State Eccentricity Vector

Derive the eccentricity vector and mathematical conic class from a nonradial two-body Cartesian state; floating values near one require physical uncertainty assessment.

Description

The two-body eccentricity vector points toward periapsis and its magnitude classifies the mathematical conic.

The two-body eccentricity vector points toward periapsis and its magnitude classifies the mathematical conic. Obtaining it from a Cartesian state requires a nonzero radius and a nonradial state with nonzero angular momentum.^[1]

Derive the eccentricity vector and mathematical conic class from a nonradial two-body Cartesian state; floating values near one require physical uncertainty assessment.

When to use State Eccentricity Vector

Use this tool to derive the eccentricity vector and mathematical conic class from a nonradial two-body Cartesian state; floating values near one require physical uncertainty assessment.

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

How to use State Eccentricity Vector and understand its results

Prepare the orbital state or model parameters in one reference frame. Use compatible distance/time units and the stated angular and gravitational-parameter conventions.
Enter position, velocity, mu using the field table and calculation rules, then run the tool.
Read eccentricityVector, eccentricityMagnitude, conicClass. 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
position	Position	list of number	Required
velocity	Velocity	list of number	Required
mu	Gravitational parameter, using the same distance and time units throughout; km³/s² for km and seconds.	number	Required

Enter finite three-component position and velocity vectors in one frame and a positive μ with consistent units. The tool rejects zero position or zero angular-momentum magnitude, evaluates the cross products and returns the three eccentricity components and their magnitude. It labels e<1 elliptic, e>1 hyperbolic and exactly e=1 parabolic, without a numerical tolerance.

h = r×v; e_vector = (v×h)/μ − r/|r|; e = |e_vector|

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

Output field	Meaning	Value type
eccentricityVector	Eccentricity vector	list of number
eccentricityMagnitude	Eccentricity magnitude	number
conicClass	Conic class	string

json

{
  "input": {
    "position": [
      1,
      0,
      0
    ],
    "velocity": [
      0,
      1,
      0
    ],
    "mu": 1
  },
  "output": {
    "eccentricityVector": [
      0,
      0,
      0
    ],
    "eccentricityMagnitude": 0,
    "conicClass": "elliptic"
  }
}

Limitations and assumptions

This is a two-body instantaneous-state calculation, without perturbation propagation or measurement uncertainty. Floating-point results near one need physical interpretation rather than relying on exact equality. Radial states are rejected, and the conic class does not assess collision or surface clearance.

Numeric inputs must be finite. Follow the calculation-specific rules for overflow, underflow and exceptional cases. Numeric lists are limited to 100,000 entries; the rules above may require smaller limits. Missing required fields, extra fields, wrong value types and violations of the calculation rules are input errors.

Alternative or Complementary methods

Use Specific Orbital Energy: Evaluate specific two-body orbital energy v²/2−μ/r in consistent units; this does not determine angular momentum or collision safety. For a complementary task, use Tisserand Orbital Parameter: Compute a circular restricted-three-body Tisserand parameter in the supplied planet frame; no universal comet taxonomy cutoff is embedded.

References

State Eccentricity Vector — tool.io

Similar or alternative tools

Specific Orbital Energy
Evaluate specific two-body orbital energy v²/2−μ/r in consistent units; this does not determine angular momentum or collision safety.
Tisserand Orbital Parameter
Compute a circular restricted-three-body Tisserand parameter in the supplied planet frame; no universal comet taxonomy cutoff is embedded.