Computing the Mass of Jupiter 1 - COSMOS - UC Davis

Computing the Mass of Jupiter 1 

Computing the Mass of Jupiter 

Based on the Orbit of Jupiter’s Four Main Moons 

Virginia Marcus 

COSMOS Cluster 9: Introduction to Astrophysics 

July 26, 2009


Abstract 

I attempted to find the mass of Jupiter based on the orbits of its four main moons. First I reduced 

the data I extracted from several images taken of Jupiter but since the photos lacked stars, I was 

unable to find the right ascension and declination of the moons. So instead I had to use a 

program called Planetarium to find their coordinates which I then entered into two different 

orbital computation programs, Findorb and Codes. Unfortunately, they were unable deduce the 

correct orbital elements due to their strong inclinations toward heliocentric orbits. However, 

when I input archival data about the semi-major axis and the period of Jupiter’s moons into 

Newton’s orbit equation, I calculated an average mass of1.9010 x10 kg which only differed by 

 

0.1285% from Jupiter’s actual mass of 1.8986 x10 kg.


Computing the Mass of Jupiter Based on the Orbit of Jupiter’s Four Main Moons 

When one thinks of orbits, one usually imagines a stationary object being circled by 

another object. In fact, the first presumption is not true and the second is only sometimes true. 

Orbits actually involve the movement of both objects and can also be any one of four conic 

sections (named thusly since their shapes can all be achieved by slicing a cone in different 

directions): parabolic, hyperbolic, elliptical, and circular. Kepler’s and Newton’s laws not only 

explain this, but also discuss how mass affects an orbit and how one can can use the orbit to 

calculate mass. Jupiter is a perfect example of an object whose orbit is affected by the mass 

surrounding it. The planet has a total of 63 moons orbiting it, causing Jupiter to slightly wobble 

as it rotates the Sun. (Frosty Drew Observatory, n.d.) Based on several of Kepler’s and Newton’s 

laws, if one observes a few key elements about these moons, one can determine Jupiter’s mass. 

Johannes Kepler’s second law, the only one Newton left unchanged, states that planets 

will sweep out equal areas in the same amount of time as they orbit the Sun. Planetary orbits are 

in the shape of ellipses with the Sun as one focus (Kepler’s first law) so one would expect that 

planets would sweep out a greater area farther away from the Sun. However, planets do not 

travel at a consistent rate since when they are close to the Sun, the gravity of the Sun’s mass is 

enough to increase the planets’ velocities. (Braeunig, R.A., 2008) Kepler’s first law also explains 

how the Sun is orbiting around the planets as well as vice versa. However this revolution focuses 

on the center of mass between the Sun and the planets orbiting it. Since the Sun has such a great 

mass, the center of mass falls within the Sun itself, creating the illusion that only the planets are 

in orbit. (Egler, R. A., 2006) 

Kepler’s third law is one of the most important for orbital computation and it states that 

the planets’ periods for revolving around the Sun depend on the size of their orbits as well as


inversely on their mass and gravity. Newton also derived the law of universal gravitation from 

Kepler’s work. The law explains Kepler’s third law by stating that the smaller the distance to the 

Sun from the planet and the greater the mass and gravitational pull of the planet, the more 

gravitational force will be there. The greater the gravitational force, the more the velocity 

increases. (Freedman, R. A., & Kaufmann III, W. J., 2005) 

Though there are six orbital elements necessary to describe an orbit mathematically (the 

semi-major axis, eccentricity, the longitude of the ascending node, the time of periapsis, and the 

argument of periapsis), in order to calculate the mass of Jupiter, I only need to know two 

elements. (Braeunig, R.A., 2008) The first is the semi-major axis (the longest radius of an 

ellipse) and the period of Jupiter’s moon around the planet, both of which I shall obtain after 

feeding my observational data into Findorb, an orbital computation program. I shall then input 

them into Newton’s equation P π 

GM ♃ 

a 

M ♃ 

4π2 a 3 

which can be easier used to calculate mass 

as 

GP 2 . In these equations, P is period, G is the gravitational constant, M 

♃ 

is the 

mass of Jupiter, and a is the semi-major axis. Using this Newtonian equation, I should be able to 

find Jupiter’s mass. 

Data 

The photos I used to help me calculate the mass of Jupiter were taken on July 21, 2009 

from 1:45 A.M. to 4:04 A.M. on the top of the Physics building at U.C. Davis. The telescope 

used to take them was a Takahashi casegrain with a 10 inch diameter. Four photos were taken 

about every fifteen minutes for the entire time period. However, the 1:45 A.M timeslot did not 

contain Jupiter and the 4:04 A.M timeslot was missing Jupiter’s moons (or rather they were


washed out by the brightness of Jupiter). Although flats were gathered for both the visual and the 

B filters, the pictures themselves were only taken using the B filter. The telescope was on 

autodark as well, so no subtraction or darkening was needed when I did data reduction. Since 

Jupiter was so bright, only around a half second exposure could be done, so no stars are visible 

and it was impossible to do astrometry to help me find the orbit of Jupiter’s moons. Had there 

been stars, I would have used a program called TheSky which uses stars as reference points to 

find the RA and dec of the desired object. 

Analysis 

My first action upon receiving my data was to separate images into folders based on the 

time they were taken. I then aligned and median combined each folder so I had seven images of 

Jupiter and its moons, each taken fifteen minutes apart. However, two images in the 3:49 A.M 

timeslot were flipped, creating a double image. I solved this by selecting the moon closest to 

Jupiter Io and marking it as the centroid, then aligning the centroids (and then combining the 

images again). After this, I proceeded to divide the flats into folders depending on whether they 

were taken through a B or visual filter and then median combined each folder (although the 

visual filter folder was unnecessary because, as previously stated, photos were only taken using 

the B filter). I then aligned all the flatfield images by marking Jupiter as the centroid. This was 

risky since it was extremely bright, which may cause pixel overflow, but the moons moved in 

every picture and there were no stars in any of the pictures due to the short exposure time. I then 

centroid aligned the images and converted them from .fits to .jpeg so they could be saved as a 

movie. 

