We now understand, pretty well, how to tell which part of the sky is going to be
visible, at any given time, at any given location on Earth.
we have a missing ingredient, which is this mysterious sidereal time.
We need to understand sidereal time. Associated with this is another small
omission. We've discussed the position of all of
the stars on the celestial sphere but we've forgotten one star.
One that might have some impact on our lives.
Namely our local star, the sun. The sun, like everything else that is not
on earth is somewhere on the celestial sphere.
In the sense, that it rises in the east and sets in the west as the earth spins,
or as the celestial sphere spins. And so we can locate the sun somewhere on
the celestial sphere. But I haven't told you where.
And where that is, this is somewhat interesting because if you put the sun
somewhere on the celestial sphere there's a whole section of sidereal time.
A whole period of the 24 hour rotation of the earth.
When the sidereal time is close to the right extension of the sun.
When the stars overhead are, include the Sun, and we do not do astronomical
observations. At least not in visible wavelengths.
And so it's somewhat important to figure out where the Sun is.
And this will turn out to be at the root of this, sidereal time issue.
And so let's see how that works. This simulation might help us to explain
what's going on. We know that in addition to spinning
around its axis daily, the Earth also orbits the sun once a year.
Equivalently, if you prefer a stationary earth, the sun orbits the earth once a
year from, again, from the point of view of who sees what when.
the two are completely equivalent. the earth orbits the sun in the same
sense in which it spins about its axis. And what that means is, that as we see.
It, the sun also orbits us in the same sense in which the Earth spins about its
axis, in other words, the sun appears to orbit Earth once a year moving from west
to east along the celestial sphere. What that tells us is that the sun cannot
be assigned affixed right ascension. Its right ascension changes over the
year. As the Earth orbits, if we start at some
point on the Earth's orbit with the sun in this direction in the sky, imagine the
celestial sphere outside this figure at celestial midnight.
Then three months later when the earth has moved.
To this point in its orbit, the Sun has changed orientation relative to the
Earth. The Sun's right ascension is now six
hours. And furthermore, three months later, the
Sun's right ascension is twelve. And three months further on.
It is eighteen hours and after a complete rotation, the sun will have made a
complete circuit of the celestial sphere moving to the east in the direction of
increasing right ascension that the sun's right ascension goes through a full 24
hours in the course of the year. You can't assign the sun affixed position
on the celestial sphere, rather, it moves along the celestial sphere to the east at
the rate of one rotation per year. Now note the celestial sphere is moving
from east to west as it rotates around the Earth.
The Sun is moving along the celestial sphere from west to east.
This is important so let's think about it again.
As the earth spins about its axis once a day, it also orbits the sun once a year
moving in the same direction. Which means, as seen from Earth,
the sun orbits us once a year. So that means the sun moves along the
celestial sphere moving from west to east in the direction of increasing right
ascension to the east. And it completes one revolution per year
around the celestial sphere. This means that which stars are invisible
because they're only up at the same time as the sun.
Well, that changes over the course of a year.
So all stars get their chance to shine, so to speak.
That is good, we get to see the entire sky.
It also means, that since the sun is moving across the sky from west to east
while the celestial sphere is rotating from east to west.
The sun is carried by the celestial sphere, so it rises in the east and sets
in the west. But its motion across the sky is a little
bit slower than the motion of the stars. How much slower?
Over the course of a year the Sun moves backwards along the celestial sphere by
one complete revolution. Let's see that extra day one more time.
This simulation will help us to understand the consequences of the fact
that as the Earth spins it's also orbiting the Sun.
And therefore the Sun is moving along the celestial sphere.
to make things a little more clear, we have pretended that the sidereal day, or
a day is not 24 but 240 hours long. This will make the effect much clearer.
So imagine that we begin our simulation with the sun directly overhead for this
tallest observer on the Earth. So this observer will presumably think
that it is noon. And now, let a sidereal day go by.
The Earth will have completed a 360 degree rotation.
Note it's, it's oriented in exactly the same way it was before.
However, it is not yet noon. The time it takes from noon back to noon
again is longer than the sidereal day because in the course of this rotation,
exaggerated by a factor of ten the earth has moved along its orbit.
There's this extra bit of rotation required to get to noon.
From the point of view of earth this is a consequence of the fact that over the
course of the day as earth spun about its axis but also moved the sun moved in this
direction to the east along the celestial sphere.
We can do that again. 360 degrees rotation and a little bit
required to realign us to the sun. Indeed,
the time from noon to noon is longer than the time it takes earth to rotate by 30,
360 degrees, by about one over 365 of a day because the sun moves back along the
celestial sphere by one over 365 of it's complete rot, annual rotation of the
celestial sphere. That's about four minutes.
So, which of these is 24 hours do you think?
