Coordinate systems in Astronomy

Varun Bhalerao

Overview

• Need for astronomical coordinate systems

• Local and global coordinate systems– Altitude – azimuth– Right ascension – declination

• Conversion of coordinates

• Spherical trigonometry

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

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

“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)

“Look” at a star…

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)

“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.

“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”

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

Coordinate systems - local

• Basic elements of the celestial sphere

Coordinate systems - local

• Altitude

Coordinate systems - local

• Azimuth

Coordinate systems - global

• The celestial sphere

Coordinate systems - global

• Diurnal circles

(Path followed by the star in

the sky during one rotation of

earth)

Coordinate systems - global

• Hour circles – Equal right ascension

Coordinate systems - global

• Declination

Coordinate systems - global

• Right ascension, declination

Right Ascension

Coordinate systems - global

• Right ascension & hour angleHour angle

Right Ascension at the meridian

=hour angle of vernal equinox

= sidereal time

vernal equinox

North Celestial Pole

star

Horizon

Celestial Equator

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.

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

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

• 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

Spherical Triangles

90o

90o

90o

Formulae

• Corresponding to formulae in plane trigonometry, there are more generalized formulae in spherical trigonometry

• Sine rule :sin a = sin b = sin csin A sin B sin C

• Cosine rule :cos A = -cos B cos C + sin B sin C cos acos a = cos b cos c + sin b sin c cos A

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.

Coordinate Conversions

vernal equinox

North Celestial Pole

star

Horizon

Celestial Equator

Zenith Sides :

• 90o – latitude

• 90o – altitude

• 90o - declination

Angles :

1. 360o – azimuth

2. Hour angle

3. Unknown (not required)

2 1

3

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

Review

Coordinate systems :

• Local : Altitude – azimuth

• Semi-local : Hour angle – declination

• Global :– Right Ascension – declination– Ecliptic– Galactic

Review

• Spherical triangles :• Sides are great circles, represented by angles• Sum of angles > 180o

• Formulae :– Sine rule :

sin a = sin b = sin csin A sin B sin C

– Cosine rule :cos A = -cos B cos C + sin B sin C cos acos a = cos b cos c + sin b sin c cos A