essay about planetary motion

• History & Society
• Science & Tech
• Biographies
• Animals & Nature
• Geography & Travel
• Arts & Culture
• Games & Quizzes
• On This Day
• One Good Fact
• New Articles
• Lifestyles & Social Issues
• Philosophy & Religion
• Politics, Law & Government
• World History
• Health & Medicine
• Browse Biographies
• Birds, Reptiles & Other Vertebrates
• Bugs, Mollusks & Other Invertebrates
• Environment
• Fossils & Geologic Time
• Entertainment & Pop Culture
• Sports & Recreation
• Visual Arts
• Demystified
• Image Galleries
• Infographics
• Top Questions
• Britannica Kids
• Saving Earth
• Space Next 50
• Student Center

Understanding Kepler’s Laws of Planetary Motion

In the early 17th century, German astronomer Johannes Kepler postulated three laws of planetary motion . His laws were based on the work of his forebears—in particular, Nicolaus Copernicus and Tycho Brahe . Copernicus had put forth the theory that the planets travel in a circular path around the Sun . This heliocentric theory had the advantage of being much simpler than the previous theory, which held that the planets revolve around Earth . However, Kepler’s employer, Tycho, had taken very accurate observations of the planets and found that Copernicus’s theory was not quite right in explaining the planets’ motions. After Tycho died in 1601, Kepler inherited his observations. Several years later, he devised his three laws.

Planets move in elliptical orbits.

An ellipse is a flattened circle. The degree of flatness of an ellipse is measured by a parameter called eccentricity. An ellipse with an eccentricity of 0 is just a circle. As the eccentricity increases toward 1, the ellipse gets flatter and flatter. A major problem with Copernicus’s theory was that he described the motion of the planet Mars as having a circular orbit. In actuality, Mars has one of the most eccentric orbits of any planet, with an eccentricity of 0.0935. (Earth’s orbit is quite circular, with an eccentricity of only 0.0167.) Since planets orbit in ellipses, that means they aren’t always the same distance from the Sun, as they would be in circular orbits. Since a planet’s distance from the Sun changes as it moves in its orbit, this leads to…

A planet in its orbit sweeps out equal areas in equal times.

Consider the distance that a planet travels over a month, for example, during which it is closest to and farthest from the Sun. One can in a diagram form a roughly triangular shape with the Sun as one point of the triangle and the planet at the beginning and end of the month as the other two points of the triangle. When the planet is close to the Sun, the two sides that have the Sun as the vertex will be shorter than those same sides of the triangle when the planet is far from the Sun. However, both of these triangular shapes will have the same area. This happens because of the conservation of angular momentum . When the planet is closer to the Sun, it moves faster than when it is farther from the Sun, so it travels a greater distance in the same amount of time. Therefore, the side of the triangle connecting the two positions of the planet when it is closer to the Sun is longer than it is when the planet is farther from the Sun. Despite the distance to the Sun being shorter, the fact that the planet travels a longer distance in its orbit means that the two triangles are equal in area.

T 2 is proportional to a 3 .

The third law is a little different from the other two in that it is a mathematical formula, T 2 is proportional to a 3 , which relates the distances of the planets from the Sun to their orbital periods (the time it takes to make one orbit around the Sun). T is the orbital period of the planet. The variable a is the semimajor axis of the planet’s orbit. The major axis of a planet’s orbit is the distance across the long axis of the elliptical orbit. The semimajor axis is half of that. When dealing with our solar system, a is usually expressed in terms of astronomical units (equal to the semimajor axis of Earth’s orbit), and T is usually expressed in years. For Earth, that means a 3 / T 2 is equal to 1. For Mercury, the closest planet to the Sun, its orbital distance, a , is equal to 0.387 astronomical unit, and its period, T , is 88 days, or 0.241 year. For that planet, a 3 / T 2 is equal to 0.058/0.058, or 1, the same as Earth.

Kepler proposed the first two laws in 1609 and the third in 1619, but it was not until the 1680s that Isaac Newton explained why planets follow these laws. Newton showed that Kepler’s laws were a consequence of both his laws of motion and his law of gravitation .

Sciencing_Icons_Science SCIENCE

Sciencing_icons_biology biology, sciencing_icons_cells cells, sciencing_icons_molecular molecular, sciencing_icons_microorganisms microorganisms, sciencing_icons_genetics genetics, sciencing_icons_human body human body, sciencing_icons_ecology ecology, sciencing_icons_chemistry chemistry, sciencing_icons_atomic & molecular structure atomic & molecular structure, sciencing_icons_bonds bonds, sciencing_icons_reactions reactions, sciencing_icons_stoichiometry stoichiometry, sciencing_icons_solutions solutions, sciencing_icons_acids & bases acids & bases, sciencing_icons_thermodynamics thermodynamics, sciencing_icons_organic chemistry organic chemistry, sciencing_icons_physics physics, sciencing_icons_fundamentals-physics fundamentals, sciencing_icons_electronics electronics, sciencing_icons_waves waves, sciencing_icons_energy energy, sciencing_icons_fluid fluid, sciencing_icons_astronomy astronomy, sciencing_icons_geology geology, sciencing_icons_fundamentals-geology fundamentals, sciencing_icons_minerals & rocks minerals & rocks, sciencing_icons_earth scructure earth structure, sciencing_icons_fossils fossils, sciencing_icons_natural disasters natural disasters, sciencing_icons_nature nature, sciencing_icons_ecosystems ecosystems, sciencing_icons_environment environment, sciencing_icons_insects insects, sciencing_icons_plants & mushrooms plants & mushrooms, sciencing_icons_animals animals, sciencing_icons_math math, sciencing_icons_arithmetic arithmetic, sciencing_icons_addition & subtraction addition & subtraction, sciencing_icons_multiplication & division multiplication & division, sciencing_icons_decimals decimals, sciencing_icons_fractions fractions, sciencing_icons_conversions conversions, sciencing_icons_algebra algebra, sciencing_icons_working with units working with units, sciencing_icons_equations & expressions equations & expressions, sciencing_icons_ratios & proportions ratios & proportions, sciencing_icons_inequalities inequalities, sciencing_icons_exponents & logarithms exponents & logarithms, sciencing_icons_factorization factorization, sciencing_icons_functions functions, sciencing_icons_linear equations linear equations, sciencing_icons_graphs graphs, sciencing_icons_quadratics quadratics, sciencing_icons_polynomials polynomials, sciencing_icons_geometry geometry, sciencing_icons_fundamentals-geometry fundamentals, sciencing_icons_cartesian cartesian, sciencing_icons_circles circles, sciencing_icons_solids solids, sciencing_icons_trigonometry trigonometry, sciencing_icons_probability-statistics probability & statistics, sciencing_icons_mean-median-mode mean/median/mode, sciencing_icons_independent-dependent variables independent/dependent variables, sciencing_icons_deviation deviation, sciencing_icons_correlation correlation, sciencing_icons_sampling sampling, sciencing_icons_distributions distributions, sciencing_icons_probability probability, sciencing_icons_calculus calculus, sciencing_icons_differentiation-integration differentiation/integration, sciencing_icons_application application, sciencing_icons_projects projects, sciencing_icons_news news.

• Share Tweet Email Print
• Home ⋅
• Science ⋅
• Physics ⋅

How Does Newton Explain Planetary Motion?

Gravity (Physics): What Is It & Why Is It Important?

The ancients believed that planets and other celestial bodies obeyed a different set of laws from ordinary physical objects on the Earth. By the 17th century, however, astronomers had realized that the Earth itself was a planet and that -- rather than being the fixed center of the universe -- it revolves around the sun like any other planet. Armed with this new understanding, Newton developed an explanation of planetary motion using the same physical laws that apply on Earth.

Sir Isaac Newton

Newton was born in Lincolnshire, England, in 1642. At the age of 27 he was appointed professor of mathematics at Cambridge University. His particular interest was the application of mathematical methods to the physical sciences. Planetary motion was one of the most hotly debated topics of the time, and Newton devoted much of his effort to developing a mathematical theory of this. The result was his law of universal gravitation, which was first published in 1687.

The Motion of the Planets

In Newton's time, everything that was known about planetary motion could be summarized succinctly in three laws attributed to Johannes Kepler. The first law states that planets move around the sun on elliptical orbits. The second law states that a planet sweeps out equal areas in equal times. According to the third law, the square of the orbital period is proportional to the cube of the distance to the sun. These are purely empirical laws, however. They describe what happens without explaining why it happens.

Newton's Approach

Newton was convinced the planets must obey the same physical laws that are observed on Earth. This meant there must be an unseen force acting on them. He knew from experiment that, in the absence of an applied force, a moving body will continue in a straight line forever. The planets, on the other hand, were moving in elliptical orbits. Newton asked himself what sort of force would make them do this. In a stroke of genius, he realized that the answer was gravity -- the very same force that causes an apple to fall to the ground on Earth.

