The Inverse Square Law (2000) | Hacker News Reader

The Inverse Square Law (2000)

hyperphysics.phy-astr.gsu.edu

48 pointstucaz33 comments

While it makes intuitive sense, I find the inverse square law for gravity really hard to comprehend when I think about the solar system, for example Uranus, or even the earth, orbiting the sun. When you look at how far apart they are (for example https://news.ycombinator.com/item?id=21735528), the radius of the orbits relative to the sizes of the sun & planets, for me it seems to defy the inverse square law that there is still enough force at those distances to keep the planets orbiting.

If you built a scale model of the solar system out in the middle of space, would you have the same orbital periods? Like with a scale model of the moon, would a tennis ball sized rock orbit a soccer ball sized rock from a distance of ~25 feet in about a month?

Long story short: yes.

To expand a bit: lowering the masses by so many orders of magnitude would make the allowable range of vectors much more precise, and the degree of perturbation which the system could withstand would be much lower.

I wouldn't expect such a system to be robust with scale-model comets flying around it, for example.

That's interesting; I wouldn't mind hearing a spoiler of the long story. Is there something in the system besides time that doesn't scale relatively? comets at this scale would probably be dust-particle sized, right? Based on the soccer ball / tennis ball scale, my estimate is that a 1km diameter comet would be ~0.02 millimeters in the scale model.

Often overlooked in such representations is just how much larger and more massive the sun is relative to the planets.

Uranus is "only" about 20 times further from the sun than the earth, which according to the inverse square law means the suns gravity (and light) is 1/400th of what the earth experiences.

But the sun is about 330000 times more massive than the earth, while Uranus is "only" 14 times more massive.

Uranus getting 1/400 the gravity that earth does is a good example. And the speed Uranus is moving relative to the sun is not much slower than the speed the earth is moving relative to the sun.

Yes, the sun is surprisingly big indeed, so I can appreciate the reminder, but I’m not entirely sure it’s relevant here how big the sun is relative to the planets?

The surprise I’m talking about comes from the ratio of the sun’s radius to the radius of the orbit, and that applies to an orbiting mass of any size, from Halley’s Comet to Jupiter. What seems surprising to me is not that the sun is large, it’s that it can attract and keep in orbit something that is at a distance 10,000 times it’s radius (that’s Pluto’s aphelion). The distances between the sun & planets is so much mind-bogglingly larger than the sun that when you see the scale of the sun in context, when you compare the size of the sun to the distance between the sun and the earth, the sun looks incredibly tiny... so tiny that it’s hard to imagine how the gravitational influence is keeping the earth in orbit, especially when thinking about the inverse square law.

You're looking for Kepler's third law, which is a result of inverse square law and Newton's laws:

https://en.wikipedia.org/wiki/Kepler%27s_laws_of_planetary_m...

For all planets, the ratio of the (cubed) distance from the sun to (squared) period is a constant which only depends on the mass of the sun, regardless of the mass of the planets (all assuming that the sun is far massive than any other planet.)

Hyperphysics is a delight. I've worked in the physical sciences for years but keep going back to refresh myself on the basics. I always learn something again for the first time.

That space is three dimensional is one of those propositions drilled into us as schoolchildren, and which most of us accept on faith.

Yet, since it is a statement of the real world, it can only be validated by experiment and not by a mathematical proof. Trying to contort one's fingers or sticks in various mutually perpendicular orientations isn't satisfactory since we could be limited by lack of imagination of the dexterity of our hands.

The existence of physical phenomena satisfying the inverse square laws is one of those observations which constrains the geometry of the world.

On long distances, anyway. The evidence for inverse square laws over short distance is much weaker, and so there is room for curled-up "extra dimensions" we sometimes hear about, both in extensions to the standard model and in stringy ideas.

„Long distance“ here includes anything longer than a 1000th of the classical radius of the hydrogen atom, which is to a very high accuracy described by an inverse square law.

Gravity might have extra dimensions on a much larger scale than that, potentially up to the millimeter scale or so.

Magnets -- not EM fields but actual magnets -- sort of follow an inverse cubed law.

