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Jan. 28 |
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Last lecture I talked about our place in the Universe. Today I'm going
to try to go back to just what we can see in the sky, relating
that a little to the big picture.
Well, what we see depends also on how we move. Let's not trace the
history--once upon a time everyone thought the Earth stood still, but it's
easier to explain motion in the sky by how we now know things to be.
How we move:
Earth rotates on an axis through the
North and South poles, once
every 24 hours. This causes the Sun to appear to rise and set, it causes
the stars to appear to move in the night sky.
We imagine the stars all on a sphere surrounding the Earth called
the celestial sphere. This is because we don't have depth
perception of stars.
As the Earth rotates, everything in the sky, which we can imagine is on
this sphere, appears to move in a circle.
A couple of things people don't always realize: there are still stars in
the sky in daytime, you just can't see them because the Sun is too bright.
Also, not all stars rise and set. In fact the Sun doesn't
always rise and set for some places on the Earth.
See, the Sun and the stars appear to move in circles as the
"celestial sphere" rotates. (Really they stay still and we turn around.)
When those circles cross our horizon, then the object rises or sets.
Everything appears to be moving in a circle about the celestial
pole, which is a point we imagine on the celestial sphere that's right
above the North Pole. Luckily for us, there's a star pretty close to that
point in the sky above the North pole. We call that the North Star, or
Polaris. There's nothing special about the star itself, other than that
it happens to be above our North pole.
We relate what we see in the sky to our horizon. Because the Earth
is round but much bigger than we are, we can't see the entire "celestial
sphere" at once, but only half of it at a time.
(So then astronomers define the zenith as the point that's
directly above you, along the line at right angles to the horizontal
plane. The meridian is the line connecting north and
south--this is why times before noon are a.m. and times afternoon are
p.m. The m. is for "meridian", when the Sun crosses that line.)
The pole star, the north Celestial pole, appears where in the sky?
Well, that depends on where you are.
If you were standing on the North Pole, then Polaris would be pretty much
straight above you all the time. All the stars would appear to go in
circles about this point straight above you. So nothing would
rise and set. None of the circles would cross the horizon. Try it out in
StarGazer by setting your latitude to 90 degrees (so that you're standing
on the North Pole.)
Ok, now what would happen if you stood on the Equator, at 0 degrees
latitude? Once again, all the stars appear to go in circles about the
North Star. But now they're going in circles about a point that's on the
horizon! So all the stars rise and set.
Here's a summary table:
So what do you think the altitude (the angle above the horizon) of Polaris
will be here in Claremont, at a latitude of 34 degrees? Well, just going
by example, it looks like the altitude of Polaris should be the same as
the latitude.
It turns out that this is right, and it's what led to the usefulness of
astronomy as a navigation tool. From finding the altitude above the
horizon of Polaris, you can find your Latitude! They are the same
number!
(Actually finding your LONGITUDE from stars is much harder, and led to the
development of accurate clocks. Dava Sobel wrote a book called Longitude
(also turned into a TV
movie) about how hard measuring longitude was.)
Well, this is not only true for a few examples, we can also prove it
mathematically with a little geometry. Understanding the following
argument is not essential, but if you're curious and can handle
the geometry, it explains why this works.
Ok, imagine you are standing on the Earth. North is straight above, to
the North Star. (Polaris is not exactly at due North, but close
enough!) So those of us in the LA area are at 34 degrees latitude. I've
drawn in lines from the center of the Earth to the equator, towards the
north pole, and towards our own zenith.
Our horizon marks the boundary beyond which we can't see. (Actually
cosmologists also use the term horizon to mean the boundary of
events in our Universe that are observable. If two galaxies collided 10
billion years ago but they were 20 billion light years away from us, their
light wouldn't have reached us yet.)
So we're asking: how high is the North Star above our horizon?
In the diagram, you see that this angle is also equal to the angle on the
other side--the other angle between the horizon line and the line going up
to Polaris.
Now, remember that a radius and a tangent of a circle meet at right
angles. So the remaining angle between the zenith line and North line
must be 90 degrees minus the altitude (the altitude is the angle of the
North star above the horizon).
The angle between the equator line and the North line is also 90 degrees.
So we're dealing with a right trangle there in red. The angles of a
triangle add up to 180 degrees. 90 are taken by the right angle, so the
angle of the latitude (that's the angle between the equator line and the
zenith line) must be 90 - (90 - altitude).
So altitude (of North Star) equals latitude. Just like switching the
letters around!
Question for thought: why do both of those parallel lines pointing
up point towards Polaris?
Here is a simulation in the SkyGazer software of the night sky,
showing the altitude of the North Star from LA:
Polaris, the North Star, can be found in the night sky (if it's dark enough)
by first finding the Big Dipper (Ursa Major, the big bear.) You follow the
two "pointer stars" of the Big Dipper, and they point to Polaris (which is
also the tail star of the Little Dipper.)
Here is an image showing the night sky from Anchorage Alaska, latitude
about 60 degrees:
In fact, astronomers use a coordinate system to describe where things are in
the sky, and the coordinate system is very similar to latitude and
longitude.
If you imagine taking the lines of constant latitude and projecting them
upon the sky, onto the "celestial sphere", then for each latitude there is a
"declination" in the sky above it.
For example, Polaris is at (or near) 90 degrees North declination. At the
zenith for someone on the equator is the "celestial equator", at 0 degrees
declination.
