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PHY 121 Astronomy, chapter 4: The Origin of Modern Astronomy
Homework: RQ. 2, 4, 7; PROBL. 2, 3
RQ 4-2
Why did classical astronomers conclude that Earth had to be motionless?
Solution:
Classical astronomers concluded that Earth had to be motionless because they could not
see any parallax on the stars. They started with the wrong premise that the stars are on a
sphere which is not too large in its diameter and so the stars were assumed to be much
closer than they actually are. Starting with this wrong assumption, they concluded that
the appearance of the constellations close to the ecliptic should change dramatically
(parallax) during one year if Earth would actually move relative to the sphere with the
stars. In addition, the fact that we do not feel any motion of Earth was taken to be as
proof that Earth is motionless.
RQ 4-4
In what ways were the models of Ptolemy and Copernicus similar?
Solution:
-Both Ptolemaic and Copernican system assumed uniform circular motion and needed
equants and epicycles.
-Ptolemy’s system required them to account for the variation in the orbital speed of a
given planet and retrograde motion. Copernicus’ system required them to account only
for the variation in the orbital speed of the planets.
-Both Ptolemy and Copernicus assumed a celestial sphere at a great distance.
RQ 4-7
Why did Tycho Brahe expect the new star of 1572 to show parallax? Why was the lack of
parallax evidence against the Ptolemaic model?
Solution:
Greek philosophy taught that “new stars” had to be closer to Earth than the Moon
because the heavens were perfect and unchangeable. This new star of 1572 should
therefore be close to Earth and show daily parallax as do the Moon and the planets.
However, Tycho Brahe observed no parallax and reasoned that it was further away than
the Moon and probably as far away as the distant stars. Also the Ptolemaic system taught
that the heavens were perfect and unchangeable. Since a new star appeared to be located
on the celestial sphere, the basic assumptions of the Ptolemaic system were not valid.
PROBL. 4-2
Galileo’s telescope showed him that Venus has a large angular diameter (61 seconds of
arc) when it is a crescent and a small angular diameter (10 seconds of arc) when it is
nearly full. Use the small-angle formula to find the ratio of its maximum distance to its
minimum distance. Is this ratio compatible with the Ptolemaic universe shown on page
45?
Solution:
Small-angle formula:
angular diameter linear diameter
=
206,265
dis tan ce
〈1〉
The farthest away an object is, the smaller its angular diameter. So when the angular
diameter is 10 seconds of arc, the distance between Venus and Earth is the farthest.
linear diameter
 10
 206,265 = dis tan ce
MAX

〈1〉 ⇒ 
 61
linear diameter

=
dis tan ceMIN
 206,265
⇒
dis tan ceMAX 6.1
=
dis tan ceMIN
1
In the Ptolemaic model, Venus is always between Earth and Sun. So according to the
Ptolemaic universe, shown on page 45, the same ratio is about 1.5 1.
PROBL. 4-3
Galileo’s telescope were not of high quality by modern standards. He was able to see the
Moons of Jupiter, but he never reported seeing features on Mars. Use the small-angle
formula to find the angular diameter of Mars when it is closest to Earth. How does that
compare with the maximum diameter of Jupiter?
Solution:
angular diameter linear diameter
=
206,265
dis tan ce
Distance between Mars and Sun: dist. MARS − SUN = 2.279 ⋅ 10 8 km
Small-angle formula:
1
Distance between Jupiter and Sun: dist. JUPITER − SUN = 7.783 ⋅ 10 8 km
Distance between Earth and Sun: dist. EARTH − SUN = 1.496 ⋅ 10 8 km
Hence,
Also,
dist.MARS − EARTH = 0.783 ⋅ 10 8 km
MARS : linear diameter = 6.796km
2
dist. JUPITER − EARTH = 6.287 ⋅ 10 8 km
JUPITER : linear diameter = 142.900km
1 ⇒ ang. diam.MARS = 206,265 ⋅
3
linear dist.MARS
2
→ ang . diam.MARS = 18 sec of arc
dist.MARS − EARTH
1 ⇒ ang . diam. JUPITER = 206,265 ⋅
linear dist. JUPITER
3
→ ang . diam. JUPITER = 48 sec of arc
dist. JUPITER − EARTH
Although Jupiter is in longer distance from Earth comparing to Mars, it looks bigger to us
for its angular diameter is 2.7 times bigger than Mars’s angular diameter. That is why
Galileo could see details on Jupiter’s surface.