The answer to this is fairly easy to determine, and should enable students to see the value of "simple mathematics" (operations which require little more than the four basic operations - addition, subtraction, multiplication, and division). To answer the question though we will have to do just a little review, and preparation.

THE INVERSE SQUARE LAW

This is one of those concepts many people seem to have a problem with for one reason or another. That is interesting since this is based on common sense. And, this law has many applications to other fields such as magnetism, gravity, as well as light. You are encouraged to develop this with you classes to develop their understanding of this extremely important concept.

You may wish to demonstrate how the Inverse Square Law works to students in any number of ways. As you may remember it is a scientific law which states that the intensity of light varies inversely with the square of the distance. Now there's a turnoff! For most people this sounds impossible to understand, so let's take a practical look at this law and what it really means.

In lay terms it means this. Light spreads out with distance. We all know that. Shine a flashlight at something a foot away and the light is both bright and makes a small circle. If you double the distance it appears dimmer and larger. Why? Simple the same amount of light is being spread out over a larger distance. It has to be dimmer! But you know scientists, they wanted to see exactly how much.

Now when scientists measured this they discovered that this was very predictable. So, if you are twice as far away - the intensity is 1/4 as great as it was originally because it covers four times as much area. If you are three times as far away the intensity if only 1/9th of that of the original distance. Four times farther away, then it is now only 1/16th as brigh! Do you see a pattern developing here? Think about it!

Perhaps this diagram below may help. It shows how the light spreads out with distance...

This can be demonstrated in the classroom by showing how the size of a flashlight beam spreads with distance. Another method is to use a slide projector with a small square actually projected from the projector. Set it up so that at a set distance it exactly fills a square 12 inches on each side. Next double that distance. What is the size of the square now? (Measure the length of each side of the projected square) Triple the distance, and so on. Or, if you have a light meter from a camera, you can actually measure the brightness of the light at specific distances. The point to go for is to show the mathematical relationship involved. What is happening to produce these answers? Have the students complete the Table below.

What we hope students will begin to understand is that we can predict answers by taking multiples of the same of units beyond our reference point and squaring them to get the answer. It is also important to note that we can then look at that answer in several different ways. If a new position was 10 times farther away, then the light would spread out and cover 100 squares each of which is only receiving 1/100th of the original light. And, the light source itself will appear 1/100th as bright as it did originally. Again, discuss familiar examples like distant car lights seen at different distances. The point to emphasize is to show how the mathematical relationships can predict the actual observation. Thus the new brightness will always be equal to the "distance units number" squared.

Next we have to establish that heat would act in the same manner (it is just another form of light - infrared). This is simple to illustrate if you use a space heater and place thermometers at the various distances. You can demonstrate the obvious that the heat rapidly decreases with distance. Does the thermometers follow the expected rule? Try it and see. It won't be absolutely perfect for several reasons. What might change the expected results? But it should show the Inverse Square Law works in the infrared as well with distance.

USING THE INVERSE SQUARE LAW TO PREDICT PIONEER 10'S "WEATHER"

Now let's apply this to Pioneer 10. Presently the 25 year old spacecraft is nearly 67 AU from the Sun. What's an AU? It's the Astronomical Unit which is the distance from the Earth to the Sun (How many miles is it? Think!) But for our uses the AU represents that "unit" we have been using to explain how light spreads with distance. Therefore it is easy to explain why Mars for example is cooler than the Earth. Because if Mars is a bit over 2 AU from the Sun, then we know it has to be colder since it is receiving only about 1/4 the heat from the Sun.

This means we can begin to accurately imagine conditions like just how bright the Sun would appear to the Pioneer spacecraft at this distance, and how cold it must be aboard Pioneer 10. Therefore, using our ideas about the Inverse Square Law, let's try and determine just how bright the Sun would appear to Pioneer 10.

Procedure: What are we going to do?

• Step 1: Take the 67 and square it. The Answer will be _________
  
• Step 2: This means that the Sun will only be 1/ ___________th as bright as it appears to us on Earth. This is going to be pretty dim.
  
• Step 3: What do we see in the night sky which might be about as bright? ___________________________________________________.
  

This also means that the heat Pioneer 10 is receiving from the Sun is also reduced by this much as well!

• Step 4: - But first the fun part - converting Farhenite temperatures to the Absolute Scale, or Kelvin (which is used by scientists). To do this we much do the conversion from F to C which uses the following formula:
  C = (f-32 times 5/9)
• Step 5: - Once you do this the next step is fairly easy. We will change this to the Kelvin scale.
  Add 273.16 to your answer.

   

• First convert to Centigrade. That answer should be _____
• Then add 273.16 to convert to Kelvin. This will be ___________ For our purposes round off to the nearest whole degree ______
• Next divided by the number you got by the square of 67 AU_________
• That answer is _________ deg K - meaning that Pioneer 10 is now very cold!

THE REAL PIONEER TODAY ...

Where Pioneer 10 is today is cold beyond belief as it approaches the very absence of temperature - Absolute Zero. However, conditions aboard the tiny, lonely spacecraft are still relatively comfortable. According with a recent interview with Pioneer Flight Director Dave Lozier, interior compartment temperatures range from -40 to -20 deg. F. This means that the spacecraft is nearly 300+ degrees warmer than the surrounding space. Partly this can be credited to the construction, insulation, and the fact that back in the 1960's when the spacecraft was designed, engineers decided to use RTG's, or radioisotope generators, for power. The RTG's are specially designed units which use the heat from radioactive plutonium to both generate energy for the spacecraft, and to use that heat to protect the ship. The two RTG's at the end of each boom are still operating at temperatures around 800 degrees F!

Still ground controllers were gravely concerned in late January when Pioneer 10 made an amazing maneuver to keep its antenna pointed towards Earth. Mission Control at NASA's Ames Research Center had to turn off the radio transmitters aboard Pioneer 10 for the first time in 25 years. Would they come back on? Or, would the delicate 1960's technology tubes burst when they came back to life? The hydrazene fuel aboard the spacecraft is supposed to freeze at -35 deg. The thermometers inside the tanks say that it is -28 deg. Would the rocket motor work, or would it fail in some way? The instructions were sent to turn off the transmitters for nearly 90 minutes while the new maneuver was made. The roundtrip radio message to do this, plus the roll, meant that it would be more than 20 hours before we knew. The controllers could only wait. With the return message Pioneer announced that it was still alive and well. The surprise and jubilation in the mission control room was evident. Pioneer 10 once again proved to be a sturdy and game little spacecraft. Perhaps the RTG's and good design really are keeping Pioneer alive as a "tropical outpost" beyond the solar system!

Pioneer 10 is alone in a very dark universe. The brilliance of the Milky Way, the colors of distant stars, and radio messages form an invisible planet, create a picture for our imagination, as Pioneer 10 continues on its journey to the stars sent from a tiny blue planet.