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Determining Distance from Light Measurements

DETERMINING DISTANCE FROM LIGHT MEASUREMENTS

OBJECTIVE: To experimentally test the relation between brightness and distance (Eq. (1) below) and then use it to estimate a distance in the lab.

INTRODUCTION:

The distances to the nearer stars can be accurately determined using parallax measurements as we will see [or have seen] in the parallax experiment. Further, any star's brightness and its spectral make-up can be measured here on Earth. With these three pieces of information (distance, brightness, and spectra) known, we can analyze the nearer stars to obtain information about their luminosities, temperatures, sizes, and compositions by using the following relations.

The brightness of a star depends on the luminosity (i.e., power output) of the star and on the distance of the star from the earth:

brightness = luminosity / (distance)2 (1)

Thus, if the brightness can be measured with a light meter and the distance determined (from parallax), the luminosity can then be calculated.

The luminosity depends on the temperature of the star and on its size. The temperature can be determined through spectroscopic analysis of the starlight. Hence, if both the luminosity and temperature are known, then the size can be calculated.

The composition of the star can also be determined to a large extent through spectroscopic analysis and knowledge of the other previously mentioned parameters. With this knowledge, special types of stars are seen to exist and a classification of stars is possible.

This classification can then be used to extend our knowledge to even farther distances. If a star can be identified as belonging to a particular class, its luminosity can be determined. If we know its luminosity and measure its brightness, then Eq. (1) can be used to calculate its distance from us.

EQUIPMENT:

In this experiment we will use a night light as our light source (star). We will use a photocell and a DMM (Digital MultiMeter) to measure the brightness. We will use a meter stick to measure distances.

The night light should be placed on a stand and should be about 30 cm or so above the table top (to minimize reflections from the table). The light shade should be removed. The photocell tube should be placed in a holder, and the holder should be placed on a force table, with the screw on the holder protruding through one of the holes in the force table. One of the three legs should be directly behind the holder so that it can be raised or lowered and thus raise or lower the aim of the light detector tube. The closed end of the tube has two metal prongs sticking out of it. The two leads from the DMM should be connected to these prongs. Make sure the leads and the alligator connecters do not touch the metal holder. Before any wires are connected, the DMM should be set to measure resistance (in the  range) at the 200k setting. For the farther distances, you may have to rotate the dial to the 2M or 20M setting.

Now play with the setup by aiming the tube (open end) at various lights and shadows. The DMM reading should decrease when it is pointed at lights, and should increase when pointed at shadows (try covering the open end with your hand to get a dark shadow). If the numbers increase beyond a certain point, the DMM will read OL (which indicates that a higher range should be used).

THEORY:

The DMM reading will depend on the amount of light hitting the photocell, that is, the DMM reading depends on the brightness. However, the DMM reading decreases with brightness:

DMM reading = some constant / brightness ,

or, solving for brightness:

brightness = some constant / DMM reading . (2)

Since we will use the same night light for all readings, its luminosity should be a constant so that Eq. (1) can be rewritten as:

brightness = some constant / (distance)2 . (3)

Combining these two equations yields a relation between DMM reading and the distance:

(distance)2 = (some constant) * DMM reading .(4)

Thus, if we plot the square of the distance versus the DMM reading we should get a straight line, and the slope of that line should be the constant in Eq. (4). Then once we know the constant in Eq. (4), we can use it to calculate the distance if we can take a DMM reading.

REVIEW:

The equation of a straight line can be written as

y = mx + b

where y and x are the variables, m is the slope of the line and is found by taking the change in y and dividing it by the change in x, and b is the yintercept (that is, the value of y when x is zero). In our case, y is the distance squared, x is the DMM reading, m should be the constant of Eq. (4), and b should be close to zero.

PART 1: Calibration using small distances

PROCEDURE:

  1. The room should be dark except for your "star" light. Position the photocell tube about 50 cm from the light source. If you hold the legs of the force table with your fingers and have your hands on the table, you can rotate the horizontal aim of the tube. You can rotate the vertical aim of the tube using the screw adjustment on the back leg and/or use the screw adjustment on the holder (which is sticking through the hole in the force table). Adjust the aim of the tube until the lowest DMM reading at that distance is obtained. Record this lowest reading and distance.
  1. Repeat this procedure for at least 4 other distances between 50 cm and 250 cm. [It is suggest that you use the following five distances: 50 cm, 100 cm, 150 cm, 200 cm and 250 cm. If you use these distances, then you can use the table below.]
  1. Make a table like the one shown below. (You do not need to use exactly these distances. Be sure to record your DMM readings.)

DMM reading (k) / distance (cm) / (distance)2 (cm2)
50 / 2500
100 / 10,000
150 / 22,500
200 / 40,000
250 / 62,500
  1. Now graph the (distance)2 values versus the DMM readings: make the vertical axis the distance squared axis and the horizontal axis the DMM axis. Plot your points and draw the best straight line. If you know how to use Excel, then use that tool. If not, you can use graph paper. (Note: the Math Center or I can help you with Excel.) Do all the points fall close to your best straight line?
  1. Now determine the slope of your straight line. To do this, choose two points on the line (not necessarily data points!). The slope of the line should then be:

slope = [(distance)2(#2)(distance)2(#1)] / [DMM(#2)DMM(#1)].

If you use Excel, you can have that program give you the equation of the best straight line.

  1. Does your best straight line go through the origin, that is, does the yintercept = 0?

REPORT:

  1. Show your data for Part I in tabular form.
  1. Show your graph with your axes clearly labeled, your data points clearly marked, and the two points used to calculate the slope clearly marked.
  1. Discuss how close your data points fall to your best straight line, and how close your line comes to going through the origin of the graph.
  1. Show your calculations for determining the slope.

PART 2: Determining large distances by extrapolation

We now have our relation: (distance)2 = slope * DMM. In this part we will use this to calculate a large distance.

PROCEDURE:

  1. Place your "star" at one end of the room (perhaps another room) and your detector at the other end. Obtain the DMM reading as before. [If you need to go to the highest range, note that the DMM reads in M, where 1 M = 1,000 k.]
  1. Now using your relation between distance squared and DMM reading, calculate the distance squared and then take the square root of this to find the distance between the "star" and the detector.
  1. To check your result, use the tape measure (or a couple of meter sticks) to measure the actual distance.

REPORT:

  1. Show your data for Part II (the long distance), and how you calculated the distance.
  1. Discuss how close your calculated distance comes to your measured distance for Part II. As for all experiments, include a discussion of sources of experimental uncertainty.
  1. Would this method work for a different "star" ? Under what conditions could you use the slope you calculated in Part 1?