We have obtained extensive photometric observations of the polluted white dwarf WD 1145+017 which has been reported to be transited by at least one, and perhaps several, large asteroids (or, planetesimals) with dust emission. We have carried out 53 observation sessions on 37 nights, totaling 192 hours, of this 17th magnitude star with small to modest size telescopes covering the interval 2015 November 1 to 2016 January 21. In all, we have detected some 237 significant dips in flux. Periodograms of the data reveal a significant periodicity of 4.5004 hours that is consistent with the dominant (A) period detected with K2. The folded light curve at this period shows there is an hour-long depression in flux with a mean depth of nearly 10%. This depression is comprised of a series of shorter and sometimes deeper dips that do not always occur at exactly the same orbital phase, and which would be unresolvable with K2. In fact, we find numerous dips in flux at other orbital phases. Nearly all of the dips associated with this activity appear to drift systematically in phase with respect to the A period by about 2.5 minutes per day with a dispersion of ~0.5 min/d, corresponding to a mean drift period of 4.4928 hours. In all, we can track approximately 15 of these drifting features. There is no detection of the B-F periods found with K2, but if they remain at the K2 levels we would not expect to have seen them. We explain the drifting motion as that of smaller bodies (`fragments) that break off from the asteroid and go into a slightly smaller orbit than that of the asteroid. If our interpretation is correct, we can use the drift rate to determine the mass of the asteroid. Under that scenario, we find that the mass of the asteroid is M_a ~ = 10^23 grams, or about 1/10th the mass of Ceres, with an uncertainty of about a factor of 2.
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