The Telescope Field of View
An Introduction to Data Analysis
Goals of the Lab
To review measurement errors, standard deviation, and random versus systematic errors.
To calculate the Field of View (FoV) of the finder scope, the 25mm eyepiece, and the 10mm eyepiece.
Requirements: a calculator, starmap SC-001.
Background and Theory
Common synonyms for error are mistake, fumble, and blunder. However, in science the term error carries a different meaning. In terms of scientific measurement error means the inevitable uncertainty that exists in all measurements. Therefore, errors are not mistakes; you cannot eliminate them by simply being careful. The best you can hope for is to ensure that errors are as small as reasonably possible and to have a reliable estimate of how large they are. For our purposes, error means uncertainty; the two words can be used interchangeably.
In lab, we will discuss two types of error: systematic and random error. Random error represents how the result may change if you repeated an observation or experiment many times. Random error typically comes from a scientist's inability to take the same observation or measurement in exactly the same way to get the exact same number. Therefore, the best way to reduce random errors is by repeating the observation or experiment many times.
Systematic error is an error not determined by chance, but introduced by an inaccuracy inherent in the system. Systematic errors are reproducible inaccuracies. Systematic errors are often difficult to detect and cannot be analyzed statistically.
As an example, consider an experiment in which we time the period of a pendulum. A random source of error will be our reaction time in starting and stopping the watch. If our reaction time were always exactly the same, these two delays would cancel each other out. In practice, our reaction time will vary. We may delay more in starting, and then underestimate the period of the pendulum; or we may delay in stopping, and overestimate the period. Each scenario is equally likely. If we repeat the measurement several times, we will sometimes overestimate and sometimes underestimate. By analyzing the spread of results statistically, we can get a reliable estimate of this type of error.
On the other hand, an systematic error would be if our stopwatch is running consistently slow. Then all of our times will be underestimates, and no amount of repetition (with the same watch) will reveal this source of error. This type of error gives us results that are always pushed in the same direction - if the watch runs slow, we always underestimate; if the watch runs fast, we always overestimate). These errors cannot be determined using statistical analysis. We would need to identify our stopwatch as the source of error, and then determine how fast or slow it ran. Only then could we correct for this error.
Note that systematic and random errors refer to problems associated with an observation or measurement. Mistakes made in calculations or in reading an instrument are not considered error analysis.
Part I: Statistical Analysis of Random Errors
As mentioned in the introduction, when we make an observation or perform an experiment some error is expected. Consequently, our result must include our best guess of this error. To express our random error, we will calculate the standard deviation. The standard deviation is defined as the average amount by which each measurement in our set differs from the mean, or average, value. It is the average uncertainty of the measurements in a given set, such as the five timings you made of Altair or Capella in the 25mm eyepiece last week. The mean is the best estimate of our measurement among all of our measured values.
Create a table with the trial number, the measured value, the deviation, and the deviation squared for each set of measurements you made for the Telescope Basics Lab. For now, leave this table blank, but make sure you have enough rows to go with the number of trials you took. Use the following table as an example. You should have three tables for: the 25mm and 10mm eyepieces for the star at zero declination the 25mm eyepiece for the star at high declination.
Trial # Measured
ValueDeviation
from MeanDeviation
Squared1 2 ... Convert all of your timings from the Telescope Basics lab into seconds. So, for instance, if you have a time which is 2m38s, that would become 158 seconds. Record these in the second column of the table above below.
Calculate the mean (that is, average) in seconds for the following measurements: the 25mm and 10mm eyepieces at for the star at zero declination and the 25mm eyepiece for the star at high declination. Write down these means.
For each trial in your tables, calculate the deviation (value minus mean value), and record it in the third column of the tables. If the value is less than the mean value for that eyepiece/star combination, then the deviation should be negative (as expected from the subtraction).
Square each of your deviations, and record the deviation squared in the fourth column of each of the tables above.
Using the values from the your table, calculate the standard deviation for each set of measurements, using the following formula (show your work):
For each measurement set, give the best estimate. This is the mean value with its appropriate error (i.e. the standard deviation). For example, if you came up with an average value of 41 seconds and a standard deviation of 4 seconds, your best estimate would look like:
41 ± 4 seconds
Part II: Systematic Errors
Compare the timings you recorded for the 25mm eyepiece at zero declination to those between 40 and 60 degrees declination. Are the timings the same for different declinations? If the timings are different, examine how they differ from each other. For example, are the timings at higher declinations bigger/smaller than those at zero declination?
