In this lab you will explore some basic issues regarding statistics in astronomy and measurement uncertainties, and finally produce an estimate of the number of galaxies in the Universe (with its associated uncertainty!). This lab was constructed for viewing with Netscape 1.1N or later versions.
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Most people rarely think about statistics, except in Las Vegas. Instead, people tend to expect the natural world to work in predictable ways - when they lean on a rock, they expect the rock to hold them up. But in science, we deal with statistical uncertainties all the time (except sometimes, as in the case of the falling through the rock, the uncertainties are exceedingly small). A measurement, which most people would think is an absolute fundamental quantity, is really a statistical quantity.
To demonstrate this, choose an object (a table, a soda can, a pencil, whatever) and measure its length (be sure to always give the units in your write-up). Try to be as precise as your ruler or tape measure will let you (i.e. if the length lies somewhere between two tick marks estimate that length in your measurement). Example 1 below shows what I mean. Have two other people measure the length of the same object (do not tell them what you think the length is). List the three measurements and compare them. Most likely the three are quite similar, but not exactly the same (perhaps, they are the same).
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Example 1: In measuring the red bar with the scale given just below it, I would say that the red bar is 3.9 ticks long. I can't quite tell that it is 3.9, but I can certainly tell that it is shorter than 4 ticks and much longer than 3.5 ticks. Someone else might say that it is 3.8 ticks long or even that it is closer to 4 than to 3.9. Therefore, I would claim that a realistic measurement would be 3.9 ticks with an uncertainty of 0.1 tick.
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In general, one would now average the various measurements of the length of whatever object you are measuring (to get a more reliable measurement) and quote an uncertainty in the length based on the differences among the measurements. To get a more precise measurement one could get more people to do the measurement and average all those measurements. The uncertainty in the final measurement is usually calculated using the following formula (which in words says that for each measurement you should calculate the square of the difference between that measurement and the mean measurement, then you should sum those differences, divide by the number of measurements (in your case divide by 3), divide by the number of measurements minus 1), and finally take the square root. The you are done and have an estimate of the uncertainty related to the average of the measurements you and your companions have made. This all sounds rather contrived, but the mathematical justification for this is beyond the scope of this course).
Calculate the average of the measurements, and the uncertainty in the manner described above.
The calculated uncertainty reflects your uncertainty in the measured length. However, it is quite possible that a different group of measurements would produce a measurement outside of the expected range. In fact, the typical uncertainty quoted in astronomy (and that calculated by the above formula) is called a 1 sigma uncertainty and it tells you that 2/3rds of the time the measurement will lie within the range covered by the mean value plus or minus the 1 sigma uncertainty.
Example 2: The graph below shows a distribution of measurements of the red bar (recall that the measurement I got was 3.9 +/- 0.1 ticks). Seven of eleven measurements (close to 2/3rds) I got my friends to do are within 0.1 tick of 3.9 (as expected if 0.1 is the 1 sigma uncertainty) and the rest of the measurements differ from 3.9 by more than 0.1. None are "wrong" measurements - we expect variations at the limit of our precision. If I had a much better "ruler", I could do a more precise job of measuring, but once again at the limit of that ruler there would be some measurement uncertainty. Note that my first estimate of 3.9 is close to the mean, but it could just as well have been farther off (afterall, other versions of the measurement gave numbers as low as 3.6).
This all seems a bit esoteric, but in fact you estimate uncertainties all the time without realizing that you are doing so. For example, if you ask someone the time and they say that its 10:30, you don't think that it is 10:30:00.00! Actually, you probably think that the time is 10:30 with an uncertainty of about 5 to 10 min (i.e. you wouldn't be surprised if it was actually 10:25, but you would be surprised if it was 10:00). Give another example of where measurement uncertainty comes up in everyday life.
Now we need to cover only one more concept about measurement uncertainties before you estimate the number of galaxies in the Universe. Combining uncertainties, for example when you calculate a quantity based on various measurements with their own uncertainties, can be tricky. All we need for this lab is to see how to combine uncertainties when multiplying a number. For example, what would be the uncertainty in the length of a pink bar that is the sum of ten copies of the pink bar shown above. The total length would be 39 (i.e. 3.9 times 10) and the uncertainty would be 1 (i.e. 0.1 times 10). Therefore, anytime you are taking a measurement and scaling it by a constant (i.e. multiplying) the uncertainty is just the original uncertainty times the multiplicative factor.
NOW WE ARE READY TO MEASURE THE NUMBER OF GALAXIES IN THE UNIVERSE!!!!
I've collected three images from which you will estimate the number of galaxies in the Universe (and the uncertainty in that number).
The three images below are sections of the Hubble Telescope Deep Field Image. This image has recorded the faintest objects ever detected (to see the wide view of this field click here). It is currently being studied by astronomers around the world (including several at UCSC) in order to understand these extremely distant galaxies. The faintest objects in these images are about 4 billion times fainter than the limit of human vision. Click on any image to see it at full resolution. Every faint smudge that looks like it is more than just a few dots is a galaxy (with billions of stars and quite possibly other life looking back at us).
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In each of the three images, count how many galaxies you see. Count the faint galaxies, but only the ones that really look like indivdual, recognizeable objects. How does one decide whether a smudge is a galaxies, or whether it is one or two galaxies? Well, you'll have to do the best you can by some consistent set of criteria (which you should explain in your write-up - this, sometimes somewhat ambiguous way of doing things, is exactly the way science is done!). Calculate the mean number of galaxies per field and the uncertainty. To get the total number of such galaxies in the Universe you need to divide your measurement by the fraction of the Universe that you have observed (which happens to be about 3e-10, or 0.0000000003, for each of the three images above). Calculate the number of galaxies in the Universe and your uncertainty.
A scientist also needs to worry about systematic errors (the uncertainties you calculated were due to random variations, but there is also the possibility that the measurement is off because of fundamental problems, such as your ruler being bad, or in this case your eyesight being poor). What possible sources of systematic errors are there in this experiment? (Hint: What might happen if we tried to estimate the number of trees on the Earth from three random photographs). How could you improve this experiment?
