The Mass Of Jupiter From Observations Of Galilean Satellites

THE MASS OF JUPITER FROM OBSERVATIONS OF GALILEAN SATELLITES N. BAILEY United States Naval Academy, Physics Department ABSTRACT Based upon Kepler’s Third Law of Planetary Motion, the mass of a central body can be calculated from information about the orbits of its satellites. In this experiment, the four Galilean moons of Jupiter were observed over the course of a month in order to determine the semi-major axes and periods of their orbits. This information was used to calculate the mass of Jupiter. 1. INTRODUCTION Jupiter, king of the planets, has been known in the skies for all of recorded history. Its moons, however, are a more recent discovery. The four brightest, most prominent moons of Jupiter—Io, Europa, Ganymede, and Callisto—are known as the Galilean moons because they were first observed by Italian astronomer Galileo Galilei. When using his new telescope, Galileo noticed four bright “stars” around Jupiter and, to his surprise, they moved relative to Jupiter like no star should have moved (Drake 1978). Galileo published his findings and his conclusion that these bright “stars” were in fact planets orbiting around Jupiter in 1610 (Kepler 1610). Johannes Kepler, a prominent astronomer of the seventeenth century, had already stated his empirical laws in his published works, and he was delighted with Galileo’s publication because it supported his findings (Kepler 1610). Kepler’s three laws were formulated

from his own observations and those of Tycho Brahe. Kepler’s third law is of importance here because it relates two measurable aspects of Jupiter’s satellites—the semi-major axis of their orbit and their period. Newton’s law of universal gravitation is the missing piece of the puzzle. The first corollary of Proposition VIII of Newton’s Principia was ground breaking because it showed how physical information about objects in a system can be gleaned from their motion by deriving the general form of Kepler’s third law (Chandrasekhar 1995). Rearranging Newton’s derivation of Kepler’s third law gives a straightforward equation for the calculation of the mass of a central object, Equation 1, where a is the semi-major axis, T is the period of the orbit, and G is the universal gravitational constant. This is true provided that the central mass is sufficiently greater than the mass of the orbiting object so that the reduced mass of the system is equivalent to the mass of the central object.

4π 2 a 3 M = GT 2

(1)

In order to determine the mass of the central body accurately, precise measurements of the semimajor axis and period of the orbiting object are needed. 2. METHOD AND EQUIPMENT The Galilean moons of Jupiter can be seen with the aid of only binoculars, but something more accurate is needed to take consistent and precise data. For our images, we used the telescope at the United States Naval Academy observatory, a refracting telescope with a Clark handmade lens of 7¾” aperture. The focal length of the telescope is 2953 mm. In order to record data, a Santa Barbara Instruments ST-10 CCD with a pixel size of 6.8 microns was attached to the telescope. During each observing session, ten images of Jupiter and all of its visible moons were taken with dark exposures incorporated by the CCD program. These darks were taken before each image exposure in order to subtract out the thermal properties of the detector. Then twenty flats were taken of a blank area of sky. These twenty flats were combined into one median flat and divided by their mean pixel intensity value in order to create a normalized median flat. Each image was divided through by this normalized median flat before analysis. Our team observed every clear night possible beginning on the

evening of September 29, 2008. The observations continued for four weeks, the last being taken on the evening of October 22, 2008, for a total of 15 different data sets. 3. OBSERVATIONS The 15 data sets were split by the team and reduced down to true distances using the Image Reduction and Analysis Facility program. For each normalized image for the observation set, the center of Jupiter and the positions of the moons were found in pixels. From this, relative distances between the moons and Jupiter were found. The arcseconds of the sky projected onto each pixel of our telescope and CCD was . 47492, as calculated in Figure 1, based on the pixel size of the CCD and the focal length of the telescope. 1 rad mm focal _ length

plate _ scale = 1 rad mm 2953 = .0003386 rad =

= 69.85 arc sec arc sec

pix = .47492

mm

* 206265 arc sec

rad

mm

= 69.85 arc sec

mm

* .0068 mm

Figure 1. Calculation of plate scale in arcseconds per pixel.

Once the separations of the moons were known in seconds of arc, the small angle approximation (Equation 2) could be used in conjunction with the known distance to Jupiter to find the actual physical

pix

separations of the moons, where s is the true distance in the sky, θ is the Date Callisto +/Io +/- Europa +/- Ganymede +/0.054 1512829 3540 125427 3540 -525910 3540 -885363 3540 3.013 -254262 5095 277778 5095 -612149 5095 1029849 5095 3.962 -880971 3814 -365800 3975 737997 4029 5.192 -1453174 8619 377175 8619 632293 8619 6.004 -1735285 4104 -372980 4104 -868972 4104 7.022 -1804738 3686 208469 4120 -549559 4012 -984447 4283 8.017 -1606038 2023 461048 1740 -362534 1914 13.096 1317503 3206 -362310 3206 -827041 3206 14.083 1686384 4310 257260 4310 -585932 4310 -980856 4310 15.013 1748523 4600 495880 4600 -348122 4600 16.179 1820676 3571 -187048 3571 345507 3571 512813 3571 20.054 -411827 4310 -452912 4310 -702488 4310 21.042 -1026575 1917 329905 3618 -620937 2079 -1073559 1325 22.958 -1750011 2276 236034 2186 599238 2344 140972 3561 23.042 -1761072 2029 130445 2976 555385 2119 211999 2074 Table 1. Observational data. Date from 0000 30Sep2008 (UT), distances in km. Negative values indicate a position east relative to Jupiter.

projected angle in radians and d is the distance from the Earth to Jupiter. The Earth-Jupiter distance was referenced from the Astronomical Almanac (2007).

s = θd

(2)

The dates and times of the observations were converted to decimals dates beginning at midnight Universal Time on the morning of September 30, 2008. The distances of each moon, in kilometers, for each observed night were compiled together into Table 1 for the team to analyze. Negative numbers were used to represent distances to the east of Jupiter and positive numbers to represent distances to the west. Error was determined in pixels by each group as they reduced the data in IRAF, converted into kilometers, and carried forward through the calculations.

