Introducing Gravity into the Heliocentric Model | Lectures in Astronomy |

[ ] roughly seventy years after the
publication of Copernicus’s book concerning the revolutions Galileo’s
observations showed that the Ptolemaic system is incorrect and that the
Copernican system is probably correct as a description of physical reality
however let’s go back a few decades to see what at least a few scientists were
thinking even before Galileo’s groundbreaking observations and to see
what kind of refinements had been made to the Copernican model there was one
alternative that combined the features of both the heliocentric or sun-centered
system and the geocentric or earth centered system and this was suggested
by the Danish nobleman Tycho Brahe he sometimes pronounced eco I’m told that
in Danish it might be closer to 2 Co anyway I’ll say Tycho cuz that’s what
I’m used to he thought that perhaps the moon orbits
the earth but everything else orbits the Sun so Mercury Venus Mars Jupiter and
Saturn orbit the Sun and then the Sun together with all of the planets that
orbit it in turn orbits the earth so basically everything except the moon
orbits the Sun and then the Sun along with its orbiting planets orbits the
earth this is essentially a change of perspective from the Copernican system
instead of everything orbiting the Sun including the earth he has everything
orbiting the Sun except the earth but then all those things orbit the earth so
it’s just a change in perspective and without actually knowing that the earth
is moving you actually can’t tell the difference between this and the
Copernican system now now it turns out now we know that the earth is moving oh
I’ll show you ways that we know later on but at that time no one really knew that
the earth was moving it was just conjectured by Copernicus that it’s
moving and this system was in a sense equivalent just a
spective but this was not a very attractive idea to either side neither
the geocentric nor the heliocentric people were particularly pleased with
this combination of motions Tycho had an interesting life at the age of fourteen
he saw a partial solar eclipse from Denmark which happened to be total as
seen from Portugal and he was very impressed that astronomers had been able
to predict this so he wanted to dedicate his life to making ever more accurate
observations of the Moon Sun and planets especially the planets and namely Mars
in order to refine the predictive power of of physics and of these models he
wanted to know exactly when eclipses and other things would occur
he was quite bright but also a rather at argumentative fellow he thought pretty
highly of himself and at the age of 20 he had a duel with a fellow student not
over a woman but over who was the better mathematician of the two he lost his
nose during that duel and thereafter wore a gold and silver amalgam and
indeed on his tombstone where there’s a relief of of Tycho’s face you can see a
line here which is supposedly the demarcation between his metal nose and
his flesh and bones Tycho discovered a supernova an exploding star in the year
1570 – now he didn’t know the physical nature of the star that it was a star at
the cataclysmic end of its life going boom and creating elements and all the
great stuff I’ll tell you about later but he but he saw that the star had
brightened to naked-eye visibility in fact it was visible during the day and
it remained visible as it faded for eighteen months and this was an
astonishing thing because Europe was just emerging from the Middle Ages and
people thought that the heavens were immutable even though the ancient Greeks
had seen you know exploding stars and other things nevertheless the idea was
that the heavens are immutable and unchanging and so Tycho’s observation
went a long way toward dispelling that belief and he
gained favor with the king and was given a rather large amount of money with
which to set up an observatory you ran a Borg on the island of Hien h ven and he
was given funds to conduct all sorts of studies and stuff so he he became quite
quite the guy and had these lavish parties and social gatherings in his
castle indeed he was thought to have had something like 1% of Denmark’s entire
wealth at one point in the 1580s so he had these great parties and lots of
social gatherings but he drank a lot and and was egotistical and argumentative
and actually kind of annoyed a lot of people he had some other interesting
characteristics he had a dwarf named JEP who was his court jester and Tycho
actually thought that JEP was clairvoyant and he kept JEP under the
dinner table and fed him scraps of food and stuff so anyway but Tycho is kind of
a weird guy but at this Observatory he made fantastic observations of the
planets he didn’t have a telescope but he had other instruments with which he
could measure the altitude of a planet above the horizon and its angular
distance from other stars and things like that so he amassed this wide and
broad body of observations which later I will discuss were analyzed by Kepler a
superb mathematician well a new king came to power in 1588 and Tycho’s
influence waned and eventually his funds were cut off and finally he he decided
to to leave Denmark I mean he had basically annoyed the king and was
really an arrogant guy and everything so he he said okay I’m done with Denmark
and he settled in 1599 in prague at the invitation of the holy roman emperor
rudolf ii in 1601 Tycho drank too much at a dinner and neglected to go
frequently enough to the bathroom so he suffered from a urinary infection and
two weeks later died of it at a rather young age more recent research suggests
that his actual cause of death was mercury poisoning whether mercury he
had taken himself to fight his urinary infection or deliberately poisoned
perhaps even by his mathematician Kepler and Kepler was rather annoyed with with
Tycho because Tycho had hired him to analyze the data yet only gave small