Universal Gravitation

Newton developed a mathematical formulation of gravity that explained both the motion of a falling apple and that of the planets. He showed that the gravitational force between any two objects is proportional to the product of their masses and inversely proportional to the square of the distance between them. When applied to the motion of a planet around the sun, this theory explained all three of Kepler's empirically derived laws.

Related Articles

How did isaac newton discover the laws of motion, list of discoveries of galileo galilei, what is inertia, the two forces that keep the planets in motion around..., two types of planetary motion, the discovery of gravity & the people who discovered..., what are the causes of perturbations discovered in..., short summary of ptolemy's discoveries, galileo galilei's invention & contributions, why the earth rotates around the sun, heliocentric model of the solar system facts, what are some examples of the laws of motion, definition of elliptical orbits, in ancient greece what shape was the earth believed..., what is the shape of earth's orbit, facts for kids about galileo, newton's laws of motion.

• University of Nebraska: Newton and Planetary Motion
• University of Texas: Kepler, Newton, and Laws of Motion
• University of Oregon: From Galileo to Newton: Physics Emerges

About the Author

Andrew May has more than 25 years of experience in academia, government and the private sector. A full-time author since 2011, he wrote "Bloody British History: Somerset" and "Pocket Giants: Isaac Newton" (to be published in 2015). He is a regular contributor to "Fortean Times" magazine, and also contributed to "30-Second Quantum Theory." May holds a Master of Arts in natural sciences from Cambridge University and a Ph.D. in astrophysics.

Photo Credits

Hulton Archive/Hulton Archive/Getty Images

Find Your Next Great Science Fair Project! GO

Empowering minds, fueling futures.

Immersive courses. Engaged faculty. Inspired scholars. These are the heart of Miami University’s diverse arts and science curriculum. The College of Arts and Science (CAS) fosters a climate that promotes intellectual discovery and personal development through the 60+ undergraduate majors, dozens of minors, and 40+ graduate degree programs. Whatever your passion, CAS will equip you to obtain a great career that enables you to live your dreams, enact meaningful change, and become the person you desire to be.

Why Arts and Science?

Explore the Scope and Diversity of Our Majors and Opportunities. Prepare for a Fulfilling Career.

Natural and biological sciences.

CAS offers a wealth of immersive programs spanning biological science, physical and environmental science, data science, and mathematics. These areas incorporate the extraction of data to examine the fundamentals of how things work, exist, and interact within both the known and theoretical universe, at all scales – from the astronomical to the nano.

Starting your first year, your professors will encourage you to join their research teams , where you can make real contributions not only in the lab but also at major conferences and in scholarly journals. These experiences will give you a strong advantage for professional or graduate school acceptance as well as rewarding careers in both the private and public sectors.

But your opportunities don’t stop there. You will find plenty of practical, career-focused experiences in such diverse offerings as our pre-healthcare preceptorships , summer geological field camps , the annual spring DataFest , and numerous internships, scholarships, and study abroad programs.

Social Sciences

The wide variety of social science programs at CAS will plunge you into the roots of human nature and society. Our students are interested in exploring individual interpersonal interactions, analyzing the behaviors of social sub-groups, and understanding larger cultural and societal norms. 

For example, Speech Pathology and Audiology majors study communication disorders, language development, and hearing. Psychology majors focus on human behavior, brain development, thought, and feelings. And Political Science majors examine issues of local and national governance, diplomacy and global politics, and social and political change. 

Our experiential learning opportunities include research, study away/abroad, internships, and scholarships. From the “inside the beltway” internship experiences at Inside Washington , to the guidance and networking resources provided by the Sue J. Henry Center for Pre-Law Education , to the on-the-ground, real-world impact of the Social Justice Internship Program , social science majors find numerous pathways to exciting careers.

Humanities students in CAS delve into a vast range of fields devoted to the examination of the vibrancy of the human experience, how it is expressed, and the ways in which the intricacies of human values have shaped the world around us. 

Whether you’re tackling a world language, analyzing the origin of ancient texts, understanding how people consume various media, exploring the perspectives of global cultures, or even  designing your own course of study , the humanities will guide you along a promising career path and sense of purpose.  They also pair well  with majors in business, science and technology, healthcare, and much more. 

Engage in research with top-notch faculty and participate in numerous resources and events with the Miami University Humanities Center . Beyond our wide variety of study abroad opportunities , you’ll discover our popular “Inside” career-focused internships in Hollywood, Washington, New York, London, and Chicago, all of which provide the chance to network with successful Miami alumni.

Latest CAS News

Miami University student spearheads Ohio’s rural renaissance

Students interning at Ohio newspapers add voices to the Opinion pages

Upcoming Events in CAS

See More Events

Submit an Event

Meet Some of our Accomplished Students

Siena Pack '25

Political Science • History

Babs Dwyer '25

Political Science • Media, Journalism, and Film

Sofia Rebull '25

Sociology and Gerontology • LSAMP • Biology • Honors College • Mallory Wilson Center for Healthcare Education

Where Passion and Talent Create Change and Opportunity

Discover why arts and science graduates are sought by top employers.

undergraduates doing research in experiential environments alongside leading faculty

of May 2022 graduates are employed or enrolled in graduate school by fall

of 2022-23 senior Miami applicants were accepted to law school (compared to 80% national average for the same period)

of first-time applicants were accepted into medical school; around 20% higher than national average

Meet Our Teacher-Scholars 

Steven conn.

W.E. Smith Professor History College of Arts and Science

" What I most enjoy about teaching at Miami is the extent to which I am able to experiment with new methods and to try out new ideas with students who are keen to try new things too. I have the flexibility to explore things that I think students will find interesting or which are driving my own scholarly interests at a given moment. I think my scholarship has enriched my teaching but I know beyond a doubt that teaching Miami students has enriched my scholarship. "

Rodney Coates

Professor Global and Intercultural Studies College of Arts of Science

"As a teacher-scholar, I actively engage my students in research, knowledge production, and critical reflection. Miami University encourages excellence in its students and its faculty. It is a place where the teacher-scholar model becomes how we foster critical inquiry and maximize student achievement. At Miami, it is not through indoctrination and memorization, but student engagement and scholarship by which learning is accomplished."

Renate Crawford

Adjunct Professor Physics College of Arts of Science

" The teacher-scholar model enables me to lift complicated concepts out of the textbook and show real-world examples to reinforce the underlying physics. Students are introduced to contemporary translational and applied research as they learn the basic principles of physics. Faculty at Miami are both teachers and scholars, transforming their classrooms through research, creating a vibrant teaching and learning environment, introducing students to what researchers and scholars are thinking about in their disciplines. "

Patrick Haney

Professor Political Science College of Arts and Science

Cameron Hay-Rollins

Chair and Professor Anthropology College of Arts and Science

"Research is an adventure. As an anthropologist, I seek to understand how people cope with suffering, illness, and inequity.  Whether in Indonesia, India, or Indiana, I venture into the unknowns of research to find evidence of how we might promote resilience, well-being, and equity in the most vulnerable moments of people’s lives. Research is an exciting activity, and Miami’s teacher-scholar model allows me to invite students into scholarly conversations and the surprising joy of pursuing research adventures."

Associate Professor English College of Arts and Science

" As faculty, it is so invigorating to find that Miami students are reflective and active participants in their learning experiences, rather than just sitting back and passively consuming the materials I’ve provided. Together, we all strive to create a knowledge-making community in the classroom, where I find myself both teaching and constantly learning from students in ways that grows my own scholarship. By challenging ourselves to become better teacher-scholars, we all can create spaces where community knowledge can happen and inclusive teaching begins. "

Wayne Nirode

Assistant Professor Math College of Arts and Science

“When I began thinking about leaving high school for the next phase of my professional career, it was important for me to go to a university that valued both teaching and research. To me, this is at the heart of the teacher-scholar model. After being at Miami University for six years, I do not think that I could have found a university that better aligns with my views on teaching and researching — in short the teacher-scholar model. I am proud to say that Miami is my professional home.”

Arnold Olszewski

Associate Professor Speech Pathology and Audiology College of Arts and Science

"As an instructor, my goal is to help future speech-language pathologists develop assessment and intervention plans using evidence-based methods. As a researcher, my goal is to work with students to develop, evaluate, and revise techniques for teaching early literacy skills. What I love about Miami is the strong emphasis on both teaching and research. Many schools prioritize one over the other, but I believe these go hand-in-hand. My favorite thing is getting a student who is scared of research to develop their own project and see the excitement they feel when they get their results."

Rosemary Pennington

Associate Professor, Journalism Area Coordinator Media, Journalism, and Film College of Arts and Science

Vaishali Raval

Professor Psychology College of Arts and Science

Elizabeth Wardle

Roger and Joyce Howe Distinguished Professor of Written Communication and Director English, Howe Center for Writing Excellence

Ellen Yezierski

Professor and Director Chemistry and Biochemistry, Center for Teaching Excellence College of Arts and Science

University Distinguished Professor Biology College of Arts and Science

"In biology at Miami, we engage students in the classroom, in the laboratory, and in the field. Our undergraduates have the rare opportunity to conduct novel and timely research under the guidance of talented graduate students and passionate faculty. Our research on freshwater invertebrates both guides and supports conservation actions by numerous state and federal agencies."

