Ep. 312: The Inverse-Square Law and Other Strangeness

Why don’t we have insects the size of horses? Why do bubbles form spheres? Why does it take so much energy to broadcast to every star? Let’s take a look at some non-linear mathematical relationships and see how they impact your day-to-day life.

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This episode is sponsored by: 8th Light, Swinburne Astronomy Online

Show Notes

  • Inverse Square Law — GSU
  • Equation of a Straight Line — Math is Fun!
  • Basic Logarithmic Properties — Purple Math
  • Inverse Square Law for Light — GSU
  • Hertzsprung-Russell Diagram — University of Oregon
  • This is how far human radio broadcasts have reached into the galaxy — Emily Lakdawalla/Planetary Society
  • How far does the gravity of the Sun extend? — Universe Today
  • How Tall Can Trees Grow? — LiveScience
  • Debunking Astrology: Mars Can’t Influence You — Universe Today
  • Transcript

    Transcription services provided by: GMR Transcription

    Female Speaker: This episode of Astronomy Cast is brought to you by Swinburne Astronomy Online, the world’s longest running online astronomy degree program. Visit for more information.

    Fraser Cain: Astronomy Cast episode 312 from Monday, June 24th, 2013: the inverse square law and other strangeness. Welcome to Astronomy Cast, our weekly facts based journey through the cosmos where we help you understand not only what we know but how we know what we know. My name is Fraser Cain. I’m the publisher of Universe Today and with me is Dr. Pamela Gay, a professor at Southern Illinois University Edwardsville and the director of CosmoQuest. Hey, Pamela. How you doing?

    Pamela Gay: I’m doing well. I’m almost as far as we’ve ever been apart doing an Astronomy Cast this week. Last week, was further but this week I’m in Lisbon while you’re in Vancouver so we’re spanning the continents here.

    Fraser Cain: We could drill a hole right through the Earth and –

    Pamela Gay: No.

    Fraser Cain: No. No. I think I’d be in the Indian Ocean if I drilled a hole through the Earth.

    Pamela Gay: Yeah. I don’t know where I’d come out.

    Fraser Cain: You should figure that out. I think there’s a website you can go and check out that is like your antipode on the Earth to find out where it would be. Great. So yeah. So you’re in Portugal continuing your European vacation, work-cation.

    Pamela Gay: No. No.

    Fraser Cain: Yeah, I know. It’s totally not a vacation. You’re working.

    Pamela Gay: Yeah. My European inflict CosmoQuest on people in other countries.

    Fraser Cain: Alright. So the European tour, the CosmoQuest European tour.

    Pamela Gay: Yeah, that I’ll agree with.

    Fraser Cain: Alright, cool. And just I wanna make one quick little promotion and mention which is that every Sunday night; we connect a bunch of telescopes into a live Google+ hangout and show you the night sky live in real time. We’ll show you planets; we’ll show you deep sky objects, clusters, the moon, the sun sometimes. It’s fun times and we often will bring in Ph.D. astronomers to explain this stuff. So if you’ve got cloudy skies and you don’t know what to do with your Sunday evening, join us on Google+. We do it whenever it gets dark on the West coast. Right now, it’s summertime so that’s 9:00p.m. Pacific DLA time and even later in the East. But it’ll get better in the summer so.

    Pamela Gay: It’ll get better in the winter.

    Fraser Cain: It’ll get better in the winter. Yes.

    Pamela Gay: For North.

    Fraser Cain: For North. That’s right.

    Pamela Gay: It’ll get better in the Southern summer.

    Fraser Cain: That’s right.

    Female Speaker: This episode of Astronomy Cast is brought to you by 8th Light Inc. 8th Light is an agile software development company. They craft beautiful applications that are durable and reliable. 8th Light provides disciplined software leadership on demand and shares its expertise to make your project better. For more information, visit them online at Just remember, that’s www. the digit 8, T H L I G H Drop them a note. 8th Light. Software is their craft.

