Stellar Photometry in the Physics Lab Room
Shawn Langan
Email: slangan@unl.edu
Phone: 402-472-2199
Fax: 402-472-6184
Zachary Smith
Email: zachary.smith@huskers.unl.edu
Phone: 701-330-8994
University of Nebraska-Lincoln
855 North 16th Street
Department of Physics and Astronomy
Lincoln, NE 68588-0299
Equipment List:
For Radiometry Counting Experiment:
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Geiger Counter
Button Source of Gamma Radiation (e.g. Cobalt-60)
Stand to support Geiger Counter above Gamma Radiation Source
Ruler
Computer with Plotting Software (Microsoft Excel) and Counting Software (DataStudio)
Computer interface box to allow Geiger Counter to interact with the Counting Software
For Stellar Photometry Experiment:
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
Computer with Astronomical Image Processing Software (Aperture Photometry Tool)
.Fitz Image File to be processed. Introductory labs may want to make use of single star
images, while more advanced labs may use images of star fields
Apparatus:
Geiger Counter
Computer
Interface Box
Computer with
Counting Software
Stand for
Geiger Counter
Figure 1: Standard apparatus for Radiometry experiment. Button Radiation Source is not shown,
but is necessary to complete the experiment
Experiment Instructions and Sample Data
The following pages include the lab manual provided to Physics 223 Lab Students and
Astronomy 224 Lab Students at the University of Nebraska-Lincoln. Following the lab manual, a
sample radiometry graph is provided.
Counting Photons – 1, 2, 3 … it’s EZ!
You will be using the Geiger-Mueller sensor during this experiment. This is an instrument to
measure radiation “hits” from a radioactive source. To prepare the experiment, first see that the GM
sensor is plugged into the interface and check that you have the correct software open. You should be
looking at a DataStudio display similar to what you have used before. You can hit start to begin data
collection, and then you will see a number come up when your GM tube has detected a “hit” of radiation.
The final product that you are interested in is “counts/time” or what is called a “count rate”. To find this
you will divide the number of counts displayed in the “digits” display, by the time interval that you took
data. A typical unit for this is “counts/second”. Since radiation comes as a single “hit”, if you find the
count rate and it is a decimal, you should round it to the nearest whole number.
1.
Today you will be using a source of radiation that emits Gamma Radiation. This is an electromagnetic wave emitted with extremely high energy. If you moved so that you were twice as far
from a Gamma source, how would your exposure to radiation decrease? Would it be half as much?
How do you know? HINT: Think about how the intensity of visible light behaves with increasing
distance!
You should make a measurement of the background radiation, as measured by this sensor in
today's conditions. Since this measurement takes time, let it run as you read the following new
information. You should run the GM sensor for about 5 minutes to get a nice average of the background
counts per 15 seconds. You will see why it’s in 15 second intervals later.
2. Write down your background count rate here when you finish the background measurement:
Fluctuations in Random Events
Suppose that you count 253 radioactive decays in a 20 sec time interval. However, when your lab
partner repeats this same measurement, he/she counts only 232 decays. Should you feel suspicious unhappy – disappointed? Accuse him/her of carelessness? Is this difference likely to be due to
experimental inaccuracy or just to "statistical fluctuations"? It is very important to have some answer to
questions of this sort when dealing with data (such as counting rates) and this lab will introduce you to
some helpful probability concepts and theory.
This question of the size of statistical fluctuations could be answered experimentally for any
given set of experimental conditions. The approach would be straightforward: You would simply repeat
the experiment a very large number of times, being careful not to commit any experimental errors. From
the number of times a given answer occurred, you could then infer (approximately) its probability.
For example, suppose your experiment is to count the number of γ particles during a 10 second
time interval. The first time this experiment is done, you obtain 4 counts; the second time, 3 counts; the
third time, 7 counts; and so on. Suppose you do the experiment a total of one hundred times (N = 100).
The data could be reported similar to the first two columns of Table I, where n is the number of counts in
a time interval and f(n) is the number of times this result was obtained (i.e., its "frequency").
