Coordinate Systems

The sky is a sphere, and we need ways to specify locations on it. This guide explains the coordinate systems astronomers use and how to transform between them.

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The Celestial Sphere

Imagine extending Earth's axis into space. The points where this axis intersects the celestial sphere are the celestial poles. The extension of Earth's equator is the celestial equator.

This conceptual sphere, centered on the observer (or Earth), is called the celestial sphere. All coordinate systems are ways of locating points on this sphere.

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ICRS (Equatorial Coordinates)

The International Celestial Reference System (ICRS) is the modern standard equatorial coordinate system.

The Two Coordinates

Right Ascension (RA or α) — The "longitude" of the sky

• Measured in hours, minutes, seconds (0h to 24h)
• Or in degrees (0° to 360°)
• Increases eastward from the vernal equinox

Declination (Dec or δ) — The "latitude" of the sky

• Measured in degrees, arcminutes, arcseconds (-90° to +90°)
• +90° is the North Celestial Pole
• -90° is the South Celestial Pole
• 0° is the celestial equator

The Vernal Equinox

The zero point of Right Ascension is where the Sun crosses the celestial equator heading north—the vernal equinox (also called the First Point of Aries, though it's now in Pisces due to precession).

Example: Polaris

The North Star has coordinates approximately:

• RA: 02h 31m 49s
• Dec: +89° 15' 51"

It's less than 1° from the North Celestial Pole.

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Parsing Coordinates in starward

starward understands multiple input formats:

# HMS/DMS format
starward coords parse "12h30m45.2s +45d15m30s"

# Colon format
starward coords parse "12:30:45.2 +45:15:30"

# Decimal degrees
starward coords parse "187.6883 45.2583"

# Mixed
starward coords parse "12h30m +45.5"

The parser is flexible—it handles signs, optional seconds, and various separators.

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Galactic Coordinates

The Galactic coordinate system is centered on the Sun, with the fundamental plane being the Milky Way's disk.

The Two Coordinates

Galactic Longitude (l) — Angle around the Galaxy

• Measured in degrees (0° to 360°)
• 0° points toward the Galactic Center
• 90° is the direction of Galactic rotation
• 180° is the anti-center

Galactic Latitude (b) — Angle above/below the plane

• Measured in degrees (-90° to +90°)
• +90° is the North Galactic Pole
• 0° is in the Galactic plane

Why Use Galactic Coordinates?

Galactic coordinates reveal the structure of the Milky Way:

• Objects with b ≈ 0° are in the Galactic plane
• Objects with |b| > 30° are "high-latitude" (away from dust)
• l tells you which direction in the Galaxy

Example: The Galactic Center is at l = 0°, b = 0° (by definition).

The Galactic Poles in ICRS

Reference Point	RA (ICRS)	Dec (ICRS)
North Galactic Pole	12h 51m 26s	+27° 07' 42"
Galactic Center	17h 45m 40s	-29° 00' 28"

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Transforming Between ICRS and Galactic

starward coords transform "12h30m +45d" --to galactic

Output:

Input (ICRS): 12h30m +45d
─────────────────────────────────────────────
Output (Galactic):
  l:  135.023680°
  b:  71.621544°

This object is at high Galactic latitude (b ≈ 72°), far from the Galactic plane.

The Transformation Math

The ICRS ↔ Galactic transformation uses spherical trigonometry. The key parameters are:

Constant	Value	Meaning
α_NGP	192.8595°	RA of North Galactic Pole
δ_NGP	27.1283°	Dec of North Galactic Pole
l_NCP	122.932°	Galactic longitude of North Celestial Pole

ICRS → Galactic:

$\sin b = \sin \delta \sin \delta_{NGP} + \cos \delta \cos \delta_{NGP} \cos(\alpha - \alpha_{NGP})$

$\tan(l_{NCP} - l) = \frac{\cos \delta \sin(\alpha - \alpha_{NGP})}{\sin \delta \cos \delta_{NGP} - \cos \delta \sin \delta_{NGP} \cos(\alpha - \alpha_{NGP})}$

To see this calculation step by step:

starward --verbose coords transform "12h30m +45d" --to galactic

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Horizontal Coordinates (Alt/Az)

Horizontal coordinates are local to an observer. They depend on:

• Your location (latitude, longitude)
• The time of observation

The Two Coordinates

Altitude (alt) — Angle above the horizon

• 0° = on the horizon
• +90° = directly overhead (zenith)
• Negative = below the horizon

Azimuth (az) — Compass direction

• 0° = North
• 90° = East
• 180° = South
• 270° = West

Related Quantities

Zenith Distance: z = 90° - altitude

Airmass: How much atmosphere light traverses. starward uses the Pickering (2002) formula:

$X = \frac{1}{\sin(alt + \frac{244}{165 + 47 \times alt^{1.1}})}$

This is accurate even near the horizon.

