Cosmological Models Review

 

1.      Know the meaning of the following vocabulary words:  direct or prograde motion, retrograde motion, conjunction, opposition, elongation, geocentric, deferent, epicycle, heliocentric, ellipse, orbital period, synodic period, ellipse, semi-major axis, focus, eccentricity, aphelion, perihelion, empirical laws.

2.      Know the significance (regarding cosmological models) of the following persons:  Aristotle, Ptolemy, Copernicus, Brahe, Kepler, Galileo, and Newton.

3.      State and explain seven points of Copernicus’ model of the cosmos.

4.      Astronomers have measured light from thousands and thousands of stars and determined that the vast majority of these are moving away from the Earth and the Sun.  Does this mean that the Earth and Sun are at the center of all the observable stars?  How would the Copernican Principle apply to these observations?

5.      Make a diagram of a heliocentric solar system showing the Sun, the Earth, and Venus at a position so that it would appear as a waxing gibbous phase.  Make a second diagram of a geocentric solar system showing the Sun, the Earth, and Venus at a position so that it would appear as a waxing gibbous phase.  How do these diagrams help support the heliocentric view of the cosmos?

6.      Epicycles in the Ptolemaic system helped to explain what observable attributes of the planets?  Epicycles in the Copernican system helped to explain what observable attributes of the planets?

7.      Which planets never reach opposition?

8.      (a) A planet at opposition has what approximate elongation?  (b) A planet at conjunction has what approximate elongation?

9.      Suppose for argument’s sake there was an additional planet orbiting the Sun at a distance between that of Earth and Mars.  As seen from here on Earth, how would it appear –  (a) Would its synodic period be greater or less than that of Mars?  (b) Would it undergo both conjunction and opposition?  (c) Would it exhibit retrograde motion?  And if so, during what part of the synodic cycle?  (d) At what point during its synodic cycle would it appear brightest?

10.  Suppose for argument’s sake there was an additional planet orbiting the Sun at a distance between that of Mercury and Venus.  As seen from here on Earth, how would it appear –  (a) Would its synodic period be greater or less than that of Venus?  (b) Would it undergo both conjunction and opposition?  (c) Would it exhibit retrograde motion?  And if so, during what part of the synodic cycle?  (d) At what point during its synodic cycle would it appear brightest?

11.  Kepler’s 2nd Law of planetary motion deals with areas of sections of ellipses – a very abstract concept.  What is the general gist of Kepler’s 2nd Law without referring to mathematical or geometric concepts?

12.  Earth has a speed of 30.3 km/s as it reaches its perihelion distance of 1.47 × 1011 m.  The eccentricity of Earth’s orbit is 0.017.  Use Kepler’s 2nd Law to determine its speed at aphelion.  (Hint:  determine the aphelion distance and draw two “skinny triangles” – one representing the motion of the Earth in one second at perihelion, the other representing the motion of the Earth at aphelion.  The area of a skinny triangle can be approximated by assuming the short side to be the base and either of the long sides to be the height.)

13.  Kepler’s 3rd Law of planetary motion can be written in the form of an equation.  What is the general gist of Kepler’s 3rd Law without referring to mathematical concepts?

14.  According to Kepler’s Laws would it be possible for a planet to orbit the Sun in a perfectly circular orbit?

15.  What are two problems with Kepler’s Laws?

16.  Use concepts from Newton’s Laws to explain why Kepler’s 3rd Law is found to be (essentially) true.

17.  Use concepts from Newton’s Laws to explain why Kepler’s 1st and 2nd Laws are found to be (essentially) true.

18.  A certain asteroid has an orbital period of 2.00 years and an aphelion distance of 1.80 A.U. as it orbits the Sun.  (a) Determine its average distance from the Sun.  (b) Determine the eccentricity.  (c) Determine the distance from the Sun to the exact center of its elliptical orbit.

19.  Suppose there was a planet Vulcan located at an average distance of 0.20 A.U. from the Sun and with eccentricity of 0.15.  (a) Determine its orbital period.  (b) Determine its maximum elongation it would ever exhibit as seen from Earth.

20.  Use the information from Table 2.1, p. 47, to answer the following:  (a) Determine the perihelion distance.  (b) Determine the aphelion distance.  (c) Construct a scale drawing of Pluto’s elliptical orbit about the Sun.  (Hint:  use an ellipse drawer like we did in class.  How long should the string be?  How far apart should the foci be?)

21.  Suppose two alien worlds are discovered to be orbiting a distant star.  Astronomers do this by measuring the small “wobble” of  that star (not by actually seeing the planets).  Judging by the wobble of the star the astronomers determine that planet B has an orbital period 8 times greater than planet A.  Using Kepler’s 3rd Law determine how much closer or farther planet B is from the star than planet A.

 

 

Selected Answers

 

1.       

2.       

3.       

4.      The Copernican Principle would state that the Earth should not occupy a special place in the universe and that there must be an alternate explanation that would explain why the stars’ receding motion appears to be “centered” on us.  (And there is an explanation!)

5.      Using a heliocentric model, Venus would appear in a gibbous phase only as it nears opposition, however Venus never reaches opposition!  If it never reached opposition in a heliocentric model it would never appear gibbous – but rather would always appear as a crescent phase.  Therefore this supports a geocentric model in which the gibbous appearance occurs near conjunction (which is what is in fact observed).

6.       

7.      Mercury and Venus

8.      a. 180° 
b. 0°

9.      a. greater
b. yes
c. yes, during opposition
d. opposition

10.  a. less
b. no – only conjunction
c. yes, during conjunction
d. at a point just before or after its maximum elongation

11.   

12.   29.3 km/s

13.   

14.   

15.   

16.   

17.   

18.  a. 1.59 A.U.
b. 0.134
c. 0.213 A.U.

19.  a. 0.0894 yrs (32.7 days)
b. 13°

20.  a. 29.6 A.U.
b. 49.3 A.U.
c.

21.  planet B is 4 times farther from the star than planet A