Effects of exoplanetary gravity on human locomotor ability
EEffects of exoplanetary gravity on human locomotor ability
Nikola Poljak, Dora Klindzic and Mateo Kruljac
Department of Physics, Faculty of Science, University of Zagreb, Croatia (Dated: August 23, 2018)
Abstract
At some point in the future, if mankind hopes to settle planets outside the Solar System, it willbe crucial to determine the range of planetary conditions under which human beings could surviveand function. In this article, we apply physical considerations to future exoplanetary biology todetermine the limitations which gravity imposes on several systems governing the human body.Initially, we examine the ultimate limits at which the human skeleton breaks and muscles becomeunable to lift the body from the ground. We also produce a new model for the energetic expendi-ture of walking, by modelling the leg as an inverted pendulum. Both approaches conclude that,with rigorous training, humans could perform normal locomotion at gravity no higher than 4 g Earth .To be published in
The Physics Teacher. a r X i v : . [ phy s i c s . pop - ph ] A ug . INTRODUCTION With the discovery of many new potentially habitable exoplanets, one needs to considerwhich ranges of planets’ physical parameters are suitable for immediate human settlement.Aside from the average temperature, insolation, pressure, atmospheric composition, etc.,all of which can be solved with spacesuits, the basic parameter of a planet is its surfaceconstant of gravity , which will determine if a person can stand upright and move safelyand at a reasonable pace from one place to another.Studies of animal sizes have already determined many physical limits the animals canreach in conditions existing on Earth. In this text, we aim to take another approach, fixingthe size of the animal, in this case a human, and determining the range of gravitationalaccelerations g in which it can stand and move. To do so, we will look at the largest g in which our skeleton still won’t fail and in which our muscles can still perform the basicmovements of standing up and walking. II. METHODS AND RESULTSA. Bone failure
Let us, for the sake of simplicity, imagine the whole human weight being supported by asingle upright bone, in this case representing the entire skeleton. The weight
M g of theentire human mass acts on the bone with a diameter D and a cross section A and producesan axial compression equal to: σ = M gA ∝ M gD ∝ gD , (1)where we used the fact that the cross section of the bone is proportional to the square ofits dimension and the mass of the human to its cube. Since we want to obtain a numericalresult, it is not enough to deal with proportionalities. Measurement data shows that foran average 50 kg mammal, which we can take as a first approximation of a human, the crosssectional area A of a tibia (which we consider to be at least under the same pressure as therest of the skeleton) equals 2 . · − m and the compressive strength σ of the bone is about170MPa, giving the maximum gravity the human can support as:2 max = σAM = 918 m / s . (2)This is indeed a large number, corresponding to more than 90 times the Earth gravity!However, this maximum value needs to be reduced since it considers only static compressivestress on the bone. Once we start moving, the dynamic stress takes over due to bendingof the bones subject to gravitational torques. It has been shown experimentally that thetotal stress increases approximately by a factor of 10 during normal running, thus reducingthe maximum gravity to ≈
10 times the Earth gravity. The same studies suggest that afactor of 10 might be too large, however, as these were conducted on larger animals (suchas cows) they can not be reliably extrapolated to human sized mammals. Further, anincreased compressive force acts on the bones due to a ground reaction force during runningor jumping. Since we accelerate vertically when running, the normal force on the bones ofthe legs is greater than the force due to gravity. However, studies show that this increasesthe stress on the bone by at most a factor of 2, so it doesn’t modify our result. B. Muscle strength
A criterion for muscle strength will be the ability of the human to get up while seated orlying down. A visual representation of the problem is given in Fig 1.
FIG. 1. A representation of the human leg. The human has a mass M . The distances denoted inthe image are discussed in the text.
