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

“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

“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)

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

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

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

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

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

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 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

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

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

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