Physics 101 - Astronomy - Spring 2019

Class notes for day 5, Jan. 29, 2019

 On day 5, I went back to the slide on Tycho Brahe and went over Kepler's Laws again, but in more detail and with a different emphasis. See the day05 Powerpoint for this material.

Kepler's Laws and Newton's Laws

Kepler used decades of Tycho’s observations in his mathematical calculations, to determine the shape of the planetary orbits, and the speed of the planets as they went around the Sun. This massive effort (doing the math by hand) took over 20 years, and resulted in three major statements about the characteristics of planetary orbits: Kepler’s three laws of planetary motion.

Kepler proposed three general laws of orbiting planets. These are:

Kepler's First Law: The orbits of the planets are ellipses with the Sun at one focus. See the text for more descriptions of ellipses.

The semimajor axis of the ellipse is half of the major axis (hence the word semi), which is the longest distance across the ellipse. Perpendicular to that is the minor axis.

The eccentricity of an ellipse is defined by e  =  c/a and for a circle e = 0. The value of e must be between 0 and 1. Near e = 1 the ellipse becomes very thin. Most planetary orbits have small eccentricity (that is, near zero), so they are close to a circle.

For orbits of planets around the Sun, the closest point in the orbit is the perihelion. "Peri" means "close" and "helion" refers to the Sun (the word helium has a similar origin related to the Sun). 

Kepler's Second Law: An imaginary line connecting the Sun to any planet sweeps out equal areas of the ellipse in equal intervals of time. Equal areas in equal time also means higher speed at closer distances as the planet swings around the Sun.

This second law basically means that a planet goes faster when it is closer to the Sun.

The second law is due to the fact that the angular momentum of the planet is constant as it goes around in its orbit. In class, I demonstrated several aspects of angular momentum.

Kepler's Third Law: The orbital period is related to the length of the semimajor axis of the orbit. (The square of the planet’s orbital period is proportional to the cube of its semimajor axis.) In other words, a planet in a larger orbit takes longer to go around its orbit, so the period of time to do this is longer.

It is worth mentioning that Kepler's laws also describe the orbits of satellites, comets, asteroids, and the Moon, provided you replace the word "planet" with "an orbiting object" and replace "Sun" with whatever large massive object that the orbiting object is orbiting around. For example, the orbit of the Moon around the Earth is an ellipse, but it has small eccentricity and is almost circular.

The Astronomical Unit is the distance from the Earth to the Sun, about 150,000,000 km. Kepler's work helps us draw a scale model of the solar system, and until recently (using radar in the 1960's) we didn't have really precise measurements of these distances.

Section 3.2 of your textbook introduces Newton's Laws. We started with the idea of inertia, which was known to Galileo. However, Newton developed the ideas of a theory of motion and now we list three basic ideas as the foundation of the Newtonian theory of mechanics.

Newton's First Law: the law of inertia. An object will remain at rest or continue moving at a constant velocity unless it is acted upon by an external net force.

Newton's Second Law: the relation between force, mass, and acceleration. This is described in words in the text, and the equation is the famous equation F = ma that is taught in high school physics classes. The acceleration of a mass is proportional to the total force acting upon it, and inversely proportional to the mass of the object (i.e., you have to re-arrange the equation to read a = F/m).

Newton's Third Law: forces occur in pairs, called action and reaction. For every force acting upon an object (action), there is a force acting on another object (reaction) which has the same magnitude (size) but points (acts) in the opposite direction.

I gave examples of these ideas which are similar to those in the text. But I also used some objects (like a pair of carts) to show some more examples of force and acceleration.

The actual orbit of one object around another causes BOTH objects to orbit around their center of mass. This might be better understood if you imagine two skaters holding hands and going around each other. They both move around a point that is midway between them. For objects with very different masses, the center of mass will be closer to the more massive object.

Gravity is a force between objects that have mass (see section 3.3). Gravitational force varies with the distance between the objects. It depends on the product of the two masses, i.e., m₁ x m₂ and on the inverse of the square of the distance between the masses (assuming they are small compared with the distance). 1/r²
The Sun’s gravity causes planets to move on a path called an orbit. These obey Kepler’s Laws, which can be explained on the basis of Newton's Laws.

The first exam is delayed due to cold weather, and will be given on Tuesday, Feb. 5.

If you are studying from the free online OpenStax book, here are the corresponding sections to study. These are in the order we discussed them in class. If you are just starting to study, just read them in order.

Section 1.1 pp. 11-13
Section 2.1 pp. 31-41
Sections 4.1 - 4.3 pp. 103-117
Section 4.5  pp. 120-124
Section 4.7  pp. 129-135
Sections 1.2 - 1.3 pp. 13-15
Section 2.2 pp. 42-49
Section 2.4 pp. 54-61
Sections 3.1-3.2 pp. 69-80
Sections 3.3-3.4 pp. 81-88

In addition, I recommend quickly reading all of Ch. 1.

At the ends of Ch. 2, 3, and 4, see the "Key Terms" to build your vocabulary, and you may want to review the "Summary" which comes just after the "Key Terms".