Shadows: Newsletter Challenge (02/07/06)

If you look at the shadow of a discrete object that has been placed in the path of light from a point light source, then the shadow's size relative to that of the object will be larger in proportion to the ratio of distance from source to object and object to shadow screen. This is based on the consideration of light as rays which radiate in all directions from the point light source.

In this example, the sun is the point light source (a reasonable assumption), the cloud the object and the ground the screen.

Looking at the distance ratios, we can see that the object to screen distance is much much smaller than the source to object distance. Therefore the enlargement factor will be very small, but real. So the shadow will be larger, somewhat.

Enough of "the science". The fun part of this questions is finding a way to measure clounds and their shadows...given that in most cases these are changing with time - both in position and shape. How can you ensure that you are measuring the same dimension on the ground and in the air? How do you take topology into account?

We need to define a frame of reference for both the cloud and its shadow and be able to project the measurements onto a plane in this frame. That should be simple (!) geometry and draughting.

The real problem will be simultaneous measurements - photos might help, but would need scaling objects in them (easier on the ground than in the air).

Anybody got any ideas?!

That nailed it! Nice explanation!

Go ahead, take the photo. Then you have a reference for the shadow on the terrain. Using the photo you can match small features on the ground where the shadow line crosses to the actual terrain. All that presumes instant photo printing from your digital camera.

The safer way would be to simply say it is bigger than a bread box.

I think that all you guys missed the real point of the question. Are there no skiers here. When you look "down the slope" a small cloud can create a shadow which can reach to the bottom of the slope. The shadow will be much larger than the cloud.

My thoughts exactly! Yes, the shadow will be the same size as the cloud if you project the shadow onto a enormous screen that is perfectly flat and perpendicular to the rays of the sun. It would even be close enough if you observed the cloud on the plains of say Kansas, but on a mountain side fat chance.

The shadow of the cloud will always be bigger than the cloud. At noon, with the sun directly overhead on a level part of the Earth, the shadow will still be larger due to the fact that the sun is 32 arc minutes across as viewed from the earth. This gives fuzzy edges to the cloud's shadow. The higher the cloud, the bigger the fuzzy edges and the bigger the shadow.

The Sun is not a point source. Objects that are very high so that their apparent dimension is less than the sun (32 arc minutes) will not seem to cast a shadow at all. If the sun was a point source, one could view the shadow of the space station racing across the landscape at 17,000 mph.

[I was not picking on "Laxman" (whose response I don't really understand) but making a general comment that we can't seem to see the forest for the trees, generating complex specious discussions that ignore simple facts - I'm not quite sure how this commentary works, but I hope this resend gets general distribution]. Seems to me we have too many engineers and not enough common sense, thereby achieving analysis paralysis. Skiing is commonly done in the winter, which season is caused by the earth's tilt and the sun at an extreme acute angle. For flat ground therefore the shadow will always be elongated and therefore always larger than the cloud. The singular case where it would not be larger is where the shadow is superimposed on an up-slope that is at a 90 degree angle to the sun's rays.

The chances of the cloud and the shadow being of equal size are nearly zero. Considering the distance from the Sun to the Earth we can consider the sun to be a point light source. Considering the distance from the Sun to the Earth in relation to the distance from the cloud to the ground the ratio is, for all intents, 1. Therefore, IF the cloud is obscuring the Sun, AT local noon, AND the ground is perfectly flat AND level, then the cloud and the shadow will be (to the extent measurable) of equal size. Now, change any of these constrains and the shadow will be larger or smaller based on the angle of incidence of the Sun's light on the ground. Just as your shadow is longer or shorter as you walk around.

Local solar noon does not mean that the sun is directly overhead (perpendicular to the ground). Perpendicularity is the true first assumption above. The other question is how are we defining the size of the cloud, in a plane parallel to the surface of the earth or is it along an axis that is in some other orientation. If we define the size of the cloud in a plane parallel to the earth, the size of the shadow will always be larger than the cloud for the reasons presented above. If we allow some other orientation then the shadow size could be any relation to the cloud (larger, smaller, or in rare circumstances the same size)

I agree about the angle of the sun or even the slope of the ground. Everything was assumed to be perpendicular and it wasn't stated one way or another in the puzzle.

