I. Overview

This document summarises the main equations of the NicheMapR endotherm model ‘endoR’, within the subroutines and functions documented in the Endotherm Components Tutorial. It begins with an overview of the governing equations of the steady state heat budget, then expands the latter terms into the respective mechanistic terms. It then shows how these equations are rearranged to allow skin and fur-air interface boundary conditions to be calculated simultaneously as a function of the internal and external conditions. All symbols used, and their respective terms in the code, are summarised in section XVII at the end of this document.

II. Governing heat balance equations

At steady state, the overall heat balance is:

$$\begin{align} \tag{1} Q_{gen,net} - Q_{evap} = Q_{fur} = Q_{env} \end{align}$$

where

$$\begin{align} \tag{2} Q_{gen,net} = Q_{gen} - Q_{resp} \end{align}$$

and

$$\begin{align} \tag{3} Q_{env} = Q_{rad} + Q_{conv} + Q_{cond} + Q_{evap,fur} - Q_{sol}. \end{align}$$

These equations are stating that the net heat generated by the body, $Q_{gen,net}$, which is the total heat generated $Q_{gen}$ less that lost by respiration $Q_{resp}$, is equal to the heat passing through the fur $Q_{fur}$ after accounting for any heat lost by evaporation from the skin surface $Q_{evap}$. The heat exchanged with the environment through the fur is equal to the heat exchanged by long-wavelength radiation $Q_{rad}$, convection $Q_{conv}$, conduction $Q_{cond}$, evaporation from the fur surface $Q_{evap,fur}$ and solar radiation $Q_{sol}$.

By convention, $Q_{resp}$, $Q_{evap}$, $Q_{rad}$, $Q_{conv}$ and $Q_{cond}$ are considered to be losses, so negative values of these terms represent situations where heat is being gained through these processes. The reverse holds for $Q_{gen}$ and $Q_{sol}$.

Thus, at steady state:

$$\begin{align} \tag{4} Q_{fur} + Q_{sol} = Q_{rad} + Q_{conv} + Q_{cond} + Q_{evap,fur} \end{align}$$

and

$$\begin{align} \tag{5} Q_{gen} + Q_{sol} = Q_{resp} + Q_{evap} + Q_{rad} + Q_{conv} + Q_{cond} + Q_{evap,fur}. \end{align}$$

That is, at steady state for a particular core body temperature, the heat production by the body and the heat gain from any solar radiation must equal the overall heat lost by the combination of respiration, evaporation, long wavelength radiation, convection and conduction. Otherwise, the animal will not be in steady state and body temperature will rise or fall. Figure 1 provides a schematic of the overall heat transfer.

III. Mechanism equations

We now expand the terms in the previous section to show their mechanistic underpinnings in relation to body dimensions, metabolic processes and the associated parameters and environmental forcing variables. These terms are further expanded in the ensuing sections that detail the subroutines and functions of the endoR program, including generalisation to different shapes; here we consider cylindrical geometry only (see also, Bird et al. 2002 for a general treatment of these kinds of problems).

The equation for net metabolic heat generation $Q_{gen,net}$ for a cylinder is:

$$\begin{align} \tag{6} Q_{gen,net} = \frac{T_c - T_s}{\frac{R^2_g}{4 k_g V_g} + \frac{R^2_g}{2 k_i V_g}\ln\left(\frac{R_s}{R_g}\right)} \end{align}$$

where $T_c$ and $T_s$ are core and skin temperature, respectively, subscript $g$ means flesh (heat generating tissue), subscript $i$ means fat (insulating tissue), subscript $s$ means skin surface, $k$ is the thermal conductivity (units $\frac{\text{W}}{\text{m} \space ^\circ \text{C}}$), and $R$ is the radial dimension ($m$).

For a hollow cylinder of fur with no internal solar heat absorption, the heat flow is:

$$\begin{align} \tag{7} Q_{fur} = \frac{2 \pi k_{fur} L_g (T_s - T_{fa})}{\ln \left(\frac{R_{fa}}{R_s}\right)} \end{align}$$

where $k_{fur}$ is the fur thermal conductivity, $T_{fa}$ is the fur-air interface temperature and $R_{fa}$ is the radial distance from the centre of the animal to the fur-air interface. The length of the cylinder of flesh plus fat is represented by $L_g$.

Heat exchange via long wavelength radiation is quantified as:

$$\begin{align} \tag{8} Q_{rad} = \epsilon h_r A_{fa} (T_{rad,fur} - T_{rad,env}) \end{align}$$

where $\epsilon$ is the long wavelength emissivity, $h_r$ is a linear approximation of the difference of the 4th powers of the respective temperatures of the fur and environment, $h_r = 4 \sigma T^3_{ave}$, where $T_{ave} = \frac{T_{rad,fur} + T_{rad,env}}{2}$, $A_{fa}$ is the fur-air interface surface area, $T_{rad,fur}$ is the effective radiant temperature of the fur and $T_{rad,env}$ is the effective radiant temperature of the environment (e.g. sky, ground, bushes, and other nearby objects). Equation 8 closely approximates the exact Stefan-Boltzmann equation, $Q_{rad} = \epsilon \sigma A_{fa} (T^4_{rad,fur} - T^4_{rad,env})$, where $T$ is in Kelvin. There is only a 1.1% error when the temperature difference between $T_{rad,fur}$ and $T_{rad,env}$ is 60 $^\circ$C. This equation is expanded below in section XIV to account for configuration factors in different directions to different objects.

Heat loss by convection is quantified as:

$$\begin{align} \tag{9} Q_{conv} = h_c A_{fa} (T_{fa} - T_a) \end{align}$$

where $T_a$ is air temperature and $h_c$ is the heat transfer coefficient (units $\frac{\text{W}}{\text{m}^{2\circ} \text{C}}$), which is usually determined experimentally and is a function of the geometric shape, wind speed and fluid properties (see section IX).

Heat lost by conduction is:

$$\begin{align} \tag{10} Q_{cond} = \frac{A_{cond} k_{sub}}{x_{sub}} (T_{fa} - T_{sub}) \end{align}$$

where $A_{cond}$ is the surface area of the animal in contact with the substrate, $T_{sub}$ is the substrate temperature, and $x_{sub}$ is the depth of the substrate from which the substrate temperature is taken.

The heat gained by solar radiation is:

$$\begin{align} \tag{11} Q_{sol} = A_{fa} \alpha I_{sol} \end{align}$$

where $\alpha$ is the fur solar absorptivity and $I_{sol}$ is the incoming solar radiation ($\frac{\text{W}}{\text{m}^2}$). This is expanded in section VIII to be explicit about diffuse and direct components, from different directions.

The heat lost by evaporation is:

$$\begin{align} \tag{12} Q_{evap} = h_d A_{wet} (\rho_s- \rho_a) \lambda \end{align}$$

where $A_{wet}$ is the surface area acting as a free-water surface, $h_d$ is the mass transfer coefficient, $\rho_s$ and $\rho_a$ are the water vapour densities at the skin surface and in the free stream air (outside the animal’s boundary layer), respectively, and $\lambda$ is the latent heat of vaporization.

Finally, the heat lost by respiration is:

$$\begin{align} \tag{13} Q_{resp} = (\dot{J}_{out} - \dot{J}_{in}) \lambda \end{align}$$

where $\dot{J}_{in}$ and $\dot{J}_{out}$ are the mass flows of water in and out of the lungs, respectively.

IV. Strategy to solve for the skin and fur-air interface temperatures

The heat balance equation (eq. 5) permits the calculation of the required metabolic rate for an animal to maintain a target body temperature under a given set of microclimate conditions. As illustrated in the solution below and in the terms of the previous section, the heat flux through each layer (core to skin; skin to fur-air interface; fur-air interface to environment) is dependent on the temperature gradient existing in each layer, the thermal conductivity of each layer, and the shape and dimensions of each layer. However, the only known temperatures for the model animals are the core temperature and the surrounding air temperature. We must calculate the skin temperature and the fur-air interface temperature in order to solve the heat balance and determine what metabolic rate is required for the animal to maintain its body temperature in its current environmental conditions.

In developing a solution for this heat budget, the following assumptions were made:

1. All solar is absorbed at the coat surface (i.e., does not penetrate into fur layer). This is a more appropriate assumption for birds than it is for mammals. The incorporation of variable penetration of solar radiation into fur is in development.

2. Evaporative heat loss $Q_{evap}$ takes place at the skin surface and thus this component of $Q_{gen,net}$ does not pass through the fur layer (i.e. equation 1), but additional evaporative heat loss may also occur from the fur surface via the term $Q_{evap,fur}$.

3. The remainder of net metabolic heat generation that is not lost through evaporation must be conducted and radiated through the fur/feather layer and then be convected, conducted and radiated away from the fur surface.

4. There is negligible free convection within the fur layer.

5. Respiratory heat loss does not affect skin or fur-air interface temperature.

From the above we can develop two equations for $T_s$ in the context of metabolic heat generation, one with and one without reference to $T_{fa}$:

From equation 6:

$$\begin{align} \tag{14} T_s = T_c - \frac{Q_{gen,net} R^2_g}{4 k_g V_g} - \frac{Q_{gen,net} R^2_g}{2 k_i V_g} \ln\left(\frac{R_s}{R_g}\right). \end{align}$$

From equation 7, but with $2 \pi L_g$ restated as $\frac{V_g}{R^2_g}$:

$$\begin{align} \tag{15} T_s = \frac{Q_{fur} R^2_g}{2 k_{fur} V_g} \ln\left(\frac{R_{fa}}{R_s}\right) + T_{fa} \end{align}$$

and from equation 1 (i.e. assumption 2 above):

$$\begin{align} \tag{16} T_s = \frac{(Q_{gen,net} - Q_{evap}) R^2_g}{2 k_{fur} V_g} \ln\left(\frac{R_{fa}}{R_s}\right) + T_{fa}. \end{align}$$

Formulating equations in relation to the temperature of the fur-air interface

The challenge is to solve for $T_{fa}$ to calculate $Q_{rad}$, $Q_{conv}$, $Q_{evap,fur}$ and $Q_{cond}$, without using $T_s$ which, along with $Q_{gen}$, is an unknown value.

One important step for doing this is to write the heat balance in terms of $T_{fa}$ without $T_s$. This was derived in Mathewson and Porter (2013), but $Q_{cond}$ was not included. Here we derive the following equation for $T_{fa}$ that includes conduction.

