Angular Calculations

Angles are the fundamental unit of positional astronomy. This guide covers how to work with angles in starward, calculate separations between objects, and find position angles.

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Understanding Astronomical Angles

Units of Angular Measure

Astronomers use several units depending on the scale:

Unit	Symbol	Size	Common Use
Degrees	°	Full circle = 360°	Large angular distances
Arcminutes	' or arcmin	1° = 60'	Galaxy sizes, Moon's diameter
Arcseconds	" or arcsec	1' = 60"	Star positions, seeing
Hours	h	Full circle = 24h	Right Ascension
Radians	rad	Full circle = 2π	Mathematical calculations

Why Hours for Right Ascension?

Right Ascension is traditionally measured in hours because the celestial sphere appears to rotate once in 24 hours. If you watch a star, 1 hour later it will have moved 15° westward (360°/24h = 15°/h).

Conversion: 1h = 15°, so 1m (time) = 15' and 1s (time) = 15"

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The Angle Class

In starward, the Angle class handles all angle representations and conversions.

Creating Angles

# In the CLI, convert between formats
starward angles convert 45.5 --from deg
starward angles convert 3.0333 --from hours

Sexagesimal Notation

Degrees-Minutes-Seconds (DMS) for declination and angular distances:

• Format: ±DD° MM' SS.ss"
• Example: +45° 30' 15.5" = 45.504306°

Hours-Minutes-Seconds (HMS) for Right Ascension:

• Format: HHh MMm SS.ss
• Example: 12h 30m 45s = 187.6875°

Parsing Angles

starward's parser handles many formats:

starward coords parse "45d30m15.5s"    # DMS
starward coords parse "12h30m45s"      # HMS
starward coords parse "45:30:15.5"     # Colon notation
starward coords parse "45.504306"      # Decimal degrees
starward coords parse "+45 30 15.5"    # Space-separated

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Angular Separation

What Is Angular Separation?

The angular separation between two points on the celestial sphere is the angle between them as seen from the center—the shortest path along a great circle.

Think of it as "how far apart do these two objects appear in the sky?"

Calculating Separation

starward angles sep "10h30m +30d" "10h35m +31d"

Output:

Point 1: 10ʰ 30ᵐ 00.00ˢ 30° 00′ 00.00″
Point 2: 10ʰ 35ᵐ 00.00ˢ 31° 00′ 00.00″
─────────────────────────────────────────────

Angular Separation:
  1° 17′ 12.34″
  = 1.28676°
  = 77.206′
  = 4632.34″

The Vincenty Formula

For small separations, simple formulas work. But for accurate results at any distance, starward uses the Vincenty formula:

$\sigma = \arctan\left(\frac{\sqrt{(\cos\phi_2\sin\Delta\lambda)^2 + (\cos\phi_1\sin\phi_2 - \sin\phi_1\cos\phi_2\cos\Delta\lambda)^2}}{\sin\phi_1\sin\phi_2 + \cos\phi_1\cos\phi_2\cos\Delta\lambda}\right)$

Where:

• $\phi_1, \phi_2$ = declinations of the two points
• $\Delta\lambda$ = difference in Right Ascension

This formula is numerically stable for any separation, from 0° to 180°.

See the Math

starward --verbose angles sep "10h +30d" "11h +31d"

┌─ Input coordinates
│  Point 1: RA = 10ʰ 00ᵐ 00.00ˢ, Dec = 30° 00′ 00.00″
│  Point 2: RA = 11ʰ 00ᵐ 00.00ˢ, Dec = 31° 00′ 00.00″
└────────────────────────────────────────
┌─ RA difference
│  Δλ = 15.000000°
└────────────────────────────────────────
┌─ Trigonometric values
│  sin(φ₁) = 0.5000000000, cos(φ₁) = 0.8660254038
│  sin(φ₂) = 0.5150380749, cos(φ₂) = 0.8571673007
│  sin(Δλ) = 0.2588190451, cos(Δλ) = 0.9659258263
└────────────────────────────────────────
┌─ Vincenty formula
│  numerator = √[(cos φ₂ sin Δλ)² + (cos φ₁ sin φ₂ − sin φ₁ cos φ₂ cos Δλ)²]
│            = √[0.2218512223² + 0.0320560402²]
│            = 0.2241552019
│  
│  denominator = sin φ₁ sin φ₂ + cos φ₁ cos φ₂ cos Δλ
│              = 0.9745534595
└────────────────────────────────────────
┌─ Result
│  σ = atan2(0.2241552019, 0.9745534595)
│    = 12.9532°
│    = 12° 57′ 11.54″
└────────────────────────────────────────

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Position Angle

What Is Position Angle?