The College of Arts and Science

Representing nearly half of all students, the College of Arts and Science (CAS) is Miami University's largest division and the centerpiece of liberal arts — the wide range of subjects in the natural sciences, social sciences, and humanities crucial for the development of key professional skills desired by employers.

100 Bishop Circle 143 Upham Hall Miami University Oxford, Ohio 45056 [email protected] 513-529-1234

Related Links

• CAS Advising
• CAS Inclusive Excellence
• Give to CAS
• Alumni Resources
• Faculty Resources  
• CAS Web Change or Profile Requests

501 E. High Street Oxford, OH 45056

• Online: Miami Online
• Main Operator 513-529-1809
• Office of Admission 513-529-2531
• Vine Hotline 513-529-6400
• Emergency Info https://miamioh.edu/emergency

1601 University Blvd. Hamilton, OH 45011

• Online: E-Campus
• Main Operator 513-785-3000
• Office of Admission 513-785-3111
• Campus Status Line 513-785-3077
• Emergency Info https://miamioh.edu/regionals/emergency

4200 N. University Blvd. Middletown, OH 45042

• Main Operator 513-727-3200
• Office of Admission 513-727-3216
• Campus Status 513-727-3477

7847 VOA Park Dr. (Corner of VOA Park Dr. and Cox Rd.) West Chester, OH 45069

• Main Operator 513-895-8862
• From Middletown 513-217-8862

Chateau de Differdange 1, Impasse du Chateau, L-4524 Differdange Grand Duchy of Luxembourg

• Main Operator 011-352-582222-1
• Email [email protected]
• Website https://miamioh.edu/luxembourg

217-222 MacMillan Hall 501 E. Spring St. Oxford, OH 45056, USA

• Main Operator 513-529-8600

Initiatives

• Miami THRIVE Strategic Plan
• Miami Rise Strategic Plan
• Boldly Creative
• Annual Report
• Moon Shot for Equity
• Miami and Ohio
• Majors, Minors, and Programs
• Inclusive Excellence
• Employment Opportunities
• University Safety and Security
• Parking, Directions, and Maps
• Equal Opportunity
• Consumer Information
• Land Acknowledgement
• Privacy Statement
• Title IX Statement
• Report an Accessibility Issue
• Annual Security and Fire Safety Report
• Report a Problem with this Website
• Policy Library

Kepler's Three Laws of Motion ( OCR A Level Physics )

Revision note.

Kepler's Three Laws of Motion

Kepler's first law.

• Kepler's First Law describes the shape of planetary orbits
• It states: 

The orbit of a planet is an ellipse, with the Sun at one of the two foci

The orbit of all planets are elliptical, and with the Sun at one focus

• Some planets, like Pluto, have highly elliptical orbits around the Sun
• Other planets, like Earth, have near circular orbits around the Sun

Kepler's Second Law

• Kepler's Second Law describes the motion of all planets around the Sun

A line segment joining the Sun to a planet sweeps out equal areas in equal time intervals

• The consequence of Kepler's Second Law is that planets move faster nearer the Sun and slower further away from it

Kepler's Third Law

• Kepler's Third Law describes the relationship between the  time of an orbit and its  radius

The square of the orbital time period T is directly proportional to the cube of the orbital radius  r

• Kepler's Third Law can be written mathematically as:
• Which becomes:
• T = orbital time period (s)
• r = mean orbital radius (m)
• k = constant (s 2 m –3 )
• In the case of our solar system,  k is constant for all planets orbiting the Sun 

You are expected to be able to describe Kepler's Laws of Motion, so make sure you are familiar with how they are worded. 

Applications of Kepler's Third Law

• Kepler's Third Law, the fact that  T 2 ∝  r 3 applies to  any body in orbit about some larger body
• The moons orbiting other planets, like the four moons of Jupiter (Io, Europa, Callisto and Ganymede)
• Exoplanets in orbit about foreign stars
• Therefore,  useful  and  interesting data about the  mass of orbital systems can be deduced from experimental data

You've read 0 of your 10 free revision notes

Unlock more, it's free, join the 100,000 + students that ❤️ save my exams.

the (exam) results speak for themselves:

Did this page help you?

• Planetary Motion
• Gravitational Potential & Energy
• Stellar Evolution
• EM Radiation From Stars
• Capacitors in Circuits
• Charging & Discharging Capacitors
• Electric Fields
• Electric Potential & Energy
• Magnetic Fields

Author: Katie M

Katie has always been passionate about the sciences, and completed a degree in Astrophysics at Sheffield University. She decided that she wanted to inspire other young people, so moved to Bristol to complete a PGCE in Secondary Science. She particularly loves creating fun and absorbing materials to help students achieve their exam potential.

Physics Bootcamp

Samuel J. Ling

Section 12.1 Kepler's Laws of Planetary Motion

Subsection 12.1.1 math of ellipse.

Ellipse plays central role in planetary motion. Here, we review some of the basics of ellipse.

Ellipse is an oval-shaped planar curve with two focal points \(F_1\) and \(F_2\) as shown in Figure 12.1.1 . The sum of the distances \(d_1\) and \(d_2\) from the focal points to the points of the ellipse is constant,

Let \(c\) denote the distance of a focus from the center, \(a\) the distance from the center to the end of the wider side, and \(b\) the distance from the center to the end of the narrower side. The quantities \(a\) and \(b\) are also called semi-major and semi-minor axes, and quantity \(c\) is called the focal distance. They are related by

The ratio \(c/a\) is called the eccentricity and denoted by \(e\text{.}\)

Ellipse in Cartesian Form

Suppose we place an ellipse with wider side along \(x\) axis and the narrower side along \(y\) axis. This will place the center of the ellipse at the origin. The foci will be symmetrically located on \(x\) axis at \((-c,0)\) and \((c,0)\text{.}\) The definition of ellipse says that any point \((x,y)\) on ellipse must obey

The points \((a,0)\) and \((-a,0)\) will be the end points of the ellipse on the \(x\) axis. Then,

By appropriately rearranging terms in Eq. (12.1.4) and squaring to remove radicals, and then using Eq. (12.1.1) we can obtain the standard form of equation of ellipse.

Ellipse in Polar Form With Origin at Focus

Sometimes, it is better to work with polar form of an ellipse that has origin at the focus and not at the center. Let \(r \) denote the radial coordinate and \(\theta\) counterclockwise from the \(x\) axis that is along the major axis. Then, it is possible to show that if origin is at the focal point on the right then

and if origin is at the focal point on the left then

If the origin was not at one of the foci, but at the center of the ellipse, we would get

Finally, we will find the formula of the area of an ellipse useful.

Subsection 12.1.2 Kepler's Laws

Johannes Kepler (1571-1630) analyzed the voluminous data of planetary positions recorded by Tycho Brahe (1546-1601) and deduced the following three laws of planetary motion. A good place to get planetary data now a days is at the NASA website .

First Law or the Law of Elliptical Orbits

Planets travel in elliptical orbits about the Sun with the Sun at one focus.

Second Law or the Law of Equal Areas

The line joining the Sun and a planet covers equal area in equal time. Thus when a planet is nearer to the Sun, it has a higher speed than when it is further out. Figure 12.1.3 illustrates this law.

Third Law or the Law of Harmonies

The ratio of square of the period of revolution about the Sun to the cube of the semi major axis of the elliptical orbit of two planets are equal to each other. Thus if \(T_1\) and \(T_2\) are time for planets 1 and 2 to go around the Sun, and \(a_1\) and \(a_2\) the semi-major axes of their elliptical orbits, we have the following equality (see Figure 12.1.4 ).

Checkpoint 12.1.5 . Earth's Orbit Around the Sun.

The closest and farthest distance of the Earth from the Sun are 0.98 AU and 1.02 AU, where, the Astronomical Unit, 1 AU = 149,598,000 km. Find (a) the semi-major axis, (b) the eccentricity of Earth's orbit, and (c) the semi-minor axis.

\(2a =\) sum of the two distances given.

(a) \(1.0\text{ AU}\text{,}\) (b) \(0.02\text{,}\) (c) \(b=0.9998\text{ AU}\approx 1.00\text{ AU}\text{.}\)

(a) We will refer to the following figure.

From the figure, we see that

(b) Therefore, the focal distance is

From \(a\) and \(c\) we get the eccentricity of the orbit.

Therefore, \(b=0.9998\text{ AU}\approx 1.00\text{ AU}\) = a. Since \(a\approx b\text{,}\) the orbit of Earth is very close to circular.

Checkpoint 12.1.6 . Mar's Orbit Around the Sun.

