Hyperbolic Anomaly Chain

Convert a physically accessible signed true anomaly to hyperbolic anomaly H and mean anomaly e·sinh(H)−H, both dimensionless.

Description

Hyperbolic true anomaly is bounded by the orbit's asymptotes.

Hyperbolic true anomaly is bounded by the orbit's asymptotes. Converting an accessible signed anomaly yields hyperbolic anomaly and mean anomaly, with signs distinguishing the sides of periapsis.^[1]

Convert a physically accessible signed true anomaly to hyperbolic anomaly H and mean anomaly e·sinh(H)−H, both dimensionless.

When to use Hyperbolic Anomaly Chain

Use this tool to convert a physically accessible signed true anomaly to hyperbolic anomaly H and mean anomaly e·sinh(H)−H, both dimensionless.

Prepare the orbital state or model parameters in one reference frame. Confirm compatible distance/time units and the stated angular and gravitational-parameter conventions.

How to use Hyperbolic Anomaly Chain and understand its results

Prepare the orbital state or model parameters in one reference frame. Confirm compatible distance/time units and the stated angular and gravitational-parameter conventions.
Enter orbitalEccentricity, trueAnomalyDegrees using the field table and calculation rules, then run the tool.
Read hyperbolicAnomaly, hyperbolicMeanAnomaly. 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
orbitalEccentricity	Orbital eccentricity	number	Required
trueAnomalyDegrees	True anomaly degrees	number	Required

Enter eccentricity strictly above one and true anomaly strictly between −180 and 180 degrees. The computed atanh argument must have absolute value below one, excluding asymptotes and inaccessible positions. Return H and hyperbolic mean anomaly as dimensionless quantities; periapsis angle zero gives both zero.

H = 2atanh[sqrt((e−1)/(e+1))tan(ν/2)]; Mh = e sinh(H)−H

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

Output field	Meaning	Value type
hyperbolicAnomaly	Hyperbolic anomaly	number
hyperbolicMeanAnomaly	Hyperbolic mean anomaly	number

json

{
  "input": {
    "orbitalEccentricity": 2,
    "trueAnomalyDegrees": 0
  },
  "output": {
    "hyperbolicAnomaly": 0,
    "hyperbolicMeanAnomaly": 0
  }
}

Limitations and assumptions

This does not return time, an epoch or a velocity. Elliptic/parabolic cases and anomalies at or beyond the hyperbolic asymptote are input errors.

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 Conic Flight Path Angle: Compute the conic flight-path angle with a physical positive-radius anomaly constraint for elliptic, parabolic, or hyperbolic eccentricity. For a complementary task, use Elliptic True To Eccentric Anomaly: Convert elliptic true anomaly to wrapped eccentric and mean anomaly while preserving the correct quadrant.

References

Hyperbolic Anomaly Chain — tool.io

Similar or alternative tools

Conic Flight Path Angle
Compute the conic flight-path angle with a physical positive-radius anomaly constraint for elliptic, parabolic, or hyperbolic eccentricity.
Elliptic True To Eccentric Anomaly
Convert elliptic true anomaly to wrapped eccentric and mean anomaly while preserving the correct quadrant.