O'Connor & Dodson, 1999

Paleobiology, 25(3), 1999, pp. 341–368

Biophysical constraints on the thermal ecology of dinosaurs Michael P. O’Connor and Peter Dodson

Abstract.—A physical, model-based approach to body temperatures in dinosaurs allows us to predict what ranges of body temperatures and what thermoregulatory strategies were available to those dinosaurs. We argue that 1.

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The huge range of body sizes in the dinosaurs likely resulted in very different thermal problems and strategies for animals at either end of this size continuum.

Body temperatures of the smallest adult dinosaurs and of hatchlings and small juveniles would have been largely insensitive to metabolic rates in the absence of insulation. The smallest animals in which metabolic heating resulted in predicted body temperatures $ 28C above operative temperatures (Te) weigh 10 kg. Body temperature would respond rapidly enough to changes in Te to make behavioral thermoregulation possible.

Body temperatures of large dinosaurs (.1000 kg) likely were sensitive to both metabolic rate and the delivery of heat to the body surface by blood flow. Our model suggests that they could adjust body temperature by adjusting metabolic rate and blood flow. Behavioral thermoregulation by changing microhabitat selection would likely have been of limited utility because body temperatures would have responded only slowly to changes in Te .

Endothermic metabolic rates may have put large dinosaurs at risk for overheating unless they had adaptations to shed the heat as necessary. This would have been particularly true for dinosaurs with masses .10,000 kg, but simulations suggest that for animals as small as 1000 kg in the Tropics and in temperate latitudes during the summer, steady-state body temperatures would have exceeded 408C. Slow response of body temperatures to changes in T e suggests that use of day-night thermal differences would have buffered dinosaurs from diel warming but would not have lowered body temperatures sufficiently for animals experiencing high mean daily Te. Endothermic metabolism and metabolic heating might have been useful for intermediate and large-sized (100–3000 kg) dinosaurs but often in situations that demanded marked seasonal adjustment of metabolic rates and/or precise control of metabolism (and heat-loss mechanisms) as typically seen in endotherms.

Michael P. O’Connor. Department of Bioscience and Biotechnology, Drexel University, Philadelphia, Pennsylvania 19104. E-mail: mike.oconnor@drexel.edu Peter Dodson. Department of Animal Biology, School of Veterinary Medicine, University of Pennsylvania, Philadelphia, Pennsylvania 19104-6045. E-mail: dodsonp@vet.upenn.edu Accepted:

15 February 1999

Introduction Dinosaurs are a testament to the ability of natural selection to push the biomechanical limits of size in terrestrial animals. Within 75 million years from their origin as small creatures (Eoraptor was only one meter long [Sereno et al. 1993]), dinosaurs such as the giant sauropods Seismosaurus and Argentinosaurus achieved maximum body lengths of 40 to 50 meters (Gillette 1991; Bonaparte and Coria 1993) and weights in tens of tons. Although the larger dinosaurs dwarf the largest known terrestrial mammals, such as Indricotherium (Alexander 1989), many dinosaurs were quite small (Pezckis 1994). Estimated body masses span five orders of magnitude. This large size q 1999 The Paleontological Society. All rights reserved.

range has prompted study of the allometric relations thought to dictate much of the engineering and physiology of these animals (Spotila et al. 1991). We use the following designations: very small, 1–10 kg estimated adult weight; small, 10–100 kg; medium-sized, 100– 1000 kg; large, 1000–10,000 kg; and very large, 10,000–100,000 kg. Using these designations, the modal size range for all dinosaurs is large, accounting for 40% of 216 genera assessed by Pezckis (1994). Sauropods alone are represented among the very large dinosaurs.

From the very beginning, physiology figured in our concept of Dinosauria (Owen 1842). Owen specifically invoked mammals as a physiological model for dinosaurs, inferring four-chambered hearts and a highly orga0094-8373/99/2503-0004/$1.00

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nized respiratory system. Although for many years dinosaurs were considered cold-blooded, in keeping with their reptilian heritage, the idea of hot-blooded dinosaurs achieved currency in the 1970s (Bakker 1971, 1972, 1975, 1986) and inspired an aptly named symposium, ‘‘A Cold Look at Hot-Blooded Dinosaurs’’ (Thomas and Olson 1980), as well as popular fiction such as Jurassic Park (Crichton 1990). More recently, certain lines of evidence including bone histology (Chinsamy 1990, 1993; Chinsamy and Dodson 1995; Chinsamy et al. 1994, 1995) and respiratory physiology (Ruben et al. 1996) have caused some re-evaluation of the concept of warm-blooded dinosaurs.

This paper is one in a series of papers attempting to apply biophysical approaches to the problem of temperature regulation in dinosaurs. As others did in earlier papers (Spotila et al. 1973, 1991; Spotila 1980), we take an engineering approach to the problem of heat transfer from a warm interior to the external environment. We model each dinosaur as a series of cylinders, with allometric equations accounting for changes in proportion of limb lengths and surface areas. The power of computers enables the use of increasingly sophisticated models for the dynamics of heat transfer. We specify a variety of physiological conditions of heat transfer as well as a variety of environmental conditions. In so doing, we make no a priori assumptions about the metabolic status of dinosaurs; we simply inquire as to which metabolic strategies are feasible under specified conditions of body size, internal physiology, and external environment.

Much of the debate over temperatures and metabolic rates of putatively endothermic dinosaurs has centered on the effect of large body size on body temperatures in ‘‘warm blooded’’ animals, the role of conductance from the animal to the environment in keeping the animal warm or cold, and the role of thermal inertia in buffering dinosaurs’ body temperatures from the vagaries of the environment (see reviews in Barrick et al. 1997; Padian 1997; Paladino et al. 1997; Reid 1997; Ruben et al. 1997). Part of the difficulty in unraveling the constraints these processes put on the temperatures and metabolic rates of di-

nosaurs is that all of the processes are interrelated and all depend on body size.

The conductance of heat from the animal to the environment depends both on conductance from the deep tissues, or core, of the animal to the surface and on conductance from the animal’s skin to the environment (Tracy 1982). Both conduction from tissue to tissue and blood flow carry heat from the deep tissues to the surface, and both depend on body size. Heat conduction from deep tissues to the surface decreases as animals get larger simply because the heat must be conducted over longer distances. Thus, either the deep tissues must be warmer in large animals, increasing the temperature difference from the deep to the superficial tissues, or another pathway (i.e., blood flow) must carry the heat. Blood travels slowly enough through capillaries to come nearly to thermal equilibrium with the tissue it perfuses. In perfusing tissues warmer than itself, blood picks up heat from those tissues (cools them) and carries that heat back to the core. There, the blood mixes with blood from all the other tissues before it is recirculated to the arteries by the heart. Thus, blood serves as a heat pump connecting the different tissues of the body. However, just as metabolic rate per gram of tissue decreases in large animals (Bennett and Dawson 1976; Bennett 1982; Calder 1984), blood flow per gram of tissue also decreases (Calder 1984). Both pathways for heat conduction from deep to superficial tissues appear more limited in large animals.

Similarly, conductance from the surface of the animal to the environment may be more limited in large animals. The major pathways of heat transfer between the skin of a terrestrial animal and the environment are convection, solar radiation, and thermal (infrared) radiation. Convection is the transfer of heat to and from a moving fluid such as air or water. ‘‘Solar radiation’’ is used as a shorthand for the shortwave (primarily visible, near-ultraviolet [UV], and near-infrared [IR]) radiation emitted by the sun and falling on the animal.

Because animals are not warm enough to emit radiation in these wavelengths, solar radiation only moves heat into the animal (Porter and Gates 1969; Tracy 1982). ‘‘Thermal radiation’’

BIOPHYSICAL CONSTRAINTS ON DINOSAURS

is also shorthand for the summation of all the radiation in the longwave IR that is emitted both by the environment and by animals at typical temperatures. Thus, thermal radiation constitutes an exchange of heat with the environment, warming or cooling animals depending on their temperatures. An animal lying on the ground can also exchange heat by conduction via the trunk and/or limbs, but the tips of limbs in a standing animal typically have small surface areas and thus are poor heat conduits to the ground.

Changes in both solar and thermal radiation are as effective in warming and cooling large animals as they are with small animals.

The same is not true for convection, however.

Before being carried away in the wind, heat lost by convection must diffuse across a layer of relatively still air known as the boundary layer. As animals increase in size, the thickness of this boundary layer increases, decreasing the rate of heat diffusion (Porter and Gates 1969). Because rates of convective heat transfer are lower in large animals than in small ones but the rates of radiative exchange remain the same, radiative exchanges become relatively more important in large than in small animals. This can be particularly important in sunny environments where animals often gain heat by radiation (sunshine) and lose heat by convection to the atmosphere. In such situations, decreased convective heat exchange in large animals results in higher body temperatures in response to the undiminished heat load from the sun.

While some treatments of boundary layers conjure images of thick laminae of becalmed air, the only truly still air is immediately adjacent to the animal’s skin. Wind speed increases asymptotically to the free stream velocity as one moves away from the animal, with most of the change in speed in the first few millimeters (Gates 1980; White 1984). The ‘‘boundary layer thickness’’ refers to the theoretical ‘‘equivalent’’ distance of totally still air through which heat would have to move by diffusion alone to match the overall conductance of heat from the skin to the airstream (Gates 1980; Monteith and Unsworth 1990).

Boundary layers need not be thick. Still air has a low conductivity to heat. Trapped air is

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responsible for much of the insulative value of fur, feathers, and many household insulators (Cena and Monteith 1975; White 1984; Monteith and Unsworth 1990). A still boundary layer that averages only 3.8 mm thick will explain the resistance to convective heat exchange for a 1000 kg animal in a wind stream at 1 m/s (Mitchell 1976). For wind speeds at 5 m/s, the boundary layer thickness drops to 1.5 mm for the same animal. Turbulence will decrease the boundary layer thickness and increase the convection coefficient by a factor of 1.7 (Mitchell 1976). These relatively still boundary layers near animal surfaces (and their dependence on body size and wind speed) have been visualized and investigated

by Schlieren photography and other techniques in both living and inanimate objects (Gates 1980; Monteith and Unsworth 1990).

