How far is it to the Sun

Do you know the energy?

Introduction

A liquid crystal thermometer shows temperatures as colors. These postcards can be used to compare the energy output of an incandescent lamp to that of the sun. The comparison can be used to measure the distance to the sun.

Material

• An Exploratorium thermal postcard(or similar thermochromic liquid crystal for example from Edmund's Scientific) optional, 3 postcards will make the experiment run more quickly.
• a meter stick
• a stopwatch
• a 200 Watt lightbulb and lamp to hold it (100 Watts will also work)
optional
• a slab of metal 1/8 inch thick or less

Assembly

Optional. You do not need an entire postcard for the experiment, you can cut the postcard into 3 smaller cards they need to be cut at least 5 cm (2”) on a side.
To slow down the heating of the postcard use spray adhesive to mount it to a slab of metal.

To Do and Notice

Place all postcards in the shade. The postcards should start out by being black.

Move a postcard into direct sunlight quickly. Hold the card perpendicular to a line pointing to the sun.

The card will warm up turning first red then yellow, green, blue and black again.
Time how long it takes from the first appearance of red to the last disappearance of blue in the center of the card.
Repeat the experiment using another card.

Turn on a 200 Watt light.(In the shade)
Hold the card 10 cm from the bulb and time how long it takes from the first appearance of red to the last disappearance of blue in the center of the card.
Be sure to hold the card to the side of the lamp so that is it heated only by radiation, not by convection.
Find the distance you must hold the card from the lightbulb so that it takes the same amount of time to change colors as the card held in the sun. Measure the distance from the center of the lightbulb to the card.
Notice that the card is much closer to the lightbulb than to the sun when it takes the same amount of time to change temperatures.

What’s Going On?

When it takes the same amount of time to change the temperature of the card with the lightbulb as with the sun, then the same amount of energy is flowing into the card from the lightbulb as is flowing into the card from the sun.
The flow of energy into the card during a time interval is inversely proportional to the square of the distance from the source of the energy. We can find the distance to the sun if we know its energy output, the distance to the lamp, and the energy flow from the lamp, which is 200 Watts. See the math root next.

Math Root

The power out of the sun or a spherical lamp bulb spreads over a sphere uniformly. The power does not vanish but simply spreads over larger and larger spheres. The area of a sphere is 4pr2. Thus the power per unit area, P/4pr2, which is what changes the temperature of the postcard, decreases proportional to the inverse square of the distance from the center of the sun or lightbulb.

When the card takes the same amount of time to change through its temperature range then the power per unit area from the lamp is equal to the power per unit area from the sun: In the following example we used 20 cm or 0.2 meter as the distance from the center of the bulb to the card.

The total power output by the sun is 2.8 x 1026 watts.

Power of lamp/distance to lamp2 = power of sun/distance to sun2

Plamp/D2lamp= Psun/D2sun
or

D2sun = PsunD2lamp/Plamp

D2sun = (2.8 x1026 W/200 W )(0.2m)2

D2sun = 2.4 x 1011 m

The actual distance to the sun is 1.5 x 1011meters

We would have done better if we corrected for the absorption of power from the sun by the atmosphere. But we got to within a factor of 2 with a \$2 postcard, so that isn't too bad.