Download Astronomy 1400: Homework 3

Astronomy 1400: Homework 3
Due in class, Friday, September 19
Name:
Instructions: To receive partial/full credit you must show your work or explain your answer thoroughly. Please circle your final answer to each problem
if it is a number.
1. In this problem, we are going to explore some properties of Venus’s motion. Venus takes
225 (Earth) days to orbit the Sun at a distance of 0.723 AU (assume Venus orbits in
a circle for simplicity). As we saw in the second lecture, Venus is only slightly smaller
than the Earth with a radius of 6051 km. However, Venus has a very strange rotation
property. Appendix E says that Venus’s rotation period is -243 (Earth) days (negative
and much longer than any other planet in the solar system).
(a) (5 points) What does the information above imply about Venus’s day and its year
(just compare the length of the day and year)?
Venus’s day is actually longer than its year!
(b) (5 points) Why is the rotation period negative (unlike Earth and most of the other
planets)?
The short answer is that Venus appears to rotate backward!
The Earth and the rest of the planets (except Venus) all rotate in the same direction
that they revolve (counter-clockwise when viewed from the Earth’s North Pole).
However, while Venus revolves counter-clockwise like the others, it rotates clockwise
(as viewed from above the north pole of the Earth). Scientists commonly denote a
different direction with a negative sign.
(c) (5 points) What does a negative rotation period have to do with the angle of the
axis tilt (hint: what range of axis tilt angles would you have that correspond to a
negative rotation period)?
If a planet is titled more than 90◦ toward the Sun (with respect to its orbit), then it
appears to be spinning backward, which corresponds to a negative rotation period.
So, negative rotation periods correspond to 90◦ < φ < 180◦ .
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(d) (15 points) How fast does Venus spin on its axis (in mi/hr)? How does this compare to what we found for the Earth in Homework 1 (make a simple quantitative
comparison; see Appendix C.5 for a guide)?
v=
©
©
d
2π(6051 ©
1 mi
km)
1©
day
=
×
=
×
©
©
©
t
−243 ©
days
1.609 ©
24 hr
km
−4.05 mi/hr
In magnitude, this is about 250× (much!) slower.
(e) (15 points) How fast does Venus orbit the Sun (in mi/hr)? How does this compare to what we found for the Earth in Homework 1 (make a simple quantitative
comparison)?
v=
©
©
2π(0.723 ©
93.0 × 106 mi 1 ©
AU)
day
d
=
×
=
×
©
©
©
t
225 ©
days
1©
24 hr
AU
78200 mi/hr
This is about 1.2× (only a little) faster.
(f) (5 points) Even though Venus rotates close to the same rate that it revolves, why
are the answers on parts (d) and (e) so different (the speed in part (e) should be
much larger than the speed in part (d))?
Parts (d) and (e) are so different because, while the times are similar, the distances
are not (the size of Venus is much smaller than its orbit); the velocity depends on
both the time and the distance: v=d/t.
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(g) (10 points (bonus)) Extra credit: If you could stand on the surface in Venus’s
northern hemisphere (north defined in the solar system from above the Earth’s
north pole) and see the Sun (rise up from a horizon, come to some height above
the southern horizon, and set on another horizon), would the direction of the Sun’s
motion through the sky be different from what we see in the northern hemisphere
on Earth? How? Qualitatively (you do not have to calculate numbers), how would
the time that the Sun spends above the horizon compare to the time spends above
the horizon on the Earth’s (e.g., similar, much longer, much shorter)? Why?
The Sun would appear to go very slowly backward (rises in the west and sets in
the east) in the sky. The time the Sun spends above the horizon during one day on
Venus would be a good fraction of an Earth year. It would move backward because
of the opposite rotation direction (compared to the Earth). Remember also that
the Sun appears to move backward in the sky through the constellations during the
year as we revolve around the Sun (in the same direction as Venus does).
2. (10 points) Uranus also has unusual orbital properties. The rotation period in Appendix E is also negative (-0.72 Earth days) and it has an unusual axis tilt of 97.92◦ .
What does this imply about Uranus’s configuration in orbit and its seasons? Explain.
Drawing a picture may help.
Uranus’s nearly 90◦ tilt implies that in its summer and winter, the poles nearly point
directly at or away from the Sun. The Sun would change its height in the sky much
more than what we see on the Earth, and, therefore, Uranus has more extreme seasonal
changes.
3. (20 points total) What did Nicholas Copernicus, Tycho Brahe, Johannes Kepler, and
Galileo Galilei each contribute to astronomy?
(a) (5 points) Copernicus:
Copernicus published work describing the Sun-centered solar system model.
(b) (5 points) Brahe:
Brahe made what is generally considered the best set of naked-eye observations ever
made that enabled Kepler to discover the laws of planetary motion.
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(c) (5 points) Kepler:
Kepler, Brahe’s apprentice, discovered the basic laws of planetary motion.
(d) (5 points) Galileo:
Amongst his many accomplishments as arguably the greatest scientist of his time,
Galileo was the first to use the telescope (he did not invent it) to take observations
of the heavens that sealed the case for the Sun-centered solar system (e.g., Venus’s
phases, Jupiter’s moons).
4. (a) (10 points) Why is astrology a pseudoscience and astronomy is not? Explain.
Pseudoscience means “false science”. To be science, a subject must conform to the
hallmarks of science listed in the text: (summed up) being able to test ideas of
how natural processes work that can, in principle, be disproven. Astrology seeks to
explain how the positions of astronomical objects affect our lives, yet such forecasts
do not stand up to scrutiny; no causual relationship has ever been established.
(b) (10 points (bonus)) Extra Credit: Name another example of a science or pseudoscience of your choice and explain why it is a science or pseudoscience. Do this in
your own words! Don’t forget the ethics policies of the course! If you
copy your answer, you will receive a zero on this problem and likely the
entire assignment.
This is assessed on a case-by-case basis.
5. (10 points) A small comet was discovered with an orbital eccentricity of e=0.6 and a
semi-major axis of 46 AU. It passed closest to the Sun in 2010. When (what year) will
it make its next close passage?
Since the object is orbiting the Sun and we want to express the answer in years, we can
use the simple form of Kepler’s third law:
p2years = a3AU ⇒ pyears = (46 AU)3/2 = 312 years
The next passage of the comet will be in 2010 + 312 =
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2322 .