Relationships between T_F and β_F derived from fitting fluxes F_ν of starless cores and protostellar envelopes (fits with free variable β). Curves plotted with squares (dark colors) and circles (bright colors) in the left and middle panels correspond to the isolated and embedded models, whereas in the right panel they correspond to the isothermal versions of the isolated starless cores and protostellar envelopes, respectively. Results for the starless cores are plotted for all masses 0.03−30 M_⊙. Results for the protostellar envelopes, shown for only selected masses (0.03,0.3, and 3 M_⊙, three sets of identical lines), span the entire range of luminosities 0.03−30 L_⊙. Thick horizontal lines indicate the true value β = 2. For clarity of the plots, error bars are not shown.