Spherical Pair Separation Angle

Compute robust great-circle separation and first-to-second position angle east of north, rejecting directionally undefined coincident/antipodal pairs.

Description

Great-circle separation measures angular distance on the sphere.

Great-circle separation measures angular distance on the sphere. Position angle additionally describes the first-to-second direction east of north and becomes undefined for coincident or antipodal directions.^[1]

Compute robust great-circle separation and first-to-second position angle east of north, rejecting directionally undefined coincident/antipodal pairs.

When to use Spherical Pair Separation Angle

Use this tool to compute robust great-circle separation and first-to-second position angle east of north, rejecting directionally undefined coincident/antipodal pairs.

Prepare the source, detector or coordinate measurements. Confirm frame, epoch, angular units, axis directions and the stated RA proper-motion convention before entering values.

How to use Spherical Pair Separation Angle and understand its results

Prepare the source, detector or coordinate measurements. Confirm frame, epoch, angular units, axis directions and the stated RA proper-motion convention before entering values.
Enter firstRaDegrees, firstDecDegrees, secondRaDegrees, secondDecDegrees using the field table and calculation rules, then run the tool.
Read separationDegrees, positionAngleDegrees. The explanation below defines the units, signs, ordering and any flags or placeholder values.
If the tool reports an input issue, correct the values and run it again. Apply the stated assumptions when interpreting a successfully returned result.

Input field	Meaning	Value type; see rules below	Requirement
firstRaDegrees	First ra degrees	number	Required
firstDecDegrees	First dec degrees	number	Required
secondRaDegrees	Second ra degrees	number	Required
secondDecDegrees	Second dec degrees	number	Required

Enter both RA/Dec pairs in degrees, with each declination within ±90. The tool constructs unit vectors, obtains separation from cross/dot products and calculates the first-to-second bearing. If the cross-product norm is below 10⁻¹⁴, it returns an input error rather than an arbitrary bearing. Separation and wrapped position angle are returned in degrees.

separation=atan2(|u×v|,u·v); positionAngle=atan2(east,north)

Units, indexing, boundary conditions and exceptional cases are defined in the calculation rules above.

Output field	Meaning	Value type
separationDegrees	Separation degrees	number
positionAngleDegrees	Position angle degrees	number

json

{
  "input": {
    "firstRaDegrees": 0,
    "firstDecDegrees": 0,
    "secondRaDegrees": 90,
    "secondDecDegrees": 0
  },
  "output": {
    "separationDegrees": 90,
    "positionAngleDegrees": 90
  }
}

Limitations and assumptions

This does not propagate epochs or identify matched sources. Pairs extremely close to coincidence or opposition are excluded by the numerical threshold; catalog errors are not propagated.

Numeric inputs must be finite. Follow the calculation-specific rules for overflow, underflow and exceptional cases. Missing required fields, extra fields, wrong value types and violations of the calculation rules are input errors.

Alternative or Complementary methods

Use Micrometer Double Star Offsets: Calibrate and rotate micrometer offsets, reporting double-star separation and position angle east of north. For a complementary task, use Convergent Point Motion Alignment: Compare a measured tangent-plane proper-motion direction with the great-circle direction to a supplied cluster apex; alignment does not establish membership.

References

Spherical Pair Separation Angle — tool.io

Similar or alternative tools

Micrometer Double Star Offsets
Calibrate and rotate micrometer offsets, reporting double-star separation and position angle east of north.
Convergent Point Motion Alignment
Compare a measured tangent-plane proper-motion direction with the great-circle direction to a supplied cluster apex; alignment does not establish membership.