First, the mechanism equation for $Q_{fur}$ needs to be modified to account for the fact that for an animal lying on the ground the portion of the surface in contact with the substrate will have its fur compressed, resulting a different fur thermal conductivity, $k_f$, and fur depth, which will in turn result in a different core-fur surface radius, $R_{fa}$.

Thus, incorporating conduction in the mechanism equation for $Q_{fur}$ described above in equation 7:

$$\begin{align} \tag{17} Q_{fur}= \left[\frac{2 \pi L_g k_{f,cmp}}{\ln\left(\frac{R_{fa,cmp}}{R_s}\right)} \left( T_s-T_{fa,cmp}\right) \right] PCOND + \left[\frac{2 \pi L_g k_f }{\ln\left(\frac{R_{fa}}{R_s}\right)} \left( T_s-T_{fa}\right) \right] \left(1 - PCOND \right) \end{align}$$

where $T_{fa,cmp}$ is the fur surface temperature for the area in contact with the substrate, $PCOND$ is the proportion of the surface area that is in contact with the substrate, $k_{f,cmp}$ is the thermal conductivity of the compressed fur, and $R_{fa,cmp}$ is the core-to-fur surface radius for the portion of the animal in contact with the substrate.

With expansion and factoring this equation becomes:

$$\begin{align} \tag{18} Q_{fur} = 2 \pi L_g \\ \left[ T_s \left(\frac{k_{f,cmp}}{\ln\left(\frac{R_{fa,cmp}}{R_s}\right)} PCOND + \frac{k_f}{\ln\left(\frac{R_{fa}}{R_s}\right)} \left(1 - PCOND\right) \right) - T_{fa,cmp} \frac{k_{f,cmp}}{\ln\left(\frac{R_{fa,cmp}}{R_s}\right)} PCOND - T_{fa} \frac{k_f}{\ln\left(\frac{R_{fa}}{R_s}\right)} \left(1 - PCOND\right) \right] \end{align}$$

where we can define new constants:

$$\begin{align} \tag{19} C_{d1} = \left(\frac{k_{f,cmp}}{\ln\left(\frac{R_{fa,cmp}}{R_s}\right)} PCOND + \frac{k_f}{\ln\left(\frac{R_{fa}}{R_s} \right)} \left(1 - PCOND \right) \right) \end{align}$$

with units $\text{W} / \text{m} ^\circ \text{C}$,

$$\begin{align} \tag{20} C_{d2} = \frac{k_{f,cmp}}{\ln\left(\frac{R_{fa,cmp}}{R_s}\right)} PCOND \end{align}$$

with units $\text{W} / \text{m} ^\circ \text{C}$, and

$$\begin{align} \tag{21} C_{d3} = \frac{k_f}{\ln\left(\frac{R_{fa}}{R_s}\right)} \left(1 - PCOND \right) \end{align}$$

with units $\text{W} / \text{m} ^\circ \text{C}$,

Modifying equation 16 to account for conduction:

$$\begin{align} \tag{22} T_s = \frac{(Q_{gen,net} - Q_{evap}) R^2_g}{2 V_g C_{d1}} + \frac{T_{fa,cmp} C_{d2}}{C_{d1}} + \frac{T_{fa} C_{d3}}{C_{d1}}. \end{align}$$

Overall, accounting for conduction, the heat balance in terms of $T_{fa}$ without $T_s$ is:

$$\begin{align} \tag{23} T_{fa} = \frac{\frac{2 \pi L_g}{D_1} (T_c C_{d1} - D_2 - T_{fa,cmp}C_{d2})-h_r A_{fa} \frac{D_3}{D_4}-h_r A_{fa} \frac{T_{fa,cmp} C_{d2}}{D_4}}{\frac{2 \pi L_g C_{d3}}{D_1}+h_r A_{fa} \frac{\left(\frac{k_f}{ln\left(\frac{R_{fa}}{R_{rad}}\right)} \left(1 - PCOND\right)\right)}{D_4}+h_c A_{fa}}\\ + \frac{h_r A_{fa} T_{env} + h_c A_{fa} T_a - C_d T_{fa,cmp} +C_d T_{sub}-Q_{evap,fur} + Q_{sol}}{\frac{2 \pi L_g C_{d3}}{D_1}+h_r A_{fa} \frac{\left(\frac{k_f}{ln\left(\frac{R_{fa}}{R_{rad}}\right)} \left(1 - PCOND\right)\right)}{D_4}+h_c A_{fa}} \end{align}$$

This is achieved by the following steps:

Step 1 Setting up the balance in terms of the temperature of the fur-air interface

Equation 3, $Q_{fur} = Q_{rad} + Q_{conv} + Q_{cond} - Q_{sol}$, can be written using the mechanism equations defined above as:

$$\begin{align} \tag{24} 2 \pi L_{g} \left(T_s C_{d1}-T_{fa,cmp} C_{d2}-T_{fa} C_{d3}\right) = \epsilon h_r A_{fa} (T_{rad,fur} - T_{env}) + h_c A_{fa} (T_{fa} - T_a) + \\ C_d (T_{fa,cmp} - T_{sub}) + Q_{evap,fur} - Q_{sol} \end{align}$$

where the constant $\frac{A_{cond} k_{sub}}{x_{sub}}$ in $Q_{cond}$ abbreviated as $C_d$, with units $\text{W} / ^\circ \text{C}$.

Step 2 Eliminating skin temperature

First, to eliminate $T_s$ from $Q_{fur}$, we know from the body that $T_s = T_c - \frac{Q_{gen,net} R^2_g}{4 k_g V_g} - \frac{Q_{gen,net} R^2_g}{2 k_i V_g}\ln\left(\frac{R_{s}}{R_g}\right)$ (eq. 14), and we also know from the steady state that $Q_{gen,net} - Q_{evap} = Q_{fur}$ (eq. 1). Thus we can say:

$$\begin{align} \tag{25} T_s = T_c - \frac{(Q_{fur} + Q_{evap}) R^2_g}{4 k_g V_g} - \frac{(Q_{fur} + Q_{evap}) R^2_g}{2 k_i V_g}\ln\left(\frac{R_{s}}{R_g}\right). \end{align}$$

Thus $Q_{fur}$ can be rewritten as:

$$\begin{align} \tag{26} Q_{fur} = 2 \pi L_{g} \left[\left(T_c - \frac{(Q_{fur} + Q_{evap}) R^2_g}{4 k_g V_g} - \frac{(Q_{fur} + Q_{evap}) R^2_g}{2 k_i V_g}\ln\left(\frac{R_{s}}{R_g}\right)\right) C_{d1}-T_{fa,cmp} C_{d2}-T_{fa} C_{d3}\right]. \end{align}$$

Then, by expanding and collecting $Q_{fur}$ terms and factoring out the $Q_{fur}$ terms, we obtain:

$$\begin{align} \tag{27} Q_{fur} \left[1 + \frac{2 \pi L_g R^2_g C_{d1}}{4 k_g V_g}+\left(\frac{2 \pi L_g R^2_g C_{d1}}{2 k_i V_g}\ln\left(\frac{R_{s}}{R_g}\right)\right)\right] \\ = 2 \pi L_g \left[T_c C_{d1} - \frac{Q_{evap} R^2_g C_{d1}}{4 k_g V_g} - \frac{Q_{evap} R^2_g C_{d1}}{2 k_i V_g}\ln\left(\frac{R_{s}}{R_g}\right) - T_{fa,cmp} C_{d2} - T_{fa} C_{d3}\right] \end{align}$$

Defining the following constants:

$$\begin{align} \tag{28} D_1 = \left[1 + \frac{2 \pi L_g R^2_g C_{d1}}{4 k_g V_g}+\left(\frac{2 \pi L_g R^2_g C_{d1}}{2 k_i V_g}\ln\left(\frac{R_{s}}{R_g}\right)\right)\right] \end{align}$$

which is dimensionless, and

$$\begin{align} \tag{29} D_2 = \left[\frac{Q_{evap} R^2_g C_{d1}}{4 k_g V_g} + \frac{Q_{evap} R^2_g C_{d1}}{2 k_i V_g}\ln\left(\frac{R_{s}}{R_g}\right)\right] \end{align}$$

which has units of $\text{W} / \text{m}$, we thereby obtain

$$\begin{align} \tag{30} Q_{fur} = \frac{2 \pi L_g T_c C_{d1}}{D_1} - \frac{2 \pi L_g D_2}{D_1} - \frac{2 \pi L_g T_{fa,cmp} C_{d2}}{D_1} - \frac{2 \pi L_g T_{fa} C_{d3}}{D_1} = \frac{2 \pi L_g}{D_1} (T_c C_{d1} - D_2 - T_{fa,cmp}C_{d2} - T_{fa} C_{d3}). \end{align}$$

Now, inserting the new definition of $Q_{fur}$ into equation 24 we obtain:

$$\begin{align} \tag{31} \frac{2 \pi L_g}{D_1} (T_c C_{d1} - D_2 - T_{fa,cmp}C_{d2} - T_{fa} C_{d3}) = h_r A_{fa} (T_{rad,fur} - T_{env}) + h_c A_{fa} (T_{fa} - T_a) + \\ C_d (T_{fa,cmp} - T_{sub}) + Q_{evap,fur} - Q_{sol}. \end{align}$$

Step 3: Writing $T_{rad,fur}$ in terms of $T_{fa}$

We can define the radius at which longwave radiation is effectively leaving the fur, $R_{rad}$, as:

$$\begin{align} \tag{32} R_{rad} = R_s + \left[ X_r (R_{fa} - R_s) \right] \end{align}$$

with $X_r$ being a fractional depth into the fur where radiant heat exchange takes place (1 - at the fur surface, 0.5 at the midpoint of the fur depth). For many animals, this may not be actual fur surface since IR can “see” into the fur for some given depth. Thus, the temperature gradient for $Q_{rad}$ may not be $T_{fa}-T_{rad,env}$. It could be something closer to $T_{s}-T_{rad,env}$. The $X_r$ term allows the user to choose where in the fur the radiant exchange takes place, and based on that choice, the $T_{rad,env}$ term is somewhere along the fur temperature profile between $T_{s}$ and $T_{fa}$.

