City 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.

Map of Ohio showing Paul’s driving tracks.

My driving in Ohio from summer 2010 through January 2021. While a lot of the state is covered, there are still plenty of towns in the gaps.

Satellite imagery from Google Earth

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:

$$F=G{\frac {m_{1}m_{2}}{r^{2}}}$$

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:

$$F={\frac {N_{1}N_{2}}{d^{2}}}$$

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

Map of the eastern United states showing lines from 100 towns and cities to Beavercreek, Ohio.

The one hundred cities with the highest gravitational force on Beavercreek, Ohio. Each black dot is a top 100 city, and thicker lines indicate higher gravitational forces. To avoid clutter, many low-distance cities do not have name labels.

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

Rank City Metro Population Distance (mi) Distance (km) Gravitational Force
1 Dayton OH 718 353 8.0 12.9 530 974 391
2 Kettering OH 54 855 5.2 8.4 95 648 149
3 Riverside OH 25 133 4.8 7.8 51 726 576
4 Fairborn OH 33 876 5.6 9.0 51 595 630
5 Cincinnati OH 1 662 691 47.1 75.8 35 744 828
6 Xenia OH 26 947 7.2 11.7 24 523 667
7 Huber Heights OH 38 154 9.3 14.9 21 169 639
8 Columbus OH 1 562 009 59.8 96.2 20 852 257
9 Centerville OH 23 703 8.0 12.9 17 611 227
10 Oakwood OH 8 936 5.9 9.6 12 089 522
11 Springfield OH 82 820 19.7 31.7 10 171 252
12 Middletown OH 97 730 22.6 36.3 9 169 352
13 Bellbrook OH 7 344 6.5 10.5 8 239 946
14 Chicago IL 8 604 203 239.0 384.6 7 193 596
15 Indianapolis IN 1 588 961 110.7 178.2 6 189 038
16 Trotwood OH 24 403 14.2 22.8 5 815 977
17 Miamisburg OH 20 143 13.1 21.1 5 613 201
18 West Carrollton OH 12 864 11.0 17.8 5 047 767
19 Vandalia OH 14 997 12.4 19.9 4 691 127
20 Detroit MI 3 506 126 190.0 305.7 4 638 581
21 Springboro OH 18 931 14.9 24.0 4 075 382
22 Wright-Patterson AFB OH 2 579 5.5 8.9 4 000 226
23 Moraine OH 6 470 10.0 16.1 3 081 966
24 New York NY 18 713 220 538.7 867.0 3 078 485
25 Cleveland OH 1 710 093 173.7 279.5 2 707 216
26 Louisville KY 1 005 654 137.7 221.5 2 533 525
27 Englewood OH 13 435 15.9 25.6 2 528 263
28 Hamilton OH 62 082 35.5 57.1 2 351 560
29 Troy OH 26 281 23.1 37.2 2 342 943
30 Clayton OH 13 222 17.1 27.5 2 166 482
31 Mason OH 33 870 28.8 46.4 1 946 363
32 Lebanon OH 20 659 22.6 36.4 1 928 342
33 Franklin OH 11 612 17.4 28.0 1 829 293
34 Washington DC 5 379 184 380.8 612.9 1 770 627
35 Yellow Springs OH 3 852 10.3 16.5 1 748 634
36 Pittsburgh PA 1 703 266 221.5 356.4 1 657 952
37 Park Layne OH 4 082 10.9 17.6 1 633 354
38 Tipp City OH 10 115 17.4 28.0 1 595 377
39 Atlanta GA 5 449 398 412.9 664.4 1 526 351
40 Fairfield OH 42 558 37.6 60.6 1 433 355
41 Fort Wayne IN 334 122 109.7 176.5 1 325 493
42 Toledo OH 482 111 135.9 218.7 1 246 265
43 Wilberforce OH 2 291 9.5 15.3 1 211 433
44 Philadelphia PA 5 649 300 473.7 762.3 1 202 115
45 New Carlisle OH 5 568 14.9 24.0 1 199 556
46 Cedarville OH 4 343 13.4 21.6 1 153 243
47 Piqua OH 21 936 30.5 49.2 1 122 739
48 Union OH 6 891 17.2 27.7 1 109 078
49 Lexington KY 317 110 118.6 190.9 1 076 145
50 Monroe OH 14 015 25.5 41.0 1 029 615
51 Richmond IN 42 373 44.5 71.6 1 020 718
52 Akron OH 565 208 162.9 262.2 1 016 444
53 Wilmington OH 12 222 24.0 38.6 1 011 793
54 Green Meadows OH 2 366 11.3 18.1 888 485
55 Carlisle OH 5 446 17.2 27.7 879 091
56 Germantown OH 5 519 17.4 28.1 865 397
57 Enon OH 2 386 11.5 18.5 857 544
58 St. Louis MO 2 024 074 339.6 546.5 837 974
59 Trenton OH 13 141 27.5 44.2 830 748
60 Waynesville OH 3 181 13.8 22.2 801 186
61 Dublin OH 49 037 55.2 88.9 767 271
62 Oxford OH 23 113 39.4 63.5 709 538
63 Grove City OH 41 820 53.5 86.1 696 748
64 Muncie IN 87 879 77.6 124.9 696 344
65 Milwaukee WI 1 365 787 306.5 493.3 693 991
66 Brookville OH 5 874 20.3 32.7 681 213
67 Lima OH 68 878 69.8 112.4 674 285
68 Sidney OH 21 148 39.0 62.7 665 358
69 Holiday Valley OH 1 390 10.0 16.1 662 500
70 Covington KY 40 341 54.0 86.8 661 477
71 Hilliard OH 36 534 52.4 84.3 636 028
72 Baltimore MD 2 106 068 398.0 640.4 634 889
73 Drexel OH 1 991 12.3 19.7 631 556
74 Nashville TN 1 081 903 287.2 462.1 626 411
75 New Lebanon OH 4 067 17.7 28.4 622 761
76 Northridge OH 7 343 23.8 38.3 618 135
77 Jamestown OH 4 038 17.7 28.5 614 811
78 Crystal Lakes OH 1 467 10.9 17.6 588 776
79 London OH 14 870 35.0 56.3 579 125
80 Forest Park OH 18 583 39.4 63.5 570 379
81 Urbana OH 11 313 30.8 49.5 570 301
82 Charlotte NC 1 512 923 359.2 578.1 559 795
83 Huntington WV 186 034 126.3 203.2 557 026
84 Washington Court House OH 15 059 36.3 58.4 545 824
85 Loveland OH 13 145 33.9 54.6 545 698
86 Upper Arlington OH 35 366 56.4 90.8 530 055
87 West Milton OH 4 828 21.0 33.9 520 398
88 Grand Rapids MI 609 314 238.0 383.1 513 357
89 Anderson IN 85 992 89.7 144.4 510 215
90 Sharonville OH 13 684 36.0 58.0 503 327
91 Canton OH 269 418 160.3 258.1 500 271
92 Ann Arbor MI 322 267 176.7 284.3 493 016
93 Newark OH 80 451 89.9 144.7 475 245
94 Norwood OH 19 776 44.6 71.8 474 626
95 Delaware OH 41 283 64.9 104.4 468 612
96 Marysville OH 24 949 50.6 81.5 464 698
97 Boston MA 4 688 346 698.9 1 124.8 458 198
98 Five Points OH 1 723 13.6 21.8 446 704
99 Westerville OH 41 103 66.3 106.7 446 414
100 White Oak OH 19 900 46.2 74.3 445 241

