Astronomy 111 Project Manual Fall 2011 Revolution of the moons of Uranus, Neptune and Jupiter, Kepler’s third law, and Newton’s law of gravity 1 Introduction This is an observing experiment in which we will determine the mass of three of the Solar system’s giant lanets, by measuring the orbital properties of their moons and analyzing their motions. Along the way we will use Jupiter and its moons, specifically, to validate Newton’s law of gravity and Kepler’s third law. This observing project will involve visits to the telescope frequently over the course of 3-4 weeks, including at least one sequence of four consecutive nights and five nights in a week. Everybody in the class will go to the telescope – in groups of three or four at a time – with the object of sharing all of the data. Each student will perform his or her own analysis and write an independent report on the results. During the first week in October, we will derive Kepler’s third law, 2 2 4π 3 P= (aa12+ ) , (1) Gm( 12+ m) from Newton’s laws of motion and law of gravity. Here P is the period of revolution of two bodies about each other, m1 and m2 are their masses, a1 and a2 are the semimajor axes of their elliptical orbits, measured from the center of mass of the two-body system, and mm1 2= aa 21. If one of the masses, say, m1 , is much larger than the other, then the more massive member can be taken to be fixed (because its semimajor axis is small), and the smaller one can be considered to do all the orbiting, with Kepler’s third law simplifying to 4π 2 Pa23= . (2) GM This approximation applies, of course, to the period and orbital semimajor axis of each planet in the solar system, taking M to be the mass of the sun. It also applies to the moons of Uranus, Neptune and Jupiter, in orbit around the much-heavier planet. In one of the first notable uses of a telescope, Galileo showed that Jupiter and its satellites look like a miniature solar system, and thus provided substantial evidence that Copernicus’ heliocentric model of the solar system was viable. In this experiment, we will monitor the motions of the satellites of Uranus, Neptune and Jupiter. From the data gathered, we will be able to measure the orbital period and semimajor axis of each of these satellites. The experiments consist of a series of observations with a CCD camera on the Mees 24-inch telescope; we will take images in which we can identify the moons and determine the distance between each moon and its host planet as a function of time. The moons of the giant planets all travel in orbits that are very nearly circular. The orbits are also very nearly coplanar; the moons appear to be lined up because we are looking almost edge-on to this plane. We can only see the projection of each moon’s orbital radius in the plane perpendicular to the line of sight between the Earth and each planet, as in Figure 1. This apparent distance is Raapparent = sinϕ , where a is the distance from the moon to the center of its host planet, and ϕ is the “azimuthal” angle of the moon’s © 2011, University of Rochester 1 All rights reserved Astronomy 111 Project Manual Fall 2011 radius with respect to the line of sight toward the planet. For each moon, the determination of the maximum value of Rapparent comprises the measurement of its orbital semimajor axis, and determination of the time between successive maxima comprises the measurement of its orbital period. But ϕ increases proportionally with time; therefore plotting Rapparent as a function of time will yield a sine curve, of amplitude a and period P, the period of the moon’s orbit. Once we know a and P in any units, we can validate Kepler’s third law, Equation 2. Note that this works even if there are time gaps in the data – as one usually gets between sunrise and sunset, or due to cloudy nights – as long as one can fit a sine function to the data. 2 Procedure 2.1 At the telescope Figure 1: geometry of observations. (From the CLEA lab You will be instructed personally by the manual) instructors in the startup and shutdown procedures for the telescope and CCD camera, in the course of which you will become familiar with the checklists we maintain for these tasks. You will also learn how to use TheSky, a planetarium program that we use to control the telescope at Mees, and CCDSoft, which controls the camera. Since this will be the first time observing for most of you, though, the initial telescope, camera and computer setup will be done for you. Here we will discuss just the observing procedures you’ll follow after everything is powered up and running. Note, throughout, that by “take an Figure 2: apparent orbital radius vs. time. (From the CLEA lab image” is meant: tell CCDSoft to take the manual). image, save this image as a file with a name that helps you remember what it is (e.g. “Neptune_V_60sec.fits”), and record © 2011, University of Rochester 2 All rights reserved Astronomy 111 Project Manual Fall 2011 in your observing notes a description of the image, and the UT date and time on which it was taken (e.g. “Neptune_V_60sec.fits: Neptune and Triton in the V filter, 60 sec, optimized for Triton, 1 September 2011, 00:45 UT”). 1. Begin by pointing the telescope at Neptune, which is the earliest riser of our three planetary targets, these days. Start by moving the diagonal mirror on the bottom of the telescope to the position that permits the eyepiece to see. Then use TheSky to move the telescope to Neptune. (One of your team should go upstairs at this point, and move the dome so that the telescope can look out the slit.) Bring the planet into the center of the field, using the telescope paddle, and adjust the focus (also on the paddle) to produce a sharp image. You will see Neptune and its brightest moon, Triton. Make sure everybody gets a chance to view Neptune and Triton through the main telescope and eyepiece. Next, return the diagonal mirror to the CCD camera’s view, and restore the Focus setting to the value last used for taking images. Take an image of Neptune with the CCD camera, using CCDSoft. By moving the telescope slightly with the guiding paddle between short exposures with the camera, make careful note of the correspondence among the directions on the CCD display (up, down, left, right) and the directions on the sky and buttons on the paddle (NSEW). Center Neptune in the CCD’s field. In doing so you will find it helpful to note that at Mees the dimensions of the ST-9 CCD’s field of view on the sky are 4.2 arcminutes square. 2. Check the focus of the telescope, using Triton and Neptune. First, set the exposure time such that the moon is easily seen. Then, using the paddle, run the telescope focus out until the images of the moons look significantly blurred. Then run it in in small increments (5-20) of the focus position reading (on the telescope status monitor), recording the diameter of the stellar image and the peak signal at every step; observe the diameter decrease to a minimum and then increase again. If at any point the peak signal is more than about half the CCD’s maximum signal, reduce the integration time, and go back to the beginning of the focus-adjustment sequence. Note the focus readout value corresponding to the minimum diameter images. Then run the focus out again, well past the minimum-image size point, and run it back in slowly until you reach the correct focus readout value. Center Neptune in the CCD’s field again. Save an image of this scene, at the exposure time you used for focusing and at a shorter time to make a nice picture of Neptune itself. The instructors may want to re-initialize the telescope position at this point; tell one of them when you reach it. 3. When this focus adjustment is complete, take note of the peak signals and the diameter of the half-peak-signal level in the images of the moons (given at the bottom of the CCD’s image window as “I”), and write the value of the diameter, in arcseconds, in the control room’s Observing notebook. Note that the plate scale is about 0.5 arcsec per CCD pixel, at Mees. The diameter of a moon’s image should be no larger than a couple of arcseconds. Neptune, on the other hand, is about 2.3 arcsec in diameter this month, so its apparent diameter will bear substantial contributions from both its real size and the blurring by the atmosphere. If the system is well focussed, the apparent angular diameter of the much smaller Triton is dominated by blurring from turbulent variations of the refractive index of the atmosphere, an effect often referred to as seeing. (That is, Triton is smaller than the image-blur from the atmosphere, and Neptune is similar in size to this blur.) On the basis of your experience in the focusing, choose two integration times: one such that the peak signal of Neptune stays well below the maximum range of the CCD, and one that brings out Triton well enough to see its location, without overexposing Neptune to the point that it no longer looks round.