The closest and farthest distance of Mars from the Sun are \(1.38\text{ AU}\) and \(1.67\text{ AU}\text{,}\) where, the Astronomical Unit, \(1\text{ AU} = 149,598,000\text{ km}\text{.}\) Data: Semimajor axis of Earth's orbit = \(1.0\text{ AU}\) and orbital period of Earth = \(365.2\text{ days}\text{.}\)

Find (a) the semi-major axis, (b) the eccentricity , and (c) the period of Mars's orbit.

The semi-major axis of Earth is \(1.0\text{ AU}\) and the orbital period is \(365.2\text{ days}\)

(a) \(1.0\text{ AU}\text{,}\) (b) \(0.02\text{,}\) (c) \(687.8\text{ days} \text{.}\)

We will refer to the following figure.

(a) From the figure, we see that

(c) From the semi-major axis of Mars and the semi-major axis and orbital period of Earth, we can find the orbital period of Mars by using Kepler's third law.

Checkpoint 12.1.7 . Orbit of Haley's Comet.

The orbit of Halley's comet is approximately elliptical with \(e=0.967\text{.}\) Halley's comet comes around every 76 years.

Find (a) the distance of the closest approach to the Sun and (b) the farthest distance the comet goes from the Sun.

Data: The semi-major axis of Earth is \(1.0\text{ AU}\) and the orbital period is \(365.2\text{ days}\text{,}\) where \(1\text{ AU} = 149,598,000\text{ km}\text{.}\)

(a) and (b) : Use third law to find \(a\) of Haley's comet first.

(a) \(0.59\text{ AU}\text{,}\) (b) \(35.29\text{ AU}\text{.}\)

First we use Kepler's third law on orbits of Haley's comet and Earth to obtain the semi-major axis of Heley's comet.

Now, we use the definition of eccentricity to find the focal distance.

Now, refer to the figure to obtain the min and max distances.

Checkpoint 12.1.8 . Predict the Orbital Period of Mercury From Aphelion and Perihelion Data on Earth and Mercury.

The perihelion and aphelion distances of Mercury are \(46.0\) and \(69.8 \) in units of \(10^{6}\text{ km}\text{,}\) and that of Earth are \(147.1\) and \(152.1 \) in the same units. What is the orbital period of Mars in Earth days if the orbital period of Earth is \(365.2\text{ days}\text{?}\) Compare your answer to the listed value of \(88.0\text{ days}\text{.}\)

Use Kepler's third law.

\(87.9 \text{ days}\text{.}\)

We can use Kepler's third law.

Here, the semimajor axis \(a\) is the average of the perihelion and aphelion distances. We don't even have to average them, we can just add them, since the dividing factor 2 will be canceled in the ratio.

This is impressively close to \(88.0\text{ days}\text{.}\)

Checkpoint 12.1.9 . Discovering Kepler's Third Law from Modern Data.

The distances of various planets from Sun and their orbital periods are given in NASA website as follows. (Unit of distance is \(10^6\text{ km} \) and the unit of time is Earth \(\text{day}\text{.}\))

The list order is [Mercury, Venus, Mars, Jupiter, Saturn, Uranus, Neptune, Pluto]. The orbital periods are [88.0, 224.7, 365.2, 687.0, 4331, 10747, 30589, 59800, 90560] and the distances from the Sun are [57.9, 108.2, 149.6, 227.9, 778.6, 1433.5, 2872.5, 4495.1, 5906.4].

You suspect that \(T \propto r^n\text{,}\) where \(T\) is the period and \(r\) the distance. But you don't know \(n\) yet. We pretend we don't know - since this has already been found by Kepler, but here we will find it our own way from the data. Find \(n \) from the given data.

Plot \(\ln T\) versus \(\ln r\text{.}\)

\(n = 1.50\text{.}\)

Since we suspect \(T\propto r^n \text{,}\) we make it an equality first and set

where \(c\) is an unknown constant - at the moment of no interest. By taking natural log of both sides you get

Thus if we plot \(\ln T\) on the \(y\)-axis and \(\ln r\) on the \(x\)-axis, and fit the data to a straight line, the slope of this line will give us \(n\text{.}\) The following figure shows the calculations and result done in Microsoft Excel.

Subsection 12.1.3 Kepler's Impact on the Development of Physics

Kepler's three laws played critical role in the discovery of the universal law of gravitation, and had major impact on the acceptance of science by the general public. By 1666 Newton had early versions of his three laws of motion, but yet did not have the law of gravitation. In that year he had an insight that Earth's gravity extended also to the Moon and was counterbalanced by the centrifugal force. Newton used the balance of the two forces to find that the gravitational force must decrease as inverse square of the distance. The calculation was, however, only for a circular motion.

Newton did not work on the planetary motion problem any further till 1679 when another physicist, Robert Hooke, which you have met in the Hooke's law of the spring force, went to see him about the elliptical orbit problem of the planets. Hooke had conjectured that a planet moving in an ellipse must be acted on by a central force by the Sun. Hooke had also come to the conclusion that this force must vary as the square of the inverse of the distance of the planet from the Sun, but he could not prove his conjectures mathematically.

After Hooke's visit, Newton went back to work on the planetary motion problem. First he showed mathematically that, if a body obeys Kepler's second law, then the force on the body must be central. Isaac Newton also showed that the angular momentum of a body is conserved if the body is acted upon by a central force. This finding demonstrated the physical basis of Kepler's second law.

Next, Newton showed that if a body is in an elliptical path, then the force must be pointed towards one of the foci and must vary as the square of the inverse of the distance from the focus. However, Newton did not publish any of these results until after the great Astronomer, Edmund Halley (1656-1742), asked him in 1684 if he could prove Hooke's conjecture. To Haley's surprise, Newton immediately replied that he had proved it five years earlier. Upon Halley's insistence Newton wrote up his treatise on mechanics and its applications to the celestial mechanics. This work is called the Philosophiae naturalis principia mathematica, or Principia and was published in 1687.

Halley used Newtonian calculations to ascertain that comets appearing in the sky in 1531, 1607 and 1682 were the same object and it was regularly appearing every 76 years. Sure enough the comet, now called Halley's comet, arrived on Christmas day in 1758 as predicted by Edmund Halley. The last time Halley's comet observable on Earth was in 1986 shown in Figure 12.1.10 .

Academia.edu no longer supports Internet Explorer.

To browse Academia.edu and the wider internet faster and more securely, please take a few seconds to  upgrade your browser .

Enter the email address you signed up with and we'll email you a reset link.

• We're Hiring!
• Help Center

Kepler's laws of planetary motion, before and after Newton's Principia: An essay on the transformation of scientific problems

1987, Studies in History and Philosophy of Science Part A

Related Papers

Frans Van Lunteren

F. van Lunteren, Framing hypotheses: conceptions of gravity in the 18th and 19th centuries (doctoral dissertation, Utrecht 1991)

Chris Smeenk

Contribution to the Oxford Handbook of the History of Physics (draft; please refer to published version).

Studies in History and Philosophy of Science Part A

Brian Baigrie

Herb Spencer

The history of modern science and its metaphysical assumptions; these were the two lifetime obsessions of Edwin Arthur Burtt, an academic historian of the first half of the 20th century, This was one of his most influential books with a huge impact. Burtt remained sympathetic to religious experience and awe and published books, like Religion in the Age of Science, Types of Religious Philosophy and The Teachings of the Compassionate Buddha, throughout his life. While completing his dissertation in philosophy at Columbia University, Burtt chose to focus his research on the pioneers of the Scientific Revolution. Burtt argued that ideas like human consciousness, lifetime purposes and religious aspirations do not fit into the mechanical worldview created by men like Copernicus, Galileo, Kepler, Boyle and especially Newton. Burtt showed that these scientists' work cannot be clearly separated out from the work of philosophers like Hobbes, Locke and Descartes. While his dissertation argued that this scientific philosophy has a positivist streak, Burtt also described and explored the rich philosophical contributions made by these men. He showed that metaphysical areas like epistemology, the philosophy of mathematics and the philosophy of physics (in particularly, space and time) were richly explored and elucidated by almost all of these noted intellectuals. He decries the newly modern view that: " nature holds a more independent, more determinative and more permanent place than mankind " with their brief finite lives. His analysis shows how the medieval synthesis of Greek philosophy and Judeo-Christian theology was split asunder by the Scientific Revolution. Aristotle's Unmoved Mover and Christian theologians, like Thomas Aquinas, had unified the ultimate causality of the Universe via Neo-Platonism. Even the understanding of reality was more obvious to ordinary people of the Middle Ages because real objects were directly perceived by our senses, while things that appeared different were simply different substances. Contrast this with the modern scientific view of the purposeless universe, with every thing (even us) built from only invisible bits of mysterious, unfeeling matter, with a guaranteed end of everyone's life:-all we gained was an empty viewpoint.

