Determining the Mass of Jupiter Based on the Orbital Characteristics of the Galilean Moons

Nathan Kelley

June 4, 2004



Abstract

The purpose of this project was to determine the mass of Jupiter based on the orbital characteristics of the four Galilean moons: Io, Europa, Ganymede, and Callisto. The planet and its moons were observed in Hα, with a visual limit of magnitude 6. From the images, the distance between each moon and Jupiter can be found and fitted to the curve describing the orbital path. From that curve the orbital period and radius can be derived, and using these orbital characteristics the mass of Jupiter is calculated to be 1.56 x 1027 +/- 2.74 x 1026 kg.

1. Introduction

While Jupiter has been known about for thousands of years, its four largest moons are a comparatively recent discovery. Discovered independently 1610 by both Galileo Galilei and Simon Marius, these large moon are the key to determining the mass of Jupiter. The mass of Jupiter is currently a critically important value for exploring our solar system and beyond. The primary method for determining the mass of Jupiter is the method of mass determination used in this project. Jupiter's mass has been critical in the deployment of both the Pioneer and Voyager satellites. Both sets used Jupiter as a gravity slingshot to save fuel and direct them towards the outer planets.

   Jupiter is the largest planet in our solar system and the majestic giant standing at the beginning of the outer solar system. Determining the mass of a body like Jupiter is a simple physics project with the right parameters, but from scratch the process is full of minute problems that have to be accounted for.

   To determine the mass of a stellar body like Jupiter, the process begins with simple gravitational physics. Jupiter pulls on the satellite with a gravitational force that must be equal to the centripetal force of the satellite moving around the planet, as shown in equation 1 (Tipler 327).

From my observations, one cannot directly measure the velocity of the satellite, but we can measure the orbital period and radius. The satellite's velocity can be written in terms of period and radius as Equation 2.
Substituting Equation 2 for the velocity, we can solve for the Mass of Jupiter. Thus, one arrives at an expression for the mass of Jupiter in terms of the orbital period and radius of its satellites (Equation 3).
To derive the orbital radius and period, the motion of the moons around Jupiter must be modeled. In a simple approach, the moons move about Jupiter in a circle. At any given time, while the moons radius remains constant, the moons x and y coordinate change with respect to the cos and sin of the angular displacement, theta (Equation 5). The Z position of the moons is defined to be 0, as they are rotating in a flat plane. The angular displacement changes with respect to time as the displacement per unit of time multiplied by the change in time since the defined start time, t = 0 (Equation 4). Obviously, equation 4 can be substituted into equation 5 for theta, resulting in equation 6. (Statler, private session).


Figure 1. This is a diagram of how the x and y positions of the moon as seen looking down the z axis can be found as a function of the angular displacement.


Figure 2. The moons do not rotate the same plane that we see, thus the coordinate system has to be rotated to match it.

However, from Jupiter’s moons do not rotate exactly on the same plane as Jupiter rotates. The moons have a declination angle that the entire system must be adjusted for. By rotating the system about the x axis, we can adjust the model to reflect the angle of declination. Since we rotated the system about the x axis, the x' coordinate of the system is the same as the old x. The y' coordinate is lost due to the lack of depth perception in the telescope, but the z' coordinate can be derived from y. The z' at a given time is equal to y times the sin of the inclination angle, i (Equation 7). From here, the distance between the satellite and Jupiter is the result of the Pythagorean theorem, with the observed radius, R', being the distance (Equation 8). Now, one takes equations 6 and 7 and substitute them into equation 8 where appropriate, and the result is the equation for the radii of the satellites (equation 9). (Statler, private session).

The equation will accurately model the results, but the picture is measured in arc seconds, while the moons orbit Jupiter at kilometers. To convert from arc seconds to kilometers. First, the distance to Jupiter must be known. For the nights in question, the distance was 4.8163 Au (Chereau). From there, 1 Au is equal to 1.496 x 10 8 km. Multiplied through, the distance to Jupiter is thus 7.377 x 10 8 km. As seen in figure 3, this becomes a problem of simple trigonometry. Solving for R' with d and phi known, produces equation 10.

Figure 3. Simple Trigonometry can be used to find the distance between each moon and Jupiter if the displacement angle between them is known.


IRAF imagining software was used to calculate the locations of the moons and Jupiter. IRAF's centroid function found the centers of each object if given an approximate value for the center. After the program calculated the position and error of each object on the frame, each moon's radial distance, R', as found with respect to Jupiter using the Pythagorean theorem. The errors in position of the moons with respect to the errors in position of Jupiter for each night was then propagated through the Pythagorean theorem. The result of each epoch was then averaged to create the points used for the fitting, and the average of each epochs error was used to calculate the error.
Click here for a table of the Data and the resultant averages

2. Observations

   Jupiter and the Galilean moons were observed using the 0.25 meter Great Ohio Telescope on both April 28 and May 3 at approximately 2 and 6 UT each night. At each time, 3 x 1 sec photographs were taken in the Hα. Neither evening was photometric, with low cirrus clouds moving across the sky throughout each night. Also, both nights at 6 UT, Jupiter rested just above several trees, which the wind would blow the branches into and out of the exposure. On April 28, the wind was very bad, causing the telescope to shake during gusts. On the night of April 28, the CCD was at -15.0 degrees C and on May 3, the temperature was -20 degrees C.

