Read Saturday Morning Breakfast Cereal. It’s great!
“BOO!” my son yells, leaping out from a shrub. And then he dissolves into a fit of laughter.
This is a game he likes to play, whenever he gets the chance. As soon as we’d parked and he got out of the car, he ran up the sidewalk towards the front door of our condo. And then he ducked back behind the hedge, lurking. The game, now, is for me to walk towards the door. Then he’ll jump out and shout “boo” and try to make me jump.
“Did you know I was there, daddy?” he asks.
Of course I did, I think. You hide in the same place every time. “Kind of,” I tell him. “I guessed where you were.”
He blows that off. “I was loud, wasn’t I?”
“Yes, you were,” I answer, unlocking the door.
“Was I loud enough for them to hear me in China?”
How Do We Hear?
Obviously, we hear with our ears.
Sound waves, which are really just pressure waves in the atmosphere, strike the outer ear and are channeled into the ear canal. These pressure waves vibrate the eardrum, which in turn vibrates the bones of the inner ear (the malleus, the incus, and the stapes), amplifying the vibrations and transmitting them into the inner ear (or cochlea). Hairs in the cochlea are stimulated by these vibrations, creating an electrical signal that transmits along the auditory nerve to the brain.
Yes, this is terribly simplified.
How Loud Are You?
Strictly speaking, “loud” is a matter of perception – the same pressure wave can result in different experiences of “loudness”. However, this perception is tied to the intensity of the pressure wave, just as the perceived pitch of a sound is tied to the frequency of the wave.
Using the above image of a wave, the intensity is how high the peaks and how low the valley is – the higher the peak, the more intense the wave. Another way to think of intensity is how much energy the wave carries – the taller the wave, the more energy (just like how bigger ocean waves hit harder than small ones). Frequency, on the other hand, is how fast the wave moves – the closer together the peaks, the faster the wave moves and the higher the frequency. Generally speaking, we perceive intensity as loudness (because the pressure wave hits the ear harder) and we perceive frequency as pitch (because the pressure wave stimulates the bones in the ear faster).
“Loudness” is measured in decibels (dB), because one decibel is the “just noticeable difference” in sound intensity for the human ear – assuming the pressure wave generated is in the 1,000 Hertz (Hz) to 5,000 Hz range we are best at hearing. Every 10 dB represents multiplying the intensity of the pressure wave by 10 – that is, a 10 dB sound is 10 times more intense than a 0 dB sound, a 40 dB sound is 10,000 times more intense than a 0 dB sound, and a 100 dB sound is 10,000,000,000 times more intense than a 0 dB sound.
We generally can’t hear anything below 0 dB, and normally speak in the 60 to 65 dB range. A jackhammer 50 feet away is about 95 dB, a power mower 3 feet away is around 107 dB, and loudness causes pain starting around 125 dB. Sounds at 140 dB and greater can cause permanent damage with even short exposure.
How Far Away Can We Hear?
This gets tricky, because the answer is “no further than when the perceived volume falls to 0 dB”. Tricky, because sound obeys the inverse square law which states that for any source power P generated at the center of a sphere, the intensity of at the surface of that sphere is P/4πr2 (although a good approximation is P/r2, since the math gets easier). According to Hyperphysics, r is pretty much always measured in meters for these purposes (because sound intensity is actually measured in watts per meter squared, so it keeps the units the same).
Since sound intensity can be transformed into decibels, it’s really not a stretch to directly apply the inverse square law to decibel measurements. So, a 60 decibel conversation would be perceived as 60 decibels at 1 meter away, 60/(2*2) = 15 decibels at 2 meters, 60/(3*3) = 6.6 decibels at 3 meters, 60/(4-4) = 3.75 decibels at 4 meters, less than 1 decibel at 8 meters, and so on. Realistically, at this point, it’s probably safe to call it “inaudible” (even though you could technically detect it).
How Loud Would You Have To Be For Someone To Hear You In China?
