«Lab 8 Parallax Measurements and Determining Distances 8.1 Introduction How do we know how far away stars and galaxies are from us? Determining the ...»
At this point, you are ready to average your distance measurements together, and compute their standard deviations. Access the plotting tool listed for this lab exercise from the GEAS project lab exercise web page (see the URL on page 2 in §8.1.2). You can use the plotting tool to create histograms of your distance measurements if you have a user account, entering the three values measured in Trials 1 through 3 for each quantity in turn. Once you enter your three values the web page will will display their mean value and associated error (you do not need to log in to see this information). Record the averaged values shown for each plot (the “mean value”, or µ), and the associated errors (σ). In order to see how good your distance measurements are, calculate these averages and errors for each of the distance estimates d recorded three times in the ﬁnal column of Table 8.3.
9. Now compare the distances that you calculated for each position using the parallax method to the distances that you measured directly at the beginning of the experiment (in Table 8.2). How well did the parallax technique work? Are the diﬀerences between the direct measurements and your parallax-derived measurements within your errors (within 2σ)? (2 points)
10. If the diﬀerences are larger than 5σ, can you think of a reason why your measurements might have some additional error in them? We might call this a “systematic” error, if it is connected to a big approximation in our observational setup. (2 points) 8.3 Calculating Astronomical Distances With Parallax Complete the following section, answering each of the four questions in turn. (Each question is worth either 3 or 5 points.) 8.3.1 Distances on Earth and within the Solar System
1. We have just demonstrated how parallax works on a small scale, so now let us move to a larger playing ﬁeld. Use the information in Table 8.4 to determine the angular shift (2α, in degrees) for Organ Summit, the highest peak in the Organ Mountains, if you observed it with a baseline 2r of not 2 feet, but 300 feet, from NMSU. Organ Summit is located 12 miles from Las Cruces. (If you are working from another location, select a mountain, sky scraper, or other landmark at a similar distance to use in place of the Organ Summit.) There are 5,280 feet in a mile. (3 points) You should have gotten a small angle!
The smallest angle that the best human eyes can resolve is about 0.02 degrees. Obviously, our eyes (with an internal baseline of only 3 or 4 inches) provide an inadequate baseline for measuring large distances. How could we create a bigger baseline? Surveyors use a “transit,” a small telescope mounted on a protractor, to carefully measure angles to distant objects.
By positioning the transit at two diﬀerent spots separated by exactly 300 feet (and carefully measuring this baseline), they will observe a much larger angular shift. Recall that when you increased the distance between your two vantage points, the angular shift increased.
This means that if an observer has a larger baseline, he or she can measure the distances to objects which lie farther away. With a surveying transit’s 300-foot baseline, it is thus fairly easy to measure the distances to faraway trees, mountains, buildings or other large objects here on Earth.
2. What about an object farther out in the solar system? Consider our near neighbor, planet Mars. At its closest approach, Mars comes to within 0.4 A.U. of the Earth. (Remember that an A.U. is the average distance between the Earth and the Sun, or 1.5 × 108 kilometers.) At such a large distance we will need an even larger baseline than a transit could provide, so let us assume we have two telescopes in neighboring states, and calculate the ratio r/d for Mars for a baseline of 1000 kilometers. Can you even ﬁnd this value in Table 8.4? (3 points) You should get a value for r/d which lies well beyond the bounds of Table 8.4, less than 5 × 10−5. For the correct value, the equivalent value of 2α is 0.00019 degrees, or 7 arcseconds (where there are 60 arcminutes per degree, and 60 arcseconds per arcminute).
8.3.2 Distances to stars, and the “parsec” The angular shifts for even our closest neighboring planets are clearly quite small, even with a fairly large baseline. Stars, of course, are much farther away. The nearest star is 1.9 × 1013 miles, or 1.2 × 1018 inches, away! At such a tremendous distance, the apparent angular shift is extremely small. When observed through the two vantage points of your two eyes, the angular shift of the nearest star corresponds to the apparent diameter of a human hair seen at the distance of the Sun! This is a truly tiny angle and totally unmeasurable by eye.
Like geological surveyors, we can improve our situation by using two more widely separated vantage points. In order to separate our two observations as far as possible from each other, we will take advantage of the Earth’s motion around the Sun. The Earth’s orbit forms a large circle around the Sun, and so by observing a star from ﬁrst one position and then waiting six months for the Earth to revolve around to the other side of the Sun, we will achieve a separation of two A.U. (twice the average distance between the Earth and the Sun). This is the distance between our two vantage points, labeled b in Figure 8.8.
An A.U. (astronomical unit) is equal to 1.5 × 108 kilometers, so b is equal to twice that, or 300 million kilometers. Even though this sounds like a large distance, we ﬁnd that the apparent angular shift (2α in Figure 8.5) of even the nearest star is only about 0.00043 degrees, or the width of a quarter coin observed from two miles away. This is unobservable by eye, which is why we cannot directly observe parallax by looking at stars with the naked eye.
However, such angles are relatively easy to measure using modern telescopes and instruments.
