<< Chapter < Page Chapter >> Page >
Kepler’s laws of planetary motion are consistent with Newtonian mechanics.

The trajectory of motion resulting from general solution of “two body” system is a conic section. Subject to initial velocities and relative mass, eccentricity of conic section can have different values. We interpret a conic section for different eccentricity to represent different types of trajectories. We have already discussed straight line and circular trajectories. In this module, we shall discuss elliptical trajectory, which is the trajectory of a planet in the solar system.

In a general scenario of “two body system”, involving elliptical trajectory, each body revolves around the common “center of mass” called “barycenter”. The two elliptical paths intersect, but bodies are not at the point of intersection at the same time and as such there is no collision.

Two body system

Elliptical trajectories

In this module, however, we shall keep our focus on the planetary motion and Kepler’s planetary laws. We are basically seeking to describe planetary motion – particularly that in our solar system. The trajectory of planet is elliptical with one qualification. The Sun, being many times heavier than the planets, is almost at rest in the reference frame of motion. It lies at one of the foci of the elliptical path of the planets around it. Here, we assume that center of mass is about same as the center of Sun. Clearly, planetary motion is a special case of elliptical motion of “two body system” interacted by mutual attraction.

We should, however, be aware that general solution of planetary motion involves second order differential equation, which is solved using polar coordinates.

Ellipse

We need to learn about the basics of elliptical trajectory and terminology associated with it. It is important from the point of view of applying laws of Newtonian mechanics. We shall, however, be limited to the basics only.

Conic section

Conic section is obtained by the intersection of a plane with a cone. Two such intersections, one for a circle and one for an ellipse are shown in the figure.

Conic sections

Two conic sections representing a circle and an ellipse are shown.

Elliptical trajectory

Here, we recount the elementary geometry of an ellipse in order to understand planetary motion. The equation of an ellipse centered at the origin of a rectangular coordinate (0,0) is :

x 2 a 2 + y 2 b 2 = 1

where “a” is semi-major axis and “b” is semi-minor axis as shown in the figure.

Ellipse

Semi major and minor axes of an ellipse

Note that “ F 1 ” and “ F 2 ” are two foci of the ellipse.

Eccentricity

The eccentricity of a conic section is measure of “how different it is from a circle”. Higher the eccentricity, greater is deviation. The eccentricity (e) of a conic section is defined in terms of “a” and “b” as :

e = 1 k b 2 a 2

where “k” is 1 for an ellipse, 0 for parabola and -1 for hyperbola. The values of eccentricity for different trajectories are as give here :

  1. The eccentricity of a straight line is 1, if we consider b=0 for the straight line.
  2. The eccentricity of an ellipse falls between 0 and 1.
  3. The eccentricity of a circle is 0
  4. The eccentricity of a parabola is 1.
  5. The eccentricity of a hyperbola is greater than 1.

Questions & Answers

I'm interested in biological psychology and cognitive psychology
Tanya Reply
what does preconceived mean
sammie Reply
physiological Psychology
Nwosu Reply
How can I develope my cognitive domain
Amanyire Reply
why is communication effective
Dakolo Reply
Communication is effective because it allows individuals to share ideas, thoughts, and information with others.
effective communication can lead to improved outcomes in various settings, including personal relationships, business environments, and educational settings. By communicating effectively, individuals can negotiate effectively, solve problems collaboratively, and work towards common goals.
it starts up serve and return practice/assessments.it helps find voice talking therapy also assessments through relaxed conversation.
miss
Every time someone flushes a toilet in the apartment building, the person begins to jumb back automatically after hearing the flush, before the water temperature changes. Identify the types of learning, if it is classical conditioning identify the NS, UCS, CS and CR. If it is operant conditioning, identify the type of consequence positive reinforcement, negative reinforcement or punishment
Wekolamo Reply
please i need answer
Wekolamo
because it helps many people around the world to understand how to interact with other people and understand them well, for example at work (job).
Manix Reply
Agreed 👍 There are many parts of our brains and behaviors, we really need to get to know. Blessings for everyone and happy Sunday!
ARC
A child is a member of community not society elucidate ?
JESSY Reply
Isn't practices worldwide, be it psychology, be it science. isn't much just a false belief of control over something the mind cannot truly comprehend?
Simon Reply
compare and contrast skinner's perspective on personality development on freud
namakula Reply
Skinner skipped the whole unconscious phenomenon and rather emphasized on classical conditioning
war
explain how nature and nurture affect the development and later the productivity of an individual.
Amesalu Reply
nature is an hereditary factor while nurture is an environmental factor which constitute an individual personality. so if an individual's parent has a deviant behavior and was also brought up in an deviant environment, observation of the behavior and the inborn trait we make the individual deviant.
Samuel
I am taking this course because I am hoping that I could somehow learn more about my chosen field of interest and due to the fact that being a PsyD really ignites my passion as an individual the more I hope to learn about developing and literally explore the complexity of my critical thinking skills
Zyryn Reply
good👍
Jonathan
and having a good philosophy of the world is like a sandwich and a peanut butter 👍
Jonathan
generally amnesi how long yrs memory loss
Kelu Reply
interpersonal relationships
Abdulfatai Reply
What would be the best educational aid(s) for gifted kids/savants?
Heidi Reply
treat them normal, if they want help then give them. that will make everyone happy
Saurabh
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Physics for k-12. OpenStax CNX. Sep 07, 2009 Download for free at http://cnx.org/content/col10322/1.175
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Physics for k-12' conversation and receive update notifications?

Ask