All right, here’s where the math gets… entertaining. I live in Cincinnati, Ohio, which is (according to Google) 10,969 kilometers from Beijing. Measuring along the curved surface of the Earth, that is. But, to keep things simple, we’ll ignore that. So, 10,909 kilometers is 10,909,000 meters. To be heard in Beijing, we’d have to generate enough decibels to result in a greater than 0 dB sound 10,909,000 meters away.
For laughs, let’s aim for a 60 dB sound. That way, our sound can be clearly understood. The radius is 10,969,000. So, the equation looks like this: x/10,969,0002 = 60. Solving for x gives us x = 60(10,969,0002), or x = 7,219,137,660,000,000 dB. This is a nonsensical level of perceived volume, and would render you deaf in ludicrously tiny fractions of a second.
What could generate that? Well, we’d have to reverse engineer the decibels into watts of power, which converts to 721913765999988 watts per meter, or about 721.9 terawatts of power. Now, you can roughly convert watts to Joules per second, so that’s roughly the explosion of a 200 kiloton nuclear weapon.
Assuming I did my math correctly, which I’m not guaranteeing. What I can guarantee is that there is no way you’d want to be standing anywhere near something loud enough in Cincinnati that you can hear it in China.
It’s summer, and it’s just finished raining, and we’re walking across a parking lot on our way back to the car from running an errand. My son is, as five-year-olds are wont to do, taking the opportunity to jump in puddles and laugh as they splash. Suddenly, he stops. “Look!” he cries, pointing at the ground. “There’s a rainbow!”
I go and look. Sure enough, there’s a small puddle with a thin film of oil slicking the top. I nod at him, and looks at it again. “Why is there a rainbow on the ground?” he asks.
“There’s oil on the puddle.”
He looks at it for a moment, then looks up at me. “How does oil make a rainbow?”
Yeah. You’ve just stumped me son.
So, what’s up?
Could you… elaborate? Just a little?
Light reflects upward both from the top of the oil film and from the underlying interface between the oil and the water; the path length (the distance from the reflection to your eye) is slightly different depending on whether the returned light comes from the top or from the bottom of the oil film. If the difference in path length is an integral multiple of the wavelength of the light, rays reflected from the two locations will reinforce each other, a process called constructive interference. If, however, the rays reach your eye out of step, they will cancel each other out due to destructive interference.
Sunlight contains all the colors of the rainbow–the famous ROYGBIV (red, orange, yellow, green, blue, indigo, violet). Each color of light has a different wavelength. Hence, a given disparity in the path length will cause constructive interference of certain colors, whereas other colors will not be observed because of destructive interference. Because the oil film gradually thins from its center to its periphery, different bands of the oil slick produce different colors.
Constructive and destructive interference?
Think of the classic sine curve you had to draw (we all had to draw them) when you took algebra or pre-calculus in high school. The wavy line that looks kind of like a snake. That thing. It’s the most common way to represent a wave of any sort, with the height of the “hills” of the wave representing amplitude (how intense the wave is – brightness for light, loudness for sound, and so on) and how close together the “hills” are representing frequency of the wave (how energetic it is).
Now, think hard about that math class. Do you remember what happens if you add two sine waves together? It changes the nature of the wave. Two identical waves will end up dobuling the amplitude (making it brighter or louder), while two utterly opposite waves will flatten the wave into a line. Check out the image below, if that isn’t clear.
The amplitude of the light is how bright it is, so the areas of the puddle where the reflected light interferes constructively you can see it clearly while you can barely see it when the light interferes destructively. The colors shift because the frequency of the reflected light (which determines the color) vary with the thickness of the oil on the water and the angle at which the light hits your eye.
“The car is cold!” my son declares as he climbs in. “Good!”
It’s a hot day out, although not as hot as it’s been recently. Stil, the car’s pleasantly cool because I’d just parked it a few minutes ago so I could pick him up from Kindergarten, and the air conditioning had been running. “You’re right,” I agree.
