The Kepler object KIC 12557548 shows irregular eclipsing behaviour with a constant 15.685 hr period, but strongly varying transit depth. In this paper we fit individual eclipses, in addition to fitting binned light curves, to learn more about the process underlying the eclipse depth variation. Additionally, we put forward observational constraints that any model of this planet-star system will have to match. We find two quiescent spells of ~30 orbital periods each where the transit depth is <0.1%, followed by relatively deep transits. Additionally, we find periods of on-off behaviour where >0.5% deep transits are followed by apparently no transit at all. Apart from these isolated events we find neither significant correlation between consecutive transit depths nor a correlation between transit depth and stellar intensity. We find a three-sigma upper limit for the secondary eclipse of 4.9*10^-5, consistent with a planet candidate with a radius of less than 4600 km. Using the short cadence data we find that a 1-D exponential dust tail model is insufficient to explain the data. We improved our model to a 2-D, two-component dust model with an opaque core and an exponential tail. Using this model we fit individual eclipses observed in short cadence mode. We find an improved fit of the data, quantifying earlier suggestions by Budaj (2013) of the necessity of at least two components. We find that deep transits have most absorption in the tail, and not in a disk-shaped, opaque coma, but the transit depth and the total absorption show no correlation with the tail length.
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