At night, solar radiation is, of course, absent. Thermal radiative exchanges can be complex because thermal radiation passes to the animal from the ground (usually warmer than air temperature), surrounding vegetation (usually at about air temperature), and the sky. The effective radiant temperature of the night sky ranges from approximately air temperature (under a canopy), to 5–108C below air temperature under a cloudy sky, to as much as 208C below air temperature under clear conditions (Spotila et al. 1992). The effective radiant temperature of the environment is some combination of all of these radiation sources. The usual treatment is to assume that half the animal’s surface exchanges thermal radiation with the ground and surrounding vegetation and half exchanges radiation with the sky.

A third influence of body size on conductance is the ratio of surface area to the volume of tissue heated by metabolism or warmed/ cooled by the environment. For biomechanical reasons, if no other, animals become proportionately more stout as they grow in size (Economos 1983). Thus, less area per gram of tissue is available to conduct heat from one region to the next, whether that next layer is a more superficial layer of tissue or the external environment.

Large animals also have more of a quantity called thermal inertia than small animals.

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Thermal inertia is so called in analogy to Newton’s translational inertia. It takes more heat to warm a large animal than a small animal, and similar environments can warm a small animal much more quickly than a large animal. In part, this is because there are more kilograms of animal to heat, so more heat is required. But it is also due in part to decreased surface area to volume ratios, decreased conductances, and the concomitant increase in time needed to heat first one layer of animal then another all the way to the deep tissues.

Increased thermal inertia is often seen as an advantage for large animals, protecting them from excessive changes in body temperature due to short-term fluctuations in the thermal environment. Higher thermal inertia may be a mixed blessing, however. To the extent that it limits excursions of the animal’s body temperature, it also limits an animal’s ability to use variations in the thermal environment to adjust either its body temperature or the heat load it must dissipate (see Tracy et al. 1986; Turner and Tracy 1986).

Allometric changes in external heat conductance (convection), internal heat conductance (conduction and blood flow), and thermal inertia (due to changes in mass, convection, conduction, blood flow) confound both the effects of different metabolic rates on body temperatures of dinosaurs and the thermoregulatory strategies such temperatures dictate. Our approach to disentangling these effects is to examine the behavior of a series of simple models of dinosaur body temperatures, each of which adds the effect of another confounding factor. This approach has two advantages.

First, because the models are simple and have analytic solutions, they allow us to understand how different factors limit the range of body temperatures available to dinosaurs of different sizes. Simultaneously, we can explore how and to what extent dinosaurs could alter their body temperatures by physiological means. Second, the simple models represent the physical extremes of what dinosaurs could accomplish and, thus, allow us to place bounds on the behavior of more complex models—perhaps even the real animals. So, after examining constraints on dinosaur temperatures via simple models, we present the

results obtained from a more realistic model.

Again, we ask what constraints would have existed on simultaneous combinations of metabolism, blood flow, and body size in different environments.

Models and Results We start by assuming that animals have no metabolic rate, no thermal inertia, and infinite internal conductance; they are like hollow metal statues that have only the size, shape, and external heat conductance of animals (Bakken and Gates 1975; Bakken et al. 1985).

This set of assumptions gives us a simple place to start and a baseline from which deviations in more complex models may be measured. As we shall see, because metabolic heating increases the body temperature of an animal, the temperatures of this inertia-less organism represent the minimum body temperature available to a dinosaur. Because the central axis of most dinosaurs was approximately cylindrical, we start with a cylindrical animal.

Operative Environmental Temperatures.—The temperature of our model animal (Te) is known as the operative environmental temperature (Bakken and Gates 1975; Bakken et al. 1985). Te is usually defined as the temperature of a massless (inertia-free) model of the animal with no metabolic rate. It can also be thought of as the temperature that an animal with no metabolism would eventually achieve if placed under the given environmental conditions. For small extant reptiles, the operative temperature is usually measured with physical models of the animal (e.g., Grant and Dunham 1988; Grant 1990). For dinosaurs, one must be content with predicted values of Te. Te’s are typically computed using an energy balance equation that demands that, at steady state, energy inputs to the animals must be balanced by energy outputs. The energy balance for our massless dinosaur is given by equation (1).

A a S 5 A hc (Te 2 Ta) 1 A hr (Te 2 Tr) (1) where A is the surface area of the animal (m2), a is the absorptance (absorptivity times fraction of surface area receiving solar radiation), S is the intensity of solar radiation (W/m2), hc

BIOPHYSICAL CONSTRAINTS ON DINOSAURS

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TABLE 1. Heat transfer coefficients for cylindrical animals at different masses and wind speeds.

Convection coef (W m 22 8 C 21) wind speed Mass (kg)

1 m/s

5 m/s

IR heat coef (W m22 8 C 21)

0.01 0.1 1 10 100 1000 10,000 100,000

31.6 23.2 17.1 12.6 9.2 6.8 5.0 3.7

82.9 61.0 44.9 33.0 24.3 17.9 13.1 9.7

4.6 4.6 4.6 4.6 4.6 4.6 4.6 4.6

is the convection coefficient (W m22 8C21), hr is the coefficient of infrared heat exchange (W m22 8C21), Te is the operative environmental temperature (8C), Ta is the ambient (air or water) temperature (8C), and Tr is the average radiative temperature of the environment (8C).

Often, Tr is not very different from Ta, and we assume that to be the case here. Then, Te 5

(h c 1 h r )Ta 1 aS (h c 1 h r )

(2)

or Te 5 Ta 1

aS H

(3)

where H is the combined heat transfer coefficient (hc 1 hr, W m22 8C21).

Operative temperatures of our Newtonian organism are higher than ambient air temperatures. The temperature increment depends on the amount of solar radiation falling on the animal (heat gain) and how easily the animal loses that heat to the environment by convection and thermal radiation (eq. 3).

Equation (3) also suggests which factors are important in determining operative temperatures for dinosaurs. Obviously, air temperature plays an important role, as does solar radiation. Two other factors affect the convection coefficient. Larger body size, by creating a larger boundary layer of still air near the animal, impedes loss of heat as animals increase in size. Higher wind speeds, by shrinking the boundary layer, enhance convection. Mitchell (1976) found that, for most animal shapes, the convection coefficient in air can be well approximated by equation (4).

FIGURE 1. Operative temperatures (T e [sensu Bakken and Gates 1975]) predicted for dinosaurs with masses ranging from 10 g to 100 metric tons ignoring any metabolic warming. Solar radiation level (800 W/m2) would be exceeded for most of a clear spring or summer day.

hc 5 6.8 u0.6/D0.4

(4)

where u is the wind speed (m/s), D is the characteristic dimension ( 5 volume0.33, [m]).

In small terrestrial animals, hr is small compared with hc, and the value of H is dominated by hc, but with increasing size, hc decreases and is on the same order as hr (Table 1). Thus, large animals are said to be relatively ‘‘convectively decoupled.’’

For cylindrical animals, operative temperatures increase with mass and the intensity of solar radiation and decrease at higher wind speeds (Fig. 1). Of note, with moderate solar radiant intensities, large animals (.1000 kg) can experience operative temperatures 10– 308C warmer than ambient air temperatures (eqs. 2–4, Fig. 1). Assuming that dinosaurs tolerated the same range of body temperatures as extant reptiles (#458C), air temperatures between 5 and 208C would not have allowed large dinosaurs much room for metabolic heating above their operative temperatures.

This constraint might be especially severe because their size may have prevented large dinosaurs from using burrows, crevices, and the shade of vegetation as effective refugia from solar radiation as extant reptiles do.

Under some conditions, factors not included in our simulations could alter the results of the simulations. (1) Truly large animals can find themselves in a different convective environ-

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ment than small animals because wind speeds may increase and air temperatures decrease with height above the ground (Porter and Gates 1969; O’Connor and Spotila 1992). The steepest parts of these gradients occur within a few cm of the ground (or the tops of vegetation when there is a vegetative canopy [Rosenberg 1974]). Reported air temperature and wind speed are typically measured about 1.5 m above the ground in meteorologic stations.

Thus small animals (,10 kg) usually experience higher air temperatures, slower wind speeds, and higher operative temperatures than climate stations report. Macroclimatic air temperatures and wind speeds probably represent a fair approximation of the convective environment experienced by large dinosaurs.

Because we focus here on dinosaurs, we have chosen not to use vertical wind speed gradients, but if one focuses on small reptiles, such gradients cannot be ignored. (2) Because of the high thermal conductivity and heat capacity of water, convection in aquatic organisms can be one to two orders of magnitude higher than in terrestrial animals (Spotila et al. 1992). If an animal stands in shallow water, the part of the animal in the water experiences aquatic convection. Dinosaurs standing in cool water could use that water as a heat sink and maintain lower operative temperatures. (3) Some animals can change their absorptance to solar radiation by changing melanin dispersion in their skins. In these simulations, we used an integrated solar absorptivity of 65% found in some desert reptiles, presumably those most susceptible to overheating due to absorption of solar radiation. While not the lowest absorptivity measured in reptiles, it may well be conservative in this case. Thus, we suspect that our estimates, if anything, underestimate the operative temperatures of animals in sunny environments. Equation (3) can be used to estimate the effects of changes in absorptivity.

Changing Environments.—Because the animal is assumed to have no thermal inertia, operative temperature (Te) is simply a weighted sum of environmental factors (air temperature, thermal radiative temperature, and solar radiation) and varies just as quickly as do its components. Equation (3) can be used to estimate the operative temperature at any given

FIGURE 2. Response of dinosaur body temperatures (ignoring effects of metabolism) to variation in operative temperature (Te).

time. In this paper, we are concerned mainly with environments that change cyclically (e.g., daily or seasonal variations, temperature changes as an animal shuttles in and out of the shade or as clouds obscure the sun, changes with a 3–5-day periodicity due to the passage of weather systems). We will take as an example air temperatures and solar radiation levels that vary sinusoidally (Fig. 2). Because operative temperatures are simply weighted sums of their components (eq. 3), and because such sums of sines with the same period are again sinusoidal, variations in operative temperature through time will again be approximated by sine waves. Furthermore, the mean operative temperature will be the operative temperature produced by inserting the mean air temperature and solar radiation level in equation (3).

