Astronomy Assignment #4: History and Philosophy of Western Astronomy

Astronomy Assignment #4: History and Philosophy of Western Astronomy

Solutions

Review Questions

1.  What two basic kinds of models have been proposed to explain the motions of the planets?

The two basic kinds of models have been proposed to explain the motions of the planets are a geocentric model credited to Ptolemy (circa. 150 C.E.) where planets circled the Earth on large deferents and smaller epicycles and the heliocentric model proposed by Copernicus (1543 C.E.) where all the planets and the Earth circled the Sun on orbits.

2.  Why are epicycles needed in Ptolemy's model?

Epicycles needed in Ptolemy's model to explain both retrograde motion and the asymmetric times that planets appear above and below the celestial equator.

3.  Why was the Ptolemaic model accepted for more than 1000 years?

The Ptolemaic model accepted for more than 1000 years because it worked, after a fashion, and it was consistent with the larger model of Aristotle.

4.  What is the Copernican model and how did it explain retrograde motion?

The Copernican model placed the Sun at the center of the solar system and had the planets, as well as the Earth, orbiting the Sun on circular orbits at constant speed. In addition to equating the Earth with the planets, Copernicus gave the Earth two motions (1)an eastward rotation on its 23½° tilted axis once every 23h 56m 4.03s and (2) a eastward revolution around the Sun once every 365.241 days on the ecliptic. He explained retrograde motion as an illusionary motion that occurred when the Earth passed another planet in its orbit and, due to the difference in relative speeds, the other planet would appear to move westward through the stars contrary to its normal eastward through the stars apparent motion. The illusion of retrograde motion is caused by us observing a moving planet form a moving platform – the Earth.

5.  Why did Copernicus not know the absolute distance between various planets and the Sun in his model? Explain what he would have needed to know to get the absolute distances.

Copernicus did not know the absolute distances of the planets from the Sun because he did not know how far the Earth was from the Sun. He was able to determine the relative distances the planets were from the Sun compared to the Earth’s distance. However, without knowing how many kilometers there were in an astronomical unit, Copernicus was unable to set exact absolute distances of the planets from the Sun.

6.  What important contributions did Tycho Brahe make to astronomy?

Tycho made several important contributions to astronomy. Foremost he made and recorded the most accurate naked-eye observation s of planetary positions ever attained. His naked-eye observations, with te assistance of giant surveying-like instruments he had constructed, enabled Tycho to locate a planets position in the sky to an accuracy of 0.1 degrees. Trés Formidable!

Additionally, Tycho proposed a short-lived but important model of the solar system that had Venus orbiting the Sun but all the other planets with the Sun orbited the Earth. This model was developed as a result of Galileo’s observations of the phases of Venus that unambiguously indicated that Venus must orbit the Sun and not the Earth as the prevailing model of Ptolemy and Aristotle required. Although Tycho’s model was not long-lived, it provided a bridge to aid in the acceptance of the full Copernican heliocentric model.

Finally, lest we forget…, he hired Kepler and it was Kepler working with Tycho’s data that improved the originally flawed Copernican model.

7.  What important contributions did Galileo make to modern science?

Galileo was the first person to observe and record what he saw in the heavens through a telescope. Although his observations never proved that the Earth was a planet and moved as Copernicus postulated, everything that Galileo did observe (The Moon, the Sun, Jupiter, Venus, and more) contradicted the ideas of Aristotle and Ptolemy. Galileo’s true impact was not in bolstering the Copernican model but in destroying the model of Aristotle.

8.  What were his astronomical discoveries and why was he able to make those discoveries? In particular address his observation of the Moon, Sun, Jupiter and Venus. Explicitly state what he saw, how he interpreted it and how his interpretation contradicted the prevailing Aristotelian paradigm.

