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

Question

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

The initial orbit :

Perigee radius : 6700 km

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!

Answer 1

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.