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The Orbital Motion of Uranus

Summary: See the Uranus lab for background on the planet. Uranus is in an orbit of about 19.2 AU, and takes 84 years to go around the Sun. As such, you won't see nearly as much orbital motion as you do from other planets. Indeed, much of the motion you will see will be due to the parallax of Uranus as Earth goes around in its orbit.

Goal: Whereas the other "orbital motion" labs require naked eye observations to see how the planets move relative to bright stars, Uranus is both dim enough and moves little enough that use of the telescope is required to see it and to see its motion. Knowing the field of view of your finder scope and your telescope, and observing Uranus over the course of many weeks, you will be able to see the rate at which the planet moves.

Due Date: December 4. In Fall 2003, this is an extra credit lab which is not required. You can receive up to 10 points of extra credit for successfully completing the lab.

When to Start: As soon as possible. You will not be able to successfully complete all observations for lab if you start any later than October 23.

Prerequisites: You must have successfully completed the Uranus lab in order to be able to do this lab. You will also need the results of the Field of View lab.

You will not receive extra credit for this lab unless you have performed all of the assigned "required" labs.

Procedure

1. Getting Started: As always, note the date, time and atmospheric conditions. Is the sky clear, partly cloudy, high cirrus clouds? Is the air moist? Is it windy? How does this night compare with other nights on which you have observed (better? worse? in which way?).

2. Locate Uranus: Follow the procedure of the Uranus Lab to find Uranus in the 25mm eyepiece, and confirm that you've got the right object by looking at it with the 10mm eyepiece.

   You can get a finding chart for Uranus for the Fall of 2003 here. You will want to download and print out this finding chart. You can start your star-hop by locating either Delta Capricorni or Beta Aquarii, both of which are labelled on your SC-001 star chart.

3. Sketch the field of view around Uranus: Sketch the position of Uranus among the stars as seen through both the finder-scope and the 25mm eyepiece (i.e. make two sketches). Use the generic template to make your sketches. Identify Uranus on you sketch and surrounding stars visible in each field of view. Take several minutes to look at the field of view; as you look at it, you may be able to see some dimmer stars that aren't apparent at first glance.

   Indicate the orientation of your sketch (direction of North and East in the field of view). You can determine these directions by moving the telescope around one axis at a time. If you move the telescope in declination toward the South (lower declinations), objects will drift North in the field of view. Similarly, if you move the telescope in Right Ascension (using the slow motion knob) towards the West, objects in the field of view will appear to drift East. Finally, indicate the scale of your sketch. Use the field of view of the instruments (determined in the Field of View lab) as your "measuring stick."

4. Perform steps 1-4 at least four times (ideally more, i.e. one observation a week). Each observation must be separated from the previous observation by at least a week, and the total amount of time from the first to the last observation must be at least five weeks. In the Fall of 2003, ideally you would star this lab the week of October 6 and make your last observation the week of December 1.

   Each time you sketch Uranus in the 25mm field of view, try to position it so that at least a couple of stars which were visible in your previous drawing of Uranus are visible in your current field of view. If this moves Uranus off to the edge of the field of view, you will want to do a second drawing with Uranus near the center of the field of view, so that you'll be able to get a similar overlap next time around.

Calculations and Report

Once you have collected all of the data, process the data according to the procedure below in an attempt to measure the orbital period of Uranus.

1. Measure the angular offset of Uranus from one observation to the next. Do this by measuring how far Uranus moved relative to other stars in the field of view of the 25mm eyepiece. If Uranus moved too far and you can't match stars in subsequent drawings from the 25mm eyepiece, measure the difference in the your finder drawing. Note that your 25mm field of view will not always encompass the same region of the sky from one drawing to the next, as you will have had to move the telescope to see Uranus. However, if you managed to include some stars from one observation to the next as instructed above, you will be able to figure out how far Uranus moved, even if it was greater than the field of view of the 25mm eyepiece. Refer to your results from the Field of View lab to find out the 25mm field of view in arcminutes, and use this to figure out the angular. offset of Uranus in arcminutes.

