The Planet Uranus in Opposition

Note: I hope to update this piece when I catch Uranus at the turning point, August 2018. As of right now we are in thunderstorms, so it may take a while.

Okay Let’s Cut to the Chase

Here is an animated GIF of two astrophotographs I took of the planet Uranus from my house in Virgin, Utah. The first one was taken on October 24, 2017 around 2:00 am, the next on November 19,2017 at 10pm (click on the image for full-size animated versions). See if you can spot Uranus. In the course of that month it has moved a bit, near the center of the image, so you should be able to see a blue-green dot jumping back and forth.

One-month TIme-lapse of Uranus. (Nikon D-5000, F9 3 minute exposure, equatorial mount)

If you still can’t catch it, here is an annotated versions, with labels and stuff (again, click on the images for the full screen version):

In addition to the dated labels, I have put in some graphics showing the constellation Pisces, as well as a chart, showing where computer models say Uranus should be in that part of the sky, for various dates between 2016 and 2020. I had to pull all of these other things in, just to convince myself that I really caught the planet, and not just a random earth satellite or other transient object.

It has taken me quite a bit of work to get to this final product, of which I am quite proud, and happy that it came out so cleanly, riding exactly along those predicted lines. In August of 2018 (now) I hope to capture that endpoint of maximum extent. The rest of this blog piece is the retelling of the story of this image, along with the occasional digressions into the geometry of the whole thing.

About the Planet Uranus

Uranus (Wikipedia) Voyager II photo 1986

Here is a picture of the planet Uranus, taken by the Voyager II spacecraft in 1986.

By the time I came to work at the NASA Jet Propulsion labs in ’87 the Voyager II probe had already passed by Uranus and was approaching Neptune, so I never got a chance to see these “live” images coming in. It’s not much to look at, and is best described as a large ice ball (unlike Saturn or Jupiter which are mostly gaseous). Even with a really good earthbound 8″ telescope, you’re not going to see much more than a fuzzy dot.

Though it had been seen before (even in ancient times), the object was not identified as a planet until it was observed and reported by William Herschel in 1781, who thought it might be a comet. However, after reporting it to the Astronomer Royal Nevil Maskelyne (who figures prominently in the quest to measure Longitude), Maskelyne concluded that it was probably a planet.

Other than its name (being the only one in the solar system based on the original Greek gods, and not the later Latin names of the Roman gods), Uranus is notable for having its rotational axis nearly horizontal to the orbital plane, so that for half the Uranian year (about 45 earth years) the “north” pole is in perpetual day, and the other half the year is perpetual night.

Uranus in Opposition

What started all this for me was the announcement last month that the planet Uranus was in what they call “opposition“, meaning that it was on the opposite side of the celestial sphere (as seen from Earth) as the Sun. From the Sun’s perspective, this means that the Earth and Uranus are on the same side of the Sun, and typically on closest approach to each other:

Planetary Opposition (source: Wikipedia)

The news in social media suggested that it would be so close that “it could be seen with the naked eye.” That sounded like hogwash to me, as there have been a lot of viral bogus memes around about being able to see things like the rings of Saturn and such.

Having now tracked down the planet, I can attest that — technically — it would be possible for a young person with excellent sharp eyesight to see the planet Uranus without binoculars … if they knew exactly where to look, and gazed at it out of the corner of their eye, and in a place (like where I live) with extremely dark skies and no cities nearby, but only on a cool clear night. But otherwise, forget about it.

The Plan

Barn Door Equatorial Mount

Anyway, with the announcement of the opposition of Uranus in October I decided that this was a good opportunity to do some amateur astronomy and try to capture Uranus with some very low-tech equipment, which is a Nikon D-5000 camera mounted on a crude “Barn Door” equatorial mount. Using this mount, I can take long-exposures of up to 15 or 20 minutes, without smearing of the stars due to earth’s rotation.

