Have We Found Planets Using Math?

Planets Found by Math

Urbain Le Verrier

Ancient peoples have always known about Mercury, Venus, Mars, Jupiter and Saturn because you can easily see them with you eyes. The discovery of Uranus was the first time a planet was found that needed the aid of a telescope to be see. William Herschel discovered this planet in 1781.

In the next century, another planet was discovered, and this one was discovered in a very interesting way. The credit for finding planets is usually thought to go to the first person to point their telescope at it, but this case is a little different. In the 1845, Urbain Le Verrier looked very closely at the orbit of Uranus, and discovered it to be slightly off from what was known from Kepler about planetary motion. Soon after, he had a hypothesis that another planet beyond the orbit of Uranus could account for the perturbations in the orbit he observed. Le Verrier contacted Johann Gottfried Galle at the Berlin Observatory and told him to point his telescope at a particular location at a particular time to look for this eighth planet.

Johann Gottfried Galle

Sure enough, when Galle peered through the telescope lens on September 23^rd in 1846, there sat Neptune, 1° away from Le Varrier's predicted position. Interestingly, the director of the Cambridge observatory, James Challis, later realized he too had seen Neptune on two separate occasions before that, but failed to recognize it as a planet.

This, however, was not the only planet found by math. Later on, small perturbations were noticed in the orbit of the planet Mercury, again by Le Verrier. A small planet was hypothesized, this time inside the orbit of mercury, too close to the sun to see. This theorized planet was given a name – Vulcan.

Yes, the one and same, though the hypothesized planet came before the Star Trek series by more than a century (and was the Roman god of fire, volcanoes, and metalworking well before that). This planet does not actually exist. We have since sent spacecraft close enough to the sun to see any potential Vulcanoids, and to date have found none. So what of the perturbations of the Mercurial orbit? The answer is relativity. Because Mercury is so deep in the sun's gravity well, it experiences relativistic affects, and this accounts perfectly for the precession observed in its orbit.

Image: Terry Virts

LLAP

Cheers,

    - Scott

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How Random is a Deck of Cards?

A Humble Deck of Cards

Here is a short post about a humble deck of cards. Statistically speaking, after 7 riffle shuffles (below) of a deck of cards, a deck of cards is appropriately shuffled, meaning that the deck could now be in any one of 10⁶⁸ possible arrangements.

Image: Johnny Blood

To put that number in perspective. If there were a quadrillion galaxies (10¹⁵), each galaxy with a trillion suns (10¹²), each sun with a trillion planets (10¹²), each planet with a trillion inhabitants (10¹²), and each inhabitant shuffled a deck of cards every second, and had been doing this since the beginning of the universe (10¹⁷), we would only now be expecting to be getting repeats.

Every shuffled deck of cards, to an extraordinary significance can be said to be truly unique, that is no deck of cards has ever been in that order before, nor will any ever be in that order again.

Except the order that new decks come in. That happens all the time.

Cheers,

     - Scott

What Can We Learn From Twinkling Starlight?

When you look up at the night sky, the stars twinkle. This is interesting, but not what I am talking about today. If you are curious why the stars twinkle from here on Earth check out this video:

There ya go. Now, the 'twinkling starlight' I'll be mainly talking about is related to exoplanets, or planets orbiting stars other than our own. If you look carefully at stars from outside our atmosphere, or correct for atmospheric effects, the stars still twinkle, but for what I think is a much more interesting reason.

Do you want to know something interesting about starlight?

Great!

For the longest time, it was thought that our planet and our solar system were pretty unique. Aristotle laid down the thinking about many topics including astronomy for many centuries, and as it turns out he was wrong about a fair bit of it. While understandable for his time, by the 1600's, times were changing. In the early 1600's Galileo looked up at the moon with a telescope he made (didn't invent) and observed the moon's terminator, the area where light met dark. In the shadows he saw craters, bumps, and ridges; the moon wasn't a perfect celestial orb, it was its own world with its own unique features. Couple that with his discovery of moons orbiting Jupiter, and we were on our way to discovering other worlds, inferred from points of light.

*IC6.G1333.610s, Houghton Library, Harvard University

Both the cratering of the moon and the motion of the Jovian
moons were published by Galileo in 1610 in this pamphlet.

We have now sent spacecraft to nearly all of the thirty or so largest bodies in the solar system. With missions visiting the asteroids Vesta and Ceres, and the upcoming mission to Pluto, New Horizons, our curiosities about other worlds just took steps much farther afield.

