Week 12 - Parallax and Triangulation

Position 1 Position 2 Distant Objects Nearby Object Distance between observations Distance to Object Parallax Name : Key Parallax and Triangulation Triangulation Triangulation is a way to measure distance. In triangulation, the location of a nearby object is measured compared to distant background objects. The observer then moves and measures the location of the nearby object again. The nearby object will have appeared to move compared to the background objects! By measuring the angle that the nearby object moved (we call this angle the parallax), the distance to the nearby object can be calculated. Distance to Object = Distance Between Observations x 57.3 v Parallax (in degrees) 1) You make two observations of a nearby object. The observations are 20 meters apart and you notice a parallax of 2 o . What is the distance to the nearby object? Distance to Object = 20 meters x 57.3 = 573 meters 2 2) You notice that a nearby tree lines up exactly with a distant building. When you move 20 meters to your right you notice that the tree is now 5 o away from the distant building. Calculate the distance to the tree. Distance to Object = 20 meters x 57.3 = 229.2 meters 5 o 3) As the parallax of an object increases, does the distance to the object increase or decrease? 2 o parallax = 573 meter distance; 5 o parallax = 229 meter distance. Both had the same distance between observations (20 meter). As parallax increases, the distance decreases.

4) It is impossible to measure parallax angles if they are too small. From Earth looking into space the smallest angle we can measure is about 5.6 x 10 -6 degrees. What is the main reason why we can’t measure smaller angles? The turbulence in Earth’s atmosphere means that we can’t measure angles in space that are too small – our image of the star is too blurred by the turbulence to measure its position that accurately. 5) What’s the farthest distance you can measure for an object in space if you move 1 kilometer between the two measurements? Hint: Assume that the parallax angle is 5.6 x 10 -6 since this is the smallest angle that can be measured. Show your work. Distance = 1 kilometer x 57.3 = 10,232,142 kilometers 5.6 x 10 -6 6) The Moon is about 385,000 km away. Based on your answer to Question 5, could you measure the distance to the Moon with two measurements that are 1 kilometer apart? Explain. Yes – from Question 5, the farthest distance we can measure with two measurements that are 1 kilometer apart is 10,232,142 kilometers. The Moon is only 385,000 kilometers away, so we can measure its distance with two measurements that are 1 kilometer apart. 7) What is the only way to measure larger distances if we can’t measure parallaxes smaller than 5.6 x 10 -6 degrees? Hint: Look at the equation for parallax on the previous page and see what variables you could change. If we cannot change the smallest parallax that can be measured, the only way to measure larger distances is to increase the distance between the two measurements (i.e. have a longer baseline). 8) The maximum distance between two measurements on Earth is 12,756km. What is the largest distance you can measure with triangulation from Earth? Hint: Assume that the parallax angle is 5.6 x 10 -6 since this is the smallest angle that can be measured. Show your work. Distance = 12,756 km x 57.3 = 1.3 x 10 11 km 5.6 x 10 -6 9) How many AU can we measure distances out to using triangulation from Earth? How many light years? 1 AU = 1.5 x 10 8 km; 1 ly = 9.5 x 10 12 km. 1.3 x 10 11 km x 1 AU = 866.7 AU; 1.3 x 10 11 km x 1 ly = 0.014 ly 1.5 x 10 8 km 9.5 x 10 12 km 10) Are each of these statements true or false? Explain your reasoning. a) We can measure the distance to almost everything in our solar system using triangulation from Earth. True – 866.7 AU takes us out past the orbit of the Kuiper Belt. Everything we have seen in our solar system is < 866.7 AU away. b) We can measure the distance to Proxima Centauri and other close stars using triangulation from Earth. Proxima Centauri is 4.2 ly from Earth. False – we can only measure out to 0.014 ly with triangulation from Earth. Proxima Centauri is 4.2 ly or ~300 times farther away, so we cannot measure the distance to Proxima Centauri.

January 1st July 1st Star A Star B Smaller because all other stars are farther away than Proxima Centauri, and the farther away you are the smaller the parallax. 9) The smallest stellar parallaxes that can be measured from Earth are around 0.02”. Find the largest distances that can be measured from Earth. Distance = 6.5 / 0.02 = 325 ly 10) In the pictures below, you see two pictures of the same part of the sky taken 6 months apart. Star A and B are both nearby and have moved compared to the background stars because of stellar parallax. Which of the two stars has the largest parallax? Which star is closest? Largest Parallax: A Closest: A Measuring Distances Using Triangulation We will now use triangulation to measure the distance to three different objects outside. Record all your data in the Table on the next page. 1. Record the names of the three objects to be observed in the table. 2. Answer these questions: Which object appears to be the farthest away? Which object should have the larger parallax shift? The closest object should have the largest parallax shift. 3. Make a guess at the distance to each object and record your guesses in the table. For each object do the following: a) Stand so that the object lines up with a distant landmark. Record a description of the landmark in the table. b) Move 4 meters to your left or right. Be sure to use the ruler to help you measure. c) The object will have moved compared to your distant landmark. d) Using your hand, thumb, and pinky estimate the angular distance between the object and the distant landmark. This is the parallax! Record this in the table. 4. Answer the following questions: Which object had the largest parallax? Is this what you expected from step 2? The closest object should have the largest parallax. 5. Calculate the distance to each object in cm using the parallax equation: Distance to Object = Distance Between Observations x 57.3 v Parallax (in degrees) Record your answers in the table.

6. Answer the following question: Looking at your distances, do the distances match what you would expect based on the parallaxes? In other words, do the shorter distances have larger parallaxes, etc.? 7. Answer the following question: Describe how accurate your calculated distances were to the true distances. What are some reasons why the distances you calculated may not agree with the true distance? Probably the biggest reason that your distances might be off is that the measurement of the parallax angle was off since you were just estimating it with your hands. Also, if your distant landmark is too close to your object, then this will mess up the distance measurement. Object 1 Object 2 Object 3 Name of Object Guess at Distance to Object Distant Landmark Used Parallax for the Object Calculated Distance to the Object Actual Distance to the Object