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:

[sc name=”block_box” text=”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.” width=”500″]

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.

The Radiant

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.

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:

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.”

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 wo resent the idea that there is a depressingly slow speed limit imposed on us by nature that 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 (more precisely, it rotates or twists spacetime coordinates).

Tech Note 1: In the Star Trek original series, their “Warp N“ actually refers to 3-dimensional space warp, and so goes as the cube of my linear-scaled “Warp”. So technically, when Trekkies refer to Warp 2, it corresponds to my Warp $2^3$ = Warp 8. In Next Generation Star Trek, the exponent was 10/3. Go figure. For my discussion, I will stick with the linear scale.

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

Let’s start with Einstein’s equation:

$$E=mc^2$$

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.

Tech Note 2: “Effective” velocity is non-standard terminology. In the literature, this would be the velocity as measured by the passenger’s Proper Time.

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.