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 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.

b-plane for Apophis 2029 encounter

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.

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.

  • Yes, the encounter should lie along the MOID line, however to see if the points located near that region actually impact, I need to numerically propagate them, which is where my problem is, since I only have the b-plane coordinates of those points, they should be transformed into heliocentric state vector,. I am not able to find such transformations. I tried by using the inverse of rotation matrix initially used to obtain the first b-plane but that does not give me the same result as authors propose. Besides I already developed a n+1 body model for more accurate results – Astroynamicist Dec 17 '15 at 18:46
  • @Astroynamicist Using the inverse transformation matrix for the b-plane does only give you the results from the Keplerian approximation again, if I understand your procedure until now. However, can you please give some more information in your question in order to make it clear what exactly is the problem? If you already have an n+1 body model, I can not see what is actually missing. – Hohmannfan Dec 17 '15 at 19:23
  • added some details hope that helps. Once I finish the figures it might be easier to unerstand. – Astroynamicist Dec 17 '15 at 19:37
  • @Astroynamicist That helps a lot! I must only refresh how you actually solved this. I can update my answer soon. – Hohmannfan Dec 17 '15 at 19:45
  • Thanks, I'll look into Laplace distribution in the mean time, to see if I can obtain $\vec{r}$. Concerning the velocity it seems to get complicated, I'll start with my first guess by taking $V_{infinity}$ which is used to obtain this b-plane, then move to Hill potential – Astroynamicist Dec 17 '15 at 20:18

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