Physics 101 - Astronomy - Spring 2019

Class notes for day 4, Jan. 24, 2019

We started by discussing the nature of science. I made some remarks based on this outline of the scientific method:

Science is two things: a body of knowledge and a process of learning about nature (called the scientific method).

Knowledge is acquired by observations and experiments.

Scientific method is a process for gaining more knowledge, that can be tested and accepted by everyone.

Scientific theory is an explanation of observations or experimental results that can be described quantitatively and tested.

The theory must make testable predictions that can be verified by new observations or experiments, and can possibly be refuted. These predictions depend on basic assumptions and hypotheses, which might need to be changed if they do not lead to predictions that agree with observations.

Theories can be modified and should be the simplest version that explains the observations (This idea is called Occam’s razor).

Six words that summarize this are: Observe, hypothesize, predict, test, modify, economize.

In Astronomy, we have a good example of a major change in theory which can be understood in the terms outlined above. This revolutionary change in outlook is called the Copernican Revolution - a heliocentric model of the solar system was proposed to replace the old geocentric system of the Greeks.

The development of the current model of the solar system began with careful measurement of the movement of the Sun and the Moon. To understand this, we reviewed the motion and the phases of the Moon. When we watch the Moon, it’s shape changes from one night to the next. See this Astronomy Picture of the Day for a multiple-photo display of a month's change in the Moon. The display probably went too fast in your computer, so try this version. This phenomenon is called lunation. These are the phases of the Moon, and we will need to keep these in mind when we look at the phases of Venus. Richard Pogge's website at Ohio State has his summary of his lecture on the Phases of the Moon but this contains more than I intend to cover in our course.

To ancient peoples, the motion of the Moon and Sun seemed fairly simple, almost like they were moving in circles around the Earth. The Moon moves from west to east on the celestial sphere in a very orderly way. Each night you see it in a different position with a somewhat different phase.

Five other objects in the sky did NOT move in this simple way. They are the planets, the wanderers in the Heavens. The five visible planets that the ancients saw were Mercury, Venus, Mars, Jupiter, and Saturn. The planets usually move from west to east on the celestial sphere, but … not always. The most perplexing aspect of the planets’ motion is motion in the opposite direction, from east to west, called retrograde motion, which occurs on a regular basis. Mars, Jupiter, and Saturn usually move from West to East in the celestial sphere, but occasionally move from East to West. Retrograde motion occurs over several weeks, and involves motion to the west, as compared to prograde motion, which is to the east (relative to the stars of the ”celestial sphere”). To understand this unusual motion of the planets, we considered the Retrograde motion of Mars.

Fig. 2.13 (p. 48) in your textbook shows the retrograde motion of Mars in the sky. During this retrograde motion, the planet is also brightest and closest to the Earth.

The Greeks (in particular, an astronomer and philosopher named Ptolemy) tried to interpret this as due to epicycles (small circles) moving with their centers on larger circles called deferents. The Ptolemaic model of a single planet moving on an epicycle is shown in your text in Fig. 2-14 on p. 49.

Ptolemy's model was centered on the Earth (so it is called geocentric) and complicated, but it managed to describe the motion of the planets. Ptolemy’s model of planetary motion used deferents (big circles) and epicycles (little circles centered on a point that moves on the deferent). This involved up to 80 circles to describe 7 objects! And it couldn't explain why there would be small circles, it just described the motion without answering the question: Why would the planets move along these complicated paths?

Copernicus proposed a heliocentric model, centered on the Sun, which explains the motions using fewer circles, but still has problems. But since it is simpler, and closer to our current understanding, we might think of this as a starting point for modern Astronomy.

Galileo Galilei used a telescope (which he built in 1609) to observe the sky. Four observations are mentioned because they are particularly important.

1. The Moon has mountains, valleys, and craters (perhaps the Moon is like Earth?).

2. Sunspots move around the Sun about once per month (the Sun rotates every 25 days).

3. Jupiter has moons (Galileo saw 4), and it looked like a miniature solar system.

4. Venus has a cycle of phases (like the Moon), implying it revolves around the Sun, as in the model of Copernicus. The observed phases of Venus are NOT in agreement with the geocentric model (see the figures in my power point). In particular, in the geocentric model of the Greeks, Venus would never appear with a gibbous phase. However, if you look at it with a telescope, you WILL see Venus in a gibbous phase for part of its orbit, so this contradicts the Greek model. This is a refutation of the geocentric theory.

Galileo’s observation that Jupiter has moons provided support to the idea that Jupiter and the Earth were both planets in orbit around the Sun, each with moons of their own.

In addition to your textbook, for this material I suggest that you look at chapter 5 on Motion of the Planets from the Open Course: Introduction to Astronomy. The material is very similar to the emphasis in our course (at least for this section).

Geocentric vs. Heliocentric theories - an example of a scientific revolution (change in accepted theory)

Both theories described the positions and movement of the Sun, Moon, and 5 visible planets, as seen without a telescope. The geocentric theory was too complicated (80 circles!). (Occam’s razor could be invoked to seek a simpler way to describe planetary motion.) Once the telescope was used to observe Venus, the geocentric theory could not explain the phases of Venus. The heliocentric theory of Copernicus explained many of Galileo’s observations, but also used circular orbits. More accurate measurements did not agree with the simple theory of Copernicus (circles had to be replaced by ellipses in the newer theory of planetary motion). However, after replacing the circular orbits with elliptical orbits, Kepler was able to make the theory of Copernicus (heliocentric theory) agree very well with the observational data, and it was simpler than the geocentric theory of the Greeks.

Kepler's Laws and Newton's Laws

We continued with a summary of the observations of Tycho Brahe (commonly called Tycho, 1546 - 1601), who made careful observations of the planets over a 20 year period with an accuracy of 1 arcmin (1/60 of a degree), which is much better than observers had done before. He also used parallax measurements to determine that a nova (seen in 1572) was very distant, and probably was a star. This impressed the King and he was given an island, on which he built an observatory and castle, called Uraniborg. If you are interested in a somewhat long biography of Tycho Brahe see this web site: http://www-groups.dcs.st-and.ac.uk/history/Biographies/Brahe.html . Tycho had developed a theory which was a mixture of heliocentric and geocentric ideas and did not have much influence. His important contribution to Astronomy was all the careful observational data, which was analyzed by his assistant, Johannes Kepler, after Tycho died in 1601.

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. In class I drew a circle on the board to remind students that it only involves one length, the radius. I showed how to use string and a pen or chalk to draw an ellipse. The length of the string I used was 2a, and a is the length of the semimajor axis (half of the major 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.

Eccentricity e = 0 (zero) gives us a circle, and e = 1 gives us an ellipse that is collapsed down into a line segment. An ellipse with e = 0.95 looks very skinny like a cigar.

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.

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.

We ended the class here, and so we will cover the material below on Tuesday.

The textbook then 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.

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 on Thursday, Jan. 31.

In the 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".