“Why is it cold?” he asks.
“I had the air conditioning running,” I answer.
“Does it run all the time?” he asks, fastening his seatbelt.
“No,” I answer. “Only when I turn it on. You wouldn’t want it running in the winter, after all. When it’s cold out, you don’t want to make it colder.”
He nods at that, agreeing with the idea. “How does it make it cold?”
Uhm. “I… don’t know,” I answer, although I’ve got a vague idea in my head of how I think it works. Something to do with freon and fans, clearly. “I’ll have to find out.”
How do air conditioners work?
Air conditioners work, it turns out, for reasons very similar to the water cycle we all learned about as children. They have a liquid in them (the refrigerant) which absorbs heat from the environment (your house or car), evaporating into a gas in the process. This gas is warmer than it was as a liquid, but it’s still not hot. It gets hotter, though as it’s circulated into a compressor, a device designed to pressurize the gas back into a liquid. A side effect of this pressurization is heat (to see this yourself, try squeezing a rubber ball – the pressure generates heat).
The hot liquid is then passed into a condensor, where it goes through a radiator that allows the heat to dissipate. This works because, even if it’s really hot outside, the pressurized refridgerant is hotter. The cooler pressurized liquid is then passed through a narrow hole into the evaporator, where the pressure drops and the temperature drops along with it. Once it enters the evaporator, the whole cycle has started over.
This is an extremely simplified version of the process, of course. You can read a far more detailed explanation of air conditioning here.
Why does changing pressure make things hotter or colder?
According to Hyperphysics, heat is “energy in transit from a high temperature object to a lower temperature object. An object does not possess ‘heat’; the appropriate term for the microscopic energy in an object is internal energy. The internal energy may be increased by transferring energy to the object from a higher temperature (hotter) object – this is properly called heating.”
Internal energy is “defined as the energy associated with the random, disordered motion of molecules. It is separated in scale from the macroscopic ordered energy associated with moving objects; it refers to the invisible microscopic energy on the atomic and molecular scale.”
Now, let’s turn to the first law of thermodynamics, which states that “the change in internal energy of a system is equal to the heat added to the system minus the work done by the system.” Based on this, it makes sense that increasing pressure increases internal energy and decreasing pressure decreases internal energy. Why? Because when you increase pressure you’re adding energy (because work is being done to the system), and when you decrease pressure you’re removing energy (because the system is working at pushing outwards). The added energy when you increase the pressure increases the internal energy that can be transferred to other systems, increasing the heat. Likewise, the decreased energy when you reduce the pressure reduces the internal energy that can be transferred to other systems.
What’s a system? The thing in question. The refrigerant in the air conditioner, for example, is a system for this purpose. So is the air in your home, or your car.
So, why does an air conditioner work? Thermodynamics, and clever engineering.
I really wish I could remember what prompted this questions. I suspect it had to do with one of our conversations about time, and how “day” is from sunrise to sunset. There was probably something we were doing in the evening, something he was looking forward to doing, and the sun seemed to be just crawling through the sky. Whatever the reason, the question certainly seems to make sense. The sun takes all day to cross the sky, so it looks slow. It comes up gradually, takes four to six hours to reach noon, and then slowly sinks into the west.
Of course, appearances are deceiving.
How Fast Does The Sun Move Across The Sky?
This is actually sort of tricky, because the Sun isn’t actually moving around the Earth. To begin with, let’s refer back to the technical definition of sunrise and sunset:
Sunrise and sunset. For computational purposes, sunrise or sunset is defined to occur when the geometric zenith distance of center of the Sun is 90.8333 degrees. That is, the center of the Sun is geometrically 50 arcminutes below a horizontal plane. For an observer at sea level with a level, unobstructed horizon, under average atmospheric conditions, the upper limb of the Sun will then appear to be tangent to the horizon. The 50-arcminute geometric depression of the Sun’s center used for the computations is obtained by adding the average apparent radius of the Sun (16 arcminutes) to the average amount of atmospheric refraction at the horizon (34 arcminutes).
