File:Impact probability of Apophis as function of uncertainty.jpg

English: Impact probability of Apophis as function of uncertainty

This graph illustrates the results of a simulation of the detection of a threatening asteroid, carried out in December 2020 and early 2021 when the assteroid Apophis (presently having a period of 0.89 years) came near enough to Earth to be detected. This asteroid was discovered in 2004, and it was known that it would pass 38,000 km from the centre of the earth in 2029, but in the exercise this was ignored in order to see how well the network of astronomers and instruments would re-detect Apophis and estimate its likelihood of hitting the earth. During the course of the exercise, more and more observations were made, and the uncertainty of the orbital parameters decreased. The Xs near the blue curve show how the estimated probability of impact in 2029 changed as the uncertainty decreased (going from right to left in the graph). At first the probability was low, because there was a large area around the earth where it seemed that Apophis might pass. As this area became smaller, the estimated probability of an impact in 2029 grew, reaching a maximum of 16% in February, 2021. But then the area continued to shrink, to the point where it no longer contained the earth, and the probability dropped to zero.

The numbers at the top show the year, month, and day at which the values of uncertainty were attained.

The blue curve itself shows how the probability should have behaved, theoretically, if the exact orbit were known, and the fact that it would miss the centre of the earth by 38,000 km.

The red curve shows how the probability would increase (going from right to left) if in fact Apophis were going to hit the earth in 2029. Once the uncertainty gets small enough, it becomes clear that there is no chance it will miss.

The x-axis is the uncertainty in a value called ζ which is one of the two coordinates describing where the asteroid will be at closest approach, the other coordinate being called ξ. If the orbit of the asteroid is known but the timing or position of the asteroid along the orbit is not known exactly, then ξ is known but ζ is not.

The true values of ζ and ξ are around 47684 km and 9483 km, respectively.[1] ${\displaystyle {\sqrt {\zeta ^{2}+\xi ^{2}}}}$ is about 48618 km. The uncertainty in ζ is generally greater than that in ξ.

To avoid a collision, ${\displaystyle {\sqrt {\zeta ^{2}+\xi ^{2}}}}$ has to be more than about 20,865 kilometres. This is more than the radius of the earth (6378 km) because of w:gravitational focusing with an initial relative speed of 7.422 km/s.[2]. An object whose original position and velocity would have it miss the earth can still hit the earth because the earth's gravity bends its path toward the earth.