Gravitas Dei

Over on the Zompist boards they have a thread going on the effects of higher gravity. A comment was made to the effect that people growing up and living in such an environment would be shorter than people born on Earth and that the reduction in height would be proportional to the gravity. Thus was I reminded of a science fact article from the mid-december 1988 edition of Analog. The article was On Beanpoles and Drum-Men, by Martyn J. Fogg, I highly recommend grabbing it if you get the opportunity.

Now the problem isn’t the notion that human growth won’t be affected by gravity. It probably would be. The problem was in the linear relationship between gravity and length. To quote Torco of the ZBB, “Gravity isn’t the only factor.”

The first problem with the linear-relation hypothesis(heh) can be dealt with quickly. Let’s say we have four basically identical people. One is raised here on Earth. He’ll be, let’s say, six feet tall. Another one is raised on Mars under 0.376 gees, will, by this conjecture, reach a height of almost sixteen feet tall. The Martian child will seem like a virtual dwarf next to his twin raised under 0.1654 gees on the Moon. The lunar child will, if we accept the linear-relation hypothesis, be over 36 feet tall! So what of that fourth child? Aye, therein lies the rub. We’ll raise him at on a manned orbital satellite under zero gravity conditions. How tall is he? Divide by zero. Infinite at best. or just plain undefined. If nothing else restricts the growth of our free-fall baby, Einstein’s restriction to the speed of light should slow him down a bit. It didn’t take much reductio to render this hypothesis absurdum.

Of course there are a lot of factors slowing down the growth of a human well before we reach the impossibility of our Infinite Man. Gravity, to start with isn’t the only source of physical stresses. Even in zero gee a person has to overcome some degree of resistance due to inertia in order to move. There’s also the resistance of blood vessels to the passage of blood; the heart has to be strong enough to pump blood through the ridiculously extended circulatory system of a 36 foot tall person. That’s not to mention supporting the pressures required to get the blood out of the heart and running through all of that plumbing; this is getting back into the issue of inertia and momentum again.

Mr. Fogg goes on to postulate a more realistic scaling. He starts with a dimensional analysis based on a basic assumption that bodies will grow just thick enough to support their own weight:

modulus of elasticity×diameter2 = gravity×density×height3

Ed2 = gρl3

As Mr. Fogg notes E and ρ are physical properties of human tissues and essentially constant. On Earth g is a constant as well, so we have a proportionality between the square of the diameter and the cube of the length. This is one of several ways that the square-cube law is derived.

The same rule applies to the production and elimination of heat. It can be assumed first, that heat is produced in an organism proportionally to mass and radiated away proportionally to surface area. So, in order to maintain an equilibrium between heat production and elimination, surface area(d2) must be proportional to the volume of the organism(l3). For future reference in this article, we can refer to this as the square-cube law for heat. As you can already see, the square-cube law is an important rule in the estimation of the size and form of organisms. You can also see that it is not always dependent on gravity. We should keep this in mind.

Back to Martyn Fogg’s thought experiment. He starts by treating gravity as a variable, leading to the formula, g∝d2l-3. Because there is no clear way to separate d and l, he choses exponents for d and l such that diameter squared times length cubed comes to gravity to the first power. Somewhat arbitrarily, he choses l∝g-1/6 and, from that, d∝g1/4.

Based on this scaling relationship, he obtained several other parameters.

Length:     l∝g-1/6
Width: d∝g1/4
Mass: m∝g1/3
Weight: w∝g4/3
Stress: P∝g5/6
Muscular strength: S∝1/3

I have found these to be very useful rules of thumb for determining the appearance of people born and raised on other planets.

Figure 1 shows us his estimation of what people might look like under other gravitational accelerations.

Beanpoles and Drum-Men

©Martyn J. Fogg, used with permission

Mr. Fogg, himself notes that this relationship will obviously break down somewhere before gravity is equal to zero. My own gut-feeling is that people actually raised in a lunar environment will be only marginally taller and thinner than the image shown for a person raised at 0.4 gee. The tall, thin guy on the left might be what a freefall child will look like. While there may be absurd results at the upper end, they shouldn’t matter. I find it unlikely that humans (without genetic changes of some sort) will be able to function at much above 2 gees. The Helium-3 refineries in Jupiter’s atmosphere, slung under fusion-powered hot air balloons would definitely be a very short-term hardship posting. It seems unlikely that actual human-habitable planets would be large enough to generate much more than 1.5 gees in any case.

Actual growth rates and cut-off points will be governed by a lot more than just gravity. There may be a genetically intrinsic cut-off based on the number of generations in cell growth. Certainly growth will be governed by the relationship between radiative surface area and heat-producing volume. The Lunie in the image above will probably be very sensitive to chills. Actual appearance would be a compromise between gravity, temperature, food supply, genetics and probably other factors as well.

Speaking of food-supply, I suspect people raised under lower gravity conditions will likely have problems with obesity and those under higher gravity will tend to be quite lean and stringy. Even at rest muscular tissues use more calories than fatty tissues and people under lower gravity will tend to use less energy in movement. The reverse is true for higher gravity people. Given the prevalence of obesity here on Earth today, I believe that the human digestive system will readily adaptable to higher needs under gravity, but I suspect people in lower gravities will tend towards flabbiness.

I suspect that actual response to variation in gravity alone would be something of the form:

<attribute> = <zero-g value> + <constant>×<gravity><exponent>

At the moment, I feel neither the intellectual capacity nor the motivation to derive reasonable values for these numbers. Until such time as I get up the gumption to fire up Mathematica and go at it, this will have to remain an exercise for the motivated reader. Drop me a line if you work it out…

Thank you for your attention,

The Astrographer

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