Editing Hyperbolic trajectory (section)

=== Impact parameter and the distance of closest approach {{anchor|Impact parameter|Closest approach}}===
[[File:Hyperbolic trajectories with different impact parameters.png|thumb|upright=1.5|Hyperbolic trajectories followed by objects approaching central object (small dot) with same hyperbolic excess velocity (and semi-major axis (=1)) and from same direction but with different impact parameters and eccentricities. The yellow line indeed passes around the central dot, approaching it closely.]]

The [[impact parameter]] is the distance by which a body, if it continued on an unperturbed path, would miss the central body at its [[closest approach]]. With bodies experiencing gravitational forces and following hyperbolic trajectories it is equal to the semi-minor axis of the hyperbola.

In the situation of a spacecraft or comet approaching a planet, the impact parameter and excess velocity will be known accurately. If the central body is known the trajectory can now be found, including how close the approaching body will be at periapsis. If this is less than planet's radius an impact should be expected. The distance of closest approach, or periapsis distance, is given by:
 
:<math>r_p = -a(e-1)= \frac{\mu}{v_\infty^2} \left(\sqrt{1 + \left(b \frac {v_\infty^2}{\mu}\right)^2} - 1\right)</math>

So if a comet approaching [[Earth]] (effective radius ~6400&nbsp;km) with a velocity of 12.5&nbsp;km/s (the approximate minimum approach speed of a body coming from the outer [[Solar System]]) is to avoid a collision with Earth, the impact parameter will need to be at least 8600&nbsp;km, or 34% more than the Earth's radius. A body approaching [[Jupiter]] (radius 70000&nbsp;km) from the outer Solar System with a speed of 5.5&nbsp;km/s, will need the impact parameter to be at least 770,000&nbsp;km or 11 times Jupiter radius to avoid collision.

If the mass of the central body is not known, its standard gravitational parameter, and hence its mass, can be determined by the deflection of the smaller body together with the impact parameter and approach speed. Because typically all these variables can be determined accurately, a spacecraft flyby will provide a good estimate of a body's mass.

:<math>\mu=b v_\infty^2 \tan \delta/2</math> where <math> \delta = 2\theta_\infty - \pi </math> is the angle the smaller body is deflected from a straight line in its course.