Understand the relationship
The reasoning behind the result
One ellipse connects close and far distances
rp=a(1−e); ra=a(1+e); a=(rp+ra)/2; e=(ra−rp)/(ra+rp)
The semimajor axis a is half the longest width of the relative orbit; it is not the instantaneous separation. Eccentricity e controls the shape. The closest and farthest centre-to-centre distances are rp and ra. For e=0 both are a, so the orbit is circular and has no unique periapsis direction.
An altitude is measured from a surface. Add the entered spherical body's radius before using it in an orbital equation. The surface-height mode makes that conversion explicit; a geometric surface-touching result is not a viable orbit. No atmosphere, terrain or body-intersection screening is inferred for other input modes.
Energy and angular momentum determine the changing speed
ε=v²/2−μ/r=−μ/(2a); h=r vt=√(μa(1−e²)); v²=μ(2/r−1/a)
Here μ=G(M+m), r is relative separation and v is relative speed. Specific orbital energy ε has units of m²/s² and is negative for bound motion. Specific angular momentum h has units of m²/s. They are constants of this isolated two-body model even though r and v change around an ellipse.
Radial speed vr measures approach or separation; transverse speed vt is perpendicular to the radius. Their hypotenuse gives total speed. At each apsis vr is zero. Conservation of h makes the smaller periapsis radius correspond to the larger tangential speed. These are relative quantities: for two comparable masses they are not either body's separate speed around the barycentre.
Equal time steps are not equal angular steps
M=2πt/T=E−e sin E; tan(ν/2)=√((1+e)/(1−e)) tan(E/2)
True anomaly ν is the physical in-plane angle from periapsis. Eccentric anomaly E is an auxiliary ellipse angle; mean anomaly M increases uniformly with elapsed time. All equations use radians internally, while the table displays angles in degrees. The solver finds E from the entered time with a bracketed iteration, then uses a quadrant-preserving conversion to ν.
The curve shows how separation changes through the orbit, and the named points identify the inspected positions. A body spends more time near apoapsis because it moves more slowly there. Negative or multi-period times wrap onto the same ideal orbit; the table's time-since-periapsis column is the phase within one period, not the original unwrapped clock.
A velocity state must contain direction information
a=−μ/(2ε); e cosν=r vt²/μ−1; e sinν=r vr vt/μ
A known radius plus signed radial and positive transverse speed determines the current bound ellipse in this planar model. The radial sign distinguishes outbound from inbound motion. A total speed alone would leave that direction unresolved, so the workbench does not guess it.
The separate vis-viva mode accepts just radius and semimajor axis. It can determine total speed, energy and a bound-orbit period, but not eccentricity, angular momentum, velocity components or phase. Those quantities remain explicitly undetermined instead of being supplied by an arbitrary circular-orbit assumption.
Positive or zero specific energy describes an unbound trajectory and is outside this ellipse explorer. Purely radial motion is also excluded. An orbital plane's orientation in the sky, an epoch, perturbations and an absolute reference frame are additional information needed for an actual ephemeris.
Follow the numbers
An entered circular reference orbit
- Let μ=400,000 km³/s² and r=a=10,000 km. Eccentricity is zero, so periapsis and apoapsis are both 10,000 km.
- Speed is √(400,000/10,000)=√40=6.32455532 km/s. The period is 2π×10,000/6.32455532=9,934.58827 s.
- Specific energy is −400,000/(2×10,000)=−20 km²/s²=−20,000,000 m²/s². Specific angular momentum is 10,000×6.32455532=63,245.5532 km²/s.
- Every inspected angle has zero radial speed and the same 6.32455532 km/s transverse speed. At 90°, one quarter of the period has elapsed from the chosen zero direction.
The identical circular values provide a reference against which the varying radius and speed of an eccentric orbit can be checked.
Quick guide
How to use this calculator
- Select the geometry or a complete in-plane velocity state. A radius and total speed alone do not determine eccentricity or orbital phase.
- Enter a gravitational parameter or both masses, then choose distance and time units. Surface heights require a separate body radius.
- For geometry inputs, add named true anomalies or times measured from periapsis. In a circular orbit, the zero direction and time origin are arbitrary.
- Compare the position ledger with periapsis, apoapsis and conserved energy/angular momentum. These are relative two-body results, not a spacecraft ephemeris.
Calculation method
Calculation and interpretation
Keep the shape of a bound orbit, its conserved quantities and the changing position and velocity connected in one inspectable calculation.
p=a(1−e²); r=p/(1+e cosν); vr=√(μ/p)e sinν; vt=√(μ/p)(1+e cosν); T=2π√(a³/μ); ε=−μ/(2a); h=√(μp).
Worked example
An entered circular reference orbit
The identical circular values provide a reference against which the varying radius and speed of an eccentric orbit can be checked.
p=a(1−e²); r=p/(1+e cosν); vr=√(μ/p)e sinν; vt=√(μ/p)(1+e cosν); T=2π√(a³/μ); ε=−μ/(2a); h=√(μp).
Supported inputs
Precision and limits
Bound, isolated Newtonian model
Only circular and elliptical relative two-body motion is supported. Drag, oblateness, tides, thrust, third bodies, relativity, collision checks and orbit determination from observations are excluded.
Entered scales, not live positions
No body data or current orbital elements are fetched. G=6.67430×10⁻¹¹ m³/(kg·s²) is a measured constant used only in mass mode; a supplied μ avoids deriving it from separately rounded masses.
No absolute orientation or mission result
The phase origin is periapsis, or an arbitrary reference direction for a circle. No sky coordinates, ground track, launch window, safe altitude or spacecraft capability is inferred.
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