  High-metallicity pollution is common in white dwarf (WD) stars hosting
remnant planetary systems. However, they rarely have detectable debris
accretion discs, possibly because much of the influx is fast steeply-infalling
debris in star-grazing orbits, producing a more tenuous signature than a slowly
accreting disk. Processes governing such deposition between the Roche radius
and photosphere have so far received little attention and we model them here
analytically by extending recent work on sun-grazing comets to WD systems. We
find that the evolution of cm-to-km size (a_0) infallers most strongly depends
on two combinations of parameters, which effectively measure sublimation rate
and binding strength. We then provide an algorithm to determine the fate of
infallers for any WD, and apply the algorithm to four limiting combinations of
hot versus cool (young/old) WDs with snowy (weak, volatile) versus rocky
(strong, refractory) infallers. We find: (i) Total sublimation above the
photosphere befalls all small infallers across the entire WD temperature (T_WD)
range, the threshold size rising with T_WD and 100X larger for rock than snow.
(ii) All very large objects fragment tidally regardless of T_WD: for rock, a_0
>= 10^5 cm; for snow, a_0 >= 10^3 -- 3x10^4 cm across all WD cooling ages.
(iii) A considerable range of a_0 avoids fragmentation and total sublimation,
yielding impacts or grazes with cold WDs. This range narrows rapidly with
increasing T_WD, especially for snowy bodies. Finally, we discuss briefly how
the various forms of deposited debris may finally reach the photosphere surface
itself.
