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What the inverse-square law means to our observations

Submitted by RST on Sat, 09/17/2011 - 16:39
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The inverse-square law, then, is the missing piece that clears up why our initial rankings of energy--based on apparent brightness--were so mistaken.

Remember that if everything were correct, each of the values in each row would be equal. But since the values in column 3 don't all match the other values in its own row, there is an error somewhere. 

Star

Brightness ranking
(1 = brightest, 2 = middle, 
3 = least bright)

Energy level prediction
(1 = most energy, 2 = middle,
3 = least energy)

Actual energy level,
backed up by astronomical data

Sun

1

1

3

Rigel

3

3

1

Sirius

2

2

2

 

The problem is that we used apparent brightness--how each star looked to my naked eyes--to rank their energy levels. But apparent brightness does not take into account how far away each star is.

The sun is about 93,000,000 miles away from the earth, a distance called 1 astronomical unit (AU). Light from the sun takes about 8 minutes to reach the earth, so we say that the sun is about 8 light-minutes away from, or distant from, the Earth. If you're reading this in the daytime, then the light that is hitting us as you read these words left the Sun about 8 minutes ago.

By contrast, Rigel is somewhere between 700 and 900 light-years away--let's take 772 as our working number [1]--which means that the light you see when you look at Orion's foot left the star 772 years ago. Rigel is so much further away from the earth than the sun is, that light--which takes only 8 minutes to travel 93,000,000 miles from the sun to the earth--takes a full 772 years or so to travel from Rigel to the earth.

Just take a moment to try to visualize just how immensely, vastly far away that is!

(In fact, if something happened to Rigel--say it exploded or something--we have no way of knowing it until the light from that event reaches us some 772 years or so later. So the star you observed for part of this exercise may not even exist anymore--we can't know what it looks like today, only what it looked like some 772 years ago.)

Sirius is much closer to us--the light we see now left Sirius only about 8-1/2 years ago [2]. That's still quite a distance--about 525, 000 times as far away as the sun is at 93,000,000 miles.

So both Sirius and Rigel are immensely further away from us than the Sun is. From the inverse-square law, we've seen that brightness decreases as the square of the distance increases. That means that it drops off relatively fast. So the fact that, at those vast distances, we can see Sirius and Rigel at all must mean they are tremendous in size, and intensely bright.

In fact, Sirius is 25 times more luminous than the sun [2], and Rigel is about 85,000 times as luminous as the sun [1]--nothing at all like what our observations showed us!

In the following image, the sun is indicated by its other name, Sol (as in "solar"). You can barely see it; it's just a pinpoint of light at the bottom left of the image. Look how much bigger Sirius, next to it in the image is.

Source: http://files.abovetopsecret.com/images/member/626ccaf88607.jpg accessed 18 September 2011

 

We haven't talked about the next two stars to the right, Pollux (34 light-years away in the constellation Gemini [3])

 

 
 

 

 

and Arcturus (36.7 light-years away in the constellation Boötes [4]),

 

 

 

 

but then there's a big jump in size and brightness to Rigel, compared to both Sol and Sirius.

If, instead of the sun being 8 light-minutes away, Sirius being about 8-1/2 light-years away, and Rigel being 772 light-years away, they were all about (I'm guessing) 50 light-years away at the same time, then they'd look like they do in the preceding image, where all the stars were lined up by size. Instead of comparing apples to oranges, because the distances are so different, we'd be comparing apples to apples.

And even at that, Rigel, at 85,000 times the luminosity of the Sun, is not the biggest and brightest star in the sky by a long shot. Look how much bigger Aldebaran (65 light-years away in the constellation Taurus [5]),

 

 
 
 

 

 

 

Betelgeuse (about 640 light-years away, also in Orion [6]),

 

 
 

 

 

and Antares (about 600 light-years away in the constellation Scorpius [7]) are.

 

 

 

The next time you're out looking at the night sky, think about how so very far away the stars are--so far away that, although many of them are much bigger and brighter and burn hotter than our sun--they look like little pinpoints of light.

The reason they look so much smaller and dimmer than the sun is in large part because of the inverse-square law--a foundational property of physical energy.

 
 
References

[1] http://en.wikipedia.org/wiki/Rigel accessed 18 September 2011

[2] http://en.wikipedia.org/wiki/Sirius accessed 18 September 2011

[3] http://en.wikipedia.org/wiki/Pollux_(star) accessed 18 September 2011

[4] http://en.wikipedia.org/wiki/Arcturus accessed 18 September 2011

[5] http://en.wikipedia.org/wiki/Aldebaran accessed 18 September 2011

[6] http://en.wikipedia.org/wiki/Betelgeuse accessed 18 September 2011

[7] http://en.wikipedia.org/wiki/Antares accessed 18 September 2011

How the inverse-square law works

Submitted by RST on Sat, 09/17/2011 - 16:38
Posted in

The inverse-square law is a description of how forces and energy in its various forms--gravity, electricity, sound, radio waves, radiation, heat, light, and so forth--behave as they travel further away from their source.

