O'Connor & Dodson, 1999

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q 1999 The Paleontological Society. All rights reserved. 0094-8373/99/2503-0004/$1.00 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 pre- dict what ranges of body temperatures and what thermoregulatory strategies were available to those dinosaurs. We argue that 1. 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. 2. 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 an- imals in which metabolic heating resulted in predicted body temperatures $ 28C above oper- ative temperatures (T e ) weigh 10 kg. Body temperature would respond rapidly enough to changes in T e to make behavioral thermoregulation possible. 3. 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 thermoregu- lation by changing microhabitat selection would likely have been of limited utility because body temperatures would have responded only slowly to changes in T e . 4. 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 di- nosaurs 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 T e . 5. 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 mecha- nisms) as typically seen in endotherms. Michael P. O’Connor. Department of Bioscience and Biotechnology, Drexel University, Philadelphia, Penn- sylvania 19104. E-mail: [email protected] Peter Dodson. Department of Animal Biology, School of Veterinary Medicine, University of Pennsylvania, Philadelphia, Pennsylvania 19104-6045. E-mail: [email protected] 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 crea- tures (Eoraptor was only one meter long [Ser- eno 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 range has prompted study of the allometric relations thought to dictate much of the engi- neering and physiology of these animals (Spo- tila et al. 1991). We use the following desig- nations: 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 repre- sented among the very large dinosaurs. From the very beginning, physiology fig- ured in our concept of Dinosauria (Owen 1842). Owen specifically invoked mammals as a physiological model for dinosaurs, inferring four-chambered hearts and a highly orga-

description

Abstract.—A physical, model-based approach to body temperatures in dinosaurs allows us to pre- dict what ranges of body temperatures and what thermoregulatory strategies were available to those dinosaurs. We argue that imals in which metabolic heating resulted in predicted body temperatures 2C above oper- nosaurs with masses 10,000 kg, but simulations suggest that for animals as small as 1000 kg 1999 The Paleontological Society. All rights reserved. Accepted: 15 February 1999

Transcript of O'Connor & Dodson, 1999

Page 1: O'Connor & Dodson, 1999

q 1999 The Paleontological Society. All rights reserved. 0094-8373/99/2503-0004/$1.00

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 pre-dict what ranges of body temperatures and what thermoregulatory strategies were available tothose dinosaurs. We argue that

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

2. Body temperatures of the smallest adult dinosaurs and of hatchlings and small juveniles wouldhave been largely insensitive to metabolic rates in the absence of insulation. The smallest an-imals in which metabolic heating resulted in predicted body temperatures $ 28C above oper-ative temperatures (Te) weigh 10 kg. Body temperature would respond rapidly enough tochanges in Te to make behavioral thermoregulation possible.

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

4. Endothermic metabolic rates may have put large dinosaurs at risk for overheating unless theyhad adaptations to shed the heat as necessary. This would have been particularly true for di-nosaurs with masses .10,000 kg, but simulations suggest that for animals as small as 1000 kgin the Tropics and in temperate latitudes during the summer, steady-state body temperatureswould have exceeded 408C. Slow response of body temperatures to changes in Te suggests thatuse of day-night thermal differences would have buffered dinosaurs from diel warming butwould not have lowered body temperatures sufficiently for animals experiencing high meandaily Te.

5. Endothermic metabolism and metabolic heating might have been useful for intermediate andlarge-sized (100–3000 kg) dinosaurs but often in situations that demanded marked seasonaladjustment of metabolic rates and/or precise control of metabolism (and heat-loss mecha-nisms) as typically seen in endotherms.

Michael P. O’Connor. Department of Bioscience and Biotechnology, Drexel University, Philadelphia, Penn-sylvania 19104. E-mail: [email protected]

Peter Dodson. Department of Animal Biology, School of Veterinary Medicine, University of Pennsylvania,Philadelphia, Pennsylvania 19104-6045. E-mail: [email protected]

Accepted: 15 February 1999

Introduction

Dinosaurs are a testament to the ability ofnatural selection to push the biomechanicallimits of size in terrestrial animals. Within 75million years from their origin as small crea-tures (Eoraptor was only one meter long [Ser-eno et al. 1993]), dinosaurs such as the giantsauropods Seismosaurus and Argentinosaurusachieved maximum body lengths of 40 to 50meters (Gillette 1991; Bonaparte and Coria1993) and weights in tens of tons. Althoughthe larger dinosaurs dwarf the largest knownterrestrial mammals, such as Indricotherium(Alexander 1989), many dinosaurs were quitesmall (Pezckis 1994). Estimated body massesspan five orders of magnitude. This large size

range has prompted study of the allometricrelations thought to dictate much of the engi-neering and physiology of these animals (Spo-tila et al. 1991). We use the following desig-nations: very small, 1–10 kg estimated adultweight; 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 byPezckis (1994). Sauropods alone are repre-sented among the very large dinosaurs.

From the very beginning, physiology fig-ured in our concept of Dinosauria (Owen1842). Owen specifically invoked mammals asa physiological model for dinosaurs, inferringfour-chambered hearts and a highly orga-

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342 MICHAEL P. O’CONNOR AND PETER DODSON

nized respiratory system. Although for manyyears dinosaurs were considered cold-blood-ed, in keeping with their reptilian heritage,the idea of hot-blooded dinosaurs achievedcurrency in the 1970s (Bakker 1971, 1972,1975, 1986) and inspired an aptly named sym-posium, ‘‘A Cold Look at Hot-Blooded Dino-saurs’’ (Thomas and Olson 1980), as well aspopular fiction such as Jurassic Park (Crichton1990). More recently, certain lines of evidenceincluding bone histology (Chinsamy 1990,1993; Chinsamy and Dodson 1995; Chinsamyet al. 1994, 1995) and respiratory physiology(Ruben et al. 1996) have caused some re-eval-uation of the concept of warm-blooded dino-saurs.

This paper is one in a series of papers at-tempting to apply biophysical approaches tothe problem of temperature regulation in di-nosaurs. As others did in earlier papers (Spo-tila et al. 1973, 1991; Spotila 1980), we take anengineering approach to the problem of heattransfer from a warm interior to the externalenvironment. We model each dinosaur as a se-ries of cylinders, with allometric equations ac-counting for changes in proportion of limblengths and surface areas. The power of com-puters enables the use of increasingly sophis-ticated models for the dynamics of heat trans-fer. We specify a variety of physiological con-ditions of heat transfer as well as a variety ofenvironmental conditions. In so doing, wemake no a priori assumptions about the met-abolic status of dinosaurs; we simply inquireas to which metabolic strategies are feasibleunder specified conditions of body size, inter-nal physiology, and external environment.

Much of the debate over temperatures andmetabolic rates of putatively endothermic di-nosaurs has centered on the effect of largebody size on body temperatures in ‘‘warmblooded’’ animals, the role of conductancefrom the animal to the environment in keep-ing the animal warm or cold, and the role ofthermal inertia in buffering dinosaurs’ bodytemperatures from the vagaries of the envi-ronment (see reviews in Barrick et al. 1997; Pa-dian 1997; Paladino et al. 1997; Reid 1997;Ruben et al. 1997). Part of the difficulty in un-raveling the constraints these processes put onthe temperatures and metabolic rates of di-

nosaurs is that all of the processes are inter-related and all depend on body size.

The conductance of heat from the animal tothe environment depends both on conduc-tance from the deep tissues, or core, of the an-imal to the surface and on conductance fromthe animal’s skin to the environment (Tracy1982). Both conduction from tissue to tissueand blood flow carry heat from the deep tis-sues to the surface, and both depend on bodysize. Heat conduction from deep tissues to thesurface decreases as animals get larger simplybecause the heat must be conducted over lon-ger distances. Thus, either the deep tissuesmust be warmer in large animals, increasingthe temperature difference from the deep tothe superficial tissues, or another pathway(i.e., blood flow) must carry the heat. Bloodtravels slowly enough through capillaries tocome nearly to thermal equilibrium with thetissue it perfuses. In perfusing tissues warmerthan itself, blood picks up heat from those tis-sues (cools them) and carries that heat back tothe core. There, the blood mixes with bloodfrom all the other tissues before it is recircu-lated to the arteries by the heart. Thus, bloodserves as a heat pump connecting the differenttissues of the body. However, just as metabolicrate per gram of tissue decreases in large an-imals (Bennett and Dawson 1976; Bennett1982; Calder 1984), blood flow per gram of tis-sue also decreases (Calder 1984). Both path-ways for heat conduction from deep to super-ficial tissues appear more limited in large an-imals.

Similarly, conductance from the surface ofthe animal to the environment may be morelimited in large animals. The major pathwaysof heat transfer between the skin of a terres-trial animal and the environment are convec-tion, solar radiation, and thermal (infrared)radiation. Convection is the transfer of heat toand from a moving fluid such as air or water.‘‘Solar radiation’’ is used as a shorthand forthe shortwave (primarily visible, near-ultra-violet [UV], and near-infrared [IR]) radiationemitted by the sun and falling on the animal.Because animals are not warm enough to emitradiation in these wavelengths, solar radiationonly moves heat into the animal (Porter andGates 1969; Tracy 1982). ‘‘Thermal radiation’’

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343BIOPHYSICAL CONSTRAINTS ON DINOSAURS

is also shorthand for the summation of all theradiation in the longwave IR that is emittedboth by the environment and by animals attypical temperatures. Thus, thermal radiationconstitutes an exchange of heat with the en-vironment, warming or cooling animals de-pending on their temperatures. An animal ly-ing on the ground can also exchange heat byconduction via the trunk and/or limbs, butthe tips of limbs in a standing animal typicallyhave small surface areas and thus are poorheat conduits to the ground.

Changes in both solar and thermal radia-tion are as effective in warming and coolinglarge animals as they are with small animals.The same is not true for convection, however.Before being carried away in the wind, heatlost by convection must diffuse across a layerof relatively still air known as the boundarylayer. As animals increase in size, the thick-ness of this boundary layer increases, decreas-ing the rate of heat diffusion (Porter and Gates1969). Because rates of convective heat transferare lower in large animals than in small onesbut the rates of radiative exchange remain thesame, radiative exchanges become relativelymore important in large than in small ani-mals. This can be particularly important insunny environments where animals oftengain heat by radiation (sunshine) and lose heatby convection to the atmosphere. In such sit-uations, decreased convective heat exchangein large animals results in higher body tem-peratures in response to the undiminishedheat load from the sun.

