Coordinate systems in Astronomy: Coordinate systems in Astronomy Varun Bhalerao
Overview: Overview Need for astronomical coordinate systems
Local and global coordinate systems
Altitude – azimuth
Right ascension – declination
Conversion of coordinates
Spherical trigonometry
Slide3: 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…
Slide4: “Look” at a star…
Constellation Shapes and Boundaries: 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)
Slide6: “Look” at a star…
Slide7: “Look” at a star…
Coordinate systems: 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: Coordinate systems - local Basic elements of the celestial sphere
Coordinate systems - local: Coordinate systems - local Altitude
Coordinate systems - local: Coordinate systems - local Azimuth
Coordinate systems - global: Coordinate systems - global The celestial sphere
Coordinate systems - global: Coordinate systems - global Diurnal circles (Path followed by the star in the sky during one rotation of earth)
Coordinate systems - global: Coordinate systems - global Hour circles – Equal right ascension
Coordinate systems - global: Coordinate systems - global Declination
Coordinate systems - global: Coordinate systems - global Right ascension, declination
Coordinate systems - global: Right Ascension Coordinate systems - global Right ascension & hour angle Hour angle Right Ascension at the meridian
=hour angle of vernal equinox
= sidereal time vernal equinox North Celestial Pole star Horizon Celestial Equator
Spherical trigonometry: 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: 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 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 Spherical Triangles 90o 90o 90o
Formulae: 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: 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: Coordinate Conversions vernal equinox North Celestial Pole star Horizon Celestial Equator Zenith Sides :
90o – latitude
90o – altitude
90o - declination
Angles :
360o – azimuth
Hour angle
Unknown (not required) 2 1 3
Other systems: 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: Review Coordinate systems :
Local : Altitude – azimuth
Semi-local : Hour angle – declination
Global :
Right Ascension – declination
Ecliptic
Galactic
Review: Review Spherical triangles :
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