RESULTS AND DISCUSSION

Although living ammonoids are considered to have been only very slightly negatively buoyant (Westermann 1996 and references therein), an accurate model of an immature ammonite should be positively buoyant in order to allow for the presence of cameral liquid in the youngest chambers of the phragmocone. As discussed in Ward (1987), as each new chamber is added to the phragmocone of Nautilus, it is initially filled with cameral liquid, which remains until the newly secreted septum is strong enough to withstand pressure at depth. Once the septum reaches about one half of its ultimate thickness, emptying of the chamber begins. In addition to this, the penultimate chamber still contains some liquid, which can be on the order of about one third of the chamber volume. In the current paper, models 2 and 3 both represent immature ammonites and thus, significant cameral liquid was probably present in the last two phragmocone chambers. However, when considering the total mass of the models relative to the mass of the displaced water, model 2 (DMAX = 300 mm) is 1.02 % negatively buoyant if cameral membranes are not included and 2.26 % negatively buoyant if cameral membranes are included (Table 4) whereas model 3 (DMAX = 54 mm) is 1.10 % positively buoyant if cameral membranes are not considered and 0.11 % negatively buoyant if cameral membranes are considered (Table 6). In both cases, the models are too heavy. This discrepancy may be due to inaccuracies in the densities used for individual components, diagenetic alteration of the original material that affected the measurements or human error. Alternatively, as in most studies of ammonoid hydrostatics, we considered the body chamber to be completely full of soft tissue, which may not be accurate (Kröger 2002). Finally, the indirect method used to ascertain the mature body chamber length as discussed above may have led to an overestimate in length. Juvenile Ammonitina generally have a longer body chamber than adult shells (Westermann 1971; Lehmann 1981) whereas in our models, the body chamber length was held constant at 259 degrees. Although model 2 does not represent a mature specimen, it does represent a specimen that is much larger than is represented in model 3, and it is possible that it would have had a shorter body chamber. This may explain the increased error in model 2 compared to model 3. Whatever the cause, these discrepancies in mass are not very large and do not have a significant effect on the results of our study.

Model 1 suggests a counterbalance mechanism may have been present, which allowed the animal to remain upright with its plane of bilateral symmetry vertical in the water column. When only the siphuncle and septum are considered, the moment created by the septum offsets about 295 % of the moment created by the siphuncle (Table 2). In this case, the model leans away from the direction of the siphuncle. When the optional cameral sheets are included in the model, the moment created by the septum offsets about 36 % of the moment created by the siphuncle and cameral sheets (Table 2). In this case, the model leans toward the siphuncle. In both cases, although the correction is not perfect, it does suggest a counterbalance may have been in effect.

In the large model of the complete ammonite (model 2), each component of the animal has a very different centre of mass (Table 3; Figure 11). However, the masses of the asymmetric components in the ammonite phragmocone are very low when compared to the mass of the entire animal. When cameral sheets are not considered, the siphuncle is less than 0.1 %, and the septa are about 1.4 % of the total mass of the animal (Table 3-Table 4). When cameral sheets are considered, the siphuncle is less than 0.1 %, the septa are about 1.4 % and the cameral sheets are approximately 1.2 % of the total mass of the animal (Table 3-Table 4). The offset of the centre of mass in the Z-plane (indicated by the final Z-coordinates in Table 4) is very small (about -0.01 mm if cameral membranes are not considered and approximately 0.02 mm if cameral membranes are considered). The size of these numbers is partially due to the correction discussed in model 1 where the position of the centroid of the siphuncle (and the centroid of the optional cameral sheets) is counterbalanced by the position of the centroid of the septa in the Z-plane (Table 2). Nevertheless, in terms of the whole model, removing the offset mass of the septa only changes the flotation angle by approximately 0.20 degrees. This figure is within our range of error and demonstrates that the effects of the asymmetries in the components of the phragmocone on flotation angle are virtually negligible at large shell diameters.

Although each component of the animal has a very different centre of mass in the small ammonite model (model 3; Table 5), the masses of the asymmetric components in the ammonite phragmocone are very low when compared to the mass of the entire animal, similar to the situation in the large model. When cameral sheets are not considered, the siphuncle is just over 0.1 %, and the septa are slightly over 0.9 % of the total mass of the animal (Table 5-Table 6). When cameral sheets are considered, the siphuncle is just over 0.1 %, the septa are just over 0.9 % and the cameral sheets are approximately 1.2 % of the total mass of the animal (Table 5-Table 6). The final Z-coordinates are very small at virtually 0 mm if cameral membranes are not considered and about 0.01 mm if cameral membranes are considered (Table 6). Once again, the size of these numbers is partially due to the correction discussed in model 1 where the position of the centroid of the siphuncle (and the centroid of the optional cameral sheets) is counterbalanced by the position of the centroids of the septa (Table 2). However, in terms of the whole model, removing the offset mass of the septa only changes the flotation angle by approximately 0.14 degrees. Similar to the large model, this figure is within our range of error and demonstrates that the affects of the asymmetries in the components of the phragmocone on flotation angle are virtually negligible at small shell diameters.

One potential issue that is not considered in models 2 and 3 is the possibility of asymmetries in the soft tissues of the body chamber. Hengsbach (e.g., 1979 and references therein) suggested that asymmetries in the suture line may be a consequence of asymmetries of the internal organs within the body chamber. If this were the case, it is possible that heavier soft body parts may have influenced the hydrostatics of the animal. Currently it is not possible to consider this directly in the model, as the specifics of the structure of the soft body of B. columbiae are unknown. However, it is possible to use an indirect approach to look for asymmetries. Checa et al. (2002) demonstrated that in cases where an epizoan infested one side of an ammonite, the whorl would shift in subsequent growth, to 'correct' for the permanently tilted ammonite and allow the aperture to grow upwards within the vertical plane. Over time, this gave rise to trochospiral coiling of the shell. Checa et al. (2002) suggested that ammonites contain morphogenetic instructions that constrain growth direction to the vertical plane. By analogy, if the mass of the soft body of the modeled B. columbiae were unevenly distributed in the living chamber, it would be expected that the shell would show some degree of trochospiral coiling. However, we examined over 500 complete or nearly complete specimens of Badouxia, which consistently showed asymmetry in their siphuncle and septa. No appreciable deviation from planispiral coiling was evident (Longridge et al. 2006). This suggests that asymmetries within the body chamber of B. columbiae were not a significant factor.

Another consideration is how sensitive the models are to small changes of mass in the individual components. To test the sensitivity, we altered model 2 (including cameral sheets) in two different ways. Firstly, to consider the affects of a change in density, we altered the density of the septa and shell from 2.67 g/cm³ to 2.62 g/cm³ (the density of Nautilus shell as given in Reyment 1958 and Saunders and Shapiro 1986). This change altered the masses of the septa and shell by 0.75 g and 10.97 g, respectively, and had only a very small effect on the centre of mass of the animal, with a change of 0.06 mm in the X-plane, 0.04 mm in the Y-plane and no change in the Z-plane. Secondly, to consider the affects of a change in volume, we altered the length of the body chamber to 285 degrees. This alteration changed the volume of the body chamber by 67601.80 mm³, increasing the mass of the animal by 71.32 g. This change altered the position of the centre of mass of the animal in the X-plane by 0.34 mm, in the Y-plane by 2.7 mm and had no effect in the Z-plane. Based on these analyses, the model seems reasonably robust to minor changes in the densities and volumes of individual components, particularly in the Z-plane.