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# Mountain climbing on the Moon

27 Apr 2022
Taken from the April 2022 issue of Physics World, where it appeared under the headline "Walking on the Moon". Members of the Institute of Physics can enjoy the full issue via the Physics World app.

One night in November 1609, Galileo turned his telescope to the Moon and realized that the changing shadows implied it had mountains. I find myself wondering: what would climbing them involve?

For my expedition, I have chosen the central peaks of the spectacular Theophilus crater in the south-east. The summit is around 1800 m above the crater floor and after eons of meteor bombardments the peak has modest gradients so it should be a walk without any rock climbing. Despite a significant altitude change, my initial thought was that with the reduced gravity of the Moon, it would be relatively easy to accomplish the ascent in a reasonable time – but would it?

First, let’s consider terrestrial hill walking. On Earth we have Naismith’s rule (1892) for estimating walking times: 1 hour for 5 km of distance plus 1 hour for every 600 m of ascent. With a summit 1800 m above us and a round trip of 15 km, the rule would give us a journey time of 6 hours (excluding rests and food breaks) if the climb was on Earth.

The gravity on the Moon is one sixth of that on Earth but I would have to wear a heavy space suit with a life-support pack that adds 80 kg, so overall my weight would be about one third of my terrestrial weight. Consequently, I thought that for the same energy expenditure I could travel three times faster on the Moon, giving a speedy ascent of just 2 hours. However that means I would be progressing at an average speed of 7.5 km/h – I’d have already adopted a running gait if on Earth. With lower gravity reducing traction of the foot on the lunar regolith, the resultant “scree-running” down the mountain would most likely cause falls, risking tears in the space suit, which would be fatal. While the Apollo 17 crew experimented with kangaroo hops down a steep slope, the motto would have to be “walk don’t run”.

So what is the fastest I could walk on the Moon? Looking at the data from the Apollo missions (see “Lunar gaits”, Apollo 11 Lunar Surface Journal, November 2010), the astronauts were walking at 2.2 km/h on the flat, although Neil Armstrong achieved 3.6 km/h by adopting a loping gait. The low walking speeds and tendency to lope may have been due to the inflexibility of the suits impeding limb movement. While there are no data on sustaining a reasonable pace over a long distance, Buzz Aldrin thought the peculiar movement of the body within the suit would make distance walking tiring. But let’s assume that nowadays space suits could be more suitably designed for longer walks.

A simple mechanical model of walking is based on an inverse pendulum. One’s centre of mass (m kg) is assumed to rotate at velocity v m/s in an arc of radius r m about the contact point of the foot with the ground. The downward force due to weight is mg (on Earth g = 9.81 m/s2) and this supplies the centripetal force (Fc) to keep the foot in contact with the ground Fc = mv2/r. Hence v2/rg (a dimensionless ratio dubbed the Froude number) has to be less than unity otherwise the foot lifts off the ground and we break into a run. For a 1 m “leg” to the body’s centre of mass we thus have a limiting terrestrial walk velocity of 11 km/h. Now, while Olympic speed walkers can achieve such speeds they do so by adopting a special gait, rotating their hips and taking many short steps. However, with a natural gait empirically most people break into a run at a Froude number of about 0.5 (around 8 km/h). While less than unity, walking faster causes discomfort in the ankle and associated musculature, and though running is initially less favourable energetically, it is more comfortable at these speeds.

In the reduced gravity of the Moon (g = 1.62 m/s2) a Froude number of 0.5 would give a maximum walking speed of only 3 km/h, so applying a modified Naismith’s rule (pessimistically assuming the same distance/ascent ratio applies) our expedition would take 10 hours, which exceeds the extra vehicular activity (EVA) times of the Apollo missions (7 hours). But does the walk-to-run transition occur at the same Froude number on the Moon?

Treadmill tests have been done both with simulated lunar gravity, using overhead suspension, and real lunar gravity, as experienced in 30 s time slots during the parabolic flight phase of a DC-9 (J. Exp. Biol. 217 3200). The researchers got similar results from both tests and found that the volunteers actually only broke into a run at a Froude number of 1.25 ± 0.37 (average of both sets of results), corresponding to a speed of about 5 km/h on the flat. The greater-than-unity Froude number was thought to be due to the down forces generated by swinging arms and the free leg being more significant in reduced gravity than on Earth, thereby making the inverse pendulum model inaccurate.

The Apollo moonwalkers in their inflexible space suits opted to lope rather than walk at speeds lower than that corresponding to the Lunar Froude walk/run transition number. However, if more flexible space suits can be designed, in the reduced gravity on the Moon, a walking speed of the same value as used in the terrestrial Naismith’s rule should be possible. The time to climb Theophilus peak would therefore be only 6 hours, but with rests it would probably exceed the allowed Apollo EVA times and use up the consumables (oxygen, water) and battery life in the Apollo support packs, which was limited to 8 hours. Perhaps modern technology could improve these time-limiting factors to take the climb beyond the realms of speculation when we once again walk on the Moon.