How to correct my real orbit and turn it into predicted ideal orbit?

Question

I want to launch my ship to the purple ellipse orbit. But because of non-ideal world I have non-ideal ship orientation and non-ideal thrust impulse. So, my initial speed is a bit bigger and it's direction has a tiny error. But it leads to huge error in the final destination (point of intersection of ellipses with circle).

The purple ellipse is ideal orbit what I want to reach. The cyan ellipse is my real orbit.

How to calculate correction impulse to turn my real orbit into ideal before spacecraft left the cian circle (which is my destination orbit that I should arrive in the given position where purple ellipse intersects with cyan circle)?

I can measure my real velocity and position and calculate the needed velocity and position of ideal orbit at any moment. But I don't know how to turn one to another.

Answer 1

If you want to do it in one impulse, you have to do it at the two intersections (assuming this is 2D).

To find those intersections, you can make use of the fact that the distance to the central mass is equal for the two orbits at those points.

Radius, in terms of argument of periapsis ($\theta$), periapsis distance ($r_P$) and apoapsis distance ($r_A$)

$$r(\theta) =\frac{2r_Ar_P}{(r_A - r_P)\cos(\theta) + r_A + r_P}$$

If we say the relative argument of periapsis between them (angle between the apsis lines) is $\omega$, you can solve for:

$$r_{ideal}(\theta) = r_{real}(\theta + \omega)$$

At those distances, you can calculate the velocities, and the radial and tangential components of both orbits:

$$v_(r) = \sqrt{\mu\left(\frac{2}{r} - \frac{2}{r_A + r_P}\right)}$$

$$v_{tangential}(r) = \frac{\mu r_P\left(\frac{2}{r_P} - \frac{2}{r_A + r_P}\right)}{r}$$

$$v_{radial}(r) = \sqrt{v_(r)^2 - v_{tangential}(r)^2}$$

For a final burn of:

$$\Delta v = \sqrt{\left(v_{tangential_{real}} - v_{tangential_{ideal}}\right)^2 + \left(v_{radial_{real}} - v_{radial_{ideal}}\right)^2}$$

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Multiple impulses

The above equations should be sufficient to calculate these aproaches as well.

Some easy to calculate routes:

1. Remove relative angle, by doing an impulse at $pi + \omega/2$
2. Lower periapsis.
3. Lower apoapsis

Or, even easier to calculate as it has only periapsis and apoapsis burns:

1. Circularise at apoapsis
2. Lower periapsis
3. Lower apoapsis

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Advanced: general bi-tangential transfers

Answer 2

Finally, I found nice solution:

To correct my orbit and turn it into ideal non-perturbed orbit I need to know my real radius-vector $\vec{R_{real}}$ and velocity $\vec{V_{real}}$

Also, I need to know my ideal radius-vector $\vec{R_{ideal}}$ and ideal velocity $\vec{V_{ideal}}$ for current moment. It can be calculated from the keplerian params of needed orbit.

After that I can find the velocity that I should to have at the current moment for orbit correction:

$\vec{V_{needed}}$ = $\vec{V_{ideal}}$ + ($\vec{R_{ideal}}$ - $\vec{R_{real}}$)

And then I can calculate the direction and amount of impulse for orbit correction:

$\vec{P_{correction}}$ = $\vec{V_{needed}}$ - $\vec{V_{real}}$

This vector gives direction in which I should to accelerate and time of acceleration impulse can be given by

$t$ = $\frac{\vec{P_{correction}}*m_{spacecraft}}{F_{thrust}}$