ASTR 5610, Majewski [SPRING 2016]. Lecture Notes

## COORDINATE SYSTEMS

### CELESTIAL SPHERE COORDINATE SYSTEMS

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

• Any spherical coordinate system is defined by some primary great circle -- obviously we have an infinite variety of coordinate systems we can select based on the infinite number of possible great circles -- but in reality only a few of them are sensible to adopt.

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:

• A secondary great circle is a great circle that goes through the poles of a primary great circle.

 coordinate system principal great circle "prime meridian definition" - secondary great circle coordinates horizon or observer's observer's horizon north-south meridian altitude, azimuth 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

### SPHERICAL TRIGONOMETRY

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:

### COORDINATE TRANSFORMATIONS

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, http://mathworld.wolfram.com/EulerAngles.html.
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

where

 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:

• : rotation about z axis
• β: rotation about the new (intermediate) x' axis
• γ: rotation about the new z' axis
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 mathworld.wolfram.com 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).

### THE OBSERVER, EQUATORIAL AND ECLIPTIC COORDINATE SYSTEMS

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):

• Equatorial Coordinate System
##### From Mihalas & Binney, Galactic Astronomy.
• Right ascension, α, measured from the vernal equinox (defined below).

Traditionally measured in hours, minutes of time, seconds of time, but trend now towards use of degrees (e.g., all-sky catalogues like 2MASS).

• Declination , measured from the celestial equator to the celestial poles.

• Observer's or Horizon Coordinate System

##### From Mihalas & Binney, Galactic Astronomy.
Key horizon coordinates for an object, measured with respect to the zenith and North point are:

• Altitude, h, from the horizon along the vertical circle (the great circle including the zenith and the object).

• Zenith distance, z = 90o - h, (important for determining the atmospheric extinction and airmass).

• Azimuth, A, measured from the North Point (intersection of the meridian with the true horizon in the north) becoming positive toward East.

The meridian is the vertical circle including the zenith and celestial poles.

The angles of the important Astronomical Triangle are:

• Hour angle, t (an important angle for observers).

• Zenith angle, Z (note that the name "zenith angle" is often misapplied to zenith distance, even, frustratingly, among supposedly seasoned astronomers!).

• Parallactic angle, P (important, e.g., in spectroscopy and other applications where one needs to understand the effects of atmospheric refraction, which are always along the vertical circle).

##### From Kaler, The Ever-Changing Sky. Note that the angle φ shown corresponds to your latitude.

• Ecliptic Coordinate System
##### From Mihalas & Binney, Galactic Astronomy.
Key points:

• The ecliptic is the path of the Sun in the sky (Earth orbital plane).

• The obliquity of the ecliptic is 23o 27'.

• The moon orbits within 5o of the ecliptic.

• Planets orbit within 7o 0' of the ecliptic (except Pluto 17o 09').

• Zodiacal light, zodiacal dust, lie along this plane, often a problem in IR maps.

• Intersection of the ecliptic and celestial equator define the vernal equinox (location of the Sun on March 21) and autumnal equinox (location of the Sun on Sep. 22).

• The Vernal equinox defines the zero-point of the right ascension coordinates.

• Ecliptic longitude, λ, measured along the ecliptic increasing to the east with zeropoint at the vernal equinox.

• Ecliptic latitude, β, measured from ecliptic to ecliptic poles.

• Ecliptic Coordinate System commonly used in solar system studies.

### POSITION ANGLE

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:

• The angle between a binary star and its secondary is given by the PA.

##### From Kitchin, Astrophysical Techniques.
• One declares the angle of a spectrographic slit on the sky by its PA in equatorial coordinates. (Typically, to account for atmospheric refraction and minimize light loss in the slit due to this effect, one adopts for the slit PA the parallactic angle).

• The proper motions (angular motions) of stars on the sky are often given as an amount (in arcsec) and a direction (PA).

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

### GALACTIC COORDINATE SYSTEMS

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

• The Galactic equator is chosen to be that great circle on the sky approximately aligned with the Milky Way mid-plane.

• This plane is inclined by 62o 36' to the celestial equator.

##### From Mihalas & Binney, Galactic Astronomy.
• The North Galactic Pole is at α = 12h 49m, δ = +27° 24' in 1950 equinox, and best observed at night in the Northern Hemisphere spring (a highly demanded time for extragalactic observers...as you might expect; consider when the same coveted time in the southern hemisphere might be).

