gravitational focusing

The concept of gravitational focusing describes how the gravitational attraction between two objects increases the probability that they will collide. Without gravitational force, the likelihood of a collision would depend on the cross-sectional area of the two objects. However, the presence of gravity can cause objects that would have otherwise missed each other to be drawn together, effectively increasing the size of their cross-sectional area.{{Citation |last=Barnes |first=Rory |title=Gravitational Focusing |date=2011 |url=https://doi.org/10.1007/978-3-642-11274-4_670 |encyclopedia=Encyclopedia of Astrobiology |pages=692 |editor-last=Gargaud |editor-first=Muriel |place=Berlin, Heidelberg |publisher=Springer |language=en |doi=10.1007/978-3-642-11274-4_670 |isbn=978-3-642-11274-4 |access-date=2023-01-01 |editor2-last=Amils |editor2-first=Ricardo |editor3-last=Quintanilla |editor3-first=José Cernicharo |editor4-last=Cleaves |editor4-first=Henderson James (Jim)|url-access=subscription }}

Assuming two bodies having spherical symmetry, a collision will occur if the minimum separation between the two centres is less than the sum of the two radii. Because of the conservation of angular momentum, we have the following relationship between the relative speed when the separation equals this sum, $v\_\{\backslash text\{max\}\}$ and the relative speed when the objects are very far apart $v\_\{\backslash text\{rel\}\}$:

:$v\_\{\backslash text\{max\}\}(r\_1+r\_2)=v\_\{\backslash text\{rel\}\}\backslash Delta$

where $\backslash Delta$ is the minimum separation that would occur if the two bodies were not attracted one to the other. This means that a collision will occur not only when

:

and the cross-sectional area is increased by the square of the ratio, so the probability of collision is increased by a factor of $v\_\{\backslash text\{max\}\}^2/v\_\{\backslash text\{rel\}\}^2.$ However, by the conservation of energy we have

:$v\_\{\backslash text\{max\}\}^2=v\_\{\backslash text\{esc\}\}^2+v\_\{\backslash text\{rel\}\}^2$

where $v\_\{\backslash text\{esc\}\}$ is the escape velocity. This gives the increase in probability of a collision as a factor of $1+v\_\{\backslash text\{esc\}\}^2/v\_\{\backslash text\{rel\}\}^2.$ When neither body can be treated as having a negligible mass, the escape velocity is given by:

:$v\_\{\backslash text\{esc\}\}^2=2G(M\_1+M\_2)/(r\_1+r\_2)$

When the second body is of negligible size and mass, we have:

:$v\_\{\backslash text\{esc\}\}^2/v\_\{\backslash text\{rel\}\}^2=\backslash frac\; 83\backslash pi\; G\backslash rho\; r\_1^2/v\_\{\backslash text\{rel\}\}^2$

where $\backslash rho$ is the average density of the large body.

The equation of conservation of energy can be developed into

$$\backslash Delta^2/r\_\{\backslash text\{min\}\}^2=2GM/(r\_\{\backslash text\{min\}\}v^2\_\{\backslash text\{rel\}\})+1$$

where $r\_\{\backslash text\{min\}\}$ is the minimum separation between the centres and $M$ is the total mass.

Instead of using this to find $\backslash Delta$ for a given $r\_\{\backslash text\{min\}\},$ we can solve for $r\_\{\backslash text\{min\}\}/\backslash Delta$ given $\backslash Delta,\backslash \; M,$ and $v\_\{\backslash text\{rel\}\}:$

$$r\_\{\backslash text\{min\}\}/\backslash Delta=\backslash frac\; 1\{GM/(v\_\{\backslash text\{rel\}\}^2\backslash Delta)+\backslash sqrt\{(GM/(v\_\{\backslash text\{rel\}\}^2\backslash Delta))^2+1\}\}$$

When $r\_\{\backslash text\{min\}\}$ is less than the combined radii of the bodies, there will be a collision.

The eccentricity of the hyperbolic trajectory is:

$$\backslash epsilon=1+v\_\{\backslash text\{rel\}\}^2r\_\{\backslash text\{min\}\}/(GM)\; =1+\backslash frac\{v\_\{\backslash text\{rel\}\}^2\backslash Delta\}\{GM\}\backslash frac\; 1\{GM/(v\_\{\backslash text\{rel\}\}^2\backslash Delta)+\backslash sqrt\{(GM/(v\_\{\backslash text\{rel\}\}^2\backslash Delta))^2+1\}\}$$

When there is no collision, the trajectories turn by $2\backslash operatorname\{arccsc\}\backslash epsilon$ in the centre-of-mass fame of reference. The relative velocity vector changes by $2v\_\{\backslash text\{rel\}\}/\backslash epsilon,$ the velocity of the lighter body changing more and of the more massive body less. The relative speed goes asymptotically back down toward $v\_\{\backslash text\{rel\}\}.$