Supplemental Information for calculations, including Figures S1-S3 from Induced flow cools hovering bumble bees

Fig. S1. Cooling trials used to estimate convective heat transfer coefficients. (A) Pre-heated bees cooled inside the vertical wind tunnel, and we tracked thoracic temperature as it approached ambient air temperature. We restricted analyses to biologically relevant thoracic temperatures (28-42°C). (B) For each bee, we plotted the natural log of the ratio between the instantaneous thorax–air temperature difference and the initial temperature difference against time (s). The slope of this linear relationship provides the cooling constant used to estimate the convective heat transfer coefficient (hc), after accounting for body mass, specific heat capacity, and long-wavelength radiative heat loss. Fig. S2. Relationship between body mass and flight metabolic heat production. We estimated metabolic heat production during hovering flight by relating measured body mass to flight metabolic rates from the literature. Each point shows an individual bee, and the solid line shows the fitted linear model (±95% confidence interval). Metabolic heat production increased predictably with increasing body mass (W), providing the mass-dependent heat gain used in the heat-balance model. Fig. S3. Components of the heat-balance model across body size. (A) Metabolic heat gain (W) as a function of body mass. We used the empirical relationship shown in Fig. S2 to estimate metabolic heat gain for bees of different sizes. Each point represents an individual bee, and the solid line shows the fitted linear model. Larger bees generated more metabolic heat during hovering. (B) Solar heat gain (W) under different solar irradiance levels. We estimated solar heat gain for bees of different body sizes under three irradiance regimes (200, 600, and 1000 W m-2). Solar heat gain increased with body mass and with irradiance intensity. (C) Longwave radiative heat loss (W) at different air temperatures. We estimated longwave radiative heat loss across bee body sizes at three ambient temperatures (Tair = 15, 25, and 35°C). Radiative heat loss became more negative (greater heat loss) as body mass increased, and loss was strongest at lower air temperatures. (D) Convective heat loss (W) at different air temperatures. We estimated convective heat loss for bees of different sizes based on measured convective heat-transfer coefficients (hc) derived from cooling trials (Fig. S1). Convective heat loss increased in magnitude with body mass and was strongly temperature dependent. At cooler air temperatures (Tair = 15°C), bees experienced substantially greater convective heat loss compared to warmer conditions (Tair = 25 or 35°C).