    Fraser Cain: Why don’t we have insects the size of horses? Why do bubbles form spheres? Why does it take so much energy to broadcast to every single star in the universe? Let’s take a look at some non linear mathematical relationships and see how they impact your day to day life. Alright. Now, you threw this topic on the list so you had a plan and this is not my topic and so I am going to just grill you to get to the bottom of this. But right. So let’s sort of set the stage here. When you’re talking about non linear relationships, what are you talking about?

    Pamela Gay: So most of the time, we are used to thinking of things that if you double them, you end up with four, if you triple them; you end up with nine so the numbers, in that case, that’s not linear. Linear is if you increase this one by one, this one increases by one. You increase this one by two, this one goes up by two. So if you think about the difference between Johnny and Sallie’s ages, it will always be the same because they grow up at the same rate and so the separation stays the same and if you do a plot of Johnny’s age versus Sallie’s age, there will be a line that’s straight through the plot of those.

    Fraser Cain: And this is really the only kind of mathematical concept that our poor human brains are really able to rock intuitively.

    Pamela Gay: In general, yes.

    Fraser Cain: You know what I mean? That we evolved on the savannah and if there’s one more lion, we should bring one more hunter. Right?

    Pamela Gay: And in school, we all learned the whole Y equals MX plus B. There might be a slope to the line but it’s still a line. So it may be that you add one here and you add two thirds here. If you add two here, you add four third here but it’s still a linear relationship. So that’s what we’re used to is that straight line when you put the things on the two axes.

    Fraser Cain: Right. Okay. And so, but the universe sure doesn’t like to play by the rules and it doesn’t like to play by our pathetic human concepts. It makes its own rules. So then what are the other kinds of the non linear relationships that can be out there?

    Pamela Gay: Well, so they basically come in polynomial form. This is where if I have my X value over here, so like how fast my car is going versus how much gas goes into it, the gas going into it hopefully isn’t a power law but you can imagine that it’s Y squared, Y cubed, Y to the four thirds. Anytime X equals Y to the something that’s a power, that’s a power law.

    Fraser Cain: Right. And I think you’re guess in knowledge is a great one, right, because you have the situation that the harder you push down on the pedal, you’re not just using more fuel. You’re using, in many cases, the engine is actually gonna require a multiplier of the fuel.

    Pamela Gay: Right. And that’s because the friction goes up, the heat goes up, it gets more difficult to accelerate for mechanical reasons. Yeah.

    Fraser Cain: Yeah. Yeah, exactly. That friction, harder to push through the air resistance as you’re driving down the highway until it becomes just like a brick wall of air.

    Pamela Gay: Right. Right. So then the other class of relationships that we tend to have are logarithmic relationships. So these are things that are powers of ten or inverse powers of ten which is where you get a logarithm, basically. Inverse isn’t quite the right way to think of it but it’s a short hand.

    Fraser Cain: And this is where you’ve kind of pulled into my background, which is computer science. Right? We deal with the logarithmic scales all the time. When you search or sort an algorithm and if you’re gonna sort a bunch of numbers, that actually follows a logarithmic scale. So you can sort a million numbers. It’s not a million times as hard as to sort one number or 1,000 numbers. Right? Every time you go up, it actually follows this logarithmic curve so what is it? It doubles every ten? How does the logarithmic function work?

    Pamela Gay: So zero is to one, one is to ten, two is to 100, three is to 1,000.

    Fraser Cain: Right, right. And so the point is is it’s a way that if your numbers are going up in stupid amounts, you can kinda flatten the curve down and show it in a much sort of reasonable way as opposed to just this line that you can just go straight up.

    Pamela Gay: Yes, exactly.

    Fraser Cain: You can actually bring it down and show it in a little bit more of a reasonable way.

    Pamela Gay: And what’s interesting is how many biological things or at least the two senses of most importance that we have are our ability to see and our ability to hear, both work on logarithmic scales. The sounds around us vary by powers of ten, not by linear relationships. So when you turn your dial up to 11, it’s not 11 times louder. It’s tens and tens and tens of times louder.

    Fraser Cain: Right. And so it gets a lot louder a lot more quickly. But at the same time, you’ve got this other problem and indicates that that sound, right, which it that the sounds trying to expand outward from the speakers is hitting, I guess what we call the inverse square law. Right?