Given the fur temperature profile, where the temperature of the fur $T_{fur}$ at a depth in the profile $r$ is

$$\begin{align} \tag{33} T_{fur} = \frac{\frac{(Q_{gen,net}-Q_{evap}) R^2_g}{2 V_g} + T_{fa,cmp} C_{d2} + T_{fa} \left(\frac{k_f}{\ln\left(\frac{R_{fa}}{r}\right)} \left(1-PCOND \right) \right)}{C_{d2} + \frac{k_f}{\ln\left(\frac{R_{fa}}{r}\right)} \left(1-PCOND \right)} \end{align}$$

then

$$\begin{align} \tag{34} T_{rad} = \frac{\frac{(Q_{gen,net}-Q_{evap}) R^2_g}{2 V_g} + T_{fa,cmp} C_{d2} + T_{fa} \left(\frac{k_f}{\ln\left(\frac{R_{fa}}{R_{rad}}\right)} \left(1-PCOND \right) \right)}{C_{d2} + \frac{k_f}{\ln\left(\frac{R_{fa}}{R_{rad}}\right)} \left(1-PCOND \right)} \end{align}$$

where $Q_{gen,net} - Q_{evap} = Q_{fur} = \frac{2 \pi L_g}{D_1}(T_c C_{d1} - D_2 - T_{fa,cmp}C_{d2} - T_{fa} C_{d3})$, thus

$$\begin{align} \tag{35} T_{rad} = \frac{\left(\frac{2 \pi L_g}{D_1}\left(T_c C_{d1} - D_2 - T_{fa,cmp} C_{d2} - T_{fa} C_{d3} \right)\right) R^2_g}{2 V_g \left(C_{d2} + \frac{k_f}{\ln\left(\frac{R_{fa}}{R_{rad}}\right)} \left(1-PCOND \right) \right)} \\ + \frac{T_{fa,cmp} C_{d2}}{C_{d2} + \frac{k_f}{\ln\left(\frac{R_{fa}}{R_{rad}}\right)} \left(1-PCOND \right)}+ \frac{T_{fa}\left(\frac{k_f}{\ln\left(\frac{R_{fa}}{R_{rad}}\right)} \left(1-PCOND \right) \right)}{C_{d2} + \frac{k_f}{\ln\left(\frac{R_{fa}}{R_{rad}}\right)} \left(1-PCOND \right)} \end{align}$$

where we can define new constants:

$$\begin{align} \tag{36} D_3 = \frac{\left(\frac{2 \pi L_g}{D_1}\left(T_c C_{d1} - D_2 - T_{fa,cmp}C_{d2} - T_{fa} C_{d3} \right)\right) R^2_g}{2 V_g} \end{align}$$

with units $\text{W} / \text{m}$,

$$\begin{align} \tag{37} D_4 = C_{d2} + \frac{k_f}{\ln\left(\frac{R_{fa}}{R_{rad}}\right)} \left(1-PCOND \right) \end{align}$$

with units $\text{W} / \text{m} ^\circ \text{C}$,

thus

$$\begin{align} \tag{38} Q_{rad} = h_r A_{fa} \left( \left( \frac{D_3}{D_4} + \frac{T_{fa,cmp} C_{d2}}{D_4} + \frac{\frac{k_f}{\ln\left(\frac{R_{fa}}{R_{rad}}\right)} \left(1-PCOND \right)}{D_4} \right) - T_{rad,env}\right) \end{align}$$

and so $Q_{fur} = Q_{rad} + Q_{conv} + Q_{cond} + Q_{evap,fur} - Q_{sol}$ can be written with $T_s$ removed and $T_{rad,fur}$ in terms of $T_{fa}$:

$$\begin{align} \tag{39} \frac{2 \pi L_g}{D_1} (T_c C_{d1} - D_2 - T_{fa,cmp}C_{d2} - T_{fa} C_{d3}) = \\ h_r A_{fa} \left( \left( \frac{D_3}{D_4} + \frac{T_{fa_cmp C_{d2}}}{D_4} + \frac{\frac{k_f}{\ln\left(\frac{R_{fa}}{R_{rad}}\right)} \left(1-PCOND \right)}{D_4} \right) - T_{rad,env}\right) + h_c A_{fa} (T_{fa} - T_a) + \\ C_d (T_{fa,cmp} - T_{sub}) + Q_{evap,fur} - Q_{sol}. \end{align}$$

Finally, expanding, rearranging and solving equation 39 for $T_{fa}$ we obtain equation 23.

This overall setup, i.e. equations 14, 22 and 23, are used in the subroutine SIMULSOL below to iteratively guess a $Q_{gen}$, $T_s$ and $T_{fa}$ that balances the heat budget for a given $T_c$.

Note: when $X_R = 1$, $\text T_{r} = T_{fa}$ and $\text R_{rad} = R_{fa}$. To avoid zeros in the denominator of some terms, eq. 31 is rewritten as:

$$\begin{align} \tag{40} \frac{2 \pi L_g}{D_1} (T_c C_{d1} - D_2 - T_{fa,cmp}C_{d2} - T_{fa} C_{d3}) = h_r A_{fa} (T_{fa} - T_{env}) + h_c A_{fa} (T_{fa} - T_a) + \\ C_d (T_{fa,cmp} - T_{sub}) + Q_{evap,fur} - Q_{sol}. \end{align}$$

And rearranging and solving for $\text T_{fa}$:

$$\begin{align} \tag{41} T_{fa} = \frac{\frac{2 \pi L_g}{D_1} (T_c C_{d1} - D_2- T_{fa,cmp}C_{d2})+h_r A_{fa} T_{env} + h_c A_{fa} T_a - C_d T_{fa,cmp} +C_d T_{sub} - Q_{evap,fur} + Q_{sol}}{\frac{2 \pi L_g C_{d3}}{D_1}+h_r A_{fa}+h_c A_{fa}} \end{align}$$

V. IRPROP Subroutine

This subroutine tests for the presence of fur and coordinates the computation of the fur properties. It ultimately obtains the parameter $k_{eff}$, the effective conductivity of the fur, from which $k_{fur}$ is determined, and the average absorption coefficient $\beta$ and optical depth $\beta_1$, via the subroutine GETKFUR.

The IRPROP subroutine obtains these properties for the dorsal and ventral sides of the organism, as well as their weighted average based on the user input of the maximum percentage ventral fur. Note that the actual fraction of ventral fur may change as the organism changes from a curled to an uncurled posture, so the user may alter the percentage of ventral fur, $p_{vent}$, accordingly. In the SIMULSOL calculations, the entire body is first assumed to be ‘dorsal’ and then assumed to be ‘ventral’. This is accounted for in this subroutine by doubling the ventral percentage when weighting the ventral fur properties.

The $p_{vent}$ value is used in the weighted mean calculation of the fur density $\rho_{hair}$, hair diameter $D_{hair}$, hair length $L_{hair}$, fur depth $z_{fur}$ and absorptivity of the fur $\alpha_{fur}$.

The IRPROP function loops through the calculated average, dorsal and ventral fur properties and calls GETKFUR to obtain the conductive ($k_{eff}$) and longwave radiation ($\beta$ and $\beta_1$) fur properties for each and returns them. It also computes the conductivity of compressed fur for computing conduction to the substrate.

VI. GETKFUR Subroutine

This subroutine takes as input the density and depth of the fur, its length, diameter and thermal conductivity, and computes $k_{eff}$, $\beta$ and $\beta_1$ as follows (from Conley and Porter 1986, modified from Kowalski 1976). Note that the optical depth, $\beta_1$, can be used to inform the value the user inserts for where in the fur is the effective fur temperature that is radiating to the environment.

The conductivity of the fur is computed along the horizontal direction $x(k^{\prime}_x)$ as a surface area weighted mean of air ($k_{air}$) and hair conductivity ($k_{hair}$):

$$\begin{align} \tag{42} k^{\prime}_x = A_{hair} k_{hair} + (1 - A_{hair}) k_{air} \end{align}$$

where

$$\begin{align} \tag{43} A_{hair} = \rho_{hair} \pi \left(\frac{D_{hair}}{2}\right)^2 \end{align}$$

and $k_{air} = 0.02425+7.038\text{e-}5 \space T_a$, from the DRYAIR subroutine described in Properties of Air.

In the vertical direction $y(k^{\prime}_y)$, conductivity is expressed as

$$\begin{align} \tag{44} k^{\prime}_y = \rho_{hair,eff}^{\frac{1}{2}} k_{air} (\rho^{-\frac{1}{2}}_{hair,eff}-D_{hair}) + \frac{D_{hair} k_{hair} k_{air}}{D_{hair} k_{air} + (\rho^{-\frac{1}{2}}_{hair,eff}-D_{hair})k_{hair}} \end{align}$$

where the effective hair density is

$$\begin{align} \tag{45} \rho_{hair,eff} = \rho_{hair} (L_{hair} / z_{fur}). \end{align}$$

Then, the effective conductivity of the fur is the average of these:

$$\begin{align} \tag{46} k_{eff} = \frac{k^{\prime}_x + k^{\prime}_y}{2}. \end{align}$$

The average absorption coefficient, used in SIMULSOL to compute the conductivity to longwave radiation $k_{rad}$, is

$$\begin{align} \tag{47} \beta = \frac{0.67}{\pi} \rho_{hair,eff} D_{hair} \end{align}$$

and the optical thickness is

$$\begin{align} \tag{48} \beta_1 = \beta z_{fur}. \end{align}$$

The subroutine ensures that $k_{eff}$ does not go below $k_{air}$ as well as correcting for situations where hair diameter is impossibly large for the specified density, in which case $k_{eff}$ is capped at $k_{hair}$.