Nicolae Sfetcu

The classic example of a successful research program is Newton's gravitational theory, probably the most successful Lakatosian research program. Initially, Newton's gravitational theory faced a lot of "anomalies" ("counterexamples") and contradicted the observational theories that supported these anomalies. But supporters of the Newtonian gravity research program have turned every anomaly into corroborating cases. Moreover, they themselves pointed to counterexamples which they then explained through Newtonian theory . DOI: 10.13140/RG.2.2.32489.85601

Peter Harrison

THE SHORT HISTORY OF SCIENCE - or the long path to the union of metaphysics and empiricism

Tuomo Suntola

This book reviews the development of the natural sciences and the picture of reality produced by science from the philosophical conceptions in antique metaphysics to the picture outlined from the mathematical models produced by empirical research in modern physics. Also, the book studies the possibility of a re-evaluation of the picture of reality from a holistic perspective with a closer connection between phil-osophical and empirical aspects.

Paolo Casini

Leibniz' criticism of Newton's Principia during and after his stay in Rome (1684), his attitude concerning the Papal ban against Copernicus' system of the world, and his new science of "Dynamics" as an antidote against the corpuscular philosophy

Loading Preview

Sorry, preview is currently unavailable. You can download the paper by clicking the button above.

RELATED PAPERS

Edward Remler

Ayyadevara Ravishankar Sarma

Eric Schliesser

Luciano Boschiero

Encyclopedia of Early Modern Philosophy and the Sciences

Ori Belkind

Reading Newton in Early Modern Europe

Marius Stan

Paolo Pecere

Jeffrey Symonette

Biographical Encyclopedia of Astronomers

Alexander Gurshtein

Candra Lesmana

Hylarie KOCHIRAS

CPMMI (to be significantly shortened for)

Slobodan Nedic

Pacific Philosophical Quarterly

Matias Slavov

Francis Joseph

Historical Studies in the Natural Sciences

Massimiliano Badino

Robert Rynasiewicz

Kant Yearbook

Kenneth R Westphal

Shahriar Kiarash

Panagiotis Papaspirou

Karin Verelst

Charles Stevens

Ridwan Hidayat

Zvi Biener , Chris Smeenk

A. Bowdoin Van Riper

The Science of Nature in the 17th Century: Patterns of Change in Early Modern Natural Philosophy (2005, Springer), eds. P. Anstey and J. Schuster

John Schuster

The Divine Order, the Human Order, and the Order of Nature: Historical Perspectives, ed. Eric Watkins (New York: Oxford University Press, 2013), pp. 127-48.

A Companion to Kant, ed. Graham Bird (Oxford: Blackwell), 40-60

Martin Schonfeld

Mario G Acosta Alarcon

Philip Catton

The Oxford Handbook of the History of Physics (Oxford, OUP), pp.56-95 , eds. Buchwald and Fox

Studies in History and Philosophy of Science

Þorsteinn Vilhjálmsson

otago.ac.nz

Stanford Encyclopedia of Philosophy. Later updates authored by Michael Thompson.

Martin Schonfeld , Michael Thompson

RELATED TOPICS

•   We're Hiring!
•   Help Center
• Find new research papers in:
• Health Sciences
• Earth Sciences
• Cognitive Science
• Mathematics
• Computer Science
• Academia ©2024

