Proper Motion of a Star GM

Page 1 of 7 Group Members: Gabrielle Miller Proper Motion of a Star In this exercise you will measure the proper motion of Barnard's Star and use it, with a given parallax and radial velocity, to determine the space motion of the star. Background information In 1718 Edmund Halley had noted that some stars were not fixed, but appeared to move in the sky relative to other stars. Arctures in Bodtes and Sirius in Canis Major were the first stars detected by Halley to have proper motion. Proper motion is the angular change in position of a star across our line of sight, measured in arc seconds per year, and symbolized with the Greek letter “mu” p. Proper motion is generally measured by taking photographs several years apart and measuring the movement of the image of a star with respect to more distant background stars over that time period. Usually decades must elapse between successive photographs before a reliable measurement can be made. The star with the largest proper motion was discovered by E. E. Barnard in 1916 at Yerkes Observatory. This star, now called Barnard’'s Star, is a 9.5 magnitude star located in the constellation Ophiuchus. Its proper motion is so much larger than that of any other star that it is considered to be virtually a “runaway” star. Even so, the angular velocity is small enough that measurements must be made from photographs taken many years apart. The negatives of Barnard's Star provided in this lab were taken in 1924 and 1951 respectively. Since proper motion is an angular velocity, we also need to know the star’'s distance to find its real velocity across our line of sight, called its tangential velocity (v;). Two stars with the same proper motions can have vastly different tangential velocities, as shown in the example in figure below. v, - 988 kin/sec Observer v. =47 4km/sec ~y 3ec of are p gt L] | = 3 5pe | 10 pe This exercise is adapted from one developed by D. Scott Birney at Wellesley College

Page 2 of 7 Since stellar parallaxes are usually tabulated rather than stellar distances, tangential velocity can be calculated from the following equation 4.75 x v, = T,U Equation 1 where v is the tangential velocity in km/sec M is the proper motion in seconds of arc/year p is the parallax in seconds of arc Radial velocity (v;) describes the motion of a star along our line of sight. A negative radial velocity indicates motion toward us, and positive motion away. We still need one more piece of information to know the velocity and direction a star moves in space, called its space motion or space velocity (v). The radial velocity (vr) is added vectorially to the tangential velocity (v (at right angles) to give us the total space velocity (v). The magnitude of special velocity (v) can be calculated using the Pythagorean Theorem. — 2 2 - V =4y, 1Y, Equation 2 NG i v T i Ve Observer i O e ————————— e » r Space motion allows us to study the dynamics and interactions of groups of stars, and to see how the arrangement of our local group of stars is changing over the centuries. This exercise is adapted from one developed by D. Scott Birney at Wellesley College

Page 4 of 7 Procedure Part 1. PROPER MOTION Place a transparency of one of the two images taken of Barnard's Star and the surrounding field on top of the other and align stars 1, 2, 3, and 4. Note that the stars were exposed more than once; therefore use only the right-hand image of each star. While holding the transparency in place, measure the distance between the two positions of Barnard’s Star to a tenth of a millimeter. D= 100 mm D = 10cm = 100mm Convert this distance to the seconds of arc using the following conversion factor: 24.52 seconds of arc = Tmm. Show your calculations. D= 2452.00 seconds of arc 24.52 x 100 = 2452.00 These two photographs were taken exactly 26.96 years apart. Calculate the proper motion of Barnard’s Star in seconds of arc per year. Show your calculations. U= 90.9 seconds of arc/year _ D(in seconds of arc) _ 2452/26.96 = 90.9 seconds of arc/year 26.96 years Part 2. SPACE MOTION The parallax of Barnard's Star has been measured to be p = 0.545 seconds of arc. Using Equation 1, find the tangential velocity of Barnard’s Star in km/sec. Show your calculations. V= 90.64 km/sec. (4.75 x 10.4)/0.545 =90.64 This exercise is adapted from one developed by D. Scott Birney at Wellesley College

Page 5 of 7 Barnard's Star has a radial velocity of V.= -108 km/sec. Calculate the magnitude of the star’s space velocity by using Equation 2. Show your calculations. V= 141 km/sec ((90.64)2+(-108)"2] = 141 km/sec Part 3. CLOSEST APPROACH to the SUN Since Barnard’s Star's negative radial velocity indicates that it is moving generally toward us, we can find the epoch of its closest approach to the Sun and discover just how close that approach will be. On a piece of graph paper from a point representing the Sun, draw a line to represent the direction of Barnard's Star. Calculate the distance to Barnard's Star in parsecs from the parallax-distance relation (distance = 1/parallax). d=1p= 13 pc. Using a convenient scale (such as 10 ecm = 1 pc), mark a point at the proper distance form the Sun to represent Barnard's Star. Along the line and from Barnard’'s Star, draw an arrow representing the radial velocity of the star (a scale of 2 mm =1 km/sec works well). Again from Barnard’s Star draw another arrow (at right angles) representing the tangential velocity, using the same scale as for V. Complete the rectangle to find the direction and magnitude of the space velocity of Barnard's Star. Measure the magnitude of the space velocity of Barnard's Star and converts it to km/sec using the same scale as you used for the radial and tangential velocities (2 mm =1 km/sec). VA2 = (2mm)*2 + (4mm)*2 V= 5.66 km/sec VA2 = 32mm*2 V = 5.66km/sec How does this value of the space velocity compare to the one calculated in part 2 by using formula 2. This exercise is adapted from one developed by D. Scott Birney at Wellesley College

Page 7 of 7 Questions 1. At present the closest star to the Sun is Promixa Alpha Centauri at 1.33 pc. How will Barnard's Star compare to this at its closest approach? The Barnard star is at a distance of 1.83 pc from the sun. This is 5.98 light years from the sun. =183 pc-1.33pc =05pc Barnard star is 0.5 pc far than the Proxima Centauri. 2. How do we know that the four reference stars don’t have proper motions of their own? The stars which are moving towards or away from us may have the proper motions of their own. So, the reference stars are moving towards or away from us. This exercise is adapted from one developed by D. Scott Birney at Wellesley College