Anyway, Principia Mathematica's big conclusion was that an inverse squared force will result in elliptical orbits.

Does the law still work perfectly at very large distances considering single or zero photons (or gravitons or whatever) hitting a surface?

I guess is there a mathematical difference between a completely continuous and smooth distribution of energy vs. “quantized” energy at large distances?

Intuitively I'd think yes but only averaged over time. Waves in quantum fields should pretty much be perfectly matched to the inverse square law over any distance, but the actual observation of particles (by collapse or whichever interpretation you prefer) will obviously have variance which will only smooth out to an inverse square flux over time.

The photons (let's say) that cause the inverse square law of the E+M force aren't on-shell photons [on-shell]. So, they're not photons that are "traveling" or are "emitted" in the same way that radiation is---they don't carry any energy away. Similar with or whatever. The 'force law' isn't susceptible to low-intensity counting issues.

In the last case---radiation---the photons are on-shell (so you can, for example, count them) and inverse square does degrade when you hit low counts. However, time-averaging restores it on average.

[on-shell] https://en.wikipedia.org/wiki/On_shell_and_off_shell

So when transmit power is small enough to emit single photons, I suppose the antenna radiation pattern is a probability density function. I suppose a wave function for different modes on the antenna, as the photon has more energy (higher frequency).

I've never heard of the skunk on a flagpole example (does the smell drop off according to the inverse square law), but surely this can be refuted by pointing out that the other examples are expressions of energy transmitted from a point source and the smell is directionally emitted from the skunks butt.

For gravity and EM fields the inverse square law is a consequence of a massless force carrier particle and Lorentz invariance. Quantum Field Theory in a Nutshell by Zee talks about this early on.

An interesting but rather overcomplicated seeming explanation. The posted article basically is saying inverse square laws results from conversation of flux (of any kind) in 3d. That seems like a pretty good explanation to me

But does the inverse square law hold in non-euclidean space?

It doesn't. As far as we can tell though, our universe doesn't have much curvature, so the problem doesn't come up.

It comes up in our solar system. The anomalous motion of Mercury can explained only if the gravity does not follow inverse square law.

That's a short-distance effect, though, not a long-distance one.

For radiation, yes.

But for force laws, having a massless carrier is critical for an inverse-square law. With massive carriers (like the carriers of the weak force, W+Z bosons) the range of the force scales like 1/M; the force law is more like exp(-Mr)/r ---> 1/r as M-->0. The diminishing of the force with exp(-Mr) means flux isn't conserved. (note I worked in units where hbar = 1 = c, so that the W's mass ~= 80 GeV/c^2 corresponds to 1/M << 1 fm)

But also things that aren't radiation or high energy physics. Inverse square laws are ubiquitous in hydrodynamics, heat flow, electrostatics. Good ole fashioned 19th century physics. And I think that's its most natural arena.

Does the power of a laser beam follow the inverse square law?

Yes:

https://www.quora.com/Is-the-light-from-lasers-reduced-by-th...

ADDED. Yes at distances from the source greater than 3.3 meters

This post [1] disagrees:

> (...) if you measure the intensity of a beam with a photodiode (say) 1 inch from the laser and then 10 inches from the laser, you will not find 1/100th of the light with your fixed area detector. If you do this same experiment with an incandescent light bulb and make your closest measurement far enough away (much, much greater than the size of the filament), you will observe this square law decrease.

[1] https://groups.google.com/forum/#!topic/sci.optics/P-8VmDlY4...

That post is wrong though.

The real answer is, as usual, "it's complicated". But not very so.

The beam width is roughly constant with distance close to the focal point, so there the inverse square law does not apply. Far from the focal point, it's the 3D angle that's roughly constant, which means that the law does apply there.

Yes, makes sense.

The first sentence of the piece says that "Any point source which spreads its influence equally in all directions without a limit to its range will obey the inverse square law."

A laser beam has a direction, as such doesn't spread its influence equally in all directions.

Okay, but if A implies B, then NOT A does not necessarily imply NOT B.

Touché.

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