VII. GEOM_ENDO Subroutine

This subroutine computes various lengths, areas, masses and volumes given the shape chosen, which presently includes a cylinder, sphere or ellipsoid. Specifically it computes:

• $V_{body}$, the total body volume, $\text{m}^3$
• $V_{fat}$, the volume of body fat, $\text{m}^3$
• $V_{g}$, the volume of body flesh, $\text{m}^3$
• $M_{fat}$, the mass of body fat, kg
• $z_{fat}$, fat layer thickness, m
• $D$, characteristic dimension for convection, m
• $L_{body}$, body length (flesh plus fat), m
• $W_{body}$, body width (flesh plus fat), m
• $H_{body}$, body height (flesh plus fat), m
• $R_s$, shape-specific core-skin radius in shortest dimension, m
• $R_{fa}$, shape-specific core-fur radius in shortest dimension, m
• $A_{maj}$, a semimajor axis length (ellipsoid), m
• $B_{min}$, b semiminor axis length (ellipsoid), m
• $C_{min}$, c semiminor axis length (ellipsoid, currently only prolate spheroid), m
• $H$, height, m
• $A_{tot}$, total area at fur/feathers-air interface, $\text{m}^2$
• $A_{sil}$, silhouette area for solar radiation, $\text{m}^2$ (may be normal, parallel or in between)
• $A_{sil,n}$, silhouette area normal to sun, $\text{m}^2$
• $A_{sil,p}$, silhouette area parallel to sun, $\text{m}^2$
• $A_{s}$, area of skin, $\text{m}^2$
• $A_{evap}$, area of skin for evaporation (total skin area - hair area), $\text{m}^2$
• $A_{conv}$, area for convection (total area minus ventral area, as determined by $p_{cond}$), $\text{m}^2$

These calculations are noted for each specific shape type in the sub-sections below. The relative dimensions for each shape are determined by the input parameters $S_b$ and $S_c$ which are ratios between axes of a given shape which, together with the specified mass $M$ and density $\rho_{body}$ and hence body volume ($V_{body} = \frac{M}{\rho_{body}}$), can be converted into absolute dimensions for each axis.

The mass of fat is $M_{fat} = p_{fat} M$ where the proportion fat $p_{fat}$ is a user-specified parameter. The volume of fat is thus $V_{fat} = M_{fat} / \rho_{fat}$ where fat density is assumed to be 901 $\text{kg}/\text{m}^3$ (Going, 1996).

For all shapes, the characteristic dimension for convection $D$ is assumed to be the cube root of the volume following Mitchell (1976):

$$\begin{align} \tag{49} D = V_{body}^{\frac{1}{3}}. \end{align}$$

Also, for all shapes, once $A_{s}$ is computed, if fur is present the area for evaporative water loss is computed by subtracting the hair area using the weighted average hair properties between the dorsal and ventral sides:

$$\begin{align} \tag{50} A_{evap} = A_{s} - A_{s}\pi \left(\frac{D_{hair}}{2}\right)^2 \rho_{hair}. \end{align}$$

Otherwise $A_{evap} = A_{s}$.

Similarly, the area for convection is corrected for the proportion of the area conducting to the substrate $p_{cond}$, i.e.

$$\begin{align} \tag{51} A_{conv} = A_{tot} - A_{tot} p_{cond}. \end{align}$$

Otherwise $A_{conv} = A_{tot}$.

VIII. SOLAR_ENDO Subroutine

This subroutine computes absorbed solar radiation $Q_{sol}$ for both the direct (point source) and diffuse (scattered) components reaching the dorsal and ventral halves of the organism. The direct solar beam radiation is

$$\begin{align} \tag{74} Q_{sol,dir}=\alpha_{dors} A_{sil} (1-p_{dif}) Q_{sol,norm} (1 - p_{shd}) \end{align}$$

where $\alpha_{dors}$ is the solar absorptivity of the dorsal area, $A_{sil}$ is the silhouette area (calculated above in GEOM), $Q_{sol,norm} = I_{sol}/\cos{Z}$ where $Z$ is the solar zenith angle in radians, $I_{sol}$ is the inputted horizontal plane solar radiation flux ($\text{W}/\text{m}^2$) and $p_{shd}$ is the fractional shade level.

The diffuse component from the sky is

$$\begin{align} \tag{75} Q_{sol,sky}=\alpha_{dors} F_{sky} A_{tot} p_{dif} Q_{sol} (1 - p_{shd}) \end{align}$$

where $F_{sky}$ is the configuration factor from animal to sky.

The diffuse component from the substrate is

$$\begin{align} \tag{76} Q_{sol,sub}=\alpha_{vent} F_{sub} A_{tot} p_{dif} (1-\alpha_{sub}) Q_{sol} (1 - p_{shd}) \end{align}$$

where $\alpha_{vent}$ is the solar absorptivity of the ventral area, $F_{sub}$ is the configuration factor from animal to substrate and $\alpha_{sub}$ is the solar absorptivity of the substrate.

Thus the total diffuse radiation absorbed $Q_{sol,dif} = Q_{sol,sky} + Q_{sol,sub}$, the dorsal solar absorbed is $Q_{sol,dors} = Q_{sol,dir} + Q_{sol,sky}$, and the ventral solar absorbed is $Q_{sol,vent} = Q_{sol,sub}$.

IX. CONV_ENDO Subroutine

This subroutine computes the heat exchange by convection, $Q_{conv}$, as sum of the free (driven by the surface surface-free stream air temperature gradient) and forced (driven by wind speed) components. As indicated in equation 9, this requires calculation of the heat transfer coefficient

$$\begin{align} \tag{77} h_c=\frac{Nu \space k_{fluid}}{D} \end{align}$$

where $D$ is the characteristic dimension (calculated in GEOM, eq. 49), $Nu$ is the Nusselt number and $k_{fluid}$ is the thermal conductivity of the fluid. The mass transfer coefficient $h_d$ can be computed from $h_c$ and used to quantify evaporative heat transfer, as described at the end of this section.

This subroutine allows the fluid to be air, fresh water or sea water. The thermal properties of dry air (which depend on temperature and pressure), including thermal conductivity, are computed by a call to the function DRYAIR which is part of the NicheMapR package and is fully described in Properties of Air. The thermal properties of fresh water and sea water (which depend on temperature) are obtained from the functions WATER and SEAWATER, respectively, described below.

The heat transfer coefficient is determined differently for the forced and free components.

X. WATER Subroutine

This subroutine uses regressions fitted to relationships documented in List (1966).

For the specific heat capacity of freshwater:

$$\begin{align} \tag{99} C_{p,water}=4220.02-4.5531 \space T_{water}+0.182958 \space T_{water}^2 - 0.00310614 \space T_{water}^3 + \\ 1.89399\text{e-}5 \space T_{water}^4. \end{align}$$

For the thermal conductivity:

$$\begin{align} \tag{100} k_{fluid}=0.551666+0.00282144 \space T_{water}-2.02383\text{e-}5 \space T_{water}^2. \end{align}$$

For dynamic viscosity:

$$\begin{align} \tag{101} \mu_{fluid}=0.0017515-4.31502\text{e-}5 \space T_{water}+3.71431E-7\text{e-}7 \space T_{water}^2. \end{align}$$

For density ($\text{kg} / \text{m}^3$), water temperature is capped at 60 $^{\circ}\text{C}$, with

$$\begin{align} \tag{102} \rho_{fluid}=1017-0.6 T_{water} \end{align}$$

for $T_{water}\ge30^{\circ}\text{C}$ and $\rho_{fluid}=1000$ for $T_{water}<30^{\circ}\text{C}$.

XI. SEAWATER Subroutine

For the specific heat capacity of seawater:

$$\begin{align} \tag{103} C_{p,water}=3913.1369+1.0211143 \space T_{water}. \end{align}$$

For the thermal conductivity:

$$\begin{align} \tag{104} k_{fluid}=6.751631+0.018687 \space T_{water}. \end{align}$$

For dynamic viscosity:

$$\begin{align} \tag{105} \mu_{fluid}=6.751631-2.6\text{e-}5 \space T_{water}. \end{align}$$

For density:

$$\begin{align} \tag{106} \rho_{fluid}=1029.82-0.27907 T_{water}. \end{align}$$

XIII. SEVAP_ENDO Subroutine

This subroutine computes aspects of evaporative heat loss including the term $Q_{evap}$ from equation 12 in Section III above. It takes as input the relative humidity $\text{RH}$, barometric pressure $P_{bar}$, air temperature $T_a$, the wind speed $v$, as well the mass transfer coefficient computed in CONV_ENDO $h_d$ (eq. 98, section IX above) and the skin surface area $A_s$ computed in GEOM. The WETAIR subroutine and its sub-function VAPPRS, described in Properties of Air is used to compute the skin (saturation) and air vapour densities, $\rho_s$ and $\rho_a$.

Evaporation may occur from the skin but also from the eyes, if they are assumed to be open, and from the fur or feathers, if wet from rain, licking or other sources. Evaporation from furred/feathered skin can occur via free convection-driven mass transport only, whereas both forced and free convection will drive evaporation from bare skin or from the fur/feather surface. Thus the effective area of the skin for evaporation, $p_{wet} A$, will vary for these different scenarios.

Thus, cutaneous water loss is partitioned as

$$\begin{align} \tag{107} \dot{M}_{\text{H}_2\text{O},furskin} = h_{d,free} p_{wet} (1 - p_{bare}) A_s (\rho_s - \rho_a). \end{align}$$

and

$$\begin{align} \tag{108} \dot{M}_{\text{H}_2\text{O},bareskin} = h_{d} p_{wet} p_{bare} A_s (\rho_s - \rho_a) \end{align}$$

where $p_{bare}$ is a user input determining the proportion of the free-water surface (not total skin surface) that is bare skin. Thus $\dot{M}_{\text{H}_2\text{O},cut} = \dot{M}_{\text{H}_2\text{O},furskin} + \dot{M}_{\text{H}_2\text{O},bareskin}$.

If the organism is in a state of flight (bird/bat), it is assumed that forced and free mass transport drive evaporation (i.e., effectively $p_{bare} = 1$).

If the eyes are open, an additional amount contributes to non-respiratory evaporation:

$$\begin{align} \tag{109} \dot{M}_{\text{H}_2\text{O},eyes}=h_d p_{eyes} A_s (\rho_s - \rho_a). \end{align}$$

Finally, the fraction of the fur/feather surface area that is wet is determined by the user input $p_{wetfur}$ and

$$\begin{align} \tag{110} \dot{M}_{\text{H}_2\text{O},wetfur} = h_{d} p_{wetfur} A_{tot} (\rho_s - \rho_a). \end{align}$$

The subroutine reports the separate mass losses of water as well as the heat lost by evaporation from the skin/eyes and the fur.

XIV. SIMULSOL Subroutine

This subroutine simultaneously solves for the skin and fur-air interface temperature that balances the heat budget for a non-respiring part of the body, as described above, accounting for dorsal and ventral differences and evaporation from the skin. It is the core of the endotherm model.