(Note that the ancient Greeks used the lack of observable parallax angles to argue that the Sun rotated around the Earth rather than the reverse, having greatly underestimated the distances to the nearest stars.) Let us now discuss the idea of angles that are smaller than a degree. Just as a clock ticks out hours, minutes, and seconds, angles on the sky are measured in degrees, arcminutes, and arcseconds. A single degree can be broken into 60 arcminutes, and each arcminute contains 60 arcseconds. An angular shift of 0.02 degrees is thus equal to 1.2 arcminutes, or to 72 arcseconds. Since the angular shift of even the nearest star (Alpha Centauri) is only 0.00042 degrees (1.5 arcseconds), we can see that arcseconds will be a most convenient unit to use when describing them. Astronomers append a double quotation mark ( ) at the end of the angle to denote arcseconds, writing α = 0.75 for the nearest star. When astronomers talk about the “parallax” or “parallax angle” of a star, they mean α.
The small angle approximation
Figure 8.9: This ﬁgure plots the angle α on the y-axis and the height to width ratio r/d on the x-axis.
On the left, we see the relationship for angles between zero and 80◦, while on the right we restrict our range to run from zero to two degrees. In each case, the yellow line represents an attempt to ﬁt a straight line to the points. The slope of the yellow line (m) is listed on each plot. For which values of α is there a linear relationship between α and r/d?
Based on the data shown in Figure 8.9, is a straight line a good ﬁt to either set of data (for α ≤ 80◦, or just up to 2◦ )? What is the slope, in the linear region? Are you now comfortable using this approximation to shift between α and r/d? (2 points)
The word parsec comes from the phrase “parallax second.” By deﬁnition, an object at a distance of 1 parsec has a parallax of 1. How far away is a star with parallax angle of α = 1, in units of light-years? It is is 3.26 light-years from our solar system. (To convert parsecs into light years, you simply multiply by 3.26 light-years per parsec.) An object at 10 parsecs (32.6 light-years) has a parallax angle of 0.1, and an object at 100 parsecs has a parallax angle of 0.01. Remember that the farther away an object is from us, the smaller its parallax angle will be. The nearest star has a parallax of α = 0.78, and is thus at a distance of 1/α = 1/0.75 = 1.3 parsecs.
You may use the words parsec, kiloparsec, megaparsec and even gigaparsec in astronomy.
These names are just shorthand methods of talking about large distances. A kiloparsec is 1,000 parsecs, or 3,260 light-years. A megaparsec is one million parsecs, and a gigaparsec is a whopping one billion parsecs! The parsec may seem like a strange unit at ﬁrst, but it is ideal for describing the distances between stars within our galaxy.
4. Let’s work through a couple of examples. (6 points) (a) If a star has a parallax angle of α = 0.25, what is its distance (in parsecs)?
(b) If a star is 5 parsecs away from Earth, what is its parallax angle (in arcseconds)?
(c) If a star lies 5 parsecs from Earth, how many light-years away is it?
8.4 Final (Post-Lab) Questions
1. How does the parallax angle of an object change as it moves away from us? As we can only measure angles to a certain accuracy, is it easier to measure the distance to a nearby star or to a more distant star? Why? (3 points)
2. Relate the experiment you did in the ﬁrst part of this lab to the way that parallax is used to measure the distances to nearby stars. Describe the process an astronomer goes through to determine the distance to a star using the parallax method. What did your two vantage points represent in the experiment? (5 points)
3. Imagine that you observe a star ﬁeld twice, with a six-month gap between your observations, and that you see the two sets of stars shown in Figure 8.10:
Figure 8.10: A star ﬁeld, viewed from Earth in January (left) and again in July (right).
Which star do you think lies closest to Earth?
The nearby star marked P appears to move between the two images, because of parallax. Consider the two images to be equivalent to the measurements that we made in our experiment, where each image represents the view of an object relative to a distant object as seen through one of your eyes. None of the stars but P change position; they correspond to the distant objects in our experiment.
If the angular distance between Stars A and B is 0.5 arcseconds then how far away would you estimate that Star P lies from Earth?
First, estimate how far Star P has moved between the two images relative to the constant distance between Stars A and B. This tells you the apparent angular shift of P (2α). You can then use the parallax equation (d = 1/α) to estimate the distance to Star P. (3 points)
4. Astronomers like Tycho Brahe made careful naked eye observations of stars in the late 1600s, hoping to ﬁnd evidence of semi-annual parallax shifts for those which were nearby and so weigh in on the growing debate over whether or not the Earth was in motion around the Sun. If the nearest stars (located 1.3 or more parsecs from Earth) were 100 times closer to us, or if the resolving power of the human eye (0.02 degrees) was improved by a factor of 100, could he have observed such shifts? Explain your answer. (4 points)
8.5 Summary Summarize the important concepts discussed in this lab. Include a brief description of the basic principles of parallax and how astronomers use parallax to determine the distances to nearby stars. (25 points)
Be sure to think about and answer the following questions:
• Does the parallax method work for all of the stars we can see in our Galaxy? Why, or why not?
• Why is it so important for astronomers to determine the distances to the stars which they study?
Use complete sentences, and be sure to proofread your summary. It should be 300 to 500 words long.
8.6 Extra Credit Use the web to learn about the planned Space Interferometry Mission (SIM). What are its goals, and how will it work? How accurately will it be able to measure parallax angles? How much better will SIM be than the best ground-based parallax measurement programs? Be sure that you understand the units of milliarcseconds (“mas”) and microarcseconds, and can use them in your discussion. (5 points)