Name: Partner(s): Lab #3 Celestial Navigation Due 7/2 Objectives In this lab you will take measurements of the sun s motion around noon and the north star s position in the sky. You will use this data to find your latitude and longitude. You will get practice in collecting and interpreting data and estimating uncertainty. Materials Part 1: Gnomon (a thin, round stick at least a foot long and capable of being put into the ground or stood up vertically) Compass (to find north) Big, blank piece of paper Protractor Watch Ruler Procedure Find a level, smooth, sunny location where your gnomon can be stuck or placed vertically upright. If the gnomon cannot be easily stuck in the ground (as on asphalt), then a small stand will help. A toilet plunger turns out to be ideal! Secure the piece of paper under the gnomon so you can mark your data directly on it and it won t blow away. A possible setup is: Astronomy 101 3 1 Introduction to Astronomy
Use a compass to find true north (note: compass needles point to magnetic north, which is approximately 18 degrees east of geographic or true north in the Seattle area) and draw a north-south line on the ground through the gnomon on your large piece of paper. If you can t locate a compass, estimate a north-south line (in Seattle, avenues run north-south). Results From about 11:30 am to about 2:30 pm PDT (or 10:30-1:30 PST) accurately mark the position of the shadow s tip and the time of observation - every 15-20 minutes is sufficient except for 12:30-1:30 PDT (11:30-12:30 PST) when every 5 minutes is required. The tip s shadow will be fuzzy: you should try to mark the point halfway between the end of the dark shadow and the end of the fuzziness. Measure the gnomon s length above the ground, L 1. (1 pt) Distance in meters: Astronomy 101 3 2 Introduction to Astronomy
Make a table giving for each observation: (1) time, (2) length of shadow, (3) azimuth of shadow, and (4) altitude angle of Sun. Azimuth is the angle between the sun s position in the sky and the direction of north. This should be measured using a protractor, measuring the angle between each shadow and the north-south line. Due north should be zero degrees azimuth. Altitude of the sun can be calculated using the relation tan(a) = L d where a is the altitude of the sun, L is the length of the gnomon, and d is the length of the gnomon s shadow. 2. (2 pt) Attach your table to the back your lab. Plot the altitude angle versus time and draw a smooth curve through your data points to estimate the time of maximum solar altitude, which is called local noon. Label the time of local noon on the plot. 3. (6 pt) Attach your plot to the back your lab. Plot azimuth versus time and draw a smooth curve through the data points to get a second estimate of the time of local noon. Label the time of local noon (when the shadow crosses the north-south line) on the plot. This value of local noon may not be exactly the same as the one derived from the sun s altitude angle. 4. (6 pt) Attach your plot to the back your lab. Determine the time of local noon: in the previous two steps you found two values for local noon (one with altitude and one with azimuth). If your estimate of the direction of true north is very uncertain you might opt to trust the former; but if you are taking data on a non-level surface you might choose to trust the latter measurement of local noon. If you trust your setup completely you might opt to average the two values. Note clearly in your lab the value of local noon you are using for your calculations. 5. (3 pt) State your determination of local noon and explain how you found it. You will need to make two corrections to your value of local noon. Because the earth does not orbit the sun in a perfect circle, a correction for local noon, called the Equation of Time correction, must be made. The correction for the equation of time is supplied for you. The Equation of Time table shows, for each date, how many minutes after clock noon that true noon would happen in the center of the time zone. Using your Astronomy 101 3 3 Introduction to Astronomy
observation date and the table, subtract off the appropriate number of minutes from your value of local noon. You will also need to correct for Daylight Savings Time (subtract one hour from your value of noon if the measurement was taken during daylight savings). To clarify, the corrections you need to make to go from your value of local noon to true noon are: true noon = local noon - EOT correction - 1hr where true noon is the corrected value of noon that you will use to find your longitude, EOT correction is the correction due to the equation of time, and the 1hr correction at the end is for daylight savings. 6. (3 pt) What is your corrected value for local noon? After these two corrections, any difference between your value for noon and 12:00 is now due to the fact that you are not at the center of the time zone (120 longitude West). You can therefore use this time difference to actually calculate the longitude of your location, remembering that for each four minutes that your local noon is observed to be later than local noon for the time zone, your location must be 1 of longitude farther west. Longitude = 120 + 7. (4 pt) What is your longitude? true noon 12 : 00 4 To calculate your latitude, start with your measured value of the maximum altitude, a max, of the Sun. Because of the earth s tilt, the sun s declination (height relative to the celestial equator) changes throughout the year from 23.5 (northern winter solstice) to +23.5 (summer solstice). A table of the sun s declination gives you daily values for declination in units of degrees. The value you get from this table is used as the declination in the calculation below 8. (4 pt) What is your latitude? Latitude = 90 a max + declination Astronomy 101 3 4 Introduction to Astronomy
Estimate the uncertainty in your measurement of longitude. To get uncertainty in longitude think about (1) what your biggest source of error was in your measurement of noon, (2) by how many minutes it could have affected your measurement, and (3) how that would propagate into an uncertainty in longitude. Quote your final value of longitude as a number plus/minus the uncertainty you calculate. 9. (3 pt) What is your uncertainty in longitude? Estimate the uncertainty in your measurement of latitude. Think about (1) what your biggest source of error was in your measurement of a max, (2) by how many degrees it could have affected your measurement, and (3) how that would propagate into an uncertainty in latitude. Quote your final value of latitude as a number plus/minus the uncertainty you calculate 10. (3 pt) What is your uncertainty in latitude? Look up your actual latitude and longitude on a map. Quantitatively compare your results for latitude and longitude with the accepted (map) values. Use your uncertainty estimates in your comparison. 11. (5 pt) How do your results compare? Astronomy 101 3 5 Introduction to Astronomy
Appendix 1: Equation of Time and Declination of the Sun date time shift dec shift (degrees) 06/25/09 2m20s 23.4 06/26/09 2m33s 23.4 06/27/09 2m45s 23.4 06/28/09 2m57s 23.3 06/29/09 3m09s 23.3 06/30/09 3m21s 23.2 06/31/09 3m33s 23.2 07/01/09 3m33s 23.2 07/02/09 3m45s 23.1 07/03/09 3m57s 23 07/04/09 4m08s 22.9 07/05/09 4m19s 22.9 07/06/09 4m29s 22.8 07/07/09 4m39s 22.7 07/08/09 4m49s 22.6 07/09/09 4m58s 22.4 07/10/09 5m07s 22.3 07/11/09 5m16s 22.2 07/12/09 5m24s 22.1 07/13/09 5m32s 21.9 07/14/09 5m39s 21.8 07/15/09 5m46s 21.6 07/16/09 5m52s 21.5 07/17/09 5m58s 21.3 07/18/09 6m03s 21.1 07/19/09 6m08s 21 07/20/09 6m12s 20.8 07/21/09 6m15s 20.6 07/22/09 6m18s 20.4 07/23/09 6m20s 20.2 07/24/09 6m22s 20 07/25/09 6m24s 19.8 07/26/09 6m25s 19.6 07/27/09 6m25s 19.4 07/28/09 6m24s 19.1 07/29/09 6m23s 18.9 07/30/09 6m21s 18.7 07/31/09 6m19s 18.4 08/01/09 6m16s 18.2 08/02/09 6m13s 17.9 08/03/09 6m09s 17.7 08/04/09 6m04s 17.4 Astronomy 101 3 6 Introduction to Astronomy