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.