Transforming to Horizontal Coordinates

starward coords transform "12h30m +45d" --to altaz --lat 34.05 --lon -118.25

This shows where the object appears in the sky from Los Angeles right now.

Output:

Input (ICRS): 12h30m +45d
─────────────────────────────────────────────
Output (Horizontal):
  Altitude:   45.234567°
  Azimuth:    287.123456°
  Zenith:     44.765433°
  Airmass:    1.41

Interpretation: The object is about 45° up, in the WNW, and you're looking through 1.4× the minimum atmosphere.

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The Horizontal Transformation

Converting ICRS → Horizontal requires:

1. Calculate Local Sidereal Time (LST) from the current time and longitude
2. Calculate Hour Angle: HA = LST - RA
3. Apply spherical trigonometry for the observer's latitude

Hour Angle is how far west an object is from the meridian:

• HA = 0h → on the meridian (highest point)
• HA > 0 → west of meridian (setting)
• HA < 0 → east of meridian (rising)

The formulas:

$\sin(alt) = \sin(\delta) \sin(\phi) + \cos(\delta) \cos(\phi) \cos(HA)$

$\tan(az) = \frac{-\sin(HA)}{\tan(\delta) \cos(\phi) - \sin(\phi) \cos(HA)}$

Where φ is the observer's latitude.

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Coordinate Format Reference

Input Formats Accepted

Format	Example	Notes
HMS/DMS	12h30m45s +45d15m30s	Sexagesimal
HMS/DMS	12h30m45.2s -45d15m30.5s	With decimals
Colon	12:30:45 +45:15:30	Common in catalogs
Decimal	187.5 45.25	Degrees
Mixed	12h30m +45.5	RA in hours, Dec in degrees
Galactic	l=135.0 b=71.6	For --from galactic

Output Formats

Results are shown in the most appropriate format:

• ICRS: RA in HMS, Dec in DMS
• Galactic: l and b in degrees
• Horizontal: Alt and Az in degrees

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Practical Examples

"Is M31 up tonight?"

Find M31's current altitude from your location:

starward coords transform "00h42m44s +41d16m09s" --to altaz --lat 40.7 --lon -74.0

If altitude > 0, it's above the horizon!

"Where is the Galactic Center?"

starward coords transform "l=0 b=0" --from galactic --to icrs

Output: RA ≈ 17h45m40s, Dec ≈ -29°00'

"How high does Vega get from my latitude?"

An object's maximum altitude equals 90° - |latitude - declination|.

Vega: Dec ≈ +38.8° From latitude +40°: max alt ≈ 90° - |40 - 38.8| ≈ 88.8° (nearly overhead!) From latitude -40°: max alt ≈ 90° - |-40 - 38.8| ≈ 11.2° (barely above horizon)

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Python API

from starward.core.coords import ICRSCoord, GalacticCoord, HorizontalCoord
from starward.core.angles import Angle

# Create ICRS coordinates
coord = ICRSCoord.from_string("12h30m +45d")
print(f"RA: {coord.ra.to_hms()}")
print(f"Dec: {coord.dec.to_dms()}")

# Transform to Galactic
gal = coord.to_galactic()
print(f"l = {gal.l.degrees:.4f}°")
print(f"b = {gal.b.degrees:.4f}°")

# Transform to Horizontal (needs location and time)
from starward.core.time import jd_now
jd = jd_now()
horiz = coord.to_horizontal(
    latitude=Angle(degrees=34.05),
    longitude=Angle(degrees=-118.25),
    jd=jd
)
print(f"Alt: {horiz.alt.degrees:.2f}°")
print(f"Az: {horiz.az.degrees:.2f}°")
print(f"Airmass: {horiz.airmass:.2f}")

# Create from Galactic and convert to ICRS
gal = GalacticCoord.from_degrees(l=0, b=0)  # Galactic Center
icrs = gal.to_icrs()

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Summary Table

System	Coordinates	Reference	Use Case
ICRS	RA, Dec	Vernal equinox, celestial equator	Standard catalog positions
Galactic	l, b	Galactic center, Galactic plane	Milky Way structure
Horizontal	Alt, Az	Local horizon, North	Observing, telescope pointing

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Further Reading

• Smart, W.M. "Textbook on Spherical Astronomy" — The classic reference
• Meeus, J. "Astronomical Algorithms" — Chapters 12-13 (Coordinate transformations)
• The Hipparcos and Tycho Catalogues (ESA SP-1200) — ICRS definition

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Next: Angular Calculations — Measure distances on the sky