To determine which moon was which, I inputted the correct date, time, and place that the 

images were taken into Stellarium (a virtual planetarium) and learned that from closest to farthest


from Jupiter, they were: Io, Europa, Ganymede, and Callisto. Since Findorb requires stars to find 

certain variables of the orbit like the RA and dec (also known as the right ascension and 

declination which are basically the respective longitude and latitude of a celestial sphere) , I 

checked Stellarium again and found the RA and dec of each of Jupiter’s moons at every 

increment of time a photo was taken). Since I was using Stellarium, not my own data, the 

coordinate epoch (the moment an observation was made) was J2000 (measured in Julian years 

which are 365.25 days). I then transcribed the data I received into 4 different notepad files (one 

for each moon). I was required to type the name, date the image was taken (minutes and hours 

were had to be calculated as a percent of the day), RA, and dec. I also had to add a random line 

of data to the end of the data set because Findorb always ignored the last line. Another problem I 

ran into was that Findorb had trouble calculating Europa’s orbit (probably since it moved the 

least and there wasn’t as much data on the complete orbit as the others had). Therefore I tried to 

input Europa’s correct eccentricity (how much the shape of an object’s orbit differs from that of a 

circle) and inclination (the angle of the tilt between the plane the orbit is on and the plane that 

Earth’s orbit is on or another plane of reference). However, Findorb still had trouble computing 

the orbit of Europa so I had to turn to Stellarium to gather even more data about the moon. I 

found the RA and dec of different epoch spread out over the course of three days and this proved 

successful and Findorb managed to find an orbit. Unfortunately, the orbits it found were all 

heliocentric or on a collision course. I tried again with data from Stellarium over the course of a 

month to get an even more complete orbit, but this too failed. 

As a last resort I turned to Codes (Comet/Asteroid Orbit Determination and Ephemeris 

Software) which used the Gauss method and therefore only required three observation points as 

well as the MPC observing code of where the images were taken (K01). However no matter


which data I inputted into the program, it failed to find a decent orbit, giving me instead orbits 

with a large uncertainty or a negative semi-major axis. 

So in the end I had to search archival databases (U.S. Geological Survey, 2007 and 

NASA, n.d.) for the correct semi-major axes and periods of Jupiter’s moons. I then inserted these 

into Newton’s equation M ♃ 

 

GP . 

Semi-major axis Period Mass of Jupiter 

Io 422,600 km 1.769137786 days 

 

1.9108 x10 

Europa 670,900 km 3.551181041 days 

 

1.8975 x10 

Ganymede 1,070,000 km 7.15455296 days 

 

1.8964 x10 

Callisto 1,883,000 km 16.6890184 days 

 

1.8995 x10 

0 

Jupiter’s actual mass turned out to be 1.8986 1 kg, an amount which was extremely 

similar to all of my calculated masses. I then took averaged the calculated masses (1.9010 x10 

kg) and subtracted Jupiter’s actual mass from it to get a difference of 2.44 x10 kg. Next I 

divided the difference by Jupiter’s real mass to get the percent error of 0.1285%. 

Conclusion 

There were many setbacks in my attempt to find Jupiter’s mass. However the two biggest 

were probably the brightness of Jupiter at the epoch and the inability of orbital computation 

programs to calculate an orbit. Had Jupiter been brighter, the astronomers could have taken 

longer exposures without fear of pixel overflow (too many photons hitting one pixel) and 

therefore taken images with stars. Had there been stars, I could have used the data the 

astronomers took instead of data from Stellarium. Even if had gotten data from the observations,


I couldn’t have used it since both Findorb and Codes seemed unable to calculate a Jovian-centric 

orbit and want to impose a heliocentric one on my data. I also noticed that I had to continually 

add observations to my list until they were spread out over a month, a fact which would be wise 

to remember if I repeated this experiment and were to re-make observations. However, upon 

gathering the official semi-major axis and period for Jupiter’s moons, Newton’s equation worked 

and calculated almost the exact mass of Jupiter. The average mass I calculated for Jupiter using 

the four moons was 1.9010 x10 kg which only had a percent error of 0.1285% compared to 

Jupiter’s actual mass of 1.8986 x10 kg.


References 

Braeunig, R.A., Orbital mechanics. (2008). Retrieved July 12, 2009, from Rocket and Space 

Technology, website: 

Egler, R. A., A brief explanation of orbital elements. (2006, March 13). Retrieved July 19, 2009, 

from North Caroline State University, Department of Astrophysics, website: 

 

Freedman, R. A., & Kaufmann III, W. J. (2005). Universe (7th ed.). New York: W. H. Freeman 

and Company. 

Jupiter. (2007, August 17). Retrieved July 28, 2009, from U.S. Geological Survey, website: 

 

Jupiter: Moons. (n.d.). Retrieved July 28, 2009, from NASA, website: 

 

Orbits for amateurs. (n.d.). Retrieved July 12, 2009 from Frosty Drew Observatory, website: 


Acknowledgements 

I thank the U.C. Davis COSMOS program for providing the opportunity to explore astrophysics. 

I would also like to thanks the talented and helpful people who made this possible: 

-Abrahamse, Augusta 

-Dutra, Brent 

-Fassnacht, Christopher 

-Feldstein, Paul 

-Johnson, Jim 

-Margoniner, Vera 

-Taylor, Chris

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