Parallactic Angle Field Rotation

Evaluate parallactic angle and its analytic time derivative from a supplied hour-angle rate; the zenith singularity and exact geographic poles are excluded.

Description

Parallactic angle describes the sky orientation relative to the local vertical.

Parallactic angle describes the sky orientation relative to the local vertical. Its time derivative contributes to field rotation and becomes singular where that orientation is undefined.^[1]

Evaluate parallactic angle and its analytic time derivative from a supplied hour-angle rate; the zenith singularity and exact geographic poles are excluded.

When to use Parallactic Angle Field Rotation

Use this tool to evaluate parallactic angle and its analytic time derivative from a supplied hour-angle rate; the zenith singularity and exact geographic poles are excluded.

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 Parallactic Angle Field Rotation 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 latitudeDegrees, declinationDegrees, hourAngleDegrees, hourAngleRateRadiansPerSecond using the field table and calculation rules, then run the tool.
Read parallacticAngleDegrees, rotationDegreesPerSecond. 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
latitudeDegrees	Latitude degrees	number	Required
declinationDegrees	Declination degrees	number	Required
hourAngleDegrees	Hour angle degrees	number	Required
hourAngleRateRadiansPerSecond	Hour angle rate radians per second	number	Required

Enter latitude strictly within ±90 degrees, declination within ±90, finite hour angle degrees and a signed hour-angle rate in radians/second. The denominator is the sum of squares of atan2 arguments and must be positive. Return signed angle degrees and signed rotation degrees/second.

q=atan2(sinH,tanφ cosδ−sinδ cosH); dq/dt=[tanφ cosδ cosH−sinδ]/denominator × dH/dt

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

Output field	Meaning	Value type
parallacticAngleDegrees	Parallactic angle degrees	number
rotationDegreesPerSecond	Rotation degrees per second	number

json

{
  "input": {
    "latitudeDegrees": 45,
    "declinationDegrees": 0,
    "hourAngleDegrees": 0,
    "hourAngleRateRadiansPerSecond": 0.0001
  },
  "output": {
    "parallacticAngleDegrees": 0,
    "rotationDegreesPerSecond": 0.005729577951308234
  }
}

Limitations and assumptions

Exact geographic poles and an exactly zero orientation denominator are rejected. No observing time, sidereal-rate conversion, refraction or telescope geometry is inferred; near singularities the result can be highly sensitive.

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 Equatorial Horizontal Coordinates: Transform geometric hour angle and declination to altitude and azimuth north-through-east; no refraction or time-to-hour-angle conversion is inferred. For a complementary task, use Declination Altitude Visibility: Compute geometric daily altitude extrema and visibility relative to a supplied horizon boundary, including both circumpolar and never-rising cases; refraction is excluded.

References

Parallactic Angle Field Rotation — tool.io

Similar or alternative tools

Equatorial Horizontal Coordinates
Transform geometric hour angle and declination to altitude and azimuth north-through-east; no refraction or time-to-hour-angle conversion is inferred.
Declination Altitude Visibility
Compute geometric daily altitude extrema and visibility relative to a supplied horizon boundary, including both circumpolar and never-rising cases; refraction is excluded.