Rainbows Inside and Out (Part1)

Consider a rainbow, such as the one pictured at right. Perhaps you already know that the colors of the rainbow are caused by raindrops’ decomposition of white light into its various colors.  But, how does that happen?  In this post I want to talk about how raindrops partner with sunlight to make these colors and, perhaps more importantly, to tell you about a way in which you can bring rainbows indoors to be enjoyed on any sunny day, not just on rainy ones.  But, because of the length of this post, you'll just have to hold on until the next to find out how to accomplish that.

But, first, where do rainbows come from?  This question reduces to "How does light interact with a raindrop?"  The problem is that a raindrop is a little sphere and a sphere is a rather complicated shape with which to start the discussion.  So, let's take a stepwise approach and begin with a simpler model, a cube of glass.  Suppose we shine a narrow (say, 1/16" diameter) ray of white light from the Sun onto the surface of our glass cube.  The cube diagram, at left, shows the light ray arriving at the surface of the cube at an angle θ₁ to the “normal” to the plane.  [The symbol, θ, is Greek letter theta – for some reason, scientists like to use Greek letters whenever possible.]  The normal is a line perpendicular to the plane of the cube face.  When the light ray enters the glass medium the ray suddenly finds itself in the presence of a much larger concentration of electrons than was in the air from which it came.  The drag of the interaction of the ray with these electrons causes the velocity of the ray to decrease by 1/3 from c = 186,000 miles per second to v = 124,000 mps.  This large and sudden change in velocity rotates the light ray toward the normal line.  The ratio c/v is a property of the material and is called the index of refraction, n.  For glass, n = 186,000/124,000 = 1.50.  Most remarkably, the index of refraction is slightly different for different frequencies (colors) of light.  For Red light (in glass), n = 1.51 and for Violet, n = 1.53.  The angle of "bending" of the ray is governed by Snell's Law which predicts this angle for each frequency based on the individual indices of refraction.  Since the Sun's white light is composed of all the colors of the spectrum these colors begin to separate (disperse) as the ray passes through the cube.  Higher frequency Violet light bends more toward the normal than the lower frequency Red.  Snell's law predicts that if θ₁ = 45^o, Violet light is bent to 27.5^o and Red light to 27.9^o.  When the light rays emerge from the other side of the cube, the colors remain separated and pass from the glass medium back into the air.  Since the normal to the exit surface is parallel to the normal at the incident surface the colored rays bend away from the normal (because air is less dense than glass) back to the incident angle, θ₁.  Hence, though the colored rays are separated, they are parallel.  Because there is no further divergence of the colored rays once they return to the air, a cube is not an efficient dispersing shape.  We can calculate that if the glass cube above is 10 inches on an edge that when the refracted rays reach the opposite wall the separation between the Red and Violet colors is about 1/16 inch.  Remember, your beam was 1/16" in diameter, so the total separation of colors falls within the beam.  With a cube you can accurately measure the angle of bending of the light ray, but you won’t see rainbows.

A prism, at left, is a slightly more complicated object which does generate observable dispersion of the constituent components of white light.  It has a triangular shape when viewed along its long axis as shown in the diagram at right.  A ray of white light striking a surface of a glass equilateral prism at an angle θ₁ is bent toward the normal with Violet bent more than Red.  However, when the dispersed colors encounter the opposite wall of the prism they find a normal that is not parallel to the incident surface normal.  The colored rays thus emit from the prism not parallel but divergent.  The further you back away from the prism the greater is the separation of colors.  For θ₁ = 45^o the Red and Violet rays arrive at the opposite wall separated by 0.4^o - just as they were in the cube.  However, when they emerge from the prism they are diverging by almost 2^o.  At 10" from the prism surface Red and Violet colors are separated by ~3/16" and they are still diverging.

