There is a lot of talk about Mars right now, and understandably so. The flight of Ingenuity today was awesome. As Daniel Oberhaus pointed out on Twitter,

… the atmosphere on the surface of Mars is so thin that it’s the equivalent of flying at ~100k feet on Earth.

No rotorcraft, piloted or uncrewed, has ever broken 50k on Earth.

When I heard that gravity on Mars is about 1/3 of that of Earth, that sounded too small to me. My thinking was that gravity on the moon is about 1/6 of Earth, and Mars is much bigger than the moon, so gravity on Mars ought to be closer to gravity on Earth

Where I went wrong was my assessment that Mars is “much” bigger than the moon. The radius of Mars is only about twice that of our moon; I would have guessed higher.

Surface gravity is proportional to mass over radius squared. If the density of two balls is the same, then mass goes up like radius cubed, and so gravity would increase in proportion to radius. The density of Mars and the moon are about the same, and so the object with twice the radius has about twice the surface gravity.

Let’s put some numbers to things. We’ll let *m* and *r* stand for mass and mean radius. And we’ll let subscripts *E*, *M*, and *L* stand for Earth, Mars, and Luna (our moon).

*r*_{E} = 6371 km

*r*_{M} = 3390 km

*r*_{L} = 1738 km

The radius of Mars is approximately the geometric mean of the radii of the Earth and the moon.

(*r*_{E} *r*_{L})^{½} = 3327 ≈ 3390 = *r*_{M}

To calculate surface gravity we’ll need masses [1].

*m*_{E} = 5.972 × 10^{24} kg

*m*_{M} = 6.417 × 10^{23} kg

*m*_{L} = 7.342 × 10^{22} kg

The mass of Mars is also approximately the geometric mean of the masses of the Earth and the moon [2].

(*m*_{E} *m*_{L})^{½} = 6.6 × 10^{23} ≈ 6.4× 10^{23} = *m*_{M}

The ratio of Martian gravity to lunar gravity is

(*m*_{M} / *r*_{M}²) / (*m*_{L} / *r*_{L}²) = 2.2968

The ratio of Earth gravity to Martin gravity is

(*m*_{E} / *r*_{E}²) / (*m*_{M} / *r*_{M}²) = 2.6140

so saying surface gravity on Mars is a third of that on Earth underestimates gravity on Mars a little but not too much.

## More Mars-related posts

[1] I’m assuming mass is uniformly distributed for each body. It’s not exactly, and this makes a difference if you’re planning satellite trajectories, but it doesn’t make much of a difference here.

[2] This is not a consequence of the relationship between the radii because the bodies have different densities. The moon and Mars have similar densities, but both are significantly less dense than Earth.

(r_mars/r_earth)^2 = (3390/6371)^2 = 0.283 and Earth is 29.2% land, 70.8% water

There is as much land on Mars as Earth!