# Solar Eclipse Maths and the Cosmic Coincidence of the Saros Cycle

a solar eclipse is an absolute delight to behold and so i'm going to show you
the mathematics behind how we can predict when they're going to happen
i'm recording this before the 2015 eclipse
but don't worry the information I'm giving you now remain valid for at least
the next nine hundred and eighty-five years
as you may have noticed the moon is not transparent
this means that blocks the light from the Sun and it casts a shadow occasion
that shadow moves across the earth and if you're standing with a shadow fall's
your witness a total solar eclipse from our point of view
the moon has moved in front of the Sun and completely blocked its light for a
solar eclipse to work though
the moon has to be big enough to block out the Sun and believe me this diagram
is not to scale
in reality the moon is 400 times smaller than the Sun but in a fantastic cosmic
coincidence
it's approximately three hundred and eighty nine times close-up on average
and that's close enough
it means from our point of view the moon is almost exactly the same size as the
Sun in the sky
this means it will line up and it will completely block the disc of the Sun if
the moon with any smaller or any further away
it wouldn't be big enough from my point of view to cause total solar eclipses
but of course it could be bigger and it could be closer and would still get
eclipses but fantastically because the apparent size of the moon matches
exactly the apparent size of the sun's photosphere the bright surface of the
sun it blocked out the bulk of the light from the Sun but it doesn't block the
Sun Thank atmosphere as you can see from this photograph taken in a total solar
eclipse the Sun has a very faint atmosphere with fantastic structure in
it this is normally swamped by all the photons coming from the rest of the Sun
only because our eclipses line up so well
do we ever get to see it we are incredibly lucky that from our point of
view the size of the moon matches exactly the size of the Sun
I mean the probability of that happening is literally astronomical
it may look like the Sun the Moon move around randomly in the sky but that's
not the case
eclipses occur based on the some earth-moon system the earth orbits
around the sum and the moon orbits around the earth and its when those two
orbits come into alignment and bring the moon between the Sun and the earth that
we get a solar eclipse
although i should point out of course once again that diagram was in no way to
scare in fact there's pretty much nothing in this video
that's to scale to illustrate what happens during a solar eclipse here i
have the earth and you're going to be the Sun so your line of sight of the
rays of sunlight coming out of your face or whichever part of your body you shoot
the Sun shines out of and then it hits the earth except of course when the moon
gets in the white as it orbits around the earth
it causes eclipses and the moon orbits the earth every 29.5 3059 days and so
there you have it
this is why we get a solar eclipse every 29 days 12 hours 44 minutes and three
seconds
unfortunately it's not that simple of course physics isn't that simple
there's actually more than one type of lunar month poor exactly the same orbit
so for a start
the moon doesn't go around the earth in a circle actually follows an ellipse
which means at different points
the moon is different distances from the plant and its closest point it's 360
3396 kilometers away from the earth and its greatest distance its 405,000 504
now the month i mentioned before the 29.5 three days
that's the time the moon takes to be from the point where its closest to the
Sun back to being closest to the Sun again a different month is 27.5 five
days and that's the time the moon takes from being at the closest point to the
earth back to being at the closest point to the earth and the difference there is
almost two days and the reason for that two day discrepancy is because the
elliptical shape orbit
during the movies 21.5 five-day lap around the earth the earth hasn't stood
still
it's orbiting the Sun it will have gone for about 30 degrees of its orbit which
means even though the moon is back at the same point relative to the earth
it's no longer the same point relative to the Sun it has to go a little bit
further to go between Sun and the earth once more and that's where the expert
two days comes from but it's not always the same amount of time because the
moon's elliptical orbit tracks with the earth
the direction is pointing changes relative to the Sun so different points
in the year it takes more or less time for the moon to once again go between
the Sun and the earth that time I gave you before the 29.5 three days
that's just an average but it's okay we can factor this all in we can do the
calculations on the elliptical orbit of the moon as the earth goes around the
Sun and we can calculate how often the moon goes between the Sun and the earth
unfortunately it's not that simple yep
there was a third type of lunar month so you're the son here we have the earth
the earth is orbiting around you in a fixed plane
the man however orbits around the earth in a different plane
its orbit is tilted at 5.1 degrees from the earth orbit around the sun and this
is why we don't get an eclipse every single month
the moon has to both be going between the Earth and the Sun as well as either
going down through the Earth's all that or coming back up again and that is our
third type of lunar month
the amount of time between the moon going down through the Earth's orbit
- when we get back to the same . and there is 27 . - 12 21 days the shortest
month yet and this explains why eclipses are so rare we only get just over to a
year in fact we average 238 solar eclipses per century but now we've got
everything sorted we can calculate when they are unfortunately it's not that
simple
remember them
and elliptical orbit around the Earth but that doesn't stay still
either it actually drift by 40 points 7 degrees every year so it rotates once
every eight point eight five years
yes the moon's orbit orbits and so the two points where it crosses the earth or
but drift around the earth over the course of eight point eight five years
and only when the moon goes between the Earth and the Sun while one of these
