1. Coordinate systems in Astronomy
Varun Bhalerao

2. Overview Need for astronomical coordinate systems
Local and global coordinate systems
Altitude – azimuth
Right ascension – declination
Conversion of coordinates
Spherical trigonometry

3. “Look” at a star… Which star do we choose ?
A first look at the night sky : this is the sky as will be seen from Mumbai at around 8 p.m. in late January
Which star do we choose ?
For centuries, people have been gazing at the heavens, and have uncovered numerous facts about them. We choose to begin our journey in such a way that we can go a rather long way, discovering as many features as we can. We choose …. ALGOL

4. “Look” at a star…
Lets try to make things simpler by naming the stars and constellations. We are taking a big leap, which took mankind centuries - we begin classifying the stars. Note the constellation shapes (thin white lines) and boundaries (green lines)

5. Constellation Shapes and Boundaries
The shapes come from ancient times, as easy-to-remember patterns in the sky
Modern constellations like telescopium etc were not named after patterns they seem to form, but named after objects
Constellation shapes (stick figures) may change from chart to chart, but two main systems followed – astronomical and ray’s
Constellation boundaries standardized by IAU (International Astronomical Union)
Boundary lines parallel to RA / dec lines (RA and dec are explained later)

6. “Look” at a star…
We zoom on to some region, in this case - Algol. We are seeing only a region 45 o across as compared to the normal 100o field.

7. “Look” at a star…
The same field, with stars and constellations labeled.
This gives a better view of the stick figure and boundary of the constellation “Perseus”

8. Coordinate systems Rising and setting
Local coordinates – basic reference to a star in the sky
Layman’s representation like above the building – about halfway to overhead etc is not good enough
More standard representation required
System used is the Alt-Az system

9. Coordinate systems - local
Basic elements of the celestial sphere

10. Coordinate systems - local
Altitude

11. Coordinate systems - local
Azimuth

12. Coordinate systems - global
The celestial sphere

13. Coordinate systems - global
Diurnal circles
(Path followed by the star in the sky during one rotation of earth)

14. Coordinate systems - global
Hour circles – Equal right ascension

15. Coordinate systems - global
Declination

16. Coordinate systems - global
Right ascension, declination

17. Coordinate systems - global
Right ascension & hour angle
North Celestial Pole
Hour angle
Right Ascension at the meridian
=hour angle of vernal equinox
= sidereal time
star
Right Ascension
vernal equinox
Celestial Equator
Horizon

18. Spherical trigonometry
A great circle is made by a plane passing through the center of a sphere.
Equator, lines of RA are great circles.
Other than equator, other lines of declination are not great circles.

19. Spherical Triangles
Triangles made by intersecting great circles are spherical triangles.
The sides of these triangles are the arcs on the surface of the sphere
The angles are the angles as measured at the vertex, or angle between the planes which make those great circles
Angle of triangle – represented by A, B, C
Side of triangle – represented by a, b, c

20. The sides of spherical triangle
The length of the side is related to the angle it subtends at the center by s = r * theta
Angles subtended at center can hence be used to represent sides
Esp. in astronomy, we can measure angles in sky but they don’t necessarily relate to distances between the objects
theta
side s

21. Spherical Triangles
We can imagine that the angles of a spherical triangle need not add to 180o
For example, consider an octant cut out of a sphere… the sum of angles is 270o !
In fact, the sum must be greater than 180o and the sum of angles – 180o is called the spherical excess
90o
90o
90o

22. Formulae
Corresponding to formulae in plane trigonometry, there are more generalized formulae in spherical trigonometry
Sine rule : sin a = sin b = sin c sin A sin B sin C
Cosine rule : cos A = -cos B cos C + sin B sin C cos a cos a = cos b cos c + sin b sin c cos A

23. Coordinate Conversions
Given a star, to convert from equatorial to alt-az (or any one system to another):
First draw the celestial sphere showing the lines for both coordinate systems
Consider the spherical triangle with the star and poles of the two systems as vertices
Apply the spherical trigonometry formulae.

24. Coordinate Conversions
Zenith
Sides :
90o – latitude
90o – altitude
90o - declination
Angles :
360o – azimuth
Hour angle
Unknown (not required)
North Celestial Pole 1 2 3
star
Celestial Equator
Horizon
vernal equinox

25. Inter-conversions to be done by spherical trigonometry formulae
Other systems
Ecliptic
Reference circle : ecliptic plane
Reference point : vernal equinox
Galactic
Reference circle : galactic plane
Reference point : direction of centre of galaxy
Inter-conversions to be done by spherical trigonometry formulae

26. Review Coordinate systems : Local : Altitude – azimuth
Semi-local : Hour angle – declination
Global :
Right Ascension – declination
Ecliptic
Galactic

27. Review Spherical triangles : Formulae :
Sides are great circles, represented by angles
Sum of angles > 180o
Formulae :
Sine rule : sin a = sin b = sin c sin A sin B sin C
Cosine rule : cos A = -cos B cos C + sin B sin C cos a cos a = cos b cos c + sin b sin c cos A