To calculate the variation of operative temperatures in time, we assume that the animals have no mass and thus no thermal inertia. Although the assumption is unrealistic, and we would not expect the body temperatures of real animals to track their operative temperatures precisely through time, the variation of the operative temperature is an important concept.

Newtonian Dinosaurs.—We take a first step toward biological reality by allowing our model animals to have an appropriate body

347

BIOPHYSICAL CONSTRAINTS ON DINOSAURS

mass. We still assume that the animal has infinite internal conductance, so that temperatures everywhere within the animal are the same. Heat transfer theorists call such animals ‘‘Newtonian objects’’ (White 1984), and we will call them Newtonian dinosaurs. In a static environment, with stable air temperatures and radiation levels, the body temperatures of Newtonian dinosaurs will equal operative environmental temperatures, as long as the animal’s metabolic rate is small enough to be ignored. Thus, in a static environment, the body temperatures of Newtonian dinosaurs without metabolism would be identical to the operative temperatures presented in Figure 1.

Warming and Cooling in Newtonian Dinosaurs: Thermal Inertia.—Although the steady-state temperatures of Newtonian dinosaurs are identical to operative temperatures, the thermal inertia due to the mass of the dinosaur will cause body temperatures to vary more slowly than operative temperatures. To see how body temperatures will vary in a varying environment, we examine the response of a Newtonian animal’s body temperature to a sudden change in the thermal environment (measured as the change in Te). Then, we examine cyclically varying environments (O’Connor 1999).

To predict how a Newtonian dinosaur might warm or cool in the face of varying environmental temperatures, one must rearrange equation (1) and add a storage term. In this more complete energy balance, heat that enters an animal by one pathway (e.g., solar radiation) must either exit by another pathway (e.g., convection) or be stored in the animal’s body. Stored heat results in a rise in the animal’s temperature. mC p

dTb 5 AaS 2 Ah c (Tb 2 Ta ) dt

2 Ah r (Tb 2 Tr )

(5)

where m is the mass of the animal (kg), Cp is the specific heat of tissue (J kg21 8C21), Tb is the body temperature (8C), Ta is the ambient (air or water) temperature (8C), Tr is the average radiative temperature of the environment (8C), and dTb/dt is the rate of change of body temperature (8C/s). Using the same approxi-

mations used to simplify equation (1) into equation (3), we simplify equation (5) as AH d(Tb 2 Te ) 52 (Tb 2 Te ) dt mC p

(6)

where d(Tb 2 Te)/dt is the rate at which Tb approaches Te (8C/s), H is the combined heat transfer coefficient (hc 1 hr, W m22 8C21), and Te is the operative temperature as given in equation (3).

The solution of this differential equation, for a constant environment, is

1

Tb 5 Te 1 (Tb 2 Te )t50 exp 2

2

AH t mC p

or Tb 5 Te 1 (Tb 2 Te )t50 exp(2lt)

(7)

where (Tb 2 Te)t 5 0 is the initial difference between body and operative temperatures (8C), exp(x) is the exponential function (ex), and l is the time coefficient ( 5 2AH/mCp, s21).

Equation (7) represents the classic exponential cooling curve in which body temperature approaches operative temperature with time.

The larger the absolute value of the time coefficient, l, the more rapidly the animal heats (or cools) and approaches its operative environmental temperature (Te).

When the operative temperature (i.e., the thermal environment) changes continually and cyclically, the animal’s body temperature is constantly ‘‘approaching’’ a moving target.

What is needed is some measure of response of the animal’s body temperature to a varying environment. We chose to examine the range of predicted body temperatures of a reptile in an environment whose operative temperature varies sinusoidally. Figure 2 shows a typical response of body temperature to varying operative environmental temperature. Several features are worth noting. First, if the operative temperature varies sinusoidally, so does the body temperature. Second, body temperature lags behind operative temperature. The mass and specific heat of the animal imbue it with ‘‘thermal inertia’’; i.e., because heat flow (which takes time) is required to heat the animal, the animal’s body temperature lags behind the operative environmental tempera-

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imal and environment—specified by the time coefficient, l. Thus, Famp 5

FIGURE 3. Predicted responses of body temperatures of Newtonian dinosaurs (masses 10 g–100,000 kg) to sinusoidal variation in operative temperatures with varying cycle periods. Period is the time between successive peaks in environmental temperature in Figure 2. Fractional amplitude is ratio of body temperature amplitude to environmental temperature amplitude in Figure 2.

Solid lines 5 wind speed 1 m/s. Dashed lines 5 wind speed 5 m/s).

ture. Third, the mean body temperature is the mean operative temperature. Thus, body temperature (in this admittedly simple case) will vary around the mean operative temperature.

In the more complicated, and more biologically reasonable, simulations that follow, body temperature varies around the mean steadystate temperature. Fourth, variation in body temperature is smaller than that in operative temperature. We call the ratio of the range of body temperatures to the range of operative temperatures the ‘‘fractional amplitude.’’ Predicted fractional amplitudes vary from nearly zero for rapidly changing operative temperatures, when the ‘‘target’’ operative temperature varies much more rapidly than body temperature can follow, to near unity for slowly changing environmental temperatures. What constitutes fast or slow variation of operative temperatures varies with the animal’s mass (Fig. 3). Large dinosaurs have more mass and smaller surface to volume ratios than smaller animals. Thus, large animals have more thermal inertia, their body temperatures vary more slowly, and their body temperatures can respond only to slower variations in operative temperature, i.e., those with longer periods.

For Newtonian animals, the factors that determine what variations are fast and slow are the rate at which operative temperature changes and the heat transfer characteristics of the an-

l Ïl 1 v 2 2

(8)

where Famp is the fractional amplitude, l is the time coefficient (s21) as in equation (7), v is the rate of sinusoidal temperature change ( 5 2 p/ period, s21). When v is much smaller than l, i.e., when the change in operative temperature is slow compared with the response time of the animal’s body temperature, body temperature can respond almost as quickly as the environmental temperature varies, and the fractional amplitude will be almost one. Alternatively, when the variation in operative change is fast compared with the animal’s thermal response time, and v is much larger than l, then the fractional amplitude will be almost zero— body temperature will remain near the mean operative temperature with only small variations through time. What variation in body temperature does occur will lag significantly behind variations in operative temperature.

Because a dinosaur’s mass affects its time coefficient, it affects how strongly body temperature will respond to different types and speeds of environmental temperature variations.

By delineating which environmental variations will or will not perturb a dinosaur’s body temperature, fractional amplitudes allow us to understand thermal inertia in dinosaurs, and to quantify it in an ecologically meaningful way. For our Newtonian dinosaurs, there are three characteristics of the animal that affect inertia: mass, surface area, and external heat conductance (measured by the heat transfer coefficient, H, eqs. 6 and 7). For large dinosaurs, masses are higher, surface area to volume ratios are lower, and convection coefficients are lower than in smaller animals, each contributing to a smaller value of l, slower changes in body temperature, and increasing inertia.

Figure 3 suggests that small terrestrial organisms (mass # 1 kg) would be very sensitive to daily variations in operative temperatures and variably sensitive to thermal effects that occur on shorter timescales (e.g., due to

BIOPHYSICAL CONSTRAINTS ON DINOSAURS

movement, thermoregulatory shuttling or other behavior, or the sun being obscured by scattered clouds). Very large dinosaurs (.104 kg), however, would be largely insensitive to daily variations in the thermal environment and would maintain body temperatures near the mean daily operative temperature with little diel variation in body temperature. At higher wind speeds, the predicted sensitivity to environmental temperature variation increases (Fig. 3). Because of the convective decoupling that occurs in large animals, however, the effect of a change in wind speed is less than in smaller animals (Table 1). Nonetheless, dinosaurs may have been able to constrain or increase the variation in their body temperatures behaviorally by selecting environments with different wind speeds. Moving from an environment with a wind speed of 1 m/s to one where the wind blows at 5 m/s can almost double the predicted range of temperatures experienced by the animal (Fig. 3).

Effects of Metabolism.—Much of the debate about endothermy and body temperatures in dinosaurs turns on the contribution of metabolic heat to steady-state body temperatures.

Thus, it becomes important to estimate the extent to which metabolic rate can raise steadystate body temperatures above those for animals without metabolism (i.e., above operative environmental temperatures).

We take the resting metabolic rates predicted by Bennett and Dawson (1976) for reptiles at 308C as our baseline and as representative of resting metabolic rates of ectotherms (Reptilian Resting Metabolic Rate, RRMR). We refer to the ratio of the metabolic rate used for a simulation to that predicted by the Bennett and Dawson relation for the same-sized animal as ‘‘Mscope.’’ Here, we examine the effects of metabolic rates 1, 2, 5, 10, and 20 times that predicted by the Bennett and Dawson allometric predictions (Mscope 5 1, 2, 5, 10, 20).

At rest, endotherms typically have metabolic rates 5–10 times those of equivalent-sized ectotherms at the same temperatures (Bennett and Dawson 1976; Calder 1984). In addition, both endotherms and ectotherms often have field metabolic rates 2–3 times higher than resting rates (Nagy 1987, 1989) owing to locomotion and other activities. Thus, our 20-

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FIGURE 4. Elevations of body temperature above operative temperature (Te) for Newtonian dinosaurs. ‘‘Mscope’’ is ratio of metabolic rate to allometric predictions from Bennett and Dawson’s (1976) equation for reptiles at 308 C. Solid lines 5 dinosaurs in terrestrial habitat with wind speed 5 1 m/s. Dashed line 5 dinosaurs in aquatic environment with water flow velocity of 0.1 m/s.

fold range of metabolic rates should span most of the resting and mean field metabolic rates of both endotherms and ectotherms.