The position angle (PA) describes the direction from one object to another on the celestial sphere. It's measured:

• From North (0°)
• Through East (90°)
• Increasing counterclockwise as seen on the sky

Position angles are important for:

• Binary star observations
• Jet directions in AGN
• Proper motion directions
• Extended source orientations

Calculating Position Angle

starward angles pa "10h30m +30d" "10h35m +31d"

Output:

Position Angle: 32.45° (N through E)

This means: starting from Point 1, Point 2 is roughly northeast.

The Formula

$PA = \arctan\left(\frac{\sin(\alpha_2 - \alpha_1)}{\cos\delta_1\tan\delta_2 - \sin\delta_1\cos(\alpha_2 - \alpha_1)}\right)$

The result is normalized to [0°, 360°).

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Angle Conversions

Using the CLI

# Degrees to all formats
starward angles convert 45.5 --from deg

# Hours to degrees
starward angles convert 12.5 --from hours

# Radians to degrees
starward angles convert 0.7854 --from rad

Conversion Table

From	To Degrees
1 hour	15°
1 minute (time)	0.25° = 15'
1 second (time)	0.004167° = 15"
1 radian	57.2958°
1 arcminute	0.01667°
1 arcsecond	0.000278°

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Working with Small Angles

For very small angles, the small-angle approximation is useful:

$\sin\theta \approx \theta \quad \text{and} \quad \tan\theta \approx \theta \quad \text{(in radians)}$

This is accurate to 1% for angles < 14° and to 0.1% for angles < 4.5°.

Arcseconds Per Radian

One of our constants:

$\frac{180° \times 3600"}{\pi} = 206264.806..."$

This is useful for converting small angles:

$\theta_{arcsec} = \theta_{rad} \times 206264.8$

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Common Angular Scales

Object	Angular Size
Moon / Sun	~30' = 0.5°
Jupiter	30-50"
Mars (at opposition)	~25"
Hubble resolution	~0.05"
Ground-based seeing	0.5-2"
Full sky (hemisphere)	20,626 square degrees

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Practical Examples

How far apart are the Pointer Stars?

Dubhe and Merak in the Big Dipper "point" to Polaris.

• Dubhe: 11h 03m 43s, +61° 45' 03"
• Merak: 11h 01m 50s, +56° 22' 56"

starward angles sep "11h03m43s +61d45m03s" "11h01m50s +56d22m56s"

Answer: About 5.4° apart.

What's the position angle from Albireo A to B?

Albireo is a famous double star. PA tells you the orientation.

starward angles pa "19h30m43.3s +27d57m34.8s" "19h30m45.4s +27d57m54.9s"

Angular size of an object

If you know an object's physical size (d) and distance (D):

$\theta = \arctan\left(\frac{d}{D}\right) \approx \frac{d}{D} \text{ (radians, for small angles)}$

Example: The Moon is 3,474 km diameter at 384,400 km distance: $\theta = \frac{3474}{384400} = 0.00904 \text{ rad} = 0.518° = 31'$

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Python API

from starward.core.angles import Angle, angular_separation, position_angle

# Create angles various ways
a1 = Angle(degrees=45.5)
a2 = Angle(hours=3.0333)
a3 = Angle(radians=0.7854)
a4 = Angle.from_dms(45, 30, 15.5)
a5 = Angle.from_hms(12, 30, 45)
a6 = Angle.parse("45d30m15.5s")

# Access in different units
print(f"{a1.degrees}° = {a1.hours}h = {a1.radians} rad")
print(f"{a1.arcminutes}' = {a1.arcseconds}\"")

# Format nicely
print(a4.to_dms())  # "45° 30′ 15.50″"
print(a5.to_hms())  # "12ʰ 30ᵐ 45.00ˢ"

# Arithmetic
total = a1 + a2
diff = a1 - a2
scaled = a1 * 2
half = a1 / 2

# Normalize to a range
normalized = a1.normalize(center=180)  # [0, 360)

# Trigonometry
print(a1.sin(), a1.cos(), a1.tan())

# Angular separation
from starward.core.coords import ICRSCoord
c1 = ICRSCoord.from_string("10h30m +30d")
c2 = ICRSCoord.from_string("10h35m +31d")
sep = angular_separation(c1.ra, c1.dec, c2.ra, c2.dec)
print(f"Separation: {sep.to_dms()}")

# Position angle
pa = position_angle(c1.ra, c1.dec, c2.ra, c2.dec)
print(f"PA: {pa.degrees}°")

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Key Formulas Reference

Calculation	Formula
Hours ↔ Degrees	$h \times 15 = °$
DMS → Decimal	$° + '/60 + "/3600$
Small angle (rad)	$\theta_{arcsec} / 206264.8$
Vincenty	See section above
Position angle	See section above

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Further Reading

• Green, R.M. "Spherical Astronomy" — Comprehensive treatment
• Meeus, J. "Astronomical Algorithms" — Chapter 17 (Angular separation)
• "The Observer's Handbook" — Practical observing information

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Next: Astronomical Constants — The numbers behind it all