Download Homework #1 10 points Question #1 (2 pts) Even in ancient times

Homework #1
10 points
Question #1 (2 pts)
Even in ancient times, astronomers knew that planets vary in brightness over the course
of several months or even years. Explain, why this observation can not be used to rule out
the geocentric model, in which all planets and the Sun orbit the Earth on circular orbits.
Ignore the epicycles, i.e. assume that the distance from a planet to the Earth does not
change with time.
There are actually two different explanations, each is sufficient for the full credit.
• In this model the planets stay at the same distance from the Earth, but since the Sun
moves in space, they will be at different distances from the Sun at different times, and
therefore the amount of sunlight falling on them will change with time. Since planets do
not emit their own light, only reflect the sunlight, their brightness will vary with time as
the amount of sunlight they can reflect varies with time.
• The planets in this model will also have different phases at different times (similar to
the phases of the Moon): the sun only shines on one half of a planet, and from the Earth
we can only see one half of the planet (like one side of the Moon). But those two halves
do not have to be the same. Sometimes we will see the fully lit half of a planet (like a full
Moon), sometimes we will see a half which is only partially lit (like a crescent Moon), and
occasionally we will see the dark side of a planet (like a new Moon). Thus, the planet will
appear to change its brightness with time just as Moon brightness changes with time.
Question #2 (2 pts)
A theory of a ”7am Universe” claims that the whole universe with everything in it, from
most remote galaxies to our own bodies, was instantaneously created today at 7am including the light arriving from distant stars and galaxies at the Earth, fossil bones in
the ground, tree rings in the old trees, and history books with words describing the past.
None of us ever had childhood, all our childhood memories were created with us at 7am
this morning. Is this theory falsifiable? If yes, describe an experiment that could, at least
in principle, falsify it.
It is not hard to persuade oneself that one can always construct a ”7am universe” so
that any experiment by anyone anywhere in the world would be consistent with the same
experiment done in a ”real” universe that existed before 7am. In other words, if you
imagine the universe as traveling along the time axis, one can always chop off the past at a
given point (7am today or 4000BC), and since the physics is causal (or, at least, we think
it is), then the future evolution of the universe is not affected by this chopping off the past.
In scientific terms, the ”7am universe” theory is not falsifiable, and so cannot be considered
a valid scientific theory. One can believe in it, but it does not belong to a science classroom.
Question #3 (2 pts)
Asteroid A has a semimajor axis of its orbit of 25 AU and a semiminor axis of 5 AU.
Asteroid B has the same semimajor axis as Asteroid A, but a semiminor axis of 20 AU.
Sketch the orbits of both asteroids around the Sun.
Note: You must sketch both orbits together on one sketch, with the Sun location clearly
marked - the whole point is to see how they relate to each other. Please, try to be
reasonably careful in your sketch - at least make sure that the semiminor axis of A is
about 4 times smaller than the semiminor axis of B, but the semimajor axes are reasonably
similar on your sketch.
An orbit of an asteroid is an ellipse. Asteroid A ellipse is ”fat”, and B ellipse is ”skinny”.
Skinny ellipse has its foci farther apart (and thus closer to the edge) than a fat one. The
Sun is at the focus of each orbit, thus two ellipses have one common focus. Thus, B comes
closer to the Sun in its closest approach than A, but moves farther from the Sun in its
farthest point.
Question #4 (2 pts)
When we are having a lunar eclipse on Earth, what would an astronaut on the Moon see?
A d paralunar eclipse is when the Earth casts a shadow on the Moon. Hence, an astronaut
on the Moon will see the Earth blocking the Sun, i.e. a solar eclipse.
Question #5 (1 pts)
Compute the distance from the Sun to a star that has a parallax of 1 second of arc (measure
it in astronomical units, it’s easier that way). This is an important distance - it is used as
the main distance measure in astronomy and is called ”parsec” as a shorthand for ”par allax
of one second of arc”. Again, you have to compute the distance, not to get a number off
the web or convert from one system of units to another.
The tricky part of this problem is that I actually did not define what the parallax angle was
- it was your task to find that out. Here is a picture from Wikipedia that shows it.
From the triangle formed by purple, yellow, and one of the blue points
tan(p) = 1 AU/d,
hence
d = 1 AU/ tan(p).
For small angles (as angles on the sky are), tan(p) ≈ p with very high accuracy, so 1 parsec
is 3600 × 180/π ≈ 206, 265 astronomical units.
Question #6 (1 pts)
Compute the periods of rotation around the Sun for the two asteroids from Question #3.
Which one takes longer to make one full orbit? These numbers may help you check whether
your sketch is correct!
From the Kepler’s second law,
P 2 = R3 .
Since both asteroid have the same semimajor axes, their periods are also the same, P =
R3/2 = 253/2 = 125 AU.