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Transcript
Marian Physics! Parallax Flat Prairie Publishing Parallax 1 Measuring the distance to a star is a pretty difficult task. Direct measurement using a meter stick or long tape measure is not possible. For stars that are close to us parallax used. Section 1 Parallax Parallax: One of the most difficult problems in astronomy is determining the distances to objects in the sky. There are five basic methods of determining distances: radar, parallax, cepheid variables, standard candles, and the Hubble Law. Each of these methods is most useful at certain distances, with radar being useful nearby (e.g., the Moon), the Hubble Law being useful at the farthest distance (e.g., galaxies far, far away). In this exercise, we investigate the use of the measured parallax method to determine distances to nearby stars, those within about 650 light years from the Sun. Even when observed with the largest telescopes, most stars are still just points of light. Although we may be able to tell a lot about a star through its light, these observations do not give us a reference scale to use to measure its distance. We need to rely on a method that you are familiar with: the parallax of an object. 2 Extend your arm directly away from your body. Now take one finger and point it at a distant object. Alternately close each eye. When you close one eye the position of your finger with respect to the distant object will change. This is parallax. This is because the centers of your eyes are 5 – 6 centimeters apart, so each eye has a different point of view. As a side note most people have a dominant eye. If you close your left eye and your finger does not move, your right eye is dominant and verse visa. Since the earth is in orbit around the sun, we can look at a nearby star six months apart and measure the angle. This angle is called the parallax angle. The further the star is away from us the smaller the parallax angle. the angle is large The actual parallax angle is the angle as indicated below. This allows us to use simple trig to calculate the distance to a star. As you can see the opposite side to the parallax angle is 1AU (Astronomical Unit, the average distance from the earth to the sun). The sin function will then give us the distance to the star. 3 Of course astronomers are not ones to use simple methods. They have goobered up this measurement just like the ones with magnitude. So keep reading and the enumeration as to how astronomers use parallax to measure the distance to a star will be given. Except for our sun, the stars are pretty far away. Because of this the parallax angle is extremely small. Astronomers use arc seconds (one second of one minute of one degree of angle). The arc second is 1/3600 of a degree or .00103 degrees. Take a protractor and try to draw 3600 lines between each degree mark. Now you know how hard that small of an angle is to measure. Instruments that could measure such a small angle were not available until the mid 1800’s. Around 1830, Thomas Henderson, working in South Africa, Friedrick Wilhelm Struve, and Friedrick Wilhelm Bessell (both in Europe) started working in earnest on measuring the distance to a star. Henderson actually got the jump on the others, making measurements of Alpha Centauri. In 1833, he packed up and went back to England, along with his data. He was in no hurry to reduce his data, so it languished for years. When he finally did get to looking at his measurements, he found that there did seem to be what may have been a parallax shift in Alpha Centauri, but he did not trust his data. He had only 19 measurements and the instrument that he had been using had been damaged in shipping to South Africa. He decided to wait for better measurements. As it turns out, Henderson’s data were approximately correct. However, he didn’t know that, so he held off publishing his findings until he had more data. In 1837, Struve announced that he had measured the distance to Vega. He reported a parallax angle of 0.125 arc seconds. Bessel was working on measuring the parallax angle for 61 Cygni. He measured a parallax angle of 0.314 arc seconds and reported it in 1838. Now we should say the Struve was the first to measure the parallax angle of a star, but most authorities give that honor to Bessel. Why, well read on. In 1938, Struve was convinced that he had made an error in his first measurement to Vega and reported a parallax angle twice that of his first report. Everyone was more than a little concerned over the validity of Struve’s results and Struve himself indicated that Bessel was first with a believable parallax angle. It turns out that Struve was close to the actual parallax angle to Vega with his first paper and that the error was in his second paper. So it goes. As a side not Struve had 12 children by his first wife. After she died he married Johanna Bartels, the daughter of a prominent mathematician. He had six more children with Johanna. 4 Parallax the Astronomers Way: 3.26 Light Years (ly) Astronomers measure the parallax angle in arc seconds. One of the amazing things about trig is that for very small angles the tangent of an angle in radians is equal to the measure of the angle in radians. So through some algebraic legerdemain the astronomers come up with a simple equation as follows. (the sweat and bother section after this will explain where all of this comes from). 3.086x1013 km An Example: The star alpha Centauri has a parallax angle of 0.742 arc seconds. What is the distance to alpha Centauri in parsecs and in light years. Where d is the distance to the star in a unit called “Parsecs” and p is the parallax angle in arc seconds. Parsecs: A new fundamental unit of distance in Astronomy "A star with a parallax of 1 arcsecond has a distance of 1 Parsec." 1 parsec (pc) is equivalent to: 206,265 AU Some problems: 1. Using Struve’s reported parallax angle of 0.125 arc seconds to Vega, how far is Vega in parsecs and in light years? 2. Using Bessels reported value of 0.314 to 61 Cygni how far is it to 61 Cygni in parsecs? 5 3. The smallest parallax angle that can be accurately measured from the ground is 0.01 arc seconds. If the angle is smaller than 0.01 arcseconds we do not have a protractor good enough to measure the angle. You measure a star to have a parallax angle of 0.01 arc seconds. How far away is that star in par secs and in light years? (Any star that more than this distance away from us we cannot use ground based parallax measurements to measure the distance to that star). 4. The Hipparcos satellite measures precision parallax angles to about 0.001 arc seconds. Hipparcos measures the parallax angle to a star as 0.001 arc seconds, how far away is that star? Hipparcos has given us parallax measurements on more than 100,000 stars. (this is the current limit to our measurements using parallax). 5. NASA had proposed a Space Interferometry Mission (SIM) which would measure parallax angles to .000004 arc seconds. What would be the furthest star that could be measured by this instrument? After spending almost $200 million on the instrumentation for the mission, the project was cancelled in 2010. of a human hair at 600 miles. What will be the furthest star that can be measured with this instrument? Sweat and Bother: Those of you who are thinking are now wondering how we can measure an angle, take its inverse and wind up with a distance and what the heck are parsecs anyway. So we start with the right triangle The tangent of angle p is opposite over adjacent so: 6. In 2013 the European Space Agency plans on launching Gaia. This satellite will be able to measure parallax angles to 0.00001 arc seconds. The plans are for it to measure the parallax angle to 1,000 million stars. It can see stars down to magnitude 15. That is the same as measuring the diameter 6 Now, the small angle approximation tells us that the tangent of an angle is equal to the angle for very small angles and the angle is measured in radians. So: The tangent of p is equal to p for small angles if we measure p in radians, but we want to measure in arc seconds. We have to then convert radians to arc seconds. If I measure p arcseconds, how many radians is that. (7) Now, astronomers, like most scientists are lazy individuals they then simplify. So they define the parsec such that it makes all the numbers go away in the equation above. This leaves the simple equation. Ok some algebra: with p as the parallax angle measured in arcseconds but p(radians) is calculated from the measured p in arc seconds by equation 7 above. and d will be the distance in parsecs. You have to remember that when these things were done in the 1800’s we did not have calculators so doing the proper calculations using tangents etc would be difficult. Using this method makes the calculations easier but provides one messy derivation. Combining everything we get. 7