Now, an arcminute is 1/60th of a degree, so the day begins when the center of the sun is 5/6ths of a degree below the eastern horizon and ends when the center of the sun is 5/6ths of a degree below the western horizon. Assuming perfect viewing conditions, et cetera, et cetera. So, that means that the sun has to cover 181 2/3 degrees in a single day. Here in Cincinnati, sunrise on the day this article is published is(August 19, 2016) is 6:55 am, and sunset is 8:27 pm. So, it will require 13 hours and 33 minutes (813 minutes) to cover that distance. That works out to, let’s see… 181.6666 / 813 = 0.22345 degrees per minute, or 13.407 degrees per hour.
Now, let’s do some more math. Cincinnati is at 39.1031 degrees north. According to Ask Dr. Math, you “just multiply the equatorial circumference by the cosine of the latitude, and you will have the circumference at that latitude.” The equatorial circumference of the earth is 24,901 miles, so 24,901 * cos(39.1031) = 19323.48 miles. The Sun crosses 181.666/360 = 0.5046 of that distance in 813 minutes, so that’s 9751.197 miles in 813 minutes. That’s essentially 12 miles a minute or 719.65 miles an hour.
But that’s the speed at which the sun passes over the Earth at that latitude on August 19. It’s not how fast it appears to an observer on the ground! To an observer standing on the surface of the Earth, the distance to the horizon is approximately 2.9 miles. That means that my son, observing the motion of the Sun, is standing at the center of a perceptual circle with a circumference of 2(π)2.9 = 18.22 miles. Which means that he sees the Sun appear to take 813 minutes to cover 18.22 * 0.5046 = 9.19 miles. That works out to a perceived speed of 0.0113 miles per minute, or 0.678 miles an hour.
No wonder it looks so slow to him. When we’re out on walks, my son and I hit a pace almost four and a half times faster!
How Fast Does The Sun Move Through The Galaxy?
Of course, the speed of the sun gets even trickier. Because, although it doesn’t move through the sky (we move, creating the illusion), it still orbits Sagittarius A*, the supermassive black hole at the center of the Milky Way. This monster is some 26,000 light years from earth and weighs in at around 4,000,000 M☉. Our Sun travels in a roughly circular orbit around this distant behemoth at a speed between 217 amd 250 kilometers per second – let’s take the average of those five figures and call it 230.4 kilometers per second (143.16392 miles per second). That’s 829,440 kph (515,390.112 mph).
That sounds impressive, doesn’t it?
Here’s something to consider, though: the speed of light is 299,792,458 meters per second, or 299,792.458 kilometers per second. That means the Sun (and it’s attendant planets and dwarf planets and other detritus) are moving at 0.07685% of the speed of light. Remember that line above, the one that reads “this monster is some 26,000 light years from earth”? That means that our Sun orbits Sagittarius A* in a circle approximately 163,360 light years in circumference. As a result, it will take about 163.360/0.07685% = 212.5 million years to complete an orbit. (Actual calculations from real astronomers come in at between 225 and 250 million years, which makes sense – they have access to more accurate figures, and the sun would actually describe an ellipse instead of a circle.)
So, why is the sun moving so slow? It isn’t. It’s tearing through space at a pace five times faster than the New Horizons probe at it’s maximum velocity – the fastest ship ever built by humanity (although the Sun’s gravity had slowed it to ‘only’ 14 kilometers per second by the time it passed Pluto). We just don’t notice, because it’s a huge universe and we have a very small frame of reference.
This week, it seems, I’ll be writing about the sun. Why? Well, when we were walking home from preschool yesterday, I asked my son if he knew what day it was. “It’s the first day of summer!” he declared. And then he asked me a whole bunch of questions, one right after the other:
- “Why doesn’t the sun melt? It’s really hot!”