Just as we could see other worlds in our own solar system, we can now look for worlds orbiting other stars using several methods of analyzing flickering starlight from their home star. With few exceptions, we cannot just take pictures of the planets because their star outshines them by many orders of magnitude, what we can see it the influence they have on their star.

Transit method -

One way to detect exoplanets is too find a planet that passes directly between its home star and us here on earth. A bit like a solar eclipse. When this happens, the planet blocks a little bit of the light, and the star dims. We can track the stars brightness and if it dims consistently and periodically we can tell that there is probably a planet orbiting that star. Here is what one of these dips looks like:

Image: Nikola Smolenski

The transit method is currently by far the most common way to detect exoplanets, but it has its drawbacks. Due to the fact that the planet has to pass between the star it is orbiting and the observer here on earth, it biased toward planets that orbit "edge - on" to us here on earth. Imagine flipping a coin, and taking a picture when the coin is exactly edge-on. Most of the pictures are going to show at least some of either the heads side or the tails side. This is roughly the same probability as a particular star system appearing exactly edge on to ours so the planet passes in front of the sun.

This method tends to be biased in finding large planets orbiting close to their stars. The larger and closer to its star that a planet is, the more likely it is to cross in front of the star and dim the light we see. These planets are known as "hot Jupiter" because they tend to be larger than Jupiter and closer to their star than Mercury is to our sun. This flies in the face of how we think planets developed, suggesting that hot Jupiters are quite rare. If this is the case, there could many, many more planets out there than we can currently find using this method.

It is important to note at this point that we cannot see the outline of the planet in front of the star. The only thing we can detect from here on earth is the slight dimming from a distant point of light.

Other Methods

Two more ways I'll briefly touch on on the radial velocity method and something called astrometry.

To describe the radial velocity method I first have to talk about the Doppler effect.

<sidenote>
I always thought "Christain Doppler and the Effects" would make a great band name
</sidenote>

There are plenty of great video about how this works, so I'll only go into it very briefly here. When a noise-making object approaches you, the sound waves "stack up" and compress on their way to your ears. This registers as a higher pitch. When the noise-making object moves away from you the sound waves "stretch out" and you register this as a lower pitch. This is why cars passing you make the characteristic "weeeee-yahhhhhh" sound.
The same goes for light. When an object is moving toward you, you register the compression of the waves as a "blueshift," the object literally looks a bit bluer. When the object travels away, the light looks redder, a "redshift."

TL;DR: Stars look bluer moving toward you, redder when they're moving away.

Alright, on to radial velocity. A large planet orbiting a star will cause the star to wobble a little bit, as seen below:

This is because the planet gravitationally tugs on the star, just as the star tugs on the planet. Notice how the star moves up and down. If we look at this star from earth, we can see it getting redder and bluer as it travels farther and closer to us, and from that, infer the presence of a planet by looking at the rate of the wobble.

On to Astrometry!

If you imagine looking at the system above from earth just as it is portrayed, you would see the star travelling in a little circle. If you look at both the foreground star as well as background objects, you can see the motion of the star and from that find out characteristics of the planet orbiting it.


This topic is difficult to convey through writing alone, so if you're interested check out YouTube for some great videos about exoplanets and exoplanet detection. Here are a few of my favorites:

Overall, it is truly amazing what we can discover merely by looking at twinkling starlight.


Cheers,

   - Scott



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Bonus!

On Gravity

When thinking about the force of gravity, it is tempting to think that it is a very strong force. After all, it seems to be quite dominant in our lives. Rockets struggle against it, it limits the feats of many athletes to fewer home runs, fewer field goals and fewer slam dunks. It is the force that in massive amounts can break the very fibers of spacetime with black holes.

However, this force is minuscule compared to the forces at work on an atomic scale. In an atom, there are neutrons and protons in the nucleus and electrons that surround the nucleus. The neutrons repel each other and yet are held together by strong nuclear forces.

If we existed at this scale, we would most likely not know about gravity. Gravity at this scale only accounts for about one ten thousand trillion trillion trillionths of the force you feel. That is 40 orders of magnitude.

It begs the question: what forces are out there that we cannot detect at our scale?

Relevancy of Relativity

This week, rather than posting a piece written specifically for this blog, I am going to go green and recycle. Hopefully the topic is interesting, although it does not perfectly match the format of what I have been posting. I'll be back next week with content written just for you.


This is a paper I wrote to the prompt "Is Einstein's Theory of Relativity relevant in today's culture?"