While some treatments of boundary layersconjure images of thick laminae of becalmedair, the only truly still air is immediately ad-jacent to the animal’s skin. Wind speed in-creases asymptotically to the free stream ve-locity as one moves away from the animal,with most of the change in speed in the firstfew millimeters (Gates 1980; White 1984). The‘‘boundary layer thickness’’ refers to the the-oretical ‘‘equivalent’’ distance of totally stillair through which heat would have to move bydiffusion alone to match the overall conduc-tance of heat from the skin to the airstream(Gates 1980; Monteith and Unsworth 1990).

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

responsible for much of the insulative value offur, feathers, and many household insulators(Cena and Monteith 1975; White 1984; Mon-teith and Unsworth 1990). A still boundarylayer that averages only 3.8 mm thick will ex-plain the resistance to convective heat ex-change for a 1000 kg animal in a wind streamat 1 m/s (Mitchell 1976). For wind speeds at 5m/s, the boundary layer thickness drops to1.5 mm for the same animal. Turbulence willdecrease the boundary layer thickness and in-crease the convection coefficient by a factor of1.7 (Mitchell 1976). These relatively stillboundary layers near animal surfaces (andtheir dependence on body size and windspeed) have been visualized and investigatedby Schlieren photography and other tech-niques in both living and inanimate objects(Gates 1980; Monteith and Unsworth 1990).

At night, solar radiation is, of course, ab-sent. Thermal radiative exchanges can be com-plex because thermal radiation passes to theanimal from the ground (usually warmer thanair temperature), surrounding vegetation(usually at about air temperature), and thesky. The effective radiant temperature of thenight sky ranges from approximately air tem-perature (under a canopy), to 5–108C belowair temperature under a cloudy sky, to asmuch as 208C below air temperature underclear conditions (Spotila et al. 1992). The ef-fective radiant temperature of the environ-ment is some combination of all of these ra-diation sources. The usual treatment is to as-sume that half the animal’s surface exchangesthermal radiation with the ground and sur-rounding vegetation and half exchanges ra-diation with the sky.

A third influence of body size on conduc-tance is the ratio of surface area to the volumeof tissue heated by metabolism or warmed/cooled by the environment. For biomechanicalreasons, if no other, animals become propor-tionately more stout as they grow in size(Economos 1983). Thus, less area per gram oftissue is available to conduct heat from one re-gion to the next, whether that next layer is amore superficial layer of tissue or the externalenvironment.

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

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344 MICHAEL P. O’CONNOR AND PETER DODSON

Thermal inertia is so called in analogy toNewton’s translational inertia. It takes moreheat to warm a large animal than a small an-imal, and similar environments can warm asmall animal much more quickly than a largeanimal. In part, this is because there are morekilograms of animal to heat, so more heat isrequired. But it is also due in part to decreasedsurface area to volume ratios, decreased con-ductances, and the concomitant increase intime needed to heat first one layer of animalthen another all the way to the deep tissues.Increased thermal inertia is often seen as anadvantage for large animals, protecting themfrom excessive changes in body temperaturedue to short-term fluctuations in the thermalenvironment. Higher thermal inertia may be amixed blessing, however. To the extent that itlimits excursions of the animal’s body tem-perature, it also limits an animal’s ability touse variations in the thermal environment toadjust either its body temperature or the heatload it must dissipate (see Tracy et al. 1986;Turner and Tracy 1986).

Allometric changes in external heat conduc-tance (convection), internal heat conductance(conduction and blood flow), and thermal in-ertia (due to changes in mass, convection, con-duction, blood flow) confound both the effectsof different metabolic rates on body temper-atures of dinosaurs and the thermoregulatorystrategies such temperatures dictate. Our ap-proach to disentangling these effects is to ex-amine the behavior of a series of simple mod-els of dinosaur body temperatures, each ofwhich adds the effect of another confoundingfactor. This approach has two advantages.First, because the models are simple and haveanalytic solutions, they allow us to under-stand how different factors limit the range ofbody temperatures available to dinosaurs ofdifferent sizes. Simultaneously, we can ex-plore how and to what extent dinosaurs couldalter their body temperatures by physiologicalmeans. Second, the simple models representthe physical extremes of what dinosaurs couldaccomplish and, thus, allow us to placebounds on the behavior of more complexmodels—perhaps even the real animals. So,after examining constraints on dinosaur tem-peratures via simple models, we present the

results obtained from a more realistic model.Again, we ask what constraints would haveexisted on simultaneous combinations of me-tabolism, blood flow, and body size in differ-ent environments.

Models and Results

We start by assuming that animals have nometabolic rate, no thermal inertia, and infiniteinternal conductance; they are like hollowmetal 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 simpleplace to start and a baseline from which de-viations in more complex models may be mea-sured. As we shall see, because metabolicheating increases the body temperature of ananimal, the temperatures of this inertia-lessorganism represent the minimum body tem-perature available to a dinosaur. Because thecentral axis of most dinosaurs was approxi-mately cylindrical, we start with a cylindricalanimal.

Operative Environmental Temperatures.—Thetemperature of our model animal (Te) isknown as the operative environmental tem-perature (Bakken and Gates 1975; Bakken etal. 1985). Te is usually defined as the temper-ature of a massless (inertia-free) model of theanimal with no metabolic rate. It can also bethought of as the temperature that an animalwith no metabolism would eventually achieveif placed under the given environmental con-ditions. For small extant reptiles, the operativetemperature is usually measured with physi-cal models of the animal (e.g., Grant and Dun-ham 1988; Grant 1990). For dinosaurs, onemust be content with predicted values of Te.Te’s are typically computed using an energybalance equation that demands that, at steadystate, energy inputs to the animals must bebalanced by energy outputs. The energy bal-ance for our massless dinosaur is given byequation (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 frac-tion of surface area receiving solar radiation),S is the intensity of solar radiation (W/m2), hc

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345BIOPHYSICAL CONSTRAINTS ON DINOSAURS

TABLE 1. Heat transfer coefficients for cylindrical ani-mals at different masses and wind speeds.

Mass(kg)

Convection coef (W m22 8C21)wind speed

1m/s

5m/s

IR heat coef(W m22 8C21)

0.010.11

10100

100010,000

100,000

31.623.217.112.6

9.26.85.03.7

82.961.044.933.024.317.913.1

9.7

4.64.64.64.64.64.64.64.6

FIGURE 1. Operative temperatures (Te [sensu Bakkenand Gates 1975]) predicted for dinosaurs with massesranging from 10 g to 100 metric tons ignoring any met-abolic warming. Solar radiation level (800 W/m2) wouldbe exceeded for most of a clear spring or summer day.

is the convection coefficient (W m22 8C21), hr

is the coefficient of infrared heat exchange (Wm22 8C21), Te is the operative environmentaltemperature (8C), Ta is the ambient (air or wa-ter) temperature (8C), and Tr is the average ra-diative temperature of the environment (8C).

Often, Tr is not very different from Ta, andwe assume that to be the case here. Then,

(h 1 h )T 1 aSc r aT 5 (2)e (h 1 h )c r

or

aST 5 T 1 (3)e a H

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

Operative temperatures of our Newtonianorganism are higher than ambient air temper-atures. The temperature increment dependson the amount of solar radiation falling on theanimal (heat gain) and how easily the animalloses that heat to the environment by convec-tion and thermal radiation (eq. 3).

Equation (3) also suggests which factors areimportant in determining operative tempera-tures for dinosaurs. Obviously, air tempera-ture plays an important role, as does solar ra-diation. Two other factors affect the convec-tion coefficient. Larger body size, by creatinga larger boundary layer of still air near the an-imal, impedes loss of heat as animals increasein size. Higher wind speeds, by shrinking theboundary layer, enhance convection. Mitchell(1976) found that, for most animal shapes, theconvection coefficient in air can be well ap-proximated by equation (4).

hc 5 6.8 u0.6/D0.4 (4)

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

In small terrestrial animals, hr is small com-pared with hc, and the value of H is dominatedby hc, but with increasing size, hc decreasesand is on the same order as hr (Table 1). Thus,large animals are said to be relatively ‘‘con-vectively decoupled.’’

For cylindrical animals, operative tempera-tures increase with mass and the intensity ofsolar radiation and decrease at higher windspeeds (Fig. 1). Of note, with moderate solarradiant intensities, large animals (.1000 kg)can experience operative temperatures 10–308C warmer than ambient air temperatures(eqs. 2–4, Fig. 1). Assuming that dinosaurs tol-erated the same range of body temperaturesas extant reptiles (#458C), air temperaturesbetween 5 and 208C would not have allowedlarge dinosaurs much room for metabolicheating above their operative temperatures.This constraint might be especially severe be-cause their size may have prevented large di-nosaurs from using burrows, crevices, and theshade of vegetation as effective refugia fromsolar radiation as extant reptiles do.

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

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346 MICHAEL P. O’CONNOR AND PETER DODSON

FIGURE 2. Response of dinosaur body temperatures (ig-noring effects of metabolism) to variation in operativetemperature (Te).

ment than small animals because wind speedsmay increase and air temperatures decreasewith height above the ground (Porter andGates 1969; O’Connor and Spotila 1992). Thesteepest parts of these gradients occur withina few cm of the ground (or the tops of vege-tation when there is a vegetative canopy [Ro-senberg 1974]). Reported air temperature andwind speed are typically measured about 1.5m above the ground in meteorologic stations.Thus small animals (,10 kg) usually experi-ence higher air temperatures, slower windspeeds, and higher operative temperaturesthan climate stations report. Macroclimatic airtemperatures and wind speeds probably rep-resent a fair approximation of the convectiveenvironment experienced by large dinosaurs.Because we focus here on dinosaurs, we havechosen not to use vertical wind speed gradi-ents, but if one focuses on small reptiles, suchgradients cannot be ignored. (2) Because of thehigh thermal conductivity and heat capacityof water, convection in aquatic organisms canbe one to two orders of magnitude higher thanin terrestrial animals (Spotila et al. 1992). If ananimal stands in shallow water, the part of theanimal in the water experiences aquatic con-vection. Dinosaurs standing in cool watercould use that water as a heat sink and main-tain lower operative temperatures. (3) Someanimals can change their absorptance to solarradiation by changing melanin dispersion intheir skins. In these simulations, we used anintegrated solar absorptivity of 65% found insome desert reptiles, presumably those mostsusceptible to overheating due to absorptionof solar radiation. While not the lowest ab-sorptivity measured in reptiles, it may well beconservative in this case. Thus, we suspectthat our estimates, if anything, underestimatethe operative temperatures of animals in sun-ny environments. Equation (3) can be used toestimate the effects of changes in absorptivity.

Changing Environments.—Because the ani-mal is assumed to have no thermal inertia, op-erative temperature (Te) is simply a weightedsum of environmental factors (air tempera-ture, thermal radiative temperature, and solarradiation) and varies just as quickly as do itscomponents. Equation (3) can be used to es-timate the operative temperature at any given

time. In this paper, we are concerned mainlywith environments that change cyclically (e.g.,daily or seasonal variations, temperaturechanges as an animal shuttles in and out of theshade or as clouds obscure the sun, changeswith a 3–5-day periodicity due to the passageof weather systems). We will take as an ex-ample air temperatures and solar radiationlevels that vary sinusoidally (Fig. 2). Becauseoperative temperatures are simply weightedsums of their components (eq. 3), and becausesuch sums of sines with the same period areagain sinusoidal, variations in operative tem-perature through time will again be approxi-mated by sine waves. Furthermore, the meanoperative temperature will be the operativetemperature produced by inserting the meanair temperature and solar radiation level inequation (3).

To calculate the variation of operative tem-peratures in time, we assume that the animalshave no mass and thus no thermal inertia. Al-though the assumption is unrealistic, and wewould not expect the body temperatures ofreal animals to track their operative temper-atures precisely through time, the variation ofthe operative temperature is an importantconcept.

Newtonian Dinosaurs.—We take a first steptoward biological reality by allowing ourmodel animals to have an appropriate body

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347BIOPHYSICAL CONSTRAINTS ON DINOSAURS

mass. We still assume that the animal has in-finite internal conductance, so that tempera-tures everywhere within the animal are thesame. Heat transfer theorists call such animals‘‘Newtonian objects’’ (White 1984), and wewill call them Newtonian dinosaurs. In a staticenvironment, with stable air temperaturesand radiation levels, the body temperatures ofNewtonian dinosaurs will equal operative en-vironmental temperatures, as long as the an-imal’s metabolic rate is small enough to be ig-nored. Thus, in a static environment, the bodytemperatures of Newtonian dinosaurs with-out metabolism would be identical to the op-erative temperatures presented in Figure 1.

Warming and Cooling in Newtonian Dinosaurs:Thermal Inertia.—Although the steady-statetemperatures of Newtonian dinosaurs areidentical to operative temperatures, the ther-mal inertia due to the mass of the dinosaurwill cause body temperatures to vary moreslowly than operative temperatures. To seehow body temperatures will vary in a varyingenvironment, we examine the response of aNewtonian animal’s body temperature to asudden change in the thermal environment(measured as the change in Te). Then, weexamine cyclically varying environments(O’Connor 1999).

To predict how a Newtonian dinosaurmight warm or cool in the face of varying en-vironmental temperatures, one must rear-range equation (1) and add a storage term. Inthis more complete energy balance, heat thatenters an animal by one pathway (e.g., solarradiation) must either exit by another pathway(e.g., convection) or be stored in the animal’sbody. Stored heat results in a rise in the ani-mal’s temperature.

dTbmC 5 AaS 2 Ah (T 2 T )p c b adt

2 Ah (T 2 T ) (5)r b r

where m is the mass of the animal (kg), Cp isthe specific heat of tissue (J kg21 8C21), Tb is thebody temperature (8C), Ta is the ambient (airor water) temperature (8C), Tr is the averageradiative temperature of the environment(8C), and dTb/dt is the rate of change of bodytemperature (8C/s). Using the same approxi-

mations used to simplify equation (1) intoequation (3), we simplify equation (5) as

d(T 2 T ) AHb e 5 2 (T 2 T ) (6)b edt mCp

where d(Tb 2 Te)/dt is the rate at which Tb ap-proaches Te (8C/s), H is the combined heattransfer coefficient (hc 1 hr, W m22 8C21), andTe is the operative temperature as given inequation (3).

The solution of this differential equation, fora constant environment, is

AHT 5 T 1 (T 2 T ) exp 2 tb e b e t50 1 2mCp

or

T 5 T 1 (T 2 T ) exp(2lt) (7)b e b e t50

where (Tb 2 Te)t 5 0 is the initial difference be-tween body and operative temperatures (8C),exp(x) is the exponential function (ex), and lis the time coefficient ( 5 2AH/mCp, s21).Equation (7) represents the classic exponentialcooling curve in which body temperature ap-proaches operative temperature with time.The larger the absolute value of the time co-efficient, l, the more rapidly the animal heats(or cools) and approaches its operative envi-ronmental temperature (Te).

When the operative temperature (i.e., thethermal environment) changes continuallyand cyclically, the animal’s body temperatureis constantly ‘‘approaching’’ a moving target.What is needed is some measure of responseof the animal’s body temperature to a varyingenvironment. We chose to examine the rangeof predicted body temperatures of a reptile inan environment whose operative temperaturevaries sinusoidally. Figure 2 shows a typicalresponse of body temperature to varying op-erative environmental temperature. Severalfeatures are worth noting. First, if the opera-tive temperature varies sinusoidally, so doesthe body temperature. Second, body temper-ature lags behind operative temperature. Themass and specific heat of the animal imbue itwith ‘‘thermal inertia’’; i.e., because heat flow(which takes time) is required to heat the an-imal, the animal’s body temperature lags be-hind the operative environmental tempera-

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348 MICHAEL P. O’CONNOR AND PETER DODSON

FIGURE 3. Predicted responses of body temperatures ofNewtonian dinosaurs (masses 10 g–100,000 kg) to si-nusoidal variation in operative temperatures with vary-ing cycle periods. Period is the time between successivepeaks in environmental temperature in Figure 2. Frac-tional amplitude is ratio of body temperature amplitudeto environmental temperature amplitude in Figure 2.Solid lines 5 wind speed 1 m/s. Dashed lines 5 windspeed 5 m/s).

ture. Third, the mean body temperature is themean operative temperature. Thus, body tem-perature (in this admittedly simple case) willvary around the mean operative temperature.In the more complicated, and more biologi-cally reasonable, simulations that follow, bodytemperature varies around the mean steady-state temperature. Fourth, variation in bodytemperature is smaller than that in operativetemperature. We call the ratio of the range ofbody temperatures to the range of operativetemperatures the ‘‘fractional amplitude.’’ Pre-dicted fractional amplitudes vary from nearlyzero for rapidly changing operative tempera-tures, when the ‘‘target’’ operative tempera-ture varies much more rapidly than body tem-perature can follow, to near unity for slowlychanging environmental temperatures. Whatconstitutes fast or slow variation of operativetemperatures varies with the animal’s mass(Fig. 3). Large dinosaurs have more mass andsmaller surface to volume ratios than smalleranimals. Thus, large animals have more ther-mal inertia, their body temperatures varymore slowly, and their body temperatures canrespond only to slower variations in operativetemperature, i.e., those with longer periods.For Newtonian animals, the factors that deter-mine what variations are fast and slow are therate at which operative temperature changesand the heat transfer characteristics of the an-

imal and environment—specified by the timecoefficient, l. Thus,

lF 5 (8)amp 2 2Ïl 1 v

where Famp is the fractional amplitude, l is thetime coefficient (s21) as in equation (7), v is therate of sinusoidal temperature change ( 5 2 p/period, s21). When v is much smaller than l,i.e., when the change in operative temperatureis slow compared with the response time ofthe animal’s body temperature, body temper-ature can respond almost as quickly as the en-vironmental temperature varies, and the frac-tional amplitude will be almost one. Alterna-tively, when the variation in operative changeis fast compared with the animal’s thermal re-sponse time, and v is much larger than l, thenthe fractional amplitude will be almost zero—body temperature will remain near the meanoperative temperature with only small varia-tions through time. What variation in bodytemperature does occur will lag significantlybehind variations in operative temperature.Because a dinosaur’s mass affects its time co-efficient, it affects how strongly body temper-ature will respond to different types andspeeds of environmental temperature varia-tions.

By delineating which environmental varia-tions will or will not perturb a dinosaur’sbody temperature, fractional amplitudes al-low us to understand thermal inertia in di-nosaurs, and to quantify it in an ecologicallymeaningful way. For our Newtonian dino-saurs, there are three characteristics of the an-imal that affect inertia: mass, surface area, andexternal heat conductance (measured by theheat transfer coefficient, H, eqs. 6 and 7). Forlarge dinosaurs, masses are higher, surfacearea to volume ratios are lower, and convec-tion coefficients are lower than in smaller an-imals, each contributing to a smaller value ofl, slower changes in body temperature, andincreasing inertia.

Figure 3 suggests that small terrestrial or-ganisms (mass # 1 kg) would be very sensi-tive to daily variations in operative tempera-tures and variably sensitive to thermal effectsthat occur on shorter timescales (e.g., due to

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349BIOPHYSICAL CONSTRAINTS ON DINOSAURS

FIGURE 4. Elevations of body temperature above oper-ative temperature (Te) for Newtonian dinosaurs.‘‘Mscope’’ is ratio of metabolic rate to allometric pre-dictions from Bennett and Dawson’s (1976) equation forreptiles at 308C. Solid lines 5 dinosaurs in terrestrialhabitat with wind speed 5 1 m/s. Dashed line 5 di-nosaurs in aquatic environment with water flow velocityof 0.1 m/s.

movement, thermoregulatory shuttling orother behavior, or the sun being obscured byscattered clouds). Very large dinosaurs (.104

kg), however, would be largely insensitive todaily variations in the thermal environmentand would maintain body temperatures nearthe mean daily operative temperature with lit-tle diel variation in body temperature. Athigher wind speeds, the predicted sensitivityto environmental temperature variation in-creases (Fig. 3). Because of the convective de-coupling that occurs in large animals, how-ever, the effect of a change in wind speed isless than in smaller animals (Table 1). None-theless, dinosaurs may have been able to con-strain or increase the variation in their bodytemperatures behaviorally by selecting envi-ronments with different wind speeds. Movingfrom an environment with a wind speed of 1m/s to one where the wind blows at 5 m/s canalmost double the predicted range of temper-atures experienced by the animal (Fig. 3).

Effects of Metabolism.—Much of the debateabout endothermy and body temperatures indinosaurs turns on the contribution of meta-bolic heat to steady-state body temperatures.Thus, it becomes important to estimate the ex-tent to which metabolic rate can raise steady-state body temperatures above those for ani-mals without metabolism (i.e., above opera-tive environmental temperatures).

We take the resting metabolic rates predict-ed by Bennett and Dawson (1976) for reptilesat 308C as our baseline and as representativeof resting metabolic rates of ectotherms (Rep-tilian Resting Metabolic Rate, RRMR). We re-fer to the ratio of the metabolic rate used for asimulation to that predicted by the Bennettand Dawson relation for the same-sized ani-mal as ‘‘Mscope.’’ Here, we examine the ef-fects of metabolic rates 1, 2, 5, 10, and 20 timesthat predicted by the Bennett and Dawson al-lometric predictions (Mscope 5 1, 2, 5, 10, 20).At rest, endotherms typically have metabolicrates 5–10 times those of equivalent-sized ec-totherms at the same temperatures (Bennettand Dawson 1976; Calder 1984). In addition,both endotherms and ectotherms often havefield metabolic rates 2–3 times higher thanresting rates (Nagy 1987, 1989) owing to lo-comotion and other activities. Thus, our 20-

fold range of metabolic rates should spanmost of the resting and mean field metabolicrates of both endotherms and ectotherms.

In the simulations presented here, we do notconsider the effects of body temperatures onmetabolic rates. Metabolic rate is assumedconstant regardless of body temperature. Wedo so because metabolism does not depend onbody temperature in a linear fashion. This re-quires that one specify the dependence of me-tabolism on body temperature over severaltemperature ranges, upper and lower criticaltemperatures, and the breadth of the range oftemperatures over which metabolic rate is at amaximum. None of these factors is known fordinosaurs. Further, simulations suggest thatthe results presented here for high and lowmetabolic rates define the limits of the possi-ble behaviors of these more complicated mod-els (O’Connor 1999).

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

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350 MICHAEL P. O’CONNOR AND PETER DODSON

FIGURE 5. Predicted fraction of heat generated by me-tabolism that is lost by respiratory evaporation from amodel similar to that of Welch and Tracy (1977). Majordeterminants of respiratory evaporation in model arebody temperature, relative humidity of local atmo-sphere, and the fraction of oxygen removed on averagefrom each volume of inspired air (oxygen extraction).

means of the metabolic rates simulated here.This is true for extant reptiles of this size (Bar-tholomew 1982). For Newtonian dinosaurswith masses greater than 1000 kg, and with‘‘endothermic’’ metabolic rates (Mscope 5 5–20), metabolism raised predicted body tem-peratures 9–1028C above operative environ-mental temperatures (Fig. 4). Using windspeeds of 5 m/s cuts these increments byabout 50% (data not shown). Nonetheless,these simulations suggest that to avoid over-heating, large dinosaurs with endothermicmetabolism will require cool environments,tolerance of body temperatures over 408C, oradaptations or behaviors that allow the animalto shed heat, such as heat exchangers (e.g.,sauropod necks and tails or possible sails onAmargasaurus, Spinosaurus). Alternatively,lower metabolic rates (Mscope 5 1–2), wouldyield much lower requirements for heat loss(Fig. 4). From 5 to 1000 kg, Mscopes of 1–10led to elevations of predicted body tempera-tures of 1–208C above operative temperatures(Fig. 4). This suggests that in this range met-abolic modulation of body temperature mighthave been effective without necessarily im-posing large risks of overheating in temperateenvironments.

Aquatic environments result in much higherconvective heat loss than do terrestrial habi-tats, and predicted increments in body tem-perature due to metabolism were much small-er in all sizes of animals in water than on land.We will not consider aquatic or marine ani-mals extensively here, but the usefulness ofwater as a heat sink (e.g., for large dinosaurswith legs in shallow water) should not be ig-nored. It is possible that dinosaurs with highmetabolic rates could survive if they lived inor had access to standing water as a heat sink,under conditions in which the same dinosaurson land would overheat (Fig. 4).

In vertebrates, metabolism is powered, ei-ther immediately or ultimately, by oxidativephosphorylation. Increased metabolism thusrequires increased ventilation and increasedrespiratory evaporation and consequent evap-orative heat loss. We used a model similar tothat of Welch and Tracy (1977) to predict thefraction of metabolic heat that might be lost byrespiratory evaporation. Important variables

for the model included air and body temper-ature, atmospheric relative humidity, and thefraction of oxygen extracted from inhaled airbefore exhalation. All exhaled air was as-sumed to be saturated with water at bodytemperature. Under most simulated condi-tions, less than 20% of metabolic heat was lostthrough respiratory evaporation with bodytemperatures of 408C, although if one as-sumed low oxygen extraction and low atmo-spheric humidity, the fraction rose to about25% (Fig. 5). At higher body temperaturesthan those tolerated by extant reptiles or birds(508C) and under some assumptions, as muchas 40% of metabolic heat might be lost throughrespiratory evaporation (Fig. 5). In general,however, the fraction of metabolic heat lost byevaporation will be small. Thus, metabolicheating is unlikely to be offset to any signifi-

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351BIOPHYSICAL CONSTRAINTS ON DINOSAURS

FIGURE 6. Predicted differences between midline tem-perature and surface temperature of dinosaurs with noblood flow to aid in heat transfer—i.e., all metabolicallygenerated heat must be conducted to the surface of theanimal through tissue. ‘‘Mscope’’ is the ratio of meta-bolic rate to allometric prediction from Bennett andDawson’s (1976) equation for reptiles at 308C. Windspeed 5 1 m/s.

FIGURE 7. Heat transfer dynamics in core-shell modelof a dinosaur. A thermally homogeneous core exchangesheat with the shell by conduction and by blood flow. Inaddition, the shell exchanges heat with the environmentby convection and radiation. Both core and shell areheated by metabolically generated heat.

cant extent by the required respiratory evap-oration in the absence of heat-loss mecha-nisms such as panting.

Effects of Conductance.—Up to this point, wehave assumed that dinosaurs had sufficient in-ternal heat conductance to keep body temper-atures the same throughout the body—i.e., in-finite internal conductance. By doing so, wehave assumed that metabolism cannot heatthe core of the animal to any greater extentthan its surface, that metabolic heat generatedanywhere in the body can immediately be lostto the environment at the skin, and that chang-es in the temperature of the skin are imme-diately transmitted to the rest of the body. Inthis sense, Newtonian models predict the min-imum possible body temperature and the fast-est possible response to changes in the envi-ronment. We must now ask what effects the fi-nite conductance of real animals might haveon steady-state and transient body tempera-tures of dinosaurs.

The thermal conductivity of muscle, al-though several times that of fat, is not highenough to maintain all parts of the organismat the same temperature under most condi-tions (Turner and Tracy 1983, 1985). If blooddid not carry heat around the body, many rep-tiles would experience core body tempera-

tures substantially above those predicted forNewtonian animals (Fig. 6). Blood flow, how-ever, serves as an effective heat pump and lim-its the temperature gradients that exist withinthe body. In so doing, it determines where ac-tual organisms fall on a continuum from aNewtonian organism (Tcore 2 Tsurf 5 0) to thesorts of gradients predicted in Figure 6.

Vascular heat exchange and the low thermalconductivity of most tissues also affect rates ofheating and cooling and the thermal inertia ofan animal. Consider the cylindrical lizard inFigure 7, with a thermally homogeneous coreand a surrounding shell (an outside layerthrough which heat is transferred). When achange occurs in the external environment, itcannot affect the temperature of the core di-rectly. A change in the operative temperaturemust first effect a change in the shell temper-ature. A change in the shell temperature canthen be communicated to the core via conduc-tion through the tissues or heat exchange viablood flow. Thus, such an animal will heat or

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352 MICHAEL P. O’CONNOR AND PETER DODSON

FIGURE 8. Predicted time coefficients (ln—see eq. 9) forcore-shell model of dinosaurs with masses 10 g–105 kgas mass-specific blood flow varies from 0.01 to 100 timesallometrically predicted rates (Qscope).

cool more slowly than an otherwise similarNewtonian organism; i.e., it will have higherthermal inertia.

The rate at which heat is conducted betweenthe core and shell depends on the conductivityof the tissues and the geometry of the animal,neither of which is easily changed. The rate ofheat transfer via the blood, however, dependsprimarily on the rate of blood flow to the shell,which reptiles can control, as Cowles elegant-ly showed (1958). So we are interested in theeffects of blood flow on heating and cooling.We assume (1) that the resting cardiac outputof reptiles scales allometrically proportionate-ly to mass0.75 as do metabolic rates, (2) that thecardiac outputs measured by Baker and White(1970) and Baker et al. (1972) on extant reptilesare representative for dinosaurs, and (3) thatblood flow is distributed to all tissues evenlyon a gram-by-gram basis. We call the ratio ofcardiac output simulated to the predicted rest-ing cardiac output ‘‘Qscope’’ and simulateQscopes from 10% of predicted to ten timesthat predicted for a given animal (Qscopes of0.1, 0.2, 0.5, 1, 2, 5, and 10). We use additionalsimulations using Qscopes of 0.01 and 100 tocover extremely low and high blood flowrates.

The equivalent of equation (7), describingthe temperatures of the core during a warm-ing or cooling curve for this core/shell ani-mal, 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 theconstants that depend on core and shell tem-peratures at time zero (8C). The two time co-efficients tell us different things about heatingand cooling. l1 plays roughly the same role asl does in equation (7). It shows the approachof Tc to Tc,ss after initial transients die away.We call l1 the dominant time coefficient. l2, onthe other hand, describes how quickly the ini-tial transients die away and is called the sub-dominant time coefficient. As with l in equa-tion (7), the values of both time coefficientswill be negative (otherwise, we would predictinfinite temperatures after a long time), andlarger absolute values correspond to faster de-cays. Time coefficients for animals with dif-

ferent masses and blood flow rates are pre-sented in Figure 8, and the ratios of the dom-inant time coefficients (l1) to the time coeffi-cients of Newtonian animals (l 5 lmax) inidentical situations are shown in Figure 9.Simulations were also performed using veryhigh blood flow rates (Qscope 5 100) to ap-proximate the time coefficients for Newtoniananimals. Three important patterns emerge.First, as with Newtonian animals, body tem-peratures of small animals have larger timecoefficients and will respond more quicklythan those of larger animals (Fig. 8). Second,in all cases, the idealized Newtonian dino-saurs were predicted to warm or cool fasterthan animals with the more realistic internalconductances. As predicted, higher bloodflow enhances vascular heat exchanges, in-creases the absolute values of the time coeffi-

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353BIOPHYSICAL CONSTRAINTS ON DINOSAURS

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

FIGURE 10. Response of core temperature of core-shellmodel of a dinosaur to variation in environmental tem-perature. Period and fractional amplitude defined as inFigure 3. Symbols indicate mass of dinosaur (1–104 kg).Line style indicates cardiac output in multiples of allo-metrically predicted values (Qscope). Wind 5 1 m/s forall simulations.cients, and hastens the response of body tem-

peratures to changes in the thermal environ-ment (Figs. 8, 9), converging on values forNewtonian animals (lmax) at high blood flowrates. Third, body temperatures of large ani-mals are predicted to be more sensitive toblood flow and to respond to lower levels ofblood flow than would those of smaller ani-mals (Figs. 8, 9). This prediction is consistentwith the findings of Grigg et al. (1979) on ex-tant lizards. The sensitivity of predicted dom-inant time coefficients (l1) to blood flow in-creases in environments more conducive toconvection (e.g., those with higher windspeeds or aquatic environments) (Turner1987, O’Connor 1999). As with dominant timecoefficients, fractional amplitudes are moresusceptible to change by alterations of bloodflow in large animals than they are in smallanimals (Figs. 8, 9, 10).

The time coefficients help us understand theheating and cooling rates and the sensitivityof those rates to blood flow. As with Newto-nian animals, the response of the animal’sbody temperature to varying environmentsplaces these effects in an ecological perspec-tive. The effect of blood flow is to shift therange of periods of temperature changes towhich the animal responds strongly, makingthe animal respond like a larger or smaller an-

imal (Fig. 10). With increased (Qscope 5 10)or decreased (Qscope 5 0.1) blood flow to theshell, predicted fractional amplitudes overlapthose of animals with 2–10 times more or lessmass than the animal. For example, a 100 kgreptile, with blood flow to the shell reduced to10% of that predicted by allometric equations,has approximately the same amplitude vs. pe-riod plot as a 1000-kg animal with shell flowat 100% of predicted levels (Fig. 10). Thus, bychanging blood flow rates, dinosaurs couldhave altered their thermal inertia and therange of body temperatures they experienced(see Tracy et al. 1986; Turner and Tracy 1986).The extent to which dinosaurs could alterthermal inertia depends on the extent towhich they could alter blood flow levels. Be-cause the extent of this control is unknown,the actual ability to modify thermal inertiacan only be estimated for a range of possiblevalues as in Figure 10. Nonetheless, by se-questering blood centrally, dinosaurs could ef-fectively insulate themselves from their ther-mal environments and decrease the range ofbody temperatures experienced. Alternative-ly, by flushing blood to the periphery, theycould increase heating and cooling rates and

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354 MICHAEL P. O’CONNOR AND PETER DODSON

FIGURE 11. Effect of heat exchangers on predicted ratesof heating and cooling in dinosaurs. A, Ratio of domi-nant time coefficient (l1—see eq. 9) to that for a New-tonian cylindrical dinosaur (lmax) of the same mass butwithout limbs or sail-like heat exchanger. B, Response ofcore temperature of core-shell model of a dinosaur tovariation in environmental temperature. Period andfractional amplitude defined as in Figure 3. Cardiac out-put at allometrically predicted level (Qscope 5 1), windat 1 m/s in all simulations.

increase thermal responsiveness to the envi-ronment.

Heat Exchangers.—All predictions to thispoint have been for cylindrical dinosaurswithout any limbs. Limbs can be important tothe thermal biology of archosaurs (Turner andTracy 1983, 1985) and represent a class ofpathways of biological heat exchange that welump together as ‘‘heat exchangers.’’ Besideslimbs (and necks and tails), some dinosaurspossessed plate-like heat exchangers such asthe dorsal plates of Stegosaurus, possible sailsbased on elongate dorsal neural spines (e.g.,Ouranosaurus or Spinosaurus [but see Bailey1997]), or cranial frills as in ceratopsians(Wheeler 1978). In addition, the peripheral tis-sues and skin of an animal, which we simplycalled a shell previously, can be thought of asa heat exchanger. The common feature of eachof these heat exchangers is that heat is trans-ferred back and forth between the core of theanimal and the heat exchanger primarily byblood flow. We now examine the effects ofsuch heat exchangers on both the steady-stateand transient temperatures of dinosaurs.

In the first set of simulations, we examinethe effects of appending various types of ex-changers onto the Newtonian cylinders we ex-amined earlier. Limbs, with an allometry wehad developed for dinosaurs (Spotila et al.1991), were appended to a Newtonian cylin-der of a given mass. In another set of simula-tions, a pelycosaur-like sail with the allometryused by Tracy et al. (1986) was appended tothe same Newtonian cylinder. Finally (as forFigs. 7–10), the outer layer of the cylinder wasconsidered a shell exchanging heat with thecore by blood flow and conduction. The heat-ing and cooling of each of the heat exchangermodels can be described by equation (9). Ineach case, the main cylinder was the same sizeand had the same convection coefficient. Thepurpose was to compare the Newtonian timecoefficient of each of the model animals withthat of a Newtonian cylinder with the dimen-sions of the main cylinder.

The dominant time coefficients (l1) of eachheat exchanger model (for Qscope 5 1, windspeed 5 1 m/s) are presented in Figure 11A.In each case, the time coefficient (l1) is nor-malized by dividing by the time coefficient for

a Newtonian cylinder (lmax) under the sameenvironmental conditions. If the heat exchang-er does not change the time coefficient (l1 5lmax), the normalized time coefficient is 1. Ifthe heat exchanger enhances heat exchange,the normalized value is greater than 1. Limbsincreased the absolute value of time coeffi-cients and rates of heating and cooling morestrongly in large animals. As discussed above,core/shell models predicted smaller time co-efficients than for a Newtonian cylinder, butthe effect was less marked in large animals(Figs. 9, 11A). Pelycosaur-like dorsal sails, invery large animals (300–30,000 kg) speed

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355BIOPHYSICAL CONSTRAINTS ON DINOSAURS

model heating and cooling even more thanlimbs (Fig. 11A). The decreased effectivenessof a sail as a heat exchanger predicted for an-imals smaller than 300 kg was due to theshrinking size of the sail, which shrinks fasterthan the mass of the model organism (see Tra-cy et al. 1986 for discussion). Decreases in sailexchanger function in the largest animalswere due to two factors. First, just as sail al-lometry dictates that the sail will shrink fasterthan total mass in small animals, sail massgrows faster than cylinder mass in large ani-mals, and the sail takes on a thermal inertia ofits own. Second, in this simulation, air flow isassumed to travel parallel to the plane of thesail. As the sail grows in size (i.e., in larger an-imals), the thickness of the boundary layeralong the trailing edge grows, limiting heattransfer from the downstream side of the sail.In effect, there is a point of diminishing re-turns as the sail gets larger. The effects of heatexchangers on the time coefficients of modeldinosaurs is reflected in the fractional ampli-tudes of those animals. Heat exchangers shiftamplitude vs. period curves and change ther-mal inertia to a greater extent in large animalsthan in small animals (Fig. 11B).

We should note that several types of dorsalheat exchangers occurred in large animals. In-stead of a single large sail, stegosaurs pos-sessed a staggered series of smaller plates. Inmany ways, this sort of arrangement has ther-moregulatory advantages over the single largesail of pelycosaurs (Farlow et al. 1976). Othertypes of plates could also have been used asheat exchangers if they were well vascularized(even a local area of skin that was highly per-fused). We do not focus on the properties orrelative merits of different types of heat ex-changer, but on the heat transfer properties ofheat exchangers as a pathway for heat ex-change.

In a second set of simulations, we usedmore complex models to estimate the steady-state temperatures reached with heat exchang-ers. We still use the same three exchangers, theshell or peripheral tissues of the animal, thelimbs, and a flat, plate type of exchanger. Buthere we examine the effect of a second type ofplate-like heat exchanger with the allometry ofelephant ears instead of a pelycosaur sail as

the plate heat exchanger. In this case, however,we abandon the requirement that the centralbody cylinder be the same size and mass asthe cylinders we have used previously. All an-imals have a central core, a cylindrical shell,limbs, and plate exchangers equivalent to el-ephant ears. All body parts (except the ele-phant ears) are scaled according to the allom-etry of Spotila et al. (1991), and the total massof the organism (rather than the mass of themain cylinder) is set to the desired size. Eachheat exchanger exchanges heat with the corevia blood flow. The shell, in addition, exchang-es heat with the core via conduction. For thesimulations presented here, each exchanger iseither perfused on a gram-by-gram basis withthe same blood flow rates as the core or leftalmost totally unperfused (blood flow 5 0.1%of that in the core).

There were several major procedural differ-ences between these simulations and thosepresented above. Instead of estimating one ofthe components of body temperature, here weestimate the total elevation of core body tem-perature above ambient air temperature. Thisallowed us to reexamine the importance of so-lar radiation to heat balance in dinosaurs. Inaddition, we introduce an important model-ing assumption: Because the delivery of theoxygen that serves as fuel for metabolism is amajor role of blood flow, we assumed thatmetabolic rate and blood flow, as indicated byMscope and Qscope, were correlated. Here,we present results with metabolic rate andcardiac output both at ten times their allo-metrically predicted rest levels for reptiles(Qscope 5 Mscope 5 10).

Results of the simulations are presented inFigure 12. Three major patterns emerge. First,as with previous simulations, higher temper-atures are expected with larger animals. Sec-ond, high metabolic rates (Mscope 5 10) in acalm (wind speed 5 1 m/s), sunny environ-ment led to core-to-air temperature gradientsof 20–508C in animals with masses over 1000kg, despite the use of several heat exchangers.Although not shown in Figure 12, decreasingmetabolic rates (Mscope 5 5) or moving intowindy environments (wind speed 5 5 m/s)decreased body temperatures, especially inlarger animals (as in Figs. 1 and 4). Third, the

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356 MICHAEL P. O’CONNOR AND PETER DODSON

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

utility of the three exchangers modeled variedwith mass. In all simulations, the effects of ex-changers on core temperatures increased withanimal mass. Hypoperfusion of the limbs ledto the largest changes in temperature in ani-mals smaller than 1 kg, but to the smallesttemperature changes in animals larger than300 kg (Fig. 12). Changes in the perfusion ofthe peripheral tissues (‘‘surf,’’ Fig. 12) and ofthe plate heat exchanger resulted in similar,large (158C at 3000 kg) changes in core tem-perature for animals larger than 300 kg.

Thus, even with physiological heat-loss ad-aptations, such as the use of limbs as heat ex-changers, or anatomical heat-loss adaptations,such as plate-like heat exchangers, large di-nosaurs would have been in danger of over-heating in warm environments if they hadmetabolic rates typical of extant endotherms.Heat exchangers would have had little effecton the steady-state temperatures of small an-imals. Again, animals with masses of 10–1000kg, endothermic metabolism, and the abilityto dump heat via heat exchangers may have

been able to modulate temperature via metab-olism.

Multiple Layers.—All of the models used tothis point are simple models built to under-stand the effects of body size, internal heatconductance, and metabolism on body tem-peratures of dinosaurs. These simple modelshave the virtue that simple, analytic equationspredict both steadystate body temperaturesand the response to varying environmentaltemperatures. In a word, the simple modelsare simple to understand.

But simple models often sacrifice realismfor tractability. To investigate whether thetemperatures and dynamics predicted by sim-ple models accurately describe the thermal be-havior of more complicated, real animals, weused an ‘‘onion-skin’’ model. In this model,both the torso of the animal and its append-ages are represented as a series of concentriccylindrical layers like the skin layers of an on-ion. Each layer exchanges heat with adjacentlayers by conduction and with a central vas-cular pool by blood flow. Steady-state andtransient body temperatures were then pre-dicted by computer simulations. In all simu-lations presented here, blood flow was eitherdistributed evenly to each gram of tissue (evenblood flow condition), or tissues in the outer-most 10% of the radius of the animal—bothtorso and appendages—were deprived ofblood flow, with blood flow distributed evenlyto other tissues (dead surface condition). SeeO’Connor (1999) for details of the onion-skinmodel.

We investigated three questions with theseonion-skin models. Two are the same ques-tions we have asked of each of the simplemodels: How do predicted, steady-state‘‘core’’ temperatures vary in different environ-ments and with different blood flow and met-abolic rates? How do transient body temper-atures vary with these same conditions? In ad-dition, we ask, Is there is an isothermal coreof the animal that can be thought of as havinga single core temperature (as in our simplemodels), or does body temperature gradeevenly from the center to the surface of themodel animal? We will address this thirdquestion 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 varyingsizes 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, Qscope5 0.1. B, QScope 5 1. C, QScope 5 10. D, Potential differences between core and surface temperatures in animalswith 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 isheated by the current it carries. In this casesteady-state temperature falls off from themidline to the surface. The difference betweenthe temperature at any spot and that at thecenter of the wire is proportional to the squareof the distance from the center. Onion-skinmodels make similar predictions about thetemperatures of dinosaurs (Fig. 13). With lowblood flow (Fig. 13A), temperatures show theparabolic distribution expected in wires. In-creased blood flow (Fig. 13B,C) increases thevolume of the nearly isothermal core andsteepens the temperature gradient near thesurface. Large animals appear to have more oftheir mass in an isothermal core than dosmaller animals (Fig. 13A–C). In part, this isdue to the increased sensitivity of large ani-mals to blood flow, as we have already dis-cussed (Figs. 9, 10). In part, however, this pat-tern is deceiving. Temperature gradients in

Figure 13A–C are presented in terms of frac-tions of the total core-to-surface temperaturedifference. But the-core-to-surface tempera-ture difference is much larger in large animalsthan in small ones (Fig. 13D). Thus, if an iso-thermal core is that portion of the animalwithin some number of degrees of the coretemperature, small animals will have largerisothermal cores than Figure 13A-C suggestsimply because the total core-to-surface ther-mal gradients are small. Regardless of howthe core is defined, substantial proportions ofa model dinosaur’s mass is likely to be close tocore temperature, particularly with predicted(or higher) blood flow rates. Thus, it is rea-sonable to think in terms of a nearly isother-mal core.

Core-operative temperature differences foronion-skin models are consistent with predic-tions from simpler models with shell and limbheat exchangers (Fig. 14, compare Fig. 12).

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358 MICHAEL P. O’CONNOR AND PETER DODSON

FIGURE 14. Predicted elevations of body temperatureabove operative temperature (Te) due to metabolic heat-ing in onion-skin models of dinosaurs ranging in sizefrom 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 eachsimulation. Solid line 5 even distribution of blood to alltissues. Dashed line 5 blood flow falls linearly from fullperfusion at 90% of the way from midline to surface tono perfusion at surface (‘‘dead surface’’).

FIGURE 15. Response of core temperature to variationin environmental temperature in onion-skin models ofdinosaurs. Period and fractional amplitude defined as inFigure 3. Symbols represent dinosaurs of different mas-ses, from a 1.98-kg Compsognathus to a 30,000-kg Apato-saurus. Lines represent responses of animals withQscope 5 1. Symbols not on line represent responses foranimals of the same size but with Qscope 5 0.1–10 timesthe predicted levels. Blood flow is evenly distributed toall tissues in all simulations. Mscope 5 0–10.

Limiting blood flow to peripheral tissues re-duces the effectiveness of the heat exchangersand increases the predicted body tempera-tures, particularly in large animals, which aremore sensitive to blood flow (Fig. 14). Forlarge dinosaurs (.3000 kg), metabolic rates 5–10 times those predicted for reptiles yield pre-dicted core-operative temperature differencesof 10–308C. Predicted body temperatures ofsmall dinosaurs (,100 kg) differed from op-erative temperatures by less than 78C regard-less of metabolic rate. High metabolic rates(Mscope 5 5–10) in intermediate-sized dino-saurs would have produced core temperatures5–128C above operative temperatures.

The predictions about dinosaur heating andcooling rates made by onion-skin models arein general agreement with those of simplermodels. Blood flow affects heating and cool-ing rates and thermal inertia in large dino-saurs to a greater extent than in smaller ani-mals (Fig. 15).

The effects of blood flow on time coeffi-cients in model dinosaurs are reflected in thepredicted effects of fluctuations in environ-

mental temperatures on dinosaur body tem-peratures (Fig. 15). Increased blood flows in-crease response to environmental variationsand make the animal ‘‘thermally smaller.’’ De-creased blood flow rates, overall (Fig. 15) orspecifically to the skin (data not shown), de-crease responses to environmental variationsand make the animal appear ‘‘thermally larg-er.’’ Regardless of blood flows, large model di-nosaurs (.1000 kg) respond minimally to en-vironmental variations with daily periods.Their body temperatures varied as if to an av-erage daily operative temperature. Small ani-mals (,10 kg) track daily temperatures fairlywell (fractional amplitude . 0.95, Figure 15).Model dinosaurs with intermediate massestrack daily temperatures with varying degreesof fidelity depending on blood flow (Fig. 15).Over the range of simulated values, bloodflow patterns and levels can adjust the rangeof experienced temperatures by a factor of 2–3 but have their largest effects over the rangeof periods to which the animal responds withfractional amplitudes of 20–80% (Fig. 15).

Cretaceous Climates.—To assess the effects ofenvironmental and metabolic heat sources onthe body temperatures and thermal ecology ofdinosaurs, we calculated expected core tem-peratures for different-sized dinosaurs withvarying 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 105 kg) at different latitudesduring January under the Cretaceous climate reconstructions of Crowley and North (1991). Environmental condi-tions used are the average air temperature and solar radiation over the course of a 24-hour period. Metabolic rateand 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 simulatedsteady-state body temperatures for dinosaurs(mass 10 g–105 kg) experiencing the daily av-erage temperatures at different latitudes anddifferent seasons. Only results for January andmasses greater than 1 kg are presented (Fig.16). We simulated four combinations of bloodflow and metabolic rate (Mscope 5 Qscope 51, 2, 5, 10). Despite varying assumptions aboutwind speed (results not shown), these simu-lations suggest that large dinosaurs (.1000–5000 kg depending on assumptions), at trop-ical to middle latitudes, with mammalianmetabolic rates (Mscope 5 5–10) would ex-perience body temperatures above 408C dur-ing the summer. Those animals would be indanger of overheating in the absence of spe-cific heat-loss adaptations. Animals with met-

abolic rates predicted by the Bennett andDawson equations would be predicted to havebody temperatures between 308 and 408C insummer over a broad range of body sizes andlatitudes. Considering animals in the winterhemisphere and at latitudes above 408, onlythe very large animals (.10,000 kg) with met-abolic rates above 5 times predicted would beexpected to have body temperatures above308C. In very small adult dinosaurs (,10 kg)and in very small hatchlings and juveniles (10g–1 kg) metabolic rates have relatively little ef-fect on predicted body temperatures.

Five caveats are necessary to interpret thesepredictions. (1) The multipliers for metabolicrates should not be considered exact. The al-lometric predictions for mass-specific meta-bolic rate at 308C used here (Bennett and Daw-son 1976) have a smaller scaling coefficient

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360 MICHAEL P. O’CONNOR AND PETER DODSON

(20.17) than most other predictions or, in-deed, predictions about metabolic rate at othertemperatures from the same study (ø2 0.25).Furthermore, over the range of masses be-tween those used to estimate the allometriesand those of the large dinosaurs modeledhere, this difference in scaling coefficientscould result in a 2–3 fold difference in pre-dicted metabolic rate. Thus, ‘‘endothermic’’metabolism may be modeled better by meta-bolic rates 3–53 the prediction rather than 5–103. On the other hand, when mean field met-abolic rates are measured in vertebrates, thoserates usually range 2–33 resting metabolicrates (Nagy 1987, 1989). (2) The allometricpredictions are being extrapolated consider-ably beyond the range of masses for whichmeasurements are available. (3) These predic-tions are for a Cretaceous paleoclimate moreequable ( 5 warmer) than Recent climates(Crowley and North 1991). (4) Finally, the pre-dictions here are for animals without partic-ular heat-loss adaptations. (5) These predic-tions must be applied carefully to animalssmall enough to change temperatures signifi-cantly by taking advantage of diel variation inoperative temperatures (,200–300 kg, Fig.15).

Given the first two of these caveats, it isworth inverting the analysis of Figure 16 toask what metabolic rates would be required tomaintain specified body temperatures in thispaleoclimate (Fig. 17). For animals withoutheat-loss adaptations at the equator, there is arange of masses (2–1000 kg) at which meta-bolic rates 1–103 allometric predictions allowmaintenance of body temperatures between308 and 408C in both winter and summer; athigher masses, lower metabolic rates would berequired (Fig. 17A). At higher latitudes, as theseasonality of the climate becomes more pro-nounced, there is no single metabolic rate thatallows maintenance of body temperatures inthe 30–408C range during both winter andsummer, and seasonal acclimation to main-tain those body temperatures would requirelarge factorial changes in metabolic rate (min-imum ø 40-fold change between summer andwinter, Fig. 17B,C). Also the wintertime met-abolic rate would need to be fairly preciselycontrolled to avoid overheating or excessive

cooling (Fig. 17B,C). These predictions are notsensitive to errors in the extrapolation of pre-dicted metabolic rates. The curves represent-ing metabolic requirements in Figure 17 donot depend on the allometric predictions ofmetabolism.

Discussion

The major implication of the simulationspresented here is not that any particular com-bination of body mass and metabolic rate can-not be tolerated by a dinosaur. Rather, bodymass and metabolic rate constrain the envi-ronment, blood flow pattern, tolerated bodytemperatures, and heat-loss adaptations of theorganism. A 30,000-kg sauropod with a met-abolic rate ten times that predicted for reptilesliving in a sunny terrestrial environment(without access to shade or standing water)with daily mean summer air temperaturesnear 258C would need to tolerate body tem-peratures of about 808C, well above body tem-peratures tolerated by extant vertebrate en-dotherms or ectotherms. However, in a colderenvironment (air temperature ø 108C), with awell-perfused heat exchanger, especially oneprotected from the sun, or with limbs in coolwater, the animal’s body temperature could bewithin the bounds of currently tolerable bodytemperatures. Increased heat loads due to sev-eral sunny days or extended bouts of exercise,however, might result in dangerous levels ofbody temperature even in this animal. Alter-natively, a 2-kg Compsognathus could easily tol-erate metabolic rates 10–20 times those ex-pected for a reptile, as long as the environmentwas not very hot (i.e., operative temperaturesdid not exceed 40–458C), but would be unlike-ly to derive any thermal benefit from the ex-penditure of the extra energy unless it alsohad some way of conserving the generatedheat (e.g., fur, feathers, substantial subepider-mal fat, or some other insulation).

The pattern of body temperatures outlinedin Figures 1, 12, and 14 suggests that verylarge dinosaurs (.10,000 kg), principally sau-ropods, are unlikely to derive any thermoreg-ulatory benefit from metabolic rates abovetwice the expected reptilian level in warm ter-restrial 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 308 or 408C in dinosaurs living in the Cre-taceous climates reconstructed by Crowley and North (1991). Separate predictions are made for summer and winterand for animals at the equator, in temperate regions, and in polar regions. Conditions: wind speed 5 1 m/s, evenblood flow to all tissues.

sids, iguanodontids, ankylosaurians, andmany theropods) in cool environments couldtolerate, and benefit from, higher metabolicrates without risking overheating. Smaller di-nosaurs (,1000 kg; e.g., basal ornithopods,basal ceratopsians, basal iguanodontians,maniraptoran theropods, juveniles of manydinosaurs) could probably tolerate any of themetabolic rates simulated here, unless theyfound themselves in very warm environ-ments. The extent to which these smaller an-imals could benefit from high metabolic rateswould depend on the conductance from theanimal to the environment. In these cases, therisk of overheating and thermal benefit of highmetabolic rates can be roughly estimated withsimple models like those presented here.

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

us to estimate when thermal inertia can beused as a refuge from overly warm or cool en-vironmental conditions. In all cases simulat-ed, rapid variations, i.e., those with short timeperiods (ø 1 min) result in little to no changein body temperatures. The operative temper-ature to which the animal responds in thesecases is the time-averaged mean operativetemperature. On the other hand, variationswith periods of seasonal length (ø 2000 h) al-low body temperatures to vary through thefull available range. In this case, variations inbody temperature will track those of mean op-erative temperatures fairly closely. Betweenthese two extremes lies a range of variationperiods to which body temperatures respondmoderately (fractional amplitudes 0.2–0.8) inthe absence of thermoregulation. It is in thisrange that changes in wind speed, cardiac out-

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362 MICHAEL P. O’CONNOR AND PETER DODSON

put, blood flow distributions, and use of heatexchangers have their major effects in terres-trial animals (Figs. 10, 11, 15).

Figure 15 suggests that for environmentalvariations with periods shorter than a week(ø 168 h), large dinosaurs (.3000 kg) wouldexperience relatively small changes in bodytemperature. There are two important caveatsto such a statement. First, these large animalsare precisely the ones that will experience thelargest variations in operative environmentaltemperatures due to solar radiation (Fig. 1)and whose steady-state body temperaturescan most easily be altered by controllingblood flow (Figs. 12, 14). Thus, although thefractional amplitudes may be low, the actualchanges in temperature may still be severaldegrees. Second, although not apparent fromFigure 15, the ratio of fractional amplitude atthe highest blood flow rates to that at the low-est blood flow rates is largest for periods re-sulting in small fractional amplitudes (frac-tional amplitude , 0.1). For a 30,000-kg sau-ropod in Figure 15, the ratios of maximal(Qscope 5 10) to minimal (Qscope 5 0.1) frac-tional amplitude for sinusoidal temperaturevariations with periods of two days, one week,and three weeks are 3.4, 2.0, and 1.3, respec-tively. To understand why this is important,we must recall that predicted steady-statetemperatures are not realized body tempera-tures, but rather measures of available bodytemperatures in a hypothetically stable envi-ronment. Thus, increasing blood flow allowsfor faster response to environmental changes,and changing blood flow may allow large fac-torial changes in the range of body tempera-tures experienced by the animal. In these cas-es, it is important to know the total range ofoperative temperatures to interpret the biolog-ical importance of the fractional amplitude.For instance, for a two-day cycle, increasingcardiac output from 0.2 times predicted rest-ing levels to 10 times predicted increases pre-dicted fractional amplitude from 0.0364 to0.124. The predicted range of body tempera-tures is the fractional amplitude times therange of operative environmental tempera-tures. If that range is 18C, changing blood flowis unlikely to have a biologically significant ef-fect. If the range of operative temperatures is

408C, however, blood flow alterations couldconceivably increase the range of experiencedbody temperatures from 1.48C to 58C, whichmay be important to the biology of the dino-saur.

Despite these limitations on the analysis offractional amplitudes, Figure 15 allows us tomake some statements about thermal inertiaand the buffering effect of mass on body tem-perature. For very large animals such as sau-ropods, diurnal variations in temperaturewould not strongly affect body temperatures.On the other hand, even these very large an-imals would have little inertial defenseagainst seasonal changes (ø 2000 h) in tem-perature. For environmental variations withperiods between days and seasons (e.g.,changes due to weather systems, cold fronts,etc.), fractional amplitudes would be inter-mediate, affected by blood flow, and thus, un-der the animal’s control to some extent. Anappropriately sized plate heat exchangercould provide additional control over steady-state and transient body temperatures. Suchexchangers occur both in dinosaurs (e.g., theenlarged 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 notknown for sauropods. The extent to which ahighly perfused area of skin could serve thesame function would depend on perfusionrates, the location and areal extent of the ‘‘skinexchanger,’’ and the external environment(e.g., could the skin be wetted and allowed tocool by evaporation).

A similar analysis for a 2-kg Compsognathussuggests that the animal’s body temperaturewould respond only slightly to minute-to-mi-nute changes in the environment, but wouldtrack diurnal temperatures rather closely. Fur-ther, the animal could capitalize on any ther-moregulatory shuttling among environments(e.g., between sun and shade) by using vas-cular control of heating and cooling rates as itmoved on an hourly basis (Fig. 15). For ani-mals between 2 kg and 20,000 kg, the tem-poral scale of the thermal variations thatwould or would not affect body temperaturesand of which the animal could take advantagewould vary with body size (Fig. 15), the use

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363BIOPHYSICAL CONSTRAINTS ON DINOSAURS

of heat exchangers (Fig. 11), and insulation.Such temporal scales suggest that ‘‘inertial ho-meothermy’’ would buffer most dinosaursfrom daily temperature excursions and allowthem to tolerate (for hours or even days, Fig.15) operative temperatures that would be le-thal if the animal ever came to equilibriumwith that environment (Paul 1991). The sametimescale considerations, however, show thatthermal inertia would not effectively bufferbody temperatures from the effects of season-al variations in operative temperatures (Fig.15) of the type used to generate our steady-state temperature predictions (Fig. 16).

Simulations of steady-state body tempera-tures (in response to daily mean operativetemperatures, Fig. 16) suggest the mean val-ues around which body temperatures wouldvary. They suggest that animals larger than1000 kg with high metabolic rates would be indanger of overheating during the summer attropical and middle latitudes. Very large sau-ropods (.10,000 kg), especially those with hy-poperfused peripheral tissue layers, would bemost susceptible to overheating (Fig. 16). Suchanimals would have required either cool hab-itats or a means of cooling themselves, such asa heat exchanger (e.g., on neck and/or tail), inorder to be active during the summer. Forthese large animals, diel temperature differ-ences would not have served as refugia fromthe warm environment, because of the slightresponses of body temperature to daily tem-perature variation (Fig. 15). Alternatively,high metabolic rates might have providedsome protection from cool winter tempera-tures in these large dinosaurs (Figs. 16, 17).

At the other end of the size scale, metabolicand circulatory adjustments had little effecton the predicted body temperatures of dino-saurs with masses below 10 kg (Fig. 16). Thus,metabolic warming is unlikely to have playeda large role in the thermal physiology of theseanimals unless and until they developed in-sulation sufficient to retain metabolically gen-erated heat. But these are the animals whosebody temperatures could vary dramaticallyover the course of a day and could benefitfrom microhabitat selection, basking, behav-ioral thermoregulation, and the use of day-time or nighttime thermal refugia. By such

mechanisms, small dinosaurs might have af-fected the ‘‘average’’ operative temperaturethey experienced over any given day or at aparticular time of year, much as modern liz-ards do.

