We present deep imaging observations, orbital dynamics, and dust tail model analyses of the double-component asteroid P/2016 J1 (J1-A and J1-B). The observations were acquired at the Gran Telescopio Canarias (GTC) and the Canada-France-Hawaii Telescope (CFHT) from mid March to late July, 2016. A statistical analysis of backward-in-time integrations of the orbits of a large sample of clone objects of P/2016 J1-A and J1-B shows that the minimum separation between them occurred most likely $sim$2300 days prior to the current perihelion passage, i.e., during the previous orbit near perihelion. This closest approach was probably linked to a fragmentation event of their parent body. Monte Carlo dust tail models show that those two components became active simultaneously $sim$250 days before the current perihelion, with comparable maximum loss rates of $sim$0.7 kg s$^{-1}$ and $sim$0.5 kg s$^{-1}$, and total ejected masses of 8$times$10$^{6}$ kg and 6$times$10$^{6}$ kg for fragments J1-A and J1-B, respectively. In consequence, the fragmentation event and the present dust activity are unrelated. The simultaneous activation times of the two components and the fact that the activity lasted 6 to 9 months or longer, strongly indicate ice sublimation as the most likely mechanism involved in the dust emission process.
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