This is what I wrote:

Einstein's famous theories of special and general relativity live on in our modern culture beyond academia and the fact that “E=mc²” is a wonderfully elegant equation. Technologically, they account for the accuracy of GPS, keep the ISS on time, and relativistic effects between Earth and Mercury was discovered to be the slight perturbation in its orbit, rather than the delightfully mysterious “planet Vulcan” at Earth's L3 point. At least the latter lives on in Star Trek.

While people count on GPS, many don't care what time it is on the ISS, or spend time thinking about what is lurking on the other side of the sun, messing up the Mercurial procession. If you ask (pester?) your friends about relativity, as I have recently done, they will generally have the name Einstein come out of a holster, then mumble something about the speed of light, and maybe even mutter something about time moving slower. The science-y ones may know the name “Lorenz” has something to do with all of it. This is a bit of a shame, because since the early 20^th century there has been a profound change in many part of the scientific community that had led to many breakthroughs, and it all involved leaving your common sense and intuition at the door. Relativity is a large part of this new phenomenon.

Special relativity posits that as you speed up, you shrink down relative the person who stayed still. It also posits that things don't happen at the same time at different speeds, that time moves slower when you speed up and general relativity posits that being in a gravity well changes everything you thought was real as well.

None of this is at all intuitive. Abandoning intuition is an extremely valuable trait among scientists, and can be useful in everyday life. It allowed nearly every great advancement of human scientific thinking. Thinking that there exists an invisible army of creatures living inside you is preposterous enough as to be worried for whomever suggested it, yet it's true. The idea that the earth is hurtling through space at many kilometers per second is immediately dismissed by the evidence that we don't feel it, yet it's true. Opening oneself up to these strange ideas had proven beneficial in the past, and retains its place today.

Relativity is an excellent tool to train yourself to mistrust intuition. Learning about how a 10 meter ladder can fit in an 8 meter barn with both doors closed forces you to rethink the intuitive answer. There are other areas in which this is an invaluable tool. If a loved one faces a difficult medical choice between invasive surgery or an alternative remedy, one's intuition screams to avoid surgery where it may be the only effective option. If vaccinating your children sounds like child abuse to you, abandoning intuition may indeed help your child live longer.

These examples are not direct effects of relativity in modern culture, and the truth of the matter is that few people are interested in what they think of as Einstein's century-old dusty theories and equations. Many people are much more concerned about if they need a second mortgage, or if they need to take away their teenagers keys after a speeding ticket, even if it means shutting them to their high school. Even so, those who have spent some time with their brain in the kitchen mixer of scientific theory and inquiry can influence those around us. Family members asking for advice or friends facing tough decisions can really benefit from a bought of critical thinking and cognitive dissonance. Science is much more than math, equations, and physicists talking about Star Trek's holodeck or Alcubierre's warp drive in a faculty cafeteria, it is a way to move through life, evaluating the best course of action and persuading others to do the same, a way to avoid fooling yourself, or allowing others to fool you.

Einstein’s relativity has a knack for kick-starting this train of thought with just a small initial investment of research required, or a friend willing to talk about it.

Cheers,

  - Scott

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Random Facts

In my research for this blog as well as my own musings and curiosity, I run across some little factoids that aren't substantive enough to make a post all on their own, so I'll talk about a few of those here.

Faster Than Light Poke

I was thinking about the idea that "information can not travel faster than the speed of light," and I thought I was quite clever when I came up with a way to do it, theoretically at least. I imagined a wooden pole, say one light-year long. If you wanted to transmit information across a light year, it would take at least 1 year to send a radio signal. That's where my pole came in. If one person 'poked' in Morse code then the person one light year away would be able to receive that information immediately. I thought I had cracked it, information traveling faster than light.

It turns out I made an incorrect assumption however. I assumed the pole would all move as one, as with a broom. Poke someone with a broom and they feel it immediately. This is not so with exceptionally large brooms (nor is it with small ones, but the delay is so short it is all but immediate).

There is a 'speed of poke' or a 'poke wave' in the broom. The speed of poke turns out to be the speed of sound through that medium. Your poke would travel at about 3500 meters / second if your pole was made of wood, brass, or concrete. This is far less than the speed of light. It turns out it's about how fast GPS satellites orbit the Earth. The 'poke wave' (I love that phrase) comes about because when you move the pole, individual atoms in the pole have to move to alert their neighbors that they’re moving. Each atom must push its neighbor which must in turn push its neighbor and this is how the poke is transmitted through the object. So no faster than light pokes, science won this round.