There are two sections to this subroutine, one for the case that fur/feathers are present, and one for when the skin is bare. The most complex solution is for when fur is present, as there are three unknowns to solve for, as explained in section IV above. The approach taken in SIMULSOL is as follows (we do not describe the bare skin approach in detail here as it is a simpler version of the approach below):

XIV. RESPFUN

This subroutine is a molar balance for computing water loss from breathing, as represented in Figure 3. It uses the oxygen demand for maintaining a core temperature to compute the amount of air flowing in and out of the lungs and hence the heat lost by evaporation. The user can specify the ambient gas levels so that, for instance, conditions in a burrow can be simulated.

The required $\text{O}_2$ consumption rate at standard temperature and pressure, $\dot{V}_{O_{2},stp}$, is determined from the value of $Q_{gen}$ passed to the function via the SOLVENDO and ZBRENT subroutines (see below). If that value is lower than the specified minimum metabolic rate, $Q_{basal}$, then $Q_{basal}$ is used instead.

The conversion to litres of $\text{O}_2$ per second depends on the respiratory quotient, $\text{RQ}$ (ratio of moles of $\text{CO}_2$ produced per mole of $\text{O}_2$ consumed), as follows:

$$\begin{align} \tag{130} \dot{V}_{O_{2},stp}= \frac{Q_{gen} X_{act}}{4.185 \cdot 1000} X_{RQ} \end{align}$$

where $X_{act}$ is a user-specified activity multiplier, the denominator is a conversion factor from kilocalories to joules, and $X_{RQ}$ is a conversion factor for litres of $\text{O}_2$ per kilocalorie taken from Kleiber (1961) where, if $\text{RQ} = 1$, $X_{RQ} = 1/5.057$ (carbohydrate diet), if $\text{RQ} \le 0.7$, $X_{RQ} = 1/4.7$ (fat diet), or else if $X_{RQ} = 1/4.5$ (protein diet).

This value is then converted to actual volume consumed per second given the current air temperature and barometric pressure ($P_{bar}$):

$$\begin{align} \tag{131} \dot{V}_{O_2}= \frac{\dot{V}_{O_{2},stp} P_{bar,std}}{273.15} \frac{T_{a}+273.15}{P_{bar}}. \end{align}$$

The moles of $\text{O}_2$ consumed is then

$$\begin{align} \tag{132} \dot{J}_{\text{O}_2,con} = \frac{\dot{V}_{O_2} P_{bar}}{R(T_{a} + 273.15)} \end{align}$$

Where $R$ is the universal gas constant (= 8309.28 $\frac{\text{Pa} \space \text{Litres}}{\text{mol} \space \text{K}}$).

Then the moles of $\text{O}_2$ entering the respiratory system is at imaginary surface 1 in Fig. 3 is

$$\begin{align} \tag{133} \dot{J}_{\text{O}_2,1} = \frac{\dot{J}_{\text{O}_2,con}}{\kappa_{\text{O}_2}} \end{align}$$

where $\kappa_{\text{O}_2}$ is the fractional oxygen extraction efficiency of the lungs.

To convert this $\text{O}_2$ requirement into the molar flow of air through the lungs,

$$\begin{align} \tag{134} \dot{J}_{\text{air},1} = \dot{J}_{\text{O}_2,1} \frac{p_{\text{N}_2} + p_{\text{O}_2} + p_{\text{CO}_2}}{p_{\text{O}_2}} \frac{p_{\text{O}_2,ref}}{p_{\text{O}_2}} \frac{P_{\text{O}_2,ref}}{P_{\text{O}_2}} X_{pant} \end{align}$$

where $p_{\text{O}_2}$, $p_{\text{CO}_2}$ and $p_{\text{N}_2}$ are the proportions of each gas in the animal’s environment and $p_{\text{O}_2,ref}$, $p_{\text{CO}_2,ref}$ and $p_{\text{N}_2,ref}$ are the respective reference atmospheric proportions, $P_{\text{O}_2} = p_{\text{O}_2} P_{bar}$ is the partial pressure of $\text{O}_2$ and $P_{\text{O}_2,ref} = p_{\text{O}_2} P_{bar,ref}$ is the reference partial pressure of $\text{O}_2$. The atmospheric proportions of $\text{O}_2$ and $\text{N}_2$ are fixed at 0.2095 and 0.7902, respectively, while that of $\text{CO}_2$ is a user input to account for anthropogenic emissions changes. $X_{pant}$ is a multiplier to allow for enhanced respiratory heat loss via panting as part of the thermoregulatory response.

The volume of air flowing through the lungs per second is then

$$\begin{align} \tag{135} \dot{V}_{air} = \dot{J}_{\text{air},1} R \frac{273.15}{101325}. \end{align}$$

The moles of water entering the respiratory system can then be computed as

$$\begin{align} \tag{136} \dot{J}_{\text{H}_2\text{O},1} = \dot{V}_{air} \frac{P_{\text{H}_2\text{O},sat} \frac{\text{RH}}{100}}{P_{bar} - P_{\text{H}_2\text{O},sat} \frac{\text{RH}}{100}} \end{align}$$

where $P_{\text{H}_2\text{O},sat}$, the saturation water vapour pressure, is computed via the WETAIR subroutine and its sub-function VAPPRS, as described in Properties of Air.

The moles of $\text{O}_2$ exiting the respiratory system (imaginary surface 2, Fig. 3) is

$$\begin{align} \tag{137} \dot{J}_{\text{O}_2,2} = \dot{J}_{\text{O}_2,1} - \dot{J}_{\text{O}_2,con}. \end{align}$$

The moles of air at exit (imaginary surface 2) will be approximately the same at entrance, since the moles of $\text{O}_2$ removed is nearly the same as the moles of $\text{CO}_2$ added. The molar air flow rate at exit is

$$\begin{align} \tag{138} \dot{J}_{\text{air},2} = (\dot{J}_{\text{O}_2,2} + \dot{J}_{\text{CO}_2,2}) \frac{p_{\text{N}_2} + p_{\text{O}_2}}{p_{\text{O}_2}} \frac{p_{\text{O}_2,ref}} {p_{\text{O}_2}} \frac{P_{\text{O}_2,ref}}{P_{\text{O}_2}} X_{pant} \end{align}$$

where $\dot{J}_{\text{CO}_2,2} = \text{RQ} \cdot \dot{J}_{\text{O}_2,2}$.

The moles per second of water leaving the organism is

$$\begin{align} \tag{139} \dot{J}_{\text{H}_2\text{O},2} = \dot{J}_{\text{air},2} \frac{P_{\text{H}_2\text{O},sat}}{P_{bar} - P_{\text{H}_2\text{O},sat}}. \end{align}$$

The mass of water lost in kilograms per second is then

$$\begin{align} \tag{140} \dot{M}_{\text{H}_2\text{O},2} = 18 \cdot (\dot{J}_{\text{H}_2\text{O},2} - \dot{J}_{\text{H}_2\text{O},1}) \cdot 1000 \end{align}$$

where the value 18 is the molar mass of water.

Finally, the respiratory heat lost is

$$\begin{align} \tag{141} Q_{resp} = \lambda \dot{M}_{\text{H}_2\text{O},2} - Q_{air} \end{align}$$

where $\lambda$ is the latent heat of vaporisation, computed as $2.5012\text{e+}06 - 2.3787\text{e+}03 \cdot T_{lung}$, with $T_{lung}$ assumed to be the average of the current skin and core temperature, and $Q_{air} = C_{p,air} \dot{J}_{\text{air},1} \cdot 0.0289647 \cdot (T_{a}-T_{lung})$ is the heat taken by the air where 0.0289647 is the the molar mass of air (kg/mol).

XV. SOLVENDO and ZBRENT

The entire process of solving the heat budget is coordinated by the SOLVENDO subroutine, which receives the input data from the R environment via the ‘endoR’ wrapper function. A ‘while loop’ is run until the value of $Q_{gen}$ obtained by the procedure is less than the specified minimum possible metabolic rate $Q_{basal}$, which in the extreme is basal metabolic rate but may be a higher value depending on the locomotary and physiological activities assumed to be undertaken by the animal.

Within the ‘while loop’, after the initial calls to IRPROP, GEOM_ENDO, SOLAR_ENDO, CONV_ENDO, another loop calls SIMULSOL twice, once for the dorsal and once for the ventral side. For each call, the whole animal is simulated to have the fur/feather properties of the side being considered, and to be exposed in all directions to the environment of that side (i.e. sky or ground). Then, a final averaged value is obtained for $Q_{gen,net}$, $T_s$ and $T_{fa}$, weighted by the configuration factors to the the sky plus vegetation ($F_{sky} + F_{veg}$), versus the remainder ($1 - (F_{sky} + F_{veg})$).

The weighted mean guess of $Q_{gen,net}$ from the SIMULSOL calls is finally passed to RESPFUN via the root finding function ZBRENT (Brent, 2002) to calculate $Q_{resp}$ by guessing for a value of $Q_{gen}$ that balances the relationship in equation 2, i.e. that $Q_{gen,net} = Q_{gen} - Q_{resp}$. This guessing process is needed because $Q_{resp}$ is dependent on total $Q_{gen}$, not simply $Q_{gen,net}$. Once a suitable value for $Q_{resp}$ is found, it can be added to the value of $Q_{gen,net}$ to determine $Q_{gen}$ and the value of $Q_{gen,net}$ can be checked against the user-provided minimum value to see whether the while-loop can be terminated.

Various behaviours can be invoked as part of this overall procedure, to help find a solution, including changes in posture (shape), flesh thermal conductivity, allowing core temperature to rise, panting and sweating. This subroutine serves as an example setup, where the latter behaviours are attempted sequentially, in the order just given, but many other variants are possible such as changing the order or allowing processes to happen in parallel. The R function ‘endoR_devel’ and the Fortran subroutine ‘SOLVENDO’ are equivalent, with the latter provided as a way to easily develop behavioural algorithms which should then be converted into an equivalent Fortran version for maximal performance (the speed of the calculations may increase over 100-fold when run as a Fortran subroutine).

XVI References

Brent, R. (2002) Algorithms for minimization without derivatives, Dover Publications.

Conley, K.E. & Porter, W.P. (1986) Heat loss from deer mice (Peromyscus): evaluation of seasonal limits to thermoregulation. Journal of Experimental Biology, 126, 249–269.