Observation Target / What Galileo Saw / How he Interpreted it / How this contradicted Aristotle.
Moon / The Moon, as seen through a telescope, has definite landforms like mountains, valleys, plains, and craters. / Living in Italy, Galileo was familiar with landforms like mountains, valleys, plains and volcanic craters (calderas). Seeing these features on the Moon made him believe that the Moon was made of terrestrial-like matter (rocks and stuff). / Aristotle postulated that all objects above the atmosphere were not made of terrestrial matter but out another element called celestial matter. If the Moon was apparently made of terrestrial matter as Galileo concluded, it could not be celestial matter and it called into question the very existence of celestial matter.
Sun / The Sun, as seen through a telescope and an appropriate filter, will show dark spots on its surface that we now call sunspots. These spots were seen to change with time; changing shape disappearing and reappearing over a several day period. / Galileo interpreted these spots in two ways. First they were blemishes on the Sun’s surface – imperfections. Second, they were not a constant part of the Sun but changed over time. / Aristotle postulated that the celestial matter was perfect – unchanging and eternal. If these sunspots were indeed blemishes then the Sun was not perfect celestial material and since the spots changed over time, the Sun was not an unchanging eternal body. Again, the very existence of celestial matter is called into question by Galileo’s observations.
Jupiter
/ Jupiter looks like a star to the naked-eye. However, through even a small telescope Jupiter resolves itself into a disk. Around this disk, after several weeks of observing, Galileo had charted four tiny “stars” orbiting Jupiter. These objects were called the Medicean stars (after Cosimo Medici the ruler of Galileo’s Florence) for centuries, until the modern name of the Galilean Satellites was adopted. We now recognize these objects as the four largest moons of Jupiter. Two of them are larger thatn our own Moon and the other two are larger than the planet Mercury. / Galileo interpreted that these stars or satellites were orbiting Jupiter. That’s all he needed to see. / Aristotle and Ptolemy had all motion around a fixed and central Earth. There was no contingency for “stars” to orbit a planet in that geocentric model. Thus the foundational premise of the Aristotelian model that all motion is around the Earth is nullified.
Venus / Galileo observed Venus over several months. He saw that Venus had phases similar to the Moon’s phases and that these phases were correlated with its distance from the Earth: Full when far and Crescent when close. / As explained in the next question, Galileo concluded that Venus must orbit the Sun. These observations could not be explained using the geocentric model of Ptolemy. See the next question for more details. / Aristotle and Ptolemy had all the planets circling the Earth. To definitively prove that one of those planets circled the Sun called into question the Aristotelian postulate that any planets orbited the Sun. Thus another mortal wound to the natural philosophy of Aristotle. The seventeenth century was not kind to Aristotle.

9.  What observation finally disproved the Ptolemaic model?

Galileo’s observations of Venus going through phases that were correlated to its distance and planetary configurations could only be interpreted as a result of Venus orbiting the Sun. The figure to the right constrasts the model predictions of the Ptolemaic (geocentric) and Copernican (heliocentric) systems. Next to the figure of model preditions is an actual photographic representation of the phases of Venus. In the Ptolemaic model Venus would never appear in the near full phases, but only as a crescent and it would not vary much in apparetn diameter (angular size) since it was always in front of the Sun. However, the actual photographs of Venus pahses show it in phases near full and in those phases Venus must be far away because it has a smaller apparent diameter. The photographic evidence is much more consistent with the Copernican mode that predicts that Venus will be in a crescent phase and large when on the same side of the Sun as we are and near full phase and small when on the opposite side of the Sun. These observations were a lethal wound to the Ptolemaic geocentric model of planetary motion because they could only be interpreted as Venus orbiting the Sun.

10.  Why is Galileo sometimes referred to as the first ``modern scientist''?

Galileo is sometimes referred to as the first ``modern scientist'' because he was the first to extensively employ experiment and observation to support and test scientific models and hypotheses.