The Barn Door mount is a clever contraption which anybody can build with $20 of parts from Ace Hardware or Walmart. The idea is simple, you just have two boards attached with a hinge, one board fixed to a tripod. You line up the hinge with Polaris (the north start), and mount the camera to the board that moves. But before setting up my rig, I first had to track down the current position of the planet, which was said to be somewhere inside the constellation Pisces (the fish). Credit here must be given to Martin J Powell’s website NakedEyePlanets.com, which has this great chart of the path of Uranus: Navigating the Stars For anyone interested in astronomy, one way to begin is to learn how to find your way around the sky visible to the naked eye, without aid of telescopes and such. To do this, you need to learn some old-school tricks, in the form of stories. For example, to find Polaris, you first find the Big Dipper (Ursa Major) and follow the line traced by two of the stars in the “pan” of the dipper. Rant: With the latest GPS-enabled telescopes, it is far too easy to track down stars, planets and other bodies. These days, all you need to do is type in the name of the object (e.g. Uranus), and the telescope’s computer will use its GPS to determine where the telescope itself is located, as well as the current date/time, and then guide the telescope to the place in the sky where the object may be found. Or you can use one of the “Planets” apps on smart phones, which you can hold up in the sky and see what you are looking at. The Greatest Women Mathematicians I have to admit a reluctance to putting the modifier “Women” in this post, because it would seem to imply that on an absolute scale the mathematicians I mention here are not intrinsically great. Perhaps a better title would be The Greatest Mathematicians (who happen to be Women). In any case, it has always bothered me when I see girls and women either discouraged from or outright forbidden from becoming mathematicians. One way or another, it is I think a sign of our times that “mathematicians you’ve never heard of” is kind of redundant. I present these mathematicians in no particular order, for many reasons. Among those reasons is my personal opinion that the field of mathematics itself is not (again popular notions notwithstanding) like a vertical ladder, where first you learn counting, arithmetic, then algebra, geometry, trig, calculus and so on. In fact Mathematics, like Art, is more of a tree, with many branches, and many ways of thinking and seeing things. Some math is visual, some verbal, even some tactile. The fields these women pursued were likewise in many different areas, and their peculiar genius or accomplishment in each was profound. I won’t talk about all of the women pictured above, just the ones about which I would like to make a point. If you like, google “Greatest Women Mathematicians” for a very long and interesting list. Maryam Mirzakhani When I was writing this piece this morning I was shocked and saddened to see that had just died last year (2017) of breast cancer. She was only 40, but had already done some profound work in geometry, especially Riemannian geometry — used by physicists in general relativity and elsewhere. She won the Fields Medal for her work in 2014, and became the first woman in history to win this award, described as the “Nobel Prize in Mathematics”. Maryam was born in Iran, and upon news of her death, a number of Iranian newspapers broke the taboo of printing a picture of her (a woman) with her hair uncovered. Cathleen Synge Morawetz Just one month after Maryam Mirzakhani died, we also lost Cathleen Morawetz (1923-2017), Professor Emeriti at New York University. Unlike most of the other mathematicians in this list, I had the great fortune to meet and get to know Cathleen in the 1980’s, while doing postdoc work at the Courant Institute in New York, where she at the time was the Director. I had gone to Courant to continue my studies of nonlinear wave equations, and Cathleen had made much of her own fame in that area, studying compressible fluids and shock waves. She was also the creator of the “Morawetz Inequality(ies)”, which have proven to have many uses, even to the understanding the stability of Black Holes. Cathleen was a very smart and jovial woman, and I will miss her. Emmy Noether Going back a bit, it would be difficult to convey just how profound and far-reaching was the work done by Emmy Noether, who lived from 1882 to 1935, and whose work touched many different branches of the tree of mathematics, including abstract algebra, geometry, and dynamical systems. One of the most profound theorems she proved (actually two with her name) is now known as Noether’s Theorem. What Noether’s (first) Theorem says is that for any conservation Law (such as energy, momentum, charge, etc), there is a fundamental geometric symmetry in the universe that corresponds to it. To express this poetically, Emmy proved that in mathematical physics, Truth (Law) is Beauty (Symmetry). Emmy’s Theorem resolved questions that Einstein had not been able to solve (!), and Einstein lobbied with Göttingen University (where she worked without pay or title) to promote her to a professorship. Eventually she was made professor, but with the rise of Nazi Germany soon had to leave the country for the US, due to her Jewish ancestry. Sofie Kovalevskaya Sofia “Sofie” Kovalevskaya lived from 1850 to 1891, and like Emmy Noether made substantial contributions to mathematical physics. She was a true pioneer, the first European female to earn a PhD in modern times. Together with Augustin Cauchy, she proved the Cauchy-Kovalevskaya Theorem, regarding the solutions to many equations in physics, especially those governing waves (light waves, sound waves, matter waves etc). Without her work I likely would not have had a job. Sofie was good in math but unlucky in love, her heart often broken. She had married and had children early on, and occasional star-crossed relationships later, but was also an early radical feminist and maintained a close and possibly romantic relationship with playwright Anne Charlotte Edgren-Leffler, the sister of Gosta Mittag-Leffler. Besides her main theorem, she was also the discoverer of what is now called the Kovalevskaya Top, an exact solution to a spinning top that completed work begun long ago by Euler and Lagrange. She was also a writer, and wrote “Nihilist Girl”, a semi-autobiographical work. “It is impossible to be a mathematician without being a poet in soul.” –Sofie Kovalevskaya. Florence Nightingale (Yes that Florence Nightingale) Diagram of Causes of Mortality (click to enlarge) Besides being the founder of modern Nursing, Florence Nightingale had a knack for mathematics and especially statistics, and made great contributions in the visual display of quantitative information, a field which later was made popular by Edward Tufte in his seminal works. Ms. Nightingale was one of the first to make use of the Pie Chart, making clear causes and relationships in mortality among WWI soldiers. Hypatia There are so many others, such as Maria Gaetana Agnesi (the first woman appointed as full professor, but who died and like Mozart was buried in a pauper’s grave) but on my short list I have saved Hypatia for last. No likeness has ever been found, but she was said to be as beautiful as she was smart. The first documented female mathematician, Hypatia lived in 400 AD in Alexandria, and is considered by many to be the patron saint of mathematics. And a martyr. She was the daughter of the mathematician Theon, and inherited from him the position of Director of the Library of Alexandria, the ancient repository of world knowledge. Though Theon was considered a great geometer and wrote many treatises on Euclid, Hypatia was said to have surpassed her father in mathematics and astronomy, made astrolabes, and wrote many other works and commentaries on geometry. None of Hypatia’s works have survived, nor much of Library, whose destruction was considered one of the great tragedies in intellectual history. Hypatia was brutally assassinated by christian extremists, opposed to the “practice of sorcery, witchcraft, and mathematics”. Ironically, she was also a great teacher, and one of her most devoted students was Synesius, who studied under Hypatia as a neoplatonist, but eventually he converted to christianity and became a bishop, and contributed to the understanding of the doctrine of the Trinity. Hypatia fought hard to save the Library, but the world was changing and she could not stop it. So much was lost when the Library fell. With it was lost much knowledge, and science, and wisdom, that we will never recover. The fall of the Library presaged the Dark Ages. Had the Library stood, some have said, we might have landed on the moon in 1492, not just Florida. To be a woman. To be a scientist. To be a mathematician. All