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Orbits

Newton’s canon – how to get objects into orbit

Dr. William Romanishin, emeritus professor of astronomy at the U. of Oklahoma, writes:

Imagine shooting a cannonball from a cannon parallel to the surface of the Earth from a high mountain. The faster the cannonball moves, the further it would travel before hitting the Earth. At a certain speed, the cannonball would fall around the curving Earth, and would come back and hit the cannon. Such a path is called an orbit.

If the cannonball were above the Earths atmosphere, so that there was no friction with the air, it would just go around and around forever. The cannonball is in “free fall”- it is falling around the Earth- but it never gets any closer to the surface! The cannonball (or Space Shuttle or whatever) does not need any further “push” once it gets into an orbit.

A simple way to think about what is going on is to think about tying a string around a rock and swinging it around in a circle. The string exerts a force on the rock which causes it to keep moving in the circle, rather than in a straight line. (If the string breaks, the rock flies off in a straight line.)

In an orbit, *gravity* provides the “string” that holds the orbiting body in place. So, a body in an orbit is constantly being pulled by gravity into a curved path. By Newton’s first law, a curved path requires a force- if there were no force then the object would move in a straight line.

http://hildaandtrojanasteroids.net/A1504-11feb11.html

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More details

Planets seem to orbit the Sun in a circle, with the sun standing perfectly fixed

But that’s not exactly correct.

As a planet orbits a star, the star also moves in its own small orbit around the system’s center of mass.

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The next few paragraphs are from “Science Curriculum by Aaron Keller”

Pictures of the Solar System tend to show all the orbits of the planets as circles centered on the Sun (see image at left). The big object in the center of the circle represents the Sun. The smaller object on the circle represents a planet. Notice that the Sun is exactly in the center of the orbit. No orbit in the solar system is perfectly round.
In reality, the planets orbit the Sun traveling along an oval path (see image at right). The mathematical term for this shape is an ellipse. This ellipse is a little more stretched out than real planetary orbits but it has been exaggerated on purpose to demonstrate the shape more clearly. Notice that the Sun in this picture is not right in the center. The Sun is at one of the two ‘centers’ of the ellipse. These are called foci (plural for focus). The closer these foci are together, the more circular the orbit. The orbit of Venus is the closest to a circle of any planet in the Solar System. Scientists have a name to describe just how much like an ellipse an orbit is. This is called eccentricity and is a measure that uses numbers between 0 and 1. If an orbit has an eccentricity close to 1 then the ellipse is so long as to be more cigar-shaped than round. Comets tend to have very elongated, high-eccentricity orbits. The closer the eccentricity is to zero, the more circular the orbit.

Ellipse = the big oval shape.

Has a major axis (the longer axis) and a minor axis (the shorter one). \

Has two foci: in the case of planetary orbits one focus is the Sun.

All the points in an ellipse are defined in relation to the foci.

The sum of the distances from each point on the ellipse to both foci is constant for all points on the ellipse.

Point on an orbit nearest the Sun is called perihelion.

Point farthest from the Sun is called aphelion.

A circle is special case of an ellipse

Introductory Astronomy: Ellipses

Learning Standards

Mathematics: Precalculus

MASSACHUSETTS CURRICULUM FRAMEWORK FOR MATHEMATICS
Expressing Geometric Properties with Equations G-GPE Translate between the geometric description and the equation for a conic section.

3. (+) Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant.

MA.3.a. (+) Use equations and graphs of conic sections to model real-world problems. 

Expressing Geometric Properties with Equations G-GPE Translate between the geometric description and the equation for a conic section. 3. (+) Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant. MA.3.a. (+) Use equations and graphs of conic sections to model real-world problems. 

Analytic geometry. The branch of mathematics that uses functions and relations to study geometric
phenomena, e.g., the description of ellipses and other conic sections in the coordinate plane by quadratic
equations.

Earth Science

ESS1. Earth’s Place in the Universe

8.MS-ESS1-2. Explain the role of gravity in ocean tides, the orbital motions of planets, their moons, and asteroids in the solar system.

ESS1.B Earth and the solar system – The solar system contains many varied objects held together by gravity. Solar system models explain and predict eclipses, lunar phases, and seasons.