• Galactic latitude, b, is measured from the Galactic equator to the Galactic poles (as seen from the Earth!).

##### From http://www.astr.ua.edu/ay102/Lab9/Lab_9_Coord.html.
• Galactic longitude, l, is measured eastward around the equator in degrees.

##### From http://www.astr.ua.edu/ay102/Lab9/Lab_9_Coord.html.
• The definition of l = 0o is given by the location of the Galactic Center.

l = 90o in the direction of the motion of the Sun in its rotation about the Galactic Center.

l = 180o is called the anticenter direction.

l = 270o is sometimes called the anti-rotation direction.

• It is common to use the shorthand of Galactic quadrants when discussing directions of the Galaxy:

• First Quadrant: 0 < l < 90o

• Second Quadrant: 90o < l < 180o

• Third Quadrant: 180o < l < 270o

• Fourth Quadrant: 270o < l < 360o

(Do not get confused by the Galactic quadrants of Star Trek!).

• Note that you may find in some older references use of a previous definition of the Galactic coordinate system used before 1958, where Galactic longitude was defined by one of the intersections of the Galactic equator with the celestial equator.

• This older system is sometimes referred to as (l I, b I).

• This older system has the Galactic Center at (l I, b I) = (327o 41', -1o 24').

• Our presently used Galactic Coordinate system is sometimes referred to as (l II, b II).

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

• Maps, like those below from Ridpath's Norton's 2000.0 Star Atlas and Reference Handbook, showing the conversion can be handy for quick reference:

Galactic Quadrants 1 and 2

Galactic Quadrants 3 and 4

• Analytically, using an Eulerian transformation, we find:

 sinb = sinδ sinδNGP - cosδ cosδNGP sin(α - α0) cos(l - l0) cosb = cos(α - α0) cosδ sin(l - l0) cosb = sinδ cosδNGP + cosδ sinδNGP sin(α - α0)

where the following values for the equatorial coordinates of the North Galactic Pole, the ascending node of the Galactic equator in equatorial coordinates (α0,0), and the node of the celestial equator in Galactic coordinates (l0,0) are:

 equinox αNGP δNGP α0 l0 1950 12:49.0 = 192.25o 27:24 18:49.0 = 282.25o 33.00o 2000 12:51.4 = 192.85o 27:08 18:51.4 = 282.85o 32.93o

• The IRAF astutil.galactic routine makes the transformation back and forth between Galactic and equatorial 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:

• The Sun has (XGC , YGC , ZGC ) = (0,0,0)

• The Galactic Center has coordinates of (XGC , YGC , ZGC ) = (Ro ,0,0), where Ro is the distance to the Galactic Center (values of 7.0 to 8.5 kpc are commonly adopted).

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

• we have the coordinate flip XGC --> -XGC.

• the Galactic Center has coordinates of (XGC , YGC , ZGC ) = (-Ro ,0,0).

Again, and importantly, here "GC" stands for "Galactic Coordinates" (not "Galactic Center")!

### GALACTIC VELOCITY COORDINATE SYSTEMS

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

• One is a cylindrical coordinate system, with coordinates (Π, Θ, Z ) are defined as in the figure below.

##### The cylindrical system as defined for the solar neighborhood. From Mihalas & Binney, Galactic Astronomy.

• Warning, the above is a left-handed system as defined.

Right-handed versions of the same coordinate system, with a flip in the Π orientation, Π --> -Π, are also commonly used.

One needs to figure out the handedness from context -- it is important always for the user to define this for their reader.

• The zero-point of these velocity coordinates is generally in the Galactic rest frame.

• A Cartesian velocity system is one that mimics the (XGC , YGC , ZGC ) defined above.

• Motions in these cardinal directions are often written as (u, v, w ).

• Yet again, beware of left-handed versus right-handed versions of this.

• Also, beware that it is standard for this coordinate system to be defined with respect to the Local Standard of Rest (LSR).

(You can think of this as the system in which the old definitions of "high velocity" and "low velocity" stars makes sense!)

• The LSR is defined to be the velocity a star would have at the position of the Sun if it were following a perfectly circular orbit around the Galactic Center.

