Equations of motion for post-mortem sinking of cephalopod shells and the sinking of Nautilus

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Abstract
Consideration of the physics of sinking of hollow, rigid bodies leads to equations of motion for sinking cephalopod shells. We have derived equations of motion for three post-mortem sinking situations: sinking with a fixed amount of water in the phragmocone; rapid phragmocone filling (no siphuncular tube); and slow phragmocone filling (siphuncular tube intact). In all three cases sinking speed can be closely approximated by the terminal velocity calculated from the total weight, buoyancy, and drag parameters of the shell.Experiments on modern Nautilus shells yield sinking velocities in agreement with calculated values. The experiments also show that orientation of a sinking Nautilus shell varies as the phragmocone fills with water. With small negative buoyancy the shell sinks with its plane of symmetry upright, but as it fills, it begins to rock from side to side and leans over and sinks with its plane of symmetry horizontal when the camerae are about 55% full.The maximum sinking speed of upright adult Nautilus shells is approximately 30 cm/sec, which appears to be too small for embedding in the bottom upon impact. The maximum depth to which Nautilus sinks in the upright position ranges from about 7 m for rapidly filling shells to as much as 600 m for slowly filling shells. In the latter case, the shell will continue to fill after coming to rest on the bottom, and the stability of the vertical orientation will be removed within 1 or 2 days. Thus, primary vertical preservation of cephalopod shells indicates water depths less than about 10 m.