In the simulations presented here, we do not consider the effects of body temperatures on metabolic rates. Metabolic rate is assumed constant regardless of body temperature. We do so because metabolism does not depend on body temperature in a linear fashion. This requires that one specify the dependence of metabolism on body temperature over several temperature ranges, upper and lower critical temperatures, and the breadth of the range of temperatures over which metabolic rate is at a maximum. None of these factors is known for dinosaurs. Further, simulations suggest that the results presented here for high and low metabolic rates define the limits of the possible behaviors of these more complicated models (O’Connor 1999).

Under these assumptions, for cylindrical, Newtonian dinosaurs with mass less than 5 kg, metabolic rates had little effect on predicted steady-state body temperatures (Fig. 4). Without effective insulation such as the fur or feathers of mammals and birds, it seems unlikely that these small animals could substantially alter their body temperatures by

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means of the metabolic rates simulated here.

This is true for extant reptiles of this size (Bartholomew 1982). For Newtonian dinosaurs with masses greater than 1000 kg, and with ‘‘endothermic’’ metabolic rates (Mscope 5 5– 20), metabolism raised predicted body temperatures 9–1028C above operative environmental temperatures (Fig. 4). Using wind speeds of 5 m/s cuts these increments by about 50% (data not shown). Nonetheless, these simulations suggest that to avoid overheating, large dinosaurs with endothermic metabolism will require cool environments, tolerance of body temperatures over 408C, or adaptations or behaviors that allow the animal to shed heat, such as heat exchangers (e.g., sauropod necks and tails or possible sails on Amargasaurus, Spinosaurus). Alternatively, lower metabolic rates (Mscope 5 1–2), would yield much lower requirements for heat loss (Fig. 4). From 5 to 1000 kg, Mscopes of 1–10 led to elevations of predicted body temperatures of 1–208C above operative temperatures (Fig. 4). This suggests that in this range metabolic modulation of body temperature might have been effective without necessarily imposing large risks of overheating in temperate environments.

Aquatic environments result in much higher convective heat loss than do terrestrial habitats, and predicted increments in body temperature due to metabolism were much smaller in all sizes of animals in water than on land.

We will not consider aquatic or marine animals extensively here, but the usefulness of water as a heat sink (e.g., for large dinosaurs with legs in shallow water) should not be ignored. It is possible that dinosaurs with high metabolic rates could survive if they lived in or had access to standing water as a heat sink, under conditions in which the same dinosaurs on land would overheat (Fig. 4).

In vertebrates, metabolism is powered, either immediately or ultimately, by oxidative phosphorylation. Increased metabolism thus requires increased ventilation and increased respiratory evaporation and consequent evaporative heat loss. We used a model similar to that of Welch and Tracy (1977) to predict the fraction of metabolic heat that might be lost by respiratory evaporation. Important variables

FIGURE 5. Predicted fraction of heat generated by metabolism that is lost by respiratory evaporation from a model similar to that of Welch and Tracy (1977). Major determinants of respiratory evaporation in model are body temperature, relative humidity of local atmosphere, and the fraction of oxygen removed on average from each volume of inspired air (oxygen extraction).

for the model included air and body temperature, atmospheric relative humidity, and the fraction of oxygen extracted from inhaled air before exhalation. All exhaled air was assumed to be saturated with water at body temperature. Under most simulated conditions, less than 20% of metabolic heat was lost through respiratory evaporation with body temperatures of 408C, although if one assumed low oxygen extraction and low atmospheric humidity, the fraction rose to about 25% (Fig. 5). At higher body temperatures than those tolerated by extant reptiles or birds (508C) and under some assumptions, as much as 40% of metabolic heat might be lost through respiratory evaporation (Fig. 5). In general, however, the fraction of metabolic heat lost by evaporation will be small. Thus, metabolic heating is unlikely to be offset to any signifi-

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FIGURE 6. Predicted differences between midline temperature and surface temperature of dinosaurs with no blood flow to aid in heat transfer—i.e., all metabolically generated heat must be conducted to the surface of the animal through tissue. ‘‘Mscope’’ is the ratio of metabolic rate to allometric prediction from Bennett and Dawson’s (1976) equation for reptiles at 308C. Wind speed 5 1 m/s.

cant extent by the required respiratory evaporation in the absence of heat-loss mechanisms such as panting.

Effects of Conductance.—Up to this point, we have assumed that dinosaurs had sufficient internal heat conductance to keep body temperatures the same throughout the body—i.e., infinite internal conductance. By doing so, we have assumed that metabolism cannot heat the core of the animal to any greater extent than its surface, that metabolic heat generated anywhere in the body can immediately be lost to the environment at the skin, and that changes in the temperature of the skin are immediately transmitted to the rest of the body. In this sense, Newtonian models predict the minimum possible body temperature and the fastest possible response to changes in the environment. We must now ask what effects the finite conductance of real animals might have on steady-state and transient body temperatures of dinosaurs.

The thermal conductivity of muscle, although several times that of fat, is not high enough to maintain all parts of the organism at the same temperature under most conditions (Turner and Tracy 1983, 1985). If blood did not carry heat around the body, many reptiles would experience core body tempera-

FIGURE 7. Heat transfer dynamics in core-shell model of a dinosaur. A thermally homogeneous core exchanges heat with the shell by conduction and by blood flow. In addition, the shell exchanges heat with the environment by convection and radiation. Both core and shell are heated by metabolically generated heat.

tures substantially above those predicted for Newtonian animals (Fig. 6). Blood flow, however, serves as an effective heat pump and limits the temperature gradients that exist within the body. In so doing, it determines where actual organisms fall on a continuum from a Newtonian organism (Tcore 2 Tsurf 5 0) to the sorts of gradients predicted in Figure 6.

Vascular heat exchange and the low thermal conductivity of most tissues also affect rates of heating and cooling and the thermal inertia of an animal. Consider the cylindrical lizard in Figure 7, with a thermally homogeneous core and a surrounding shell (an outside layer through which heat is transferred). When a change occurs in the external environment, it cannot affect the temperature of the core directly. A change in the operative temperature must first effect a change in the shell temperature. A change in the shell temperature can then be communicated to the core via conduction through the tissues or heat exchange via blood flow. Thus, such an animal will heat or

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cool more slowly than an otherwise similar Newtonian organism; i.e., it will have higher thermal inertia.

The rate at which heat is conducted between the core and shell depends on the conductivity of the tissues and the geometry of the animal, neither of which is easily changed. The rate of heat transfer via the blood, however, depends primarily on the rate of blood flow to the shell, which reptiles can control, as Cowles elegantly showed (1958). So we are interested in the effects of blood flow on heating and cooling.

We assume (1) that the resting cardiac output of reptiles scales allometrically proportionately to mass0.75 as do metabolic rates, (2) that the cardiac outputs measured by Baker and White (1970) and Baker et al. (1972) on extant reptiles are representative for dinosaurs, and (3) that blood flow is distributed to all tissues evenly on a gram-by-gram basis. We call the ratio of cardiac output simulated to the predicted resting cardiac output ‘‘Qscope’’ and simulate Qscopes from 10% of predicted to ten times that predicted for a given animal (Qscopes of 0.1, 0.2, 0.5, 1, 2, 5, and 10). We use additional simulations using Qscopes of 0.01 and 100 to cover extremely low and high blood flow rates.

The equivalent of equation (7), describing the temperatures of the core during a warming or cooling curve for this core/shell animal, is equation (9).

Tc 5 Tc,ss 1 c1exp(l1t) 1 c2exp(l2t)

(9)

where Tc,ss is the steady-state core temperature (8C), ln are the time coefficients (s21), cn are the constants that depend on core and shell temperatures at time zero (8C). The two time coefficients tell us different things about heating and cooling. l1 plays roughly the same role as l does in equation (7). It shows the approach of Tc to Tc,ss after initial transients die away.

We call l1 the dominant time coefficient. l2, on the other hand, describes how quickly the initial transients die away and is called the subdominant time coefficient. As with l in equation (7), the values of both time coefficients will be negative (otherwise, we would predict infinite temperatures after a long time), and larger absolute values correspond to faster decays. Time coefficients for animals with dif-

FIGURE 8. Predicted time coefficients ( ln—see eq. 9) for core-shell model of dinosaurs with masses 10 g–10 5 kg as mass-specific blood flow varies from 0.01 to 100 times allometrically predicted rates (Qscope).

ferent masses and blood flow rates are presented in Figure 8, and the ratios of the dominant time coefficients (l1) to the time coefficients of Newtonian animals (l 5 lmax) in identical situations are shown in Figure 9.

Simulations were also performed using very high blood flow rates (Qscope 5 100) to approximate the time coefficients for Newtonian animals. Three important patterns emerge.

First, as with Newtonian animals, body temperatures of small animals have larger time coefficients and will respond more quickly than those of larger animals (Fig. 8). Second, in all cases, the idealized Newtonian dinosaurs were predicted to warm or cool faster than animals with the more realistic internal conductances. As predicted, higher blood flow enhances vascular heat exchanges, increases the absolute values of the time coeffi-

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FIGURE 9. Ratio of the dominant time coefficient ( l1— see eq. 9) for a core-shell model of a dinosaur to the time coefficient (lmax) for a Newtonian animal, i.e., one with infinite internal conductance, in the same environment.

cients, and hastens the response of body temperatures to changes in the thermal environment (Figs. 8, 9), converging on values for Newtonian animals (lmax) at high blood flow rates. Third, body temperatures of large animals are predicted to be more sensitive to blood flow and to respond to lower levels of blood flow than would those of smaller animals (Figs. 8, 9). This prediction is consistent with the findings of Grigg et al. (1979) on extant lizards. The sensitivity of predicted dominant time coefficients (l1) to blood flow increases in environments more conducive to convection (e.g., those with higher wind speeds or aquatic environments) (Turner 1987, O’Connor 1999). As with dominant time coefficients, fractional amplitudes are more susceptible to change by alterations of blood flow in large animals than they are in small animals (Figs. 8, 9, 10).