these things require more of one than any of us could ever know. Be brave, these women tell us. Be brave. Relativity, Simplified. Really. I’ll make this (mercifully) short. Consider this statement, which I call Ω:  Everything travels through the Cosmos at exactly the speed of light. Ω is Relativity. That’s it. Really, it is. And Ω is true not only for Special Relativity, but the General one too. All you have to know is that the “Cosmos” means both space and time. If you were to ask somebody what Relativity says, they would probably say something like “everything is relative.” The problem with that is, it’s not precise. In fact, it’s not even true. Some things in physics (and the world) are thought to be absolute. So I’ve been trying to come up with a good version of Relativity that can be justified mathematically, and that little box is what I came up with. If you understand what each word means exactly, everything follows from this statement, and if you like you can stop reading now and get on with life. Because that really is what relativity says. The rest of this short post is just me rambling about why I like it. In a later post I’ll defend it. Also, here is a cool picture of a galaxy for no reason. But it’s cool (*). Why I like This Version of Relativity One of the things I like about this version (Ω) is that it is simple and precise, sounds a little strange — but just enough strange to be right — and answers immediately a number of other questions people ask about relativity, matter, speeds etc. For example: Q: Will we ever be able to go faster than light? A: NO. The reason you can’t go faster than the speed of light is that you can’t go any slower either. Nothing in the universe can go any speed in spacetime but c, the speed of light. The only thing you can ever change is the direction in space-time that you are going. Q: Why does Relativity say that when you go faster, time slows down? A: Because you are always going exactly at the speed c, so if you go faster in the space directions, you have to go slower in the time direction so it still adds up to exactly c. See? Details later, film (and math) at 11. But really, it is true. Ask a physicist. He’ll scratch his head, then say yeah, that works, then go have a beer. (*) Photo: (Image Credit: X-ray: NASA/CXC/CfA/R.Kraft et al.; Submillimeter: MPIfR/ESO/APEX/A.Weiss et al.; Optical: ESO/WFI) Automotive Math Calculus is taught all wrong and too late. The basics could be taught in the car on the way to Kindergarten. Here is how it goes. Any kid who’s been in a car knows that the speedometer tells you how fast you are going, and the numbers in the odometer (mileage) tells you how far you’ve gone. Another way to say that is that the speedometer shows how quickly the odometer is changing. And another way to say that is that the speedometer is the “differential” of the odometer. To be fancy, we can use a small “∂” for differential and so we say: $$Speedometer = ∂\,{Odometer}$$ That’s differential calculus. How things change. Subtraction. Put another way again, the odometer tells you what the Sum total distance was after going all those speeds that the speedometer indicated. Instead of saying “Sum” with a big S we stretch it out into a long skinny S like this:$\int$and we say $$Odometer = \int { Speedometer }$$ That’s integral calculus. How changes accumulate. Addition. Subtraction undoes addition. That is the Fundamental Theorem of Calculus. Duh. Next stop, Rocket Science ! THE END. From Euclid to Euler to Einstein It is of course the holiday tradition this time of year, to exchange gifts and ponder over how you would explain modern mathematics to the ancient Greeks. In line with the latter part of that tradition, I’ve been sketching out a diagram to explain Euler’s number$e$(2.71828…) to Euclid. It turns out that even though the classic Greek mathematicians knew all about the number π (3.1415…), they never knew about or defined the number$e$. Which is a shame, because they could have. And had they done so, they could have beaten Einstein to the punch 2500 years earlier. Just a quick note here: for those of you who have not heard of$e$, it pops up all over the place in science, and especially when things are growing or accelerating. For example, suppose you just crossed the state line, and for some reason you thought that the mile-markers were actually speed limits, so that at the one-mile marker you slowed down to go at one mile an hour, and so on. Suppose that there were a lot of mile markers along the way, and so you were continuously speeding up with each marker. Obviously you would be going pretty slow, but at least you are speeding up. It turns out that if you obeyed the signs to the letter, by the end of one hour from mile marker one you will be at the$e$mile marker, and would be going$e$miles an hour. In any case, after much fiddling around and fanfare, here is the diagram I came up with that I think would make Euclid happy. It is a “proof without narrative”, and simply uses the classically understood conic sections (e.g. circles, and hyperbolas) to show how the numbers π and$e$may be used to relate areas of pie-shaped sectors in two conic sections, to the linear measurements along their respective curves: One of the things I like about this diagram is that on the one hand it shows how these two numbers are similar, in that they both provide a ratio relating the area of a sector in each type of conic section, with a linear measure, but on the other, we see how these two numbers differ in a fundamental way with successive sectors. For circles of radius 1, its area compares with its radius squared by a ratio of π (so the pie-slices are each π/8). For the hyperbola, drawing a line from the center to the vertex of the hyperbola, a sector of area one is made by drawing a second line whose x-axis length differs from the area by a ratio of$e$. In both cases we have a ratio relating a linear measure to an area. But at this point the similarity ends. For as we go to successive circular arcs, the areas remain in fixed linear ratios, so to produce a quarter of a circle, you have an arc-length of π/4, and so on. But for the hyperbola, to produce a sector of area 2, you need to draw a line segments whose x-axis length is not$2 * e$, but$e$to the power of 2, in other words$e^2$. For an area of three, you need to use$e^3$, and so on. So what we see is that the number π seems to be most commonly used as a linear factor or ratio, having to do with rotational symmetry in space, while the number$e$seems to be used as the base of an exponent, and is involved with things that grow exponentially over time. Which brings us to light, waves, and Einstein’s space-time. What do cones, planes and conic sections have to do with spacetime? Suppose you turn a flashlight on and off quickly. The light pulse from that event travels out in all directions at the same speed,$c$, the speed of light. Einstein (and Minkowski) suggested that we view the event where time plays the role of a fourth dimension. If we toss out one of our three dimensions, and make the time dimension the z-axis, we can visualize the light propagating out. So in the picture on the right, the horizontal plane represents space at time$t=0$, and the vertical dimension is time, with the “up” direction representing the future, and “down” representing the past. The flashlight has just gone off at time zero, but now the light wave is expanding out in a circle, getting larger with time. And so as it grows over (upward) time, the expanding circular wave traces out the “future light cone”. Conversely, all of the light from the past that reaches us can only come from the region below the plane, marked by the “past light cone”. The thing to note is that these “space-like” planes are always horizontal, though they may tilt a little due to relativistic motion of the observer. Space-like planes can be identified by the fact that their “normal” line (the one perpendicular to the plane) are pointing roughly up, in a time-like direction. Space-like planes can only intersect light-cones in circles or ellipses. In no case can an observer’s “plane” ever become vertical, so that its normal vector is pointing in a space-like direction outside of the light cone. Such planes are called “time-like”, and have the property that they always intersect light cones in hyperbolas. So I am hoping that you are starting to see how I think these two numbers$pi$and$e$are related, but also very different. Somehow, the number$pi$is related more to space, and to circular rotation in space, while$e$seems to be related to time, hyperbolic curves, and exponential growth over time. It turns out that we can even be very specific about how$e$and$pi$are related to each other, but it requires the introduction of a number that the ancient Greeks would have no concept of, and that is the number$i\$, the square root of negative one.

