Counting the Stars
One of the hurdles I had to face in the 1960's, as a graduate student at Cornell, was the PhD qualifier exam. After a year or two of study you had to prove that you were worthy of continuing by facing an oral exam by the members of your graduate committee made up of three or four professors from your areas of major and minor study. An interesting aid to this ordeal was a book where students that had undergone this rite of passage recorded their experiences. Naturally the entries of most interest were those in which the inquisitors matched the members of your own committee. Thus I was particularly concerned with one case where one of the professors was Phillip Morrison , a man known for his keen mind and interesting history. Morrison had the reputation of asking questions that called for imagination and that even reflected his views of how one should think about science. In this case the question asked was; What is the number of visible stars? The student was supposed to arrive at an answer to this question using whatever data he had at hand, and it had to be reasoned out, he couldn't just give a number he recalled from reading a book on astronomy. As I recall the student didn't answer the question to Morrison's satisfaction and Morrison's own answer met with an argument from another member of the committee, the mathematician Mark Kac (Kac is pronounced kahts). Kac was as much a "character" as Morrison and according to the student, they became so involved in the argument that they forgot all about him.
The argument involved Morrison's use of probability. The hint he gave the student was that there is a North Pole Star but no South Pole Star. Now consider how close two objects can get and still be distinguished as individuals by the human eye. In optics the distinguishablity of objects is calculated by taking account of diffraction, a property due to the wave nature of light. The English physicist Rayleigh suggested a measure of this, now called the Rayleigh Criterion. The angular separation in radians that is separable by a lens of diameter D is, according to this, given approximately by L/D, where L is the wavelength of the light. This corresponds to a discernible area on a unit sphere of π(L/D)² . Now the area of a unit sphere is 4π , so dividing the total area by the discernible area gives the number of objects that can be distinguished in the entire sky by a particular lens for a specific light wavelength. The wavelength of visible light is roughly 400 to 500 nanometers and the size of the eyes lens is about one centimeter. But the human visual system is not quite up to the Rayleigh Criteria and we can adjust for this by decreasing the effective size of the lens to account for this (same as increasing the distance for an object to be observable. Doing this by something like 400 gives about 20,000 visible stars that could be seen. But the "experimental result" that there is only a North Pole star says that we only observe half of them (in one out of the two experiments a star is observed). If light pollution is minimized this is about what is obtained by other methods. The other method most often suggested for counting visible stars is very similar to Morrison's procedure. Take a cardboard tube (say from a paper towel roll) and point it at a number of places in the sky as randomly as possible. Count the stars seen in each case and average the result. Divide the area of a sphere with the radius of the tube by the area of the opening and multiply by the average to get your result. There are a number of web sites that discuss this method.
Given all this you can see why an argument developed. The small number of two observations makes the statistics questionable, and this is what drew Kac's attention. On the other hand getting the right result also depended on making an estimate of the eye's acuity. Maybe the experiment is more of a way of measuring this than the number of visible stars? Still I think the problem does show a way of thinking when one has limited data. The acuity of the eye could presumably be determined by another experiment. Frankly I have not looked into this.
By the way, Morrison gave me another problem at my exam. It was easier in a way. He asked me to guess an answer to the three body problem of computing the positions of the Earth, Moon and Sun system. But I'll leave that for another day.
The argument involved Morrison's use of probability. The hint he gave the student was that there is a North Pole Star but no South Pole Star. Now consider how close two objects can get and still be distinguished as individuals by the human eye. In optics the distinguishablity of objects is calculated by taking account of diffraction, a property due to the wave nature of light. The English physicist Rayleigh suggested a measure of this, now called the Rayleigh Criterion. The angular separation in radians that is separable by a lens of diameter D is, according to this, given approximately by L/D, where L is the wavelength of the light. This corresponds to a discernible area on a unit sphere of π(L/D)² . Now the area of a unit sphere is 4π , so dividing the total area by the discernible area gives the number of objects that can be distinguished in the entire sky by a particular lens for a specific light wavelength. The wavelength of visible light is roughly 400 to 500 nanometers and the size of the eyes lens is about one centimeter. But the human visual system is not quite up to the Rayleigh Criteria and we can adjust for this by decreasing the effective size of the lens to account for this (same as increasing the distance for an object to be observable. Doing this by something like 400 gives about 20,000 visible stars that could be seen. But the "experimental result" that there is only a North Pole star says that we only observe half of them (in one out of the two experiments a star is observed). If light pollution is minimized this is about what is obtained by other methods. The other method most often suggested for counting visible stars is very similar to Morrison's procedure. Take a cardboard tube (say from a paper towel roll) and point it at a number of places in the sky as randomly as possible. Count the stars seen in each case and average the result. Divide the area of a sphere with the radius of the tube by the area of the opening and multiply by the average to get your result. There are a number of web sites that discuss this method.
Given all this you can see why an argument developed. The small number of two observations makes the statistics questionable, and this is what drew Kac's attention. On the other hand getting the right result also depended on making an estimate of the eye's acuity. Maybe the experiment is more of a way of measuring this than the number of visible stars? Still I think the problem does show a way of thinking when one has limited data. The acuity of the eye could presumably be determined by another experiment. Frankly I have not looked into this.
By the way, Morrison gave me another problem at my exam. It was easier in a way. He asked me to guess an answer to the three body problem of computing the positions of the Earth, Moon and Sun system. But I'll leave that for another day.
posted by Marty Lesser at 06:24
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