The time coefficients help us understand the heating and cooling rates and the sensitivity of those rates to blood flow. As with Newtonian animals, the response of the animal’s body temperature to varying environments places these effects in an ecological perspective. The effect of blood flow is to shift the range of periods of temperature changes to which the animal responds strongly, making the animal respond like a larger or smaller an-

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FIGURE 10. Response of core temperature of core-shell model of a dinosaur to variation in environmental temperature. Period and fractional amplitude defined as in Figure 3. Symbols indicate mass of dinosaur (1–104 kg).

Line style indicates cardiac output in multiples of allometrically predicted values (Qscope). Wind 5 1 m/s for all simulations.

imal (Fig. 10). With increased (Qscope 5 10) or decreased (Qscope 5 0.1) blood flow to the shell, predicted fractional amplitudes overlap those of animals with 2–10 times more or less mass than the animal. For example, a 100 kg reptile, with blood flow to the shell reduced to 10% of that predicted by allometric equations, has approximately the same amplitude vs. period plot as a 1000-kg animal with shell flow at 100% of predicted levels (Fig. 10). Thus, by changing blood flow rates, dinosaurs could have altered their thermal inertia and the range of body temperatures they experienced (see Tracy et al. 1986; Turner and Tracy 1986).

The extent to which dinosaurs could alter thermal inertia depends on the extent to which they could alter blood flow levels. Because the extent of this control is unknown, the actual ability to modify thermal inertia can only be estimated for a range of possible values as in Figure 10. Nonetheless, by sequestering blood centrally, dinosaurs could effectively insulate themselves from their thermal environments and decrease the range of body temperatures experienced. Alternatively, by flushing blood to the periphery, they could increase heating and cooling rates and

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increase thermal responsiveness to the environment.

Heat Exchangers.—All predictions to this point have been for cylindrical dinosaurs without any limbs. Limbs can be important to the thermal biology of archosaurs (Turner and Tracy 1983, 1985) and represent a class of pathways of biological heat exchange that we lump together as ‘‘heat exchangers.’’ Besides limbs (and necks and tails), some dinosaurs possessed plate-like heat exchangers such as the dorsal plates of Stegosaurus, possible sails based on elongate dorsal neural spines (e.g., Ouranosaurus or Spinosaurus [but see Bailey 1997]), or cranial frills as in ceratopsians (Wheeler 1978). In addition, the peripheral tissues and skin of an animal, which we simply called a shell previously, can be thought of as a heat exchanger. The common feature of each of these heat exchangers is that heat is transferred back and forth between the core of the animal and the heat exchanger primarily by blood flow. We now examine the effects of such heat exchangers on both the steady-state and transient temperatures of dinosaurs.

In the first set of simulations, we examine the effects of appending various types of exchangers onto the Newtonian cylinders we examined earlier. Limbs, with an allometry we had developed for dinosaurs (Spotila et al.

1991), were appended to a Newtonian cylinder of a given mass. In another set of simulations, a pelycosaur-like sail with the allometry used by Tracy et al. (1986) was appended to the same Newtonian cylinder. Finally (as for Figs. 7–10), the outer layer of the cylinder was considered a shell exchanging heat with the core by blood flow and conduction. The heating and cooling of each of the heat exchanger models can be described by equation (9). In each case, the main cylinder was the same size and had the same convection coefficient. The purpose was to compare the Newtonian time coefficient of each of the model animals with that of a Newtonian cylinder with the dimensions of the main cylinder.

The dominant time coefficients (l1) of each heat exchanger model (for Qscope 5 1, wind speed 5 1 m/s) are presented in Figure 11A. In each case, the time coefficient (l1) is normalized by dividing by the time coefficient for

FIGURE 11. Effect of heat exchangers on predicted rates of heating and cooling in dinosaurs. A, Ratio of dominant time coefficient (l1—see eq. 9) to that for a Newtonian cylindrical dinosaur (lmax) of the same mass but without limbs or sail-like heat exchanger. B, Response of core temperature of core-shell model of a dinosaur to variation in environmental temperature. Period and fractional amplitude defined as in Figure 3. Cardiac output at allometrically predicted level (Qscope 5 1), wind at 1 m/s in all simulations.

a Newtonian cylinder (lmax) under the same environmental conditions. If the heat exchanger does not change the time coefficient (l1 5 lmax), the normalized time coefficient is 1. If the heat exchanger enhances heat exchange, the normalized value is greater than 1. Limbs increased the absolute value of time coefficients and rates of heating and cooling more strongly in large animals. As discussed above, core/shell models predicted smaller time coefficients than for a Newtonian cylinder, but the effect was less marked in large animals (Figs. 9, 11A). Pelycosaur-like dorsal sails, in very large animals (300–30,000 kg) speed

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model heating and cooling even more than limbs (Fig. 11A). The decreased effectiveness of a sail as a heat exchanger predicted for animals smaller than 300 kg was due to the shrinking size of the sail, which shrinks faster than the mass of the model organism (see Tracy et al. 1986 for discussion). Decreases in sail exchanger function in the largest animals were due to two factors. First, just as sail allometry dictates that the sail will shrink faster than total mass in small animals, sail mass grows faster than cylinder mass in large animals, and the sail takes on a thermal inertia of its own. Second, in this simulation, air flow is assumed to travel parallel to the plane of the sail. As the sail grows in size (i.e., in larger animals), the thickness of the boundary layer along the trailing edge grows, limiting heat transfer from the downstream side of the sail.

In effect, there is a point of diminishing returns as the sail gets larger. The effects of heat exchangers on the time coefficients of model dinosaurs is reflected in the fractional amplitudes of those animals. Heat exchangers shift amplitude vs. period curves and change thermal inertia to a greater extent in large animals than in small animals (Fig. 11B).

We should note that several types of dorsal heat exchangers occurred in large animals. Instead of a single large sail, stegosaurs possessed a staggered series of smaller plates. In many ways, this sort of arrangement has thermoregulatory advantages over the single large sail of pelycosaurs (Farlow et al. 1976). Other types of plates could also have been used as heat exchangers if they were well vascularized (even a local area of skin that was highly perfused). We do not focus on the properties or relative merits of different types of heat exchanger, but on the heat transfer properties of heat exchangers as a pathway for heat exchange.

In a second set of simulations, we used more complex models to estimate the steadystate temperatures reached with heat exchangers. We still use the same three exchangers, the shell or peripheral tissues of the animal, the limbs, and a flat, plate type of exchanger. But here we examine the effect of a second type of plate-like heat exchanger with the allometry of elephant ears instead of a pelycosaur sail as

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the plate heat exchanger. In this case, however, we abandon the requirement that the central body cylinder be the same size and mass as the cylinders we have used previously. All animals have a central core, a cylindrical shell, limbs, and plate exchangers equivalent to elephant ears. All body parts (except the elephant ears) are scaled according to the allometry of Spotila et al. (1991), and the total mass of the organism (rather than the mass of the main cylinder) is set to the desired size. Each heat exchanger exchanges heat with the core via blood flow. The shell, in addition, exchanges heat with the core via conduction. For the simulations presented here, each exchanger is either perfused on a gram-by-gram basis with the same blood flow rates as the core or left almost totally unperfused (blood flow 5 0.1% of that in the core).

There were several major procedural differences between these simulations and those presented above. Instead of estimating one of the components of body temperature, here we estimate the total elevation of core body temperature above ambient air temperature. This allowed us to reexamine the importance of solar radiation to heat balance in dinosaurs. In addition, we introduce an important modeling assumption: Because the delivery of the oxygen that serves as fuel for metabolism is a major role of blood flow, we assumed that metabolic rate and blood flow, as indicated by Mscope and Qscope, were correlated. Here, we present results with metabolic rate and cardiac output both at ten times their allometrically predicted rest levels for reptiles (Qscope 5 Mscope 5 10).

Results of the simulations are presented in Figure 12. Three major patterns emerge. First, as with previous simulations, higher temperatures are expected with larger animals. Second, high metabolic rates (Mscope 5 10) in a calm (wind speed 5 1 m/s), sunny environment led to core-to-air temperature gradients of 20–508C in animals with masses over 1000 kg, despite the use of several heat exchangers.

Although not shown in Figure 12, decreasing metabolic rates (Mscope 5 5) or moving into windy environments (wind speed 5 5 m/s) decreased body temperatures, especially in larger animals (as in Figs. 1 and 4). Third, the

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FIGURE 12. Effect of heat exchangers on predicted steady-state body temperatures of dinosaurs (mass 10 g–105 kg). Each of the heat exchangers (shell layer of a core-shell model [‘‘surf’’], limbs [‘‘limb’’], and a platelike heat exchanger [‘‘plate’’] received either the same mass specific blood flow as the core or 0.001 times that blood flow (e.g., ‘‘no plate’’ means plate exchanger received low blood flow and other exchangers received same flow as core). See text for explanation. Solar radiation 5 800 W/m2. Wind 5 1 m/s.

utility of the three exchangers modeled varied with mass. In all simulations, the effects of exchangers on core temperatures increased with animal mass. Hypoperfusion of the limbs led to the largest changes in temperature in animals smaller than 1 kg, but to the smallest temperature changes in animals larger than 300 kg (Fig. 12). Changes in the perfusion of the peripheral tissues (‘‘surf,’’ Fig. 12) and of the plate heat exchanger resulted in similar, large (158C at 3000 kg) changes in core temperature for animals larger than 300 kg.

Thus, even with physiological heat-loss adaptations, such as the use of limbs as heat exchangers, or anatomical heat-loss adaptations, such as plate-like heat exchangers, large dinosaurs would have been in danger of overheating in warm environments if they had metabolic rates typical of extant endotherms.