The relationship was itself discovered by Euler himself, and has come to be known as Euler’s Equation, and has also been called (at least by mathematicians), The Most Beautiful Equation in the World. I hope some time in a future post to try to explain what the equation means, but for the moment, we will just display it here and be done with it.

$$e^{i\pi} + 1 = 0$$

And yes, this is how I spend my holiday vacations. Having Fun ! Happy new year !

The Geometry of Meteor Showers

Whenever a meteor shower is coming up, the news gives details on how to find the constellation in which the “radiant” can be found. Don’t bother trying to find the constellation. Too much work. Here is all you really need to do:

 On the night of the shower, go outside around 2am. Look eastward, toward where the sun has been rising, and halfway up the sky, along the path the sun takes. That's the center ('the radiant'). Further away from this point the meteor trails will be longer.

That’s it. The rest of this post is just my rambling about the geometry (or astrometry as it were) that makes this all work. You won’t need it. If you come out sooner, around midnight, the radiant will be close to the horizon, and as it gets closer to sunrise the radiant will be almost overhead.

If you study the pattern of meteors in the picture above, it looks like we are flying through a bunch of stars very quickly, and that the center point where all those stars appear to be streaking from is simply the direction that we are flying.

It turns out that is exactly what you are seeing. The center point (in the upper left quadrant of the picture) is called the Radiant of the meteor shower, and it is the current direction in which the earth is moving, as it travels along its orbit around the Sun.

The Picture

Here is a (simplified) picture describing the general situation. To keep things simple, I’ve put the little guy (who’s supposed to be us) right on the earth’s equator, around 3am his time. We are looking down at the earth from above the North Pole, and the earth is rotating counter-clockwise on its axis. Meanwhile, the earth is travelling around the sun at 18.6 miles a second, going from right to left in the picture.

The comet dust in the picture was left behind by a comet years before, and now is for the most part not moving much. The earth however is plowing through the dust trail at 18.6 miles/second, and so the relative motion of the dust to the observer is likewise 18.6 miles a second, or about 30km/s.