    Pamela Gay: Right. So here’s where we start to get into where do these things matter and an inverse square law is a power law. It’s just a power law with a negative sign in it. So if you have something that’s radiating out, it has to basically radiate out over an entire surface. Sound coming out of a speaker, it’s not laser beamed to your ear with all the power from the speaker focusing in the sound on your ear drum.

    Instead, it’s radiating away in a sphere and so the energy gets spread out over the surface of the sphere and for every distance that it goes, the area proportionately goes up by the square. So take the energy, divide it by the area, and if the area’s going up by the square, that means the energy per small area on the surface is going down by the square.

    Fraser Cain: Alright. Now, let’s bring it home to astronomy, which is, I think, what people are hoping to hear. So give us some examples then in astronomy where we deal with this power law. It feels very mathy and I know a lot of people, their eyes are starting to glaze over with the math but this is core to a lot of the calculations that astronomers have to do with gravity and light and brightness and luminosity and sound and earthquakes and all this kinds of stuff. It all follows this exact same system.

    Pamela Gay: Right. I’d argue that earthquakes are probably more planetary science than astronomy but let’s tie it back down to physics –


    Fraser Cain: Geologist? Sure.

    Pamela Gay: Physics astronomy space science. So the most fundamental place that we worry about it in astronomy is light as it radiates away from the star has to fill this entire sphere of space and fill the surface, not fill the inside. You can have a flash of light and that’s a shell that radiates away. So these apparent brightness of a star decreases by that power law, that square, the distance squared as you get further and further away. So if I double my distance to a star, the star will appear four times fainter. If I triple my distance to a star, it will appear nine times fainter.

    And so things get faint fast and that’s problematic when you’re trying to look at things far away just because it is this every increasing thing that you’re dealing with.

    Fraser Cain: Yeah. When we think about the stars that we can see out in the sky at night, most of those stars are bright, really bright stars because in many cases, they’re mostly just far away. Things like Betelgeuse and Rigel, they’re just big bright stars and that’s what it takes to put up that much energy over such great distances.

    Pamela Gay: And this is that square law again working in a different way. When you look at the Hertzsprung-Russell Diagram, the color magnitude diagram that we talked about in a past episode, you see that the brightest stars are also physically the biggest stars and this is because they have more surface area for the light to come out of. Our own sun, in the future, it’s going to cool down. It’s going to become colder but it’s also going to become much, much larger and that much, much larger star is going to be significantly more luminous and it will be visible from a much bigger distance.

    So bigger stars with more surface area, surface area goes as the square of the radius of the star, so you double the size of a star, you multiply by four the surface area that’s giving off light and you end up with a much brighter star. There’s other factors that have to do with temperature and stuff like that but the basic idea holds.

    Fraser Cain: Right. And people don’t realize. Like Betelgeuse, even though it’s a gigantic star, would envelop Jupiter, it’s actually cooler than the sun surface.

    Pamela Gay: Yeah. Yeah. It would still kill you quite effectively.

    Fraser Cain: Oh, sure. Yeah, no. Absolutely. Okay. So that’s a good example. Right? And I think that exact same thing sort of comes back to this concept of searching for extra terrestrials that if you were attempting to broadcast a signal out into space, to reach any distance, you’re gonna require a ton of power.

    Pamela Gay: Right. And this is one of those things. If you’ve ever walked too far away from your Wi-Fi antenna at home, you know how quickly you go from five bars to zero on your laptop. Well, think about how quickly you go from detectable to not detectable as you move away from the planet Earth. Yeah, we sent episodes of, “I Love Lucy.” Yes, we sent the opening of the games in Germany from the Olympics with Hitler into outer space but those signals are so amazingly weak that they’re going to be very quickly imperceptible against the background noise of the universe.

    So trying to detect aliens from radius signals, the only way we could probably ever succeed is if they were purposely beaming a message into space and we just happened to look the right place at the exact right time.

    Fraser Cain: So you actually think the power requirements are so high that it would take an alien beaming a signal directly at us? It would require … I guess when you think about it, there are stars, there are red dwarf stars that are ten light years away from us that you require a really powerful telescope to even discern them.

