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.
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.
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
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.
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.
Sun
Jupiter
Venus
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 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: rP  rA  2a . So, the semi-major axis is just
a
1
rP  rA   1 300,000 km  500,000 km   400,000 km .
2
2
Moon
Planet
Perigee
Apogee
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
r
little messy but then cleans up at the end. We start with rP  rA  2a and e  1  P . The goal will be to
a
combine these two equations, eliminating a and ending with a relationship between rP, rA and e. Hang
on…here we go.
rP
r
. This can be rearranged to a  P . Now put this expression for a in the
a
1 - e 
r
equation rP  rA  2a . You get rP  rA  2 P . Now we just have to manipulate this relation until we
1  e
r
get the ratio P on the left-hand side. Each of the steps below is one simple algebrtaic change. See if
rA
you can follow them.
First start with e  1 
rP
1  e 
rP  1  e   rA  1  e   2rP
rP  rA  2
rP  1  e  rA  1  e 
r

2 P
rA
rA
rA
rP 
1  e   1  e   2 rP
rA
rA
1  e   2 rP
rA

rP 
1  e 
rA
1  e   rP 2  1  e 
rA
1  e   rP 2  1  e
rA
1  e   rP 1  e
rA
1  e   rP
1  e  rA

rP 1  e 

rA 1  e 
rP 1  e 

. Now we can use this relation for any value of e to
rA 1  e 
calculate e the ratio of the perihelion and aphelion distances. For an astronomer this is real cool.
So the result we are looking for is
Eccentricity
0.1
0.5
0.8
0.99
Ratio of rP/rA
0.818
0.333
0.111
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
2
equal the planet's orbit period expressed in years and cubed. Mathematically Kepler’s 3rd Law is pyears
 aau3 .
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:
2
p years
 aau3
0.241 yr 2  0.387 AU 3
.0581  .0580
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:
2
p years
 aau3
29.4 yr 2  9.6 AU 3
864  884
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?
The planet with a smaller average distance (a.k.a. smaller semi-major axis) will have the shorter orbital
period. This is consistent with Kepler’s 3rd Law.
18. What is the semi-major axis of an asteroid orbiting the Sun with a period of 64 years? (Kepler's third law
works for any object orbiting the Sun.)
If the asteroid has an orbital period of 64 years, then, according to Kepler’s 3rd Law, 642 should equal a3.
So that a, the semi-major axis, is just a  3 64 2  64 2 3  64 2 
major axis of the asteroids orbit is 16.0 AU
1
Instructor Assigned Topic: None this assignment
0.3333
 4,096 0.3333  16 AU . The semi-