Furthermore, a 2-D shadow and a 3-D object's size have no correlation to each other in terms of the clouds size, if you want to get technical. The cast shadow is simply a silhouette and represents one view of many possible views that can exist as a cross-section of a cloud. No one defined what "size" means in the question. Is it the volume, breadth, width, or height? How do you measure something that has no clear defined boundary? That is, where does the boundary of a cloud start and the atmosphere end?

Maybe we are getting too technical here. Let's ask the question "what is the heart of the puzzle?" Many times the intended question and its answer are obscured by technical flaws. I think we are getting obscured by clouds.

My guess is that the root of this puzzle'question has to do with the relative distances of the sun, the cloud, and the ground and not the refraction index of the air, slope of the ground, and the azimuth of the sun. Just a guess.

I wouldn't agree with the analogy comparing to a person's shadow growing longer.. other people have commented on size of clouds too. I would make assumption that cloud is parallel to surface of earth, and in that context the size of shadow would approximately be equal to size of cloud. The analogy I would give would be placing a flat object on the ground and measuring the shadow size. No matter what time of the day it is, shadow size will be equal to object size. Now if you move the object away from the ground but not change the angle, the shadow size will be approximately the same (because sun's rays are approximately parallel, no matter what time of the day). I am not arguing against any minor changes in size other people have commented about.

The only answer I can come up with that has a quality equal to that of the question is: "Close."

Ditto. Good one bill.

The cloud is described as overhead, and you are looking down the mountain to see the shadow. The implication (not clearly stated though) is that the hill still slopes away from you where the shadow lies. This would make the area of the shadow larger than that of one side of the cloud - which I take to be the intent of the question. Of course, the cloud has ripples and depth and two sides, so total area will be at least twice the apparent single-sided value. Take your pick...

You have to explain that seeing is not always believing. A simple demonstation will help her understand. 1. Place your hand, gently, over her eyes and ask her what she can see and how big is your hand. Answer: dark or just the palm of your hand, and 'it's huge,Daddy.' 2. Ask her to count out ten paces and hold up your hand and ask her what she can see and how big is your hand. Aswer: everything and 'it's small, Daddy.' 3. ask her if the actual size of your hand has changed just because she walked ten paces away. She will see that distance changes the relative size of an object, but not its actual size. Apply to cloud and sun. 1. Which is nearer? Answer: cloud 2. Which seems bigger? Answer:cloud 3. Which is really bigger? Answer: sun 4. the shadow of the cloud will always be smaller than its real size, as the light source creating the shadow is far larger, though a long way away; the light rays will shrink the shadow, to a greater or lesser degree, dependant on the height of the cloud. No need to confuse her at this stage with the penumbra... If any doubt still exists show her the pictorial explanation of an eclipse in her school atlas: substitute moon for cloud

The sizes of the cloud and its shadow can be the same if: 1. The sun is not considered as a point source. Even though the sun is far away, it is still not that far and the size of the sun is too large. I would rather think of it as a parellel rays of light. 2. The size of the cloud is not too large. If it is a very large one, the shadow will be casted onto the earth which is curved, which has a greater distance compared to a flat surface. If it is small, the earth can be considered flat. 3. And diffraction of light does not take place, assuming that the cloud is a distinct one and not surrounded by other clouds which has a very tiny distance of separation. But who can see this? :) Just some thoughts and they may be wrong. But I would appreciate some comments so that i can learn.

I beg to differ on point #1. The Sun is, on average, 93,000,000 miles away. Let's say the cloud has an altitude of 1 mile (just over 5,000 feet). Let's say the cloud is 1/5 of a mile across, for the sake of this argument. Let us also consider the Sun as a point source for the moment. Now the math…

We need to calculate the angular separation, in degrees for the clouds width at a distance of 93,000,000 miles from the radius. That is, two lines 93,000,000 miles long, separated by 1/5 mile (0.2 miles) at their ends. For brevity we will consider the resulting triangle as a right angle since the error is so small.

tan = rise/run = 0.2 / 93,000,000 = 2.15054 e-9

Ang = arctan (2.15054 e-9) = 1.2322 e-7 degrees or 0.00000012322 degrees for dramatic effect.