    Pamela Gay: Right. And –

    Fraser Cain: And that’s the whole power of a star.

    Pamela Gay: Right. So how are we going to pick up the accidentally leaked into the atmosphere television signals? At best, we might pick up the type of signal that’s being sent to one of their own space crafts somewhere else in their solar system but even then, we just use enough energy for Voyager to pick it up when we send messages to Voyager. We’re not trying to reach elfis and tory or anything else out there.

    Fraser Cain: Yeah. Okay. So then now what about gravity? How does gravity sort of follow a similar role?

    Pamela Gay: Well, gravity goes as the radius of the planets squared but inversely again. So if you have a planet of a given mass, as that mass is held constant, so you assume the same mass all the time and you make the planet bigger and bigger and bigger, the gravity you feel while standing on its surface goes down as the square of that radius again. So it works the same way as light. So just like the lights gets you double the radius, the light goes down by a fourth, you double the radius of a planet, gravity goes down by a fourth.

    Now, the thing is you’re fundamentally changing a planet when you take a rocky planet and you try and blow it up with some unknown, weightless nothing but at the same time, one interesting thing about this is you can end up with different combinations of mass and radius that lead to the same surface gravity. So if you try to stand on the surface of Saturn, which is so not dense, it would float on water, it has almost the same surface gravity as Earth so you would weigh almost the same amount on the surface of Saturn.

    So even though Saturn’s this monstrously huge planet, because its mass is spread out over such a large volume, and that’s the key. Volume goes up as a power of three. So this is the insects that size of horses don’t exist example you tried to give at the beginning. If I have a sphere of material and I double its radius, its surface area goes up by a factor of four and its volume goes up by a factor of eight which is the cube of that two. So two times two times two is eight. So you’re getting massive very quickly. This is the reason that someone who’s twice as tall may weigh eight times as much, which is kinda creepy to think about.

    Fraser Cain: Right. Well, we talk about those super earths sometimes, how planets have been found with twice the mass of the Earth and you would think that if you were standing on the surface as you mentioned before, that you would experience twice the gravity but actually, it all just depends on the size of it correspondingly. Right? Is it a bigger planet? Is it a smaller planet?

    Pamela Gay: Yeah, and we’re still figuring those things out.

    Fraser Cain: Yeah, because we only get those measurements through the radial, either the radial velocity so we only can detect their mass. We can’t detect their size.

    Pamela Gay: Exactly.

    Fraser Cain: Yeah. Right. So I definitely wanna talk about that bug example here. Right? So as your bug increases in volume, right, in size, in volume, it has to be correspondingly stronger.

    Pamela Gay: And this is where the legs very quickly no longer become strong enough to hold it up. If you look at the radius of the legs on a bug, they have these hair sized legs until you start looking at tarantulas which finally have legs more like match sticks or French fries. And those little, tiny legs are able to hold up the bugs because they’re small. They don’t have that much volume. They don’t have to hold that much stuff inside of them.

    As you start to get to birds, which are another very lightweight animal, you see that the legs get a little bit bigger. As you start to get to small mammals, you have legs that are proportionately a little bit bigger compared to the body. Humans next but then when you get to elephants, you suddenly have these, compared to the size of the body, these massively large radius legs and that increase in the radius of the skeleton bones in the legs is needed to support that power of three increase in the volume as the animals get bigger and bigger and bigger.

    Fraser Cain: Right. Dinosaurs and so on. What was it? The California redwoods are pretty much at the largest size you can get for trees. Just that they can’t pull water anymore up to the top. If you got any bigger, the hydrostatic system they use to bring water out just doesn’t work any more and there’s no sort of bigger possible tree. So nature has tested that.

    And the other example I used in my introduction with the balloon, the thing I love about a balloon or like a bubble, when you blow a bubble, is you’re seeing the math made for you right there. You get this bubble. You know what the radius of the bubble is because you can see it and you can see the size and surface area and the volume of the bubble and it’s following this exact same rule as it’s trying to match its volume versus its surface area to the minimum size.

