The Relationship between Mid-Infrared and Sub-Millimetre Variability of Deeply Embedded Protostars

We study the relationship between the mid-infrared and sub-mm variability of deeply embedded protostars using the multi-epoch data from the Wide Infrared Survey Explorer ($WISE$/NEOWISE) and the ongoing James Clerk Maxwell Telescope (JCMT) transient survey. Our search for signs of stochastic (random) and/or secular (roughly monotonic in time) variability in a sample of 59 young stellar objects (YSOs) revealed that 35 are variable in at least one of the two surveys. This variability is dominated by secular changes. Of those objects with secular variability, 14 objects ($22\%$ of the sample) show correlated secular variability over mid-IR and sub-mm wavelengths. Variable accretion is the likely mechanism responsible for this type of variability. Fluxes of YSOs that vary in both wavelengths follow a relation of $\log_{10} F_{4.6}(t)=\eta \log_{10} F_{850}(t)$ between the mid-IR and sub-mm, with $\eta=5.53\pm0.29$. This relationship arises from the fact that sub-mm fluxes respond to the dust temperature in the larger envelope whereas the mid-IR emissivity is more directly proportional to the accretion luminosity. The exact scaling relation, however, depends on the structure of the envelope, the importance of viscous heating in the disc, and dust opacity laws.