Understand the relationship
The reasoning behind the result
An impulse changes a velocity vector
Δv=|v⃗2−v⃗1|; Δv²=v1²+v2²−2v1v2 cosδ
The burn must change both the magnitude and direction of velocity. The cosine rule applied to the two velocity vectors gives its magnitude. Rewriting the expression as (v2−v1)²+4v1v2 sin²(δ/2) retains small-angle precision and makes the speed-only and angular contributions explicit.
With unchanged speed, the two endpoints lie on one velocity circle and the impulse is its chord: 2v sin(δ/2). A 180° reversal costs 2v, while a zero-angle unchanged-speed case costs zero. A small plane change at high orbital speed can still need a substantial velocity budget.
Inclination and node together define plane separation
cosδ=cos i1 cos i2+sin i1 sin i2 cos(Ω2−Ω1)
Inclination i measures the angle of the orbital angular-momentum direction from the reference pole; ascending-node longitude Ω sets that plane's orientation around the pole. Two orbits with the same inclination can therefore have different planes. Their angular-momentum vectors determine the true plane separation δ.
For two 30° inclinations with nodes 180° apart, δ is 60°, not zero. If the nodes coincide, δ reduces to the absolute inclination difference. All plane angles are displayed in degrees; trigonometric calculations use radians internally. This geometry does not determine where or when a spacecraft reaches the mutual node.
An inverse burn magnitude has a feasible interval
|v2−v1| ≤ Δv ≤ v1+v2
The minimum possible impulse for fixed speed magnitudes occurs when the two velocities point in the same direction. The maximum occurs for opposite directions. A requested exact burn outside that interval has no corresponding angle and is rejected.
Inside the interval, the inverse returns the unique angle from 0° to 180°. This is the angle for an exact magnitude, not an automatic maximum plane change from a mission budget: location, other maneuvers, finite thrust and reachable orbital states still matter. Zero speed is excluded because it does not define an orbital velocity direction.
A combined vector burn differs from two separate burns
For comparison, the table shows changing speed along the existing direction and making the plane change as a separate impulse at the lower of the two specified speeds. The comparison is the geometric sum |v2−v1|+2 min(v1,v2) sin(δ/2). It is not an optimized trajectory, and separate execution may not be dynamically feasible at a chosen location.
The single combined vector impulse is no larger than this same-point two-step geometric comparison. The difference explains why adding a speed-change number to a plane-change number can overstate one combined impulse. A useful orbital strategy still requires a full trajectory and node analysis outside this calculator.
Follow the numbers
Rotating a 1,000 m/s velocity by 60°
- Before and after speed magnitudes are both 1,000 m/s. The plane angle is δ=60°, so sin(δ/2)=sin30°=0.5.
- The vector impulse is Δv=2×1,000×0.5=1,000 m/s. Neither the original nor the final total speed has doubled.
- The inverse uses sin(δ/2)=Δv/(2v)=0.5 and returns δ=2 asin0.5=60°. Its feasible exact-burn interval is 0 to 2,000 m/s.
The burn changes direction while preserving the speed magnitude. Speed, angle and impulse remain three different quantities.
Quick guide
How to use this calculator
- Choose whether the velocity magnitudes are equal, unequal or being combined with a known exact burn magnitude.
- Use positive transverse speed magnitudes at a common orbital-plane node. The generalized unequal-speed model requires zero radial components at the burn point.
- If starting with inclinations, enter the ascending-node difference as well. Inclination difference alone generally does not give the actual plane angle.
- Inspect the before/after speeds, actual plane angle, vector burn and the separate-maneuver comparison. The result does not locate the node or establish a complete transfer.
Calculation method
Calculation and interpretation
Calculate the vector change required at a common node while distinguishing orbital-plane angle from inclination difference and delta-v from total speed.
Δv=√[(v2−v1)²+4v1v2 sin²(δ/2)]; for v1=v2=v, Δv=2v sin(δ/2). cosδ=cos i1 cos i2+sin i1 sin i2 cosΔΩ.
Worked example
Rotating a 1,000 m/s velocity by 60°
The burn changes direction while preserving the speed magnitude. Speed, angle and impulse remain three different quantities.
Δv=√[(v2−v1)²+4v1v2 sin²(δ/2)]; for v1=v2=v, Δv=2v sin(δ/2). cosδ=cos i1 cos i2+sin i1 sin i2 cosΔΩ.
Supported inputs
Precision and limits
A common node and transverse velocities
The pure plane change assumes unchanged radial motion; this workflow's generalized unequal-speed model is restricted to zero radial components at a common node/apsis. Arbitrary three-dimensional position/velocity changes are not solved.
Instantaneous impulse
Burn duration, propulsion losses, gravity during firing, required propellant and mission feasibility are excluded. The tool does not locate nodes or plan a maneuver sequence.
Comparison is geometric only
The separate-burn figure is a same-point velocity comparison, not proof that a two-step orbit transfer is feasible or optimal.
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