4. ANALYSIS AND RESULTS In order to determine the semimajor axis and period of each moon, we modeled the data for each orbit as a sine wave as shown in Figures 2-5. Using LoggerPro software, we were able to determine best fit values for the semi-major axis and period of each moon’s orbit. We were able to determine a margin of error from how well the data matched up to the fit line, which was important because the error resulting from our observations was negligible, as shown by the plotted error bars. The best fit values for each orbit and the calculated mass of Jupiter from those parameters are shown in Tables 2-5. We also calculated the statistical coefficients of determination, R2, to evaluate the closeness of the fits, as seen in Table 6.

Callisto

Millions

Observed Data

Line Fit to Data

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Distance from Jupiter (km)

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-1.0 -1.5 -2.0 -2.5 Day

Figure 2. Callisto.

a (m) T (s) M (kg)

(1.8412 ± 0.0142) x 109 (1.4485 ± 0.0049) x 106 (1.7602 ± 0.0425) x 1027 Table 2. Callisto.

Millions

Io 0.5 0.4

Distance from Jupiter (km)

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Figure 3. Io.

a (m) T (s) M (kg)

(4.1460 ± 0.1340) x 108 (1.5275 ± 0.0016) x 105 (1.8072 ± 0.1753) x 1027 Table 3. Io.

Ganymede

Distance from Jupiter (km)

Millions

Observed Data

Line Fit to Data

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Figure 4. Ganymede.

a (m) T (s) M (kg)

(1.0548 ± 0.0248) x 109 (6.2049 ± 0.0231) x 105 (1.8034 ± 0.1278) x 1027 Table 4. Ganymede. Europa

Distance from Jupiter (km)

Millions

Observed Data

Line Fit to Data

0.8 0.6 0.4 0.2 0.0 0

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Figure 5. Europa.

a (m) T (s) M (kg)

(6.9660 ± 0.1474) x 108 (3.0481 ± 0.0066) x 105 (2.1526 ± 0.1370) x 1027 Table 5. Europa.

Callisto Ganymede Europa Io

0.980 0.709 0.978 0.983

Table 6. Coefficients of determination.

Using the amplitudes and periods of the fitted sinusoid and Equation 1, a mass for Jupiter was determined for each moon with an uncertainty based on how well the sine curve fit our data. These were then averaged together to arrive at a combined value for Jupiter’s mass. However, we realized that some of the masses were likely to be more accurate due to how well the individual sine curves fit each moon. Our final mass of Jupiter was calculated as an average of the four moons’ calculated value weighted by their correlation coefficient. Using this method we found the mass of Jupiter to be (1.8864 ± 0.1753) x 1027 kg. 5. DISCUSSION There were a few difficult spots during this experiment. The observations themselves had to be well taken, which meant that the CCD had to be in focus. The extension of the barrel to achieve good focus was tricky to find because we couldn’t observe how the image changed as we moved it. Instead we had to move the barrel, attempt a picture, see how it turned out and adjust accordingly. Once the focal length was established, however, we were able to mark it and use the same length every time under the assumption that the ambient temperature was the same.

Getting observations every night for a good length of time was vital to successful data. Unfortunately the weather does not always cooperate with astronomical desires, and so many nights of data were lost in this way. This is especially problematic for the two moons closest to Jupiter, Io and Europa, whose positions change drastically from night to night. Some of the most useful data points in the sine fitting were the last two data points, which we took 3 hours apart. In order to get a better fit and more accurate results, many such spaced observations would be crucial. When we first collected all of our data, there was an odd problem. The periods evaluated for each of the four moons were consistent with previous values, but the semi-major axes were too short by a factor of about 10% for every moon. We examined our process and our data in order to determine the source of what appeared to be a systematic error, but nothing could account for such a big discrepancy. Inclination, which we had assumed to be negligible, was calculated and found to not be a factor. Our math was correct, and each part of the team had independently arrived at semimajor axes which were uniformly too short. After much searching, it was discovered that a parameter of the CCD, the microns per pixel, had been misreported, which introduced an error of a factor of 1.1146 which perfectly accounted for our discrepancy. After this was applied, all of our measured values became consistent with accepted values.

6. CONCLUSION We conclude that the mass of Jupiter is (1.8864 ± 0.1753) x 1027 kg. This value concurs with 1.8988 x 1027 kg, the accepted value of the mass of Jupiter from previous experiments as reported by the Astronomical Almanac (2007). Much acknowledgement and many thanks go to my partner, Cody Forsythe, who worked with me for all our assigned observations, data reduction, and analysis, and whose help was invaluable. Thanks go to the rest of the team as well: N. Backstrom, S. Lozano, C. Navarro, K. Devers-Jones, J. Walker, and E. Knebel. REFERENCES Chandrasekhar, Subrahmanyan 1995, Newton’s Principia for the Common Reader, Clarendon Press, New York, 373. Drake, Stillman 1978, Galileo at Work: His Scientific Biography, University of Chicago Press, Chicago, 146-156. Kepler, Johannes 1610, Conversation with Galileo’s Sidereal Messenger, Johnson Reprint Corporation, New York, 14.

The Astronomical Almanac for the Year 2008, Nautical Almanac Office United States Naval Observatory 2007, Washington, E4, E29.