Lab Title: Parallax and Astronomical Distances Equipment: Sextant Meter sticks (or tape measures) Calipers Magnetic compasses Introduction: Since we cannot travel to most celestial objects in order to learn about their characteristics, we must gather information about them remotely from Earth by relying on the information contained in their transmitted and reflected radiation. As we shall see throughout the course, understanding the characteristics of these objects is a slow but steady process with each new measurement relying on observations and measurements that have come before. The first of these observations is simply the position of celestial objects and the corresponding local time. The second is a measurement of the distance to nearby objects. The problems of determining the distances to far away objects will run throughout our studies in astronomy. This is of great importance because we can always measure the amount of energy received at the Earth from a star, but to figure out how much energy is actually emitted by a star, we must somehow determine its distance too! The Parallax Method One reason ancient philosophers discounted the notion that the Earth moved about the Sun was that the stars showed no shift in their angular position as the Earth "allegedly" moved in orbit about the Sun. In reality, their angular positions did (and still do) shift, but by amounts too small to be detected by the naked eye. Plato, Socrates, Aristotle, Ptolemy, and their contemporaries would have been startled to learn of the actual fantastic distances to even the nearest stars! Using photography and telescopes, however, the distance to nearby stars can be determined, using the Earth's orbit as a baseline: -1-
Earth's orbit about the Sun June 1 A.U. 1 A.U. Distance to nearby star in pc L θ To distant "fixed" star Nearby star Dec To distant "fixed" star In the diagram above, the angle θ is called the parallax angle. The angle 2θ is called the parallax displacement, or parallax shift. In actual practice, this measurement requires astronomers to take two photographs of the same region of the sky six months apart. During that time interval, nearby stars will shift their position relative to the more distant stars (see the star chart and transparency overlay in Part II). In this lab, however, we cannot wait six months, so we will measure much smaller distances and use a correspondingly smaller baseline for our measurements. In Part II and Part III, we will make use of astronomical data to determine parallax angles and stellar distances. All methods of determining parallax rely on the very simple geometrical relationship between an arc length (d), a subtended angle (θ), and a radial distance (L): L = d/θ (1) θ 2θ θ L L d L L θ d If d is much smaller than L (such as in the triangle on the right) then the arc length is approximately a straight line. Equation (1) can then be used to determine L if both d and θ are known. Throughout the course, we will be using angular (arc) measurements. In the conventional system, there are 360 in a full circle. Each circular degree ( ) contains 60 arc minutes (') and each arc minute contains 60 arc seconds (''). With these definitions we see that: -2-
1 = 60' = 3600'' 360 = 21,600' = 1,296,000'' It will also be necessary to use a unit of angular measure known as radians. Radians are simply another unit of angular measure defined by the ratio of the arc length of a portion of a circle to its radius. For example, the arc length (circumference) of a full circle of radius R is 2πR. The ratio of the circumference to the radius for a complete circle is 2πR/R = 2π. So, there are 360 degrees in 2π radians or 57.3 degrees in one radian. For example, to compute the number of radians equal to 2 degrees, 20 minutes of arc, we do the following: 2o 20' = 2 + (20/60) = 2 + 0.33 = 2.33 degrees 2.33 (1 radian/57.3 ) = 0.041 radians -3-
Part I. Distance Determination Using the Sextant Procedure: 1. Review the instructions on how to use the sextant and first practice using it indoors before heading outside. Make sure you understand how to read the angle in degrees ( ) and minutes (') off the scale. 2. In the classroom, stand on one side of the room, and sight on an object on the opposite side. This object (draw one with chalk if necessary) will be your "nearby star". 3. Using the meter sticks or tape measures, mark off a total baseline of 2 meters (1 meter on either side of the line to the object). Make sure that the baseline is perpendicular to the line-of-sight to the object. Tables and countertops can serve as references. 4. Stand on one end of the baseline (i.e., the left side), and sight straight ahead (perpendicular to the baseline). The nearby star should not be in view. The angular measure on the sextant should be set at zero. Now, rotate the sight until the nearby star comes into view. The angle that you must rotate it will be your "angle left". Record this angle in degrees and minutes of arc in Data Table 1. 5. Move to the other end of the baseline and repeat the procedure in Step 4. This angle will be your "angle right" in Data Table 1. 6. Repeat Step 2 through Step 5 for two other objects of different distances. One should be closer and one considerably farther (i.e., 20-50 meters). For the more distant object, you will have to go outside. Enter all of your results in Data Table 1. 7. Using the meter sticks or tape measures, measure the actual distances to each of your three objects. 8. Complete Data Table 1 and compare your distances determined using the parallax method to the distances you actually measured using the tape measures. Make use of the conversion between degrees and radians (1 radian = 57.3 degrees), and equation (1) to determine the experimental distance (L). -4-
Date: Instructor's Signature: Lab Partners: Data Table 1. Distances Object #1 (Near) #2 (Medium) #3 (Far) Baseline Length (m) Angle Right ( and ') Angle Left ( and ') Avg. Angle (θ R + θ L )/2 ( ) Parallax Angle (radians) Exptl. Distance (m) Actual Meas. Distance (m) Percent Difference* * Percent Difference = Experimental distance Measured distance Measured distance 100% -5-
