# On the moon there is mountaineering

Galileo turned his telescope to the Moon one night in November 1609 and noticed that the changing shadows showed that there were mountains. I’m curious what it would entail to climb them.

I chose the central peaks of the beautiful Theophilus crater to the southeast for my adventure. The summit is about 1800 meters above the crater floor, and the summit has minor inclines due to meteor bombardment ages, so it should be a hike rather than a climb. Despite the considerable altitude difference, I assumed that due to the Moon’s reduced gravity, it would be fairly straightforward to complete the ascent in a reasonable amount of time – but would it be?

Let’s first consider overland hiking. On Earth, we have Naismith’s rule (1892) for estimating walking times: 1 hour for 5 km of distance plus 1 hour for 600 m of elevation gain. With a summit 1800m above us and a 15km round trip, the rule would give us a travel time of 6 hours (excluding rest and meal breaks) if the ascent was on Earth.

Gravity on the Moon is one-sixth that on Earth, but I would have to wear a heavy spacesuit with a survival bag that adds 80kg, so overall my weight would be about one-third of my Earth weight. Therefore, I thought that for the same energy expenditure, I could travel three times faster on the Moon, a rapid ascent of only 2 hours. However, this means that I would progress at an average speed of 7.5 km/h – I would have already adopted a running gait if I were on Earth. With lower gravity reducing foot traction on the lunar regolith, the resulting “scree run” up the mountain would most likely cause falls, possibly tearing the spacesuit, which would be fatal. As the Apollo 17 crew experimented with kangaroo hops down a steep incline, the motto had to be “walk, don’t run.”

A simple mechanical model of walking is based on an inverse pendulum. The center of mass (m kg) is assumed to rotate at speed vm/s in an arc of radius rm around the point of contact of the foot with the ground. The downward force due to weight is mg (on Earth g = 9.81 m/s2) and this provides the centripetal force (Fc) to keep the foot in contact with the ground Fc = mv2/r. Therefore, v2/rg (a dimensionless ratio dubbed the Froude number) must be less than unity, otherwise the foot lifts off the ground and we begin a run. For a “leg” of 1 m at the center of mass of the body, we therefore have a limit ground speed 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 start running at a Froude number of about 0.5 (about 8 km/h). Although less than unity, walking faster causes discomfort in the ankle and associated musculature, and although running is initially less energetically supportive, it is more comfortable at these speeds.

So what’s the fastest I can walk on the moon? Looking at the data from the Apollo missions (see “Lunar paces”, Apollo 11 Lunar Surface Journal, November 2010), the astronauts walked at 2.2 km/h on the flat, while Neil Armstrong reached 3.6 km/h in adopting a loping gait. Low walking speeds and the tendency to lope may be due to the stiffness of the suits impeding limb movement. Although there is no data on maintaining a reasonable pace over a long distance, Buzz Aldrin believed that the particular movement of the body in the suit would make distance walking tiring. But suppose nowadays spacesuits might be better designed for longer walks.

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, thus applying a modified Naismith’s rule (assuming pessimistically that 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-run transition occur at the same Froude number on the Moon?

Treadmill tests were performed with both simulated lunar gravity, using aerial suspension, and real lunar gravity, as experienced in 30 s time slices during the parabolic flight phase of a DC-9 (J. Exp. Biol. 217 3200). The researchers obtained similar results for both tests and found that the volunteers only started a run at a Froude number of 1.25 ± 0.37 (average of the two sets of results), corresponding to a speed of about 5 km/h on the flat. The Froude number greater than unity was thought to be due to the downward forces generated by the swinging arms and free leg being greater in reduced gravity than on Earth, thus rendering the inverse pendulum model inaccurate.

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

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- On the moon there is mountaineering
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