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ASTR 5610, Majewski [SPRING 2016]. Lecture Notes

ASTR 5610 (Majewski) Lecture Notes



In astronomy we are constantly dealing with spherical coordinate systems, in particular when exploring the celestial sphere.

We have five principal coordinate systems defined for the celestial sphere, each defined by a particular pair of principal and secondary ("prime meridian") great circles:

coordinate system principal great circle "prime meridian definition" -
secondary great circle
horizon or observer's observer's horizon north-south meridian altitude,
equatorial or celestial projection of Earth's equator head of Aries -- vernal equinox right ascension,
declination, δ
ecliptic plane of Earth's revolution head of Aries -- vernal equinox ecliptic longitude, λ
ecliptic latitude, β
Galactic plane of the Milky Way Galactic center Galactic longitude, l
Galactic latitude, b
supergalactic apparent planar concentration
of galaxies in local Universe
intersection with Galactic plane supergalactic longitude, SGL
supergalactic latitude, SGB


It is useful to be familiar with some basic concepts in spherical trigonometry, since they are key for understanding and converting between coordinate systems.

From Kaler, The Ever-Changing Sky.
As in plane geometry, we have a spherical law of cosines

and a law of sines:

Also useful is the five parts rule:


It is also useful to have an understanding of Euler rotations and Euler Angles, which are a way to determine the transformation from one coordinate system to another defined on the celestial sphere.

From Eric Weisstein's Mathworld,
Any random 3D rotation, A, can be described by a succession of three rotations D, C, and B (with Euler rotation angles ψ, θ, φ, respectively) about pre-defined cardinal axes:

A = B C D


D =
C =
B =
For example, if we convert each spherical coordinate system defined above to its corresponding Cartesian coordinates (e.g., with the z axis defined by the pole of the spherical system, etc.), then the coordinates x1, y1, z1 in system 1 can be converted to the coordinates x2, y2, z2 in the second system using rotations, e.g., similar to the matrices above:

After combining each individual rotation operator, we end up with a single operator that looks like this:

If you have not seen Euler transformations before in your undergraduate math and physics courses, please consult or some other source.

Of course, the modern astronomy student has access to several utilities (in IDL or in the IRAF astutil package) that already have encoded the most common transformations (e.g., from equatorial to Galactic coordinates).


It is assumed that you are already familiar with the following coordinate systems
(if you are not, please read Section 2.1 of Mihalas & Binney, and/or Section 2.1 of Binney & Merrifield):


Another frequently used concept is that of the position angle, which is used to describe relative positions on the sky (usually in the equatorial system, but not necessarily).

For example:

By convention, PA is defined to be 0o to the North, 90o to the East, etc.


A more natural coordinate system to use in the analysis of the Milky Way (and even extragalactic objects) is the Galactic coordinate system, where:

It is useful to be able to transform between equatorial and Galactic coordinates:

There is a corresponding Galactic rectangular coordinate system:

From Binney & Merrifield, Galactic Astronomy. Note that the Z axis is positive towards the North Galactic Pole.
From the usual transformation from spherical to Cartesian coordinates, we have:

XGC = d cos(b ) cos(l )

YGC = d cos(b ) sin(l )

ZGC = d sin(b )

Here "GC" stands for "Galactic Coordinates" (not "Galactic Center").

Note that in this right-handed system:

Unfortunately, a left-handed version of the Cartesian system is also in common use, where:


There are two commonly used velocity coordinate systems in Milky Way studies.


The supergalactic coordinate system has its equator aligned with the apparent supergalactic plane, the planar distribution of nearby galaxy clusters in the local Universe (e.g., the Virgo cluster, Pisces-Perseus supercluster).

All sky galaxy distribution in the 2MASS survey shown in supergalactic coordinates. The blue swaths represent the Milky Way disk. From apod/ap030917.html.


The precession of the Earth is a 25,800 year periodic wobble of the direction of the Earth's axis of rotation.

This is a major effect that can be detected nightly, and which has a large effect on coordinates over the period of years.


Because the Earth spins, it is in fact a little fatter around the equator by one part in 298.

Because the Moon orbits the Earth in a plane that is within 5 degrees of the ecliptic, typically the Moon is not aligned with the Earth's equatorial bulge (unless the Moon is on the Celestial Equator).


There are other effects (e.g., nutation, aberration of starlight, proper motion, refraction) that affect the observed positions of stars. For those interested, these topics are discussed here.

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All material copyright © 2003,2006,2008,2010,2012,2014,2016 Steven R. Majewski. All rights reserved. These notes are intended for the private, noncommercial use of students enrolled in Astronomy 551 and Astronomy 5610 at the University of Virginia.