That speed, by the way, adds a lot of energy to the situation. Many of the comet dust particles are small, some just grains of sand. But if we take a quarter-inch piece of iron, with a mass of one gram say, and compute its kinetic energy when the earth hits it, we get

$$E = \frac{1}{2}mv^2 =\frac{1}{2}(1gm)(30km/s)^2 = 450,000 Joules$$

Now a Joule is the amount of Energy to drive a one Watt light bulb for a second, which is about how long a meteor flare takes. So the light that our quarter inch piece of iron is putting out during that second is close to a half a megawatt of power. Impressive.

Another Picture Closer In

So here is a much closer picture. We’ve now rotated the picture so that the little guy is on “horizontal” ground, and we only see a small slightly curved part of the earth. The atmosphere is a very thin shell not more than 70 miles above the earth (1 percent of the earth’s diameter), and the shaded part is what the little guy can see from where he’s standing. It is a flat lens shaped piece of atmosphere, and the comet dust is coming in at about a 45 degree angle, about to slam into that circular lens. I’ve drawn a cylinder around all of the dust that will hit the part of the sky that the guy can see.

Now if you look at the cylinder of comet dust coming at you from the little guy’s perspective, the rays of dust look like this:

Which looks just like the photo of the Geminid meteor shower. So, the reason that showers look like rays flying away from the Radiant is simply a matter of perspective, and the Radiant itself is just the direction that we are are flying, along earth’s orbit.

A Bigger Picture, Further Out

Just to tie everything together, here is a diagram showing the geometry of a typical meteor shower, arising from a regularly reappearing comet such as Halley’s comet:

In the case of Halley’s comet, the diagram shows how the orbit of the comet may intersect the Earth’s orbit in two places. In the current picture, the Earth is passing through one of the intersections, and is going in the direction of the constellation Orion (bottom left). This is the Orionid meteor shower, which this year (2016) will be visible from October 2 to November 7. The other intersection occurs when the Earth is heading in the direction of Aquarius, which happens around May 5-6, during the Eta Aquarid’s meteor shower. Not all comets have orbits which intersect Earth’s orbit twice, but Halley’s does.

Rainbow, Part II: Yellow Is An Idea

This is part II of my discussion of color which began with Part I, “The Infinite Piano”. In the first part I explained that the colors of the rainbow are single “notes” on an infinite piano whose keys are pure “tones” of light, and the “sheet music” for a more complex color such as PINK can be written as a 3-note chord composition in RED, GREEN, and BLUE. This composition can be written out over the color piano keyboard with three vertical bars, each indicating the loudness or softness of each of the three keys we need to play, using ranges from 0 to 255, like this:

We can further shorten this musical notation by saying (Red,Green,Blue) = (255, 192, 203). Now you may think that I just made up those particular numbers, but in fact if you check with Wikipedia, the internet standard for color on computer displays has exactly these three values for the color pink. They chose the range 0 to 255 because it is easy to express using 8 bits — which makes computers happy.

We live in the computer age, and this (R,G,B) system is now used to define all the colors that you can see on a computer monitor. So, it sounds like color is three dimensional, and you can represent any color in nature (or at least in a photo of nature) using just three colors. But is this true ?

Anyone who has tried to match paint colors may doubt this. Each paint manufacturer has their own system of specifying colors, and complex formulas of mixing their “component” pigments into Salmon, Chestnut, or other copyrighted name and color. There are many systems of defining color, such as Munsell and CIE-Lab, which are 3-dimensional, like this:

3D Munsell color space (Wikipedia – credit)

These systems are oriented toward luminance-based applications such as TV’s and computer monitors that emit their own light. There are also CMYK (Cyan-Magenta-Yellow-BlackKey) and Pantone™ systems, which are effectively 4 dimensional dimensional and used mostly in pigment-based applications such as printing and paint. But Pantone also had a six-dimensional version called Hexachrome, which add Orange and Green to form a CMYKOG space (now discontinued), and there is also a CcMmYK system used in six-color inkjet printers. These latter are called “subtractive” systems, because the pigments effectively absorb colors from white light to give you their indicated color.

So clearly something must be going on. Why do we even think color is three dimensional, when there are so many color systems using more than three. What’s up?

The Yellow That Isn’t There

Let’s take a closer look at this Wikipedia computer color thing. If you look up Yellow in wikipedia, you’ll see that standard Yellow is defined in color coordinates by (R,G,B) = (255, 255, 0). But if we plot that “musical chord” out on our piano we get this:

Now this is crazy, because there is clearly a “yellow” key halfway between green and red, and we aren’t hitting that key at all. Instead we are leaning with a strong 255 “forte” on both RED and GREEN. Indeed, in the same Wikipedia entry for Yellow, it indicates that the “spectral” coordinates of Yellow is 570–590 nanometers. This is the wavelength of the light which is colored yellow in the rainbow spectrum.

To understand what is going on requires an understanding of human beings more than the color spectrum, and how we evolved. Modern humans perceive color with the use of three kinds of cells in the retina of our eyes, called cones. These cones come in three types, each of which respond only to specific “chords” in the color spectrum. The three chords look something like this (approximately):

What this says is that we have in our eyes three kinds of cells (not counting rods which detect brightness), which respond to “color chords” that are centered (roughly) around the blue, green and red keys. There is no cell that responds just to “yellow” chords, and so the way that we “see” the yellow color is that our brains get strong positive signals from both the Green and the Red cones.

