Understanding gravitational keyhole analysis for Near-Earth Objects

Question

I've been studying the resonant return and gravitational keyhole analysis for NEOs and using the extension of Öpik's theory of close encounters (aka Öpik-Valsecchi theory, Valsecchi et al., Astron Astrophys 408:1179–1196, 2003), developed to identify the keyholes on the encounter b-plane for an asteroid. Understanding the analytical part is relatively easy. However it's numerical implementation got me confused;

Once I have the locus of Earth for next encounter corresponding to a particular semi-major axis, using the Minimum orbit intersection distance (MOID), I can identify if the impact or close encounter is possible.

Analytical theory suggests that my keyholes for next encounter should lie near the intersection of the circle and MOID line represented on b-plane (target plane). They have shown numerical results corresponding to this but did not mention how the transformations were obtained.

I tried taking the coordinates of the intersection for various resonances and propagate them under both restricted (n+1) body motion and Keplerian model, but have not been able to obtain similar results.

How can this particular part can be verified?

The problem is to identify if the points that lie on my first encounter b-plane near the MOID line will impact on next encounter. To do that, those points have to be numerically propagated from current encounter to next. That requires heliocentric state vector of those points $(\vec{r}, \vec{v})$. And I do not have/know the transformations that can convert my b-plane coordinates $(\xi, \eta, \zeta)$ that lie near the MOID to $(\vec{r}, \vec{v})$.

The case here is when my original set of VAs do not touch the resonance circle but lie close to it, but my MOID intersect those circles, unlike the case where I can use rotation matrix to obtain reverse transformations.

As you can see in the image, for 99942 Apophis (2004MN4), I do not have any impact with the Earth (red circle) in 2029. But My MOID (black line) intersects with 6:7 resonance circle indicating the existence of keyholes. I am stuck at obtaining the heliocentric coordinates of the point lying near the intersection of black line and blue circle.

Answer 1

Your encounter must by definition occur somewhere around the MOID line, if you do not consider to significantly change the trajectory. A solution can be obtained using Keplerian analysis, but that is only an approximation, due to the hard-to-restrict three body nature of the problem. If I understand your question correctly, you want a higher degree of accuracy.
In this particular case, you are not left with many other possibilities than to brute force it with numerical integration, because of the general lack of analytical solutions to n-body problems. (except for certain exceptions)

Response to the edit:
Until I can find something better, it seems to me that you can use the Laplace parameters to calculate a position (Either in a co-rotating reference frame or other). This can easily by transformed to a position state vector $(\vec{r})$. Using that again, you can obtain the velocity state vector $(\vec{v})$from the resulting change in the Hill potential from before the encounter. This is however a rather annoyingly complicated way of doing it. I am still looking for a more intuitive solution.