The intensity of a star's light falls off with distance according to a simple mathematical law. We will test this law in the lab, and illustrate its key applications in astronomy.

In everyday life we describe light subjectively; for example, light
is `good' if it enables us to do what we want to do, and `bad' if it
doesn't. But light can be measured and described numerically. In
particular, we can measure the *intensity* of light; if a given
source produces one unit of light, two such sources will produce twice
as much light, ten sources will produce ten times as much, and so on.
Thus it makes sense to talk about the intensity of light in
mathematical terms.

In this class we will need to measure the intensity of light in two
different ways. First, we must consider the total amount of light a
source - say, a star, or a light-bulb - gives off. Second, we must
consider the amount of light from a source which reaches our location.
The difference between these two kinds of intensity is part of
everyday experience. For example, a 100 watt light-bulb is a
fairly powerful source of light; placed a few feet from your desk, it
provides plenty of reading light. But even a 1000 watt
light-bulb won't provide enough light to read by if it's located a few
*hundred* feet away.

It helps to give different names to these two ways of measuring
intensity. The total amount of light a source emits is called its
* luminosity*. A light-bulb's luminosity is roughly
proportional to the number of watts it consumes. (This is not an
exact relationship because light-bulbs are not 100% efficient: in
addition to light, they give off lots of heat.) The amount of light
we receive from a source is called its

A simple experiment illuminates (pun intended) the relationship
between luminosity, brightness, and distance. As shown in the diagram
below, we will set up a light-bulb, and on one side of the bulb we
will set up a wall with a small hole. The light from the bulb spreads
out in all directions. A certain amount of light passes through the
hole and falls on a movable screen which is parallel to the wall. The
*total* amount of light passing through the hole and falling on
the screen does not depend on where we put the screen. But as we move
the screen further away, this fixed amount of light must cover a
larger area, and the brightness on the screen decreases.

To be specific, suppose we are using a 200 watt light-bulb.
According to the manufacturer, this bulb has a light output of about
4000 lumens. (A * lumen* is a unit of luminosity -
the proper definition would take a while to explain, but you can get a
rough idea from the fact that this rather bright bulb is putting out
4000 of them.) Let's put the wall 1 foot away from the center of
the light-bulb, and make the hole a square 1 inch on a side.
Imagine a sphere with a radius of
1 foot = 12 inches centered on the light-bulb.
This sphere has a surface area of 1,810 square inches; in
other words, it would take 1,810 squares, each 1 inch on a side,
to cover the entire sphere. The 4000 lumens put out by the
light-bulb spreads evenly over the entire surface of the sphere, so
each square inch gets just 4000 / 1810 = 2.2 lumens,
which is also the amount of light passing through the 1 inch hole
we've cut in the wall.

The inverse-square law in action. A certain amount of light passes through the hole at a distance of 1 foot from the light-bulb. At distances of 2 feet, 3 feet, and 4 feet from the bulb, the same amount of light spreads out to cover 4, 9, and 16 times the hole's area, respectively. |

Now consider the light passing through the hole and falling on the
screen. If we put the screen up right next to the hole, this light
falls on a square 1 inch on a side. This square receives a total
of 2.2 lumens, spread over 1 square inch, so the
*brightness* of the light on the screen is 2.2 lumens /
1 square inch = 2.2 lumens per square inch. If we
move the screen to a distance of 2 feet from the light-bulb, the
light passing through the hole now falls on a square which is
2 inches on a side. The area of this square is 2 inches
× 2 inches = 4 square inches, so the brightness
on the screen is now 2.2 lumens / 4 square inch =
0.55 lumens per square inch. Moving the screen even further
away spreads the light out more and reduces the brightness of the
light even further. The numerical results for this simple experiment
are summarized in the table below. In every case, the last column is
just 2.2 lumens divided by the area of the illuminated
square.

