# The math behind Gravity’s Rainbow (Part 1): The speed with which Earth rotates

The speed with which Earth is rotating is not something we think about every day. The only time I really noticed it was on a nightly hike in the mountains. It was a starry night, and at some point, the full moon was rising up behind a mountain massif. With the mountains as a reference point, it was possible to see the relative speed difference between earth and that rock in the sky. After watching for a few moments, I got dizzy. I realised with which incredible speed we are moving through space in spite of not feeling any movement at all. It was like simulator sickness, the feeling you get by, e.g. taking a rollercoaster ride in virtual reality. The whole way up to the mountain shelter, I couldn’t shake this unsettling feeling that something is wrong.

This little anecdote came back to me after reading the following passage from Gravity’s Rainbow:

No sign at all of Marvy’s plane. Schnorp adjusts the flame.

They begin to rise. Toward sundown, Schnorp gets thoughtful. “Look. You can see the edge of it. At this latitude the earth’s shadow races across Germany at 650 miles an hour, the speed of a jet aircraft.” The cloud-sheet has broken up into little fog-banklets the color of boiled shrimp. The balloon goes drifting, over countryside whose green patchwork the twilight is now urging toward black: the thread of a little river flaming in the late sun, the intricate-angled pattern of another roofless town. The sunset is red and yellow, like the balloon. On the horizon the mild sphere goes warping down, a peach on a china plate. “The farther south you go,” Schnorp continues, “the faster the shadow sweeps, till you reach the equator: a thousand miles an hour. Fantastic. It breaks through the speed of sound somewhere over southern France—around the latitude of Carcassonne.” The wind is bundling them on, north by east. “Southern France,” Slothrop remembers then. “Yeah. That’s where I broke through the speed of sound. . . .”

Pynchon, Thomas. Gravity’s Rainbow (Classic, 20th-Century, Penguin) (S.336). Penguin Publishing Group. Kindle-Version.

Slothtrop aka “Rocketman” thinking about breaking the speed barrier is undoubtedly related to his connection with the V2 rockets. But this blog post is not about deep metaphors, but just about fact-checking the math.

Carcassonne has a latitude of ~43.2 and the earth radius is 6371km, so the situation looks like this:

If we want to know how fast stuff is moving in Carcasonne, we need to calculate its distance from the rotational axis. This can be done by using this trigonometric function:

sin(A) = opposite/hypotenuse <=>

opposite = sin(A)*hypotenuse <=>

distance_from_rotational_axis = sin(46.8°)*earth_radius = sin(46.8°)*6371km = 4664km

Now we can calculate the distance Carcasonne travels in a day using the formula for the circumference:

circumference = 2*pi*radius <=>

travel_distance = 2*pi*4664km = ~29305km

As a day has 24 hours, Carcasonne moves with a speed of 29305km/24h = 1221km/h.

The speed of sound is 1235km/h, so this is close enough in my book. It makes me pretty confident that Thomas Pynchon actually did the math.

Additionally, we could take into account that the Earth is rotating not only around itself but also around the sun. If Earth had no self-rotation, it would take 365 days for the daylight border to revolve around the planet once. So in Carcasonne, the border would move with the following speed:

speed = travel_distance / hours_in_year = 29305km/(24h*365) = ~3.3km/h

Earth is a prograde planet, which means that it moves in the same direction as it rotates. So we have to add the two results to get a final speed of Earth’s shadow in Carcasonne: 1221km/h + 3.3km/h = 1224.3kmh