• TPC and eLearning
• What's NEW at TPC?
• Read Watch Interact
• Practice Review Test
• Teacher-Tools
• Request a Demo
• Get A Quote
• Subscription Selection
• Seat Calculator
• Ad Free Account
• Edit Profile Settings
• Metric Conversions Questions
• Metric System Questions
• Metric Estimation Questions
• Significant Digits Questions
• Proportional Reasoning
• Acceleration
• Distance-Displacement
• Dots and Graphs
• Graph That Motion
• Match That Graph
• Name That Motion
• Motion Diagrams
• Pos'n Time Graphs Numerical
• Pos'n Time Graphs Conceptual
• Up And Down - Questions
• Balanced vs. Unbalanced Forces
• Change of State
• Force and Motion
• Mass and Weight
• Match That Free-Body Diagram
• Net Force (and Acceleration) Ranking Tasks
• Newton's Second Law
• Normal Force Card Sort
• Recognizing Forces
• Air Resistance and Skydiving
• Solve It! with Newton's Second Law
• Which One Doesn't Belong?
• Component Addition Questions
• Head-to-Tail Vector Addition
• Projectile Mathematics
• Trajectory - Angle Launched Projectiles
• Trajectory - Horizontally Launched Projectiles
• Vector Addition
• Vector Direction
• Which One Doesn't Belong? Projectile Motion
• Forces in 2-Dimensions
• Being Impulsive About Momentum
• Explosions - Law Breakers
• Hit and Stick Collisions - Law Breakers
• Case Studies: Impulse and Force
• Impulse-Momentum Change Table
• Keeping Track of Momentum - Hit and Stick
• Keeping Track of Momentum - Hit and Bounce
• What's Up (and Down) with KE and PE?
• Energy Conservation Questions
• Energy Dissipation Questions
• Energy Ranking Tasks
• LOL Charts (a.k.a., Energy Bar Charts)
• Match That Bar Chart
• Words and Charts Questions
• Name That Energy
• Stepping Up with PE and KE Questions
• Case Studies - Circular Motion
• Circular Logic
• Forces and Free-Body Diagrams in Circular Motion
• Gravitational Field Strength
• Universal Gravitation
• Angular Position and Displacement
• Linear and Angular Velocity
• Angular Acceleration
• Rotational Inertia
• Balanced vs. Unbalanced Torques
• Getting a Handle on Torque
• Torque-ing About Rotation
• Properties of Matter
• Fluid Pressure
• Buoyant Force
• Sinking, Floating, and Hanging
• Pascal's Principle
• Flow Velocity
• Bernoulli's Principle
• Balloon Interactions
• Charge and Charging
• Charge Interactions
• Charging by Induction
• Conductors and Insulators
• Coulombs Law
• Electric Field
• Electric Field Intensity
• Polarization
• Case Studies: Electric Power
• Know Your Potential
• Light Bulb Anatomy
• I = ∆V/R Equations as a Guide to Thinking
• Parallel Circuits - ∆V = I•R Calculations
• Resistance Ranking Tasks
• Series Circuits - ∆V = I•R Calculations
• Series vs. Parallel Circuits
• Equivalent Resistance
• Period and Frequency of a Pendulum
• Pendulum Motion: Velocity and Force
• Energy of a Pendulum
• Period and Frequency of a Mass on a Spring
• Horizontal Springs: Velocity and Force
• Vertical Springs: Velocity and Force
• Energy of a Mass on a Spring
• Decibel Scale
• Frequency and Period
• Closed-End Air Columns
• Name That Harmonic: Strings
• Rocking the Boat
• Wave Basics
• Matching Pairs: Wave Characteristics
• Wave Interference
• Waves - Case Studies
• Color Addition and Subtraction
• Color Filters
• If This, Then That: Color Subtraction
• Light Intensity
• Color Pigments
• Converging Lenses
• Curved Mirror Images
• Law of Reflection
• Refraction and Lenses
• Total Internal Reflection
• Who Can See Who?
• Lab Equipment
• Lab Procedures
• Formulas and Atom Counting
• Atomic Models
• Bond Polarity
• Entropy Questions
• Cell Voltage Questions
• Heat of Formation Questions
• Reduction Potential Questions
• Oxidation States Questions
• Measuring the Quantity of Heat
• Hess's Law
• Oxidation-Reduction Questions
• Galvanic Cells Questions
• Thermal Stoichiometry
• Molecular Polarity
• Quantum Mechanics
• Balancing Chemical Equations
• Bronsted-Lowry Model of Acids and Bases
• Classification of Matter
• Collision Model of Reaction Rates
• Density Ranking Tasks
• Dissociation Reactions
• Complete Electron Configurations
• Elemental Measures
• Enthalpy Change Questions
• Equilibrium Concept
• Equilibrium Constant Expression
• Equilibrium Calculations - Questions
• Equilibrium ICE Table
• Intermolecular Forces Questions
• Ionic Bonding
• Lewis Electron Dot Structures
• Limiting Reactants
• Line Spectra Questions
• Mass Stoichiometry
• Measurement and Numbers
• Metals, Nonmetals, and Metalloids
• Metric Estimations
• Metric System
• Molarity Ranking Tasks
• Mole Conversions
• Name That Element
• Names to Formulas
• Names to Formulas 2
• Nuclear Decay
• Particles, Words, and Formulas
• Periodic Trends
• Precipitation Reactions and Net Ionic Equations
• Pressure Concepts
• Pressure-Temperature Gas Law
• Pressure-Volume Gas Law
• Chemical Reaction Types
• Significant Digits and Measurement
• States Of Matter Exercise
• Stoichiometry Law Breakers
• Stoichiometry - Math Relationships
• Subatomic Particles
• Spontaneity and Driving Forces
• Gibbs Free Energy
• Volume-Temperature Gas Law
• Acid-Base Properties
• Energy and Chemical Reactions
• Chemical and Physical Properties
• Valence Shell Electron Pair Repulsion Theory
• Writing Balanced Chemical Equations
• Mission CG1
• Mission CG10
• Mission CG2
• Mission CG3
• Mission CG4
• Mission CG5
• Mission CG6
• Mission CG7
• Mission CG8
• Mission CG9
• Mission EC1
• Mission EC10
• Mission EC11
• Mission EC12
• Mission EC2
• Mission EC3
• Mission EC4
• Mission EC5
• Mission EC6
• Mission EC7
• Mission EC8
• Mission EC9
• Mission RL1
• Mission RL2
• Mission RL3
• Mission RL4
• Mission RL5
• Mission RL6
• Mission KG7
• Mission RL8
• Mission KG9
• Mission RL10
• Mission RL11
• Mission RM1
• Mission RM2
• Mission RM3
• Mission RM4
• Mission RM5
• Mission RM6
• Mission RM8
• Mission RM10
• Mission LC1
• Mission RM11
• Mission LC2
• Mission LC3
• Mission LC4
• Mission LC5
• Mission LC6
• Mission LC8
• Mission SM1
• Mission SM2
• Mission SM3
• Mission SM4
• Mission SM5
• Mission SM6
• Mission SM8
• Mission SM10
• Mission KG10
• Mission SM11
• Mission KG2
• Mission KG3
• Mission KG4
• Mission KG5
• Mission KG6
• Mission KG8
• Mission KG11
• Mission F2D1
• Mission F2D2
• Mission F2D3
• Mission F2D4
• Mission F2D5
• Mission F2D6
• Mission KC1
• Mission KC2
• Mission KC3
• Mission KC4
• Mission KC5
• Mission KC6
• Mission KC7
• Mission KC8
• Mission AAA
• Mission SM9
• Mission LC7
• Mission LC9
• Mission NL1
• Mission NL2
• Mission NL3
• Mission NL4
• Mission NL5
• Mission NL6
• Mission NL7
• Mission NL8
• Mission NL9
• Mission NL10
• Mission NL11
• Mission NL12
• Mission MC1
• Mission MC10
• Mission MC2
• Mission MC3
• Mission MC4
• Mission MC5
• Mission MC6
• Mission MC7
• Mission MC8
• Mission MC9
• Mission RM7
• Mission RM9
• Mission RL7
• Mission RL9
• Mission SM7
• Mission SE1
• Mission SE10
• Mission SE11
• Mission SE12
• Mission SE2
• Mission SE3
• Mission SE4
• Mission SE5
• Mission SE6
• Mission SE7
• Mission SE8
• Mission SE9
• Mission VP1
• Mission VP10
• Mission VP2
• Mission VP3
• Mission VP4
• Mission VP5
• Mission VP6
• Mission VP7
• Mission VP8
• Mission VP9
• Mission WM1
• Mission WM2
• Mission WM3
• Mission WM4
• Mission WM5
• Mission WM6
• Mission WM7
• Mission WM8
• Mission WE1
• Mission WE10
• Mission WE2
• Mission WE3
• Mission WE4
• Mission WE5
• Mission WE6
• Mission WE7
• Mission WE8
• Mission WE9
• Vector Walk Interactive
• Name That Motion Interactive
• Kinematic Graphing 1 Concept Checker
• Kinematic Graphing 2 Concept Checker
• Graph That Motion Interactive
• Two Stage Rocket Interactive
• Rocket Sled Concept Checker
• Force Concept Checker
• Free-Body Diagrams Concept Checker
• Free-Body Diagrams The Sequel Concept Checker
• Skydiving Concept Checker
• Elevator Ride Concept Checker
• Vector Addition Concept Checker
• Vector Walk in Two Dimensions Interactive
• Name That Vector Interactive
• River Boat Simulator Concept Checker
• Projectile Simulator 2 Concept Checker
• Projectile Simulator 3 Concept Checker
• Hit the Target Interactive
• Turd the Target 1 Interactive
• Turd the Target 2 Interactive
• Balance It Interactive
• Go For The Gold Interactive
• Egg Drop Concept Checker
• Fish Catch Concept Checker
• Exploding Carts Concept Checker
• Collision Carts - Inelastic Collisions Concept Checker
• Its All Uphill Concept Checker
• Stopping Distance Concept Checker
• Chart That Motion Interactive
• Roller Coaster Model Concept Checker
• Uniform Circular Motion Concept Checker
• Horizontal Circle Simulation Concept Checker
• Vertical Circle Simulation Concept Checker
• Race Track Concept Checker
• Gravitational Fields Concept Checker
• Orbital Motion Concept Checker
• Angular Acceleration Concept Checker
• Balance Beam Concept Checker
• Torque Balancer Concept Checker
• Aluminum Can Polarization Concept Checker
• Charging Concept Checker
• Name That Charge Simulation
• Coulomb's Law Concept Checker
• Electric Field Lines Concept Checker
• Put the Charge in the Goal Concept Checker
• Circuit Builder Concept Checker (Series Circuits)
• Circuit Builder Concept Checker (Parallel Circuits)
• Circuit Builder Concept Checker (∆V-I-R)
• Circuit Builder Concept Checker (Voltage Drop)
• Equivalent Resistance Interactive
• Pendulum Motion Simulation Concept Checker
• Mass on a Spring Simulation Concept Checker
• Particle Wave Simulation Concept Checker
• Boundary Behavior Simulation Concept Checker
• Slinky Wave Simulator Concept Checker
• Simple Wave Simulator Concept Checker
• Wave Addition Simulation Concept Checker
• Standing Wave Maker Simulation Concept Checker
• Color Addition Concept Checker
• Painting With CMY Concept Checker
• Stage Lighting Concept Checker
• Filtering Away Concept Checker
• InterferencePatterns Concept Checker
• Young's Experiment Interactive
• Plane Mirror Images Interactive
• Who Can See Who Concept Checker
• Optics Bench (Mirrors) Concept Checker
• Name That Image (Mirrors) Interactive
• Refraction Concept Checker
• Total Internal Reflection Concept Checker
• Optics Bench (Lenses) Concept Checker
• Kinematics Preview
• Velocity Time Graphs Preview
• Moving Cart on an Inclined Plane Preview
• Stopping Distance Preview
• Cart, Bricks, and Bands Preview
• Fan Cart Study Preview
• Friction Preview
• Coffee Filter Lab Preview
• Friction, Speed, and Stopping Distance Preview
• Up and Down Preview
• Projectile Range Preview
• Ballistics Preview
• Juggling Preview
• Marshmallow Launcher Preview
• Air Bag Safety Preview
• Colliding Carts Preview
• Collisions Preview
• Engineering Safer Helmets Preview
• Push the Plow Preview
• Its All Uphill Preview
• Energy on an Incline Preview
• Modeling Roller Coasters Preview
• Hot Wheels Stopping Distance Preview
• Ball Bat Collision Preview
• Energy in Fields Preview
• Weightlessness Training Preview
• Roller Coaster Loops Preview
• Universal Gravitation Preview
• Keplers Laws Preview
• Kepler's Third Law Preview
• Charge Interactions Preview
• Sticky Tape Experiments Preview
• Wire Gauge Preview
• Voltage, Current, and Resistance Preview
• Light Bulb Resistance Preview
• Series and Parallel Circuits Preview
• Thermal Equilibrium Preview
• Linear Expansion Preview
• Heating Curves Preview
• Electricity and Magnetism - Part 1 Preview
• Electricity and Magnetism - Part 2 Preview
• Vibrating Mass on a Spring Preview
• Period of a Pendulum Preview
• Wave Speed Preview
• Slinky-Experiments Preview
• Standing Waves in a Rope Preview
• Sound as a Pressure Wave Preview
• DeciBel Scale Preview
• DeciBels, Phons, and Sones Preview
• Sound of Music Preview
• Shedding Light on Light Bulbs Preview
• Models of Light Preview
• Electromagnetic Radiation Preview
• Electromagnetic Spectrum Preview
• EM Wave Communication Preview
• Digitized Data Preview
• Light Intensity Preview
• Concave Mirrors Preview
• Object Image Relations Preview
• Snells Law Preview
• Reflection vs. Transmission Preview
• Magnification Lab Preview
• Reactivity Preview
• Ions and the Periodic Table Preview
• Periodic Trends Preview
• Chemical Reactions Preview
• Intermolecular Forces Preview
• Melting Points and Boiling Points Preview
• Bond Energy and Reactions Preview
• Reaction Rates Preview
• Ammonia Factory Preview
• Stoichiometry Preview
• Nuclear Chemistry Preview
• Gaining Teacher Access
• Task Tracker Directions
• Conceptual Physics Course
• On-Level Physics Course
• Honors Physics Course
• Chemistry Concept Builders
• All Chemistry Resources
• Users Voice
• Tasks and Classes
• Webinars and Trainings
• Subscription
• Subscription Locator
• 1-D Kinematics
• Newton's Laws
• Vectors - Motion and Forces in Two Dimensions
• Momentum and Its Conservation
• Work and Energy
• Circular Motion and Satellite Motion
• Thermal Physics
• Static Electricity
• Electric Circuits
• Vibrations and Waves
• Sound Waves and Music
• Light and Color
• Reflection and Mirrors
• Measurement and Calculations
• About the Physics Interactives
• Task Tracker
• Usage Policy
• Newtons Laws
• Vectors and Projectiles
• Forces in 2D
• Momentum and Collisions
• Circular and Satellite Motion
• Balance and Rotation
• Electromagnetism
• Waves and Sound
• Atomic Physics
• Forces in Two Dimensions
• Work, Energy, and Power
• Circular Motion and Gravitation
• Sound Waves
• 1-Dimensional Kinematics
• Circular, Satellite, and Rotational Motion
• Einstein's Theory of Special Relativity
• Waves, Sound and Light
• QuickTime Movies
• About the Concept Builders
• Pricing For Schools
• Directions for Version 2
• Measurement and Units
• Relationships and Graphs
• Rotation and Balance
• Vibrational Motion
• Reflection and Refraction
• Teacher Accounts
• Kinematic Concepts
• Kinematic Graphing
• Wave Motion
• Sound and Music
• About CalcPad
• 1D Kinematics
• Vectors and Forces in 2D
• Simple Harmonic Motion
• Rotational Kinematics
• Rotation and Torque
• Rotational Dynamics
• Electric Fields, Potential, and Capacitance
• Transient RC Circuits
• Light Waves
• Units and Measurement
• Stoichiometry
• Molarity and Solutions
• Thermal Chemistry
• Acids and Bases
• Kinetics and Equilibrium
• Solution Equilibria
• Oxidation-Reduction
• Nuclear Chemistry
• Newton's Laws of Motion
• Work and Energy Packet
• Static Electricity Review
• NGSS Alignments
• 1D-Kinematics
• Projectiles
• Circular Motion
• Magnetism and Electromagnetism
• Graphing Practice
• About the ACT
• ACT Preparation
• For Teachers
• Other Resources
• Solutions Guide
• Solutions Guide Digital Download
• Motion in One Dimension
• Work, Energy and Power
• Chemistry of Matter
• Names and Formulas
• Algebra Based On-Level Physics
• Honors Physics
• Conceptual Physics
• Other Tools
• Frequently Asked Questions
• Purchasing the Download
• Purchasing the Digital Download
• About the NGSS Corner
• NGSS Search
• Force and Motion DCIs - High School
• Energy DCIs - High School
• Wave Applications DCIs - High School
• Force and Motion PEs - High School
• Energy PEs - High School
• Wave Applications PEs - High School
• Crosscutting Concepts
• The Practices
• Physics Topics
• NGSS Corner: Activity List
• NGSS Corner: Infographics
• About the Toolkits
• Position-Velocity-Acceleration
• Position-Time Graphs
• Velocity-Time Graphs
• Newton's First Law
• Newton's Second Law
• Newton's Third Law
• Terminal Velocity
• Projectile Motion
• Forces in 2 Dimensions
• Impulse and Momentum Change
• Momentum Conservation
• Work-Energy Fundamentals
• Work-Energy Relationship
• Roller Coaster Physics
• Satellite Motion
• Electric Fields
• Circuit Concepts
• Series Circuits
• Parallel Circuits
• Describing-Waves
• Wave Behavior Toolkit
• Standing Wave Patterns
• Resonating Air Columns
• Wave Model of Light
• Plane Mirrors
• Curved Mirrors
• Teacher Guide
• Using Lab Notebooks
• Current Electricity
• Light Waves and Color
• Reflection and Ray Model of Light
• Refraction and Ray Model of Light
• Teacher Resources
• Subscriptions