3. Reductions

   The data reduction for the project is fairly straight forward. From the first night, the 14 zeros were used with 3 dark frames to dark and zero correct the flats and objects, then the Hα flats were combined to further reduce the object frames. The second night followed a similar pattern path, with the 20 zeros and 3 darks being used to dark and zero correct the flats and object frames.

After the data on Jupiter was reduced, it was discovered that due to the time and filter of the exposures, chosen to reduce the amount of light from Jupiter to a minimum. This prevented Jupiter from overexposing onto the CCD. However, it created a problem because no stars from the background appeared in the image. Thus, in order to find the scale, I was forced to rely on other data taken each night to find the scale of the CCD. One frame of datum from another project was reduced for each night. Then 4 bright stars in each frame were chosen and the displacement between them was determined. Then, they're displacements were compared that with the known displacement using the images from US Naval Observatory. After comparing the displacement on the Naval Observatory frames to the data, the scale was determined to be 0.7132 arcsec/pixel on the first night, and 0.7136 arcsec/pixel on the second night.

4. Results

   The following images show the photographic results of each night. The first two were taken on April 28th, 2004 at 2:30 and 6:30 AM UT. Images three and four were taken on May 3rd, 2004 at 3:45 and 5:45 UT.

Figure 4. This image is of the first night. From left to right, the moons are Callisto, Europa, Io, and Ganymede. Jupiter and all of the moons are actually much smaller, but have a light radius as shown in the image.


Figure 5. Europa is not visible in this image, it has moved behind Jupiter. Io is the small nub growing out of the side of Jupiter. IRAF was able to separate Io and Jupiter, because of the difference in magnitude between Io and Jupiter's halo. Callisto and Ganymede were are in relatively the same locations as before.


Figure 6. . This is the first image form the second night. From left to right, the moons are Europa, Io, Ganymede, and Callisto. By this time, Io and Europa have made at least one rotation, making calculating the rotational periods less precise.


Figure 7. . This image is from the second set take that was taken May 3rd. All of the moons are in very similar locations to the previous.

I was able to locate and plot the positions of the satellites with respect to Jupiter. Then, I applied the rotational curve calculated above in the introduction to the data. This produces a series of curves representing the rotational period of the various moons.
Figure 8. This is a graph of Io's period curve. The values of the variables are R = 3.8 x106 +/- 6.65 km, P = 1.74 +/- .055 days, i = 9.8 x10-7 +/- 2.6 x10-8 o.


Figure 9. This is a graph of Europa's period curve. The values of the variables are R = 4.81 x106 +/- 9.43 km, P = 3.07 +/- .021 days, i = 3.2 x10-2 +/- 1.6 x10 o-2.


Figure 10. This is a period graph of Ganymede period. The values of the variables are R = 9.2 x106 +/- 7.05 km, P = 5.48 +/- .29 days, i = 1.8 x10-5 +/- 8.6 x10-6 o


Figure 11.. This is the period graph of Callisto's period. The values of the variables are R = 1.8 x107 +/- 3.82 km, P = 16.1 +/- .36 days, i = 4.2 x10-2 +/- 5.4 x10-3 o

The overall results of the fitting of the graphs are believable rotational maxima and to a lesser extent, reliable periods. The error bars are included with the data, but they are so small in comparison to the scale, they cannot be seen.

From each of these, the mass of Jupiter can be determined from each moon individually as:
MJ Io= 1.45 x1027 +/- 7.71 x1026 kg.
MJ Europa= 9.36 x1026 +/- 1.49 x1024 kg.
MJ Ganymede= 2.06 x1027 +/- 3.23 x1026 kg.
MJ Callisto= 1.78 x1026 +/- 2.47 x1021 kg.

Averaged, they produce Jupiter's mass to be 1.56 x 1027 +/- 2.74 x 1026 kg.

5. Discussion

   The published mass of Jupiter is 1.90 x 10 27 kg (Carrol, appendix A-3). Comparing my data to the published result produces 18% error. While this gives a good model for the mass, the loss in accuracy is rather severe. The best remedy for this is more data points. Io and Europa require more points over shorter period of time, while more data in general would assist Ganymede and Callisto. With the limited available data, I was able to get a good estimate of my goal, which makes this project and over all success.

  

References

Chereau, Fabien; Stellarium v 51; 2002
Carrol, Bradley; Ostile, Dale; An Introduction to Modern Astrophysics ; Addison-Wesley: 1996
Tiplar, Paul A.; Physics W.H. Freeman; 1999