Now we are ready to talk about rainbows.  The formation of colors in a rainbow can be explained by reference to the classic diagram at right. A ray of white light coming over the Picasso-like figure’s shoulder strikes a spherical raindrop at an angle θ₁ to a line perpendicular to the surface of the rainbow at the point of contact. This is the normal line.  When the ray enters the aqueous interior it gets refracted, as before, by interaction of the ray with the denser medium - for water, n = 1.33.  The wave front of the slowed ray is turned toward the normal to a new, smaller angle, θ₂.  Higher frequency (Violet) is slowed more than lower frequency (Red) and the colors become separated as they pass though the aqueous interior of the raindrop.  Since the light rays producing a rainbow must come from behind us, the light we see is that which reflects off the back of the raindrop and comes back toward us as separated colors.  The normal line at the exit point is not parallel to the incident ray normal and the colored rays emerge from the raindrop not parallel, just as was the case for the prism.  Since we are several thousand feet from the raindrops that produce a rainbow we see a separation of colors many feet wide.  Remember the old saying that there is a bucket of gold at the end of the rainbow?  If you go to the spot where the rainbow forms, you won't see gold and you won't see a rainbow because the width collapses to almost zero.  Wise investment practice is probably a better wealth creation vehicle than rainbow chasing.

That explains rainbows outside, so what about bringing rainbows inside the house?  Well, our discussion of refraction has gone on for awhile, so we’ll have to talk about inside rainbows in the next post.  Hint - this doesn't involve creating a mist in your house.  Instead we will make use of the object at right – an Asfour Crystal Ball Prism.  See you then.

Transit of Venus Adventures

The 2012 Transit of Venus (TOV) attracted quite a bit of media attention because it was the last such transit we will see until 2117, and, thus, the last we will experience in our lifetimes.  For those who missed the media coverage, a TOV occurs when Venus passes between the Earth and the Sun.  During such events the planet appears as a dark circular spot on the face of the sun.  You can think of a transit as a sort of a mini-eclipse of the Sun by a planet or asteroid.  A TOV is a relatively rare event - the 2012 TOV was only the eighth one since the invention of the telescope around the year 1600.  This year's TOV lasted for six hours.  In New Jersey the transit began at 6:05 PM, but the Sun set (in NJ) before the transit was complete.  I wanted to see the TOV from my home which has a nice view of the western sky.  But first, I needed to invent something so that I could safely observe and record the event.  As it happens my telescope nicely projects an image of the Sun so I simply needed to provide a projection "screen".  My $0 invention (not counting the cost of the telescope) is shown here, at right. It consists of a box (obtained from a liquor store) with the top and one long side cut away.  For the "screen" a glossy white piece of paper was taped to the inside bottom of the box.  The box was then turned upside down and taped to a thin piece of scrap plywood.  A small section of the plywood was cut away so that the unit could sit snugly on top of the telescope, as shown.  On the morning of the TOV (6/5) the sky was relatively clear so I collected an image of the sun (at left as viewed on the "screen" from the floor).  A blowup of the silky cloud covered solar image obtained that morning is also shown at left.  Note that a number of sunspots were visible.  To give some perspective of scale, the bottom-most sunspot was some 27,000 miles long, more than three times the Earth's 8,000 mile diameter.  The Sun's photograph had an egg-like appearance because I was unable to get under the telescope tripod enough to be perpendicular to the projection.  In spite of this limitation, I concluded that the apparatus was working sufficiently well.  All I needed now was cooperation from the weather.  That cooperation was not forthcoming for most of the day - thick rain clouds covered the sky into the evening.  I was not able to personally observe the beginning of the TOV but, fortunately, NASA provided a live feed from Mauna Kea in Hawaii.  At right is the screen shot of the NASA feed just after Venus had fully "ingressed", i.e., its shadow was completely within the visible disk of the sun.  The NASA image was exciting, verifying that the TOV was indeed in progress.  It was also gratifying that NASA was seeing the same pattern of sunspots that I had observed in the morning.  But, I wanted to personally see the TOV on my home-made apparatus.  I spent the day going up and down the stairs about a thousand times, checking and rechecking weather maps, grousing and complaining, and testing mightily a saintly spousal patience.  Finally, at 7:37 PM, just after sitting down to supper, the sun broke through the clouds.  I immediately raced upstairs to record many images, all of them of a partially cloud covered Sun.  At left is one showing Venus having moved northwest (since the NASA image at 6:30 PM) across the Sun's face.

