# Earth Cross Section for Meteoroids

For a projectile to hit a target it normally must impact within an area which
is the cross-sectional area of the target. For a spherical body, this area is
r^{2} where r is the radius of the sphere. However, in the case of a meteoroid
or an asteroid impacting the Earth (or another planet), the appropriate cross-
sectional radius is not the physical radius of the Earth, but it gravitational
radius, which depends upon the impactor velocity.

This is because the Earth's gravity will attract an impactor that might be
approaching on a near miss orbit. Imagine we have an impactor that in the
absence of gravity has a space velocity v, and is on a trajectory with a
distance R_{g} from the Earth's centre. Because of gravity the trajectory
will curve toward the Earth and impact with a velocity V.

This impact will be tangential if R_{g} is the gravitational radius of the Earth
(for the space velocity v (ie the body just impacts the planet). We can use the
laws of conservation of momentum aned conservation of energy to determine R_{g}.

v R_{g} = V R_{p} (conservation of momentum)

v^{2}/2 = V^{2}/2 - G M_{e} / Rp (conservation of energy)

where G is the universal gravitational constant and M_{e} is the mass of the Earth.
Solving these two equations simultaneously we find that:

R_{g} = R_{p} √[ 2 G M_{e} / ( v^{2} R_{p} ) + 1 ]

The table below presents values of R_{g} (as a function of the space velocity), both in
kilometres and as a fraction of the Earth's radius. Also shown are the fractional
area, and the collision velocity.

Space vel(km/s) |
Collision vel(km/s) |
Effective radius(km) |
Fractional radius |
Fractional area |
---|---|---|---|---|

2 | 11 | 36237 | 5.68 | 32.26 |

4 | 12 | 18942 | 2.97 | 8.81 |

5 | 12 | 15630 | 2.45 | 6 |

6 | 13 | 13494 | 2.12 | 4.47 |

8 | 14 | 10965 | 1.72 | 2.95 |

10 | 15 | 9571 | 1.5 | 2.25 |

15 | 19 | 7958 | 1.25 | 1.56 |

20 | 23 | 7309 | 1.15 | 1.31 |

25 | 27 | 6989 | 1.1 | 1.2 |

30 | 32 | 6809 | 1.07 | 1.14 |

35 | 37 | 6698 | 1.05 | 1.1 |

40 | 42 | 6625 | 1.04 | 1.08 |

45 | 46 | 6574 | 1.03 | 1.06 |

50 | 51 | 6538 | 1.02 | 1.05 |

Material prepared by John Kennewell