- “What is the hottest star? Is it the biggest?“
- “What if the sun turned into a black hole? Would it swallow the earth and all the planets?”
- “Is the earth bigger than the sun?“
To be honest, I felt like I’d been hit by an avalanche of curiosity. But they’re all great questions.
What is melting?
Let’s start with the Dictionary.com, and a definition of melt that states:
- to become liquefied by warmth or heat, as ice, snow, butter, or metal.
- to become liquid; dissolve
- to pass, dwindle, or fade gradually (often followed by away)
- to pass, change, or blend gradually (often followed by into)
- to become softened in feeling by pity, sympathy, love, or the like
- Obsolete. to be subdued or overwhelmed by sorrow, dismay, etc.
From a more technical perspective, ‘melting’ is a first-order phase transition in which a material’s latent heat increases and it’s density or volume decreases sufficiently that it moves from the solid phase to the liquid phase.
Wait. Phases? Phase transitions? What now?
You’ve probably seen a diagram like this before:
That is a phase diagram for water, showing the different states (solid, liquid, or vapor) that water can be in based on temperature and pressure. Under one atmosphere of pressure (1 bar or 100 kPa), it is a solid at or below 0 degrees Celsius, a liquid between 0 and 100 degrees Celsius, and a vapor above 100 degrees Celsius. Phase is just the technical term for these states, and a phase transition is simply where the material changes (or makes a transition) from one phase to another.
While researching these phase changes, I ran into the terms “first-order phase transitions” and “second-order phase transitions” a lot. I’ll be honest and say I don’t fully understand them, because the best definitions all seem to involve a whole lot more physics than I understand. But here’s my best attempt at an explanation, after reading several articles and staring hard at Wikipedia:
- A first-order phase transition is driven by heat, and the material transitions from one phase to another at a set rate based on the energy added to the system. Think of melting ice, or boiling water.
- A second-order transition is also called a continuous phase transition, and appears to a uniform change across the material – imagine a block of ice instantly becoming water, for example.
The first-order phase changes are the ones we’re all familiar with. There are a whole lot of other types of phase change, so have fun reading up on them some time.
Right. So what does all of this have to do with a star?
All of this is a long walk to the answer I originally gave my son, when he asked me why the sun didn’t melt. “The sun can’t melt,” I told him. “Melting is when a solid turns into a liquid, and the sun is made of gas and plasma. It’s way hotter than melting.” Which is more or less true, and also led to the next question he asked me. But we’ll handle that tomorrow.
What is a star?
I’ll be honest here. I thought the International Astronomical Union would have a formal definition of a star, much like they do for planet and dwarf planet and the like. But, if they do, I couldn’t find it. So, in brief, a star is “a luminous sphere of plasma held together by its own gravity”. It can’t melt, because “melt” is not a phase transition available to plasma. Anything in it that could have melted has already melted, then vaporized, then ionized around 4.6 billion years ago.
My son loves mirrors. Not to the point that he spends hours in front of the mirror, mind, but he finds them interesting. He also thinks they work because of electricity, because both my car and my wife’s car are OnStar capable and so have a thick cable that connects the rear view mirror to the roof of the car. I haven’t been able to disabuse him of this notion, because he insists that I’m wrong when I tell him that electricity has nothing to do with reflection.
To tell the truth, I’ve been looking forward to this one myself. A lot. Because mirrors fascinate me, too. I’ve spent a lot of time in front of them, craning my head and discovering that I can see things in them that I wouldn’t have expected to be able to see reflected. You know, like how you can see into the living room a little by craning your head as you look into the bathroom mirror. They seem spooky at times, even though I know there must be a rational explanation.
Really, there is. It starts with the mechanics of how we see, and ends with physics.