  

 So, why do we care about all this?  Historically, transits of Venus were very important because they were used to determine the actual distance between the Earth and the Sun.  This distance is called the "astronomical unit (AU)".  We had long known that Venus was 0.72 AU from the sun, but we didn't know the value of the AU.  During a TOV the planet Venus is at its closest approach to Earth.  This distance is 0.28 AU (1.00-0.72), so if we are able to measure that distance we would know the value for the AU and, therefore, the size of the Solar System. This determination can be accomplished during a TOV by accurately plotting the paths of Venus across the Sun's face observed from two locations north and south of each other and calculating the distance to Venus by parallax.  For details of this calculation using simulated data from Anchorage and Honolulu see the excellent article by Dr. Sten Odenwald at http://sunearthday.nasa.gov/2012/articles/ttt_75.php. I used Dr. Odenwald's "measurements" and determined a value of the AU of 100 million miles, less than 6% different than the currently accepted value.  Once we know the AU what else do we know?  Well, for one thing we can immediately calculate the diameter of the sun from it's angular diameter (0.525 degrees - there are 180 degrees from horizon to horizon).  Simple trigonometry using the AU value of 100 million miles gives a solar diameter of 900 thousand miles (cf. tabulated value 865 thousand miles).  We can also easily measure the angular diameter of Venus (0.016 degrees corresponding to 7900 miles, accepted value 7500 miles) from the photo just above .  All this leads to the distances to and diameters of all the planets as well as the distances from a planet and all its moons - only Mercury and Venus don't have at least one moon.  So, that's why we care - and it's why you should go ahead and put the date of the next TOV, 12/10/2117, on your Outlook calendar.  That way, the big event won't sneak up on you.

Oscillating Chemical Reactions

Some chemical reactions proceed from reactant (Species A) to product (species B) in a steadfast manner, viz.,

                                                                                    A → B

That is, as reactant A is consumed, B is produced.  In general, however, most reactions proceed through a series of steps which lead to the observed products.  These reaction steps are referred to as the mechanism of the reaction.  For example, the Lotka-Volterra reaction converts species A to species B  by means of the following mechanistic steps:

(1)        A + X → 2X

(2)        X + Y → 2Y

(3)        Y → B

If you add all these equations together (you should treat the arrows in chemical reactions as equals [=] signs) you see that the X's and Y's all cancel out and the overall reaction is just

A → B.  This mechanism predicts that the intermediate species X (blue) and Y (amber) will oscillate in concentration as depicted in the accompanying graph.  The x-axis in the plot is time and the y axis is the concentrations of species X and Y.

This type of reaction is called an oscillating reaction, an example of which is the Briggs-Rauscher Reaction shown in a YouTube video posted by kviht:

http://www.youtube.com/watch?v=-RFb8T2ED5E

If you're still awake after that soothing video you may have noticed that the blue species and the amber species alternately dominated in concentration.  You may also have noticed that I labeled the two species as Rabbits (blue) and Foxes (amber) in the above plot.  This is because Alfred Lotka, who proposed the above reaction mechanism as a theoretical chemical exercise in 1920, realized a few years later that this mechanism would also describe the population dynamics of a closed predator-prey animal system.  Vito Volterra independently came to the same conclusion at about the same time.  Thus, in the mechanism above, species A becomes rabbit food (grass, clover, whatever) present in great abundance, species X corresponds to the rabbits who reproduce after eating A, Y denotes the foxes who reproduce after eating rabbits, and B stands for foxes that have passed on by natural causes.  As the population of foxes increases it becomes a bad day for bunnies and the rabbit population plunges.  When the greedy foxes deplete the rabbit population, the foxes fade away.  Not so many foxes allows the rabbit population to recover, then spike, and so on.

 So, the next time you're walking through the woods and you spot a sly fox slinking about, warn him about Lotka-Volterra and the dangers of rabbit over-indulgence.

What's in the Sky Tonight (3/28) - Explanatory

What's in the Sky Tonight (3/28/12)?

What's in the Sky Tonight (3/28/12)?