Distance from bulb to screen |
Size of square on screen |
Area of square on screen |
Brightness in square |

1 foot (12 inches) | 1 inch × 1 inch | 1 square inch | 2.20 lumens per square inch |

2 feet (24 inches) | 2 inches × 2 inches | 4 square inches | 0.55 lumens per square inch |

3 feet (36 inches) | 3 inches × 3 inches | 9 square inches | 0.244 lumens per square inch |

4 feet (48 inches) | 4 inches × 4 inches | 16 square inches | 0.138 lumens per square inch |

We're now ready for the last step, which is to take away the wall
between the light-bulb and the screen! When we do this, the
*brightness* of the light falling on the screen does not change.
The wall with its central hole helped us define the amount of light
falling on the screen, and the bright outline of the hole helped us to
see how that fixed amount of light spreads over a greater area as the
screen is moved further from the bulb. But the light passing through
the hole on its way to the screen `had no idea' that the wall was
there, so it produces the same brightness on the screen no matter
what. When we take away the wall, more of the screen is illuminated,
but the brightness remains the same. The brightness depends on only
two things: the *luminosity* of the light-bulb, and the
*distance* from the bulb to the screen.

We can express the relationship between luminosity, brightness, and
distance with a simple formula. Let *L* be the luminosity of a
source which emits light in all directions, and *D* be the
distance from the source to the point where we want to calculate the
source's brightness. Then the brightness is

To test the inverse-square law, we need a way of *measuring*
brightness. With modern technology, brightness can be measured
electronically. Unfortunately, it's not easy to explain *how*
this technology works; we would have to discuss the nature of
electricity, some mysteries of quantum mechanics, and the physics of
electromagnetic fields. So we will fall back on an earlier technology
which can be understood at an intuitive level without a lot of extra
explanation.

A * null-photometer* is a device for comparing the
brightness of two light sources. It can't provide a direct
measurement of brightness, but it can tell you when two sources have
the

The operation of a null-photometer is illustrated in the diagram
below. To begin with, you orient the photometer so each side is
pointing directly at one ot the two light sources you want to compare;
the light must strike the wax slabs squarely, and not at an angle.
Thus one side is illuminated by one source, and the other side is
illuminated by the other source. You then look through the window.
If both sources have the same brightness, both halves of the sandwich
will be equally bright; this is called a `null' reading (hence the
term *null*-photometer). If one source is brighter than the
other, the corresponding side of the sandwich will be brighter than
the other side. You eyes are pretty good at judging relative
brightness; with a little care, you can determine a null reading quite
accurately.

A null-photometer in operation. (a)
With more light (arrows) coming from the left than from the right,
the left half of the photometer's window is brighter. (b)
With equal amounts of light coming from both sides, the two halves
of the window have the same brightness. |

To test the inverse-square law using a null-photometer, we need to
express the law in a slightly different way. A null-photometer tells
you if two light sources provide equal brightness; in mathematical
terms, that is *B*_{a} = *B*_{b},
where *B*_{a} is the brightness produced by light source
`a' and *B*_{b} is the brightness produced by light
source `b'. Let's say that source `a' has luminosity
*L*_{a} and is at distance *D*_{a}, while
source `b' has luminosity *L*_{b} and is at distance
*D*_{b}. Then if
*B*_{a} = *B*_{b}, we must have

or |

The basic procedure for our laboratory test of the inverse-square law is shown in the diagram below. We will set up two lights of known luminosities. The null-photometer is placed between the lights, and moved to the point where both halves of the window are equally bright. The distances from the photometer to the lights are then measured. Finally, the luminosities and distances are substituted into the equation just derived; if the law is correct, the two sides should be equal, or nearly equal if we allow for experimental error.

Experimental measurement. The null-photometer is placed between the two lights and moved until both halves of the window have the same brightness. |

To test the law properly, we will set up several pairs of lights, with each pair separated from the others to avoid confusion. You will find this experiment easier if you work with a partner; one person can hold the photometer in position, while the other measures the distances to the lights. However, you and your partner should switch jobs so that everyone gets a chance to do every measurement.

