Astronomy 100 Names: Exercise 8: Intensity and distance (and color) The method of standard candles and the inverse-square law of brightness From everyday experience you know that light sources become brighter when you move closer to them. The apparent brightness of the source is related to your distance from it. However, moving a light source twice as close to you does not make it twice as bright, though it may seem this way. Astronomically, this correlation between distance and intensity is of great importance, because it allows the determination of distances to distant stars and galaxies. You will determine the relationship experimentally and then use it to answer some astronomical questions. For this experiment, you will use a light detector (photocell) which enables you to make light intensity measurements that are more precise than those made by the human eye. The photocell converts light intensity to an electrical current, which can be measured with a current meter (ammeter). The current produced by the photocell is directly proportional to the amount of light falling on it. If the light intensity doubles, the meter reading will also double. Procedure (work in groups of two or three): A. Obtain a Vernier datalogging device and a Vernier photocell detector. B. Connect the photocell to the datalogger through the CH 1 port and make sure that the datalogger display shows light intensity. C. Obtain two meter sticks, a high intensity (standard bulb size) light source and a low intensity (small bulb) light source. Place the high intensity source on a level flat surface and plug it in. Do not turn it on yet. Lay out the meter sticks in a line directly away from the light source; make sure you are not in line with some other group's light source! D. Place the photocell 50 cm away from and pointing toward the unlit bulb and turn out the room lights. Take a measurement and write it down in Table 8.1 under the "" column. Don t forget to record the luminosity (wattage) of the bulb. 1. a. Does the intensity change for the depending on distance from the unlit bulb? If not, do you have to measure the for every distance you will be using? E. Turn on the bulb. Take measurements of source intensity from 30, 45, 60, 75 and 90 cm away. Take more measurements if you have time.
1. b. Should you enter this data into the "source" or "source and " column of Table 8.1? Then how do you calculate the "source" number? Table 8.1: High-intensity source W bulb distance (cm) source and light intensity (lumens) source F. Now replace the high intensity light bulb with one of a lower wattage. Follow steps D and E for the lower intensity bulb and record your data in table 8.2. Don t forget to record the luminosity (wattage) of the bulb. Table 8.2: Low-intensity source W bulb distance (cm) source and light intensity (lumens) source
2. a. Error analysis with the light bulb on, place the photocell at 30 cm and slowly swing the photocell so that it is not directly pointing at the light bulb. What happens to the intensity? b. Estimate the number of degrees away from the bulb you can point the photocell before an error of 10% is introduced. 3. Plot both your data sets on the same graph. Distance from source (cm) should be along the x-axis and light intensity (arbitrary units) should be along the y-axis. Draw in two smooth curves (use two different colors for the two different intensities) that best fit each set of data points; don't just draw straight line segments. Label the low-intensity and high-intensity curves. Attach the plot to the back of this exercise. 4. Looking at the graph: is the relationship between light intensity and distance linear? Is the relationship between light intensity and distance inverse? 5. a. Using the high-intensity curve on the graph, find the intensity of light at d = 40 cm. Divide the intensity by four. Find this new intensity on the high-intensity curve on the graph; what distance does this new intensity correspond to? Compared to 40 cm, how would you mathematically describe this new distance? b. Using the low-intensity curve on the graph, find the intensity of light at d = 40 cm. Divide the intensity by four. Find this new intensity on the low-intensity curve on the graph; what distance does this new intensity correspond to? Compared to 40 cm, how would you mathematically describe this new distance? c. In both these cases, quartering (multiplying by 1/4) the intensity led to what change in the distance? 6. Write a simple equation connecting I (the intensity of light) and d (the distance); use k for a proportionality constant. What you will have written is an example of an inverse-square law.
7. To test your new law, find the distance d at which the detector must be placed in order for the intensity to be one-ninth as much as the intensity at d = 100 cm. Hint: you may want to set up a ratio of two I s and cancel out the k s. 8. If one star is at a distance of 100 light years, how far away would a second star, of exactly the same luminosity (wattage) as the first, have to be to appear at one-ninth the brightness of the first? 9. If one star is at a distance of 100 light years, how far away would a second star, that is 9 times as luminous, have to be, to appear at the same brightness as the first? 10. Would you expect the inverse-square law to always hold accurately for the stars in the sky? Why or why not? 11. a. From your two measurements of intensity made at d = 30 cm, find the ratio of intensities between the high and low wattage bulbs, dividing the lower intensity by the higher intensity. Repeat this calculation for the d = 60 cm and d= 90 cm distances; are all the results nearly (within 20%, let's say) of each other? In other words, do you have consistency between the two curves? b. Now divide the luminosity (wattage) of the low intensity bulb by the luminosity (wattage) of the high intensity bulb. Is this fraction the same as the fraction you calculated in part a? NO. Why not? In other words, why don t the light intensities of the two bulbs scale simply as a function of the luminosity?
