Timing Analysis of the Periodic Radio and Optical Brightness Variations of the Ultracool Dwarf, TVLM 513-46546

We describe the arrival time measurements and timing modeling of the periodic radio flares and optical brightness variations of the M9 ultracool dwarf, TVLM 513-46546. We confirm the stability of the observed period and determine its best-fit value to be 7054.468(+/-)0.007 s over the last 7 years, based on both the new and archival radio observations and the archival optical data. The period, when measured separately for the radio flare and the optical periodicities, is the same to within (+/-)0.02 s. We show that the radio flares are out of phase with respect to the optical brightness maxima by 0.41(+/-)0.02 of the period. Our analysis also reveals that, on shorter timescales, the period varies with the amplitude of (+/-)1-2 s about its long-term average and that these variations are correlated between the radio and the optical wavelengths. These results provide further evidence that TVLM 513-46546 is equipped with a stable, approximately dipolar magnetic field which powers the activity of the star observed over a wide wavelength range, and that the active area has been maintaining its identity and positional stability over no less than 7 years. A stepwise decline of the apparent radio flaring period of TVLM 513-46546 deduced from timing observations with the Arecibo radio telescope in late 2012 and early 2013 suggests that this effect may be the manifestation of differential rotation of the star.