parcels of data at a time he wouldn’t give Kepler this huge set of
observations and so Kepler felt frustrated he couldn’t really discern
what was going on from fragments of data so Kepler in a sense was happy when
Tycho died because he was able to get his hands on a data after a bunch of
battles with Tycho’s relatives who wanted to confiscate the the data but
Kepler managed to manage to get away with it so you then analyzed the data so
here’s Kepler you had a lot of interesting ideas he thought that the
planets are on these spheres that circumscribe the five perfect Platonic
solids and stuff and he spent years trying to prove this and it just didn’t
work out because turns out it’s not true but in any case he analyzed Tycho’s data
and he refined the Copernican model with three important empirical laws now they
were empirical in that Kepler had no physical explanation for them he just
found that quantitatively they appeared to be true but he didn’t know why they
were true that will come with Newton the first law that he figured out in the
year 1604 is that the planetary orbits are ellipses not circles and the Sun is
not at the center of the ellipse but rather is at one focus of the ellipse
now there are two foci and there’s nothing at the other focus in Kepler’s
system the Sun is at one focus nothing as of at the other focus here you can
see in this diagram that an ellipse is defined to be the set of points such
that the sum of the distances from two other points the foci is a constant so
distance one let’s call it distance to is B and ellipse is the set
of points such that a plus B is equal to a constant and it’s easy to draw an
ellipse using this in mind by taking a cardboard sheet like this putting a loop
of string around it you pull the string tight with the span and then draw the
ellipse around the two tax keeping the pen in such a way that the string is
tight and the sum of the distances is clearly a constant there we go there’s
an ellipse anyway that gives you an ellipse Kepler’s first law then says
that the Sun is at one focus and the earth and indeed other planets orbit the
Sun along an elliptical not a circular trajectory with the Sun at one focus
nothing at the other focus may be Kepler’s ghost or something like that
let’s discuss ellipses in a little bit more detail here’s an ellipse with the
two foci marked the long axis is known as the major axis the short axis is the
minor axis half of the major axis is just the semi-major axis and similarly
for the semi semi minor axis here’s a set of ellipses having different
eccentricity but the same major axis now the eccentricity is defined to be the
distance between the foci divided by the length of the major axis so in all these
cases that major axis has the same length but as I put the foci farther and
farther apart the ellipse becomes more and more highly eccentric here it’s a
very highly eccentric ellipse here it’s an eccentricity of only 0.3 so it looks
more circular it turns out that most of the planets
have orbits that are nearly circular that ellipses have only a very small
eccentricity and this as I briefly mentioned in lecture 13 is the reason
that Copernicus’s system with circular orbits worked quite well recall that
Copernicus had circular orbits like that of Mars but whose was a little bit
offset from the Sun well a circle whose Center is offset from the Sun looks
almost identical to an ellipse with a small eccentricity the dots the circle
the dashed line is the ellipse you see that they are nearly the same and so
Copernicus a system worked so well because though the planetary orbits
really are elliptical not circular the ellipses are nearly circular the
eccentricities are very low the second law that Kepler came up with is that a
line between the Sun and a planet sweeps out equal areas in equal times let me
show you what I mean here’s the Sun with the elliptical orbit of a planet
surrounding it what we’re saying is that in going from position 1 to position 2
along its orbit the line between the planet and the Sun sweeps out a pie
shaped area like this that area is the same as the area swept up elsewhere in
the orbit when the planet goes from position 3 to position for as long as
the time interval it took to go from 3 to 4 was exactly the same as the time
interval from 1 to 2 equal areas in equal times let’s look at a whole bunch
of sectors here all of these sectors have equal areas and so if you make T
one which spans two of these sectors equal to t2
Kepler’s law his second law tells you that the planet travels a lot less of a
distance way out here far from the Sun than it does close to the Sun now here
the the elliptical orbit has been greatly exaggerated but if the planet
had such an eccentric orbit it would travel more slowly out here far from the
Sun then when it’s close to the Sun because the area it sweeps out is the
same in each case and when it’s close to the Sun it has to move more in order to
sweep out an area equal to that swept out in this long skinny triangle so I’ve
set up here a planet whose orbit has an eccentricity of 0.6 and it’s orbiting
the Sun and what’s let’s watch it move it moves slowly when it’s far from the
Sun and more quickly when it’s near the Sun slow fast that kind of a thing okay
it changes its speed and quantitatively you can figured out by how much by using
Kepler’s second law if we go back and make the eccentricity zero you will
notice that the speed of the orbit is unchanging with time but it changes the
more eccentric the trajectory is this applies most visibly to comets
such as this one in particular comets like Halley’s Comet have periodic orbits
that is they come around the Sun every once in a while as is shown here some
comets come in from very far away and sweep past the Sun only once but the
so-called periodic comets sweep past the Sun of a few decades or so and they
spend most of their time way out here far from the Sun and then they zip past
