The Near-Earth Objects Coordination Centre predicts future close approaches with Earth, for example these approaches for 2010RF12. For example, the time uncertainty of the 2022-09-23 closest approach is 276.7 minutes, with a distance of 3.834E-4 AU, in 2024-03-01 uncertainty of 21.3 minutes and 4.456E-4 AU, and in 2034-07-16 uncertainty of 119.3 minutes and 3.82E-5 AU.

My question is why do those numbers not monotonically increase? If their precision is such that they only state with an uncertainty of 4 hours for 2022, how do they get that down to 20 minutes for a prediction two years away?


1 Answer 1


I have a couple of guesses, but they're fairly generic, and not closely tuned to this particular case. I have been fiddling around with the ephemeris generation tools on the source page, but I have not been able to prove either of my suppositions. With enough queries, you may be able to check up on them.

My first guess is it's related to the relative velocity. In particular, the slower the relative speed, the more time the conjunction event takes, and the more uncertainty there is in estimating the exact moment of closest approach, because the difference in distance from one moment to the next is so small. Conversely, the faster the relative speed, the less time the conjunction event takes, and the less uncertainty there is in estimating the exact moment of closest approach, because the difference in distance from one moment to the next is relatively large. Or at least, those things ought to be true in general, but I don't have evidence that they explain this specific case.

Another interesting possibility is that when treated carefully, some errors sometimes subtract rather than add. This specific object has only been observed on five consecutive days in 2010. That is a very small fraction of a complete orbit, so the program that solved for the closest-fit orbit probably reported significant correlations among the fit parameter estimates. In such a case, the simple "root sum square" (RSS) error propagation can be very misleading. For example, if we are interested in a quantity $Q$ which equals some function $f$ of the three variables $u$, $v$, and $w$, then the square of the standard error of estimating $Q$ is $$\newcommand{\d}{\partial} \newcommand{\s}{\sigma} \left( \frac{\d f}{\d u} \right)^{\!2} \!\s_u^2 + \left( \frac{\d f}{\d v} \right)^{\!2} \!\s_v^2 + \left( \frac{\d f}{\d w} \right)^{\!2} \!\s_w^2 \\ + 2 \frac{\d f}{\d u} \frac{\d f}{\d v} \rho_{uv}\s_u\s_v + 2 \frac{\d f}{\d u} \frac{\d f}{\d w} \rho_{uw}\s_u\s_w + 2 \frac{\d f}{\d v} \frac{\d f}{\d w} \rho_{vw}\s_v\s_w $$ The first three terms are the familiar RSS, and are necessarily positive, but the other three each contain three things (two derivatives and a correlation) that might be negative, and might partly cancel each other or the squared parts. Further, even without correlation, the exact contribution of each of three supposedly independent errors is scaled by the derivatives of the estimated state with respect to that variable, so for example a slowly increasing $\s_u$ might be paired with a large $\d f/\d u$ in two months but a much smaller $\d f/\d u$ in two years.


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