One of the interesting consequences is that it is possible to make a person “see” yellow even if there is no yellow in the light at all. All you have to do is to take a pure green and red light (such as from two distinct lasers), and shine them on the same spot on the wall:

Our retinas will report to the brain that where they intersect it is getting a strong green and red signal, and the brain will interpret that as yellow — even though a light spectrometer pointed at the wall will report that there is no yellow there at all. It is a color optical illusion !

Here is the take-away from all this: the color YELLOW is an IDEA, as are all other colors. It is something unique that our brain thinks — a state of mind — in response to what the outside world is doing. In the case above, the YELLOW our brain “sees” is entirely in our own heads. Now most of the time, in nature, there really is a yellow frequency light wave “out there”, and we know from the yellow in the rainbow that this frequency of light actually exists. You can create a pure yellow by simply dropping salt into a flame (sodium ions radiate at that color). But the idea of yellow must be distinguished from the light that usually triggers it.

And so, YELLOW as a specific color of light must be understood as a separate dimension from RED, GREEN, and BLUE. So how many dimensions does color really have? We will explore this further in the next post, “Shadows of The Infinite.”

The Colors of the Rainbow, Part I: The Infinite Piano

The phrase “All the colors of the rainbow” is often used to refer to every imaginable color that you can see. What is interesting is that almost the exact opposite is true: With the exception of the rainbow itself, you almost never see the colors of the rainbow in nature, and indeed almost all of the colors that you do see are NOT in the rainbow.

Look closely at the rainbow spectrum above. Try to find Pink. Or Brown. Or Teal. Or Chartreuse, Mauve, Vermillion, etc etc… You won’t and you can’t. So what’s going on?

Think of it this way: picture the rainbow spectrum above stretched out over the keys of a piano. But not just any piano will do, and 88 keys are nowhere near enough. You will need a piano where the keys are infinitely thin, and there are an infinite number of keys, so the keyboard looks like this:

So the idea is, each color in the rainbow is just a single (very thin) key, a single note on the piano, and as you run your finger along the piano, playing a glissando, you are really just playing just one note at a time. But in our world, the colors that we see are each a chord, made up of many of these keys played together. You will need a lot of fingers, and a hand-reach far beyond that of even Rachmaninov, covering the entire piano for some colors.

And it has to be a real piano, not just a harpsichord where strings a plucked. Remember, the reason a piano is called a piano is that you can play each note soft or loud (piano e forte = soft and loud), depending on how hard you hit the key or step on a pedal. So, in the real world, if you see a green leaf, for example, most likely what is being “played” is a very strong solid GREEN fortissimo note, with millions of close “greenish” unison notes nearby but more pianissimo, kind of like this:

Just to explore this piano metaphor a bit further, we should note that light is a wave just like sound, and has specific frequencies and wavelengths. But one difference is that we can hear a very wide range of frequencies of sound, across roughly ten octaves. Since the speed of sound and light are so different, let’s put it in terms of wavelengths. Each octave is half the wavelength of the previous one, and so for sound the range of wavelengths goes from 17 meters (low pitch 20 Hertz) to 1.7 cm (20,000 Hertz). The standard piano covers about seven of those musical octaves. By comparison, the wavelengths of light we can see go from deep red, about 700 nanometers (billionths of a meter), to deep violet, about 400 nanometers. In other words, the color/light piano usable to humans is just short of covering a single octave of light. Not much opportunity for harmonizing, although some shades of violet could be a perfect fifth above deep red.

(I should apologize for one mistake in my piano picture: to make the analogy exact, the RED should be at the left, as it is a deep low-frequency bass, while violet should be at the right, a high-frequency treble. So let’s call this a left-handed piano get on with life.)

So where are all of our more familiar colors located? Some of them are fairly complicated chords. For example, you might play a RED note loudly, a GREEN note softer, and a BLUE note just a bit more strongly … and if you did, the name of that chord is — guess what? —  PINK.

The “sheet music” for this single 3-note chord composition could be written out over the keyboard with three vertical bars, each indicating the loudness or softness of each of the three keys we need to play, like this:

We could even assign numbers to each of these loudness values, say, from 0 being absolute quiet (ie, don’t touch the key), to 255 being the LOUDEST you can hit the key. In the case of “PINK”, it would look something like this:

We could even shorten this musical notation by saying (R,G,B) = (255, 192, 203). Now you may think that I just made up those particular numbers, but in fact if you check with Wikipedia, the internet standard for color on computer displays has exactly these three values for the color pink.

So, the take-away from this first part of my blog is that the universe of color is much larger than the single keys on the rainbow piano. You’ve got to play chords. But even then it gets complicated, and more interesting, which we’ll see in part two, “Yellow is An Idea“.

Bottom Line

For those with limited attention spans: yes, in this universe, with a powerful enough rocket you really can go anywhere in the universe as quickly as you like, in your own lifetime, without resorting to any medical tricks like suspended animation. Einstein’s theory of relativity won’t stop you from getting there, the same day even. The hard part is just getting enough energy — and working out the math.