Going, S.B. (1996) Densiometry. Human Body Composition (ed. by A.F. Roche), S.B. Heymsfield), and T.G. Lohman), pp. 3–24. Human Kinetics Press, Champaign, IL.

Kleiber, M. (1961) The Fire of Life. An Introduction to Animal Energetics.

Kreith, F. (1965) Principles of Heat Transfer, International Textbook Co, Scranton, Pa.

Kowalski, G.J. (1978) An Analytical and Experimental Investigation of the Heat Loss Through Animal Fur.

List, R.J. (1966) Smithisonian Meteorological Tables, 6th Ed. Smithonian Institute, Washington, D.C.

Mathewson, P.D. & Porter, W.P. (2013) Simulating Polar Bear Energetics during a Seasonal Fast Using a Mechanistic Model. PLoS One, 8, e72863.

Mitchell, J.W. (1976) Heat transfer from spheres and other animal forms. Biophysical Journal, 16, 561–569.

Porter, W.P., Budaraju, S., Stewart, W.E., Ramankutty, N. (2000) Calculating climate effects on birds and mammals: impacts on biodiversity, conservation, population parameters, and global community structure. American Zoologist, 40, 597-630.

Stahl, W.R. (1967) Scaling of respiratory variables in mammals. Journal of Applied Physiology, 22, 453–460.

Walsberg, G.E. & King, J.E. (1978) The Relationship of the External Surface Area of Birds to Skin Surface Area and Body Mass. Journal of Experimental Biology, 76, 185–189.

