RE: Mass estimates

[Andy Farke <andyfarke@hotmail.com>]

> From: "Jerry D. Harris" <jharris@dixie.edu>
> Reply-To: "Jerry D. Harris" <jharris@dixie.edu>
> To: DINOSAUR Mailing List <dinosaur@usc.edu>
> Subject: Mass estimates
> Date: Fri, 22 Apr 2005 08:56:50 -0600
> 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.

To follow up on the talk of assumptions in mass estimates. . .an assumption 
which gets very, very little attention is the scale accuracy of the model 
itself. Just for fun, I did some calculations based on a Triceratops model I 
have at home here (I'm not sure what company did it or anything--I found it 
for cheap in the AMNH gift shop a few years ago). At any rate, the model is 
a reasonable restoration (not perfect) of Triceratops. I did a quick and 
dirty volume measurement, and calculated a model volume of about 360 mL.

No scale was listed on the model, so I measured the skull and calculated the 
scale relative to known skull lengths for various Triceratops. I then made 
the somewhat hazardous assumption that Triceratops scaled geometrically. 
Assuming a skull length of around 2.4 m (a scale of about 1:25.6), I 
calculated an estimated live mass (assuming a density of 1 g/mL) of 5.87 
tonnes. Assuming a skull length of about 3 m (a scale of about 1:32), I 
estimated a mass of 11.4 tonnes! Assuming a skull length of about 1.8 m (a 
scale of about 1:19.2), I estimated a mass of 2.48 tonnes. Within these 
biologically reasonable values of skull length, I get a range of masses 
between 2.48 and 11.4 tonnes--a spread of 8.92 tonnes! Of course, some of 
this could be due to body proportion errors in the model I used, but a 
similar spread would be seen, regardless.

One might say: "Andy is just playing around with drastically different 
scales to make a point." So what if the errors in scale are less drastic? At 
a scale of 1:25, I calculate a mass of 5.47 tonnes. At a scale of 1:26, I 
calculate a mass of 6.15 tonnes. This is a difference of 0.68 tonnes (680 
kg, or 1500 pounds)! What would this difference of scale look like on a 
physical model? A Triceratops measuring 7.6 m in length during life would, 
at a scale of 1:25, measure 304 mm in length. At a scale of 1:26, it would 
measure 292.3 mm in length. If it had a skull measuring 2.43 m in length, 
the model skull length would be sculpted at 101.3 mm and 97.2 mm, 
respectively. That ain't much.

I am not accusing anyone of having badly scaled models. If done carefully, I 
trust that a model could get pretty darned close. But. . .I think this is a 
problem which isn't given enough attention in the literature. Even at a 
relatively decent scale of 1:10, even the smallest problems in proportion 
(whether it's a rib cage just a little too bulky, or a tail just a little 
too small) get cubed to frighteningly large proportions! ;-)

So four main comments I might add here, on top of what Jerry and others have 
said:

1) Yes, mass estimates based on bone circumferences have a wide margin of 
error. But, mass estimates based on even the best models could have a pretty 
big margin of error, too. This will be compounded as one sculpts bigger 
animals at smaller proportions (i.e., it could be much worse for a sauropod 
at 1:50 than a little ornithopod at 1:10).

2) Even the best models are typically based on a single specimen, and do not 
account for the biological range of sizes that would have been present in a 
single population. One could go out on a limb, assume isometry, and 
calculate this.

3) Computer modeling techniques are not immune to these problems. Garbage 
in, garbage out.

4) None of the volumes and mass estimates that I've listed above should be 
taken as anything close to my informed scientific opinion. They're strictly 
for illustrative purposes.

Andy