Speed Limits

One big downer — if you can call it that — most people take away from Einstein’s theory of special relativity is that nothing can go faster than light. We are a species that likes to explore, after all, and the idea that there is a depressingly slow speed limit imposed on us by nature makes it very difficult to journey through the galaxy.

Science fiction often addresses this either by inventing a device to “warp” space-time (Star Trek), or by adding a few extra dimensions to the universe and bypassing normal space by jumping into hyperspace (Star War).

A lot of this comes, I think, from a confusion about how the universe actually works, as described by Einstein’s theory of relativity. (Note: given the recent creationist attempt to color the word “theory” as meaning something tentative, I prefer to use Richard Dawkin’s coined word “theorum” — similar to theorem — as indicating a theory that is so well established by overwhelming evidence that it might as well be an undebatable mathematical theorem).

The Confusion

Here is the deal: while it is true that to people watching from earth a spacecraft can never be observed to go faster than light, that doesn’t mean that the passengers on the spacecraft have the same experience. In fact, what Einstein’s theory would say is that as far as the passengers can tell, it seems like they can go as fast as they like. Due to the relativistic “warp” of space-time as you approach the speed of light $c$, the passengers experience time much more slowly and their “effective” speed as they travel through space appears to be much greater than light.

Let’s do the numbers.

Some Terminology

In the “Star Trek” series they used the “warp $N$” terminology to refer to “effective” speeds that were $N$ times the speed of light $c$, so that “Warp Two” for example was twice the speed of light or $2c$. In that series they had a special “warp drive” that bent space-time around so that they would go faster, but the actual fact is that in our everyday world just the mere act of going faster by any means actually warps space-time.

The Warp Equation

I plan on using the trekkie terminology (and standard relativity) to state and prove the following interesting fact:

 The Warp Equation If you have a payload with mass $m_{payload}$, and a means of converting matter into kinetic energy with 100% efficiency, then the mass $m_{fuel}$ of fuel needed for you to travel at an effective speed of Warp $\omega$ where $\omega > 0$ is given by$$m_{fuel} = {\omega}^2 m_{payload}$$

So for example, in order to travel at Warp 2, a person of mass 80 kilograms would require 320 kilograms of (say) a proton-antiproton fuel in order to travel at that effective speed. That is roughly equivalent to 6,400 Megatons of TNT. Coincidentally, that is almost exactly the combined explosive power of all nuclear weapons now on our planet. That is a hell of a lot of energy, but the point to be made is that is within the bounds of our current technology.

The fact that you have to square the warp factor to get the amount of energy to go that speed makes perfect sense. Even in classical Newtonian physics, the energy related to going at velocity $v$ is given by

$$E = \frac{1}{2}mv^2$$

so doubling the velocity $v$ on the right hand side multiplies the energy by four. The fact that the energy happens to be equivalent to four times your payload’s mass comes from Einstein.

The way in which we’ll prove this is to first calculate how much matter is needed to attain an observed velocity v, and then figure out what the relationship is between the observed velocity, and what effective velocity the passenger actually experiences. Note: I have no doubt that there is probably an easier way to derive this formula. But this is the one I came up with and it isn’t all that complicated.

Conversion of Matter to Kinetic Energy

$$E=mc^2$$

What we are going to do is to use this equation, together with the law of conservation of energy, to compute how much matter it takes to accelerate a payload $m$ to (observed) velocity $v_{o}$. Now as the observed velocity $v_o$ approaches the speed of light, the relativistic mass of the payload becomes:

$$m_{relative} = \frac{m_{payload}}{\sqrt{1-(\frac{v_o}{c})^2}}$$

Now Einstein’s equation for energy represents both the energy of the mass at rest, together with the (kinetic) energy of the mass in motion. And so, if this mass was put into motion by the conversion (at rest) of a certain mass $m_{fuel}$, where

$$m_{fuel} = \alpha m_{payload}, where \alpha > 0$$

Then since energy is conserved we can relate the conversion of the mass $m_{fuel}$ into motion $v_o$ by:

$$(m_{payload}+m_{fuel})c^2 = E_{rest} = E_{moving} =\frac{m_{payload}}{\sqrt{1-(\frac{v_o}{c})^2}} c^2$$

so dividing both sides by $m_{payload}c^2$

$$1 + \alpha = \frac{1}{\sqrt{1-(\frac{v_o}{c})^2}}$$

squaring both sides and solving for $v_o$ we get the following rule:

 Matter to Velocity Conversion For a payload of mass $m$ and a ratio $\alpha > 0$, if fuel $m_{fuel}=\alpha m$ is converted to kinetic energy, the observed velocity $v_o$ of the body will be$$v_o = (\sqrt{\frac{\alpha}{1+\alpha}})c$$

This jibes with what Einstein said about observed velocities, as the right hand side will never be greater than the speed of light $c$. As the ratio $\alpha \rightarrow \infty$, the velocity goes to $c$, so we can get as close to $c$ as we like — but no further.