nodal points is also between the Earth and the Sun will we get a solar eclipse
unfortunately it's not that simple
any second now unfortunately it's not that simple
there's so much more to take into account the Earth's orbit of course
that's not a circle
it's no lips as well and a lip that also moves it rotates once every 20 and a
half thousand years and the u.s. gravity is a nice and tidy all know
for a start the earth is in a perfect sphere it's an oblate spheroid on top of
that it's covered in water and the title drag messes with everything and then
there's the gravity between the moon and the Sun so far already take into account
the gravity between the Sun and the year and then separately between the Earth
and the moon but of course it's a gravitational pull between the moon and
the Sun in fact the mutual gravitational attraction between the Sun and the moon
is over twice that between the Earth and the moon
the direction the moon is traveling relative to the Sun makes a big
difference on its speed and that is constantly changing in fact every single
number i have given you in this video every direction every speed every angle
every time they're all only approximations their averages the whole
system is constantly changing it is a wonder we have eclipses at all
what this means is we can't make any meaningful proper long-term predictions
about solar eclipses but for the reasonably near future we can and NASA
have gone through all of the years between negative 1999 and positive three
thousand they have analyzed all 60 1841 lunar months that's from New Moon to New
Moon and they've checked all eleven thousand eight hundred and ninety-eight
eclipses and put all the data up on their website but to the undrained I
just watching it happening
we're back to a random movement of the Sun and the moon in the sky and
occasionally they happen to cross but then out of all this chaos
another cosmic coincidence steps forward to save the date
just like the moon's apparent size happens to match the sun's apparent size
there is one special combination of the three different types of Luna
and that happens to line up perfectly
if you take 220 three new moon months so from closest to the Sun the closest to
the Sun and you compare that to 239 of the month between the closest point of
the earth to the closest point of the earth and then you compare those two 242
of the months between when the moon goes through the Earth's orbit back to win
our next goes through the earth orbit
all three of those are pretty much exactly the same length of time they all
come out spot on 18 years 11 days and eight hours
and so it means if out of all this random movement you somehow have a solar
eclipse
you know that 18 years 11 days and eight hours later we will have had an integer
number of all the different types of months in bold and so we will return to
pretty much exactly the same geometry
there will be another very similar eclipse 18 years 11 days and eight hours
later
it's called the sarah cycle and allows us to predict when eclipses are going to
happen in the reasonably near future
except of course unfortunately it's not that simple
thanks you're welcome
in 2009 july twenty second i was lucky enough to witness the total solar
eclipse when the moon shadow came across the Pacific Ocean through Japan across
mainland China and then out through India because of the Saros cycle
I know I can enjoy the next equivalent to clips in 18 years 11 days and eight
hours time and it will look like this in the year 2000 27 on august second
except those eight hours are about a third of the day and so the Earth's
rotation won't line up everything else will
but the earth will have spun about another third of a rotation so the
eclipse will actually happen over here and then another Sarah cycle later in
2045 on august 12
it will be another third of a rotation across and it will go through America
and this cycle can also be worked backwards before the 2009 eclipse is one
in 1991 then 1973 1955 1937 1919 and 1901 you notice that over time the
eclipses are actually working their way up the planet and that is because of the
slight mismatch of the Saros cycle with the rotation of the moon's an elliptical
orbit around the Earth and it means that the eclipses gradually drift further and
further north
this particular Sarah cycle Sarah cycle 136 actually began on the fourteenth of
june in the year 1360 on that day there was an eclipse right down near the South
Pole is now gradually been working its way up across the planet and the very
last one will occur on the thirtieth of july in the year two thousand six
hundred and twenty-two right up at the North Pole
there are currently 41 active Sarah cycles working their way up or down the
planet
so there you have it the maths and solar eclipses is an absolute mess because
orbits are really complex
cated however we do know solar eclipses are amazing to watch if you can go and
see one do of course be safe and don't blind yourself
and bear in mind that they're not gonna be around forever partly because the
moon is gradually moving further away because of that title drag actually
drifts about 3.7 centimeters throw away from the earth every year which means it
about point six billion years it will be too far away to cast a shadow on our
planet but apart from that we have no idea what the orbits will be like that
far in the future according to our best simulations in the next five billion
years
there's a one-percent chance that mercury will leave its orbit and either
crash into Venus or the Sun in fact there was a nonzero chance that in three
point three six billion years the gravitational force from Jupiter were
not mercury out of its orbit in turn will impact on Mars that within slam
into the earth
so you know enjoy eclipses while you can
for more math behind solar eclipses and all sorts of diagrams and calculations
and plots and things that I can possibly fit in this video's google checkout
NASA's five millennium canon of solar eclipses i'll put a link to it in the
comments below is based on the fantastic work of thread s been ax and genius
where you just want to see more videos of me talking about math
some of them a lot shorter and there's much more mathematically certain then
just click on the move
that's now orbiting me
yeah