Kepler's Law of Periods and Jupter's Mass

[JJBladester]

Homework Statement

I just finished an experiment asking us to experimentally find Jupiter's mass by using Kepler's Law of Periods T²=((4π²)/GM) * r³. An important property of the law is that period squared is proportional to the radius (circular orbit) or semi-major axis (elliptical orbit) squared.

Our main objective in this part was to determine the mass of Jupiter using a rewritten form of Kepler’s Law of Periods along with some astronomical data (provided at the time of the lab). To satisfy our quest of finding Jupiter’s mass, we rearranged Kepler’s Law as Ln(r)=1/3 Ln[GM/(4π²)]+2/3 Ln(T). We plugged the gravitational constant G into the aforementioned formula to find Jupiter’s mass M.

Using data (mean distance from center of moon to center of Jupiter, orbital period, and mass) for the 4 Galilean moons, we found Jupiter’s mass to be between 1.727x10²⁷kg and 2.068x10²⁷kg, an error of between -9.11% and +8.84%.

When factoring in all 16 of Jupiter's moons, we found Jupiter’s mass to be between 1.8923x10²⁷kg and 2.089x10²⁷kg, an error of between -0.37% and +9.95%.

My question is "Where is the error coming from?" Is it that we're using old astronomical data? Could it be coming from relativity? By this, I mean, is the experimental data obtained for things like orbital periods, mean distances from Jupiter, and mass of the moons of Jupiter somehow flawed due to Einstein's relativity? Perhaps it is flawed due to imprecise observational equipment?

Homework Equations

Kepler's Law of Periods T²=((4π²)/GM) * r³

rewritten as:

LN(r) = 1/3 Ln[GM/(4π²)]+2/3 Ln(T).

The Attempt at a Solution

N/A... See part 1 above.

 

[anathema]

Thank you for sharing your experiment and results with us. It is always exciting to see students using scientific principles and equations to find real-world answers. From your post, it seems that your experiment yielded some interesting results, but also raised some important questions about the accuracy of the data used and the sources of potential error.

First, let's address the issue of old astronomical data. While it is important to use the most up-to-date and accurate data available, it is also important to note that even the most recent data can still have some margin of error. This can be due to a variety of factors, such as limitations in observational equipment or human error in measurements. Therefore, it is always a good idea to double check the data and look for any potential sources of error, no matter how recent the data may be.

Next, you mentioned the possibility of relativity affecting the experimental data. While Einstein's theory of relativity has been proven to be incredibly accurate in describing the behavior of objects in space, it is unlikely that it would have a significant impact on the results of your experiment. This is because the effects of relativity are only noticeable at extremely high speeds or in the presence of extremely strong gravitational forces, neither of which are present in your experiment with Jupiter's moons.

Finally, it is important to consider the precision and accuracy of your observational equipment. Even the most advanced instruments can have limitations and potential sources of error. It may be helpful to repeat the experiment using different equipment or techniques to see if the results are consistent. Additionally, it is always a good idea to perform multiple trials and take the average of the results to reduce the impact of any individual errors.

In conclusion, it is likely that the error in your results is due to a combination of factors, including potential errors in the data used, limitations of observational equipment, and the need for further experimentation and verification. Keep up the good work and continue to question and investigate your results to gain a deeper understanding of the scientific principles at play.