posterior distribution of the fit in Figure \[fig:posterior\_2M0501\_full\]. ]{}We adopt $15.7\pm0.2~$hr as the most likely rotation period.

Fitting the intra-pixel phase effect and astrophysical variability simultaneously
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[The pixel phase effect can in principle be covariant with astrophysical variability [@Heinze2014]. For the variable objects and , it is possible that in correcting for the pixel phase effect using Equation \[eq:pixelphase\], the variability signal may have been distorted. For these targets, we also fit their raw flux using a model that includes the pixel phase effect (Equation \[eq:pixelphase\]) and a sinusoidal model to represent the variability, following previous *Spitzer* brown dwarf variability and exoplanet transit studies [e.g. @Metchev2015a; @Delrez2018].]{}

[We find that simultaneously fitting both the pixel phase effect and astrophysical variability simultaneously does not significantly affect the variability parameters in Table \[tab:parameters\]. However, the simultaneous fit yields a higher correlation coefficient between the x and y-pixel positions and the measured flux in all cases, suggesting that including the sinusoidal fit during this step worsens the intra-pixel phase effect correction. We use the BIC framework discussed in Section \[sec:BIC\] to assess whether the extra sinusoidal parameters are warranted by the data, and find that the pixel phase effect model (i.e. Equation 