Neil Armstrong and Buzz Aldrin’s footprints on the moon will never fade. Until they get struck by a meteor or the sun expands. Well, we know the sun will expand to engulf the Earth’s current orbit in approximately 5 billion years, but how likely is it that a meteor will wipe one of these footprints off of the moon?

Let’s start with how many meteorites hit the moon at all. Should be simple, right? Wrong. Turns out, nobody really knows how many meteorites hit the Moon. But one brave man, Bill Cooke of the Marshall Space Flight Center’s Meteoroid Environment Office, is trying to find out. No, he’s not flying to the moon and looking up, he is instead analyzing data from his cushy office in Alabama. He uses data from four seismometers put on the Moon from 1969-1972 and tries to identify the sources of tremors, using nine known, man-made impacts as reference points. The seismometers indicate some 1700 confirmed meteoroid impacts (as small as four-inch, 1kg objects), but there are still 6000 unidentified tremors. If we assume that all of the unidentified tremors are indeed impacts, it puts the total number at 7700 meteorites. But the seismometers were only ever placed on the side of the Moon that faces the Earth, so the Moon actually experiences double that number. Luckily, however, we are only concerned with the near side of the Moon because that is where the footprints in question are located.

Now that we have the total number, what is the rate of meteorite impacts? The seismometers have been up there since 1969, and Bill Cooke started his analysis in 2006, so that means the 7700 impacts occurred over 37 years. This means there are roughly 208 meteorite impacts per year, or roughly one every two days.

The next questions we need to tackle are about surface area: how much area is covered by footprints on the Moon, and what is the area of half the Moon? The second question is much simpler, so let’s start with that one. A quick Wolfram Alpha search reveals that the surface area of the near side of the moon is roughly 19 million square kilometers, about equivalent to Russia. For the first question, let’s start with some basic facts and make assumptions. There were 6 manned missions that landed on the Moon: Apollo missions 11, 12, 14, 15, 16, and 17. A total of 12 astronauts spent a total of about 80 minutes walking (or hopping, or flag-planting) around on the Moon. Let’s assume there were only ever two astronauts on the surface at one time and that they walked the whole time (an over-estimate because they used rovers for the majority of their travel). Going over the Moon landing footage, it seems that they take a hop about every second, leaving two footprints. There are 4800 seconds in 80 minutes, meaning that a generous estimate for the number of footprints is 9600 (let’s round to 10000). The average US male has a size 10.5 shoe that leaves a footprint of area roughly 0.017m^{2}. But astronauts on the Moon wear astronaut shoes, so let’s increase that by 50%: 0.26m^{2}. If no two footprints overlap, that means the footprints on the Moon cover 2600m^{2}, roughly the area of Monaco. Therefore footprints make up about one part in 10 million of the surface of the near side of the Moon.

Now here comes the exciting part! If there are 208 random meteorite impacts per year, and the average meteorite wipes out an area of one square metre (even the small ones could be travelling very fast), that means about one part in a hundred billion gets cleared every year. There is a 1 in 100 billion chance that any given part of the Moon will be cleared in a year and there is a 1 in 10 million chance that any given part of Moon will be a footprint in any given year. But what are the chances that a given part of the moon will be cleared (hit my a meteoroid) AND be a footprint? Because the two events are independent, we can simply multiply the probabilities, meaning in any given year, there is a one in a quintillion chance of a footprint being wiped out. We do have 5 billion years before the sun expands though, so if we factor that in, there is about a one in 200 million chance that a footprint will be hit by a meteoroid on the Moon before the Sun becomes a Red Giant. For reference, the odds of winning the 649 lottery are about 1 in 14 million. So you have about as much chance of rolling a 5 on a 20-sided die and simultaneously winning the lottery as a footprint on the Moon does of ever being hit by a meteoroid.

Great maths!