    Pamela Gay: Right. So the most stuff that you can fit within a given surface area is fit inside of a sphere. Something like a pyramid is a extremely in efficient shape where the amount of surface area you have is greater than necessary to contain the volume.

    Fraser Cain: So what are some other situations in space, maybe in astronomy where we’re gonna see this same relationship play out? What do astronomers need to deal with or even spacecraft engineers need to deal with as they’re sending probes out to other places to be able to communicate with them to some of their trajectories, some of the gravitational fly byes, things like that?

    Pamela Gay: Well, as we look at all of the different orbital mechanics equations, there’s always a to the power of the square root, there’s a cube, a square. So just looking at Kepler’s equations, we have the period, the amount of time that it takes an object to go around its central mass squared is proportional to its semi major axis distance. You take the ellipse, you cut it in half on the long side, you measure that length, is equivalent to that semi major axis cubed.

    So here, you have one thing that’s squared related to something else that’s cubed, now plot and so you end up with some fabulous curves and what’s awesome is if you’re ever digging through an old professor’s drawers finding the fancy curve devices they had to try and draw all of these old plots back in the days when you had to turn in hand drawn plots to journals.

    Fraser Cain: Really?

    Pamela Gay: Yeah, yeah.
    Fraser Cain: Well, what did one of these things look like?

    Pamela Gay: Well, they just have on them the various, “Here’s a square. Here’s a cube,” and so they had these tightening cubes that represent the tightening curves rather that represent the different mathematical relationships. So you found the part that you needed and fitted it to the points on your paper.

    Fraser Cain: Like a Spiro graph kind of?

    Pamela Gay: No. The Spiro graphs were the things you did this with. You had the circle that you put inside the other circle with a gear and you made flowers basically. No, this was like a stencil that you might use to draw circles today, except part of the stencil was it had these curves and then you’d have to figure out where on this tightening curve the particular curve you needed happened to fall.

    Fraser Cain: And it’s like finding an old slide rule.

    Pamela Gay: Yeah, yeah.

    Fraser Cain: Do you know how to use a slide rule?

    Pamela Gay: No. No clue. Someone showed me once when I was in the 11th grade and I knew how to do it for about a week and then forgot.

    Fraser Cain: I think another good example of this situation is, for example, people always talk about the planets, astrology, and they talk about how the planets are aligning and it’s gonna have some nonsense effect on you. But the reality is is that you’re experiencing a zillion, I believe is the right math, times more from just the gravity of the Earth and the moon but even that’s not very much. You’re feeling more gravity from mountains. Right? Because the distances.

    Pamela Gay: And the more important thing is if you calculate the gravitational pull of an average massed semi truck driving past you, its pull on you as it drives past is gonna be greater than that of any of the planets outside of Earth. So unless you’re plotting the movements of the local fleet of shipping trucks and trains, I don’t think you can correctly compensate for their effects relative to all of the other planets.

    Fraser Cain: Yeah, it’s like the tides. Right? The majority of the tides comes from the moon, not the sun, even though the moon, it’s just because the moon is closer even though the sun has much, much more mass than the moon, it’s just that the moon is closer and so we experience more gravitational effect from it. Cool. Okay. Well, thank you very much, Pamela.

    Pamela Gay: It’s my pleasure. Thank you.

    Male Speaker: Thanks for listening to Astronomy Cast, a non profit resource provided by Astrosphere New Media Association, Fraser Cain, and Dr. Pamela Gay. You can find show notes and transcripts for every episode at You can email us at Tweet us @astronomycast. Like us on Facebook or circle us on Google+. We record our show live on Google+ every Monday at 12:00p.m. Pacific, 3:00p.m. Eastern or 2000 Greenwich Mean Time. If you miss the live event, you can always catch up over at

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    [End of Audio]

    Duration: 27 minutes

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    One Response to Ep. 312: The Inverse-Square Law and Other Strangeness

    1. Randy July 31, 2014 at 2:59 pm #

      Those curved stencils used to draft complex curves are called French curves. We as drafters still use them today when Drafting curves in 2d and isometric drawings.

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