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. 

Some Science Stocking Stuffings

Einstein's Time Dilation

Recall that equation for time dilation is:

Δt = Δt0 / √(1 - u^2/c^2)

Where:
Δt = time observed by the person standing still
Δt0 = time observed by the traveler (in the observer's frame of reference)
u = speed of the moving object , which contains the traveler
c = speed of light in a vacuum = 299,792,458 m/s

In short, the traveler will note observe and note that time Δt0 has passed, which the person standing still observes that time Δt has passed. The closer someone goes to the speed of light, that less person she/he experiences. You will need to go super fast. Driving at 65 mph (29.0756 m/s) on the freeway won't cut it and here's why:

For 1 unit of time to pass for the person driving 65 mph (Δt0 = 1), the change of time for the observer (Δt) is:

Δt = 1/√(1 - 29.0756^2/299,792,458^2) ≈ 1.0000000000000047 (that is fourteen zeroes between the decimal point and the 4) (The Wolfram Alpha app was used for this calculation).

Virtually the same time passes.

Same deal regarding observing an airplane flying at 600 mph (268.224 m/s). For the person watching the plane, for a plane's passenger, pilot, or cocktail and peanut server to observe one unit of time, us watchers observe 1.0000000000004 (twelve zeroes) units of time.

If you want to be on an object where the people observing you experience twice the time (Δt = 2) you do (Δt0 = 1), you will need to travel (c*√3)/2 or 259,627,884.491 m/s (580,771,037.247 mph). That is 86% the speed of light!

Archer's Paradox (E.J. Rendtroff, 1913)

In archery, archers aim their arrows slightly to the side, instead of directly at the target. On the surface, that seems crazy. This is where the Archer's Paradox comes into effect.

Basically, when the arrow is shot, it travels in an "S" curve. Drawing a string makes the arrow bend such that he tip is pointed away from the target. As the arrow is fired and the string returns to the bow, the arrow bends the other way, turning the arrow back to the target.

Other factors to making accurate shots include the stiffness of the arrow, brace height, and wind conditions. I found this video by Billgsgate helpful:

http://youtu.be/bNlx6MBlymw

Galactic Coordinates

Galactic Coordinates are spherical coordinates that are set such as:

* The center is our sun.
* The coordinates are (l°, b°), where l is the longitude (0° to 360°, flat angle) and the latitude (-90° to 90°, height).
* Pointing "due east", l = 0° and b = 0° is pointing towards the center of the Milky Way. "Due west", l = 180° and b = 0° points away from the center of the Galaxy.

Using Wolfram Alpha ( http://m.wolframalpha.com ) and an online coordinator converter ( http://ned.ipac.caltech.edu/forms/calculator.html ), here are the 30° longitude markers when latitude is 0° (b = 0°):

* 0° points towards the constellation Sagittarius (as it should, it having the Milky Way center)
* 30° points towards Aquila the Eagle
* 60° points towards the Vulpecula the Fox
* 90° points towards Cygnus the Swan
* 120° points towards Cassiopeia the Vain Queen
* 150° points towards Perseus the Hero
* 180° points towards Auriga the Charioteer
* 210° points towards Monoceros the Unicorn
* 240° points towards Puppis (a ship's poop deck)
* 270° points towards Vela (a ship's sails)
* 300° points towards Crux, The Southern Cross
* 330° points towards Norma (a carpenter's square (measuring tool))

Spherical Lenses (see figure 2 below)

Variables:
P = place of object (with distance s)
P' = place of image (with distance s')
C = center of curvature
V = vertex of the lens

General relations:

tan α = h/(s - δ)
tan β = h/(s' - δ)
tan Φ = h/(R - δ)

However, if α < π/2 ( α < 45° ), we are dealing with paraxial rays:

1/S + 1/S' = 2/R

Surprisingly, the object is far from the vertex.


Forgive me if I repeat things. Instead of going out to clobber everyone last Black Friday for that extra 10% off, I stayed home and cracked open some books that I have been meaning to look at for months. I hope you find this enjoyable and insightful.

The best always,

Eddie

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P.S. I was thinking about adding a section (or blog entry) about the age of Aquarius and some of the head scratchers I have about it. I am not sure if this subject would be appropriate for this blog. If you have any thoughts about this, or anything else, go and ahead and comment! - E.S.

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This blog is property of Edward Shore. 2014