• Newton's Laws
• Einstein's Theory of Special Relativity
• About Concept Checkers
• School Pricing
• Newton's Laws of Motion
• Newton's First Law
• Newton's Third Law

Kepler's Three Laws

• Kepler's Three Laws
• Circular Motion Principles for Satellites
• Mathematics of Satellite Motion
• Weightlessness in Orbit
• Energy Relationships for Satellites

Kepler's three laws of planetary motion can be described as follows:

• The path of the planets about the sun is elliptical in shape, with the center of the sun being located at one focus. (The Law of Ellipses)
• An imaginary line drawn from the center of the sun to the center of the planet will sweep out equal areas in equal intervals of time. (The Law of Equal Areas)
• The ratio of the squares of the periods of any two planets is equal to the ratio of the cubes of their average distances from the sun. (The Law of Harmonies)

The Law of Ellipses

The law of equal areas.

Kepler's second law - sometimes referred to as the law of equal areas - describes the speed at which any given planet will move while orbiting the sun. The speed at which any planet moves through space is constantly changing. A planet moves fastest when it is closest to the sun and slowest when it is furthest from the sun. Yet, if an imaginary line were drawn from the center of the planet to the center of the sun, that line would sweep out the same area in equal periods of time. For instance, if an imaginary line were drawn from the earth to the sun, then the area swept out by the line in every 31-day month would be the same. This is depicted in the diagram below. As can be observed in the diagram, the areas formed when the earth is closest to the sun can be approximated as a wide but short triangle; whereas the areas formed when the earth is farthest from the sun can be approximated as a narrow but long triangle. These areas are the same size. Since the base of these triangles are shortest when the earth is farthest from the sun, the earth would have to be moving more slowly in order for this imaginary area to be the same size as when the earth is closest to the sun.

The Law of Harmonies

Kepler's third law - sometimes referred to as the law of harmonies - compares the orbital period and radius of orbit of a planet to those of other planets. Unlike Kepler's first and second laws that describe the motion characteristics of a single planet, the third law makes a comparison between the motion characteristics of different planets. The comparison being made is that the ratio of the squares of the periods to the cubes of their average distances from the sun is the same for every one of the planets. As an illustration, consider the orbital period and average distance from sun (orbital radius) for Earth and mars as given in the table below.

Observe that the T 2 /R 3 ratio is the same for Earth as it is for mars. In fact, if the same T 2 /R 3 ratio is computed for the other planets, it can be found that this ratio is nearly the same value for all the planets (see table below). Amazingly, every planet has the same T 2 /R 3 ratio.

( NOTE : The average distance value is given in astronomical units where 1 a.u. is equal to the distance from the earth to the sun - 1.4957 x 10 11 m. The orbital period is given in units of earth-years where 1 earth year is the time required for the earth to orbit the sun - 3.156 x 10 7 seconds. )

Kepler's third law provides an accurate description of the period and distance for a planet's orbits about the sun. Additionally, the same law that describes the T 2 /R 3 ratio for the planets' orbits about the sun also accurately describes the T 2 /R 3 ratio for any satellite (whether a moon or a man-made satellite) about any planet. There is something much deeper to be found in this T 2 /R 3 ratio - something that must relate to basic fundamental principles of motion. In the next part of Lesson 4 , these principles will be investigated as we draw a connection between the circular motion principles discussed in Lesson 1 and the motion of a satellite.

How did Newton Extend His Notion of Gravity to Explain Planetary Motion?

Newton's comparison of the acceleration of the moon to the acceleration of objects on earth allowed him to establish that the  moon is held in a circular orbit by the force of gravity  - a force that is inversely dependent upon the distance between the two objects' centers. Establishing gravity as the cause of the moon's orbit does not necessarily establish that gravity is the cause of the planet's orbits. How then did Newton provide credible evidence that the force of gravity is meets the centripetal force requirement for the elliptical motion of planets?

Recall from  earlier in Lesson 3  that Johannes Kepler proposed three laws of planetary motion. His Law of Harmonies suggested that the ratio of the period of orbit squared ( T 2 ) to the mean radius of orbit cubed ( R 3 ) is the same value  k  for all the planets that orbit the sun. Known data for the orbiting planets suggested the following average ratio:

Newton was able to combine the law of universal gravitation with circular motion principles to show that if the force of gravity provides the centripetal force for the planets' nearly circular orbits, then a value of  2.97 x 10 -19  s 2 /m 3  could be predicted for the  T 2 /R 3  ratio. Here is the reasoning employed by Newton:

Consider a planet with mass M planet  to orbit in nearly circular motion about the sun of mass M Sun . The net centripetal force acting upon this orbiting planet is given by the relationship

This net centripetal force is the result of the gravitational force that attracts the planet towards the sun, and can be represented as

Since F grav  = F net , the above expressions for centripetal force and gravitational force are equal. Thus,

Since the velocity of an object in nearly circular orbit can be approximated as v = (2*pi*R) / T,

Substitution of the expression for v 2  into the equation above yields,

By cross-multiplication and simplification, the equation can be transformed into

The mass of the planet can then be canceled from the numerator and the denominator of the equation's right-side, yielding

The right side of the above equation will be the same value for every planet regardless of the planet's mass. Subsequently, it is reasonable that the  T 2 /R 3  ratio would be the same value for all planets if the force that holds the planets in their orbits is the force of gravity. Newton's universal law of gravitation predicts results that were consistent with known planetary data and provided a theoretical explanation for Kepler's Law of Harmonies.

