Long-period Comet C/2013 A1 (Siding Spring) will experience a close encounter with Mars on 2014 October 19. As of 2013 October 21, the distance of closest approach between the two is projected to be between 89,000 km and 173,000 km, with a nominal value of 131,000 km. Thus, a collision between the comet and the planet has been ruled out, but the comet's coma may very well envelop Mars and its man-made satellites. We present a simple analytic model of the dust component of cometary comae that describes the spatial distribution of cometary dust and meteoroids and their size distribution. We find that this model successfully reproduces, to within an order of magnitude, particle fluxes measured by spacecraft Giotto in the coma of 1P/Halley and by spacecraft Stardust in the coma of 81P/Wild 2. We apply our analytic model to C/2013 A1 (Siding Spring) and compute the expected total fluence of potentially damaging particles at Mars at the time of closest approach between the two bodies; we obtain a nominal fluence of 0.15 particles per square meter. We conduct numerical simulations of particle ejection from the comet's nucleus and compare the resulting spatial distribution with that of our analytic model, and conclude that our spherically symmetric analytic model is adequate for order-of-magnitude fluence estimates.