3e consider the entire mass M of the human to be located at their center of gravity. Thequadriceps connects the massless femur and tibia and is responsible for getting up. In orderto do so, the force with which the quadriceps must pull has to be at least: F ≥ M gL D , (3)in order to at least balance the gravitational torque on mass M . In this expression, D isthe torque arm of the muscle force and L is the torque arm of the gravitational force. Thefactor 2 arises since humans have 2 legs and each needs to lift only half of the total mass.Another way to express the maximum force the quadriceps produces is with the help of themaximum isometric stress σ m muscles can produce: F = σ m A m = σ m M m ρ m L m , (4)where the indices m denote the muscle and A is the muscle cross section, expressed in thesecond equation with the help of muscle mass M , its density ρ and and its length L . Onceagain, the values for these parameters are known for a mammal of 50 kg . Finally, thetorque arm L is longer than the muscle length and approximated by L = 1 . L f , where L f is the femur length. Plugging in all the numerical values, we obtain the limiting value of g : g max = 2 σ m M m D M ρ m L m L f = 10 . / s . (5)Note that this result is reproduced for a “worst-case scenario” - it is very difficult to getup from a sitting position based only on muscle strength without bending forward or pushingyour feet slightly backwards. Still, we think this is an extraordinary result, showing how wellour muscle system is adapted to life on Earth! It would appear that living on planets withincreased gravity would be very difficult due to our relatively weak muscles. However, wecould either try to increase our muscle strength or move around with the help of technology.Evidence from Earth suggests that increase in muscle strength is possible within some limits.Looking at isometric squat standards for 50 kg men , one can see that an average personcan lift about 36 kg, while an elite athlete can lift 145 kg, which is an increase by a factor of ≈
4. Of course, we also use our arms and other muscles when getting up, which we have tofactor in and choose as an extra 20-50%. We choose to take the lower limit to this estimatesince we believe that an increase of factor of 4 that we included earlier could be a slight4verestimate since not everyone can become an elite athlete. Thus, one could assume thatwith rigorous training, we could get up at gravity values of ≈ . · · . / s ≈ g Earth .One could argue that we could have used the world squat record for a 50 kg man, butthat’s a result only one person can achieve, while there are quite a few elite athletes. Thelatter is far more useful information if we wish to colonize a planet. Since the quoted strengthincrease factor of ≈ , 5 g Earth should be the maximumgravity for them as well.We can see that increased gravity induces stress on the muscular system much more sothan on the skeletal system. This was to be expected, since we all know it is relativelyeasier to get strong than to break a bone. The limit on the surface gravitational constanttherefore arises mostly from the ability to get up from the floor using your muscles. However,assuming that vehicle-assisted transportation is unacceptable for long-term settlement onplanets, we must also examine the energetic expenditure of walking.
C. Locomotion
Walking can be accurately represented by the ’ inverted pendulum gait ’, as in figure 2.The leg that supports the weight is stiff and behaves like an inverted pendulum, with thebody’s center of mass on top. The free leg swings forward as a free pendulum, although inreality, it’s not a completely conservative system - we know we have to use energy to swingit. In an ideal scenario, though, once an organism started walking in such a stance, it wouldnot require any energy to maintain the walk. Therefore, walking in the inverted pendulumgait is the most energy-efficient means of limb-assisted locomotion, and since it appears inevery land animal on Earth, we can assume that it will be present in extraterrestrial life,too.Now, we may consider that most of the work is spent on making the center of mass (CM)’bob’ up and down . In that case, the CM repeatedly oscillates between maximum kineticenergy, at the bottom of the movement arch, and maximum gravitational potential energyat the top. The ratio of those two energies is proportional to F = v gL , (6)where v is the traveling speed, g is the surface gravitational constant and L is the limb5 IG. 2. Whilst walking, one leg behaves as an inverted pendulum, and the other as a swingingpendulum. length. The number F is called the Froude number , and studies have conclusively foundthat animals with nearly the same F exhibit the same gait . That is, the ratio of kineticand potential energy of the CM determines if an animal will walk, run, trot or gallop at acertain speed, regardless of species.How does this apply to the walk of bipedals, quadrupedals, or other creatures with aneven number of legs? At the high point of the CM ’bobbing’ at each pair of legs, there is acentrifugal force lifting it upwards, and at the same time, a gravitational force downwards.If the centrifugal force exceeds it, it will become impossible to keep the foot on the ground.The condition for this is: mv L < mg, or F < . (7)When the Froude number exceeds 1, walking becomes impossible. In reality, nature neverpushes physics to the limits, so humans and animals consistently adopt a trot or run at6round F = 0.5. A study has shown that the transition from walking to running occurs atF ≈ θ (figure 3) is equal to the difference ofgravitational potential energies of the feet, W s = mgL (1 − cos θ ) . (8)To avoid dealing with angles and make a connection with the Froude number, we will usethe step length s as a parameter, so we identify sin θ = s/L , and s = v t s , where t s is thetime required for one step. Since the most energy-efficient way for the swinging leg to moveis like a stiff pendulum, we can take the step time to be one quarter of the period of such apendulum.Using equation (6) and the natural period of the human leg, 4 t s = T = 2 π (cid:112) L/ g , weget: W s = mgL (cid:32) − (cid:114) − π F (cid:33) . (9)Let’s put this to the test. According to medical information used by the Wolfram Alphacomputational engine, a woman of 65 kilograms and 20 years of age, briskly walking at 1.4m/s, makes 100 steps per minute and spends 15 kJ of energy. That is roughly 150 Joulesper step. Her Froude number should be around 0.3. By plugging all this into equation (9),we can estimate her leg size as 0.8 meters. This turns out to agree with the average leglength for the female population of this age , as well as the popular ’45% of total height’estimate.Notice that for equation (9) to hold, the expression under the square root has to bepositive, i.e. F ≤ .