XVII Symbols

Symbol	units	Variable Name	Description
$\alpha_{dors}$	$\text{none, ratio}$	$\text{ABSAND}$	solar absorptivity, dorsal fur
$\alpha_{fur}$	$\text{none, ratio}$	-	solar absorptivity, fur
$\alpha_{sub}$	$\text{none, ratio}$	$\text{ABSSB}$	solar absorptivity, substrate
$\alpha_{vent}$	$\text{none, ratio}$	$\text{ABSANV}$	solar absorptivity, ventral fur
$\beta$	$\text{m}^{-1}$	$\text{BETARA}$	average infrared absorption coefficient
$\beta_1$	$\text{m}$	$\text{B1ARA}$	fur/feather optical depth
$\delta_T$	$\text{K},^\circ \text{C}$	$\text{DELTAT}$	temperature difference between animal surface and surrounding air/fluid
$\epsilon$	$\text{none, ratio}$	$\text{EMIS}$	longwave emissivity
$\gamma$	$\text{K}^{-1}$	$\text{BETA}$	inverse of the fluid temperature $T_{fluid}$ in Kelvin
$\lambda$	$\text{J} \space \text{kg}^{-1}$	$\text{HTOVPR}$	latent heat of vaporization
$\mu$	$\text{kg} \space \text{m} \space \text{s}^{-1}$	$\text{VISDYN}$	dynamic viscosity of air or fluid surrounding the animal
$\mu_{fluid}$	$\text{kg} \space \text{m} \space \text{s}^{-1}$	$\text{VISDYN}$	dynamic viscosity of fluid surrounding the animal
$\rho_{body}$	$\text{kg} \space \text{m}^{-3}$	$\text{ANDENS}$	body density
$\rho_{fat}$	$\text{kg} \space \text{m}^{-3}$	$\text{FATDEN}$	fat density
$\rho_{fluid}$	$\text{kg} \space \text{m}^{-3}$	$\text{DENSTY}$	fluid density
$\rho_{fur}$	$\text{hairs} \space \text{m}^{-2}$	$\text{RHO}$	weighted mean calculation of pelage hair density
$\rho_{hair,eff}$	$\text{hairs} \space \text{m}^{-2}$	$\text{RHOEFF}$	effective pelage hair density
$\rho_{hair}$	$\text{hairs} \space \text{m}^{-2}$	$\text{RHO}$	pelage hair density
$\rho_a$	$\text{kg} \space \text{m}^{-3}$	$\text{AIRVD}$	vapor density of surrounding air
$\rho_s$	$\text{kg} \space \text{m}^{-3}$	$\text{SURFVD}$	vapor density at the skin surface
$\sigma$	$\text{W} \space \text{m}^{-2} \text{K}^{-4}$	$\text{SIG}$	Stefan-Boltzmann constant
$A_{cond}$	$\text{m}^2$	$\text{AV}$,$\text{AREACND}$	area of body part in contact with substrate for conduction
$A_{conv}$	$\text{m}^2$	$\text{SURFAR}$,$\text{CONVAR}$	area for convection (total area minus ventral area in contact with substrate, as determined by $p_{cond}$)
$A_{evap}$	$\text{m}^2$	$\text{CONVSK}$,$\text{AREASKIN}$	area of skin for evaporation (total skin area - hair area)
$A_{fa}$	$\text{m}^2$	$\text{CONVAR}$	fur-air interface area
$A_{hair}$	$\text{m}^2$	$\text{AHAIR}$	cross-sectional fur area that is hair (rather than air)
$A_{maj}$	$\text{m}$	$\text{ASEMAJ}$	semimajor axis length (ellipsoid)
$A_{s}$	$\text{m}^2$	$\text{CONVSK}$,$\text{AREASKIN}$	area of skin, $\text{m}^2$
$A_{sil,n}$	$\text{m}^2$	$\text{ASILN}$	silhouette area normal to sun
$A_{sil,p}$	$\text{m}^2$	$\text{ASILP}$	silhouette area parallel to sun
$A_{sil}$	$\text{m}^2$	$\text{ASIL}$	silhouette area for solar radiation (may be normal, parallel or in between)
$A_{tot}$	$\text{m}^2$	$\text{AREATOTL}$,$\text{AREA}$,$\text{ATOT}$	total area at fur/feathers-air interface
$A_{wet}$	$\text{m}^2$	$\text{EFFSUR}$	surface area acting as a free-water surface
$B_{min}$	$\text{m}$	$\text{BSEMIN}$	semiminor axis length (ellipsoid)
$C_{min}$	$\text{m}$	$\text{CSEMIN}$	semiminor axis length (ellipsoid, currently only prolate spheroid)
$C_d$	$\text{W} \space {^\circ \text{C}}^{-1}$	$\text{CD}$	constant $\frac{A_{cnd} k_{sub}}{x_{sub}}$ used to simplify equation notation
$C_{d1}$	$\text{W} \space \text{m}^{-1} \space {^\circ \text{C}}^{-1}$	$\text{CD}$	constant $\left(\frac{k_{f,cmp}}{\ln\left(\frac{R_{fa,cmp}}{R_s}\right)} PCOND + \frac{k_f}{\ln\left(\frac{R_{fa}}{R_s} \right)} \left(1 - PCOND \right) \right)$ used to simplify equation notation
$C_{d2}$	$\text{W} \space \text{m}^{-1} \space {^\circ \text{C}}^{-1}$	$\text{CD}$	constant $\frac{k_{f,cmp}}{\ln\left(\frac{R_{fa,cmp}}{R_s}\right)} PCOND$ used to simplify equation notation
$C_{d3}$	$\text{W} \space \text{m}^{-1} \space {^\circ \text{C}}^{-1}$	$\text{CD}$	constant $\frac{k_f}{\ln\left(\frac{R_{fa}}{R_s}\right)} \left(1 - PCOND \right)$ used to simplify equation notation
$C_f$	$\text{W} \space {^\circ \text{C}}^{-1}$	$\text{CF}$	constant $\frac{2 \pi L_g k_{f,cmp} PCOND}{\ln\left(\frac{R_{fa,cmp}}{R_s}\right)}$ used to simplify equation notation
$C_f1$	$\text{W} \space {^\circ \text{C}}^{-1}$	$\text{CF1}$	constant $\frac{2 \pi L_g k_{f,cmp}}{\ln\left(\frac{R_{fa,cmp}}{R_s}\right)}$ used to simplify equation notation
$C_{p,air}$	$\text{J} \space \text{kg}^{-1} \space {^\circ \text{C}}^{-1}$	$\text{CP}$	specific heat of air
$C_{p,fluid}$	$\text{J} \space \text{kg}^{-1} \space {^\circ \text{C}}^{-1}$	$\text{CP}$	specific heat of fluid
$C_{p,water}$	$\text{J} \space \text{kg}^{-1} \space {^\circ \text{C}}^{-1}$	$\text{CP}$	specific heat of water
$d$	$\text{m}^{2}\space\text{s}^{-1}$	$\text{DIFVPR}$	mass diffusivity of water vapor
$D$	$\text{m}$	$\text{D}$	characteristic dimension for convection
$D_{hair}$	$\text{m}$	$\text{DHAIR}$	hair diameter
$D_1$	-	$\text{DV1}$	constant $\left[1 + \frac{2 \pi L_g R^2_g C_{d1}}{4 k_g V_g}+\left(\frac{2 \pi L_g R^2_g C_{d1}}{2 k_i V_g}\ln\left(\frac{R_{s}}{R_g}\right)\right)\right]$ used to simplify equation notation
$D_2$	$\text{W} \space \text{m}^{-1}$	$\text{DV2}$	constant $\left[\frac{Q_{evap} R^2_g C_{d1}}{4 k_g V_g} + \frac{Q_{evap} R^2_g C_{d1}}{2 k_i V_g}\ln\left(\frac{R_{s}}{R_g}\right)\right]$ used to simplify equation notation
$D_3$	$\text{W} \space \text{m}^{-1}$	$\text{DV3}$	constant $\frac{\left(\frac{2 \pi L_g}{D_1}\left(T_c C_{d1} - D_2 - T_{fa,cmp}C_{d2} - T_{fa} C_{d3} \right)\right) R^2_g}{2 V_g}$ used to simplify equation notation
$D_4$	$\text{W} \space \text{m}^{-1} \space {^\circ \text{C}}^{-1}$	$\text{DV4}$	constant $C_{d2} + \frac{k_f}{\ln\left(\frac{R_{fa}}{R_{rad}}\right)} \left(1-PTCOND \right)$ used to simplify equation notation
$D_5$	-	$\text{DV5}$	constant $\left[1 + \frac{C_{f1} R^2_g}{4 k_g V_g} + \frac{C_{f1} R^2_g}{2 k_I V_g}\ln\left(\frac{R_{s}}{R_g}\right)\right]$ used to simplify equation notation
$E_{tot}$	$\text{none, ratio}$	$\text{E2}$	eccentricity of ellipsoid for fur-air interface surface area calculations
$E_s$	$\text{m}$	$\text{none, ratio}$	eccentricity of ellipsoid for skin surface area calculations
$F_{bsh}$	$\text{none, ratio}$	$\text{FATOBJ}$	configuration factor from animal to surrounding vegetation or objects
$F_{sky}$	$\text{none, ratio}$	$\text{FASKY}$	configuration factor from animal to sky
$F_{sub}$	$\text{none, ratio}$	$\text{FAGRD}$	configuration factor from animal to substrate
$F_{veg}$	$\text{none, ratio}$	$\text{FAVEG}$	configuration factor from animal to overhead vegetation
$G$	$\text{m} \space \text{s}^{-1}$	$\text{GRAV}$	acceleration due to gravity
$Gr$	$\text{none, ratio}$	$\text{GR}$	Grashof number
$H$	$\text{m}$	-	height
$H_{body}$	$\text{m}$	$\text{AHEIT}$	body height (flesh plus fat)
$h_{d,forced}$	$\text{m}\space \text{s}^{-1}$	$\text{HDFORC}$	forced convection mass transfer coefficient
$h_{d,free}$	$\text{m}\space \text{s}^{-1}$	$\text{HDFREE}$	free convection mass transfer coefficient
$h_{r,bsh}$	$\text{W} \space \text{m}^{-2} \text{K}^{-1}$	-	radiative heat transfer coefficient between animal and surrounding vegetation or objects
$h_{r,sky}$	$\text{W} \space \text{m}^{-2} \text{K}^{-1}$	-	radiative heat transfer coefficient between animal and sky
$h_{r,sub}$	$\text{W} \space \text{m}^{-2} \text{K}^{-1}$	-	radiative heat transfer coefficient between animal and substrate
$h_{r,veg}$	$\text{W} \space \text{m}^{-2} \text{K}^{-1}$	-	radiative heat transfer coefficient between animal and overhead vegetation
$h_c$	$\text{W} \space \text{m}^{-2} \text{K}^{-1}$	$\text{HC}$,$\text{HCCOMB}$	convective heat transfer coefficient
$h_d$	$\text{m}\space \text{s}^{-1}$	$\text{HD}$	mass transfer coefficient
$h_r$	$\text{W} \space \text{m}^{-2} \text{K}^{-1}$	-	radiative heat transfer coefficient
$I_{sol}$	$\text{W} \space \text{m}^{-2}$	-	incoming solar radiation
$\dot{J}_{\text{air},1}$	$\text{mol} \space \text{s}^{-1}$	$\text{AIRML1}$	moles of air entering the respiratory system per second
$\dot{J}_{\text{air},2}$	$\text{mol} \space \text{s}^{-1}$	$\text{AIRML2}$	moles of air exiting the respiratory system per second
$\dot{J}_{\text{CO}_2,2}$	$\text{mol} \space \text{s}^{-1}$	$\text{CO2MOL2}$	moles of carbon dioxide exiting the respiratory system
$\dot{J}_{\text{H}_2\text{O},1}$	$\text{mol} \space \text{s}^{-1}$	$\text{WMOL1}$	moles of water entering the respiratory system per second
$\dot{J}_{\text{H}_2\text{O},2}$	$\text{mol} \space \text{s}^{-1}$	$\text{WMOL2}$	moles of water exiting the respiratory system per second
$\dot{J}_{\text{O}_2,1}$	$\text{mol} \space \text{s}^{-1}$	$\text{O2MOL1}$	moles of oxygen entering the respiratory system per second
$\dot{J}_{\text{O}_2,2}$	$\text{mol} \space \text{s}^{-1}$	$\text{O2MOL2}$	moles of oxygen exiting the respiratory system per second
$\dot{J}_{\text{O}_2,con}$	$\text{mol} \space \text{s}^{-1}$	$\text{O2MOLC}$	moles of oxygen consumed per second
$\dot{J}_{in}$	$\text{mol} \space \text{s}^{-1}$	-	mass flow of water vapor into the lungs
$\dot{J}_{out}$	$\text{mol} \space \text{s}^{-1}$	-	mass flow of water vapor out of the lungs
$k^{\prime}_x$	$\text{W} \space \text{m}^{-1} {^\circ \text{C}}^{-1}$	$\text{KX}$	thermal conductivity of fur in the horizontal direction
$k^{\prime}_y$	$\text{W} \space \text{m}^{-1} {^\circ \text{C}}^{-1}$	$\text{KY}$	thermal conductivity of fur in the vertical direction
$k_{air}$	$\text{W} \space \text{m}^{-1} {^\circ \text{C}}^{-1}$	$\text{THCOND}$,$\text{KAIR}$	thermal conductivity of air
$k_{eff}$	$\text{W} \space \text{m}^{-1} {^\circ \text{C}}^{-1}$	$\text{KEFF}$	effective conductive thermal conductivity of the fur/feather layer (average of $k^{\prime}_x$ and $k^{\prime}_y$)
$k_{fluid}$	$\text{W} \space \text{m}^{-1} {^\circ \text{C}}^{-1}$	$\text{THCOND}$	thermal conductivity of fluid
$k_{fur}$	$\text{W} \space \text{m}^{-1} {^\circ \text{C}}^{-1}$	$\text{KFUR}$	combined conductive and radiant thermal conductivity of the fur/feather layer
$k_{fur,cmp}$	$\text{W} \space \text{m}^{-1} {^\circ \text{C}}^{-1}$	$\text{KFURCMP}$	combined conductive and radiant thermal conductivity of the compressed fur/feather layer for the portion of the animal in contact with the substrate
$k_{hair}$	$\text{W} \space \text{m}^{-1} {^\circ \text{C}}^{-1}$	$\text{KHAIR}$	thermal conductivity of hair (assumed to be keratin)
$k_{rad}$	$\text{W} \space \text{m}^{-1} {^\circ \text{C}}^{-1}$	$\text{KRAD}$	longwave radiation component of fur/feather layer thermal conductivity
$k_{sub}$	$\text{W} \space \text{m}^{-1} {^\circ \text{C}}^{-1}$	$\text{KSUB}$	thermal conductivity of the substrate