Helicopter Speed Limit

It turns out that most helicopters have a speed limit imposed not by the FAA or the maker of the helicopter, but by physics. On helicopter, there is a “retreating blade” (#1, left side) and a “retreating blade” (#2, right side) due to the direction of the blades' rotation.

Image: AVStop.com

When the helicopter is flying, the advancing blade is traveling at the speed of the helicopter PLUS the speed of the rotor. The retreating blade is traveling at the speed of the rotor MINUS the speed of the helicopter. This creates something called "dissymmetry of lift." Because one blade has a higher airspeed than the other one, the central hub of the helicopter changes whats called the angle of attack on the retreating blade, giving it more lift.

Image: Johnathan M. Stern

The increased angle of attack compensates for the slower air speed or the retreating blade. This only works up to a point. If the retreating blade requires more lift than can be compensated for by altering the angle of attack, the helicopter pitches back and slows itself down. This gives most helicopters a physics-induced speed limit.

Image: SPC Glenn Anderson

Most helicopters

Melting Ice and Boiling Water

It is really hard to get water to change temperature. This is because of a property called specific heat capacity that I wont get into much here. Suffice it to say that it requires a whole lot of energy. But the true champion of temperature stubbornness is ice. Ice takes more energy to change by a few degrees than water takes to change by dozens of degrees.

The experiment:

Put two glasses in the microwave for about two minutes. Fill one with water and one with ice and start your microwave. Go do it, it only takes two minutes, and your dishes are still clean afterword. As it turns out the glass filled with water will boil before most of the ice has even melted. This means the water changed temperature from around 70°F to 212°F, or 142°F verses the ice which changed from the temperature in my freezer, 15°F to about 32°F, a change of 17°F. Both the ice and the water absorbed the same amount of energy from the microwave.

The way a microwave works is by creating a standing wave that oscillates about 2.5 billion times per second. This means a charged molecule like water...

...will act like a magnet and try to align itself to the current direction of the field, and this effort causes it to vibrate at about 2.5 billion times per second, increasing thermal energy, heating the water up.

Red is slightly negative and blue is slightly positive

The reason ice doesn't play along and do the same is because it's molecules are in a solid crystal arrangement:

Image: Danski14

In order to align with the field, the molecules must be free to rotate, and in a solid crystal structure, the molecules must break their bonds before they can vibrate and increase the thermal energy. This takes a lot of energy and happens very slowly, which is why the ice takes so long to change temperature and melt.

Bonus fact: When writing this I thought to myself “because fat molecules are non-polar they should heat up less in a microwave.” I tried the experiment, and I heated up identical volumes of water and vegetable oil for one minute, and sure enough, the water changed by 61°C and the oil changed by only 32°C. Science wins another round.

Falling Out of Orbit

When something in orbit loses power or is disabled in some dramatic fashion, often it is portrayed to "fall out of orbit." Almost every science fiction franchise is guilty of portraying this. Even if the writers know the physics, portraying something falling out out of orbit is climactic, while reality in this case is rather underwhelming.

Orbits themselves are often misunderstood. Getting into orbit is not about going up quite as much as it is about going sideways. Space is only 100 km away, so getting there is easy. Staying there takes some work. As you can see in this amazing video (click it) a US space shuttle launch doesn't go straight up, but turns at a steep angle a few seconds into it's flight. By the end of the main fuel tanks' supply, the shuttle will be going about 7.8 kilometers per second sideways and it will not be be changing height at all.

If the shuttle were to get shot down by aliens or lose power, it would not dramatically fall back to Earth but simply stay in orbit. The only way to get out of orbit is to change your velocity dramatically so you're going slower relative to the Earth's surface. In order to change your velocity in orbit you have to take some of your mass and accelerate it away from you. This is all rocket engines do; the fuel is the mass, and they accelerate it by igniting it and directing the outflowing material opposite the direction you want to travel. Without this process occurring you would remain in orbit because your velocity didn't change.

Put simply, it takes nearly as much energy to get out of orbit than it does to get into orbit, so losing power does send your craft careening towards the thing you are orbiting. Orbiting isn't an active process, it is a passive one.

Cheers,

   - Scott

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How Could We Have Broken Germany's Enigma Code?

I initially set out to write about the German Enigma Machine and how it worked, but as I was doing some research, two things happened: I found a video that explains it better than I ever could, with an actual WWII-era Enigma Machine, and I wound up doing some math that I thought was somewhat interesting.