Heat exchangers would have had little effect on the steady-state temperatures of small animals. Again, animals with masses of 10–1000 kg, endothermic metabolism, and the ability to dump heat via heat exchangers may have

been able to modulate temperature via metabolism.

Multiple Layers.—All of the models used to this point are simple models built to understand the effects of body size, internal heat conductance, and metabolism on body temperatures of dinosaurs. These simple models have the virtue that simple, analytic equations predict both steadystate body temperatures and the response to varying environmental temperatures. In a word, the simple models are simple to understand.

But simple models often sacrifice realism for tractability. To investigate whether the temperatures and dynamics predicted by simple models accurately describe the thermal behavior of more complicated, real animals, we used an ‘‘onion-skin’’ model. In this model, both the torso of the animal and its appendages are represented as a series of concentric cylindrical layers like the skin layers of an onion. Each layer exchanges heat with adjacent layers by conduction and with a central vascular pool by blood flow. Steady-state and transient body temperatures were then predicted by computer simulations. In all simulations presented here, blood flow was either distributed evenly to each gram of tissue (even blood flow condition), or tissues in the outermost 10% of the radius of the animal—both torso and appendages—were deprived of blood flow, with blood flow distributed evenly to other tissues (dead surface condition). See O’Connor (1999) for details of the onion-skin model.

We investigated three questions with these onion-skin models. Two are the same questions we have asked of each of the simple models: How do predicted, steady-state ‘‘core’’ temperatures vary in different environments and with different blood flow and metabolic rates? How do transient body temperatures vary with these same conditions? In addition, we ask, Is there is an isothermal core of the animal that can be thought of as having a single core temperature (as in our simple models), or does body temperature grade evenly from the center to the surface of the model animal? We will address this third question first, then return to the other two.

The classic example of a solid cylinder with

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FIGURE 13. Predicted body temperature profiles from the center of the trunk to the skin in dinosaurs of varying sizes with varying levels of blood flow (QScope 5 ratio of cardiac output to that predicted by allometric equations).

A–C, Radial temperature profiles with midline temperature scaled to 1 and skin temperature scaled to 0. A, Qscope 5 0.1. B, QScope 5 1. C, QScope 5 10. D, Potential differences between core and surface temperatures in animals with high metabolic rate (Mscope 5 10) and varying levels of blood flow, QScope, from 0.2 to 10 times predicted.

internal heat generation is the wire that is heated by the current it carries. In this case steady-state temperature falls off from the midline to the surface. The difference between the temperature at any spot and that at the center of the wire is proportional to the square of the distance from the center. Onion-skin models make similar predictions about the temperatures of dinosaurs (Fig. 13). With low blood flow (Fig. 13A), temperatures show the parabolic distribution expected in wires. Increased blood flow (Fig. 13B,C) increases the volume of the nearly isothermal core and steepens the temperature gradient near the surface. Large animals appear to have more of their mass in an isothermal core than do smaller animals (Fig. 13A–C). In part, this is due to the increased sensitivity of large animals to blood flow, as we have already discussed (Figs. 9, 10). In part, however, this pattern is deceiving. Temperature gradients in

Figure 13A–C are presented in terms of fractions of the total core-to-surface temperature difference. But the-core-to-surface temperature difference is much larger in large animals than in small ones (Fig. 13D). Thus, if an isothermal core is that portion of the animal within some number of degrees of the core temperature, small animals will have larger isothermal cores than Figure 13A-C suggest simply because the total core-to-surface thermal gradients are small. Regardless of how the core is defined, substantial proportions of a model dinosaur’s mass is likely to be close to core temperature, particularly with predicted (or higher) blood flow rates. Thus, it is reasonable to think in terms of a nearly isothermal core.

Core-operative temperature differences for onion-skin models are consistent with predictions from simpler models with shell and limb heat exchangers (Fig. 14, compare Fig. 12).

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FIGURE 14. Predicted elevations of body temperature above operative temperature (Te) due to metabolic heating in onion-skin models of dinosaurs ranging in size from a 2-kg Compsognathus to a 30,000-kg Apatosaurus.

Metabolism (Mscope) and mass-specific perfusion (Qscope) are both increased to the same extent for each simulation. Solid line 5 even distribution of blood to all tissues. Dashed line 5 blood flow falls linearly from full perfusion at 90% of the way from midline to surface to no perfusion at surface (‘‘dead surface’’).

Limiting blood flow to peripheral tissues reduces the effectiveness of the heat exchangers and increases the predicted body temperatures, particularly in large animals, which are more sensitive to blood flow (Fig. 14). For large dinosaurs (.3000 kg), metabolic rates 5– 10 times those predicted for reptiles yield predicted core-operative temperature differences of 10–308C. Predicted body temperatures of small dinosaurs (,100 kg) differed from operative temperatures by less than 78C regardless of metabolic rate. High metabolic rates (Mscope 5 5–10) in intermediate-sized dinosaurs would have produced core temperatures 5–128C above operative temperatures.

The predictions about dinosaur heating and cooling rates made by onion-skin models are in general agreement with those of simpler models. Blood flow affects heating and cooling rates and thermal inertia in large dinosaurs to a greater extent than in smaller animals (Fig. 15).

The effects of blood flow on time coefficients in model dinosaurs are reflected in the predicted effects of fluctuations in environ-

FIGURE 15. Response of core temperature to variation in environmental temperature in onion-skin models of dinosaurs. Period and fractional amplitude defined as in Figure 3. Symbols represent dinosaurs of different masses, from a 1.98-kg Compsognathus to a 30,000-kg Apatosaurus. Lines represent responses of animals with Qscope 5 1. Symbols not on line represent responses for animals of the same size but with Qscope 5 0.1–10 times the predicted levels. Blood flow is evenly distributed to all tissues in all simulations. Mscope 5 0–10.

mental temperatures on dinosaur body temperatures (Fig. 15). Increased blood flows increase response to environmental variations and make the animal ‘‘thermally smaller.’’ Decreased blood flow rates, overall (Fig. 15) or specifically to the skin (data not shown), decrease responses to environmental variations and make the animal appear ‘‘thermally larger.’’ Regardless of blood flows, large model dinosaurs (.1000 kg) respond minimally to environmental variations with daily periods.

Their body temperatures varied as if to an average daily operative temperature. Small animals (,10 kg) track daily temperatures fairly well (fractional amplitude . 0.95, Figure 15).

Model dinosaurs with intermediate masses track daily temperatures with varying degrees of fidelity depending on blood flow (Fig. 15).

Over the range of simulated values, blood flow patterns and levels can adjust the range of experienced temperatures by a factor of 2– 3 but have their largest effects over the range of periods to which the animal responds with fractional amplitudes of 20–80% (Fig. 15).

Cretaceous Climates.—To assess the effects of environmental and metabolic heat sources on the body temperatures and thermal ecology of dinosaurs, we calculated expected core temperatures for different-sized dinosaurs with varying metabolic rates in the setting of one

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FIGURE 16. Steady-state body temperatures predicted for dinosaurs (sizes from 10 g to 10 5 kg) at different latitudes during January under the Cretaceous climate reconstructions of Crowley and North (1991). Environmental conditions used are the average air temperature and solar radiation over the course of a 24-hour period. Metabolic rate and cardiac output are both presented as ratios of simulation value to allometrically predicted value. Conditions: wind speed 5 1 m/s, even blood flow to all tissues. A, Mscope 5 Qscope 5 1. B, 2. C, 5. D, 10.

hypothesized set of Cretaceous climates (Crowley and North 1991). We simulated steady-state body temperatures for dinosaurs (mass 10 g–105 kg) experiencing the daily average temperatures at different latitudes and different seasons. Only results for January and masses greater than 1 kg are presented (Fig. 16). We simulated four combinations of blood flow and metabolic rate (Mscope 5 Qscope 5 1, 2, 5, 10). Despite varying assumptions about wind speed (results not shown), these simulations suggest that large dinosaurs (.1000– 5000 kg depending on assumptions), at tropical to middle latitudes, with mammalian metabolic rates (Mscope 5 5–10) would experience body temperatures above 408C during the summer. Those animals would be in danger of overheating in the absence of specific heat-loss adaptations. Animals with met-

abolic rates predicted by the Bennett and Dawson equations would be predicted to have body temperatures between 308 and 408C in summer over a broad range of body sizes and latitudes. Considering animals in the winter hemisphere and at latitudes above 408, only the very large animals (.10,000 kg) with metabolic rates above 5 times predicted would be expected to have body temperatures above 308C. In very small adult dinosaurs (,10 kg) and in very small hatchlings and juveniles (10 g–1 kg) metabolic rates have relatively little effect on predicted body temperatures.

Five caveats are necessary to interpret these predictions. (1) The multipliers for metabolic rates should not be considered exact. The allometric predictions for mass-specific metabolic rate at 308C used here (Bennett and Dawson 1976) have a smaller scaling coefficient

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(20.17) than most other predictions or, indeed, predictions about metabolic rate at other temperatures from the same study (ø2 0.25).

Furthermore, over the range of masses between those used to estimate the allometries and those of the large dinosaurs modeled here, this difference in scaling coefficients could result in a 2–3 fold difference in predicted metabolic rate. Thus, ‘‘endothermic’’ metabolism may be modeled better by metabolic rates 3–53 the prediction rather than 5– 103. On the other hand, when mean field metabolic rates are measured in vertebrates, those rates usually range 2–33 resting metabolic rates (Nagy 1987, 1989). (2) The allometric predictions are being extrapolated considerably beyond the range of masses for which measurements are available. (3) These predictions are for a Cretaceous paleoclimate more equable ( 5 warmer) than Recent climates (Crowley and North 1991). (4) Finally, the predictions here are for animals without particular heat-loss adaptations. (5) These predictions must be applied carefully to animals small enough to change temperatures significantly by taking advantage of diel variation in operative temperatures (,200–300 kg, Fig. 15).

