Equatorial to Galactic Coordinates: Updating the Constants
Introduction
There are several coordinate systems used by astronomers to determine the placement of celestial objects, such as stars, galaxies, planets, and black holes, in our night skies.
Equatorial System
The most common system is the Equatorial System. Imagine a sphere which represents the Universe with the center of the Earth as the center. The coordinates are the right ascension (α) and the declination (δ).
Right ascension (α): The right ascension is the angular distance from the vernal equinox, going eastward traveling with the celestial equator. The 0 point is the vernal equinox. The vernal equinox, which generally takes place around March 20 or 21, is when the sun is over the Earth's equinox heading north. The lengths of daytime and nighttime are equal. For the northern hemisphere, it's the first day of spring, and for the southern hemisphere, it's the first day of autumn. Sometimes, the vernal equinox is called the First Point of Aries (♈). The range of the right ascension is from 0 hours to 24 hours. Each hour is equivalent to 15° (15 degrees).
Declination (δ): The declination is the angular distance north (above) or south (below) the celestial equator. The range of declination is from -90° to +90°.
Galactical System
The galactic coordinate system focuses on aligning with our Milky Way Galaxy, with our Sun as the center of the sphere.
Galactic Longitude (l): The galactic longitude measures the angular distance from the center of the Milky Way Galaxy, increasing in the eastward direction. The range of the galactic longitude is from 0° to 360°, with the 0° point at the Galactic Center, which lies in the constellation Sagittarius the Archer. (Sagittarius A*)
Galactic Latitude (b): The galactic latitude is the angle northward from the galactic equator, and it's range is from -90° (south pole located in the constellation Sculptor) to 90° (north pole located in the constellation Coma Berenices).
Practical Astronomy: With Updated Constants
A popular resource for astronomical calculations is the book Practical Astronomy With Your Calculator by Peter Duffett-Smith.
On page 48 of Practical Astronomy With Your Calculator, Duffett-Smith provides these equations for converting from equatorial (α, δ) to galactic (l, b) coordinates:
b = arcsin( cos δ × cos 27.4° × cos(α - 192.25°) + cos δ × cos 27.4° )
l = arctan( ( sin δ - sin b × sin 27.4° ) ÷ ( cos δ × sin(α -192.25°) × cos 27.4°) + 33°
Note that in calculation, α, δ, b, and l must be in decimal degrees. Usually the coordinates are given in hours-minutes-seconds or degrees-minutes-seconds, and the quantities must be converted before calculation.
The numerical constants? Those are the 1950.0 coordinates of the north galactic pole with α0 = 192.25° = 12h 49m and δ = 27.4° = 27°24'.
Obviously, in 2024, we would be working with the epoch J2000.0 coordinates of the north galactic pole. If we want to work the J2000.0 coordinates in the above formulas, the constants must be changed.
From the "Conversion of coordinates" page of Tobias Westmeier's webpage, the J2000.0 of the north pole are:
α0 ≈ 192.8595° (12h 51m 26.28s)
δ0 ≈ 27.1284° (27°07'42.24")
I choose to use these coordinates because it provides more decimal places than what is presented in the Galactic coordinate page of Wikipedia.
This leaves us with updated equations:
b = arcsin( cos δ × cos 27.1284° × cos(α - 192.8595°) + cos δ × cos 27.1284° )
l =
arctan( ( sin δ - sin b × sin 27.1284° ) ÷ ( cos δ × sin(α -192.8595°) × cos 27.1284°) + C
We need to determine the value of C.
I'm going to use our galactic center, Sagittarius A*, as a reference point, with the coordinates as determined by NASA/IPAC Extragalactic Database's Coordinator Calculator tool:
Sagittarius A*:
Equatorial Coordinates
α ≈ 266.41681667° (17h 45m 40.036s)
δ ≈ -29.007825° (-29°00'28.17")
Galactic Coordinates
l ≈ 359.94418679° (359°56'39.072")
b ≈ -0.04610951° (-0°2'45.994")
(Theoretically, this should be l0 = 0°, b0 = 0°).
Substituting the following data into equation for l (only the second equation has C):
l =
arctan( ( sin δ - sin b × sin 27.1284° ) ÷ ( cos δ × sin(α -192.8595°) × cos 27.1284°) + C
359.94418679° =
arctan( ( sin -29.007825° - sin -0.04610951° × sin 27.1284° ) ÷ ( cos -29.007825° × sin(266.41681667° - 192.8595°) × cos 27.1284°) + C
359.94418679° = arctan( (-0.4845538612°) ÷ (+0.7465187073°)) + C
We have to keep in mind that anytime we are working with astronomical math, we have to mind the coordinate system.
359.94418679° = atan2(0.7465187073°, -0.4845538612°) + C
359.94418679° = arg(0.7465187073° - 0.4845538612° × i) + C (where i = √-1)
Note:
atan2(0.7465187073°, -0.4845538612°) = -32.98698493°
To put this answer in the range of 0° to 360°:
-32.98698493° + 360° = 327.0130151°
359.94418679° = 327.0130151° + C
C = 32.93117169
Our final updated equations are:
b = arcsin( cos δ × cos 27.1284° × cos(α - 192.8595°) + cos δ × cos 27.1284° )
l =
arctan( ( sin δ - sin b × sin 27.1284° ) ÷ ( cos δ × sin(α -192.8595°) × cos 27.1284°) + 32.93117169
and will be used in the programs coming up this weekend.
Sources
"Equatorial coordinate system" Wikipedia. Last Edited April 10, 2023. Accessed December 10, 2023. https://en.wikipedia.org/wiki/Equatorial_coordinate_system
"Galactic coordinate system" Wikipedia. Last Edited April 21, 2023. Accessed November 23, 2023. https://en.wikipedia.org/wiki/Galactic_coordinate_system
Duffett-Smith, Peter. Practical Astronomy With Your Calculator Second Edition. Cambridge University Press: Cambridge, UK. 1981.
ISBN: 0 521 28411 2 (paperback)
National Aeronautics and Space Administration (NASA). "Coordinate Calculator" NASA/IPAC Extragalactic Database. Operated by the California Institute of Technology. 2023. Accessed November 26, 2023. https://ned.ipac.caltech.edu/coordinate_calculator?in_csys=Equatorial&in_equinox=J2000.0&obs_epoch=2000.0&ra=17h45m40.036s&dec=-29d00m28.17s&pa=0.0&out_csys=Galactic&out_equinox=J2000.0
Westmeier, Tobias. "Conversion of coordinates" Homepage of Tobias Westmeier. The University of Western Australia. Last Modified 26 September 2023. Accessed November 26, 2023. https://www.atnf.csiro.au/people/Tobias.Westmeier/index.php
Eddie
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