References:
Weidner and Sells, Elementary Modern Physics (1973), Section 8-2.
Joseph and Reynolds, Laboratory Manual for 223 (1975), Section N1.
LabNet Geiger-Müller Interface Instruction Manual, Pasco Scientific (1994).
Table I - Sample Data
N = 100 trials using identical experimental conditions
n
f(n)
Experimental Probability
P(n) = f(n)/100
(# of counts)
(# of times
[bin size = 1]
n occurred)
0
1
0.01
1
2
0.02
2
12
0.12
3
13
0.13
4
13
0.13
5
21
0.21
6
9
0.09
7
16
0.16
8
8
0.08
9
2
0.02
10
1
0.01
11
2
0.02
From these data, you would, for example, estimate a probability of 0.13 (that is, 13 chances out of
100) of obtaining 4 counts if exactly the same experiment were repeated once more. That is, P(n=4) =
0.13. What is the probability of n = 10 based on this data?
Histogram of the Data
A good way of presenting such
data is shown on the graph of frequency
as a function of number of counts (f(n)
vs. n).
For each value of n, a bar is
drawn with a height equal to the number
of times that value of n was observed.
Such a graph is called a "histogram".
Experimental Probability P(n) of Obtaining n Counts
Probability of Obtaining n Counts
0.25
0.2
P(n)
0.15
Histogram
0.1
0.05
0
0
1
The data can also be displayed
as probability data, which is shown in
the second graph (P(n) vs. n). For each
value of n, a bar is drawn with a height
equal to the experimental probability of
that value of n being observed.
2
3
4
5
6
7
8
9
10
11
n (# of counts)
Now these histograms look rather jagged. Are the irregular ups and downs also statistical
fluctuations? Indeed they are. To obtain a more regular curve, we might try doing the experiment, not a
mere 100 times, but 1000, 10,000, or 106 times. However, it is now clear that this would certainly be an
arduous way to answer the question at the beginning of this section! Is there a better way…?
A Little Probability Theory
Fortunately, probability theory provides us with an answer without all that work. A radioactive
source such as you will be using contains a very large number of unstable nuclei, each of which has a
very small probability of decaying in, say, 10 s. Under these conditions, the probability of obtaining
precisely n counts during a given time interval is:

1 1 / 2( n n )
Pn ( n )  
e
2
2


2
2

  bin size  ,


where σ is the standard deviation defined on the next page and n is the average number of counts during
such an interval (the mean value).
Using this value and the Gaussian distribution equation above, yields the theoretical
probabilities listed in the fourth column of Table II, and plotted as a smooth solid curve on the graph.
This curve gives the locus of the ends of the bars in the histogram that we would have obtained had we
done the experiment a huge number of times. (This curve might be shifted a bit due to error in our
determination of n .)
Returning to the sample data of Table I, we can calculate the mean value of n, with the result
 n f (n )
i
n 
i
100
i
 5.02 . Since there will be some experimental uncertainty in this number also, we will
round it off to n = 5.0.
Table II - Sample Data and Theoretical Estimates
N = 100 trials using identical experimental conditions
n
f(n)
Gaussian Probability
(# of times)
Experimental
Probability
P(n)=f(n)/100
(# of counts)
0
1
0.01
0.015
1
2
0.02
0.036
2
12
0.12
0.073
3
13
0.13
0.120
4
13
0.13
0.161
5
21
0.21
0.178
6
9
0.09
0.161
7
16
0.16
0.120
8
8
0.08
0.073
9
2
0.02
0.036
10
1
0.01
0.015
11
2
0.02
0.005
( n = 5)
[bin size = 1]
Experimental and Theoretical Probability P(n) of Obtaining n Counts
The fact that the Gaussian distribution is a good approximation is very useful, because tables are
easily available for the normal distribution that tell the probability of obtaining a value of n which
deviates by any given amount from the average value n . (See, for example, the Mathematical Tables
from the Handbook of Chemistry and Physics.) In such tables, the deviations are usually expressed in
terms of the "standard deviation” which is simply,  = sqrt( n ). The most important results
(worth remembering) are
Probability that ( n  n    1 standard deviation ) = 68 %
Probability that ( n - n £ 2s ) = 95 %
Probability that ( n  n  3 ) = 99.7 %
Returning to the question at the start of this section; suppose that you count 253 radioactive
decays in a 20 sec time interval. However, when your lab partner repeats this same measurement, he/she
counts only 232 decays. Is this difference likely to be due to experimental inaccuracy or just to
"statistical fluctuations?”