So to start, I think you should watch this video about the Enigma Machine:

Did you watch it? If not, I'll give you one more chance:

IF THE VIDEO DOES NOT SHOW UP CLICK

 >  HERE  <

Alright, so in the video we saw that the Enigma Machine was a device used to encrypt internal messages in the German military, and has an extraordinary number of possible arrangements, each of which will produce a different code. The Enigma code was broken through a regimented and careful application of cleverness on the part of Alan Turing and his team at Bletchley Park. I got to thinking however, “what sort of team would have to be assembled to solve the Enigma code through sheer human power and luck?”

This rabbit hole was quite fun to descend into. I started with a few assumptions for this scenario. Every person who was working on the code had their very own enigma machine to work with, they could try a new arrangement each 100 seconds, and they could work for 16 hours a day. I also assumed that the infrastructure to feed, house and give them water was in place. Alright, here is the number we started with:

158,962,555,217,826,360,000

159 quintillion. That is the number of possible arrangements the enigma machine could be in, each would output a different code, and only one of which would be correct. Oh, and it changed every single day.

In order to try every combination in the space of 24 hours, you would need to try 1,840,000,000,000,000, or 1.8 quadrillion arrangements per second.

If one person can try one arrangement in 100 seconds, that means it will take a force of 184 quadrillion people to try every arrangement in the space of 24 hours, but that only if we work them 24 hours a day. Adjusting for the workforce working 16 hour days, we can multiply 184 quadrillion by (4/3) to get 245 quadrillion people.

So we have ~ 2,350,000,000 people in the world (in 1945)

We need 245,000,000,000,000,000.

Keen-eyed readers will note that the second number is longer, so we are now moving into a very hypothetical world. As long as we're not bound by reality, lets go ahead and pack in our workforce across the entire land area of earth (minus Germany), and lets pack them in at the population density of current-day Tokyo (we are ignoring small facets like food water, adequate shelter, etc...).

Land area of Earth: 57.53 million km²

Land area of Germany: 137,903 km²

Population density of Tokyo: 1800 people per km²

1800 people per km² * ( 57.53 million km² – 137,903 km²)

This gives us a population on our Super-Earth of about 100 billion people.

This means we would need 2.45 million Super-Earths to support our workforce.

It looks like we'll need to drain the oceans to beat the Nazis.

Due to the fact that Earth's surface area is ¾ water, draining the oceans gives us 3x the surface area we had using land alone (minus Germany). We can now fit 400 billion people on each Mega-Super-Earth, and now we need a quarter as many, or a little more than 600,000 Mega-Super-Earths. While that many Earth-Sized objects can orbit many stars, we need to restrict our Enigma team to one star in order to transmit the correct code in time to implement a strategy. The nearest star to us is Proxima Centauri, over 4 light years away. That is not close enough to get information back to central in time to help the war effort.

The question now becomes “Can 600,000 Earth-sized object orbit our sun without bad things happening?” I talked to some astronomers and astrophysicists here at CU Boulder, and the consensus seems to be “no.” When you put a lot of objects in similar orbits, most of the bodies are ejected from the system, most of the rest collide with each other, and quite a few will be engulfed by the star in the middle of the system.

No video?

CLICK HERE

This is the “Nice Model” of the solar system. About 3.8 billion years ago many Kuiper belt objects were ejected from the system and Uranus and Neptune switched places (at 30 seconds in the video). This area was much more sparsely populated than our scenario would be.

This doesn't happen immediately, so to wrap up, if you’re going to use humans alone to solve Enigma, do it fast, and make it worth it.

There are of course a few problems with this plan:

On average, the code will be broken by midday. Sometimes you will get lucky and solve it early on, and sometimes you wont be so lucky, and it will take until late in the evening to break the code. Even with this workforce, there will only be an average of 12 hours to actually use the information obtained.

If you have 245 quadrillion people united to defeat the axis powers, you probably wouldn't even need to bother with the Enigma code. If each person donated one strand of hair (half a milligram), you could bury Germany in 122 billion kilograms of human hair, which would probably deter them. I wanted to figure out how many feet this would be, but there aren't reliable figures for the packing efficiency of average human hair, so I figured a head of hair (a wig) weighs about 100 grams, and I could stuff one in a one liter bottle giving it a density of 0.1 kg/liter. This means 122 billion kilograms of hair could cover Germany to a depth of ~9 meters, give or take a few orders of magnitude.

So now you know that...

If you’re curious about the Enigma Machine itself, as well as the flaw that turned out to be its undoing, there is another video that describes how Turing and his team broke the code:

Cheers,

    - Scott

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