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Atmospheric heavy elements have been observed in more than a quarter of whitedwarfs (WDs) at different cooling ages, indicating ongoing accretion ofasteroidal material, whilst only a few per cent of the WDs possess a dust disk,and all these WDs are accreting metals. Here, assuming that a rubble-pileasteroid is scattered inside a WD's Roche lobe by a planet, we study its tidaldisruption and the long-term evolution of the resulting fragments. We find thatafter a few pericentric passages, the asteroid is shredded into its constituentparticles, forming a flat, thin ring. On a timescale of Myr, tens of per centof the particles are scattered onto the WD, and are therefore directly accretedwithout first passing through a circularised close-in disk. Fragment mutualcollisions are most effective for coplanar fragments, and are thus onlyimportant in $10^3-10^4$ yr before the orbital coplanarity is broken by theplanet. We show that for a rubble pile asteroid with a size frequencydistribution of the component particles following that of the near earthobjects, it has to be roughly at least 10 km in radius such that enoughfragments are generated and $\ge10\%$ of its mass is lost to mutual collisions.At relative velocities of tens of km/s, such collisions grind down the tidalfragments into smaller and smaller dust grains. The WD radiation forces mayshrink those grains' orbits, forming a dust disk. Tidal disruption of amonolithic asteroid creates large km-size fragments, and only parent bodies$\ge100$ km are able to generate enough fragments for mutual collisions to besignificant. Hence, those large asteroids experience a disk phase before beingaccreted.

Accretion of tidally disrupted asteroids onto white dwarfs: direct accretion versus disk processing