Such a star has (Π, Θ, Z ) = (0, Θo, 0), where Θo is the circular velocity, defined by the rotation curve of the Milky Way at the solar position (i.e., the mass interior to the orbit).

Θo is not well established, with values ranging from 180 to 250 km/s in the literature, but a number like 220 km/s most commonly adopted.

In recent years, however, it seems that new measurements of things like the proper motion of Sgr A* (in the center of the Galaxy) using VLBA astrometry, is yielding a ΘLSR that is creeping up to about 240 km/s.

• A particular nearby star has, then, a peculiar velocity with respect to the LSR as:

• Note that the Sun has a peculiar velocity with respect to the LSR.

It is not firmly established, and there are various ways to define it (see a great and lengthy description of this in Chapter 6 of Mihalas & Binney).

The basic solar motion discussed there is (u, v, w ) = (-9, 11, 6) km/s in a LEFT-HANDED system, so (u, v, w ) = (9, 11, 6) km/s in a RIGHT-HANDED system.

This is a commonly used (though now outdated) value for the solar peculiar velocity.

!! However, a recent reanalysis by Schoenrich, Binney and Dehnen (2010, MNRAS, 403, 1829) yields (in right-handed units):

(U,V,W)sun = (11.1 +/- 0.74, 12.24 +/- 0.47, 7.25 +/-0.37) km/s.

Obviously all of these values suggest that the Sun is actually moving faster than the LSR.

• Note that the velocity of any star with respect to the Sun is the difference in peculiar velocities:

• Of course, all of these definitions become more complex for stars not in the solar neighborhood, because of differences in the angles with respect to the Galactic Center and because of differential rotation of the Galaxy.

### SUPERGALACTIC COORDINATE SYSTEM

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).

• This plane is almost perpendicular to the Galactic plane:

The north supergalactic pole (SGB = 90o) is at (l , b ) = (47.37, +6.32)o).

• The zero point (SGL, SGB ) = (0,0) lies at (l , b ) = (137.37, 0)o.

• This website has a supergalactic coordinate conversion program.

### PRECESSION OF THE EQUINOXES

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.

CAUSE:

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

• The Earth is 43 km larger in diameter across the equator than from pole to pole (a radius of 6378 km toward the equator compared to 6357 km toward the poles).

• Being 0.33% closer to the Earth's center at the pole, translates to 0.67% greater weights measured on the surface of the Earth at the poles than at the equator.

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).

• Thus the Moon generally is at an angle to the equatorial bulge, and tugs on the Earth's bulge.

##### From Abell's Exploration of the Universe, Ed. 3.

• There are also smaller contributions from the Sun and planets attempting gravitationally to do the same thing (the Sun is only on the celestial equator twice a year and at all other times of the year it is pulling the Earth's bulge toward the ecliptic plane).

• These external forces on the spinning Earth creates the precessional "wobble" in the Earth's motion.

EFFECTS:

• For the Earth, the precession acts to slowly change the direction that the Earth's rotational pole points.

• The direction of the Earth's North and South Celestial Poles rotate to different points on the Celestial Sphere with a 25,800 year cycle.

• The orbital axis of the Earth stays fixed in space but the rotational axis constantly changes direction.

• This means that the Ecliptic Poles are always in the same place (the North Ecliptic Pole is in Draco), but the North and South Celestial Poles are circling around the Ecliptic Poles.

• ##### The North Celestial Pole is circling counterclockwise around the North Ecliptic Pole. From Kaler's The Ever-Changing Sky.

• Presently the Earth's North Pole points to Polaris, but 14,000 years ago it pointed towards Vega. Other North "pole stars" in the 25,800 year cycle are shown below.

• The star gamma Cephei is the next northern pole star (it will be 3 degrees from the NCP in 2200 years).

• Note that it is not the location of the rotational pole on the Earth that is changing, but where that pole points on the Celestial Sphere!

• Note also that if the direction of the poles is changing, so too is the direction of the equator of the Earth as projected on the sky.

• This means that their are apparent "flows" of stars on the celestial sphere as induced by the precessional wobble.

• Just as stars appear to rotate above us is a reflection of the Earth's rotation, the wobble of the Earth is reflected in the changing positions of stars on our celestial sphere.