6. As mentioned before, it’s been shown that the transition from walkingto running occurs at F ≈ . IG. 3. A sketch of our pendulum model. The legs have length L and are separated by an angle θ before each step. back . We used the world record because there are no official elite athlete results for logcarrying, as we’ve found for squatting. Regardless, it is reasonable to assume this resultis close enough to an elite athlete so that any error in setting him as our benchmark isprobably negligible. We’ll use equation (9) to find the gravity at which the strongmans“free” walking is the same as walking with that log here on Earth. Now, he was walkingvery slowly, meaning v (cid:28)
1, i.e. F (cid:28)
1. We will assume that Froude numbers on bothplanets are small and equal and use the Taylor expansion √ − x ≈ − x in equation (9),which then reduces to: ( M man + M log ) g Earth = M man g max (10)which, for the strongmans mass of 179 kg, gives g max ≈ . g Earth .Comparing this result to the former result in subsection II B, we find the models give avery similar result for the maximum bearable gravity.8
II. DISCUSSION
Even though we found some limits on the maximal surface gravitational constant onacceptable planets, there are other considerations that have to be taken into account. Forexample, since increased gravity would be strongly pulling our blood down to our legs, theheart would have to work harder to pump it up into the brain. On the other hand, once we’resubject to low gravity, blood rushes from our legs into the face. To adapt, the heart wouldneed to work differently and regulate the blood pressure. Following this line of reasoning,we can conclude that subject to high gravity, blood goes from the chest region into the legs,resulting in larger blood volume and higher blood pressure. Because blood cells are moreeasily destroyed than created, the body’s cardiovascular system should adapt sooner to lowthan to high gravity. Until enough blood cells are created in high gravity, we could feelweak, like after donating blood, in addition to having trouble standing and walking.We also know there are health risks due to low and high blood pressure, meaninghigh-g planets could cause damage to the heart, arteries, kidneys and the brain as well asleave us with dizziness, nausea and fatigue. Studies have shown that the average humanbody could not withstand gravity greater than 5 g E without passing out , because the heartcouldn’t pump enough blood into the head, so we’ll assume that 4 g E is the maximum gravitythe human body can withstand in the long run. Other health risks in a high-g environmentcould be as simple as falling down, since our weight would be substantially increased, butwe don’t consider those in detail here. In conclusion, all of the systems governing the humanbody we examined give the maximum gravity of about 5 g E . This should only be consideredas an upper limit which could only be achieved by a handful of astronauts. It is difficult toclaim what the exact upper limit for an average person could be, but we can estimate thata number above 3.5 g E would be reasonable.Now that we have found 5 g E as the upper limit of the surface gravitational constant onacceptable planets, we can look at real data and find out how many exoplanets satisfy ourcondition. Out of 3605 confirmed exoplanets (as of January 2018), 594 have known radiiand masses, which are needed to determine g . A short calculation shows that 469 of thesefit our criterion. A chart of the distribution of g is given on figure 4. We can easily noticethe peak in the percentage of planets in the gravitational range from 0.5 g E to 1.5 g E , whichrepresent the planets we could relatively easily adjust to.9 IG. 4. Distribution of the gravitational constant g among confirmed exoplanets . The valuesdisplayed are rounded-off percentages of the total number of sub-5 g E exoplanets made up byexoplanets in a given range of surface gravity. It should be noted that planets with gravity less than 5 g E are in fact very low- g planetswhen compared to 125 planets discovered with gravity stronger than 50 g E (some of themeven go as high as 200 g E ). This shows that we can classify humans as low- g organismson the galactic scale and predict most exoplanetary life we may encounter to far outweighand outgrow us . Fortunately, the majority of discovered exoplanets fit in the low sub-5 g E group, which means a greater chance of colonization and, perhaps, meeting alien life formswhich are in some ways similar to us. IV. CONCLUSION