What is the cross-section impact probability for an extended object moving trough a field of randomly moving particles?

In trying to get an estimate of the probability for an earth orbiting (LEO) satellite to collide with small debris particles.

So I need to understand what is the impact probability for an orbiting object ($$v_o$$) with a cross sectional area of $$A_c$$, with a small particles, and assuming:

• the geometry of the large object is a Sphere. ($$A_c = const$$)
• the particles are small ($$A_c >> A_p$$) point-particles
• the velocity field of the particles is randomly distributed
• the density of the particle field is $$d_p$$ [particles/$$m^2$$]
• circular low earth orbits (LEO).

(What other assumptions are needed?)

In reality the particle field would also have a velocity bias, due to orbital insertion factors. But I'm not sure how we would incorporate that here.

The ultimate aim would be to get a mathematical formula looking something like:

$$P(A_c, v_o, d_p, h)$$

Where:

• $$P$$ = The probability of the object to collide with a particle [no. of collisions/time]
• $$A_c$$ = Cross sectional area normal to velocity
• $$v_o$$ = orbital velocity of object, relative to earth
• $$d_p$$ = density of the particle field (at height $$h$$ above earth surface)

References:

A more useful question is perhaps: What is the probability that the large object will suffer $$n$$ collisions in time $$t$$?
You can adapt the standard argument presented here and in Wiki. The cross-section of the large object traces out a circular, cylindrical tube in space. This tube, when opened and laid out in a straight line, will be of infinite length because the object keeps retracing its path forever. Then if the statistical distribution of particles at every instant obeys Poisson statistics (as the statement of your question seems to imply) then the probability of $$n$$ collisions in time $$t$$ is:
$$P(n|t) = \frac{(\lambda t)^n}{n!}e^{-\lambda t}$$
where the average collision rate $$\lambda = d_pA_cv_0$$ where $$d_p$$ is the particle number density per unit volume and $$A_cv_0$$ is the volume per unit time traced out by the large object.