# Mass estimates

`Hi All -`

`    Andy wrote:`

Something which bugs the heck out of me is the absence of information on internal architecture in dinosaur bones (although I seem to recall that Eric Snively and others >addressed this in some of their papers--please correct me if I'm wrong). Yes, ceratopsid bones are mostly solid, but they still have a less-dense area in the middle. And >most theropod bones certainly aren't solid all the way through. I think it would be *extremely* illuminating for someone to rigorously work up the cross-sectional properties >of dinosaur bones (along with other large critters). I suspect that they're probably playing by the same rules as large mammals, but who knows until someone takes a look at >it. . .that is, perhaps a ceratopsid just *looks* more robust than a comparably sized elephant, but these distinctions disappear when one starts evaluating the distribution of >bone mass across the cross-section. Perhaps this explains some of the discrepancies in Alexander's equation, too.

Well, last I looked, the usual (Alexandrian) formula for calculating the mass of an extinct animal (or anything else) was W=dv, where W = weight, d = density, and v = volume. The volume, of course, is what is measured using Archimedes' Principle of water displacement (plus the scaling factor between the model and the actual animal), and of course this is going to vary substantially depending on the model used (e.g., is the model the Ely Kish-style, grossly emaciated dinosaur, or is it the traditional, kids'-toy-style lumbering, overfed behemoth?). So that introduces a great deal of variability. Getting the value for d correct, though, is an even bigger problem, because it depends almost solely on using mean density values for modern animals. I haven't been able to track down a lot of specific values, but it appears that typical "reptiles" (i.e., lizards) have a mean density ~1 g/cm^3 (about the same as water). Crocs I _think_ (!) have an average density that is slightly lower, around 0.9-0.95 g/cm^3. Birds have average densities that range from 0.73-0.94 g/cm^3 (I don't know what the distribution of those densities is, phylogenetically, but I am assuming that the higher values are for those taxa that have secondarily lost much of their pneumatization, such as ratites, loons, penguins, etc., while absurdly air-filled taxa like pelicans are on the low end). I have had a hard time finding a mean value for mammals; I've seen it reported as low as 0.9 g/cm^3 and as high as 1 g/cm^3. So if you assume, as did most early estimates, that dinosaurs have mean densities like lizards or mammals, you could plug in a mean density of 1 g/cm^3. If we pretend that we have a nice, average-bulk scale model of _Tyrannosaurus_ that, via displacement measurements and subsequent calculations, produces a life-volume value for the animal of 6.3 * 10^6 cm^3, and you use the lizard value of 1 g/cm^3 for the density, you get a weight out of 6.3 metric tons. If you plug in a low bird value of 0.73 g/cm^3 instead, assuming that _Tyrannosaurus_ had vast degrees of pneumatization, you get a weight out of about 4.6 metric tons, almost 2 tons less! I think it was Matt Wedel who estimated more derived saurischians as having a realistic density value around 0.8 g/cm^3 (that would, in this case, produce a weight of just over 5 metric tons); how realistic that is, I couldn't say.

But Andy is correct: it isn't just the volume estimate that is the problem; the density values also need to be refined. I too have contemplated trying to find some way to quantify degrees of permineralization and/or replacement in fossils so that they could be weighed as fossils, then that value could be plugged into a nice formula to estimate its original weight...but it's the living muscle mass that is going to comprise the bulk of the animal's overall mass, and measuring all the bones in the world isn't going to fill in that gap. Within the ranges of whatever maximum stresses they can assume, it's conceivable that dinosaurs had light bones and a bird-like set of streamlined, mostly skinny muscles; or, they could have had big, heavy, mammal-like bones and thick, massive muscles. Or in between -- light bones with thicker muscles, heavy bones with skinny muscles, etc. All those assumptions go in to both the volumetric (model-building and dunking) estimates _and_ the mean density estimates. In the end, until we build time machines, all weight values will be dependent on the assumptions of the calculator.

```~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Jerry D. Harris
Director of Paleontology
Dixie State College
Science Building
225 South 700 East
St. George, UT  84770
Phone: (435) 652-7758
Fax: (435) 656-4022
E-mail: jharris@dixie.edu
and     dinogami@hotmail.com
http://cactus.dixie.edu/jharris/```

```An expert is a man who has made all
the mistakes that can be made in a very
narrow field. -- Niels Bohr```

After one look at this planet any visitor
from outer space would say "I want to
see the manager." -- William Burroughs