What's in the sky tonight?  Well, if it's cloudy then it's clouds.  But if it's clear, what's up there is a real treat whether you live in the country or in a more urban environment.  No fewer than four of the five planets visible to the naked eye, along with the crescent moon, are up there putting on a show.  If that's not enough, the brightest star in the sky, Sirius, and one of the most prominent and recognizable constellations, Orion provide additional sky candy.  If the sky is not clear, the directions below will be good for the next few days with the exception that the moon will continue to wax (that means grow) to become a half moon, then gibbous (fat), then full, etc.

So, although it depends on your exact location, go outside about 7:30 PM or just after sunset and get yourself oriented.  First, you need to know approximately where north, south, east, and west are in your environs.  Where the sun has set is obviously west (note that position of setting along the horizon) and from there you can figure out the rest.  Face south (in the Northern Hemisphere), look up to find the moon.  From the moon follow a line (let's call that line the "ecliptic") toward where the sun has set and you will encounter a bright "star" which is the planet Venus, named for the Goddess of Love.  If the sky is too bright to easily see Venus, note the sun's setting position and plan to come back out in about 30 minutes.  If you do that, again follow the ecliptic from the moon toward the setting sun position until you find Venus.  Follow the ecliptic further toward the setting sun position (about 10 moon diameters on 3/28) and you will find Jupiter.  Through a telescope you may still be able to see the moons of Jupiter, but it is becoming more difficult to do so as Jupiter each day moves toward the sun.  If you go back to the moon and continue east along the ecliptic you will find Mars, bright, reddish, and nearly at its brightest for the next two years.  Finally, if you come back out after 9:00 or so, Saturn will appear along the ecliptic further to the east.  Saturn is almost at its brightest , but is less remarkable than Venus or Mars.  But, through a telescope, there's those rings ......

The brightest star in the sky is Sirius, the dog star, easily found by going back to the moon and moving your gaze further south.  Sirius is in the constellation, Canis Major (of course), it's doggy outline easily traced.  A photo of the constellation taken last year in my back yard is shown below. Just to the right of Sirius is Orion, the hunter, with his well defined large quadrangle of stars outlining the constellation.  The hunter proudly displays his belt.  The bright red star in the upper left of the quadrangle is Betelgeuse (pronounced "beetle juice") and the bright blue one at the lower right is Rigel.

That's enough for now, but I will follow up with some explanatory details of this night's planetary observations in the next post.

Chemistry Joke

A sodium atom and a cesium atom were walking down the street when the sodium atom let out a scream.

Cesiun:  "What's wrong?"

Sodium: "I've lost an electron!!"

Cesium: "Are you sure?"

Sodium: "I'm positive!!"

First post

A few days ago my good friend of many years, Dr. Anita Brandolini, passed away suddenly.  She was a faculty member in Chemistry at Ramapo College in Oakland NJ.  She created and maintained a blog called Dr. B's Science Lab (drbssciencelab.blogspot.com) aimed primarily at the science education of children.  She had many scientific publications and wrote a children's book, "Fizz, Bubble, and Flash! Element Explorations and Atom Adventures".  I and all of her friends and students will miss her terribly, but not one will forget her.  She would be hard to forget in any case, such a force of nature was she.

I had been toying with the idea of building a science blog for some time and I decided that the time is now.  I have created this blog, similarly named to Anita's, in memory of and in fond tribute to her.  Although Anita's blog was intended for children, mine is aimed at providing more sophisticated "explanations" of  things "scientific" to the reader of any age not working in a science field.  I give the usual "layman" blog disclaimer that these reports will be non-mathematical (i.e., you don't need to be able to do calculus) but, I don't claim them to be non-arithmetical (you do need to be able to add, subtract, multiply, and divide.)  I will also refuse to shy away from the use of simple equations (for example, F=ma) completely explained in words (in this case, force equals mass times acceleration) when such equations can provide explanation that would require pages of words.

Those are the plans.  I hope these reports will prove to be entertaining and, occasionally, enlightening. I end this initial post with a graphic that will need no explanation for many of Anita's friends but would require book-length description for others.  Suffice it to say, Anita spent years of her life carrying out the experiment symbolized in this graphic