How We See
Light passes through the cornea, the clear, dome-shaped surface that covers the front of the eye. The cornea bends – or refracts – this incoming light. The iris, the colored part of the eye, regulates the size of the pupil, the opening that controls the amount of light that enters the eye. Behind the pupil is the lens, a clear part of the eye that further focuses light, or an image, onto the retina. The retina is a thin, delicate, photosensitive tissue that contains the special “photoreceptor” cells that convert light into electrical signals. These electrical signals are processed further, and then travel from the retina of the eye to the brain through the optic nerve, a bundle of about one million nerve fibers. We “see” with our brains; our eyes collect visual information and begin this complex process.
That light has to come from somewhere, of course. While there can be many different sources of the light our eyes uses, those sources are ultimately one of two different categories: luminous objects, or illuminated objects. Luminous objects generate their own light, like a lightbulb or the sun. Illuminated objects are objects that reflect light, like the moon. Or mirrors. Or, really, anything at all that you can see.
Pictured: luminous and illuminated objects
What is reflection?
Dictionary.com provides the following definition of reflection:
- the return of light, heat, sound, etc., after striking a surface.
- something so reflected, as heat or especially light.
It’s a little more complicated than that, but overly complicated. Reflection depends on something called the law of reflection, which states that “when light falls upon a plane surface it is so reflected that the angle of reflection is equal to the angle of incidence and that the incident ray, reflected ray, and normal ray all lie in the plane of incidence”.
Clear? If not, the University of Texas has you covered. Start with this image:
There’s geometry there, and if you’re like me you haven’t done any geometry since high school. But really, the concept isn’t difficult. The incident ray is the ray (light, in this case) that strikes the reflecting surface. Assuming the surface is smooth, the reflected ray bounces off at the same angle as the incident ray.
And if the surface isn’t flat?
Well, in that case, the law still holds. Imagine zooming in on a rough surface – a rough-cut block of wood, for instance. At a small enough level, there are flat surfaces. Each flat surface becomes the plane for this law, and then the light is reflected accordingly. The rougher the surface, the more chaotic the reflected rays. The smoother the surface, the more uniform the reflected rays. This is the key to the two types of reflection:
Specular reflection is simply reflection from a surface that makes the majority of the incident rays travelling in the same direction reflect from the surface in the same direction. Diffuese reflection, on the other hand, is when the reflected rays scatter in different directions. Now, clearly, no surface is perfectly reflective. But mirrors produce significantly more specular reflection than diffuse reflection.
So why do things look reversed in a mirror?
It turns out that this is nice and simple. HowStuffWorks: Science explains this pretty well, but I’ll take a stab at it myself. In essence, it has to do with the fact that the mirror is just reflecting back light that reflects off an object. The light doesn’t flip around, so you see the right side of a reflected object (your body, for instance) on the left side of the mirror.
Hmm… still not clear. Here’s what HowStuffWorks said to help clarify it further:
Take a piece of thin, translucent paper and write your name on it. Stand in front of a mirror and hold the paper up so that you can read the paper normally. Now look in the mirror. You are seeing the back of the translucent sheet in the mirror, and the word is not reversed — it looks completely normal. Now turn the paper over and look at it in the mirror. It is reversed, but so are the letters on the back of the translucent sheet. Note that you turned the paper over — you reversed it!
Explained that way, it makes perfect sense. To me, anyway.
But what about the witchcraft?
Have you ever noticed that you can see things in mirrors that look like they are way, way off to the side? Things that look like they shouldn’t be able to be seen? This always felt like witchcraft to me. I mean, I’ve always felt that there had to be an explanation that wasn’t black magic, but I didn’t know what it was. Well, it turns out to be all about lines of sight and the law of reflection. If you’re standing to one side of the mirror, you see the reflected rays that have a flatter angle of reflection. Those reflected rays are created by incident rays that have a flatter approach angle relative to the plane of the mirror. But then your line of sight “in” the mirror is a line that follows the reflected rays “through” the mirror, making it appear that the reflected ray originated from an object inside the mirror.
All in all, it’s no wonder my son is fascinated by mirrors. They just don’t do what you think they’ll do. Instead, they do what physics thinks they’ll do, and that is usually a completely different experience.