxies movement. This correction, given the components of motion in ? (?) and ? (?’), of milliarcseconds, can be simply multiplied by the number of years and added to the original coordinates.

I wonder if the TeX feature will produce the Greek letters that got turned into squares. I guess I ought to look and see, huh?

Edit: Yes, it will. Here's a link to the commands to embed that TeX will convert to the various Greek letters, upper or lower case as may be.

Mathematical Symbols.pdf