Position Angle in the Sky: North, East and Spherical Calculations

What does position angle mean in astronomy?

The reference point matters. At a normal point on the celestial sphere, imagine a small tangent plane touching the sky there. Mark increasing declination as north and increasing right ascension as east. The initial tangent of the shorter great-circle arc toward the second direction defines the bearing in that plane.

Astropy documents the east-of-north convention. The calculations below use invented directions in a fixed equatorial frame. They are geometry examples, not dated positions of planets, observing forecasts or measurements from a telescope.

Position angle, separation, azimuth and image angle

A companion can have the same position angle at several different separations. Conversely, many positions at the same separation surround a reference direction. Two numbers—bearing and separation—are needed to specify an offset, along with the original position and the coordinate convention.

On a north-up celestial image with east to the left, increasing position angle runs from the top toward the left. A rotated or mirrored image can look different. Do not infer celestial east from the right-hand side of a photograph, or treat a pixel slope as a calibrated sky bearing without the image's coordinate mapping.

Local altitude and azimuth require an observer and a time. Equatorial position angle uses the local north direction of the chosen celestial coordinate grid. “Local” here refers to the tangent plane at the reference position, not automatically to the observer's terrestrial horizon.

Put both positions in the same frame and units

Write the reference as right ascension α₁ and declination δ₁, and the target as α₂, δ₂. Match the coordinate frame and all relevant epoch or observation conventions before comparing them. Mixing a catalogue direction with an apparent direction from another system can produce a numerically tidy answer to an inconsistent question.

Right ascension may be supplied in hours. Convert it to degrees using 15° per hour before using the degree examples here. For trigonometric functions expecting radians, multiply degrees by π/180. Declination retains its north-positive sign; a negative declination is not a negative right ascension.

A pair straddling 0° RA is still close in longitude. The sine and cosine of Δα=α₂−α₁ handle the wrap automatically. For example, 359° to 1° has the same trigonometric difference as +2°, even if raw subtraction produces −358°. Keep units and signs explicit rather than repairing surprising results afterward.

Calculate east and north components before using atan2

With Δα=α₂−α₁, calculate E = cos δ₂ sin Δα and N = cos δ₁ sin δ₂ − sin δ₁ cos δ₂ cos Δα. Then P = wrap360(atan2(E,N)), converting the result to degrees if the function returns radians. wrap360 means a value in 0 ≤ P < 360.

The order is deliberate: most programming libraries write atan2(y,x), and the arguments here are E first, N second. That measures from north toward east. Using atan2(N,E) instead measures from another axis. Using an ordinary arctangent of E/N loses the quadrant and fails when the denominator is zero.

These components can be derived by projecting the target's unit vector onto the reference point's east and north basis vectors. Their length is sin θ, where θ is the separation between the directions. The bearing is the orientation of that projected vector. This explains both the formula and why a zero-length projection cannot define a bearing.

Seven bearings that reveal common mistakes

The first two rows share right ascension, so the target lies along the reference meridian. Moving from +30° to +50° declination is north, giving 0°; moving to +10° is south, giving 180°. On the equator, 359° to 1° points east and the reverse points west.

The fifth row is more subtle. From RA 150°, declination +30°, toward RA 170°, declination +30°, the initial great-circle bearing is about 84.961631°, not exactly 90°. A path along a constant nonzero declination is a small circle. The shorter great-circle route between its endpoints initially bends toward higher declination.

The last two rows check quadrants: from RA 0°, declination 0°, the target at 90°, +45° gives 45°, while 270°, −45° gives 225°. A one-argument arctangent can make these diagonally opposite orientations look alike. Test more than just a due-north example when auditing a formula.

The inverse problem: choose a bearing and a separation

Suppose the starting position, position angle P and angular distance ρ are known. The target declination satisfies sin δ₂ = sin δ₁ cos ρ + cos δ₁ sin ρ cos P. Take the appropriate latitude value with arcsine, then obtain the right-ascension increment from Δα = atan2(sin P sin ρ cos δ₁, cos ρ − sin δ₁ sin δ₂). Finally wrap α₁+Δα into a full turn.

These are spherical forward-offset equations. For ordinary nonpolar starting points they preserve the requested angular distance, even when a flat map approximation would fail. Tiny floating-point excesses beyond −1 or +1 can be clamped before arcsine, but a grossly invalid input should be investigated rather than hidden by clamping.

Astropy's directional offset operation similarly takes a position angle and separation; its corresponding bearing and separation operations recover the two components. Here the equations are used to construct an independently checkable example, without identifying any real object.

One 20° separation, many different directions

Every row has the same 20° separation from RA 150°, declination +30°. At P=0°, the target is directly north at declination +50°. At P=180°, it is directly south at +10°. Both have RA 150°, yet they are distinct positions on opposite sides of the reference.

At P=90°, the target is approximately RA 172.795877°, declination +28.024321°. It starts in the eastward tangent direction but does not remain at declination +30° after travelling twenty degrees on a great circle. Adding twenty directly to RA would be a different construction and would not satisfy both the specified bearing and distance.

The final 360° row repeats the 0° direction to close the graph. It is not an extra unique bearing. Six decimals make the rounded teaching rows reproducible; they do not imply that an observed object has been measured to that precision.

Why the reverse bearing is not always P+180°

The forward bearing uses north at the first position. The reverse bearing uses north at the second position. Those two tangent bases need not be parallel when compared along a curved sky, so adding 180° to the first number is not a general method for obtaining the second.

Use the 20° eastward example from the graph. The forward angle from (150°, +30°) to approximately (172.795877°, +28.024321°) is 90°. Reversing the unrounded endpoints gives approximately 281.170229°, not 270°. Recompute E and N with the endpoints exchanged.

The separation is still 20° in both directions. Distance is symmetric here; the reported orientation depends on the starting tangent basis. Simple equatorial or common-meridian examples can hide this distinction because their reverse bearings do happen to differ by a half-turn.

Coincident points, opposite points and the celestial poles

If the directions coincide, there is no preferred route from one toward the other: θ=0 and both projected components vanish. If they are exactly opposite, θ=180° and many equally short great-circle semicircles connect them. Again the initial bearing is not unique. A software return of zero for atan2(0,0) would be a convention of that routine, not evidence of a northward direction.

Near these cases, a tiny coordinate error can cause a large change in P. Check the separation and projected-vector length before interpreting many decimals. A threshold must match the precision and purpose of the data; a universal arbitrary cutoff is not a substitute for uncertainty analysis.

At an exact celestial pole, right ascension and the usual local east/north basis are not uniquely fixed by the position alone. A bearing can be reported relative to an explicitly chosen meridian or tangent basis, but that choice must be stated. The ordinary nonpolar examples in this guide avoid that singularity.

A reproducible way to report an orientation

Record which point is first, both coordinates and units, the common frame, the angle convention, the resulting separation and position angle, and any rounding or uncertainty. With an image, also retain its orientation or calibrated coordinate mapping. The phrase “the target is at 45°” is incomplete unless the reader knows whether it means bearing, altitude or distance.

Use the angular-separation guide for distance on the sphere and the coordinate-conversion guide when comparing equatorial and ecliptic quantities. A geometric bearing alone establishes neither an object's visibility above your horizon nor its future motion.

Frequently asked questions