Model line ratio of ¹³CO J = 6−5 to ¹³CO J = 2−1 as a function of rotational temperature for very low (purple) to very high (green) optical depths of ¹³CO 2−1. The contours highlight the line ratio for a ¹³CO J = 2−1 optical depth of 0.5, 1, and 2 (black). We note that the line ratio scales linearly with temperature if the emission is optically thin and 25 K ≲ T_rot ≲ 150 K and that it becomes independent of temperature at a value of 9 if the emission is optically thick and T_rot ≳ 40 K. The typical values for the line ratio and rotational temperature (LkCa15) and brightness temperature (HD 169142) in their dust cavities are indicated in the figure.