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

Map of the United states showing lines from 25 towns and cities to Beavercreek, Ohio.

The twenty-five U.S. cities with the lowest gravitational force on Beavercreek, Ohio.

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

Rank City Metro Population Distance (mi) Distance (km) Gravitational Force
28347 Ferry AK 16 3 049.3 4 907.4 0.082
28348 Nikolski AK 26 3 897.0 6 271.5 0.082
28349 Monowi NE 1 770.0 1 239.3 0.081
28350 Chicken AK 13 2 836.0 4 564.2 0.077
28351 Warm River ID 3 1 426.7 2 296.0 0.070
28352 Lake Minchumina AK 14 3 153.5 5 075.0 0.067
28353 Wiseman AK 13 3 093.3 4 978.1 0.065
28354 Manele HI 24 4 421.0 7 115.0 0.059
28355 Bettles AK 12 3 129.2 5 036.0 0.059
28356 Stevens Village AK 11 3 059.0 4 923.0 0.056
28357 Evansville AK 11 3 126.1 5 030.9 0.054
28358 Kenny Lake AK 9 2 916.6 4 693.8 0.051
28359 Lake Louise AK 9 2 973.1 4 784.7 0.049
28360 Chase AK 9 3 081.2 4 958.6 0.045
28361 Attu Station AK 17 4 559.6 7 337.9 0.039
28362 Lime Village AK 8 3 261.5 5 248.9 0.036
28363 Ugashik AK 8 3 366.6 5 418.0 0.034
28364 Kalaeloa HI 14 4 460.3 7 178.1 0.034
28365 Poso Park CA 2 1 895.7 3 050.8 0.027
28366 Healy Lake AK 4 2 919.8 4 699.0 0.022
28367 Livengood AK 4 3 038.2 4 889.5 0.021
28368 Four Mile Road AK 4 3 056.3 4 918.6 0.020
28369 Beluga AK 4 3 125.0 5 029.3 0.020
28370 Alatna AK 4 3 164.4 5 092.6 0.019
28371 Birch Creek AK 2 2 971.0 4 781.3 0.011

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:

Rank City Gravitational Force
1 Dayton OH 5.3 × 10+8
28371 Birch Creek AK 1.1 × 10−2

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

Scatter plot with distance (miles) on the x-axis, and gravitational force on the Y-axis. Dayton is a substantial outlier.

It’s striking how strong Dayton’s force is compared to every other city.

Scatter plot with distance (miles) on the x-axis, and gravitational force on the logarithmic Y-axis.

Even with gravitational force on a logarithmic scale, Dayton stands out.

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?

$$ \displaylines{ 530\,974\,391={\frac {(47\,741)(7.83\times10^9)}{d^{2}}}\\ d=\sqrt{{\frac {(47\,741)(7.83\times10^9)}{530\,974\,391}}}\\ d=839\text{ mi} } $$

839 miles (1 350 km), as it turns out. That’s just slightly less than the distance from Beavercreek to Dallas, TX (853 mi or 1 373 km)—so if the earth’s entire population moved to Dallas, it would still have less gravitational force on Beavercreek than Dayton does now!

Map of the United states showing an 839 mile radius circle around Beavercreek, Ohio. Dayton OH and Dallas TX are also plotted; Dallas is just outside of the circle.

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.


  1. Technically we don’t need the population of my hometown either, since if we’re comparing everything to my hometown its population will be a constant, but to make this formula more applicable to the comparison between any two towns I went ahead and left it in. ↩︎

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