How big is the Moon in Super Mario Odyssey?
The Moon is one of the major themes in Super Mario Odyssey. Power Moons are collected as you go on your journey to power Mario's ship, and eventually you will reach the Moon Kingdom, where Mario tries to stop Bowser's forced wedding with Princess Peach.
A view of the huge Moon from the Cap Kingdom
Although this isn't the first game where Mario has been into space - just looking back to Super Mario Galaxy, he's been all over the universe. In Super Mario Odyssey however, the game only takes place on Earth and the Moon, bringing these two celestial objects into focus. In many areas of the game, the Moon is visible, but it seems huge! It feels like a much bigger presence in the sky compared to what we are normally used to in real life. How big is the Moon in Super Mario Odyssey? How do the properties of this Moon compare relative to the real Moon and Earth?
In front of the New Donk City Hall, there is a depiction of the Earth and the Moon, if this is perfectly to scale, we can see that the Moon is about one third of the radius of Earth, which is actually pretty similar to our real life Earth and Moon.
A very unscientific way is to measure the amount of pixels in this screenshot, the radius of the Earth is 318 pixels, and the radius of the Moon is 100 pixels. Based on this ratio, the Moon's radius is about 31.4% that of the Earth.
Earth and Moon in front of New Donk City Hall
The real life Earth radius is 6371 km, and the real Moon radius is 1737 km. If the Super Mario Odyssey "Earth" is the same as the real Earth we live in, using that percentage from above, that means the radius of the Moon in the game is 6371 km x 31.4% = 2000 km. The radius is a difference of 263 km, in other words, the in-game Moon is 15% larger relative to the real Moon.
This is one measure of the "size", but what about how "massive" is the Moon? How heavy is the moon? To figure this out, we might need to consider a few things.
Remember when Mario arrives on the lunar surface, there's a noticeably lower gravitational pull. If you remember from your science class, acceleration due to gravity on Earth is about 9.8m/s^2, or 1g. The Moon, however, is a much smaller astronomical object, and gravity on the Moon is about 1.62 m/s^2, or approximately 0.16g.
In other words, this means that on the Moon, the gravity is 16% of that on Earth, and that it means Mario should jump 6 times higher than normal while on the Moon.
Mario seems to jump twice as high on the Moon here...
But wait, this isn't the case! Just from an eyeball test - based on the jump height to the height of the Odyssey, Mario seems to only jump about twice as high as he does on Earth, not 6 times as high.
We can deduce the gravitational pull based on how much higher Mario jumps, he's jumping twice as high, which means that the gravity would be 0.5g, half the amount of gravity that we would see on Earth. 0.5g would mean that the acceleration due to gravity is 4.9 m/s^2.
Using this Newton's Law of Gravity formula we can calculate the mass of the Moon based on the properties we observe in the game. "a" is the acceleration due to gravity, "r" is the radius of the object in metres - in this case the Moon, and G is the gravitational constant. We will convert 2000 km to metres which makes it 2,000,000 metres.
M = ar^2 / G
M = (4.9 (m / (s^2)) x ((2 000 000 m)^2)) / G
M = 2.93673435 x 10^23 kilograms
The real Moon weighs 7.34767309 x 10^22 kilograms, this means that the in-game Moon weighs significantly more than the real Moon, in fact the in-game Moon is 4 times heavier!
Logically, if you think about it this makes sense, since the gravitational pull for the in-game Moon is far higher - since Mario can't jump as high as he theoretically could jump in the real Moon. A far more dense environment leads to the force of gravity being far more stronger.
Of course, Mario games aren't real, so it doesn't always make sense, but sometimes it's fun to think about how things relate to real life!
This is fine and dandy, it's not like Mario hasn't been into space, he's been to all sorts of worlds and galaxies in Super Mario Galaxy. We've seen his increased jumping ability in low gravity environments. This isn't news.
However, on the Moon Kingdom, there are deep underground caverns that Mario needs to traverse through. But wait! Once you drop into this area, the gravity becomes "normal", and Mario's movement is exactly the same as on Earth.
To a certain extent, depending on how deep underground you go, the gravitational pull may actually does increase - but nowhere to the extent where you would be anything close to 1g as encountered on Earth. Perhaps this is something we can look at in a future follow up!
Maybe Bowser has installed some artificial gravity to mess around with Mario... Of course video games are fiction, and it seems like all the characters are able to survive on the Moon without necessarily wearing protective gear, so it's best to not overthink things too much!