Inverse-Square Law
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# Basic Information

The Inverse-Square Law is a concept in Physics that can be applied to gravity, radiation, acoustics, electrostatics, etc, so there's several different fields of physics which each have their own version of the inverse-square law. What they all have in common is that the quantity or strength of something (usually a signal or type of energy) is inversely proportional to the square of the distance from it's source.

In it's simplest form, you can remember it as "if you're twice as far away from a source, you'll experience it as being one fourth as strong."

The reason for this has to do with geometry, and the surface area of spheres. The same number of particles, photons, etc, are getting spread across a wider and wider surface area as though range further out. The surface area of a sphere is $4 \pi r^2$. When you double the radius of the sphere, there's 4 times as much surface, which means 4 times as much area to spread out the energy or particles that are being emitted.

For the sake of explaining it, lets assume that a distance of 5 feet from it's source, the force in question has a strength that we'll arbitrarily call "100" . We might think of that as a percentage, but in gaming it might as easily represent 100 points of damage.

• "100" is the strength at 5 feet. So, applying the inverse-square law, we can deduce that at 10 feet, the effect will drop to a rating of "25". (10 is twice the distance of 5, and 25 is one fourth the strength of 100.)
• If we back up to 20 feet, the strength will drop to "6.25".
• At 25 feet, it'd be "4"
• At 30 feet, it'd be "2.7"
• At 35 feet, it'd be about 2% of the strength experienced at 5 feet from the source.
• At 40 feet, it'd be just over "1.5".
• At 80 feet, it'd be down to about "0.4"

## Sources

Bibliography
4. Non-Fiction: The Greatest Show On Earth by Richard Dawkins

# Game and Story Use

• In a sci-fi game set twenty minutes into the future, the inverse-square law allows you to justify us not yet finding alien radio broadcasts before the aliens show up. The nearest star (alpha centauri) is four light years away. The strength of any signal they send us is 1/4th as strong here as it is at the half-way point between us, and 1/16th the strength it is at 1 light year away from them, and 1/64th the strength it is at 0.5 lightyears from them, and 1/256th the strength it is at a mere one and a half trillion miles from them. Etc, etc. Think of the weakest radio signal you've ever picked up, practically drowned out by static. A message from outerspace would be thousands of times weaker than that. Aliens could be broadcasting to us round the clock from the next star over, and it'd be almost impossible for us to notice.
• On a related note, the Inverse-Square Law is also the explanation behind the Goldilocks Zone - the narrow band of potentially life-sustaining worlds in a Planetary System. Too close to your star, and the sunlight is so concentrated it scorches the surface, burning off all life. Too far away, and the light's not concentrated enough to fuel photosynthesis, nor to keep water in a liquid state where it can function as a biological solvent. Unless of course, you have some sort of exotic Alien Biochemistry solve that puzzle and produce life where it's not expected.
• Useful info for post-apocalyptic scenarios, first responders to a dirty-bomb detonation in an espionage genre game, etc: You've got a Geiger counter, so you can tell the direction the gamma rays are coming from, and can tell that the radiation level is right here. How do you know if it's safe to move forward, without just walking ahead and exposing yourself to whatever radiation is there? Note the reading on your Geiger counter, and walk away from the source. When you reach the point where the reading drops to 1/4 the rads, you'll know two things:
1. How far you are from the source of the radiation. The distance from the place of your initial reading to your current position is equal to the distance from your initial reading to the source of the radiation. If you walked backwards 50 feet before the radiation dropped to 1/4 it's original rating, then you now know you are now standing roughly 100 feet away from the radioactive material.
2. A rough statement of the strength of the radiation at it's source. Calculating it exactly results in a lot more math than you'd ever want to engage in for a game (and feels a bit like Xeno's paradox), but it's certainly the sort of math your character would be willing to do if it could save their life. So, a reasonable GM would allow a PC with a Geiger counter to roll a skill like Science, Physics, Academics or Math to figure out not just where the source is, but also how dangerous it is at the source.
• Fireballs coming online! If you wanted your game to simulate reality well, the inverse-square law would definitely apply to energy weapons, lasers, sonic attacks, dragon-breath, grenades, etc.
• Taken as a general principle without rigorous math, the point to remember is that blast radius and template weapons should do significantly less damage at the outskirts of the template or radius.
• This can be used to justify sudden drop-offs in damage, and characters just barely escaping from some huge explosion.
• The damage reduction for making a saving throw or being only partly under a template should perhaps be reduced by 3/4 instead of 1/2, because that more closely matches the rate at which strength of force and density of particles declines. In this case, making your save would mean diving a little bit further away from the blast.
• Magic, by it's very nature, might be able to skip this diminishing effect, but truly Functional Magic is probably subject to this law unless the caster intentionally crafts the spell to avoid it.
• A critical hit with a breath weapon or the like at point blank range should do a lot of damage! Full-force concentrated dragon plasma can probably cut through rock.

The following are specific examples of how to model the inverse-square law on a battlemap, should very accurate simulations be your cup of tea:

Type or Template Epicenter Effects Beyond
Small explosion or burst: 4d6 damage to anyone in the square of the detonation 1d6 damage to the 8 spaces adjacent to that square. No damage beyond that.
Larger explosion or burst: The center point is a dot that is the central point of a 2x2 grid of 4 adjacent squares. Anyone in one of those squares takes 8d6 damage. The next row of squares out (the 12 squares surrounding those 4) takes 2d6 damage. The next row furthest out takes 1d6 damage. The row beyond that (a distance of 4 spaces from the central point) takes about 1d3 damage.
Short-range Cone Template: 8d6 to anyone in the adjacent space 2d6 in the second space (10 feet away), 1d6 to the third space, 1d3 to the space after that (the fourth space total) and the space to the left and right of it (because the beam is much wider here).
Potent Beam Weapon or Cone Template: Figure out at what range the width of the beam fills an entire space, and assign your damage rating of, say, 4d6. At one-half the "full space width" range, the beam is focused on a very narrow part of the victim is hit by the full force of the weapon - which in this case would be a devastating 16d6. At double the "full space width" range, the attack falls off to 1d6 damage. If this ends up being a large number of spaces, you might assign more of a curve to it, with every couple of spaces reducing the damage a little more.

In some game systems this would need adjusting, especially that devastating last row on the chart with it's 16d6 at point blank. In particular, game systems that track damage per limb and have a real threat of meaningful critical hits probably should just automatically remove a limb or have some other major critical rather than throwing a huge double-handful of dice. In Cyberpunk 2020, for example (where 8 points of damage on an unarmored limb means it's time to get replacement cyberware) it really doesn't make any sense to allow a 56-point hit to an arm. Just say the arm is gone and the victim is in critical condition, to save yourself the trouble of rolling and counting. Feel free to substitute whatever other numbers or effects strike you as valid damage ratings, but the 4/1 ratio should be preserved if you care about the modeling the real-world physics - which, admittedly, is not necessarily a high priority for all gamers or campaigns.

page revision: 4, last edited: 08 Dec 2009 20:18