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Cope's rule: did dinosaurs get bigger?
How big can a land animal get? (bone stress)
Scale the selected animal up like a photocopier. If every length grows by s, mass grows as s3 but bone cross-section only as s2, so stress grows as s.
Scale chart (length / hip height)
Body length and hip height of the nine species, to one common scale (a 1.75 m human for reference).
Read a trackway
Alexander (1976): v = 0.25 g0.5 λ1.67 h-1.17. Walking if λ/h < 2, trotting 2-2.9, running above 2.9.
Speed curve
Experiment: how fast was the trackmaker?
A mystery trackway has appeared. Estimate the speed by eye, then reveal what the formula says and which animal made it.
Your guess (m/s)
Who could outrun you?
Same gait for everyone (λ/h fixed), so only leg length matters. A fit person jogs at ~3 m/s; elite sprinters reach about 10 m/s for a few seconds.
The science
Family tree and time
Dinosaurs split into two great groups: Saurischia (theropods such as Tyrannosaurus and Velociraptor, and the long-necked sauropodomorphs) and Ornithischia (Stegosaurus, Triceratops and their kin). Birds are living theropods. First-appearance dates in the tree are approximate - the fossil record only samples a tiny fraction of past life, so real lineages are older than their oldest fossil (ghost lineages).
Weighing a fossil
M[g] = 0.0787 C^2.749 (bipeds) M[g] = 0.078 C^2.73 (quadrupeds)
C is the sum of humerus and femur shaft circumferences in millimetres (Anderson, Hall-Martin & Russell 1985; Campione et al. 2014). Bone strength scales with cross-section, so circumference tracks the load the skeleton carried. Typical scatter is ±20-30 %. The circumferences used here are rounded illustrative values consistent with published specimens, not measurements of one particular fossil.
Cope's rule and gigantism
Cope's rule says lineages tend to grow larger through time. The chart fits log10M against age for the nine species; with so few points (and species chosen for being famous) the fit shows the idea, not a statistical proof. Bone stress is estimated as
σ = kb Fpeak / (A · EMA), A = πro²(1 - 0.55²), ro = Cfemur/2π, EMA = min(1, 0.14 M0.26)
with kb = 8 (bending dominates the load), Fpeak = 1.2 body weights for walking and 2.5 for running (carried on one femur for bipeds, 30 % of weight for quadrupeds), and compressive bone strength 150 MPa. Large animals straighten their legs (EMA → 1, Biewener 1989), which is why real stress stays roughly constant with size. Pure isometric scaling gives σ ∝ s, so the safety factor falls as 1/s and reaches 1 at s = SF0 - an unrealistic extreme: real limits also include heat loss, food supply, joint cartilage and bird-like lungs and hollow bones in sauropods.
Speed from footprints
v = 0.25 g^0.5 λ^1.67 h^-1.17
R. McNeill Alexander (1976) used dynamic similarity: animals of different size move alike when their Froude number v²/(gh) and stride ratio λ/h match, which gives the formula above (SI units; λ = distance between two successive prints of the same foot, h = hip height, often estimated as 4 x footprint length). It reproduces human and elephant gaits to about 10-20 %; for dinosaurs the errors are bigger because hip height must be guessed from the foot and substrate, gait and slip are unknown, and some track makers were stepping in mud.
The impact
E = ½ m v², m = ρ (4/3) π (D/2)³
For a 10 km stony body at 20 km/s this is 2.8 x 1023 J, about 6.8 x 107 megatons of TNT or four billion Hiroshima bombs. Crater size uses the Collins, Melosh & Marcus (2005) scaling Dtc = 1.161 (ρi/ρt)1/3 L0.78 v0.44 g-0.22 sin1/3θ followed by collapse to a complex crater. The scaling gives roughly 115 km, while the real structure is about 180-200 km across - these formulas are rough (typically good to a factor 1.5-2). The seismic magnitude uses the Gutenberg-Richter relation with 10-4 seismic efficiency, an order-of-magnitude estimate.
Darkness, cold and food
Dust, soot and sulphate aerosols block sunlight: transmission is exp(-τ) with τ(t) = τ0 Σ fi e-t/Ti (three components, e-folding times 0.3, 1.5 and 2.5 years). The start value τ0 = 5 (E/Enom)1/2 is a heuristic tuned to published “years of darkness” studies, not derived from first principles. Temperature follows a two-box energy balance integrated with RK4:
C dT/dt = (S₀/4)(1-α) e^(-τ) - εσT⁴ + k (Tother - T)
(land: fast, C = 107 J m-2 K-1; ocean: slow, 3 x 109). Plants grow with a saturating light response and stop below about 5 °C; dead plants feed a detritus pool that decays over ~6 years. The survival model gives each lineage an energy reserve (∝ M1/4 days, doubled by torpor), food from plants, prey or detritus, extra cold costs reduced by feathers/fur and shelter, and mortality when reserves run out. A lineage counts as surviving if its population stays above 1000 breeding adults, using Damuth's density law N ∝ M-3/4.
What is simplified - honest limits
- The extinction model is a toy: it shows why big, plant-dependent, unsheltered animals are most exposed while small, insulated, detritus-eating or burrowing animals are best placed. It cannot prove why a particular species died.
- The known outcome is that no non-avian dinosaur survived. In this model a small feathered predator can sneak through the bottleneck at some settings; that mismatch is a reminder of what the model leaves out (nest-bound reproduction, slow maturity, local refuges and chance).
- The heat pulse from re-entering ejecta is debated; the “severity” slider lets you switch it off.
- 3D models are plausible restorations: body outlines come from skeletons, but skin colour, feathers on Velociraptor-like and Tyrannosaurus-like animals, and soft tissue are partly invented. Titles say “-like” for that reason. Not affiliated with any film or toy brand.
- Footprint prints and animations obey the formulas (stride = distance per gait cycle), but the real leg motion of extinct animals is unknown.