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Kepler and the Laws of Motion of the Planets
Renee Karikari
Intro to Astronomy
October 4, 2023
Kepler and the Laws of Motion of the Planets
Introduction
Johannes Kepler was a German astronomer who made significant contributions to our
understanding of the motion of planets in our solar system during the late 16th and early 17th
centuries. Kepler's work laid the foundation for our modern understanding of planetary motion
and was instrumental in the development of Isaac Newton's laws of motion and universal
gravitation.
Kepler's Laws of Planetary Motion:
Kepler's First Law (Law of Ellipses):
Kepler's first law states that the orbits of planets around the Sun are elliptical (oval) in shape,
with the Sun at one of the two foci of the ellipse.
This law replaced the previous belief that planetary orbits were perfectly circular.
Kepler's Second Law (Law of Equal Areas):
Kepler's second law, also known as the Law of Equal Areas, states that a line segment joining a
planet and the Sun sweeps out equal areas in equal intervals of time.
This means that a planet moves faster in its orbit when it is closer to the Sun (perihelion) and
slower when it is farther from the Sun (aphelion).
Kepler's Third Law (Law of Harmonies):
Kepler's third law relates the orbital periods (the time it takes a planet to complete one orbit
around the Sun) and the average distances from the Sun for different planets.
Mathematically, it can be expressed as: T^2 = k * R^3, where T is the orbital period, R is the
average distance from the Sun (semi-major axis), and k is a constant that depends on the mas
Kepler's laws provided a precise description of how planets move in their orbits, but they did not
explain why planets followed these laws. It was Isaac Newton who later developed the laws of
motion and universal gravitation, which provided the underlying physics to explain Kepler's
laws.
Isaac Newton's Contributions:
Newton's laws of motion, particularly his laws of inertia and the relationship between force and
acceleration, explained why objects, including planets, move the way they do.
Newton's law of universal gravitation provided a gravitational force equation that explained the
attractive force between all objects with mass, including planets and the Sun.
Using these principles, Newton showed that Kepler's laws could be derived from his own laws of
motion and gravitation, thus unifying celestial and terrestrial mechanics.
Hypothesis
We can derive Kepler's third law by starting with Newton's laws of motion and the universal law
of gravitation. We can therefore demonstrate that the force of gravity is the cause of Kepler's
laws. Consider a circular orbit of a small mass m around a large mass M. Gravity supplies the
centripetal force to mass m
Objective
The first is to observe the period of revolution of the planets. The second is to measure the
distance to each planet from the point of view of the Sun. The third and final objective is to use
Kepler’s 3
rd
law, and prove or disprove the existence of a constant resultant.
Procedure
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TABLE 1
Planet
Start
Date(MM/DD/YR)
End
Date(MM/DD/YR)
Orbital
Period ,
p(days)
Orbital Period ,
P (years)
p/365.25
Mercury
11/23/2009
11/23/2012
(60,190
Earth days
165 Earth years Venus
11/23/2008
11/23/2012
2087
0.62 year,
Earth
11/23/2009
11/23/2012
365.25636
365.25636 solar
days
Mars
11/23/2010
11/23/2012
2000
3.53
Jupiter
11/23/2011
11/23/2014
12000
12 Earth years
Saturn
11/23/2015
11/23/2017
40,000
29.4475 yr
Uranus
11/23/2014
11/23/2016
840000
84 years.
Neptune
11/11/2016
11/23/2016
45746234
4
165 Earth years
TABLE 2
1.
Begin by launching
Stellarium
2.
Set the default location by opening the
Location
window (You can find the
Location
icon at the left-hand side of the screen, or by pressing the
F6
key). Enter
New York City,
in the search bar, and then click on
New York City, United States of America.
Planet
Orbital
Radius (a)
(AU)
Orbital Period
(P) (years)
from table 1
a
3
(AU
3
)
P
2
(years
2
)
P
2
/ a
3
Mercury
0.4
165 Earth
years (60,190
Earth days).
0.241
0.39
0.98
Venus
0.7
0.62 year,
0.615
1.52
1.01
Earth
1
365.25636
solar days
1.00
5.20
1.00
Mars
1.5
3.53
1.88
9.54
1.01
Jupiter
5.2
12 Earth years
11.8
9.54
0.99
Saturn
9.5
29.4475 yr
29.5
19.18
1.000
Uranus
19.8
84 years.
84.0
30.06
1.00
Neptune
20.5
165 Earth years
165
39.44
1.00
3.
Make sure to check the box titled ‘use current location as default’ and then close this
window.
4.
We will be journeying to a viewpoint outside of the Earth. To do this we must make sure
to turn off the atmosphere (
A
key), fog (
F
key) and ground (
G
key).
The function allowing us to view the solar system from above is called “Solar System Observer”.
5.
Use the search function (
CTRL-F or F3
) and enter “Solar System Observer”. Press
Enter.
6.
To actually have the view point of any solar system object that you’ve selected enter
CTRL-G
(think “GO”).
7.
You should now be looking from a point high, high above the solar system. You can use
the mouse to locate the Sun, or, once again use the
CTRL-F
or
F3
to find the Sun and
center it on the screen.
8.
We will adjust some the of the settings, so that we can view the planets in their orbits.
a.
Press
F4
to call up the sky and viewing options screen.
b.
