City Population Gravity

I’ve visited every county in Ohio, so I’ve seen a lot of the state—but there are still a lot of nearby towns I haven’t been to.

A lot of those towns are small enough that there’d be no reason to take a long drive to visit them specifically, but if they were close enough, it might be worth swinging through. The larger a town was, the further I’d be willing to drive to visit it.

I started thinking about this relationship between population and distance, and whether there was a way to quantify which towns I should be most willing to visit. And then I realized that a force that got stronger as objects got bigger and distances got smaller sounded a lot like gravity:

F is the gravitational force between two objects, G is a constant, m₁ and m₂ are the masses of two objects, and r is the distance between them. If I treated cities like objects where their populations were their mass, then the “force” between my home city and another town would be proportional to the product of their populations divided by the square of the distance between them. (I wasn’t the only person to have this idea.)

Since we don’t need the gravitational constant for a ratio,¹ we end up with the following formula:

N₁ and N₂ are the number of people in (population of) my hometown of Beavercreek, Ohio, and the town I’m comparing it to; d is the distance between them. So my willingness to drive to a town should be proportional to the product of Beavercreek’s and that town’s population, divided by the square of their distance.

Since distance is squared, this basically means that a town would need to have four times the population for me to drive twice as far—or 100 times the population to drive ten times as far. So the highest force cities will mostly be pretty close to home, but a few further cities with large populations could potentially overcome their distance disadvantage.

Gravitational Force for Each Town

To calculate the gravitational force for every town in the United States, I needed each town’s location (latitude and longitude) and population. Fortunately, I was able to find that as a spreadsheet from the Simplemaps United States Cities Database (November 18, 2020 edition). I used the Basic (free) data, as I was fine limiting myself to only census-recognized incorporated towns.

One caveat to note is that this data uses the metro population for cities (so for larger cities, both the city proper and its suburbs are counted in the population). This does mean that suburbs are double counted—once as their own population, and once under the metro area they belong to. For what I was doing, this was fine; the draw (“force”) of a larger city really is its whole metro area population, and it would also be interesting to see if any individual suburbs were themselves large enough to exert a significant force.

This was the most obvious for Dayton, which Beavercreek is a suburb of. Dayton was both very close and included the population of everything nearby (including Beavercreek), so it became by far the most forceful city in my list.

I wrote a quick Python script which would read the location/population spreadsheet, calculate the distance from Beavercreek to every town based on their coordinates, calculate the gravitational force between Beavercreek and each town based on the population and distance, and save the whole thing to a CSV spreadsheet.

100 Highest Gravity Cities

The hundred U.S. cities with the highest demographic gravitational force on Beavercreek, Ohio, are as follows:

Though the majority of the high-gravity cities are close to Beavercreek, the forces of the high populations of Chicago (#14) and New York (#24) more than make up for their distances. But although Los Angeles has a higher population than Chicago, L.A. only ranks #202—the distance is just too great for it to have much effect.

25 Lowest Gravity Cities

The twenty-five U.S. cities with the lowest attraction to Beavercreek are tiny faraway towns, mostly in Alaska:

Of course, I wouldn’t be able to drive to the Hawaiian (or many of the Alaskan) cities anyway.

Analysis

One thing that stood out to me was the enormous difference in gravitational force between the top and bottom U.S. city:

So Dayton exerts ten orders of magnitude more gravitational force than Birch Creek—the force is about forty-eight billion times stronger!

If the earth’s current population (7.839 billion as of 14 January 2021) moved to a single city, how far away would it have to be to exert the same force on Beavercreek as Dayton?

839 miles (1350 km), as it turns out. That’s just slightly less than the distance from Beavercreek to Dallas, TX (853 mi or 1373 km)—so if the earth’s entire population moved to Dallas, it would still have less gravitational force on Beavercreek than Dayton does now!

Conclusion

Although demographic gravitational force seems pretty arbitrary, the list of cities with highest force actually lines up pretty well with where I go the most often. I spend most of my time in the towns immediately near where I live, but I tend to visit larger cities a few hours’ drive away (Chicago, Indianapolis) more often than I visit some of the smaller towns a few dozen miles from me.