Is an hour going to be a twenty-fourth of the time it takes the earth to rotate 360
degrees, or would you define an hour to be a twenty-fourth of the time from noon
to noon? Clearly, you want it to be the time from
noon to noon. Our clocks keep solar time.
The reason for this is, because as we saw, while you, the difference between a
sidereal day, 24 hours, and the time from noon to noon is only four minutes, these
four minutes accumulate over the course of a year, to a complete day.
If you tried to work with a sidereal clock, then if you started it out such
that noon fell at twelve hours sidereal, six months later, noon, the sun over
head, would fall at zero hours sidereal. And if you tried to operate by the
sidereal clock, you would want to have lunch.
When it was darkest. This does not work very well for
agriculture, though it is very well for astronomy, cause sidereal time keeps
track of the stars. And the clocks that run our life, of
course, keep not sidereal, but solar time.
We call that local time. And so what that means is that these
solar clocks. Our local clocks that keep solar time run
slower than a sidereal clock. 24 sidereal hours, one 360 degree
rotation of the earth is less than 24 solar hours, the time kept by our clocks
by about four minutes. A little less than four minutes, one over
365 of a day to be precise. So our sidereal clocks run faster than
the clocks that are. Used to measure time.
So how do you relate the two? Well, we have this issue of two, clocks
one of which runs faster than the other. They're both 24 hour clocks and so at
some point they agree. And then, the faster the clock runs
ahead. And it will remain ahead until it has
completed one full 24 hour rotation more that the other clock.
And this takes, by definition, precisely a year.
Once a year there is a day when sidereal and solar clocks agree and by convention
that day is set to happen. We'll see why in our next lesson on or
about September 21st. So on or about September twenty first
wherever you are on Earth you're sidereal time is approximately equal to your local
time. Now.
I should qualify this. Remember local time varies from position
to position continuously. Local time also varies from position to
position. If you move about fifteen degrees east in
longitude then local time is an hour later.
That's what time zones are about but for the convenience of arranging train
schedules, we don't let local time vary continuously so that each town has its
own local time. Local time is fixed over an entire
fifteen degree slice of the Earth, and then jumps by an hour.
Whereas sidereal time is defined locally so varies continuously.
So, even when I say sidereal time equals to local time I mean,
to within half an hour which if the precision that we are working is good
enough. So on September 21st to then this half an
hour in precession, sidereal time is equal to local time and then if we know
that we can compare that to add any time before or after September 21st because we
know that the sidereal clock runs faster. A solar day is four minutes longer than a
sidereal day. So, a day later, the sidereal clock will
have become, run ahead and be four minutes fast.
And D days after September 21st, the sidereal clock is ahead of the local
clock by D * four minutes. D days before September 21st,
it was behind by 4 minutes, and catching up.
Now of course this is approximate. 4 minutes is not precise.
We are in any event ignoring time zones. Beware of Daylight Savings Time.
This expression of course ignores the jump by an hour that we artificially
introduce into our clock. So this is Standard Time.
But if you want to, this is good for dates that are near September 21st.
We can rescale this at the four points of the compass if you will of Earth's orbit.
So on December respectively March respectively June 21st, you can reset
things so sidereal time is local time plus 6, 12, 18 hours and then on
September 21st the difference will become 24 hours, which is the same as zero.
So now, we know how to find sidereal time.
And. Now that we know how to find sidereal
time, let's use this to, solve an actual problem that might interest us.
So suppose that we are interested in looking at the star, bright star Vega in
the constellation Lyra. And we want to know when, Vega might be
visible in the sky as high as it can around midnight.
Now, Vega, if we look it up, has a right ascension of 18 hours and 36 minutes.
According to what we said, Vega is the highest in the sky.
It is crossing our meridian when our local sidereal time is 18 hours and 36
minutes. When our local meridian corresponds with
the meridian on which Vega lies. I want to know, when is this going to
happen at midnight? Well, this will happen when local time of
24 hours, which is the same as zero hours, corresponds with sidereal time
equals 86 hours and 36 minutes. Well, if you look back at the previous
slide, you'll realize that on a. June twenty-first we had sidereal time,
was approximately local time +18 hours and so if we wait nine days later.
Sidereal time will be local time plus 18 hours plus 4 minutes times D, if we want
4 minutes times D to be 36 minutes, D is going to be nine days.
And Vega will be high overhead at midnight, 9 days after June 21st until
about June 30th. This might explain to you why the group
of stars, that, the bright triangle of bright stars that included Vega, was
something they called the summer triangle.
And so now, we finally have a way of understanding what the sidereal time is.
Predicting which stars will be overhead in which season.
And at any given time, we can map what the picture of the sky is that we'll see.
So we have a complete solution of the mathematics problem we set out to solve,
describing which part of the sky will be visible where and when.
Congratulations.