11.  What evidence is there that the Earth is rotating and that it is revolving around the Sun?

It is difficult for the lay person to prove that the Earth moves either on its axis or around the Sun. However there are several pieces of evidence that the lay person can appreciate. First, there is a type a pendulum called a Foucault pendulum that when set swing will appear to change the direction of its swing over time. In fact, the apparent change in direction of its swing is due to the Earth rotating on its axis underneath the pendulum. We are moving, not the plane of the pendulum’s swing.

Another piece of evidence that was not successfully measured until the year 1838 is the annual parallax of stars. This exceedingly small oscillation in the relative position of the nearest stars is unambiguous evidence for the Earth’s revolution around the Sun. Visit this link to our text to see stellar parallax illustrated and explained: http://www.astronomynotes.com/starprop/s2.htm .

12.  What shape are planet orbits and where is the Sun with respect to the orbit?

Planets orbit the Sun on ellipses with the Sun not at the center of the ellipse, but at one focus of the ellipse. This is summarized in Kepler’s 1st law. See Figure to the right.

13.  What happens to a planet's orbital speed as it approaches its farthest point from the Sun and as it approaches its closest point?

As a planet's approaches its farthest point from the Sun and as it approaches its closest point to the Sun its orbital speed changes. When a planet is farthest from the Sun is speed is slightly slower than its average speed in its orbit. When a planet is closest to the Sun is speed is slightly faster than its average speed in its orbit. This is expressed in Kepler’s 2nd Law as “A line joining a planet and the Sun sweeps out equal areas during equal intervals of time“. See figure to the right.

14.  A moon's closest distance from a planet is 300,000 km and its farthest distance is 500,000 km. What is the semi-major axis of its elliptical orbit?

As shown in the figure below, the sum of the closest distance to the planet (perigee) and the farthest distance from the planet (apogee) must equal twice the semi-major axis of the moon’s orbit. In mathematical terms: . So, the semi-major axis is just

.

15.  How will the perihelion compare with the aphelion for an ellipse with eccentricity = 0.1, 0.5, 0.8, 0.99? Find the value of (perihelion/aphelion) for each of the eccentricities. [Hint: using the relation that the perihelion + aphelion = 2× semi-major axis and a little algebra, you can find that (perihelion/aphelion) = (1-e)/(1+e).]

The most elegant way to approach this problem is to do a little algebra that I’ll illustrate below. It gets a little messy but then cleans up at the end. We start with and . The goal will be to combine these two equations, eliminating a and ending with a relationship between rP, rA and e. Hang on…here we go.

First start with . This can be rearranged to . Now put this expression for a in the equation . You get . Now we just have to manipulate this relation until we get the ratio on the left-hand side. Each of the steps below is one simple algebrtaic change. See if you can follow them.

So the result we are looking for is . Now we can use this relation for any value of e to calculate e the ratio of the perihelion and aphelion distances. For an astronomer this is real cool.

Eccentricity / Ratio of rP/rA
0.1 / 0.818
0.5 / 0.333
0.8 / 0.111
0.99 / 0.005

You can see that for high eccentricity orbits, the perihelion distance is much smaller than the aphelion distance.

16.  How is the average distance between a planet and the Sun related to the planet's orbit period?

The average distance between a planet and the Sun related to the planet's orbit period through Kepler’s 3rd Law that states that the average distance between a planet and the Sun if expressed in AU’s and squared will equal the planet's orbit period expressed in years and cubed. Mathematically Kepler’s 3rd Law is. This law is not an exact mathematical relationship, but it is very close to exact. See examples below.

Mercury takes 0.241 years to complete one orbit around the Sun and the semi-major axis of its orbit is 0.387 AU. Applying Kepler’s 3rd Law:

Saturn takes 29.424 years to complete one orbit around the Sun and the semi-major axis of its orbit is 9.58 AU. Applying Kepler’s 3rd Law:

I don’t know why the relationship is not exact, but I’m going to make some inquires.

17.  Which planet has a shorter period---one with a large average distance, or one with a small average distance?