An "inverse" relationship means that two things change together, but in opposite ways--as one gets larger, the other gets smaller.

Examples of things that are in inverse relationships with each other include:

  • How old you are (how many years you already have) and how many years you have remaining to live;
  • education and unemployment--in general, the rates of unemployment are lower among people who have higher rates of education, and rates of unemployment tend to be higher among people who have lower rates of education.

"Squared" means a number is multiplied by itself, and "inverse squared" means it varies in the opposite way: 1 divided by the square of a number.

1 squared = 12 = 1 x 1 = 1. The inverse square of 1 = 1/12 = 1.
2 squared = 22 = 2 x 2 = 4. The inverse square of 2 = 1/22 = 1/4.
3 squared = 32 = 3 x 3 = 9. The inverse square of 3 = 1/32 = 1/9.
4 squared = 42 = 4 x 4 = 16. The inverse square of 4 = 1/42 = 1/16.
5 squared = 52 = 5 x 5 = 25. The inverse square of 5 = 1/52 = 1/25.
6 squared = 62 = 6 x 6 = 36. The inverse square of 6 = 1/62 = 1/36.
7 squared = 72 = 7 x 7 = 49. The inverse square of 7 = 1/72 = 1/49.
8 squared = 82 = 8 x 8 = 64. The inverse square of 8 = 1/82 = 1/64.
9 squared = 92 = 9 x 9 = 81. The inverse square of 9 = 1/92 = 1/81.
10 squared = 102 = 10 x 10 = 100. The inverse square of 10 = 1/102 = 1/100.

 

A physical law is a general description of how things work in the natural world, based on the result of many observations and experiments.

The inverse-square law is called a "law" because it's been observed to work so consistently and constantly for different forms of energy (radiation, heat, light, and so forth), as well as forces such as gravity.

 

Source: http://upload.wikimedia.org/wikipedia/commons/2/28/Inverse_square_law.svg

 

When energy (for example, light from a flashlight) leaves its source (S), it travels out in straight lines in different directions. As the straight line paths gets further from the source, they also get further from each other.

So they cover a larger area, as the squares at r, 2r, and 3r show. Each of the squares is the same size--r x r, or r2--but they get less bright as they spread out, because the same amount of brightness now has to cover a larger area. So the first square has all of the light rays; in this example, we'll call its brightness 9, because it has 9 light rays all in the same square.

By the time it gets to 2r, those same 9 rays have to cover 4 times as much space as they did at r, so the brightness of any one square at 2r is only 1/4 as bright as the square at r. At 3r, each of those squares is only 1/9 as bright as the square at r.

At 4r, which is not shown in the drawing above, each square will be only 1/16th as bright as the original square at r.

How bright will each square be at 5r? 8r? 10r?

 

This 5-minute video shows the process in an animation. 

 

Source: Derek Owens' MIT Physics course accessed 18 September 2011

 

So how did our predictions work out, and why?

Submitted by RST on Sat, 09/17/2011 - 16:37
Posted in

Using the information provided in this exercise, I filled out the table. Columns 1 and columns 2 contain my observations and my predictions.

Notice that the numbers in column 3 don't completely match my predictions in column 2.

How did that happen, when I observed what I did, and when laboratory studies provide evidence that that brightness is positively correlated with light energy?

 

Star

Brightness ranking
(1 = brightest, 2 = middle, 
3 = least bright)

Energy level prediction
(1 = most energy, 2 = middle,
3 = least energy)

Actual energy level,
backed up by astronomical data

Sun

1

1

3

Rigel

3

3

1

Sirius

2

2

2

 

Clearly, in addition to what we've taken into account so far, something else must be going on.

That something is the effects of the inverse-square law, and we'll talk about it in the next post.

 

 

Predicting the relative energy of the stars based on our brightness rankings

Submitted by RST on Sat, 09/17/2011 - 16:36
Posted in

Draw a chart on a piece of paper, or print this table to fill in your prediction of the relative energy level of the stars, according to your observations.

Use the fact that brightness depends on light energy as a base for your predictions. "Depends on" in this case means that they change in the same way as each other--as one goes up, the other goes up, and as one goes down, the other goes down.

As brightness goes up, light energy goes up. Copy your observed brightness ratings into the second column, and then write your predictions for the energy level of each star in the third column.

We'll check those predictions against actual astronomical data in the next post, to see what insight it gives us into what is going on.

 

Star

Brightness ranking
(1 = brightest, 2 = middle, 3 = least bright)

Energy level prediction
(1 = most energy, 2 = middle, 3 = least energy)

Sun

1

 

Rigel

 

 

Sirius

 

 

 

 

Ranking the stars from brightest to least bright

Submitted by RST on Sat, 09/17/2011 - 16:34
Posted in

Draw a chart on a piece of paper, or print this table to fill in the relative brightness of the stars, according to your observations. Obviously, the sun is the brightest, so I've filled in that one to get us started.