Given the first two of these caveats, it is worth inverting the analysis of Figure 16 to ask what metabolic rates would be required to maintain specified body temperatures in this paleoclimate (Fig. 17). For animals without heat-loss adaptations at the equator, there is a range of masses (2–1000 kg) at which metabolic rates 1–103 allometric predictions allow maintenance of body temperatures between 308 and 408C in both winter and summer; at higher masses, lower metabolic rates would be required (Fig. 17A). At higher latitudes, as the seasonality of the climate becomes more pronounced, there is no single metabolic rate that allows maintenance of body temperatures in the 30–408C range during both winter and summer, and seasonal acclimation to maintain those body temperatures would require large factorial changes in metabolic rate (minimum ø 40-fold change between summer and winter, Fig. 17B,C). Also the wintertime metabolic rate would need to be fairly precisely controlled to avoid overheating or excessive

cooling (Fig. 17B,C). These predictions are not sensitive to errors in the extrapolation of predicted metabolic rates. The curves representing metabolic requirements in Figure 17 do not depend on the allometric predictions of metabolism.

Discussion The major implication of the simulations presented here is not that any particular combination of body mass and metabolic rate cannot be tolerated by a dinosaur. Rather, body mass and metabolic rate constrain the environment, blood flow pattern, tolerated body temperatures, and heat-loss adaptations of the organism. A 30,000-kg sauropod with a metabolic rate ten times that predicted for reptiles living in a sunny terrestrial environment (without access to shade or standing water) with daily mean summer air temperatures near 258C would need to tolerate body temperatures of about 808C, well above body temperatures tolerated by extant vertebrate endotherms or ectotherms. However, in a colder environment (air temperature ø 108C), with a well-perfused heat exchanger, especially one protected from the sun, or with limbs in cool water, the animal’s body temperature could be within the bounds of currently tolerable body temperatures. Increased heat loads due to several sunny days or extended bouts of exercise, however, might result in dangerous levels of body temperature even in this animal. Alternatively, a 2-kg Compsognathus could easily tolerate metabolic rates 10–20 times those expected for a reptile, as long as the environment was not very hot (i.e., operative temperatures did not exceed 40–458C), but would be unlikely to derive any thermal benefit from the expenditure of the extra energy unless it also had some way of conserving the generated heat (e.g., fur, feathers, substantial subepidermal fat, or some other insulation).

The pattern of body temperatures outlined in Figures 1, 12, and 14 suggests that very large dinosaurs (.10,000 kg), principally sauropods, are unlikely to derive any thermoregulatory benefit from metabolic rates above twice the expected reptilian level in warm terrestrial environments. Typical large dinosaurs (1000–10,000 kg; e.g., hadrosaurids, ceratop-

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FIGURE 17. Metabolic rates required to maintain body temperatures of 30 8 or 408 C in dinosaurs living in the Cretaceous climates reconstructed by Crowley and North (1991). Separate predictions are made for summer and winter and for animals at the equator, in temperate regions, and in polar regions. Conditions: wind speed 5 1 m/s, even blood flow to all tissues.

sids, iguanodontids, ankylosaurians, and many theropods) in cool environments could tolerate, and benefit from, higher metabolic rates without risking overheating. Smaller dinosaurs (,1000 kg; e.g., basal ornithopods, basal ceratopsians, basal iguanodontians, maniraptoran theropods, juveniles of many dinosaurs) could probably tolerate any of the metabolic rates simulated here, unless they found themselves in very warm environments. The extent to which these smaller animals could benefit from high metabolic rates would depend on the conductance from the animal to the environment. In these cases, the risk of overheating and thermal benefit of high metabolic rates can be roughly estimated with simple models like those presented here.

The patterns of transient thermal response to changes in the environment (Fig. 15) allow

us to estimate when thermal inertia can be used as a refuge from overly warm or cool environmental conditions. In all cases simulated, rapid variations, i.e., those with short time periods (ø 1 min) result in little to no change in body temperatures. The operative temperature to which the animal responds in these cases is the time-averaged mean operative temperature. On the other hand, variations with periods of seasonal length (ø 2000 h) allow body temperatures to vary through the full available range. In this case, variations in body temperature will track those of mean operative temperatures fairly closely. Between these two extremes lies a range of variation periods to which body temperatures respond moderately (fractional amplitudes 0.2–0.8) in the absence of thermoregulation. It is in this range that changes in wind speed, cardiac out-

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put, blood flow distributions, and use of heat exchangers have their major effects in terrestrial animals (Figs. 10, 11, 15).

Figure 15 suggests that for environmental variations with periods shorter than a week (ø 168 h), large dinosaurs (.3000 kg) would experience relatively small changes in body temperature. There are two important caveats to such a statement. First, these large animals are precisely the ones that will experience the largest variations in operative environmental temperatures due to solar radiation (Fig. 1) and whose steady-state body temperatures can most easily be altered by controlling blood flow (Figs. 12, 14). Thus, although the fractional amplitudes may be low, the actual changes in temperature may still be several degrees. Second, although not apparent from Figure 15, the ratio of fractional amplitude at the highest blood flow rates to that at the lowest blood flow rates is largest for periods resulting in small fractional amplitudes (fractional amplitude , 0.1). For a 30,000-kg sauropod in Figure 15, the ratios of maximal (Qscope 5 10) to minimal (Qscope 5 0.1) fractional amplitude for sinusoidal temperature variations with periods of two days, one week, and three weeks are 3.4, 2.0, and 1.3, respectively. To understand why this is important, we must recall that predicted steady-state temperatures are not realized body temperatures, but rather measures of available body temperatures in a hypothetically stable environment. Thus, increasing blood flow allows for faster response to environmental changes, and changing blood flow may allow large factorial changes in the range of body temperatures experienced by the animal. In these cases, it is important to know the total range of operative temperatures to interpret the biological importance of the fractional amplitude.

For instance, for a two-day cycle, increasing cardiac output from 0.2 times predicted resting levels to 10 times predicted increases predicted fractional amplitude from 0.0364 to 0.124. The predicted range of body temperatures is the fractional amplitude times the range of operative environmental temperatures. If that range is 18C, changing blood flow is unlikely to have a biologically significant effect. If the range of operative temperatures is

408C, however, blood flow alterations could conceivably increase the range of experienced body temperatures from 1.48C to 58C, which may be important to the biology of the dinosaur.

Despite these limitations on the analysis of fractional amplitudes, Figure 15 allows us to make some statements about thermal inertia and the buffering effect of mass on body temperature. For very large animals such as sauropods, diurnal variations in temperature would not strongly affect body temperatures.

On the other hand, even these very large animals would have little inertial defense against seasonal changes (ø 2000 h) in temperature. For environmental variations with periods between days and seasons (e.g., changes due to weather systems, cold fronts, etc.), fractional amplitudes would be intermediate, affected by blood flow, and thus, under the animal’s control to some extent. An appropriately sized plate heat exchanger could provide additional control over steadystate and transient body temperatures. Such exchangers occur both in dinosaurs (e.g., the enlarged perfused dorsal plates of Stegosaurus, [Farlow et al. 1976]) and in other large animals (e.g., the sails of some pelycosaurs [Tracy et al.

1986] or the ears of an elephant) but are not known for sauropods. The extent to which a highly perfused area of skin could serve the same function would depend on perfusion rates, the location and areal extent of the ‘‘skin exchanger,’’ and the external environment (e.g., could the skin be wetted and allowed to cool by evaporation).

A similar analysis for a 2-kg Compsognathus suggests that the animal’s body temperature would respond only slightly to minute-to-minute changes in the environment, but would track diurnal temperatures rather closely. Further, the animal could capitalize on any thermoregulatory shuttling among environments (e.g., between sun and shade) by using vascular control of heating and cooling rates as it moved on an hourly basis (Fig. 15). For animals between 2 kg and 20,000 kg, the temporal scale of the thermal variations that would or would not affect body temperatures and of which the animal could take advantage would vary with body size (Fig. 15), the use

BIOPHYSICAL CONSTRAINTS ON DINOSAURS

of heat exchangers (Fig. 11), and insulation.

Such temporal scales suggest that ‘‘inertial homeothermy’’ would buffer most dinosaurs from daily temperature excursions and allow them to tolerate (for hours or even days, Fig. 15) operative temperatures that would be lethal if the animal ever came to equilibrium with that environment (Paul 1991). The same timescale considerations, however, show that thermal inertia would not effectively buffer body temperatures from the effects of seasonal variations in operative temperatures (Fig. 15) of the type used to generate our steadystate temperature predictions (Fig. 16).

Simulations of steady-state body temperatures (in response to daily mean operative temperatures, Fig. 16) suggest the mean values around which body temperatures would vary. They suggest that animals larger than 1000 kg with high metabolic rates would be in danger of overheating during the summer at tropical and middle latitudes. Very large sauropods (.10,000 kg), especially those with hypoperfused peripheral tissue layers, would be most susceptible to overheating (Fig. 16). Such animals would have required either cool habitats or a means of cooling themselves, such as a heat exchanger (e.g., on neck and/or tail), in order to be active during the summer. For these large animals, diel temperature differences would not have served as refugia from the warm environment, because of the slight responses of body temperature to daily temperature variation (Fig. 15). Alternatively, high metabolic rates might have provided some protection from cool winter temperatures in these large dinosaurs (Figs. 16, 17).

At the other end of the size scale, metabolic and circulatory adjustments had little effect on the predicted body temperatures of dinosaurs with masses below 10 kg (Fig. 16). Thus, metabolic warming is unlikely to have played a large role in the thermal physiology of these animals unless and until they developed insulation sufficient to retain metabolically generated heat. But these are the animals whose body temperatures could vary dramatically over the course of a day and could benefit from microhabitat selection, basking, behavioral thermoregulation, and the use of daytime or nighttime thermal refugia. By such

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mechanisms, small dinosaurs might have affected the ‘‘average’’ operative temperature they experienced over any given day or at a particular time of year, much as modern lizards do.

