Astronomical Coordinates: Altitude-Azimuth (AltAz)

Zenith: straight up!

Meridian: N/S line going through the zenith

Altitude: height above the horizon

Zenith angle: 90-Altitude

Azimuth: where great circle connecting star and zenith touches horizon, measured N through E.

Airmass or secz: another measure of altitude is Airmass, which measures path-length through the atmosphere.

For z<60, Airmass=secant(Zenith angle). Rotation of the Earth

Stars rise and set, over a 12 hour period. (Thought question: In the northern hemisphere, stars rise in the East and set in the West. What about in the southern hemisphere?)

Transit: When a star crosses the meridian, it is at its highest elevation.

Hour angle: How many hours since an object transited. e.g., HA = -2hrs means it is rising and will transit in 2 hours.

So we can think of coordinates on the sky in terms of angles, or time.

We want a coordinate system where the position of objects can be defined without respect to a specific time and place.

Astronomical Coordinates: Equatorial Coordinates (RA/Dec)

Define coordinates by the projection of the Earth's pole and equator onto the celestial sphere.

North celestial pole: projection of Earth's north pole Celestial equator: projection of Earth's equator

Declination (δ): angular distance from the celestial equator (+=north, -=south)

Right Ascension (α): angular distance along circles parallel to the equator. Define zero point to be the vernal equinox, the point where the Sun's position in the sky crosses the celestial equator as it moves north. Right ascension increases going eastward.

The Local Sky

A few orientation questions

• What is the declination of an object at zenith at the North Pole? • What is the declination of an object at zenith at the Equator? • What is the declination of an object at zenith in Cleveland? (Latitude: 41o N) • What is the altitude of the giant elliptical galaxy M87 (declination: +12o) when it transits in Cleveland? • What declination defines a circumpolar star in Cleveland? Astronomical Coordinates: Equatorial Coordinates (RA/Dec)

Declination is measured in degrees, minutes of arc, seconds of arc, or decimal degrees.

Right ascension is measured in either sexagesimal time (hr, min of time, sec of time), or in decimal time , or in decimal degrees.

Minutes and seconds of time are DIFFERENT from minutes and seconds of arc!

So, the coordinates for the galaxy M87 can be written as

(α,δ) = 12:30:49.0, +12:23:07 (or 12h30m49s, +12°23’07”) or (α,δ) = 12.51361h, +12.39619° or (α,δ) = 187.7042°, +12.3962°

How many digits of accuracy to use? 1” is roughly ground-based seeing. • 1” = 1/3600° ≈ 0.0003° (so use 4 decimal points for degrees) • 1” = 1/15 seconds of time ≈ 0.07s (use a decimal point for seconds of time but not for seconds of arc) Projections

When looking at a flat image, remember you are looking at a curved surface projected onto a plane. Distortions abound!

For small fields of view, a common projection system is gnomic or tangent plane. As you move off-center, equidistant points begin to “stretch” out. Projections

When looking at a flat image, remember you are looking at a curved surface projected onto a plane. Distortions abound!

For all sky maps (surveys, CMB, etc) many systems exist. Example: Mollweide

CMB

Earth Coordinates and angular separations

Remember: coordinate distances are not angular separations! For two objects of position (α1,δ1) and (α2,δ2), what is their angular separation Δ(θ)?

For small separations (where tan(θ)≈θ), we can say

Δ"(°) = "' − ") Δ*(°) = *' − *) × cos δ ç the dreaded “cos-dec term”

(remember, if you have right ascension measured in hours, you need to multiply by 15 to RA separations in degrees.)

Then we can treat it as cartesian: Δ/ ° = Δ*) + Δ")

But as separations get large, this rapidly becomes a bad approximation.

Note: astropy has routines that automatically calculate true angular separations given two coordinates. Computational asides

1. Remember that any time you see a “naked angle” (i.e., one outside of a trigonometric function) in an equation, that angle has units of radians unless explicitly stated otherwise.

So in the expression tan(θ) ≈ θ, we must be using radians to make this work.

2. Remember that in most computer languages, the trig functions default to the assumption that angles are given in radians, not degrees. ALWAYS CHECK.

3. Also, in most computer languages, log = natural log and log10 = base ten log. So,

WRONG WAY: m-M = 5*np.log(d)-5 RIGHT WAY: m-M = 5*np.log10(d)-5 Making your way around an astronomical image

North Orientation: unless specified otherwise, standard orientation is north up and east to the left. This is flipped from terrestrial maps.

Note also that you East can see the cos-dec term at work on this image! Solid angle

Finally, solid angle is a measure of angular area on the sky. It relates to angle the way area relates to length.

Think of a patch of area on a sphere. If that patch has an area !, the solid angle Ω it corresponds to is defined by

! Ω = $%

Units: • Steradians (4' steradians = whole sky) • Square degrees (41253 sq deg = whole sky) • Square arcsec Coordinate Epochs

The equatorial coordinate system is tied to the Earth's rotational axis. But the Earth's axis shifts with time, due to precession, over a periodic cycle of 25,800 years. This means coordinates are constantly changing!

Rate of change: 360o/25800yr = 0.14o/yr = 50"/yr. Over 50 yrs, this is about 42', > half a degree!

So every coordinate must include an epoch. "B1950" refers to coordinates based on the 1950 pole position; "J2000" refers to coordinates based on the 2000 pole position. And for precise coordinate, you must correct your coordinates to be accurate to the present day. Galactic Coordinate System The equatorial system is very useful for the mechanics of observing. But its physical meaning is tied to the Earth. What about a galactic coordinate system?

ℓ: galactic longitude b: galactic latitude

Galactic center: ℓ=0, b=0

Direction of motion: ℓ =90, b=0

The Earth's axis is tipped from the galactic plane by about 80 degrees or so, so the equatorial and galactic coordinate systems are nearly at right angles to one another. Time: Sidereal versus Solar

The Earth is spinning on its axis and orbiting the Sun. This means that a solar day (defined as noon-to-noon) is different from a sidereal day (defined as one Earth rotation).

Mean Solar day: 24hrs Sidereal day: 23hrs, 56min

This means that a fixed star rises 4 mins earlier each successive night, or two hours earlier each month.

We define the Local Sidereal Time to be the RA which is currently transiting. Depends on current time and location (CLE longitude: 81.6944deg)

Now see how LST, RA, and HA fit together: HA = LST - RA Time: Julian Dates

We want a date system which just counts days (not day/month/year/leapdays, etc). The Julian Date system does this.

• Measures days elapsed since Jan 1, 4713BC. (Why? sigh....). • Days begin at noon, not midnight. • Modified Julian Date (MJD) = JD - 2,400,000.5 (days since Nov 17 1858)

AGN variability plot: