The imaging properties and sensitivity of the facial pits of pitvipers as determined by optical and heat-transfer analysis

It has been known for some time that the eponymous facial pits of pitvipers(Viperidae: Crotalinae) are sense organs that respond to thermal infrared radiation emitted by nearby surfaces (ca. 5–30 μm wavelength), and can thereby sense the temperature of surrounding surfaces(Bullock and Cowles, 1952). Functionally, the facial pits are image-forming structures based on pinhole optics resembling the chamber-type eye of the nautilus(Fernald, 2006). We have a general sense of some behavioral functions, and various aspects of the anatomy and neurophysiology of the facial pit are well documented. But, we lack clear understanding of the physical and physiological optics of the facial pit system, and thus of what pitvipers can actually `see' with the facial pits.

Behavioral studies have established that the facial pits aid in prey acquisition (Bullock and Diecke,1956; de Cock Buning,1983; Kardong,1986; Noble and Schmidt,1937) and mediate behavioral thermoregulation(Krochmal and Bakken, 2003). Other functions have been proposed but not tested, including general navigation and predator detection (Bullock and Barrett, 1968; Greene,1992; Sexton et al.,1992). Behavioral evidence suggests that input from the facial pits compensates for visual deprivation(Kardong and Berkhoudt, 1999; Kardong and Mackessy, 1991). Thus, this sense may be particularly important on moonless nights, when both rattlesnakes and their rodent prey appear to be most active(Clarke et al., 1996), as well as when surface temperature itself is relevant, as in behavioral thermoregulation (Krochmal and Bakken,2003; Krochmal et al.,2004).

Anatomically, the facial pits are located between the eyes and nostrils(Fig. 1A). Each consists of a 1–3 mm diameter aperture expanding internally to form an asymmetric and somewhat irregular mushroom-shaped cavity(Fig. 1B). Thermal radiation entering the aperture falls on and heats a sensory membrane suspended in the back of the pit, dividing the pit cavity into an inner and outer chamber. The membrane contains a few thousand receptors that respond to membrane temperature changes of 0.003°C or less(Bullock and Cowles, 1952; Bullock and Diecke, 1956; de Cock Buning, 1983; Moiseenkova et al., 2003).

Results from neurophysiological and neuroanatomical studies indicate both image-forming and depth perception capabilities for the facial pits(Berson and Hartline, 1988). Receptor output is preprocessed in the medulla, which apparently sharpens the image (Stanford and Hartline,1980) and then is spatially mapped onto the optic tectum, where it is merged with visual signals (Hartline et al., 1978; Molenaar,1974; Newman and Hartline, 1981). The presence of neural interconnections of at least six different types in the optic tectum and a common tecto-rotundo-telencephalic pathway for thermal and visual signals strongly suggests that the facial pits function as a second pair of eyes. Thus, pitvipers apparently have a multispectral visual sense that includes both the visual color (Peterson,1992; Sillman et al.,2001) and the thermal radiation brightness (i.e. temperature) of surrounding surfaces (Berson and Hartline,1988; Hartline et al.,1978; Newman and Hartline,1982; Newman and Hartline, 1981; Stanford and Hartline,1980).

The design and interpretation of studies of both behavioral responses and neural processing requires knowledge of the temperature contrast image on the membrane, which is defined by the optical and the heat transfer properties of the facial pit. Angular resolution (the sharpness of the image) determines the`brightness' of small targets, the extent to which larger objects such as food items contrast with background clutter, and the overall quality of spatial information available for tasks such as general navigation and thermoregulation. Background and target surface temperatures as well as surface temperature contrasts are affected by air temperature, current and past solar radiation, and the heat storage capacity of the object. Thus,thermal contrast varies with habitat structure and time of day, creating spatiotemporal variation in the probability of success of behavioral activities hinging on thermal cues.

Limited understanding of the relevant optical and thermal physics of the facial pits is a common deficiency of existing studies. For example, a theoretical analysis of pit sensitivity(Jones et al., 2001) severely overestimated absorption of thermal radiation by the atmosphere and concluded that absorption limited the pit organ to a range of a few cm(Bakken, 2007). A number of researchers have presented pitvipers with thermal stimuli having surface temperatures equal to or exceeding body core temperature of typical prey items(e.g. Bullock and Barrett,1968; Goris et al.,2000; Goris and Nomoto,1967; Hartline et al.,1978; Pappas et al.,2004). However, the furred and feathered surfaces covering most of the body are actually closer to air temperature (e.g. Hill et al., 1980; Hill and Veghte, 1976; Kardong, 1986; Veghte and Herreid, 1965). Behavioral experiments have typically used a single target against a uniform thermal background. This may overestimate performance in natural habitats,because the angular resolution of the pit organ is likely poor(Otto, 1972; Stanford and Hartline, 1984). As a result, the radiation from small, warm objects is spread over a large area of the pit membrane and blended with non-uniform natural thermal backgrounds. The only experimental study known to have examined background effects (Theodoratus et al.,1997) placed test targets behind aquarium glass, which is completely opaque to thermal radiation(Hsieh and Su, 1979). Consequently, the reported responses are experimental artifacts.

The foregoing review shows that there is a need for a comprehensive study that will define the input to the sensory system of a pitviper under relevant natural situations. Such a study requires detailed knowledge of the physical optics of the facial pit and its heat transfer properties. Prior studies(de Cock Buning, 1984; Otto, 1972) examined the distribution of radiation from a point source over the pit membrane using simplified geometric models, but lacked the modern computational tools needed to translate this information into a representation of the temperature contrast image on the pit membrane. Further, these studies omitted potentially important heat transfer processes such as convection and conduction from the pit membrane.

To fill this need, we have analyzed the facial pit as an optical system and used heat transfer analysis to convert image irradiance to membrane temperatures on the basis of published physiological data. We then obtained radiometric thermograms of some realistic natural habitats to determine the typical surface temperatures and temperature contrasts present. Finally, we used the results of our optical and heat transfer analysis and image processing software to manipulate these thermograms to generate corresponding representations of the image falling on the facial pit membrane. The processed images indicate the general characteristics of the sensory input to the facial pit sensory system in various ecologically meaningful situations, and provide insights that can aid the design and interpretation of behavioral and neurophysiological studies.

The facial pit is essentially a pinhole camera consisting of a lensless aperture in front of a detector (the pit membrane) that forms the image plane(Fig. 1B). Radiation simply passes through an opening (the optical pupil) without deflection and falls on the image plane. Facial pit apertures are large enough relative to pit depth that diffraction may be neglected, and thus elementary geometric optics and photometric analysis can be used (Born and Wolf, 1970). Briefly, the light from a point on the source object that passes through the aperture irradiates a defined area on the image plane,called the point spread function. The image is formed by overlapping spread functions, and, as demonstrated later, is either sharp but dim when the optical pupil is small, or bright but blurred when it is large. We will follow de Cock Buning (de Cock Buning,1984) and model the facial pit as a circular aperture of radius r_a located a distance d from the image plane(Fig. 2). Though a simplification of the geometry in Fig. 1B, this model is adequate to illustrate the main features of the optics of the facial pit. The analysis proceeds in three steps and follows standard procedures (Born and Wolf,1970).

The first step is to define the ideal (perfectly focused) image. This is found by tracing the chief ray, i.e. the line passing from a point on the source object through the center of the pinhole aperture to the corresponding point on the image plane. All of the radiant energy from a point on the source that passes through the aperture is assumed to fall on the corresponding point of the image (Born and Wolf,1970).

We must now convert the irradiance contrast into the temperature contrast that is detected by the pit membrane receptors. The pit membrane absorbs ca. 50% of incident thermal radiation (wavelength 5–30 μm), and so half of the image irradiance is absorbed and converted to heat, forming a temperature contrast image (Bullock and Diecke, 1956; Goris and Nomoto, 1967). This is detected by sensory receptors that respond to membrane temperature changes of 0.003°C or less(Bullock and Diecke, 1956; de Cock Buning, 1983; de Cock Buning et al., 1981). There is no evidence for quantum detection(Moiseenkova et al.,2003).

The temperature at a point on the membrane is determined by the local balance between radiant heat transferred from the source object and other sources of heat gain and loss. The heat storage capacity of the membrane is important only for transient stimuli. Lateral conduction within the membrane may reduce angular resolution somewhat(DeSalvo and Hartline, 1978),but has a negligible effect on the heat balance.

A pinhole camera produces a geometrically perfect image, but the radiation from a single source point is spread over an area of the image called the point spread function. The radiance at a given point (x,y) on the membrane is the sum of the radiation from all the point spread functions that overlap (x,y). Consequently, the image on the membrane is `fuzzy' and has less contrast than the corresponding ideal image.

Mathematically, the real temperature distribution over the image plane T′(x,y), is found by convoluting the temperature distribution of the ideal image,

To investigate the consequences of variation in the spatial resolution and receptor sensitivity of real and hypothetical facial pits, we derived representations of the blurred image on the pit membrane using the above analysis. As approximations to the ideal image, we used sharply focused thermograms of various scenes recorded with a resolution of 0.1°C and an absolute accuracy of 1–2°C using a FLIR PM575 radiometric thermal imager (FLIR Inc, North Billerica, MA, USA). We computed the membrane temperature contrasts using Eqn 9, and then convoluted these images using Eqn 11 as implemented in MATLAB(The MathWorks, Inc. Natick, MA, USA). We used the `circular' option in MATLAB to reduce edge artifacts. The output is our representation of the approximate appearance of the temperature contrast images on the pit membrane.

We explored the interrelated effects of angular aperture on image sharpness and membrane temperature contrast by using spread functions withθ _i from 2.5° to 20°. The observed angular apertures for Crotalus oreganus (Fig. 1B) are ca θ_i=20–30° to the side, and ca. 10° in the forward direction (see also DeSalvo and Hartline, 1978). The importance of forward imaging is indicated by a higher density of receptors and associated blood vessels on the portion of the membrane corresponding to objects directly in front of the head(Amemiya et al., 1999; Goris and Nomoto, 1967; Goris and Terishima, 1973). Viewed from the forward direction, the external aperture is higher than it is wide (Fig. 1A) and the optical spread function is therefore sub-elliptical. To simulate this, we used elliptical spread functions with the horizontal θ_i half of the vertical θ_i.

Bullock and Diecke (Bullock and Diecke,1956) estimated the sensitivity of the pit membrane to temperature differences as <0.001°C to 0.003°C because they obtained a response to even the smallest temperature difference they could measure, 0.0025°C. They also found that neural response is roughly linear for temperature contrast up to 100× threshold sensitivity. To relate images to this membrane sensitivity, we set the temperature range and the number of colorbar steps such that each of the 30 colorbar steps would represent a membrane temperature difference equivalent to presumed temperature sensitivities within this range.

Environmental conditions have a strong effect on source temperature contrasts, particularly for mammalian and avian prey. The surface temperature of fur or feather insulation is closer to air than body core temperature (e.g. Hill et al., 1980; Hill and Veghte, 1976; Kardong, 1986; Veghte and Herreid, 1965). To illustrate, we recorded thermograms (Fig. 3A,E) of a potential prey item in a temperature cabinet at the lowest body temperature of active rattlesnakes, 15°C, and at the typical hunting temperature of 30°C (Beck,1995; Hirth and King,1969; Moore,1978). While the temperature contrast over most of the 60 g Ord's kangaroo rat Dipodomys ordii is greatest at 15°C, over most of the fur surface it is only about 6°C above air temperature. This is less than 1/3 of the 21°C difference between body core (36°C) and air temperature. At 30°C, temperature contrast is about 2.5°C, compared to the 6°C difference between core and air. Only the eyes have a surface temperature near body core temperature. The results for other bird and mammal prey items we examined are similar, although some thinly furred ground squirrels had higher fur temperatures.

The size of the angular aperture of the pit influences both resolution and signal strength. Fig. 3B–D,F–H are representations of the temperature contrast images on the membrane of a 3 mm total diameter pit with the membrane located 0.75 mm from the back wall and various θ_i from 20° to 5°. Colorbar steps correspond to membrane temperature differences of 0.001°C, near the lower end of the sensitivity range suggested by Bullock and Diecke (Bullock and Diecke, 1956). Resolution is lower but temperature contrast is greater for larger pit apertures.

The laboratory images have a uniform contrasting background, while natural thermal backgrounds are strongly patterned. To investigate the potential impact of background pattern on prey targeting, we recorded outdoor thermal images of a variety of targets. Fig. 4A is an image of two mice recorded in open scrub habitat at midnight following a mostly cloudy day. Based on Fig. 3, thermal imaging conditions were nearly optimal for the snake (air temperature, 15°C;surface temperature 11–13°C). Nevertheless, the contrast of the convoluted images (Fig. 4B–H) is low and the mice hard to detect with color steps of 0.001°C. For clarity, we exaggerated contrast by using color steps of 0.0005°C.

Fig. 4B–D are visualizations of the case where mice are viewed along the pit axis by convoluting the thermogram with various circular spread functions withθ _i from 5° to 20°, while Fig. 4E–G visualize mice directly in front of the snake by using elliptical spread functions ofθ _i from 5°×2.5° to 20°×10°(vertical × horizontal). These values are based on anatomy of the facial pit (Fig. 1B). Forθ _i=10° and 20°, the angular dimensions of the mice are much less than θ_i, and their thermal radiation is smeared over a large area of the pit membrane. Consequently, the mice are indicated not by the highest temperature, but by an overall circular or elliptical pattern superimposed on the background. The highest membrane temperatures are created by a large background area that is only slightly warmer than average. Consequently, in an outdoor environment, a pitviper cannot simply target the strongest signal.

The low membrane temperature contrasts evident in Figs 3 and 4 are largely the result of signal loss by heat conduction through air to the walls of the facial pit cavity. The dimensions of the posterior chamber are particularly important, as the membrane generally lies near the back wall of the facial pit and the smallest dimension dominates heat loss (Eqn 5). A given scene will produce less temperature contrast on the membrane of snakes with smaller facial pits or when the membrane lies nearer the pit wall (Fig. 5) because conduction is inversely proportional to distance.

The consequences of varying facial pit dimensions are visualized in Fig. 6. We computed temperature contrast images for facial pits from 1 to 4 mm total thickness with the pit membrane 25% of the total thickness from the wall of the posterior chamber and both circular (Fig. 6B–D)and elliptical (Fig. 6E–G) spread functions with θ_i=20°. Even assuming a high membrane sensitivity (color steps 0.0005°C), the mice are indistinct to invisible when the total thickness of the facial pit is 2 mm thick or less.

We have previously suggested (Krochmal and Bakken, 2003) that thermoregulation represented the initial adaptive force that drove the evolution of the facial pits because the ancestral pit was likely insensitive, and environmental features are larger than prey items and typically show greater temperature contrast in daylight and early evening. This is illustrated in Fig. 7A–D, which show a desert rodent burrow that might provide the snake with a refuge from the heat. The burrow is clearly visible even with the smallest(θ_i=5°) aperture (membrane temperature contrast 0.008°C), and is least evident for θ_i=20° due to background interference.

Another situation providing strong contrast against background is a warm-blooded prey item viewed against a clear sky, which emits little thermal radiation (Swinbank, 1963). This is illustrated in Fig. 7E–H, which shows a cardinal (Cardinalis cardinalis) viewed against the sky at an air temperature of 20°C. The 5°C radiant temperature of the sky contrasts strongly with the 30°C feather surface temperature of the bird, and creates an 0.01°C membrane temperature contrast even with the smallest angular aperture(θ_i=5°).

Our findings indicate either that the membrane is more sensitive than currently estimated, or that pitvipers obtain less information from this sense than is commonly believed. Accounting for conductive heat loss through still air in the anterior and posterior chambers reduces the temperature contrast of the image to only 5–25% of that estimated by earlier studies, which did not consider conduction through the air(Fig. 5). To achieve the observed behavioral capabilities of pitvipers(Ebert and Westhoff, 2006),the contrast sensitivity of the pit membrane needs to be at the most sensitive end of the range (0.003–0.001°C) found by Bullock and Diecke(Bullock and Diecke, 1956). Our analysis assumed the snake was on the ground, and thus did not include forced convection. Significant air movement with the exterior pit, e.g. because the snake was exposed to wind on an arboreal perch, would further reduce membrane temperature contrasts. Thus, the membrane may conceivably respond to contrasts of less than 0.001°C. The mechanism by which such sensitivity might be obtained is presently unknown.

To simplify this preliminary study, we chose not to examine the effects of either heat storage in the pit membrane or target movement. Heat storage in the pit membrane would potentially slow the time response and cause blurring of moving targets. Experimental studies have reported maximum flicker fusion frequencies of 8 Hz or less, depending on flicker contrast(Bullock and Diecke, 1956). Goris et al. (Goris et al.,2000) has suggested that neurological control of pit membrane microcirculation may serve to increase time response, but this has not been demonstrated experimentally. Membrane receptor response is commonly regarded as phasic (Barrett et al.,1970; Bullock and Cowles,1952) although some studies have reported tonic and phasic-tonic responses as well (Goris and Terishima,1973). A primarily phasic response would presumably make the facial pit more sensitive to moving targets than stationary ones, although scanning head movements could make stationary targets equally conspicuous(Goris and Terishima, 1973). Behavioral and modeling studies are needed to confirm that these conclusions apply to overall sensory performance.

The effect of conductive heat loss through the air is inversely proportional to pit size. This suggests that the facial pit may be of limited value to juvenile snakes. Possibly as a result, juvenile pitvipers are known to depend more on ectothermic prey than do adults, although body size is also a factor. The observations that juvenile Gloydius shedaoensispreferentially select arboreal ambush sites where prey are viewed against the cool sky (Fig. 7E–H), and are slower and less accurate in their strikes(Shine et al., 2002b) might be explained by the reduced sensitivity of their smaller facial pit organs. However, direct evidence with which to test this hypothesis is lacking.

Once the full angle of the facial pit (2θ_i) exceeds the angle subtended by the target, membrane irradiance is constant and the only effect of further enlarging the aperture is to impair resolution and increase background interference. In Fig. 3, the kangaroo rat at 0.5 m subtends a full angle of 10°, so that the optimal half angle θ_i is 5°, much smaller than the actual 10–22° θ_i of C. oreganus(Fig. 1B). As is evident in Figs 4 and 7, a small aperture may be advantageous even when target irradiance on the pit membrane is somewhat reduced because background interference is decreased more than the contrast between target and background. Thus, the facial pit aperture appears to be larger than optimal for detecting small objects, like prey, and it is not clear why the angular aperture of the facial pit is as large(θ_i=20–30°) as is observed. A large aperture provides more detectable contrast and might be used to reconstruct images with more angular resolution and contrast than could be produced with a smaller aperture. Similarly, the aperture might be optimized for detecting larger objects, including environmental features. This could facilitate behavioral thermoregulation, which has been proposed as the initial adaptive force that drove the evolution of the organ (Krochmal and Bakken, 2003).

Several simple mechanisms have been proposed to explain how pits may function despite apparently low spatial resolution, including edge detection or the use of both pits as a null detector (e.g. Goris and Terishima, 1973). Our results call into question the utility of such simple mechanisms. Specifically, the spread function resulting from a large aperture blurs the edges of real targets, and the warmest part of the pit membrane may not represent the target when the background is not uniform(Fig. 4B,E).

A more credible alternative is that the imaging properties of the facial pit may be improved by image sharpening during post-processing in the central nervous system. If the blurred image has sufficiently fine discrimination of irradiance levels, and if the optical spread function is known, the inverse operation of Eqn 11 can reconstruct the original image in some detail. Image sharpening routines are included in many image processing programs such as MATLAB, and a hypothetical neural network procedure has been proposed specifically for pitvipers(Sichert et al., 2006), though empirical data to support it is lacking.

The quality of the processed image is closely linked to how accurately the spread function is known [chapter 5, Gonzalez and Wintz(Gonzalez and Wintz, 1977)]. The geometry of the facial pit, and thus the spread function, varies only with growth, if at all. Thus, the snake's neural network can potentially learn a single spread function, which can then be used to sharpen images. Indeed, it has been reported that the temperature contrast image on the facial pit membrane is sharpened by neural processing in the lateral descending trigeminal tract (LTTD) of the medulla(Berson and Hartline, 1988; Stanford and Hartline,1980).

The spread function might be provided in two ways. First, neural interconnections between thermal and visual neurons have been reported in the optic tectum (Berson and Hartline,1988; Hartline et al.,1978). It has been shown that some alternative spatial senses,such as acoustic prey location in owls, are fine-tuned to match visual input(Harris, 1986; Schnitzler et al., 2003), and the same may be true for the facial pit. Second, spread functions might be determined from facial pit input by using neural mechanisms analogous to forensic computer algorithms [chapter 5, Gonzalez and Wintz(Gonzalez and Wintz, 1977)] or the autofocus mechanism found in some cameras.

Pitvipers commonly use an ambush foraging strategy (e.g. Reinert et al., 1984), and the coupled problems of thermal signal strength and angular resolution that we have documented suggest that predation effectiveness would be greater if pitvipers were to seek out ambush sites offering a contrasting and relatively uniform thermal background. Shine and Sun(Shine and Sun, 2002)conducted a 2-day study of thermal and visual background in pitvipers foraging on migrating birds and found that snakes indeed selected ambush sites that offered a cool sky background. Although snakes struck preferentially at warm targets, the importance of the facial pits was not entirely clear. The snakes generally left arboreal perches during the night and returned in the morning,and thus foraged primarily when ample visual illumination was available. Further, site selection was equally consistent with selecting areas of highest prey availability. Thermoregulation is another factor that might influence ambush site selection (Shine et al.,2002a). Clearly, this is a complex problem needing carefully designed field studies.

The ancestral facial pit likely had lower angular resolution and less temperature sensitivity than the modern organ. Thus, evolution of the facial pit is most likely to have occurred in a situation where large targets with strong temperature contrast were present, and the ability to sense them provided a selective advantage (Krochmal and Bakken, 2003; Krochmal et al., 2004). At least two situations provide high contrast(Fig. 7). First, the ability to sense thermal radiation may allow a snake to thermoregulate more effectively in a habitat characterized by strong temperature contrasts caused by solar radiation (Krochmal and Bakken,2003). This scenario also provides large targets, minimizing the demands on angular resolution. Second, perching birds contrast strongly against the sky (Shine and Sun,2002). Thus, although most extant pitvipers are heavy bodied and terrestrial in habit, the facial pits may have evolved initially to facilitate nocturnal predation on roosting birds or active bats in sparse deciduous vegetation, where they would be viewed against a clear night sky. The radiant temperature of a night sky is well below air temperature(Swinbank, 1963), so birds and mammals would contrast even if the insulating coat is at air temperature. A similar, but less extreme, contrast may exist between burrowing mammals and the burrow walls. Although most pitvipers are sedentary, many other snakes actively search burrows (Secor,1995), and so the facial pit may have evolved initially to aid in burrow searching. To test these hypotheses, studies are needed to determine how extensively extant pitvipers use the facial pit sense in habitats imposing different thermal radiation backgrounds and different thermoregulatory demands. Careful attention to paleohabitat signatures associated with any discoveries of fossils of putative ancestral pitvipers might also indicate the relative merit of these scenarios.

Our survey of the approximate imaging properties of the facial pits leads us to conclude that the imaging properties of the pit organ are critical to understanding its function as a sensory organ and its role in the behavior and ecology of the animal. While there has been significant progress, many interesting studies of field and laboratory behavior and sensory neurophysiology remain to be done before we will fully understand this novel sensory organ and its role in the ecology of pitvipers.

List of symbols

 
• Φ

  radiant flux (W)

 
• ω

  solid angle (sr; steradians)

 
• ϵ

  emittance for thermal infrared radiation (0⩽ϵ⩽1)(dimensionless)

 
• θ

  spherical coordinate (rad)

 
• θi

  half-angle of facial pit aperture as seen from (x,y) on pit membrane, θ_i=arctan (r_a/d)(rad)

 
• ϕ

  spherical coordinate (rad)

 
• (ξ,ζ)

  coordinate of ideal image point contributing energy to(x,y)

 
• (x,y)

  coordinates of an image point of interest (m)

 
• A

  area (m²)

 
• d

  distance from aperture center to image plane (m)

 
• dω

  element of solid angle (sr)

 
• dA

  element of surface area (m²)

 
• E_i(x,y)

  irradiance at image point (x,y) due to source object (W m^–2)

 
• Gr

  Grashof number (dimensionless)

 
• Q_air

  heat flow through the air to the pit wall by conduction (W m^–2)

 
• Q_pit

  background irradiance from pit walls falling on an image point (W m^–2)

 
• R

  equivalent conductance due to thermal radiation, R=4σϵ_mT_p³ (W m^–2 °C)

 
• r_a

  facial pit aperture radius (m)

 
• S(x–ξ, y–ζ)

  point spread function

 
• T

  membrane temperature at any particular point (°C or K)

 
• \(T_{\mathrm{i}}^{{^\prime}}({\xi},{\zeta})\)

  temperature of the ideal image at point (ξ,ζ)

 
• T₁`

  temperature of the source object point corresponding to T₁ (°C or K)

 
• T_o

  surface temperature of the source object (K)

 
• T_p

  pit wall temperature (°C or K)

 
• w

  effective distance from membrane to the wall of the inner (posterior)chamber (m)

 
• z

  effective distance from membrane to the wall of the outer (anterior)chamber (m)

 
• k

  thermal conductivity of air, k=0.026

 
• T₁

  temperature at point 1 on pit membrane (°C or K)

 
• B

  radiance (W m^–2 sr)

 
• B_i

  radiance of the image (W m^–2 sr)

 
• B_o

  source (W m^–2 sr)

 
• E

  irradiance (W m^–2)

 
• σ

  Stefan-Boltzmann constant, 5.67×10^–8 (W m^–2 K^–4)

We thank Marilyn Banta for the use of her laboratory to take thermal images of the kangaroo rat in her teaching collection, University of Northern Colorado IACAUC protocol 0305 and Colorado Division of Wildlife Scientific Collecting License 04-TR1014. Other thermal images were of semi-tame animals foraging on spilled food in public campgrounds. Snake anatomy studies(Fig. 1) were done by S. Colayori under Indiana State University IACAUC protocol 2-10-2006:GSB/SEC, and were supported in part by a grant from the Indiana Academy of Sciences. NSF Grant 99-70209 provided the thermal imager, and Indiana State University provided general support.