it so they go zoom zum-zum like that and they spend almost
all their time way far from the Sun here’s the orbit of Comet Hallie in fact
in 1948 it was way out here by 1977 that had come closer than 1983 then 85 and
then it’s swung past the Sun in 1986 zoomed past it very quickly and then
started slowing down and spending most of its time way out in the far reaches
of the solar system generally between Neptune’s orbit and Pluto’s orbit the
third law that Kepler came up with was much later in time in 1618 and that he
called his harmonic law and it says that the square of the orbital period of a
planet around the Sun is proportional to the cube of its average distance or more
precisely the cube of the semi-major axis of the ellipse they turn out to be
mathematically about the same thing so the planets farther out have longer
orbital periods that was already known before Kepler’s time but quantitatively
Kepler showed that the square of the orbital period is proportional to the
cube of the semi-major axis so you can write that P squared equals some
constant K times R cubed where R is the semi-major axis Kepler was ecstatic
about this law he called it his harmonic law because he was looking for the music
of the spheres he had a real musical talent and he was trying to find harmony
among the spheres and some sort of musical notes out there and this comes
the closest to that because musical notes are arranged in harmonic ways that
are not precisely like this but have some similarities so let’s write down
this equation P proportional to or P squared proportional to R cubed or you
can write it P squared equals K some constant times R cubed now it’s
convenient to use units based on the Earth’s orbit for example I
can write that the square of the orbital period of some planet is equal to the
constant K times the cube of the planets semi-major axis I can write that same
equation for the earth the square of the Earth’s orbital period is some constant
multiplied by the cube of Earth’s distance from the Sun if I then divide
one equation by the other the constant K cancels out because it’s a constant and
so we get that the orbital period of the planet divided by the Earth’s orbital
period that quantity squared is equal to not just proportional to but now equal
to the distance of the planet from the Sun divided by the distance of the Earth
from the Sun that kinetic quantity cubed so we know that the Earth’s orbital
period is one year that’s a familiar unit and the Earth’s distance from the
Sun is just some number it happens to be 93 million miles 150 million kilometres
I’ll talk in the next lecture on how that’s determined we can write 150
kilometres as 1.5 times 10 to the 8th kilometers the 10 to the 8th just tells
you how many decimal points you should move over to get the number that you’re
trying to write down you should move over eight decimal points one two three
four five six seven eight if you write that out you would get 150 million but
it’s kind of cumbersome to keep track of all those zeroes and write them all down
you might miss some so we use scientific or exponential notation of this kind if
you’re using a calculator this would show up on your calculator as 1.5 e8
okay I think most people are familiar with this kind of notation because we
all use calculators anyway that’s called an astronomical unit you can figure out
what it is through measurements so we know that the earth orbits in one year
at a distance of one astronomical unit if we then say that oh I’ve measured the
orbital period of Mars to be one point eight eight years
one can do that just look at Mars and see how long it takes to go all the way
around the Sun turns out to be one point eight eight years if you plug in one
point eight eight here take its square and then ask yourself what number when
cubed gives you the same number that is the square of one point eight eight you
find that that number is one point five two or one point five two astronomical
units that is 52% greater than the distance of the Earth from the Sun so
using ratios like this as is very useful so Mars is about 52% farther from the
Sun than Earth is and you can figure that out using Kepler’s third law only
by measuring the orbital period of Mars okay well let’s set up another animation
here where for simplicity I’ll just have circular orbits initially this one is
close to the Sun the planet is really zipping around quickly if I now stop the
animation and I put the planet farther away say 2.5 units from the Sun instead
of 1 you can see that the orbit the orbital velocity is much slower and
quantitatively what Kepler said was that the square of the orbital period is
proportional to the cube of the distance this also applies to objects orbiting
the Earth and in particular there are lots of satellites orbiting the Earth a
satellite just above the Earth’s surface goes about 230 degrees in one hour that
is about 2/3 of the full circle as the earth itself rotates about 15 degrees
per hour so the space shuttle and other near-earth satellites orbit in about an
hour and a half it turns out a satellite orbit at 3 and 1/4 Earth radii traverses
42 degrees in one hour but if you put the satellite at six-and-a-half Earth
radii then it traverses 15 degrees in one hour in other words it traverses the
angular distance as the Earth’s rotating surface and that means that this
satellite will appear stationary above a particular point on the Earth’s surface
this is the idea behind geostationary orbits here’s one shown right here if
you have a satellite above a communications tower it will remain
above that communications tower all the time if it’s orbiting at a distance of
six-and-a-half Earth radii from the middle of the earth if you put it too
close to the earth it orbits more quickly than the rotating earth and so
the same satellite antenna won’t always be in communication with the satellite
that’s okay it just happens to be a different kind of communication