Let's assume the cloud's shadow is projected onto a flat surface that is perpendicular to a line that passes from the Sun through the center of the cloud. The angular separation is still the same, the distance is no longer 93,000,000 miles, it is 93,000,001 miles. If the cloud is 0.2 miles wide, how wide is the shadow?

Again, let's assume a right triangle since the angular error is so small.

We know the angle is 0.00000012322 degrees and the length of the long side is 93,000,001 miles. We can calculate the length of the other triangle's side as follows:

Rise = run * tan = 93,000,001 * (0.2/93,000,000) = 0.20000000215053763440860215053763 miles

So, if the Sun is considered a point source and the ground is both level and perpendicular to the line drawn from the Sun through the center of the cloud, the cloud's shadow is larger than the cloud!

However, to your point, the Sun is not a point source! The diameter of the Sun is 870,000 miles, which is huge! However, the end effect can be calculated by a simple ratio and them multiply that by two and subtract the from the original calculation for the cloud's shadow. I know you are chomping at the bit, so let's go!

I love proportions, so that is what I will use. Think of a long lever that is 93,000,001 miles long with the fulcrum 93,000,000 miles away. Remember the cloud is 1 mile high. The long end moves 870,000 miles. How far does the short end move?

870,000 / 93,000,000 = X / 1.0

X = 1 * (870,000 / 93,000,000) = 0.0093548387096774193548387096774194 miles

We need to double that number since it effects both sides of the cloud's shadow and subtract it from our original answer for a point source.

Shadow Width = 0.20000000215053763440860215053763 miles – (2 * 0.0093548387096774193548387096774194 miles)

Shadow Width = 0.18129032473118279569892473118216 miles

Wow! The shadow is smaller by about 9%!

Why? Because the angular size of the Sun is larger than angle that is made by two lines drawn from the center of the Sun to each side of the cloud, so the cloud is slightly larger than its shadow under theoretical conditions.

I hope my calculations are correct!!! I am going to rest my brain now.

I should point out that although there are cases for two different cloud sizes when you consider the Sun as a point source and as a disk, the reality is that a disk source has some additional luminary properties that will create two shadows; an umbra and a penumbra. The umbra is the complete shadow (darkest part) caused by light from all points of a source (i.e., the disk of the Sun) and represents the smaller of the two shadow sizes calculated.

Please tell me that you didn't do this when you should have been buying your wife or girlfriend a box of chocolate for Valentine's Day . . .

Just checked the calendar. I still have time! :D

They'd be pretty darn close!

In my opinion returning to the first part of this discussion, defining the sun az a point source is acceptable, but in this case you should not calculate with the distance of cloud and terrain. If talking simply about sun-cloud-shadow in a line, that's perpendicular to plain terrain, than I say, becouse the sun's diameter is much larger than the cloud's the shadow will be smaller than the cloud. That would be a engeneering problem, if you would plant solar cell farms in the mountains, and the smallest shadow, covering the cells, would be the goal. But, as it has already been stated this effect is neglectable. The angle of the sun, which is always changing in the mountains, determins the size of the shadow, if thats calculated from the covered tarrain's size.

The size of the shadow will be equal to the size of the object, if the light source and the object are in a single line, i.e 180°

Seems to me we have too many engineers and not enough common sense, thereby achieving analysis paralysis. Skiing is commonly done in the winter, which season is caused by the earth's tilt and the sun at an extreme acute angle. For flat ground therefore the shadow will always be elongated and therefore always larger than the cloud. The singular case where it would not be larger is where the shadow is superimposed on an up-slope that is at a 90 degree angle to the sun's rays.

I have just given my opinion in this regard. If you have some common sense then you should have thought about the measument consistency and the mode of measurement and the parallelity of projecting surface and the exact distance between the sun and clouds and the distance between the cloud and the earth surface and the angle of the surface. Skiing will be done on the slopes of the mountains and how can you expect a flat surface there. Is this having any common sense Mr. Engineer.

The sun isn't a point source of light. All rays of light are basically parallel by the time they reach earth. Am i right? or am i right?