$k_f$	$\text{W} \space \text{m}^{-1} {^\circ \text{C}}^{-1}$	$\text{KFUR}$	thermal conductivity of fur/feather layer
$k_g$	$\text{W} \space \text{m}^{-1} {^\circ \text{C}}^{-1}$	$\text{AK1}$	thermal conductivity of flesh
$k_i$	$\text{W} \space \text{m}^{-1} {^\circ \text{C}}^{-1}$	$\text{AK2}$	thermal conductivity of subcutaneous fat
$L_{body}$	$\text{m}$	$\text{ALENTH}$	body length (flesh plus fat)
$L_{fat}$	$\text{m}$	$\text{FATTHK}$	fat layer thickness
$L_{hair}$	$\text{m}$	$\text{LHAIR}$	hair length
$L_g$	$\text{m}$	$\text{LEN}$	length of body part
$M$	$\text{m}$	$\text{AMASS}$	total body mass
$M_{fat}$	$\text{kg}$	$\text{MFAT}$	mass of body fat
$\dot{M}_{\text{H}_2\text{O},2}$	$\text{kg} \space \text{s}^{-1}$	$\text{KGEVAP}$	mass of water lost from respiratory system per second
$\dot{M}_{\text{H}_2\text{O},bareskin}$	$\text{kg} \space \text{s}^{-1}$	$\text{WCUTHF}$	water mass flux from bare skin
$\dot{M}_{\text{H}_2\text{O},cut}$	$\text{kg} \space \text{s}^{-1}$	$\text{WCUT}$	water mass flux from cutaneous surface
$\dot{M}_{\text{H}_2\text{O},eyes}$	$\text{kg} \space \text{s}^{-1}$	$\text{WEYES}$	total water mass flux from eyes
$\dot{M}_{\text{H}_2\text{O},furskin}$	$\text{kg} \space \text{s}^{-1}$	$\text{WCUTF}$	water mass flux from skin with furred or feathered covering
$\dot{M}_{\text{H}_2\text{O},wetfur}$	$\text{kg} \space \text{s}^{-1}$	$\text{WTFUR}$	water mass flux from furred/feathered surface
$Nu$	$\text{none, ratio}$	$\text{NUTOTAL}$	Nusselt number
$Nu_{forced}$	$\text{none, ratio}$	$\text{NUFORCED}$	Nusselt number for forced convection
$Nu_{free}$	$\text{none, ratio}$	$\text{NUFREE}$	Nusselt number for free convection
	$P_{\text{H}_2\text{O},sat}$	$\text{\%}$	$\text{ESAT}$
$P_{\text{O}_2}$	$\text{\%}$	$\text{PO2}$	partial pressure of oxygen in the animal’s environment
$P_{bar,std}$	$\text{Pa}$	$\text{PSTD}$	barometric pressure at standard temperature and pressure
$P_{bar}$	$\text{Pa}$	$\text{BP}$,$\text{BARPRS}$	barometric pressure
$p_{\text{CO}_2}$	$\text{none, ratio}$	$\text{PCTCO2}$	proportion of carbon dioxide in the animal’s environment
$p_{\text{CO}_2,ref}$	$\text{none, ratio}$	$\text{RPCTCO2}$	reference atmospheric proportion of carbon dioxide
$p_{\text{N}_2}$	$\text{none, ratio}$	$\text{PCTN2}$	proportion of nitrogen in the animal’s environment
$p_{\text{N}_2,ref}$	$\text{none, ratio}$	$\text{RPCTN2}$	reference atmospheric proportion of nitrogen
$p_{\text{O}_2}$	$\text{none, ratio}$	$\text{PCTO2}$	proportion of oxygen in the animal’s environment
$p_{\text{O}_2,ref}$	$\text{none, ratio}$	$\text{RPCTO2}$	reference atmospheric proportion of oxygen
$p_{bare}$	$\text{none, ratio}$	$\text{PCTBAREVAP}$	proportion of free-water surface that is bare skin
$p_{cond}$	$\text{none, ratio}$	$\text{PTCOND}$	proportion of surface area in contact with substrate for conduction
$p_{dif}$	$\text{none, ratio}$	$\text{PCTDIF}$	proportion of sunlight that is diffuse
$p_{eyes}$	$\text{none, ratio}$	$\text{PEYES}$	proportion of surface area comprised of animal’s eyes
$p_{fat}$	$\text{none, ratio}$	$\text{PCTFAT}$	body fat percentage
$p_{shd}$	$\text{none, ratio}$	$\text{ASHADE}$	fractional overhead shade level
$p_{vent}$	$\text{none, ratio}$	$\text{PVENT}$	proportion of ventral fur
$p_{wet}$	$\text{none, ratio}$	$\text{PCTWET}$	proportional skin available for evaporation
$p_{wetfur}$	$\text{none, ratio}$	$\text{FURWET}$	proportion of the fur/feather surface area that is wet
$Pr$	$\text{none, ratio}$	$\text{PR}$	Prandtl number
$Q_{air}$	$\text{W}$	$\text{QAIR}$	heat taken by the air in the respiratory tract
$Q_{basal}$	$\text{W}$	$\text{QBASAL}$	basal metabolic rate
$Q_{cond}$	$\text{W}$	$\text{QCOND}$	conductive heat flux
$Q_{conv}$	$\text{W}$	$\text{QCONV}$	convective heat flux
$Q_{env}$	$\text{W}$	$\text{QENV}$	heat flux with the surrounding environment ($\text{Q}_{rad}+\text{Q}_{conv}+\text{Q}_{cond}+Q_{evap,fur}-\text{Q}_{sol})$
$Q_{evap}$	$\text{W}$	$\text{QSEVAP}$	cutaneous evaporative heat loss
$Q_{evap,fur}$	$\text{W}$	$\text{QFSEVAP}$	fur surface evaporative heat loss
$Q_{fur}$	$\text{W}$	$\text{QFUR}$	heat flux through the fur/feather layer
$Q_{gen,net}$	$\text{W}$	$\text{QGENNET}$	net metabolic heat generation ($\text{Q}_{gen}-\text{Q}_{resp}$)
$Q_{gen}$	$\text{W}$	$\text{QGEN}$	metabolic heat generation
$Q_{r1}$	$\text{W} \space {^\circ \text{C}}^{-1}$	$\text{QR1}$	abbreviation for radiant heat transfer coefficients, surface areas, and configuration factors to simplify equation notation
$Q_{r2}$	$\text{W} \space {^\circ \text{C}}^{-1}$	$\text{QR2}$	abbreviation for radiant heat transfer coefficients, surface areas, and configuration factors to simplify equation notation
$Q_{r3}$	$\text{W} \space {^\circ \text{C}}^{-1}$	$\text{QR3}$	abbreviation for radiant heat transfer coefficients, surface areas, and configuration factors to simplify equation notation
$Q_{r4}$	$\text{W} \space {^\circ \text{C}}^{-1}$	$\text{QR4}$	abbreviation for radiant heat transfer coefficients, surface areas, and configuration factors to simplify equation notation
$Q_{rad,bsh}$	$\text{W}$	$\text{QRBSH}$	radiative heat flux with surrounding vegetation
$Q_{rad,sky}$	$\text{W}$	$\text{QRSKY}$	radiative heat flux with sky
$Q_{rad,sub}$	$\text{W}$	$\text{QRGRD}$	radiative heat flux with substrate
$Q_{rad,veg}$	$\text{W}$	$\text{QRVEG}$	radiative heat flux with overhead vegetation
$Q_{rad}$	$\text{W}$	$\text{QRAD}$	longwave radiative hat flux
$Q_{resp}$	$\text{W}$	$\text{QRESP}$	respiratory heat loss
$Q_{sol,dif}$	$\text{W}$	$\text{QSDIFF}$	total diffuse solar radiation absorbed by the animal
$Q_{sol,dir}$	$\text{W}$	$\text{QSDIR}$	direct solar radiation absorbed by the animal
$Q_{sol,dors}$	$\text{W}$	$\text{QDORSL}$	dorsal solar heat flux
$Q_{sol,norm}$	$\text{W} \space \text{m}^{-2}$	$\text{QNORM}$	direct solar radiation intensity normalized for zenith angle
$Q_{sol,sky}$	$\text{W}$	$\text{QSSKY}$	diffuse solar radiation absorbed by the animal from the sky
$Q_{sol,sub}$	$\text{W}$	$\text{QSRSB}$	diffuse solar radiation absorbed by the animal reflected off the substrate
$Q_{sol,vent}$	$\text{W}$	$\text{QVENTR}$	ventral solar heat flux reflected from substrate
$Q_{sol}$	$\text{W}$	$\text{QSLR}$	total solar heat flux
$r$	$\text{m}$	-	depth within the fur/feather layer
$R$	$\frac{\text{Pa} \space \text{Litres}}{\text{mol} \space \text{K}}$	$\text{RGC}$	universal gas constant
$R_{fa}$	$\text{m}$	$\text{RFUR}$,$\text{R2}$	shape-specific core-fur radius in shortest dimension
$R_{fa,cmp}$	$\text{m}$	$\text{RFURCMP}$,$\text{R2}$	shape-specific core-fur radius in shortest dimension for portion of animal in contact with substrate
$R_{rad}$	$\text{m}$	-	radius at which longwave radiation is effectively leaving the fur/feather layer
$R_g$	$\text{m}$	$\text{RFLESH}$	shape-specific core to surface of heat generating flesh in shortest dimension
$R_s$	$\text{m}$	$\text{RSKIN}$,$\text{R1}$	shape-specific core-skin radius in shortest dimension
$Ra$	$\text{none, ratio}$	$\text{RA}$	Rayleigh number
$Re$	$\text{none, ratio}$	$\text{RE}$	Reynolds number
$\text{RH}$	$\text{\%}$	$\text{RH}$	relative humidity
$\text{RQ}$	$\text{none, ratio}$	$\text{RQ}$	respiratory quotient (ratio of moles of carbon dioxide produced per mole of oxygen consumed)
$S_b$	$\text{none, ratio}$	$\text{SHAPEB}$	ratio of length to height of an ellipsoid
$S_c$	$\text{none, ratio}$	$\text{SHAPEC}$	ratio of length to width of a cylinder
$Sc$	$\text{none, ratio}$	$\text{SC}$	Schmidt number
$Sh$	$\text{none, ratio}$	$\text{SH}$	Sherwin number
$T_{air}$	$^\circ \text{C}$	$\text{TAIR}$	air temperature
$T_{ave}$	$\text{K}$	-	average of $\text{T}_{rad,env}$ and $\text{T}_{rad,fur}$
$T_{env}$	$\text{K}$	-	radiant temperature of relevant environmental component (sky, substrate, vegetation, etc.)
$T_{fa}$	$^\circ \text{C}$	$\text{TFA}$	temperature at the fur-air interface
$T_{fa,cmp}$	$^\circ \text{C}$	$\text{TFACMP}$	temperature at the fur-air interface for portions of animal in contact with substrate
$T_{fur}$	$^\circ \text{C}$	-	temperature of the fur at depth $r$
$T_{lung}$	$^\circ \text{C}$	$\text{TLUNG}$	estimated lung temperature
$T_{rad,approx}$	$\text{K}$	$\text{TRAPPX}$	initial approximation of fur radiant temperature ($\text{T}_{rad,fur}$)
$T_{rad,bsh}$	$\text{K}$	$\text{TBUSH}$	effective radiant temperature of the surrounding vegetation
$T_{rad,env}$	$\text{K}$	-	effective radiant temperature of the environment (e.g., sky, ground, bushes, nearby objects)
$T_{rad,fur}$	$\text{K}$	$\text{TR}$	effective radiant temperature of the fur/feather layer
$T_{rad,sky}$	$\text{K}$	$\text{TSKY}$	effective radiant temperature of the sky
$T_{rad,sub}$	$\text{K}$	$\text{TLOWER}$	effective radiant temperature of substrate
$T_{rad,veg}$	$\text{K}$	$\text{TVEG}$	effective radiant temperature of overhead vegetation
$T_{sub}$	$^\circ \text{C}$	$\text{TGRD}$	substrate temperature
$T_{water}$	$^\circ \text{C}$	$\text{TWATR}$	water temperature
$T_a$	$^\circ \text{C}$	$\text{TA}$	air temperature
$T_c$	$^\circ \text{C}$	$\text{TC}$	core body temperature
$T_s$	$^\circ \text{C}$	$\text{TSKIN}$	skin temperature
$v$	$\text{m} \space \text{s}^{-1}$	$\text{VEL}$	wind speed
$V$	$\text{m}^3$	$\text{VOL}$	total body volume
$V_{body}$	$\text{m}^3$	$\text{VOL}$	total body volume
$V_{fat}$	$\text{m}^3$	$\text{VOLFAT}$	volume of body fat
$V_g$	$\text{m}^3$	$\text{VOL}$,$\text{FLSHVL}$	volume of body flesh
$\dot{V}_{air}$	$\text{m}^3 \space \text{s}^{-1}$	$\text{AIRVOL}$	volume of air flowing through the lungs per second
$\dot{V}_{O_{2},stp}$	$\text{m}^3 \space \text{s}^{-1}$	$\text{O2STP}$	oxygen consumption rate at standard temperature and pressure
$\dot{V}_{O_2}$	$\text{m}^3 \space \text{s}^{-1}$	$\text{VO2CON}$	oxygen consumption rate at the current air temperature and barometric pressure
$W_{body}$	$\text{m}$	$\text{AWIDTH}$	body width (flesh plus fat)
$X_{pant}$	$\text{none, ratio}$	$\text{PANT}$	multiplier to allow enhanced respiratory heat loss via panting for thermoregulation
$X_{RQ}$	$\text{none, ratio}$	-	conversion factor for liters of oxygen per kilocalorie
$x_{sub}$	$\text{m}$	-	depth in the substrate from which temperature is taken for conduction calculations (assumed to be 2.5 cm)
$X_r$	$\text{proportion}$	$\text{XR}$	proportional distance from the skin surface in the fur where radiant exchange takes place (1.0 = at the fur surface; 0.5 = at the midpoint depth
$Z$	$\text{radians}$	$\text{ZEN}$	solar zenith angle
$z_{fat}$	$\text{m}$	$\text{FATTHK}$	subcutaneous fat thickness
$z_{fur}$	$\text{m}$	$\text{ZL}$	fur/feather layer thickness