   Make a table like the following:

   Date	Days From First Observation (DFO)	Uranus Offset From Previous (arcmin)	Uranus Offset From First (arcmin) (ΔUEL)
   2003/10/06	0	...	0
   2003/10/13	7	-10	-10

   with one line for each observation. (Obviously, substitute your own data for the example numbers above!) The "Days from First Observation" is the number of days that have elapsed since your first observation (surprise!). The "Uranus Offset From Previous" measurements are what you measured above: how far Uranus moved between the previous observation and the current observation. The "Uranus Offset from First" column should include the net motion of Uranus, i.e. how far it has moved since your first observation. This is, of course, the sum of all the "offset since previous" measurements. If Uranus moves to the east, record the offset in arcminutes from the previous observation. If it moves to the west, record a negative offset from the previous observation. Indicate all measurements with the number of significant figures to which you believe they are accurate. (I.e., do you think you are good to a few arcminutes? If so, indicate your answer to a few arcminutes. Do you think you are good to a few tenths of an arcminute? Then add a decimal place to the second column.)

   Look up and calculate the following numbers for the date of your first observation:

   • Uranus' Ecliptic Longitude (UEL). You will want to refer to the Uranus finding chart with ecliptic longitudes labelled.

   • The Sun's Ecliptic Longitude (SEL). Refer to the dated chart of the ecliptic on your SC-001 star chart.

   • Earth's Ecliptic Longitude (EEL). This is 180^o plus the Sun's Ecliptic Longitude. If you get a number greater than 360^o, subtract 360^o.

2. Now you will calculate how Uranus would appear to move if it were physically fixed in space relative to the Sun, and the apparent motion we see were only the parallax resulting from Earth's motion around the Sun. We will do this by calculating how much UEL (Uranus' Ecliptic Longitude) would change only as a result of Earth's orbit around the Sun. We will use the knowledge that Uranus is 19.2 AU from the Sun. (You could in principle figure out Uranus' distance as part of this lab if you were to observe Uranus longer, disentangling the effects of parallax and Uranus' orbit, but for simplicity we will use the distance to Uranus as a "revealed result" here.)

   Once we have how far Uranus would have appeared to move due to parallax, we will subtract that from the actual observed motion of Uranus; whatever is leftover should be due to the actual orbit of Uranus around the Sun.

   For each observation, you will need to calculate a value "ΔEEL", the change in Earth's Ecliptic Longitude as a result of its orbit around the Sun. The Earth moves through 360^o in 365.24 days, which means that it moves 0.986^o in one day. You can calculate ΔEEL by:

   ΔEEL = 0.986*DFO

   where DFO is from your first table above. With this number and the other numbers you have calculated, you can figure out how much you would expect Uranus to move only due to parallax. Actually figuring this out is a rather ugly couple of pages of geometry, trigonometry, and algebra. A decent approximation (good to a little better than 1 arcminute so long as you complete all the observations between 2003 October 6 and 2003 December 8) to the still-nasty result is:

   ΔUEL_expected = -60*arcsin[ (2/19.2) * sin(ΔEEL/2) * cos(EEL-UEL+ΔEEL/2) ]

   Where EEL and UEL are the numbers you calculated in the previous step for the first date of your observation. This gives ΔUEL_expected in arcminutes. Use this formula to make another table for every observation after your first one:

   Days From First Observation (DFO)	ΔUEL measured	Earth's Ecliptic Motion (ΔEEL)	ΔUEL expected	Uranus' Orbital Motion (arcmin)	Uranus' Orbital Angular Speed (arcmin/day)
   7	-10	6.90	-15	5	-0.7

   DFO comes from the first column and ΔUEL_measured comes from the last column of your table in step 1 above. Uranus' Orbital Motion is ΔUEL_measured-ΔUEL_expected, and Uranus' Orbital Angular Speed is Uranus' Orbital Motion divided by DFO.

3. How long does Uranus take to go around in its orbit? Average together all of your individual measurements of Uranus' Orbital Angular speed. Use this to figure out what the period of Uranus' orbit is. How close are you to the value you would expect knowing that Uranus is 19.2 AU away from the Sun?