# BRIGHTNESS AT VARYING DISTANCE

**o-Do Date: Apr 5 at 11:59pm**

To complete this lab please do the following:

1. READ the intro paragraphs to understand what is happening to the lab.

2. View the Prediction Demo Video first to complete questions 1-2.

3. Open the simulation and complete questions 3-11. **REMEMBER TO SHOW ALL MATH WORK!!**

4. Once #11 is completed you may view the Verification Demo Video to verify whether your conclusions are indeed correct.

6. For the 2nd data table note that B and B1 are different values from the data table in #3. Solve 13-14 based off of the method of table 2. **REMEMBER TO SHOW ALL MATH WORK!!**

7. Print out the last page of the lab and use a ruler to complete Part 2. **REMEMBER TO SHOW ALL MATH WORK!!**

## BrightnessatVaryingDistances-Lab.docx

**Brightness at Varying Distances Lab**

Image by Borb CC license: http://en.wikipedia.org/wiki/Inverse-square_law#/media/File:Inverse_square_law.svg

**Purpose:** In this lab, you will look at how light leaving a star “spreads out” and how this spreading can be used to determine the brightness of the star at different distances. While the focus of this lab is on light, your results will apply equally well to sound and the loudness of sounds at varying distances.

**Equipment:** This lab uses the optics bench, a square of aluminum foil sandwiched between two squares of paraffin wax, a lens holder to hold the wax, three incandescent light bulbs of equal wattage with bases, and three optics stands. The lab also requires access to the internet and a ruler.

Let’s start this lab by introducing the basic question that we want to answer.

**Part 1: Introducing the Question**

At the front of the class is an optics bench with two identical light bulbs on opposite sides of a wax block. In the center of the wax block is a piece of reflective foil. The foil ensures that each side of the wax is only illuminated by one of the light bulbs.

In a moment, the instructor will turn on the light bulbs and turn off the overhead light.

1. How does the brightness of each side of the wax block compare when the bulbs are both equal distances from the wax? a) Both sides of the wax are approximately the same brightness b) The left side of the wax is noticeably brighter c) The right side of the wax is noticeably brighter

Your question for this experiment is: If we add a second identical light bulb to the left side of the optics track, how far must the two light bulbs be from the wax in order to make both sides of the wax appear equally bright?

2. What is your prediction? If the single light bulb on the right side is 20 cm from the wax, how far away do you think the two light bulbs will need to be from the wax in order to produce an equal amount of brightness on their side of the wax?

**Part 2: Computer Simulation**

Open your internet browser and go to the online Flux Simulator at http://astro.unl.edu/classaction/animations/stellarprops/lightdetector.html. The simulation shows two light bulbs and two light sensors. The number on the sensors can be considered a numerical value of the brightness at that location. Take a few minutes to play around with the controls and see what you can do to increase and decrease the brightness readings.

3*. Set the wattage of the top bulb to 50 and use the simulation and your calculator to fill in the table below**. For columns 3 and 4, note that B1 is always 3.979.**

Distance from bulb |
Brightness Value |
B1/R |
B1/R2 |

R = 1.0 | B1 = 3.979 | ||

R = 2.0 | B = | ||

R = 3.0 | B = | ||

R = 4.0 | B = | ||

R = 5.0 | B = |

*Note that in the last column, only R is squared, B1 is not being squared.

4. The brightness value at R = 2.0 is:

a) approximately half of the brightness value at R = 1.0

b) significantly more than half of the brightness value at R = 1.0

c) significantly less than half of the brightness value at R = 1.0

Your answer to Question 4 tells us that the brightness does not decrease linearly with distance. The brightness decreases faster than linearly.

5*. Use your table to determine which equation below best represents the brightness at different distances.

6*. Let’s try out our equation. Calculate what you think B would be at a distance of R = 2.3. Show your work.

Move the sensor in the simulation to R = 2.3 and check that you get the same result as from your calculation above.

7*. Calculate what you think the brightness value would be at R = 7.0. Show your work.

8*. Based on your results so far, make another prediction about the opening question. Do you think the two light bulbs will need to be: a) less than 40 cm from the wax block

b) approximately 40 cm from the wax block

c) more than 40 cm from the wax block

9*. Discuss your answer above with the instructor and obtain his/her initials indicating that you have considered how your results above relate to our opening question.

Let’s try to model our opening question is the simulation. Set the top bulb to 50 Watts and the bottom bulb to 25 Watts. Try to use the simulation to answer our opening question.

10*. Describe how you used the simulation to answer our opening question. Where did you move each object? What values are you comparing?

11*. Based on your simulation results: If a single bulb is 20 cm from the wax block, what distance from the wax block should the two bulbs be in order to produce equal brightness on the wax?

12. Compare your answer for Question 11 with another group. Do your answers agree? Did you do the same thing in the simulation to determine your answers?

**Once everyone is ready, the instructor will turn off the classroom lights and demonstrate the two bulbs vs. one bulb arrangement either confirming or contradicting your answer above.