 

Star

Brightness ranking
(1 = brightest, 2 = middle, 3 = least bright)

Sun

1

Rigel

 

Sirius

 

 

 

Experiential hands-on learning: Energy and the brightness of stars

Submitted by RST on Sat, 09/17/2011 - 13:02
Posted in

It's a nice fall day here in the Seattle area, and as the change in seasons leads us forward into winter, we can also look forward to seeing some of the most prominent constellations in the night sky--weather permitting, of course.

Orion and Canis Major, winter constellations that will be easily viewable in a couple of months, and that are rich with stories from other cultures in addition to the Greek ones, will be the basis for an experiential hands-on learning exercise about the brightness and thermodynamic energy of stars.

Why you might want to know this

This exercise will permit you to directly experience the effects of the inverse-square law.

The inverse-square law is a foundational property of physical energy. Experiencing the physical universe around us is truly awesome; understanding what it is that we are experiencing only adds to the awesomeness.

If you understand the inverse-square law, you will be prepared for certain advanced physiological training, such as Vodder manual lymph drainage, which makes use of that law in the theory behind the treatment.

You will also be able to use it to analyze the validity of claims made about treatment effects of energy.

 

Click the following links to participate in the experiential hands-on learning exercise:

Energy and the brightness of stars

VERY IMPORTANT: Safety precautions for this exercise

Safety: Viewing the sun in a safe way by protecting your eyes

Safety: Backcountry stargazing

The stories behind the constellation Orion and Sirius the Dog Star

Optional field-trip component: Finding Orion and Sirius in the night sky

Ranking the stars from brightest to least bright

Predicting the relative energy of the stars based on our brightness rankings

So how did our predictions work out, and why?

How the inverse-square law works

What the inverse-square law means to our observations

What the inverse-square law means in Vodder manual lymph drainage

What the inverse-square law means as a general physical principle

 

 

Source: http://en.esimg.org/upl/2011/08/11aug25_430.jpg

 

The stories behind the constellation Orion and Sirius the Dog Star

Submitted by RST on Sat, 09/17/2011 - 10:58
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In ancient Greek legend, the twin gods Apollo (Ἀπόλλων) and Artemis (Ἄρτεμις) (called Apollo and Diana by the Romans) were the children of Zeus (Ζεύς), god of the sky, and Leto (Λητώ), daughter of Titans. The boy, Apollo, was the bright and shining god of the sun, while his sister Artemis, goddess of the moon and wild animals, preferred hunting at night.

According to some versions of the old story, Artemis fell in love with one of her companions, Orion the hunter (Ὠρίων), and Apollo was so jealous that he decided to get rid of Orion. Knowing what a skilled archer his sister was, he challenged her to hit a tiny target bobbing out on the distant water. Proud to demonstrate her prowess, she did so, finding out only afterward that the target was Orion, swimming in the distance, and she had killed him with a single arrow to the head.

 

 

Source: Diana next to the corpse of Orion (Diane auprès du cadavre d'Orion), by Daniel Seiter, 1685. http://upload.wikimedia.org/wikipedia/commons/3/3e/Diane_auprès_du_cadavre_d%27Orion.jpg

 

 

Grief-stricken at the discovery that she had killed the man she loved, Artemis was not able to bring him back to life, but she put him up in the sky to honor him as the constellation Orion the Hunter. There, he is followed by his hunting dogs Canis Major (containing the Dog Star Sirius, or Σείριος) and Canis Minor, while he does battle with Taurus the Bull, and hunts Lepus the Hare.

 

Source: [1]

Source: [2]

(Although I focused on the Greek story because that is the one that contributed the most to how we know the constellations in modern Western societies, there are many stories known from other cultures about these stars, and other stories that have been lost without ever having been known outside small groups of people. You can use the Wikipedia entries at Orion: Cultural Significance and Sirius: Etymology and cultural significance to serve as a portal to finding out about these other star stories and legends, if you are interested.)

 

References

[1] http://www.starrynighteducation.com/stargazer/images/1573OrionDogs.jpg accessed 17 September 2011

[2] http://www.starrynighteducation.com/stargazer/images/1573OrionTaurus.jpg accessed 17 September 2011

 

 

 

Safety: Viewing the sun in a safe way by protecting your eyes

Submitted by RST on Sat, 09/17/2011 - 10:53
Posted in
The Death of Semele: In Greek mythology, Semele died by incineration after she gazed directly on the overpowering radiance of the sun-god, Zeus.
 
Extremely important notice:
 
Never, ever, in any case, under any circumstances, look directly at the sun with your naked eyes, or with eyeglasses or contact lenses, or with ordinary sunglasses!
 
There are ways to protect your eyes for safe viewing of the sun, such as during an eclipse, but they require a little investment of effort to get, and they need to meet certain standards in order to protect your eyes. That information is NOT provided here.
 
If you look at the sun without special protection that is up to standard, you can permanently damage your eyesight!
 
DON'T DO IT!
 
For this exercise, you will only visualize the sun in your mind, or look at a picture of it--you will absolutely never, ever directly look at it.
 
This is the necessary thought experiment component of this exercise to ensure your safety by protecting your eyes.
 
 
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