Investigate!

Check your understanding.

1. Our understanding of the elliptical motion of planets about the Sun spanned several years and included contributions from many scientists.

a. Which scientist is credited with the collection of the data necessary to support the planet's elliptical motion? b. Which scientist is credited with the long and difficult task of analyzing the data? c. Which scientist is credited with the accurate explanation of the data?

Tycho Brahe gathered the data. Johannes Kepler analyzed the data. Isaac Newton explained the data - and that's what the next part of Lesson 4 is all about.

2. Galileo is often credited with the early discovery of four of Jupiter's many moons. The moons orbiting Jupiter follow the same laws of motion as the planets orbiting the sun. One of the moons is called Io - its distance from Jupiter's center is 4.2 units and it orbits Jupiter in 1.8 Earth-days. Another moon is called Ganymede; it is 10.7 units from Jupiter's center. Make a prediction of the period of Ganymede using Kepler's law of harmonies.

Answer: T = 7.32 days

Io: R io = 4.2 and T io = 1.8

Ganymede: R g = 10.7 Tg=???

Use Kepler's 3rd law to solve.

(T io )^2/(R io ) 3 = 0.04373;

so (T g )^2 / (R g ) 3 = 0.04373

Proper algebra would yield (T g )^2 = 0.04373 • (R g ) 3

(T g ) 2 = 53.57 so T g = SQRT(53.57) = 7.32 days

3. Suppose a small planet is discovered that is 14 times as far from the sun as the Earth's distance is from the sun (1.5 x 10 11 m). Use Kepler's law of harmonies to predict the orbital period of such a planet. GIVEN: T 2 /R 3 = 2.97 x 10 -19 s 2 /m 3

Answer: T planet = 52.4 yr

Use Kepler's third law:

(T e )^2/(R e )^3 = (T p )^2/(R p ) 3

Rearranging to solve for T p :

(T p )^2=[ (T e ) 2 / (R e ) 3 ] • (R p ) 3 or (T p ) 2 = (T e ) 2 • [(R p ) / (R e )] 3 where (R p ) / (R e ) = 14

so (T p ) 2 = (T e ) 2 • [14] 3 where T e =1 yr

T p = SQRT(2744 yr 2 )

4. The average orbital distance of Mars is 1.52 times the average orbital distance of the Earth. Knowing that the Earth orbits the sun in approximately 365 days, use Kepler's law of harmonies to predict the time for Mars to orbit the sun.

Given: R mars = 1.52 • R earth and T earth = 365 days

Use Kepler's third law to relate the ratio of the period squared to the ratio of radius cubed

(T mars ) 2 / (T earth ) 2 • (R mars ) 3 / (R earth ) 3 (T mars ) 2 = (T earth ) 2 • (R mars ) 3 / (R earth ) 3 (T mars ) 2 = (365 days) 2 * (1.52) 3

(Note the R mars / R earth ratio is 1.52)

T mars = 684 days

Orbital radius and orbital period data for the four biggest moons of Jupiter are listed in the table below. The mass of the planet Jupiter is 1.9 x 10 27 kg. Base your answers to the next five questions on this information.

5. Determine the T 2 /R 3 ratio (last column) for Jupiter's moons.

a. (T 2 ) / (R 3 ) = 3.16 x 10 -16 s 2/ m 3

b. (T 2 ) / (R 3 ) = 3.13 x 10 -16 s 2/ m 3

c. (T 2 ) / (R 3 ) = 2.87 x 10 -16 s 2/ m 3

d. (T 2 ) / (R 3 ) = 3.03 x 10 -16 s 2/ m 3

6. What pattern do you observe in the last column of data? Which law of Kepler's does this seem to support?

The (T 2 ) / (R 3 ) ratios are approximately the same for each of Jupiter's moons. This is what would be predicted by Kepler's third law!

7. Use the graphing capabilities of your TI calculator to plot T 2 vs. R 3 (T 2 should be plotted along the vertical axis) and to determine the equation of the line. Write the equation in slope-intercept form below.

T 2 = (3.03 * 10 -16 ) * R 3 - 4.62 * 10 +9

Given the uncertainty in the y-intercept value, it can be approximated as 0.

Thus, T 2 = (3.03 * 10 -16 ) * R 3

See graph below.

8. How does the T 2 /R 3 ratio for Jupiter (as shown in the last column of the data table) compare to the T 2 /R 3 ratio found in #7 (i.e., the slope of the line)?

The values are almost the same - approximately 3 x 10 -16 .

9. How does the T 2 /R 3 ratio for Jupiter (as shown in the last column of the data table) compare to the T 2 /R 3 ratio found using the following equation? (G=6.67x10 -11 N*m 2 /kg 2 and M Jupiter = 1.9 x 10 27 kg)

The values in the data table are approx. 3 x 10 -16 . The value of 4*pi/(G*M Jupiter ) is approx. 3.1 x 10 -16 .

Return to Question #6

Kepler's laws of planetary motion, before and after Newton's Principia: An essay on the transformation of scientific problems

• Studies in History and Philosophy of Science Part A 18(2):177-208
• 18(2):177-208

• University of Toronto

Discover the world's research

• 25+ million members
• 160+ million publication pages
• 2.3+ billion citations

• Metaphilosophy

• Mahmoud Jalloh
• Johannes Kepler
• Daniel Špelda

• Perspect Sci Hist Phil Soc
• Bernard R. Goldstein

• STUD HIST PHILOS SCI
• Isaac Newton
• Marie Boas Hall
• Thomas S. Kuhn

• J. L. E. Dreyer
• Thomas Heath
• Patricia Cline Cohen
• Richard J. Moss
• Curtis Wilson
• J. Edleston
• Richard S. Westfall
• I. Bernard Cohen
• Curtis A. Wilson
• Recruit researchers
• Join for free
• Login Email Tip: Most researchers use their institutional email address as their ResearchGate login Password Forgot password? Keep me logged in Log in or Continue with Google Welcome back! Please log in. Email · Hint Tip: Most researchers use their institutional email address as their ResearchGate login Password Forgot password? Keep me logged in Log in or Continue with Google No account? Sign up

• What is Kepler's First Law of Planetary Motion?

Kepler’s First Law of Planetary Motion states that the orbit of a planet is an ellipse, with the sun located on one of the two foci. Contrary to many people’s beliefs and understanding, the orbits that the planets move on are not circular. The Kepler’s First Law of Planetary explains the real shape of the orbits. A circle has fixed points that are equidistant from the center, unlike an ellipse which consists of points with a certain distance from two points, which are referred to as foci. The foci are situated on the major axis of the ellipse such that the sum of the distance between any point on the ellipse and the two foci is constant.

Reason for the Assumption

The pioneer astronomers who attempted to make models of the solar system assumed that the orbits were perfectly circular. Therefore, they made models that perpetuated perfect circular orbits. This is probably why most individuals believed that the orbits were circular. Plato’s astronomical works also highlights the orbits as perfectly circular. However, it should be noted that most of the orbits are almost circular and with varied eccentricity.

Kepler’s Discovery

Johannes Kepler, a German astronomer discovered that the circular orbits were unrealistic. Therefore, he studied the astronomical objects and their orbits and came up with laws to prove that the orbits were elliptical and not circular. Kepler was making a case study on the orbital motions of planet Mars when he discovered that the orbits were elliptical, or rather oval-like in shape. Calculations of the eccentricity of planet Mars’ orbit indicate that it is perfect ellipse shape. It is also proof enough that orbits of other planets located farther from the sun are also ellipses. He wrote to another astronomer, David Fabricius, to explain his discovery. He penned his new-found discovery on October 11, 1605, with most of his works published between 1605 and 1609.

How Effective Is Kepler’s First Law of Planetary Motion

Kepler’s first law of planetary motion came in handy in explaining the Heliocentric Theory proposed by Nicolaus Copernicus. Nicolas Copernicus was attempting to explain why the speed of the planets while moving around the sun varied. It was quite a challenge for him to explain and provide evidence for his discovery. But with the Kepler’s First Law of Planetary motion, he was able to explain his theory, which was later proved to be valid.

How Valid Is Kepler’s First Law of Planetary Motion

If the orbits were perfectly circular, then the phenomena of perihelion and aphelion could not have been possible. Perihelion and aphelion occur because the orbits are elliptical. In 1687, Isaac Newton validated all of the three Kepler’s Laws and found that they would apply in the solar system to an extent as a consequence of his own low of motion and universal gravitation. The orbits have also been found to become more elliptical as the years go by. The eccentricity of the orbits keeps on increasing after over 10 decades. This explains why the dates of perihelion and aphelion are not fixed but rather keep changing.

• World Facts

More in World Facts

The Largest Countries In Asia By Area

The World's Oldest Civilizations

Is England Part of Europe?

Olympic Games History

Southeast Asian Countries

How Many Countries Are There In Oceania?

Is Australia A Country Or A Continent?

Is Turkey In Europe Or Asia?