Part II. Distance to Nearby Stars Procedure: Included in this part of the lab are two diagrams (made six months apart) that show five nearby stars against a field of very distant stars. A scale is included on each diagram in the form of a small line. The length of this line corresponds to a given number of arc seconds on the scale of this photograph. 1. Measure this line in millimeters and divide by the given number of arc seconds to find the scale of the photograph. Your result will be the number of arc seconds per millimeter on the photograph. " = mm So the scale of the "photograph" is: 1 mm = arc seconds 2. Use a clear plastic sheet to make a copy of the second diagram and then overlay the two diagrams. Determine the linear displacement (in mm) for each of the numbered stars and record them in Data Table 2. 3. Using your scale determined in Step 1, convert your linear displacements into arc seconds and enter these into Data Table 2 as well. 4. From the Pre-lab Exercises and your text, recall that equation (1) can also be written as L = 1/θ (2) where θ is parallax angle (half of the parallax shift) in arc seconds, and L is the distance in parsecs. Use this result to complete Data Table 2. -6-
Date: Instructor's Signature: Lab Partners: Data Table 2. Stellar Parallaxes and Distances Star 1 Parallax Shift (mm) Parallax Shift (") Parallax Angle (") Distance (parsecs) Distance (light-years) 2 3 4 5-7-
Part III. Distance to the Moon Included in the lab write-up are Voyager II sky charts of the region near the Moon. These "photographs" are taken at the same time, but from two different locations on the surface of the Earth. By overlaying these two photographs you can determine the distance to the Moon. First, you will need to determine the baseline distance, which is the straight-line distance between the two locations. To do this, note that the difference in longitude between the two cities tells you what fraction of a circle it is between them. For example, if the difference in longitude is 100 degrees between two cities near the equator, then the two locations are separated by 100/360th of the Earth's circumference as measured along the curved surface of the Earth. Rermember to correct for the fact that the baseline must be the straight-line distance between the two observation points. Second, you need to overlay the two photos on top of each other and then measure the parallax displacement between the two different apparent positions of the Moon relative to the distant stars. To do this, you need to match up the stars (not the photo frames). To determine the scale of the photographs, note that the angular diameter of the Moon is 29.5 arc minutes. By using a ruler, or calipers, to measure the diameter of the Moon on the photographs, you can now convert from a linear distance scale to an angle. Remember that the parallax angle is one-half of the measured parallax shift. Finally, you can calculate the distance to the Moon from equation (1). Don't forget to convert your angular measure to radians before using equation (1)! Your classmates, and your instructor will be able to help if you are having problems at this point. Calculated distance to the Moon using equation (1) = km Percent difference from the accepted value of 400,473 km = % Show all of your work in this area: -8-
Conclusions and Comments -9-
Name: Lab Partner: Pre-lab Exercises: Parallax and Astronomical Distances 1. Hold one finger up at arm's length. View it with one eye open and the other closed. Now open the closed eye, and close your open eye. Repeat this a view times. Your finger will appear to shift back and forth. Estimate what angle (in degrees) it shifts. Recall that your fist subtends an angle of about five degrees when held at arms length. Bring your arm in to about half the distance and repeat the exercise. Now how much is the parallax shift? Repeat for objects across the room, or outside. What, do you estimate, is the smallest angular shift you can detect? 2. Refer to the sketch shown below. If the parallax angle (θ) is fairly small, astronomers, surveyors, and you can all use a simple formula to relate θ to the distance (L) to the object, and to half the baseline (d): L = d/θ (1) In Exercise 1, for example, the baseline is the distance between your two eyes. The distance d is simply half that distance. θ is half of the shift angle you estimated, and L is the distance from the center of your two eyes to your finger. Note that if d is measured in centimeters, then L must also be measured in the same units of centimeters. The angle θ must be in radians. Baseline = 2d d d L θ θ Distant Object (a) If the angle θ is measured to be 5.73 degrees, what is θ in radians? -10-
(b) If the distance (2d) between your eyes is 10 cm, how far away is the distant object (i.e., your finger)? 3. Refer to the sketch at the beginning of the lab write-up, showing the Earth's orbit about the Sun and the parallax shift of a nearby star observed from the Earth at two different times of the year separated by six months. In this case, the distance d is one Astronomical Unit. Also, the parallax angles are very tiny. Rather than convert Astronomical Units to kilometers or meters, and angles from arc seconds to radians, astronomers modify equation (1) by simply defining a new unit of distance called a parsec. Equation (1) then becomes: L = 1/θ (2) where L is the distance from the Earth to the star now measured in parsecs, and θ is the parallax angle measured in arc seconds. Using the diameter of the Earth's orbit around the Sun as a baseline, one parsec turns out to be 3.26 light-years. (a) If the parallax angle is 0.1 arc seconds, how far away (in parsecs) is the star? (b) If the parallax angle is 0.01 arc seconds, how far away (in light-years) is the star?) 4. Tycho Brahe used a parallax method similar to this to accurately measure the distance to the Moon as early as 1590. His sextants had an accuracy of 1 arc minute (this means that the smallest parallax shift that he could measure was about 1/60th of a degree). Estimate the minimum length baseline (2d) on the Earth that Tycho used to obtain an accuracy of 10% in his measurement of the distance to the Moon. You will probably want to -11-
make use of equation (1) and the fact that the distance to the Moon is about 400,000 km. SHOW ALL OF YOUR WORK BELOW. -12-