The other celestial coordinate, "right ascension" is a little more
complicated so we'll talk about it next time.
This causes:
In looking at the image below (from StarGazer) you can see that nights are
longer in the Northern hemisphere in winter. As we go around a circle
about the Earth's axis, more of that distance is in shadow than in
light.
Even more important, look at the angle above the horizon of the Sun
at noon during winter.
Seasons happen because the Earth's spin axis is tilted relative to
the plane that it is moving in for its orbit around the Sun. This tilt is
23 1/2 degrees.
Similarly for summer, the daylight is longer than 12 hours a day in
the Northern hemisphere, and the Sun's greatest altitude here in the LA area
would be 79.5 degrees.
Figuring out how planets move was complicated, because we are ourselves on
a planet that's moving. Earth's motion affects how the planets appear to
move.
The inner planets, which orbit closer to the Sun than the Earth,
and outer planets which orbit further away have different
appearances. The inner planets always appear pretty close to the Sun in
the sky (this is why Venus is the "evening star" or "morning star".) You
will never see Venus 180 degrees opposite to the Sun in the sky, but you
can see Jupiter like that.
Also, the inner planets move faster in their orbits than the outer
planets. The Earth moves faster in its orbit than Mars, for example.
Because of this, the Earth can catch up with Mars and pass it. It then
appears in the sky that Mars has "backed up" and started to move in the
opposite direction for a bit! This is called "retrograde motion."
In olden days (they times of Greek and Egyptian astronomers) this motion of
the outer planets was explained by inventing "epicycles." The planets
were thought to move in circles around other circles. All the
planets were thought to move about The Earth! What doomed this
system was that it kept getting more and more compliated. Whenever a
prediction didn't work, they added another epicycle, another circle for
the planet to move around. If a theory needs to be propped up all the
time by adding extra assumptions, it's not a good theory!
Well, stars are much further away than planets. But for the nearest
stars, we can see them appear to move because of our orbit around the
Sun!
Astronomers can use this effect to measure the distances to nearby
stars! This is actually very similar to the way that you estimate
distances to nearby objects using the stereo vision that results from
having two eyes.
If you put your finger close in front of your eyes and then look through
one eye or the other, you'll notice that it appears at a different place
relative to things further away, which don't appear to move as much.
This effect is called parallax. Here
is a very nice demonstration of this effect.
Instead of having the two vantage points be your two eyes, in parallax the
two vantage points are the Earth 6 months apart in its orbit, 2
Astronomical Units (300 million km) apart in distance.
The 2 pictures in the link above show a nearby star appearing in 2
different places, just like your finger appears in 2 places when you look
through one eye or the other.
The more a star appears to move with the Earth's motion, the closer it is.
Because even the nearest stars are so far away, the angle they move
because of the Earth's orbit is always small. 360 degrees make up a
circle, but we subdivide a degree into 60 "minutes of arc" (or "arc
minutes" or just "minutes") and each minute is divided into 60 arc
seconds. (Note: these have nothing to do with measures of time, except
they sound similar and are also based on 60.)
If a star appears to move in the sky by 1 arc sec in one direction and
then 1 arc second in the other direction over 6 months, then we define
that distance to a star as one parsec. Trigonometry (or just
geometry) using the size of the Earth's orbit tells us that 1 parsec is
about 3.3 light years. So parsecs and light years are both units of
distance.
Longitude
the book
Longitude the TV
movie
The Celestial
Sphere from Astronomy Notes dot com.
Earth's Daily Rotation
This page is being updated!
Location Latitude Altitude of Polaris What
stars do
North Pole 90 degrees 90 degrees All
circumpolar--no rising
or setting Equator 0 degrees 0 degrees All rising or
setting--no circumpolar Here (LA area) 34 degrees ? What do you
think? 
Notice that SkyGazer uses a projection system that gets 180 degrees onto
your screen, so you can see both due East and due West. As a result, some
parts of the sky may seem distorted. It's like a Mercator projection map of
the world that makes Greenland and Antarctica seem so big, even though
they're just insignificant wastelands filled with penguins. (I'm joking.
There are no penguins in Greenland.) 
You'll notice that on SkyGazer I've turned on the option to display
coordinates, and at Polaris the coordinates meet just like latitude and
longitude lines meet at Earth's North Pole.
Earth's Motion Around the
Sun--seasons
The Earth goes around the Sun once a year, once every 365.24 days.
So why do we have winter? Some people think it's because the Earth is
furthest from the Sun in winter. Actually, the Earth is about 3%
closer to the Sun in winter!
Motions of the planets
(I covered this material for the 2:45 pm class but
not
the 1:15 class, which will see it on Wednesday) 
A diagram showing how the Earth "catching up" and "passing" Mars, causing
its path in the sky appear to change direction. This is called retrograde
motion, and was better explained by the idea that planets went around the
Sun than the idea that circles within circles or "epicycles" made the
planets go backwards
Earth's Motion Around the
Sun--Parallax
So the Earth's motion around the Sun affects how we view planets. Could
it affect how we view stars? 
If something switches in the sky over 6 months from 1 arc
sec to one side, and then to 1 arc sec to the other side, then we know the
distance is 1 parsec, or about 3.26 light years. This results from the
Earth going around the Sun. Nothing is to scale in this diagram.
| Page | Problems |
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| 36 | 1-9 |
| 63 | 1-8 |
| 64 | 10,11 |
| 88-89 | 1,14 |
Due Wednesday, Jan. 30