Casio fx-9750GIII: Equatorial to Galactic Coordinates Conversions

Casio fx-9750GIII: Equatorial to Galactic Coordinates Conversions 

Introduction

Coordinate Systems

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.

Two coordinate systems are:

Equatorial System:

The equatorial system frames the University with the Earth's center as the center of the university.  

Right Ascension (α):   Customarily stated in hours, minutes, and seconds (H°M°S°).   Each hour is equivalent to 15 degrees.  The right ascension at 0 hours is aligned where the sun would be during the vernal equinox.  

Declination (δ):   Customarily stated in degrees, minutes, and seconds (D°M°S°).  

Galactic System:

The galactic system frames the University, aligning it with the Milky Way Galaxy.  The center of the system is our Sun.  The galactic longitude at 0 degrees aligns with the center of our galaxy (Sagittarius A*).

Galactic Longitude (l):  Customarily stated in degrees, minutes, and seconds (D°M°S°).  

Galactic Latitude (b):  Customarily stated in degrees, minutes, and seconds (D°M°S°).  

The program EQT2GLT converts equatorial coordinates (α, δ) to galactical coordinates (l, b).  

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°

=  atan2((sin δ - sin b × sin 27.1284°), (cos δ × sin(α -192.8595°) × cos 27.1284°)) + 32.93117169°

b is in hours, from -90 to 90 degrees.

l is in degrees, from 0 to 360 degrees.  Be sure to consider quadrants in your calculation.   

The program GLT2EQT converts galactical coordinates (l, b) to equatorial coordinates (α, δ).  

δ = arcsin( (cos b × cos 27.1284° × sin( l - 32.93117169° ) + sin b × sin 27.1284° )

α = arctan( ( cos b × cos( l - 32.93117169° ) ) ÷ ( sin b × cos 27.1284° - cos b × sin 27.1284° × sin( l - 32.93117169° ) ) + 192.8595°

= atan2((cos b × cos(l - 32.93117169°),(sin b × cos 27.1284° - cos b × sin 27.1284° × sin(l-32.93117169°)) + 192.8595°

δ is in degrees, from -90 to 90 degrees.  

α is in hours, from 0 to 24 hours.  Be sure to consider quadrants in your calculation.   Remember that 1 hour is equivalent to 15 degrees.  

The equations are from Practical Astronomy With Your Calculator by Peter Duffett-Smith, with the constants updated for J2000.0.

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")

(Sources:   Duffett-Smith, Westmeier - refer to the Source section)

For more details, please refer to the Equatorial to Galactic Coordinates:  Updating the Constants posted on January 6, 2023.   

Casio fx-9750GIII Program:  EQT2GLT

a+bi

Deg

"EQUATORIAL -> GALACTIC"

"J2000.0"

"R.A. IN H°M°S°"

?->A

"DEC. IN D°M°S°"

?->D

15*A->A

192.8595->R

27.1284->E

32.93117169->C

sin^-1 (cos D*cos E*cos (A-R)+sin D*sin E)->B

sin D-sin B*sin E->Y

cos D*sin (A-R)*cos E->X

Arg (X+Yi)->T

Y<⇒T+360->T

T+C->L

ClrText

"B="

B ▶DMS◢

"L="

L ▶DMS

Note:  The calculator is set to the following modes: degrees, and rectangular complex mode. 

Example:  

The star Hamal in the constellation Aries (Alpha Arietis) is located approximately at:

α = 2 hours, 7 minutes, 10 seconds

δ = 23°27'44"

Enter hours-minutes-seconds and degrees-minutes-seconds by pressing [ OPTN ], [ F6 ] ( > ), [ F5 ] (ANGL), [ F4 ] (°''').

b:  -36°12'22.12"

L:  144°34'33.2"

Casio fx-9750GIII Program:  EQT2GLT

a+bi

Deg

"GALACTIC_->_EQUATORIAL"

"J2000.0"

"B (LAT) IN D°M°S°"

?->B

"L (LONG) IN D°M°S°"

