How to Calculate Correct delta-V for Manoeuvre Node

[Benzyme]

I am attempting to calculate a the required dV for a circularisation burn using kOS. I am calculating the required dV for both a prograde burn, as well as the required dV to achieve the desired inclination to cancel out the rotation.

For the prograde burn I am doing:

v_old = sqrt(body:mu * (2/(orbitAlt + body:radius) - 2/(ship:periapsis + orbitAlt + 2*body:radius)))
v_new = sqrt(body:mu * (2/(orbAlt + body:radius) - 2/(2*orbAlt + 2*body:radius)))
dv_prograde = v_new - v_old

For the normal burn I am doing:

i_deg = targetInclination - ship:orbit:inclination.
dv_normal = 2 * orbitalVelocity * sin(i_deg / 2).

I then add the two together to get overall dv required. I then create a manoeuvre node using these values:

burnNode = node(time:seconds + eta:apoapsis, 0, dv_normal, dv_prograde).

This produces a manoeuvre node orbit on the map view which has the correct inclination, but the final apoapsis is way too high ( aiming for 75km circular, get 75km periapsis and 161.268km apoapsis).

I am currently getting round this in my script by checking for ship:periapsis >= orbAlt, but I would really like to know what is going wrong!

I assume that I am missing some magic around adding these two delta-V values together, so if anyone knows why this is and how to fix it I would be grateful!

Edited April 26, 2018 by Benzyme

[Snark]

1 hour ago, Benzyme said:

I assume that I am missing some magic around adding these two delta-V values together

I think the problem is that your normal burn isn't simply a pure normal burn.  For example, in stock KSP:  Put yourself in LKO.  Drop a maneuver node.  Grab the handle and drag it to give it some magnitude.  Now look at your projected Ap, post-node.  It's bigger than the original.  In other words, doing nothing but dragging on , with no component at all, you still raised your Ap.

That's because as soon as the component has any magnitude at all, it's no longer pure , because as soon as your inclination has changed a bit, the ship's path is no longer perpendicular to the original component, and therefore that original component actually has a certain amount of component in the new trajectory.

Here's how it shows up in the math:

• Your ship has some original velocity, v.  This is a vector quantity with (x,y,z) components.  It has a total magnitude |v|, equal to the Pythagorean sum of its three components.
• Usually folks deal with vectors in terms of X, Y, and Z components, e.g. v as the vector sum of v_x, v_y, v_z.  But we can use any frame of reference we like, so let's use the current direction vector as a reference, and instead refer to v as the vector sum of v, v, and v .  So, the magnitude of your initial velocity can be found as:  |v₁| = sqrt(v² + v² + v²)
  • Yes, I know that obviously the component is the only nonzero one here, and |v₁| simply equals v.  Bear with me here. 
• Now let's look at what you're doing.  You're adding a dv and a dv  into the mix.  So, after the burn, your new velocity magnitude uses (v + dv) and (v + dv) for those two components.
• Therefore, the magnitude of your new velocity is:  |v₂| = sqrt₍(v + dv)²+ v² + (v + dv)²₎
• If you crank through the algebra, you'll find that the actual |v₂| is therefore bigger than simply |v₁| + dv.

And that's why your new Ap is higher than you wanted it to be.

[ElWanderer]

What Snark said for why it doesn't work.

I had a lot of trouble getting inclination corrections right. My interim solution involved the following:

LOCAL dv_pro IS -1 * ABS(dv * SIN(ang / 2)).
LOCAL dv_norm IS dv * COS(ang / 2).

Note that doesn't work accurately if you're burning at a point in your orbit where your flight angle is non-zero (as you need a radial component to the burn too). That's not a problem in this specific case of burning at apoapsis (flight angle is 0 at an apsis or if eccentricity is 0).

My current implementation is more complicated - I calculate the desired final velocity vector, then the difference from the initial velocity vector at the node... then convert that difference vector into pro/radial/normal components.