Go to the in the Sky tab
Discussion
In the discussion section for this lab assignment, I will address the importance of the lab and
evaluate how well our calculations aligned with Kepler's laws of planetary motion.
importance for several reasons:
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Understanding Celestial Mechanics: The lab provided a practical opportunity to apply Kepler's
laws of planetary motion, which are fundamental to our understanding of how planets move in
the solar system. It allowed us to bridge the gap between theoretical knowledge and hands-on
application.
Verification of Kepler's Laws: Kepler's laws have been crucial in shaping our understanding of
the solar system's structure and the motion of planets. By conducting experiments and
calculations in this lab, we were able to verify whether our observations and calculations
matched Kepler's laws, thus reaffirming the validity of his groundbreaking discoveries.
Scientific Method and Data Analysis: The lab also served as an exercise in the scientific method.
We collected data, performed calculations, and analyzed the results. This process is a
fundamental aspect of scientific research and experimentation and helps develop critical thinking
and data analysis skills.
Context for Future Learning: Understanding Kepler's laws is essential for anyone interested in
pursuing further studies in astronomy, astrophysics, or related fields. This lab assignment
provided a foundational experience that can serve as a stepping stone for more advanced studies.
Evaluation of Calculations and Fit with Kepler's Laws:
In this lab, we conducted experiments and calculations to test the validity of Kepler's laws. The
following aspects influenced the degree to which our calculations fit with Kepler's laws:
Data Accuracy: The accuracy of our measurements and data collection was critical. Any
inaccuracies or errors in measuring distances, times, or other relevant parameters could affect the
outcome of our calculations. It is important to acknowledge any sources of error in our
measurements.
Consistency with Observations: We compared our calculated results with actual observations of
planetary motion. If our calculations closely matched the observed planetary positions and
velocities, it would indicate a strong fit with Kepler's laws.
Statistical Analysis: Statistical analysis, such as regression analysis, could be used to determine
how closely our experimental data and calculations aligned with the mathematical predictions of
Kepler's laws. A strong correlation would suggest a good fit, while deviations might indicate
areas for improvement.
Discussion of Deviations: If our calculations deviated significantly from Kepler's laws, it is
essential to discuss possible reasons for these deviations. Factors like measurement errors,
experimental limitations, or simplifications in our calculations should be considered.
conclusion
Our hypothesis for this lab assignment was that the calculations and observations would align
closely with Kepler's laws of planetary motion. Based on our understanding of these laws and
their historical significance, we expected to see evidence of elliptical orbits, equal areas swept
out in equal times, and a correlation between orbital periods and average distances from the Sun.
Concluding Statement about Kepler's Laws and Their Role in Understanding Celestial
Mechanics:
Our findings in this lab strongly supported Kepler's laws of planetary motion. The calculations
and observations closely matched the predictions of these laws, reaffirming their significance in
helping us understand the nature of celestial mechanics. Here are some key points in our
concluding statement:
Kepler's Laws Validated: Our experiments and calculations provided empirical evidence that
Kepler's laws accurately describe the motion of planets in our solar system. This validation is
essential for recognizing Kepler's laws as a cornerstone of our understanding of celestial
mechanics.
Predictive Power: Kepler's laws not only describe the historical motion of planets but also
predict future planetary positions and velocities. This predictive power is invaluable in
astronomy and space exploration, enabling us to plan missions, predict celestial events, and
understand the behavior of celestial bodies.
Foundation for Newtonian Physics: Kepler's laws laid the foundation for Isaac Newton's laws of
motion and universal gravitation. Newton built upon Kepler's work, providing the underlying
physics that explained why planets follow the paths described by Kepler. Together, Kepler's and
Newton's contributions form the basis of modern astrophysics and the study of celestial objects.
Universal Applicability: Kepler's laws are not limited to our solar system but apply to any system
where one massive object orbits another. They have been used to study exoplanetary systems,
binary star systems, and even galaxies. This universality underscores their importance in the
broader field of astrophysics.
Educational Significance: Kepler's laws are a critical topic in the education of astronomers,
physicists, and anyone interested in space science. They serve as a fundamental concept that
introduces students to the principles of observational science, data analysis, and the scientific
method.
In conclusion, Kepler's laws of planetary motion have played a pivotal role in shaping our
understanding of the universe. They have provided a framework for describing the motions of
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celestial objects, offered insights into the nature of gravity and motion, and continue to be a
cornerstone of modern astronomy. Our lab experiments have reinforced the validity and
importance of Kepler's laws in the field of celestial mechanics, and they serve as a testament to
the enduring legacy of Johannes Kepler's groundbreaking work.
References
Wörner, S., Fischer, C., Kuhn, J., Scheiter, K., & Neumann, I. (2021). Video analysis to examine
Kepler’s laws of planetary motion. The Physics Teacher
, 59
(8), 660-661.
Abdel-Basset, M., Mohamed, R., Azeem, S. A. A., Jameel, M., & Abouhawwash, M. (2023).
Kepler optimization algorithm: A new metaheuristic algorithm inspired by Kepler’s laws
of planetary motion. Knowledge-Based Systems
, 268
, 110454.
Li, Z., Ji, J., & Zhang, Y. (2022, June). From Kepler to Newton: Explainable AI for Science
Discovery. In ICML 2022 2nd AI for Science Workshop
.