For this example, you may wonder how to find n without a lot of work (such as repeating the
measurement a thousand times!). This turns out to be no problem in practice. Suppose you wish to
measure a rate to an accuracy of 10%, with a probability of 68%. (This is surely a modest accuracy.) To
do so, you must choose a time interval such that
n
 0.10 , which requires that n  100. Now your
n
first measurement will almost surely lie in the range 80  n  120. So you can use it for n and your
possible error in n is only 10% (with a probability of 95.4%).
Using this short cut and the stated values, the standard deviation,   253  16 . Therefore,
253 - 232 = 21  1.3 . Even if n were exactly 253, there would still be a sizable probability that a
second measurement would deviate by this much. So you should thank your partner for the data, add that
number to yours to obtain 485  22 (note that 485  22 ) for the number of counts in an interval of
twice the original length, and quote your combined result as 242  11 counts for the original time interval.
(Note the standard form used here; the count is reported as n  .)
Now that you have a little theoretical background, it is time to try this all out with some real
data!
The Experiment
You should not handle the radioactive sources any more than necessary. However, note that in
this experiment you can acquire at most a small fraction of the daily radiation limit recommended by the
AEC for those who handle radioactive materials regularly. Keep in mind the following precaution:
Caution:
Do not bring the source close to your eye; it can damage the cornea.
Your experimental task is:
1.
2.
3.
4.
5.
6.
7.
8.
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Obtain a radioactive γ-source from your instructor. Try out the set up until you are convinced:
(1) that you know what you are doing and (2) that it’s working properly.
Place your radioactive source under the GM tube and tape it to the table.
Move your GM tube to within 5 cm of the top of the source.
Determine the number of counts in 50 successive 15-second intervals.
You can ask your instructor for a quick way to do this in DataStudio.
Record your data in Excel and leave it in units of “counts/15 seconds”.
Next subtract off your background data. Then remember to round your answer to the nearest
integer if it is not already.
This will leave you with whole numbers to plot and that makes it MUCH easier to do the
following data analysis:
FIRST return the radioactive source to your instructor.
Plot a histogram of your experimental probability values and compute n from your data.
NOTE:
a. You may find the “COUNTIF” function of Excel helpful in doing this.
b. You should plot the decimal probability, NOT the actual number of occurrences.
3. Now compare your data to the theoretical distributions. How do your histograms compare with the
Gaussian distribution? Make the appropriate layers on ONE plot using Excel to compare your data to
these predictions. Once this is accomplished, print out the plot and tape it into your lab notebook.
4. Probability theory predicts that 68% of your data should be within 1 standard deviation of the mean
and 95% within 2 standard deviations. How well do your data support this prediction?
So you may ask, why we are considering γ-radiation when light from a star is a photon!? Well, it turns out
that γ-radiation is simply a higher energy photon than the light we observe in a CCD from a distant star.
Therefore we are actually performing “observations” of small, manufactured “stars” when we take data on
a radiation source. Both types of photons obey what are called “counting statistics”. For example, if we
were to take a picture of star with a CCD and we received 1000 photons with our camera. Then took another
picture of the star, we might receive 999 counts. Or 1001 counts! Following this trend we would find that
the readout from the CCD camera would follow a Gaussian distribution! A graph of similar data illustrating
this is below:
Here we are seeing the number of occurrences of a pixel value plotted vs. counts/pixel of the sky pixels.