For animals with masses between 100 kgand 10,000 kg, predictions suggested a gra-dient from the relative unimportance of bloodflow and metabolism to body temperature insmall animals to the much larger effect of me-tabolism on body temperature in larger ani-mals with the concomitant risk of overheatingin some situations (Fig. 16). Predicted temper-atures vary with assumptions about metabo-lism, blood flow, and environmental condi-tions. With low wind speeds (1 m/s), high me-tabolism (103 RRMR), and a cornified or hy-poperfused peripheral layer (Fig. 16D), allanimals larger than 100 kg would be at risk ofoverheating in low latitudes and all animalslarger than 1000 kg would be similarly at riskat any latitude during the summer. Alterna-tively, with higher wind speeds (3 m/s), lowermetabolic rates (53 RRMR) and even perfu-sion of all tissues only large animals (.10,000kg) at low latitudes would be at risk of over-heating, and other animals might gain somebenefit from the relatively high metabolicrates—either from a thermoregulatory pointof view or otherwise.

An interesting perspective on these predic-tions is provided by examining model predic-tions for African elephants, the largest extantterrestrial endotherm. Elephants are largetachymetabolic endotherms that are frequent-ly thought to be at risk of overheating due totheir combination of large body size, mam-malian metabolic rates, and relatively warmenvironments (Wright 1984, Wright and Luck1984, Williams 1990). We predicted the steady-state core body temperatures for a 4000-kg Af-rican elephant (Loxodonta africana) in the pa-leoenvironment used to generate Figure 16 at208 latitude during the summer (hottest avail-able mean Te). Although the paleoclimateused is warmer than current mean tempera-tures, the differences for the Tropics are only3–58C (Crowley and North 1991). Thus, in es-sence, we simulate an unusually warm sum-mer.

Elephant resting metabolic rates corre-

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364 MICHAEL P. O’CONNOR AND PETER DODSON

FIGURE 18. Core body temperatures predicted for a4000-kg elephant in July at 208 N latitude in the paleo-climate used for Figures 16 and 17. Predictions areshown for a range of metabolic rates spanning restingand active metabolic rates for elephants and for a varietyof convective environments (wind speed and turbu-lence). For the simulations shown, the elephant is as-sumed not to use its ears as heat exchangers. Qscope 5Mscope for all simulations.

spond to an Mscope (as calculated in this re-port) of approximately 3.6 with mean meta-bolic rates approximately at Mscope 5 6–9and high rates during induced exercise atMscope 5 13–14 (Benedict 1936; Wright 1984;Williams 1990; Withers 1992). Figure 16 sug-gests that a 4000-kg animal with such meta-bolic rates would be at risk of overheating.Several factors mitigate this risk. First, Africanelephants, particularly during warmer partsof the day, experience wind speeds signifi-cantly above 1 m/s (Buss and Estes 1971) andusually turbulent airflows, which increaseconvection coefficients by a factor of 1.7 overthose for laminar airflows (Mitchell 1976). As-suming higher wind speeds and turbulentconditions raises the threshold metabolic ratesfor overheating (predicted temperature .408C, Fig. 18). The most common wind speedmeasured by Buss and Estes (1971) duringwarm parts of the day was approximately 3m/s. Using a turbulent 3 m/s air flow, the on-ion-skin model we present predicts a criticalMscope of approximately 5.

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

timates that 50% of an elephant’s metabolicheat production can be lost across the ears. Tostudy this possibility, we used a model withelephant-ear-like heat exchangers (Fig. 12)and the blood flow rates measured in elephantears (Wright 1984). The model also predictsthat for a 4000-kg elephant, perfusing the earshas the same effect on steady-state tempera-ture as cutting the metabolic rate by 50% (datanot shown), effectively doubling the criticalMscope. Perhaps as important, elephants canand do use their ears to ‘‘create’’ airflows overthe ears and neck, especially under putativelystressful thermal conditions (Buss and Estes1971).

A series of factors too poorly characterizedto incorporate into our models may also re-duce the predicted temperature of elephants.Wright and Luck (1984) argue that relativelyhigh rates of cutaneous evaporation, particu-larly over the ears, may lead to substantial lossof metabolic heat, although this is difficult toquantify. In addition, behavioral thermoreg-ulation (shade seeking, use of mud and waterfor cooling) appears from behavioral studiesto be important in elephants (Buss and Estes1971; McKay 1973). Finally, heat from thehighest metabolic rates associated with in-duced exercise (and thus transient) may bebuffered by heat storage and reradiation at lat-er times.

Thus, although elephants in warm environ-ments would appear to be at risk of overheat-ing because of their high metabolic rates, a se-ries of heat-loss mechanisms (adaptations?)reduces the risk at all but the highest (andlikely transient) metabolic rates.

These simulations argue that the thermal ef-fects of high metabolic rates might vary fromdinosaur to dinosaur. Very large and verysmall dinosaurs would not be expected to de-rive thermoregulatory benefit from high met-abolic rates. For dinosaurs of intermediatesize, high metabolic rates might have yieldedbenefits in terms of high activity capacities(Reid 1997), but in many cases in which met-abolic warming would have been most usefulthermally (animals in temperate latitudes),thermally useful metabolic heating may havecarried substantial risks of overheating andwould have needed seasonal adjustment.

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What would seem to be needed are not onlyendothermic thermogenic capacities, but alsoendothermic patterns of response to externaltemperatures with lower external tempera-tures evoking an increase in metabolic rate.

Such differences in the potential thermalutility of high metabolic rates in large andsmall animals raise the question of the extentto which metabolic rates might have variedwithin individual dinosaurs and between di-nosaurs with different levels of relatedness.Available data do not clearly answer this ques-tion. Metabolic rates differ with activity level.Standard metabolic rates clearly differ withinan animal as a function of ontogeny (Mautzand Nagy 1987), combinations of season andacclimation (Gatten 1985; Tsuji 1988; Beyerand Spotila 1994), and endocrine effects (John-Alder 1983, 1990). Significant differences inmetabolic expenditures also occur among in-dividuals within a population (Konarzewskiand Diamond 1994; Burness et al. 1998),among conspecific populations (Beaupre1995), among sympatric, congeneric species(Nagy et al. 1984), and among families and or-ders within classes (e.g., mustelids andsloths). However, such differences among rep-tiles, among birds, or among mammals rarelyreach the 5–10-fold differences that commonlyexist between extant metabolic endothermsand equivalently sized reptiles. The apparentdichotomy in metabolic rates between extantreptiles and extant mammals and birds maybe due to phylogenetic constraints, to differ-ences in natural and life history (Pough 1980;Anderson and Karasov 1981), and/or to phys-iologic and pleiotropic constraints on the suiteof traits necessary to achieve and benefit fromhigh metabolic rates. To what extent the cur-rent ‘‘endotherm/ectotherm’’ differences de-pend on any or all of these factors is unclear.If, however, standard metabolic rate is highlyconserved, that would tend to limit the abilityof closely related dinosaurs of different sizesto have different metabolic rates that mightsuit their mass-specific thermoregulatoryneeds.

Thus, we suggest that a physical approachto body temperatures in dinosaurs can allowus to predict what ranges of body tempera-tures and what thermoregulatory strategies

were available to those dinosaurs. We presentsome simple tools for exploring such ap-proaches. Substantively, we argue that

1. The huge range of body sizes found withinthe dinosaurs likely resulted in very differ-ent thermal problems and strategies for an-imals at either end of this size continuum.

2. Body temperatures of the smallest adult di-nosaurs and small hatchlings and juvenileswould have been largely insensitive to met-abolic rates without insulation. For meta-bolic rates 5–103 RRMR (Mscope 5 5–10),the smallest animals in which metabolicheating would likely result in body tem-peratures more than 28C above operativetemperatures weighed 10 kg (Figs. 4, 12).Animals of this size were unlikely to ben-efit thermally from high metabolic rates,and their body temperatures would re-spond rapidly enough to changes in envi-ronmental temperature to make behavioralthermoregulation possible (Figs. 10, 15).These animals could be thought of as ther-mally like living terrestrial lizards.

3. Body temperatures of large dinosaurs(.1000 kg) were likely very sensitive toboth metabolic rate and the delivery of heatto the body surface by blood flow (Figs. 4,12). Our model suggests that they were ca-pable of adjusting body temperature by ad-justing metabolic rate and blood flow. Be-havioral thermoregulation by changing mi-crohabitat selection would likely have beenof limited utility because body tempera-tures would have responded so slowly tochanges in external temperature (Figs. 10,15). In these large dinosaurs, the availabil-ity and perfusion of heat exchangers suchas limbs or plate-like exchangers could al-ter both steady-state temperatures and therates of heating and cooling in these ani-mals (Figs. 11, 12).

4. Endothermic metabolic rates may have putlarge dinosaurs at risk of overheating un-less they had adaptations to shed the heatas necessary. Although this would havebeen true particularly for dinosaurs withmasses greater than 10,000 kg, simulationssuggest steady-state (‘‘average’’) body tem-peratures would have exceeded 408C for

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366 MICHAEL P. O’CONNOR AND PETER DODSON

animals as small as 1000 kg in the Tropicsand during the summer in temperate lati-tudes (Fig. 16), under the paleoclimateused in our simulations. Slow responses ofbody temperatures to environmentalchanges suggest that use of day-night ther-mal differences alone would have been use-ful in buffering dinosaurs from diel warm-ing if the mean daily temperature was nottoo high, but it would not have been ade-quate to lower body temperatures suffi-ciently to allow activity during the day foranimals experiencing high daily mean op-erative temperatures (Figs. 15, 16). Consid-eration must be given to the mechanismsthe animals used to control heat loss underthese circumstances. Endothermic metab-olism may have allowed activity in the win-ter in some circumstances (Fig. 16), butthose same metabolic rates would endan-ger the animal if maintained during thesummer (Figs. 16, 17). The control of met-abolic heating in large dinosaurs must beaddressed.

5. Endothermic metabolism and metabolicheating might have been useful for inter-mediate and large (100–3000 kg) dinosaursbut often in situations that would demandmarked seasonal adjustment of metabolicrates (Fig. 17) or relatively precise controlof metabolism (and heat-loss mechanisms),as typically seen in endotherms (Fig.17B,C).

Acknowledgments

We are grateful to J. E. Bentz, E. M. Dzi-alowski, S. S. Kilham, J. R. Spotila, and S.Zeitz who provided valuable criticisms dur-ing the construction of the model. E. M. Dzi-alowski, J. R. Spotila, J. Farlow, W. Hillenius,J. Ruben, A. Tumarkin, M. Lamanna, and J.R. O’Connor provided helpful criticism ofearly versions of the manuscript.

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