For animals with masses between 100 kg and 10,000 kg, predictions suggested a gradient from the relative unimportance of blood flow and metabolism to body temperature in small animals to the much larger effect of metabolism on body temperature in larger animals with the concomitant risk of overheating in some situations (Fig. 16). Predicted temperatures vary with assumptions about metabolism, blood flow, and environmental conditions. With low wind speeds (1 m/s), high metabolism (103 RRMR), and a cornified or hypoperfused peripheral layer (Fig. 16D), all animals larger than 100 kg would be at risk of overheating in low latitudes and all animals larger than 1000 kg would be similarly at risk at any latitude during the summer. Alternatively, with higher wind speeds (3 m/s), lower metabolic rates (53 RRMR) and even perfusion of all tissues only large animals (.10,000 kg) at low latitudes would be at risk of overheating, and other animals might gain some benefit from the relatively high metabolic rates—either from a thermoregulatory point of view or otherwise.

An interesting perspective on these predictions is provided by examining model predictions for African elephants, the largest extant terrestrial endotherm. Elephants are large tachymetabolic endotherms that are frequently thought to be at risk of overheating due to their combination of large body size, mammalian metabolic rates, and relatively warm environments (Wright 1984, Wright and Luck 1984, Williams 1990). We predicted the steadystate core body temperatures for a 4000-kg African elephant (Loxodonta africana) in the paleoenvironment used to generate Figure 16 at 208 latitude during the summer (hottest available mean Te). Although the paleoclimate used is warmer than current mean temperatures, the differences for the Tropics are only 3–58C (Crowley and North 1991). Thus, in essence, we simulate an unusually warm summer.

Elephant resting metabolic rates corre-

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FIGURE 18. Core body temperatures predicted for a 4000-kg elephant in July at 208 N latitude in the paleoclimate used for Figures 16 and 17. Predictions are shown for a range of metabolic rates spanning resting and active metabolic rates for elephants and for a variety of convective environments (wind speed and turbulence). For the simulations shown, the elephant is assumed not to use its ears as heat exchangers. Qscope 5 Mscope for all simulations.

spond to an Mscope (as calculated in this report) of approximately 3.6 with mean metabolic rates approximately at Mscope 5 6–9 and high rates during induced exercise at Mscope 5 13–14 (Benedict 1936; Wright 1984; Williams 1990; Withers 1992). Figure 16 suggests that a 4000-kg animal with such metabolic rates would be at risk of overheating.

Several factors mitigate this risk. First, African elephants, particularly during warmer parts of the day, experience wind speeds significantly above 1 m/s (Buss and Estes 1971) and usually turbulent airflows, which increase convection coefficients by a factor of 1.7 over those for laminar airflows (Mitchell 1976). Assuming higher wind speeds and turbulent conditions raises the threshold metabolic rates for overheating (predicted temperature .

408C, Fig. 18). The most common wind speed measured by Buss and Estes (1971) during warm parts of the day was approximately 3 m/s. Using a turbulent 3 m/s air flow, the onion-skin model we present predicts a critical Mscope of approximately 5.

Second, elephants have excellent plate heat exchangers in their pinnae. Wright (1984) es-

timates that 50% of an elephant’s metabolic heat production can be lost across the ears. To study this possibility, we used a model with elephant-ear-like heat exchangers (Fig. 12) and the blood flow rates measured in elephant ears (Wright 1984). The model also predicts that for a 4000-kg elephant, perfusing the ears has the same effect on steady-state temperature as cutting the metabolic rate by 50% (data not shown), effectively doubling the critical Mscope. Perhaps as important, elephants can and do use their ears to ‘‘create’’ airflows over the ears and neck, especially under putatively stressful thermal conditions (Buss and Estes 1971).

A series of factors too poorly characterized to incorporate into our models may also reduce the predicted temperature of elephants.

Wright and Luck (1984) argue that relatively high rates of cutaneous evaporation, particularly over the ears, may lead to substantial loss of metabolic heat, although this is difficult to quantify. In addition, behavioral thermoregulation (shade seeking, use of mud and water for cooling) appears from behavioral studies to be important in elephants (Buss and Estes 1971; McKay 1973). Finally, heat from the highest metabolic rates associated with induced exercise (and thus transient) may be buffered by heat storage and reradiation at later times.

Thus, although elephants in warm environments would appear to be at risk of overheating because of their high metabolic rates, a series of heat-loss mechanisms (adaptations?) reduces the risk at all but the highest (and likely transient) metabolic rates.

These simulations argue that the thermal effects of high metabolic rates might vary from dinosaur to dinosaur. Very large and very small dinosaurs would not be expected to derive thermoregulatory benefit from high metabolic rates. For dinosaurs of intermediate size, high metabolic rates might have yielded benefits in terms of high activity capacities (Reid 1997), but in many cases in which metabolic warming would have been most useful thermally (animals in temperate latitudes), thermally useful metabolic heating may have carried substantial risks of overheating and would have needed seasonal adjustment.

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What would seem to be needed are not only endothermic thermogenic capacities, but also endothermic patterns of response to external temperatures with lower external temperatures evoking an increase in metabolic rate.

Such differences in the potential thermal utility of high metabolic rates in large and small animals raise the question of the extent to which metabolic rates might have varied within individual dinosaurs and between dinosaurs with different levels of relatedness.

Available data do not clearly answer this question. Metabolic rates differ with activity level.

Standard metabolic rates clearly differ within an animal as a function of ontogeny (Mautz and Nagy 1987), combinations of season and acclimation (Gatten 1985; Tsuji 1988; Beyer and Spotila 1994), and endocrine effects (JohnAlder 1983, 1990). Significant differences in metabolic expenditures also occur among individuals within a population (Konarzewski and Diamond 1994; Burness et al. 1998), among conspecific populations (Beaupre 1995), among sympatric, congeneric species (Nagy et al. 1984), and among families and orders within classes (e.g., mustelids and sloths). However, such differences among reptiles, among birds, or among mammals rarely reach the 5–10-fold differences that commonly exist between extant metabolic endotherms and equivalently sized reptiles. The apparent dichotomy in metabolic rates between extant reptiles and extant mammals and birds may be due to phylogenetic constraints, to differences in natural and life history (Pough 1980; Anderson and Karasov 1981), and/or to physiologic and pleiotropic constraints on the suite of traits necessary to achieve and benefit from high metabolic rates. To what extent the current ‘‘endotherm/ectotherm’’ differences depend on any or all of these factors is unclear.

If, however, standard metabolic rate is highly conserved, that would tend to limit the ability of closely related dinosaurs of different sizes to have different metabolic rates that might suit their mass-specific thermoregulatory needs.

Thus, we suggest that a physical approach to body temperatures in dinosaurs can allow us to predict what ranges of body temperatures and what thermoregulatory strategies

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were available to those dinosaurs. We present some simple tools for exploring such approaches. Substantively, we argue that 1. The huge range of body sizes found within the dinosaurs likely resulted in very different thermal problems and strategies for animals at either end of this size continuum.

2. Body temperatures of the smallest adult dinosaurs and small hatchlings and juveniles would have been largely insensitive to metabolic rates without insulation. For metabolic rates 5–103 RRMR (Mscope 5 5–10), the smallest animals in which metabolic heating would likely result in body temperatures more than 28C above operative temperatures weighed 10 kg (Figs. 4, 12).

Animals of this size were unlikely to benefit thermally from high metabolic rates, and their body temperatures would respond rapidly enough to changes in environmental temperature to make behavioral thermoregulation possible (Figs. 10, 15).

These animals could be thought of as thermally like living terrestrial lizards.

3. Body temperatures of large dinosaurs (.1000 kg) were likely very sensitive to both metabolic rate and the delivery of heat to the body surface by blood flow (Figs. 4, 12). Our model suggests that they were capable of adjusting body temperature by adjusting metabolic rate and blood flow. Behavioral thermoregulation by changing microhabitat selection would likely have been of limited utility because body temperatures would have responded so slowly to changes in external temperature (Figs. 10, 15). In these large dinosaurs, the availability and perfusion of heat exchangers such as limbs or plate-like exchangers could alter both steady-state temperatures and the rates of heating and cooling in these animals (Figs. 11, 12).

4. Endothermic metabolic rates may have put large dinosaurs at risk of overheating unless they had adaptations to shed the heat as necessary. Although this would have been true particularly for dinosaurs with masses greater than 10,000 kg, simulations suggest steady-state (‘‘average’’) body temperatures would have exceeded 408C for

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animals as small as 1000 kg in the Tropics and during the summer in temperate latitudes (Fig. 16), under the paleoclimate used in our simulations. Slow responses of body temperatures to environmental changes suggest that use of day-night thermal differences alone would have been useful in buffering dinosaurs from diel warming if the mean daily temperature was not too high, but it would not have been adequate to lower body temperatures sufficiently to allow activity during the day for animals experiencing high daily mean operative temperatures (Figs. 15, 16). Consideration must be given to the mechanisms the animals used to control heat loss under these circumstances. Endothermic metabolism may have allowed activity in the winter in some circumstances (Fig. 16), but those same metabolic rates would endanger the animal if maintained during the summer (Figs. 16, 17). The control of metabolic heating in large dinosaurs must be addressed.

5. Endothermic metabolism and metabolic heating might have been useful for intermediate and large (100–3000 kg) dinosaurs but often in situations that would demand marked seasonal adjustment of metabolic rates (Fig. 17) or relatively precise control of metabolism (and heat-loss mechanisms), as typically seen in endotherms (Fig. 17B,C).

Acknowledgments We are grateful to J. E. Bentz, E. M. Dzialowski, S. S. Kilham, J. R. Spotila, and S. Zeitz who provided valuable criticisms during the construction of the model. E. M. Dzialowski, J. R. Spotila, J. Farlow, W. Hillenius, J. Ruben, A. Tumarkin, M. Lamanna, and J. R. O’Connor provided helpful criticism of early versions of the manuscript.

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