satellite if you put the orbit too far from the earth then you notice that the
earth is rotating more quickly than the satellite is orbiting and once again it
does not remain hovering above one and only one communication station so
geostationary satellites are important for a number of forms of communication
and weather you know monitoring the weather at any particular location or
something like that ok well the stage was now set for Isaac Newton later Sir
Isaac Newton a brilliant but rather eccentric English physicist who made
many magnificent contributions to the development of physics and astronomy and
especially his three laws of motion and his law of universal gravitation Newton
lived from 1642 to 1727 and he he is what I would call not an ordinary genius
I mean they’re lots of ordinary geniuses wandering around on college campuses and
businesses and wherever you get people who are very talented they know what to
do they do it well they do it efficiently they’re smart people but
then there are some people who live in a completely different plane of existence
and this happens in all fields Mozart and Michelangelo and the Vinci well for
physics it was Newton Newton lived in a different plane of existence and
fought in ways that most of us even those of us who happen to be ordinary
geniuses cannot comprehend Newton’s life was interesting in 1661 he went to
Cambridge University but in 1665 he had to flee Cambridge due
to the plague and he went to the countryside and that was his most
fertile time he was just in his early 20s and he developed the calculus a form
of mathematics he developed his laws of motion the law of universal gravitation
everything it was just incredible what he did while away from college you know
on his summer vacations sort of you know his vacation from the plague in 1668 he
invented a form of telescope the Newtonian telescope and the next year
became the exalted Lucasian Professor of mathematics at Cambridge University he
finally published his work in the Principia being urged by his friend
Edmund Halley to do so and partially funded by Hallie just six years later
however he had a nervous breakdown but he got out of it and then became a
warden of the mint where he took extreme pleasure in in punishing counterfeiters
he apparently was had a sort of a cool bent to him he published his work on
optics in 1704 and then the next year was knighted not mostly for his
scientific studies but rather for his government service in the Principia he
described his three laws of motion the first is that if there are no forces
acting on a body then that body’s speed and direction remain constant now
usually we think of rest as being the natural state of things and indeed
Aristotle thought that a force was needed to keep a body in motion when you
throw something and you have it dragged all across the table it comes to rest
and so people thought that rest was the natural state of objects Newton said no
motion is just as good as rest and that motion is uniform along a straight line
in a given direction as long as there are no forces acting upon the body but
rest and motion are equally natural if there
are no external forces so you could have a hockey puck for example going along
some clean very smooth ice eventually we know it’ll come to rest due to friction
but if you didn’t have the friction it would keep on going the second law known
as F equals MA is just that the force is equal to the mass times the acceleration
now acceleration is a change either in speed or in direction if I’m moving in a
circle but at a constant speed I am accelerating you can tell that when
you turn in a car on the curve you saw your body sort of knows that the turn is
being made you’re being accelerated so Newton said that accelerations are due
to forces and for a given force the acceleration is less for a massive body
than for a low-mass body because after all acceleration equals force over m so
if the force is given then a large body is accelerated by that force less than a
small body so if I kick a small ball I can get it to really go hauling but if I
try to kick a truck you know it won’t really move very much because the mass
of the truck is so much bigger than that of the ball
the third law often known as the law that says for every action there is an
equal and opposite reaction well what what he was really saying was that when
two bodies interact they exert equal and opposite forces on each other the forces
come in pairs if I push on the wall its pushing back on me okay if I hop from
this step down toward the earth the earth is indeed pulling on me but I’m
pulling on the earth as well and my measly mass pulls on the earth just as
much as the earth pulled on me the forces are equal and opposite so then
why do I fall why doesn’t the earth come up toward me ah let’s go back to the
second law the Earth’s mass far exceeds my maths and so for a given force the
earth is accelerated much much less than I am but the forces are equal and
opposite a good example of this is in rocket propulsion when the fuel burns
and a jet of gas comes shooting out of the end it’s not that the gas is pushing
on the earth and that somehow it propels the rocket upward you can have a rocket
way out in outer space and the gas propels the rocket forward because the
gas going outward has to be balanced by a force on the rocket pushing it forward overall the system has no net forces
acting on it from outside but the internal forces have to be paired the
outward push of the gas has is balanced by the inward that is balanced by the
forward motion of the rocket these laws and the subsequent law of universal
gravitation unified many seemingly disparate areas of physics into one
simple unified whole and this is one of the great goals of science to explain a
lot of different phenomena through a small number of fundamental ideas Newton
went a long way in achieving that goal


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