When you make measurement, hold the null-photometer between the
lights, and move it back and forth along an imaginary line between
them until both halves of the photometer's window appear the same.
Your partner can check to make sure the photometer really is on a line
between the two lights, and then measure the distances
*D*_{a} and *D*_{b}. Ideally, these
distances should be measured from the *center* of each light-bulb
to the *nearest* side of the photometer, as shown in the diagram.
Once both distances have been recorded, flip the photometer over so
the left face is now the right face, and *vice versa*.
Re-position the photometer between the lights, move it so both halves
appear the same, and again measure the distances. Repeat two more
times, again flipping the photometer each time. You should now have
four separate measurements of the two distances. Each set of four
measurements can be averaged to get a more precise value; you can also
look at the range of values for each measurement to get some idea of
the accuracy of your work.

Before moving on to the next pair of lights, be sure to record the
luminosities *L*_{a} and *L*_{b}.

When you write up your lab report, first compute averages for your
measurements of *D*_{a} and *D*_{b} for each
pair of lights. Summarize your results in a table, with a separate
row for each pair of lights, and separate columns for your average
values of *D*_{a}, *D*_{b},
*L*_{a} / *L*_{b}, and
(*D*_{b} / *D*_{a})^{2}.
If the last two columns of each row are equal, allowing for
experimental error, then the inverse-square law passes the test.

The same procedure can be used to measure the luminosity of a
light-bulb. We will set up a pair of lights and tell you the
luminosity of one light; your job is to calculate the luminosity of
the other. Follow the *same* procedure you used when testing the
law: position the photometer so both halves appear the same, measure
the distances to the bulbs, repeat another three times, flipping the
photometer each time. Record your measurements for
*D*_{a} and *D*_{b}, along with the known
luminosity *L*_{a}.

When you write up your lab report, compute averages for your
measurements of *D*_{a} and *D*_{b} just as
you did when testing the law. Then plug your averages and the known
luminosity *L*_{a} into the equation

(In astronomy, we sometimes know the distance to a star but not its luminosity. A measurement like this can be used to find the star's luminosity.)

A similar procedure can be used to measure an unknown distance,
given the luminosities of both light-bulbs. We will set up one last
pair of lights and tell you both luminosities. This time you will
keep the photometer fixed, and move bulb `a' back and forth until both
halves of the photometer appear the same; then measure the distance
*D*_{a}. Once again, repeat another three times,
flipping the photometer each time. Record your measurements for
*D*_{a}, along with the given luminosities
*L*_{a} and *L*_{b}.

When you write up your lab report, compute averages for your
measurements of *D*_{a}, and plug your result into the
equation

(In astronomy, stars come in a range of luminosities, and we can sometimes figure out the luminosity by measuring the star's color. If we know the luminosity, we can then use this technique to measure the star's distance.)

Do the experiments described in the section on THE INVERSE-SQUARE LAW IN THE LAB and write a report on your work. This report should include, in order,

- the general idea of the experiments,
- the equipment you used for this work,
- a summary of your experimental results, and
- the conclusions you have reached.

In somewhat more detail, here are several things you should be sure to do in your lab report:

- In your own terms, explain the difference between luminosity and brightness, and summarize the inverse-square law.
- List individual values for each
*D*_{a}and*D*_{b}you have measured. Estimate your experimental errors from the range of values you got for each distance, and check for bad data values.Example: suppose that for one set of distances you have 15.3 cm, 14.9 cm, 15.5 cm, and 21.3 cm. It's likely that the 21.3 cm measurement is wrong - people do make mistakes! In this case, you should exclude the 21.3 cm value from your analysis. The three remaining values give you an average of (15.3 cm + 14.9 cm + 15.7 cm) / 3 = 15.3 cm, with a typical error of less than 0.4 cm.

- Make a table with your results as described in the subsection on Testing the law.
- Do your results support the inverse-square law? Give reasons for your conclusion.

- If you can read comfortably by the light of a single 100 watt bulb which is 10 feet from your page, how many 100 watt bulbs would you need to provide the same brightness at a distance of 100 feet?
- Why should you flip the null-photometer over between measurements? How might that help improve your accuracy?
- Consider a laser which fires a beam of parallel light rays. If the diameter of the beam is constant regardless of the distance from the laser, does the inverse-square law apply?

Joshua E. Barnes (barnes@ifa.hawaii.edu)

Last modified: August 16, 2003

`http://www.ifa.hawaii.edu/~barnes/ASTR110L_F03/inversesquare.html`