Notice the familiar shape of the best fit line to the histogram?!
What further connections can we make to observing a star? Well, you recall that we measured the
background radiation in the room. It turns out that whenever you take data on a star, you are also taking
measurements of the photons from the earth’s atmosphere. This background noise is often referred to as
sky counts. In subtracting out the background radiation, we performed the same process as subtracting out
the sky.
There are actually many more parameters we can similar relating radiation to stars. To begin with,
some of the software you used in past labs referred to a signal/noise ratio. This is the number of photons
that you would expect to receive, or actually receive, compared to the photons of the background static.
When imaging a star, there are MANY sources of background noise, the sky being only one of them. But
let’s play around with the statistics of this S/N.
The direct equation for S/N is:
S/N = n/√(n),
where n is the number of counts you receive at the CCD/Geiger Counter. In the graph above, let’s assume
we measure 100 counts / pixel, the most common measurement. The S/N is then:
S/N = 100/√(100) = 100/10 = 10,
This is NOT a good signal to noise!! An average S/N is about 100.
5. What would be the value for n to get a S/N of 100?
However, we are making some BIG assumptions when we do this simple calculation! We assume that the
sky is part of the measurement that we REALLY want!? Sadly, that is not the case. We want to know the
signal from the star compared to the noise from the photons from the star AND the sky. We assumed
before, that for the total counts from the star, we can simply follow the basic logic of:
Cstar = Cmeasured - Csky
This makes perfect sense, and it’s true for measuring the raw counts, but not quite the whole story when we
measure the S/N. The equation for S/N when factoring in the noise from the sky becomes:
S/N = nstar/√(nstar+sky) = nstar/√(nmeasured)
6. Using the most common value of hits/15 seconds from your radiation data, calculate your signal to noise,
including the background radiation in the noise. Is it very good?
Something interesting that may come to mind at this point. What if you were to simply start the
GM tube and let it run indefinitely, would your S/N improve? Yes, it most definitely will! How would this
change behave? Let’s take a look!
The counts from a star can be put in a different way:
C = R*T,
where C is the total counts after … say 15 seconds, R is the count rate/second, and T is the time you are
measuring. If we plug this into our S/N equation, then we are left with:
S/N = T*Rstar/√(T*Rstar+sky) = √T *[Rstar/√(Rmeasured)]
So the S/N goes at √T, when it comes to integration time.
7. How much time would you need to integrate/measure over to get a S/N that is twice your current signal
to noise?
As was mentioned in previous labs, there is a certain sensitivity of each photon detector to various colors.
For example, CCD’s are VERY sensitive to red and not very sensitive to blue. They are completely useless
for UV radiation, but pretty good in the Infrared. GM tubes are good for higher energy photons, where the
CCD is useless, but GM tubes have a terrible “efficiency” in measuring photons of any energy. CCD’s
absorb about 90% of the photons in the red spectrum, while GM tubes absorb about 10% of the incoming
photons. One must take this into account when measuring light from a star with a filter on it. We call this
ability to absorb photons the Quantum Efficiency.
Not only do many detectors have a certain Quantum Efficiency, but many CCD’s and GM tubes
have what is called a gain. This usually is a multiplicative factor that accounts for the discrepancy between
photons and the electrons they generate in the circuits, which is dependent on the device being used. It is
usually small and more than 1 for a CCD. The final count of photons for star can illustrated as:
PHOTONS = (Counts*Gain)/Quantum Efficiency
8. Using your most common value of radiation hits, and knowing that the Q.E. for GM tubes is typically
10% with a gain of 2, how many photons did you actually detect for those measurements?
Note, when astronomers convert to the magnitude scale for viewing stars, they must know the incoming
photon flux, which is photons/m2. In order to calculate this, we must know the number of photons, not just
the number of electrons measured by the computer.
Please be sure you have returned the radiation source to the instructor and turn off both the interface
and computer.
Sample Radiometry Data