?->L

192.8595->R

27.1284->E

32.93117169->C

sin^-1 (cos B*cos E*sin (L-C)+sin B*sin E)->D

cos B*cos (L-C)->Y

sin B*cos E-cos B*sin E*sin (L-C)->X

Arg (X+Yi)->T

R+T->A

A>360⇒A-360->A

A÷15->A

ClrText

"R.A.="

A ▶DMS◢

"DEC="

D▶DMS

Example:

The star Regulus in the constellation Leo (Alpha Leonis) is located approximately at:  

b:  48°56'03"

l:  226°25'36"

α:  10°08'22.31"  (10 hr, 8 min, 22.31 sec)

δ:  11°58'02.14"

Note:  The results are approximated.  

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

All original content copyright, © 2011-2024.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author. 

Equatorial to Galactic Coordinates: Updating the Constants

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

All original content copyright, © 2011-2024.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author. 

Greetings from Carpinteria/UCSB - Astronomy

I am in Carpinteria, CA; soon to be headed back home (vacation goes by soooooooo fast!)

Last night was a sight to see. Unfortunately I did not catch the young moon, but I did manage to see Jupiter, and the constellations Ursa Major, Leo, Gemini, and Virgo. I can thank Google Sky for the immense help. Who knew the stars of Cancer were so faint?

My trip to the UCSB Library (Davidson Library) completed a circle for me. It was there where I got the itch to visit the mathematics section of as many libraries I can get to. I first visited the UCSB library last September.

This visit I concentrated more on astronomy than mathematics. Learning about the structure of the Milky Way has become an interest for me.

Thanks to Tycho Brahe, comets are a major part of astronomy. He also brought all the astronomical almanacs to date around 1600, all without a telescope. Telescopes were first used by Galileo Galilei. Brahe's data was eventually inherited by Johannes Kepler, who was both an astrologer and astronomer.

Kepler showed how proportional the orbits of Jupiter (inscribed circle) and Saturn (circumscribed circle) were (well, in general) by drawing two circles around an equilateral triangle. My attempt at drawing is shown above. (Freehand art was never my strength).

Unlike what astronomers believed before, Galileo showed, among other things:
1. Planets are not self-moving.
2. Stars are not close to the Earth, instead they are distant suns.

The Milky Way

Harlow Shapley was the first to locate the Earth's place in the Milky Way Galaxy (around 1919): in the galactic disc two thirds out from the center. Today, astronomers estimate that we are about 24,500 to 27,000 light years away from the center; which could be a good thing. The center of our galaxy is said to be a massive black hole.

During the time of World War II, H.C. van de Hurst suggested trying to detect radio waves emitted by hydrogen atoms. This opened the door to a more detailed map of our celestial sky.

Source: Charles A. Whitney. "The Discovery of our Galaxy" Alfred A. Knoff: New York. 1971

-------

Currently, the center of our Milky Way lies in the constellation Sagittarius (♐). The biggest identifier is a radio source named Sagittarius A*, which can't be seen by the naked eye. It lies near the boarder of Scorpio (Scorpius) and Ophiuchus.

Coordinates of Sagittarius A* (approximately)
RA (α) 17hr 45min 40.04sec (266.416833°)
Dec (δ) -29°00'28.1" (-29.007806°)

I am curious: with the precision of the equinoxes, will Sagittarius A* "move" to Scorpio and/or Ophiuchus in the far future? It stands to reason, since the point (0hr, 0°), the vernal equinox once lied in the constellation Aries 2,000 years ago, now lies in Pisces on its way to Aquarius.

It is also an explanation why the star Polaris in Ursa Minor and the star Vega in Lyra trade the honor of being the North Star approximately every 12,857 years. (Close to 13,000) Source: http://csep10.phys.utk.edu/astr161/lect/time/precession.html

Hipparchus determined that the stars rotated approximately 50 arc seconds (50".3 to 50".4) around the ecliptic pole. Hence, it takes about 25,714 2/7 years for the stars to complete one cycle. (≈ 360°/50".4 = 360°/0.014°)

Until next time, cheers!


Eddie