The zodiacal cloud is a thick circumsolar disk of small debris particles produced by asteroid collisions and comets. Here, we present a zodiacal cloud model based on the orbital properties and lifetimes of comets and asteroids, and on the dynamical evolution of dust after ejection. The model is quantitatively constrained by IRAS observations of thermal emission, but also qualitatively consistent with other zodiacal cloud observations. We find that 85-95% of the observed mid-infrared emission is produced by particles from the Jupiter-family comets (JFCs) and $10^4$ times brighter during the Late Heavy Bombardment (LHB) epoch $\approx$3.8 Gyr ago, when the outer planets scattered numerous comets into the inner solar system. The bright debris disks with a large 24-$\mu$m excess observed around mature stars may be an indication of massive cometary populations existing in those systems.
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