• The optical illusion thus makes stars appear to drift past the the North Celestial Pole as if they were on a plate turning about a point at the North Ecliptic Pole.

• The effect is most noticeable at the celestial poles, and the direction and the speed of the "flow" depends on where you are looking on the celestial sphere.

• If the position of the celestial poles and equators are changing on the celestial sphere, this means that the celestial coordinates of objects, which are defined by reference to the celestial equator and celestial poles, must also be constantly changing.

• Because of this change in the direction of the Earth's pole with time, the coordinate systems of RA and DEC that we adopt for one epoch are actually different for other epochs!

• The effects are quite noticeable, almost an arcminute a year along the ecliptic.

• Thus, it is proper (and imperative!) that when an astronomer gives the coordinates of an object she specifies the year that corresponds to those coordinates (because they will be significantly different in future years).

• This specified year for the coordinates is called the EQUINOX of the coordinates.

• NOTE: A common mistake made by even senior astronomers is to call the year of the coordinates the "epoch" of the coordinates.

THIS IS WRONG. An epoch specified with coordinates means something completely different (see proper motions below). DO NOT GET IN THE HABIT OF MAKING THIS MISTAKE!

• Astronomers tend to use "standard" years, like 1950, 2000, 2050 when they cite the Equinox of the coordinates.

Presently we see most people using "J2000.0" coordinates (e.g., the book Norton's 2000.0).

• Coming to a telescope with coordinates precessed to the wrong year is one of the most common mistakes by observers.

A mistake of 50 years in the coordinate system (most typical) will general move your object of interest off a typical CCD field of view.

• Because the plane of the Earth's orbit is fixed, the position of the ecliptic is fixed.
• But since the position of the Celestial Equator is changing, the position of the Vernal and Autumnal Equinoctes (where the Celestial Equator and the ecliptic cross) slowly shifts with time.
• ##### From Kaler's The Ever-Changing Sky.

• In the Figure above, if the NCP is coming at you, the front side of the Celestial Equator is going down and the back side of the Celestial Equator is going up.

• This means that the positions of the Vernal Equinox is sliding to the left (or, to the right from the Earth's point of view).

• Thus, the motion of the equinoctes is westward along the ecliptic because of the motion of the equator.

• Since in a 25,800 year period the Vernal Equinox will slide 360 degrees, we have that the annual motion of the Vernal Equinox (and Autumnal Equinox) is 360o/(25800 yrs) = 50.3"/yr.

• Notice how in the figure the Celestial Coordinates of the stars Deneb and Achernar are constantly changing, but the Ecliptic Coordinates of these stars remain fixed.

• KEEP THESE IDEAS CLEAR:

• From our point of view at a given latitude on Earth, the direction towards the equator, poles, and ecliptic maintain their same orientation to a given horizon every year.

It is the stars that appear to move east along the ecliptic past the equinox and solstice points.

• Since the Vernal Equinox is slipping, the dates when the Sun is in a given constellation slowly changes.

• This is why the months associated with certain "signs of the zodiac" do not match with the Sun's true position with respect to them, which is how the dates of the "houses" were originally defined.

• Another effect of precession is to complicate the definition of a year.

• A sidereal year is the time between the Sun appearing across a given star = 365.2564 days.

• A tropical year is the time between successive Vernal Equinoctes = 365.2422 days.

• The difference is because the Sun is every day moving eastward along the ecliptic while the Vernal Equinox is slipping westward.

• Thus, the Sun has less than 360 degrees to move to go from one Vernal Equinox to the next, because the VE is moving towards the Sun.
• The tropical year is 20 minutes shorter than a sidereal year because it takes 20 minutes for the Sun to move 50.3 arcsecs.

##### From Kaler's The Ever-Changing Sky.

• The 50.3"/yr precession discussed above is actually the total of all precessional affects and is called the general precession.

• The effect of lunar-solar precession is actually a westward motion of the equinoctes by 50.4"/yr on a 25,800 year period.

• The effect of planetary precession (combined gravity of all planets) is actually an eastward motion of the equinoctes by 0.1"/yr.

Planetary precession also acts to change the obliquity of the ecliptic of the Earth (between 21.5 and 24.5 degrees) over a 41,000 year cycle.

Currently the obliquity of the ecliptic is being reduced by 0.5"/yr.

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.

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.