Consider the above discussion about different types of error. Would you say that the fact that one star had different timings from the other star is from the sort of effect that would give you a random error, or from the sort of effect that would give you a systematic error? Why?
Discuss possible explanations for the difference in the times of the equatorial star and the times of the high-declination star. How could you correct for these differences? (Hint: look back at the last question qualitatively.)
Give a couple of examples of systematic errors that could contribute to your overall error in any of your average timings.
Class Discussion. At this point, draw a line, and do not erase or add to anything above the line. We will have a class discussion about the previous three questions. Make any notes you wish to make that you would want to add to your answers to the above questions. Again, add theses notes below the line you've drawn; do not go back and change any previous answers.
Comparison with class average: After the previous discussion, the TAs will instruct you to go to the following page and enter your average timings for the 25mm and 10mm eyepiece at the equator (calculated in Part I above), and your average timings for the 25mm eyepiece and the high-declination star:
From this data, the TAs will present you with two numbers for each star/eyepiece combination:
- average of class averages; this is an overall average of
the class' measurement of the timing.
- standard deviation of class averages; just as your standard deviation (random error) from Part I tells you how much your individual measurements tend to differ from each other, this tells you how much the average timings of different people in the class differ from each other.
- average of class averages; this is an overall average of
the class' measurement of the timing.
Answer the following three questions before the class discussion on them; do not change your answers during or after the class discussion.
How do your average timings compare to the class average? For each measurement, how does the difference between your timings and the class average compare to (a) your standard deviation, and (b) the standard deviation of class averages? Is your value acceptable? Explain.
Recall that the standard deviation provides an estimate of your random error. Is your estimate of your random error larger or smaller than the standard deviation of the averages for everybody in the class? Does this tell you that your random errors, or your systematic errors are bigger?
If your measurement uncertainty were dominated by pure random errors, would you expect your own standard deviation to be smaller than, similar to, or greater than the standard deviation of class averages? Explain.(Hint: Think about how the standard deviation depends on N, the number of measurements.)
Wait for class discussion of the previous three questions. After the class discussion, answer the previous three questions again; do not erase or cross out your old answers, and make very clear which answers were after the class discussion. (If you would give exactly the same answers, you can just say that.)
Part III: Calculating the Field of View
The crossing time you measured last week is the amount of time it took for the Earth to rotate just enough so that the edge of the FoV of your telescope passed from one side of the object to the other side of the object.
For the crossing times we use the fact that the earth rotates 360° in 24 hours. The FoV is then given by the following unit conversion:
where t is the time in seconds it takes to cross the field of view. This formula gives you the field of view in degrees. For small fields of view (less than 1°), it's often more useful to convert the field of view to arc minutes:
60' means 60 arc minutes. Each arc minute may be further divided into an arc second, with 1'=60".
From the above discussions, which measurements do you want to use to calculate the FoV: those at the Celestial Equator (i.e. at 0°) or those made between 40° and 60°? Explain your choice.
For your measurements, you can use unit conversions to change your time into an angular distance. The final result you obtain will represent the angular size of the piece of sky seen through the telescope with the different eyepieces or finder scope. Since the result is an angle, it is measured in degrees, arc minutes, or arc seconds.
Use the above equation to calculate the FoV, using the mean calculated in Part I. Give your results in both degrees and arc minutes. Convert your standard deviation as well, and express your final answer with its appropriate error. Remember that you only measured half of the field of view for the finder scope. Calculate a field of view for (a) the 25mm eyepiece, (b) the 10mm eyepiece, and (c) the finder scope. For the finder scope, you don't have a standard deviation in time, so you don't need a standard deviation in the field of view.
One application of the FoV calculation is that you will be able to determine how much of the SC-001 map you will be able to see through different eyepieces. This helps you better identify what you see through the telescope, with the stars on the map. Using your values for the FoV, would you be able to see both gamma and beta Lyrae in the 25 mm eyepiece? in the 10mm eyepiece? in the finder scope? (To find these stars, locate Vega, the brightest star in Lyra, on SC-001. Beta and gamma Lyra will be the two stars nearby labelled with the appropriate Greek letters– β for beta and γ for gamma.)
Would you be able to see the entire moon (approximately 0.5 degrees) in the 25 mm eyepiece, the 10 mm eyepiece, or the finder scope?