## Introduction

Before diving into the big question, can Newton’s generalization of Kepler’s third law for planetary motion be validated and if so, how, it is important look back into the development of western thought. The Greek Geocentric Model is worth looking into because it shows how and when individuals were coming up with these ideas and testing them. Individuals were also taking parts of what others thought and knew that did not work and reshaped their thinking and testing.

The most important part of being able to see everyone’s thought is to understand how Newton came up with a generalization of Kepler’s third law for Planetary Motion. Going into the development, there are a few models that have been used due to their ability to present a model that appears to work. Dating back to 150 A.D, Claudius Ptolemy was able to dispense a model that was comprehensive which was used for approximately 1400 years following his death (Koupelis, 2011).

This model was presented for a while and appeared to work because, “the moon fits the Ptolemaic Scheme perfectly. All that is necessary is that it moves around the Earth as the closest heavenly body” (Koupelis, 2011). Later on, Nicolaus Copernicus thought the celestial objects were not agreeing with positions that were observed and that Ptolemy projected a different model which could be more pleasing and that there was something else needed to explain the changes in brightness (Koupelis, 2011).

This model was revolutionary but no better at predicting the planetary positions because of the simplicity of it. Both of the models were able to explain observations of motion, but the heliocentric model was more clear therefore it became the more preferred model (Koupelis, 2011). Following Copernican model, Kepler decided to start fresh by throwing out the idea of circular orbits and began to try out other shapes that work which eventually, an ellipse appeared to be the shape for every planet (Koupelis, 2011). Kepler was able to figure out which shape applied to every planet but never really got into what caused the orbits such as an ellipse.

Later with Galileo, it was observed that, “there seemed to be no logic supporting the laws of Kepler, except that they work” (Koupelis, 2011), which led Galileo, known for the four moons Lo, Europa, Gallisto and Ganymede, to initiate a study that would look at the causes of motion. Isaac Newton was the one to expand more onto the idea of motion. Through his findings, he was able to show that with his laws, they apply to any motion and to any forces along with being able to explain the cause of motion with Kepler’s Laws.

## Theory

Newton’s Law of Universal Gravitation states, “Between every two objects there is an attractive force, the magnitude of which is directly proportional to the mass of each object and inversely proportional to the mass of each object and inversely proportional to the square of the distance between the centers of the object” (Koupelis, 2011).

This is a reduced form of Kepler’s third law in a sense where it is stated as, “the ratio of the cube of the semimajor axis of a planet’s orbit to the square of its orbital period around the sun is the same for each planet.” (Koupelis, 2011). Throughout this research paper, there will be methods explained and tested on why it is possible that Newton’s Law of Universal Gravitation is able to replace previous models even though those were validated.

## Method

The objective of this paper is to test the validity of Newton’s generalization of Kepler’s Third Law with the four Galilean moons of Jupiter. Since those moons are the most visible and the largest, there is enough data to be provided. CLEA will be the software program that will be used for all the calculations needed to test whether or not Newtons generalization can be validated. The first calculation that can be done is the orbital parameter.

This equation is a cubed over p squared. For this objective, a represents the semi-major axis which represents the average orbital distance along with becoming the radius of the orbit and p represents the years. Another way of how one will be able to test the validity is by determining the mass of Jupiter. This will be done four times with each of a Galilean satellite and an average number will be calculated to determine Jupiter’s mass.

The equation used for this is mass equals volume times density. The importance of collecting four different masses of Galilean Satellites is by finding the average, it will help determine that if the four numbers are pretty close together, then it could be shown that Newton’s generalization can be validated. Another calculation that will be done to test the validity is finding the average orbital distance.

This will be tested to see if the orbits are reasonably circular which if they are, then that could show that Newton’s generalization can also be validated. The equation for this is force of gravity equals G times M1, M2 over R squared. The G stands for a constant, the M1 and M2 stand for the masses of the objects involved, and R is the distance of their center’s mass. All of these calculations will help determine whether or not Newton’s generalization of Kepler’s Third Law can be validated or not depending on the results.

## Results

The first moon of Jupiter to look at to the find its orbital period, the calculation for this is stated in the method section, is by looking at the start time of Lo’s orbit to when it reaches back around, which takes 1.66667 days. The second moon of Jupiter’s, Europa, orbital period is 3.505 days, the third moon, Gallisto, is 7.20834 days and the fourth moon, Ganymede, orbital period is 12.04167 days.

With these numbers, we are able to become one step closer to seeing if Newton’s generalization of Kepler’s Third law can be validated or not. Another calculation that was done was determining the orbital distance for each moon which the calculation for this is stated in the methods section

. The first inner most moon, Lo, has an orbital distance of 0.0194 au. Europa has an orbital distance of .0309 au, Gallisto is .0971 au and Ganymede orbital distance is 1.411 au in which we can also see that these are relatively the same. The average of these four moons is 0.3896.

## Moon Orbital Period Orbital Distance

- Lo 1.66667 days 0.0194 au
- Europa 3.505 days .0309 au
- Gallisto 7.20834 days .0971 au
- Ganymede 12.04167 1.411 au
- Average 6.10542 0.3896 au

After discovering the average for orbital period and orbital distance, we are able to determine the mass for Jupiter by taking the distance cubed over orbital period squared in which we get 1,500 kg.

## Conclusion

From our results and calculations, we can validate Newtons Generalization of Kepler’s Third Law because we can see that all of the four moons orbital distance are really close together. We can also validate this generalization because we when we added all four orbital distances and divided them, the average are pretty close in range along with Jupiter’s mass being fairly close to what we have been given.

With these results, we can refer back to the theory section and see that Jupiter’s’ moons stay in a circular orbit because gravity is keeping them pulled towards Jupiter which is Newton’s Law of Universal Gravitation that states, “Between every two objects there is an attractive force, the magnitude of which is directly proportional to the mass of each object and inversely proportional to the mass of each object and inversely proportional to the square of the distance between the centers of the object” (Koupelis, 2011)