# Pure orbit inclination changing, why delta v differs between vector and numerical approach?

as said in the title I have different results between numerical en vector approach during inclination changing maneuver.

The initial orbit :

Eccentricity : 0.7

Inclination : 40.0 °

Ascending node : 20.0 °

Perigee Argument :10.0 °

The target orbit is the same with inclination set at 55 °, so we have a relative inclination of 15°

My vector approach:

At planes intersection I subtract final vector by initial vector and I get the following result : |delta_v| = 2608.9 m/s

And the result of my new orbit matches perfectly all expected parameters.

Now the problem comes from the numerical approach :

1. I compute delta v magnitude |delta_v| = 2.0 * velocityAtNode * sin(relativeInclination * 0.5) = 2615,65 m/s

2. My delta v vector is initialized on normalized velocity vector.

3. I compute the rotation amplitude of my normalized delta v vector : rotationAngle = 90° + relativeInclination*0.5; = 97.5 °

4. I rotate my normalized delta V vector around line of nodes axis by an angle of 97.5°

5. Then I apply delta v computed previously to my rotated normalized vector: final delta v vector = rotated normalized vector * 2615,65

6. I add this delta v vector to my initial velocity vector

In this numerical approach the orbits planes are perfectly aligned but I've a drift in others parameters :

perigee height becomes : 6675 km

eccentricity : 0.706

perigee argument : 12°

If I compare my delta v vector obtained by vector approach and by numerical approach. I notice an angle of 4.1° between these two vectors and a magnitude difference of 6.7 m/s

Any help is welcome to understand the difference between these two approaches. Thanks!

• You haven’t shared how you are performing the transformation between keplerian elements and position+ velocity vectors. Are you confident you are performing that step correctly?
– cms
Nov 10, 2021 at 12:43
• Hi @cms, I use a routine from spice toolkit provided by JPL : naif.jpl.nasa.gov/pub/naif/toolkit_docs/C/cspice/conics_c.html. The conversion to keplerian elements is perfect on the vector approach so I think it should be correct for the numerical approach.
– sl20
Nov 10, 2021 at 12:57

Instead of using the normalized velocity vector, you should first project the velocity vector onto the transverse plane (i.e., take $$\mathbf{v} - \frac{\mathbf{v}\cdot \mathbf{r}}{\mathbf{r}\cdot \mathbf{r}}\mathbf{r}$$, where $$\mathbf{r}$$ is the position vector), and then normalize this projection. Similarly, you should use the magnitude of this projection to calculate the magnitude of $$\Delta v$$. Otherwise, your $$\Delta v$$ has an extra radial component.