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 Name___________________________________________ Elementary Astronomy
Activity 4
Stellar Distances
Background There are a number of ways to measure distances to stars and galaxies. One of the most important methods for measuring stellar distances is parallax. Parallax is the apparent motion of stars as the Earth moves through its orbit around the Sun. If the parallax angle (p) is given in arcseconds then the distance (d) is given in a unit called a parsec. Then the relation between parallax angle and distance is: (1) !
𝑑 = !
One parsec is equal to 3.26 light years. Stellar parallax can only be used for stars within a few hundred light years because at distances larger than that the parallax angle is too small to be measured with current instruments. Another important distance measure is the Cepheid variable method. A Cepheid variable is a post-­‐
main sequence star that oscillates in size and thus in magnitude. The Cepheid variable method utilizes the distance modulus equation: (2) 𝑑 = 10(!!!!!)/! In this equation, m is the apparent magnitude and M is the absolute magnitude. Apparent magnitude can be easily measured from a CCD image, but absolute magnitude takes some work. Cepheid variables are useful in this way as it was found in the late 19th century that there is a direct relation between their period and their luminosity. Period is directly observable, and is simply the time it takes the star to reach the same magnitude again. The period-­‐luminosity relation can then be used to predict the luminosity or absolute magnitude. Finally Eq. 2 is used to find distance. The Big Dipper is a famous asterism that is composed of seven stars. It represents the back section of the constellation Ursa Major. An image is provided below. Name___________________________________________ Image from Stellarium Experiment 1. First an introduction to angular measure. Let’s say you are 1.7 meters tall (that’s 5 ft 7 in). If you are 10 meters away, what angle will you subtend? Consider the image below: Angle_______________________________° 2. Now what if you are 100 meters away, what angle will you subtend then? Name___________________________________________ Angle________________________________° 3. Convert the result of 2 into arcseconds, remembering there are 3600 arcseconds in 1 degree. Angle________________________________” 4. Now to use angular measure for stars: Below are given the observed parallax angles for seven stars in the Big Dipper: Star Parallax Angle Distance (arcsec) (parsecs) Dubhe 0.0026 Merak 0.0041 Megrez 0.0040 Phad 0.0039 Alioth 0.0040 Mizar 0.0042 Alkaid 0.0032 Complete the table with the distances in parsecs. Show work for one star. 5. Which stars in the Big Dipper are actually at about the same distance away from Earth? 6. Consider the plot below. This is the period-­‐luminosity relation for Cepheid variables. (Note the x-­‐axis is the log of the period.) The nearest Cepheid variable is Polaris, and it has a period of about 4 days. What is the distance to Polaris using the figure below and Eq. 2? The visual apparent magnitude of Polaris is 1.95. Show your work below. Name___________________________________________ (From http://www.astro.sunysb.edu/metchev/PHY515/cepheidpl.html) 7. The parallax angle of Polaris is 0.00754 arcsec. Find the distance to Polaris using parallax. 8. Find a percent difference for the values in 6 and 7. Percent difference is: (3) 𝑃𝑒𝑟𝑐 𝐷𝑖𝑓𝑓 =
𝑣1−𝑣2
𝑥100 𝑣1+𝑣2 /2
Name___________________________________________ Questions 1. The distance to Polaris is given as 433±6 ly. Which method got closer to this value? Why do you think this is? 2. Aristotle argued that the Earth must be stationary or else we would observe stellar parallax. Why did he end up making this wrong claim?