Let’s use our new understanding of brightness at varying distances to think about what the Sun would look like from other planets. To help us answer these questions, let’s go back and notice a pattern in your results from the simulation. Use the results in your table above to fill in the following table:

R = 2.0 (twice as far away) | B/B1 = | 1/22 = |

R = 3.0 (three times as far away) | B/B1 = | 1/32 = |

R = 4.0 (four times as far away) | B/B1 = | 1/42 = |

R = 5.0 (five times as far away) | B/B1 = | 1/52 = |

This table shows us that if an object is three times farther away, it will appear approximately 1/32 = 1/9th as bright. If an object is five times farther away, it will appear approximately 1/52 = 1/25th as bright. And so on. This pattern even works for moving objects closer. If an object is half as far away, it will appear times as bright. Use this pattern of reasoning to answer the next two questions.

13*. Neptune is 30 times farther from the Sun than the Earth. How would the brightness of the Sun viewed from Neptune compare to the brightness of the Sun viewed from Earth? Your answer should be a numerical value (1/2 as bright, 1/40 as bright, something like that).

14*. Mercury is 4/10 as far from the Sun as the Earth. How would the brightness of the Sun viewed from Mercury compare to the brightness of the Sun viewed from Earth? Your answer should be a numerical value.

Why 1/r2?

Why does the brightness behave this way; why is the equation B = B1/r2? It is because we live in a three-dimensional world. Imagine turning the bulb on for a millisecond and then turning it off again. During the millisecond that the bulb is on, it emits a flash of light in all directions. As the light leaves the bulb, it spreads out in all directions like an explosion. As the flash of light travels outward in all directions, the light (electromagnetic energy) becomes spread over the surface of a sphere. As the light moves farther away from the bulb, the sphere and its surface get larger causing the light energy to be more spread out meaning less of the total light hits your eye or the wax block. The surface area of a sphere is given by 4r2. This is where the r2 in our brightness equation comes from. If we lived in a two-dimensional world, then the light would spread out over the surface of a circle and the equation would be B = B1/r. If we lived in a four-dimensional world, then the light would spread over the surface of a hypersphere and the equation would be B = B1/r3. You could say that our experiment today proved that we live in a three-dimensional world. (Or at least a world with three dimensions large enough to be noticed.)

**Part 2: Using Apparent Brightness to Estimate Stellar Distances***

The apparent brightness of different celestial objects can vary significantly, by many orders of magnitude. In order to have smaller values and ranges of values to work with, astronomers classify the apparent brightness of stars, planets, and other celestial objects by a number called ‘apparent magnitude’. This system was introduced about 2000 years ago with the astronomer Hipparchus gave the brightest appearing stars a value of 1. Stars that appeared somewhat dimmer were given apparent magnitude values of 2 and so on so that **the larger the apparent magnitude, the dimmer the star appears. **In the mid 19th century the apparent magnitude system was revised on a more systematic basis. As a result, we now have apparent magnitude values of zero and even negative values for stars that appear very bright.

You know from the first part of this lab, that the apparent brightness of a star depends on both its distance from us and its wattage. The tables on the following page give some helpful apparent magnitude and wattage values.

*Based on *Using Apparent Brightness to Estimate Stellar Distances* from The Universe in Your Hands

Apparent Magnitude |
Object |

-26.7 | Sun |

-12.5 | Full Moon |

-2.5 | Jupiter (at its brightest) |

-1.5 | Sirius (brightest star) |

6.5 | Limit with unaided eye on darkest night |

13.0 | Limit with 8-inch telescope |

24.0 | Limit with 200-inch telescope |

28.0 | Limit with Hubble Space Telescope |

Wattage |
Object |

Watts | Sun |

Watts | Trinity atom-bomb test |

200 Watts | Light bulb |

1 Watt | Candle |

Watts | Firefly (lightning bug) |

On the last page of this activity is a nomogram. The nomogram is a chart that allows you to convert between wattage, apparent magnitude, and distance using a ruler rather than an equation. As long as you know two out of the three quantities (wattage, apparent magnitude, and distance) you can use a ruler to connect the two quantities you do know and read off the value of the third unknown quantity.

1. Use the nomogram and the apparent magnitude and wattage values above to determine the distance to the Sun. Check your result with the instructor.

2*. What would be the apparent magnitude of a 200 Watt light bulb if it were 100 m from you?

3*. What astronomical object has approximately the same apparent magnitude as the light bulb in Question 2?

4*. How far would you have to be from a burning candle for it to be barely visible to the unaided eye? (It’s surprisingly far.)

5. The distance in question 4 is surely much larger than you would have guessed. Offer two possible explanations for why the value in 4 is so surprisingly large.

6*. How far would you have had to be from the Trinity atom bomb test for it to have had the same apparent magnitude as the Sun?

7*. The distance from Earth to Sirius is about 9 light years (l.y.). What is the wattage of Sirius?

8*. How far would you have to be from Sirius for it to appear as bright as the Sun?

Howard Community College – ASTR 114 Page 8

Howard Community College – ASTR 114 Page 7