Velocity – Observed and Effective

So now we come to the idea of “effective” velocity. The weirdness of relativity comes from the fact that as the observed  velocity $v$ of ship approaches the speed of light, the passenger’s own time-scale is compressed by what’s called the Lorentz-FitzGerald contraction, according to the formula

$$t_{effective} = t_{observed}\sqrt{1-(\frac{v_o}{c})^2}$$

(From this point on we will just write $t_e$ and $t_o$ for $t_{effective}$ and $t_{observed}$ respectively) Then given a fixed distance $\Delta x_o$ as measured by the observers on earth, the effective velocity as experienced by the passengers when traversing that segment of space over their time $\Delta t_e$  is:

$$v_e = \frac{\Delta x_o}{\Delta t_e} = \frac{\Delta x_o}{\Delta t_o\sqrt{1-(\frac{v_o}{c})^2}}$$

which in turn simplifies to this formula for converting observed to effective velocity:

 Observed to Effective Velocity $$v_e = \frac{v_{o}}{\sqrt{1-(\frac{v_o}{c})^2}}$$

The important thing to note about this concept of “effective” velocity is that we are computing the ratio of observed change in our distance with our own experienced change in time. Suppose for example that over centuries the earth residents went out and placed “mile markers” along your path to your destination. Then once you got up to warp speed, not only would your sense of time be compressed, but distances that you measure out with your tape measure would also be compressed by the same Lorentz-FitzGerald contraction. Consequently, if you were going at Warp 2, for example, your sense of time elapsed going from one mile marker to the next would be cut in half, but also (this is the point) the distance between those two fixed mile markers as the earth-people laid them out would also appear to you to be half. So if you were instead to calculate your velocity by dividing your measured distance by your time, you would never get a value at or greater than c.

All Together Now

So if we start with fuel $\alpha m$ which we use to accelerate our mass $m$ to the observed velocity $v_o$, we can use the two formulas we just derived to express the effective velocity $v_e$ as a function of $\alpha$. We can rewrite the “Matter to Velocity” formula as

$$(\frac{v_o}{c})^2 = \frac{\alpha}{1+\alpha}$$

So our effective velocity formula simplifies the bottom of the fraction to

$$v_e = v_{o}\sqrt{1+\alpha}$$

and then substituting the formula again for  $v_o$ we see that our fuel mass $\alpha m$ gives us an effective velocity of:

$$v_e = c\sqrt{\alpha}$$

Thus if we have defined velocity “Warp $\omega$” to be $\omega c$, then we can write

$$\omega = v_e / c = \sqrt{\alpha}$$

So that to attain an effective velocity of Warp $\omega$ we must use a fuel-payload ratio of $\alpha = \omega^2$, ie

$$m_{fuel} = \omega^2 m_{payload}$$

which is exactly the “Warp Equation” we were to prove. QED

Let’s Do the Time-Warp Again

It should be pointed out that of course to the observers on earth, even though you are going at an effective speed of Warp $\omega$ you will never appear to be going faster than $c$ and so it will take you a long time to get where you are going. You, however, will not experience that, and so you will effectively be travelling through time much faster than your friends at home. How much faster? According to our formula above relating $t_e$ to $t_o$, and expressing that in terms of the warp factor $\omega$, we can show that the time-warp you experience will be:

$$t_e = \frac{t_o}{\sqrt{1+\omega^2}}$$

And so, in our example, the 80kg person travelling at Warp 2 will feel like they’ve reached their destination in $1/ \sqrt{5}$ of the earth time, ie getting them there in about 0.44 of the time observed on earth, and exactly twice the time it would take light to appear to get there.

So, not only can you go as fast as you like, you can also travel as far in the future as you like. For example, to travel 100 years into the future, just get in a spaceship armed with 10000 times your own mass in matter-antimatter fuel, and then travel at Warp 100 for one year. When you reach your destination, one year will have passed for you, and 100 years (plus a little bit) will have passed on earth.

Of course, then you’ll have to get back to earth, so good luck with that.

The Buckaroo Banzai Principle

 The Buckaroo Banzai Principle No matter where you go — there you are. — Buckaroo Banzai

The point of this exercise is that if you really understand what Einstein said, the idea should be that there is no absolute frame of reference. What this means is that even if you are travelling at 99.999 % the speed of light relative to the earth, as far as you know everything still looks and feels like Newton’s physics, where F = ma and you can always accelerate faster and faster. And not only that, but if you are heading for a specific location, the faster you go, the faster you will get there.

No Free Lunches

Now having said that, there are some consequences that the universe may unleash should you decide to try to go Warp 100. This is because even though the physics of your spaceship will be the same even at this insane speed, you are also surrounded by the gases in your local galaxy, as well as all of the light from stars that are visible to you. And even though from the earth much of this light is nice, low-energy visible spectrum, and even though that light will still be reaching you at the speed of light, it’s relative energy is radically different when you are plowing through that light at Warp 100. In fact, what you will be observing is a massive Doppler-shifting into the deep blue/ultraviolet of all light coming at you in the direction you are headed (and conversely, red-shifted looking back towards earth). Some of this light may be equivalent to the powerful cosmic rays that hit the earth, and which were generated by massive explosions or quasars just after the Big Bang. The energy in these photons may be enough to kill you all by themselves, especially at Warp 100. You may need a very large and thick radiation shield, along with all the extra energy to carry that shield along with you and your ship.

And so as we already should have known, there are no free lunches. At least it is nice to know that a faster-than-light lunch is available, should one choose to pay the price.