X and the City: Modeling Aspects of Urban Life (88 page)

From the earlier discussion we can identify
δV
, the relative difference in speeds, as
v
₀
and the difference in orbital radii,
δa
, as the impact parameter
b
. Using equation (A12.7) in (A12.5) we obtain (in the current notation)

The positive real root of this biquadratic equation is

In performing these calculations I found it simplest to write the term

since
a/V
=
T
/2
π
, where
T
is the period of the Earth’s orbit in seconds. We have already calculated the quantity 2
GM
/
R
from equation (A12.6). The result is
δa
≈ 8.45 × 10
⁵
km (
δa
≈ 5.63 × 10
⁻³
A.U.). In terms of the Earth’s equatorial radius
R
≈ 6.38 × 10
³
km, this is about 132 Earth radii, or approximately 2.2 times the Earth-moon semi-major axis.

From equation (A12.7) we find that the corresponding difference in orbital speed |
δV
| ≈ 0.085 km/s.

Exercise:
Show from equations (A12.2) and (A12.5) that, for a grazing collision,

This shows that the perigee speed is quite sensitive to the impact parameter, varying as its square. Using this result we calculate the impact speed at a grazing collision to be ≈11.1 km/s, just a tiny bit less than the escape speed from the Earth. Since we have shown that the Earth sweeps out a toroidal volume of radius approximately 0.056 A.U. in its path around the sun, it can be thought of as a giant vacuum cleaner with CCS (132)
²
≈ 1.74 × 10
⁴
times that of the Earth’s GCS. Therefore any object moving in a circular orbit with radius between 0.9944 A.U. and 1.0056 A.U. will collide with (be captured by) the Earth. That’s some vacuum cleaner!

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