Astronomical magnitudes G. Copy the previous data or obtain new data to fill table 8.3 below. 12. a. The formula for calculating any type of magnitude is: Magnitude = -2.5 * log10(intensity of object) Fill in the magnitude column in the table. Table 8.3: Magnitudes Bulb Low wattage placed at a distance of 30 cm High wattage placed at a distance of 90 cm Bulb Low wattage placed at a distance of 60 cm High wattage placed at a distance of 60 cm Photocell intensity Photocell intensity Apparent visual magnitude Absolute visual magnitude b. Which type of visual magnitude is used to describe stellar and other astronomical object luminosity from Earth? Why is this type misleading when trying to determine the actual stellar luminosity? Photometry is the determination and use of the color spectrum of astronomical objects to determine the objects properties. Two properties you will investigate in this exercise are distance and age. The objects you will use are stars in various clusters in the Milky Way galaxy and beyond. This is known as the Color Index method of distance determination. Figuring out a color index As seen in Wien s Law, the color of a star is related to its temperature. So to figure out how far away or how old a star is, one needs to agree to a color index which anyone can use and will not be affected by the distance to the star (remember, the star should get dimmer when it s further away, not change color!).
With the advent of colored filters in color photography, this idea became easier to implement. One could take color time-exposure photos of the stars using various filters and then compare the results. 13. a. Given a red star and a blue star of equal magnitude (apparent brightness) and given a yellow filter, which would appear brighter? b. Given a red star and a blue star of equal magnitude (apparent brightness) and given a blue filter, which would appear brighter? Hipparchus, in the second century BC, decided that a star of magnitude one (first magnitude as bright as some of the brightest stars in the sky) should be 100 times as bright as a star of magnitude six (sixth magnitude the limit of human vision). This works out to be a 2.512 times increase in brightness for every lower magnitude (Hipparchus didn t work this out this work was done in the eighteenth century AD). So the magnitude scale is a logarithmic scale, just like the Richter scale for earthquake magnitudes. In the early twentieth century, it was shown that if one measured the magnitude of star s brightness using an image of the star taken with a yellow filter (problem 1a call this the V or visual index) and if one did the same with a blue filter (problem 1b the B or blue index), then the quantity B-V (B minus V) could be related to the star s temperature. This is the B-V color index. 14. a. What color stars tended to have the hottest surface temperature? the coolest? b. What temperature stars tend to have negative B-V index numbers? What temperature stars tend to have positive B-V index numbers? A very basic correlation using the color index By the 1920 s, various astronomers had evidence that the temperature of a star was also related to the star s luminosity (the intrinsic brightness of a star). So, to quantify the luminosity without using non-standard brightness units, astronomers developed the absolute magnitude scale, which is the magnitude of a star if it could be viewed from a standard 10 parsecs away). This takes away the distance-dimming effect. When the absolute magnitude (M) was plotted against the B-V index for lots of stars, the correlation became apparent.
For instance, for a particular star cluster, these data were derived: Star Brightness Color Number in Visual B-V Magnitudes 133 14.4 1.28 165 7.6 0.12 522 11.9 0.90 980 4.2-0.10 1234 6.8-0.02 1266 8.3 0.32 1305 13.5 1.14 1355 14.0 1.19 1432 2.9-0.13 1766 9.1 0.43 1797 10.1 0.52 1040 15.8 1.44 2406 11.1 0.72 2601 15.0 1.51 15. Plot the B-V color index (x-axis) versus the absolute brightness in visual magnitudes (y-axis) on standard graph paper. One weird thing: put 0 high up on the y-axis, and number sequentially downwards. Make sure you have room for the negative B-V values. Remember to label axes and title the graph. 16. Using an appropriate diagram in your textbook, label the various areas of your diagram that classify the various stars.