Ph 344 -- Lab 3 Analysis
Jupiter's Moons and Kepler's Laws

Due: Thursday, March 27, 2003

Sinusoid Fitting Procedure: The program to fit a sinusoid to the positions of the moons as a function of time is called jupiter.exe and is located in \\Hooker\ph344data\lab3\, the same location as the results file. Double-clicking on this file in windows explorer will run it and bring up a window with a text and (shortly) a graphics subwindow. Copy the program and the two data files into a folder in your ``My Documents'' and run it there.

In the text window answer the ``Graphics device/type'' question with ``/gw" (just /gw, no quotes). This creates the graphics subwindow. Select (click on) the original window and type in ``results.txt" as the name of the position data file. This is the final set of data with incorrect dates, swapped moons, angles in the incorrect quadrant, and other more mysterious problems corrected. The name of the velocity data file is ``velocities.txt". Then type in a number to choose a moon (1=Io, ..., 4=Callisto) and go to work.

The program will show you the distance from Jupiter vs. time for the selected moon and ask for guesses of the period and time of maximum (positive) elongation. It will then plot this guess and allow you to revise the values. When you have a reasonable set of initial guesses, answer ``n'' to the ``another guess'' question and the program will fit the data. When the fit is done, it types out the best-fitting parameters (maximum elongation, period, and time of maximum elongation) and the data points and their residuals. Some of this text may have scrolled off the top of the window, but the next question is for the name of a file in which to save this same information.

You can print a plot by selecting the graphics window and then selecting ``print...'' from the ``File" menu in the upper left-hand corner of the main window. The plots can supposedly also be cut out and incorporated into Word or other documents, but don't ask me how. If you have selected the text window rather than the graphics window when you do a print, it will print out that window, using a lot of black ink in the process. If this happens, please hit the cancel button (X) on the printer to minimize wasted ink. The contents of the text window are written to a file, as discussed above, and you can print this, if you wish.

The program claims very small uncertainties for the fitted period and time of zero phase (t0). These should be taken with a large grain of salt because of the large covariance between these two parameters. This means that changing these two parameters together in a specific way results in nearly equally good fits to the data.

The final plot produced by the program shows a curve which is the expected radial velocity for the moon as a function of phase, based on the fitted size and period of the orbit. Part 9 describes how this curve is derived in more detail. The observed radial velocities are also plotted so that you can compare the expected velocities and the observations.


7. Fit each of the four moons. Print and hand in the three plots for each moon. Also hand in your fitted parameters for the sinusoids.


8. Kepler's Laws

  1. Discuss how the fitted sinusoid curves show that the Jovian moons obey Kepler's first and second laws.

  2. If the real orbits of the Jovian moons were different from circles, the residuals of the measured radii from the fitted sinusoid would vary systematically with the orbital phase. Examine the residuals given in the tables produced by your fit. Do you see any evidence for a pattern in the residuals with phase?

  3. Calculate the ratio of the period squared to the semi-major axis cubed for each moon using your fitted values and, thus, verify Kepler's third law. How accurately is the third law satisfied?


9. The orbital velocities of the moons and the distance to Jupiter:
In class I showed that the observed radial velocity of a Jovian moon with respect to Jupiter, when observed from Earth at time $t$, is

\begin{displaymath} v_r = -2 v_{orb} \cos(\beta/2) \cos\left(\frac{2\pi}{P}(t-t_0) - \beta/2\right). \end{displaymath}

Here $v_{orb}$ is the circular orbit velocity of the moon, $P$ is its period, $t_0$ is the time of zero phase (the time of maximum easterly elongation of the moon from Jupiter), $\beta$ is the angle between the Sun and Earth as seen from Jupiter, and $c$ is the speed of light. The leading factor of two is there because the moon acts as a moving mirror. Here we will adopt the known distance to Jupiter to calculate the expected orbital velocities using our measured orbital sizes and periods. Then we will compare these to the measured radial velocities.

  1. The greatest separation observed between Jupiter and a moon is the angular size of the moon's circular orbit. This greatest distance is just the amplitude of the sinusoid fit by the program jupiter.exe. The image scale for a zoom setting of 210 is $2.1588 \pm 0.0017$ arcseconds/pixel. Multiply your fitted angular sizes of the orbits (which are in pixels) by the scale to get the angular sizes in arcseconds.

  2. To calculate the orbital velocity, we must convert the observed angular sizes of the orbits to sizes in kilometers. This conversion depends on the distance. The program actually takes into account the changing distance between the Earth and Jupiter (it increased about 5% over the one month of our observations) and converts all separations into the values that would have been observed if Jupiter had been fixed at the distance it had on 0 hours Universal time on February 1: 4.327 astronomical units (an astronomical unit is $1.4960\times 10^8$ km). The angular sizes of the orbits are all small, so just convert them to radians and multiply by this distance to get the orbital sizes in km's.

  3. Use the just-derived orbital sizes with your fitted periods to calculate the orbital velocities for each of the four moons.

  4. The program jupiter.exe duplicates the calculation you just did and plots the expected $v_r$ for each moon as a curve vs orbital phase. The value of $\beta$ changed from about 4 degrees to 6 degrees during our observations and the program assumes a value of 5 degrees. The program also plots the radial velocities measured by the class. How well do the observed radial velocities agree with the expected values? Do the points agree with the curve to within their uncertainties?


10. The mass and radius of Jupiter:
Newton's form of Kepler's third law is

\begin{displaymath} P^2 = \frac{4 \pi^2 a^3}{G(M_j + M_m)}. \end{displaymath}

Here $M_j$ is the mass of Jupiter, $M_m$ is the mass of the moon, $a$ is the semi-major axis of the orbit of the moon (just the radius for a circular orbit), and $G$ is the gravitational constant. This formula, or variations on it, is used to derive almost every mass measured for celestial objects.
  1. Use Newton's form of Kepler's third law to calculate the mass of Jupiter using the data for each of the four moons. Use the orbital sizes that your derived in part 7b. Make sure that you plug numbers into the formula only after you have converted them to a consistent set of units. How well do your four values for the mass of Jupiter agree? Find the average of the four values and estimate the uncertainty of the average. How well does your value agree with the ``true'' mass of $1.899\times 10^{30}$ g?

  2. Pick one of your unsaturated observations of Jupiter (probably the shortest one) and examine the image of Jupiter using ATV. Produce a radial profile of Jupiter with the j key and use the plot to estimate the radius of Jupiter in pixels. Note that the full-width-at-half maximum (FWHM) of the image of Jupiter is a bad measure of Jupiter's radius. Why is this? Use the image scale and distance to Jupiter to calculate its radius in km.

  3. Use your observed radius of Jupiter and your measured mass to calculate its density (in g/cm$^3$). Comment on what this implies for the composition of a giant planet like Jupiter.