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1. Frontiers/Controversies in Astrophysics: Introduction
Poziom:
Temat: Edukacja
Professor Charles
Bailyn: Some things about
this course.
This is a course for
non-scientists.
That portion of the enrollment
policies is not a suggestion.
I really don't want science
majors in this class.
If you are a science major,
I'm going to notice because
that's one of the things that
appears on the class list;
what your major is.
So, don't take the course if
you're a science major.
Let me point out that freshmen
don't have major,
so it doesn't matter if you
intend to be a science major if
you're a freshman.
If you are a science major I
recommend Astro 210,
which is being given this term.
I have a little handout on all
the different introductory
astronomy courses at the front
of the room if you're
interested.
Let's see, it is also true that
this course is kind of intended
for non-science majors who have
a certain basic high school
level comfort with tenth-grade
science and math.
If you're extremely phobic
about these kinds of things,
I would say that Astronomy 120,
while it has a similar level of
math, has a somewhat shallower
learning curve and a somewhat
deeper safety net.
So, if you're the kind of
person who breaks into a sweat
when somebody writes down an
equal sign, check out 120.
Let's see, but that's not the
biggest difference between this
class and 120.
I think the biggest difference
is what the class is trying to
do.
Astronomy 120 and also 110 and
other courses in our department,
and elsewhere in the
university, are basically survey
courses.
Most introductory science
courses are survey courses.
They cover a fairly wide
subject matter.
This is--isn't that,
what this course is supposed to
do is we're going to talk about
three particular topics in very
considerable detail.
Enough detail so that by the
end of our discussion you'll
understand what's going on in
current research in this topic.
And by current,
I don't mean this decade,
I mean this week.
Astronomy is currently in a
stage of very rapid advancement,
and one of the things that's
happened every time I've taught
this course in the past,
is that at some point during
the semester someone will
publish some piece of research
which changes some aspect of the
curriculum.
I'll come in waving some paper
and everything will be changed,
and I can't guarantee that of
course,
because I can't predict the
future but it's happened every
time in the past.
So, we really are trying to get
you all the way out to the
frontiers of the subject.
I think this is actually a
better approach for non-science
majors, because after all,
we live in the Internet age.
If you want to find out a bunch
of facts about some scientific
topic you could go online and go
to Wikipedia or wherever,
look these facts up.
That's not a big problem.
The problem comes when there
are two sets of facts which
directly contradict each other.
This happens quite frequently
in scientific topics these days,
particularly those with kind of
political or moral overtones,
and you get facts that directly
contradict each other.
What are you supposed to do
about that?
What I'm hoping is that by
talking about situations in
which the facts at the moment
really aren't known yet,
you can develop some skill in
interpreting these kinds of
contradictory facts for
yourself.
If you don't do that,
then the only alternative is to
listen to the experts argue with
each other and vote for whoever
argues the loudest or looks the
best when they're doing it,
or has a degree from Harvard or
whatever it is.
You guys can do better than
that.
So, the hope is that by
practicing this kind of skill of
evaluating science when the
answer isn't fully understood,
that you can develop skills
that will stand you in good
stead when you run into
scientific controversies in a
political context or a legal
context,
or just as ordinary citizens in
the course of your lives.
It also happens that the three
particular topics I think are of
some real interest and
importance in themselves.
And I'll get to the three
topics again in just a moment
here.
Let me point out that this kind
of approach has a downside to it
and this has been pointed out
repeatedly on course
evaluations.
Because we're dealing with
stuff which ultimately the
answers are not yet understood
there's no textbook.
There can't be a textbook.
We haven't figured out what to
put in the textbooks yet.
And the problem with that is
that that makes the lectures
very important because that's
the only information you're
going to get.
There's a whole bunch of online
readings and stuff but they tend
to have a point of view,
and so it's really the lectures
that are the basis of the
course.
The problem with that is that
I've chosen to give this course
at the ungodly early hour of
9:30 in the morning,
and you guys are going to have
to show up and so here's the
deal.
I'll make a deal with you:
Your job is to get to class by
9:30 in the morning.
My job is to keep you awake
once you're here,
and so if we both succeed in
cooperating in this sense we'll
probably be okay.
But seriously,
if you're anticipating regular
difficulties in getting to class
this is not actually a great
class to take because there's no
backup in the form of a
textbook.
All right, the particular
topics that are under
discussion, I've listed them
here in green.
The first of them are
extra-solar planets,
by which I mean planets around
stars other than the Sun.
It's well known that there are
many, many of these planets.
All you have to do is watch
Star Trek or something like that
and you'll find many,
many examples and this has been
a staple of science fiction for
quite a long time.
Oddly enough,
until ten years ago,
there was absolutely no
evidence for this.
We assumed that,
because the stars are normal
stars and there are many other
stars--the Sun is a normal star
and there are many other stars
like the Sun that there must be
many planets of the same kind as
the planets in our own Solar
System circling around all these
other stars.
But until 1995 there was not
one bit of evidence to support
that idea.
Since 1995, this has become a
huge growth industry and
research, and we now know of
literally hundreds of planets,
all of them discovered in the
last ten years.
So, this is a situation in
which, what ten years ago was
science fiction,
has become science fact and
we're very rapidly trying to
figure out exactly what kinds of
planets these are,
whether there are Earth-like
planets out there,
and that has some bearing on
what the science fiction people
refer to when they say as,
"life as we know it."
And so, that's currently one of
the hottest topics in astronomy.
The next topic is going to be
black holes, and this is a
similar situation.
Fifteen, twenty years ago black
holes were sort of poised
precariously on the boundary
between theoretical physics and
science fiction.
A boundary that is more porous
than you might believe.
But again, in the past fifteen
years or so this has been
converted into a standard topic
in observational astronomy.
There are dozens,
probably hundreds of objects we
can point to in the sky and say,
"yes those things are black
holes."
And so now, the current topic
of research is do these things
that we are pretty sure are
black holes actually behave in
the incredibly bizarre,
science-fictiony manner that
the theoretical physicists have
been talking about for the past
thirty or forty years.
So, to what extent are these
very exotic behaviors actually
manifested in real life?
Finally, I want to talk a
little bit about cosmology.
Cosmology is the study of the
Universe as a whole.
That's too big a topic to go
into in depth,
so I've picked one piece of it.
The piece I've picked is the
existence, which was discovered
in the late 1990s,
of something called "dark
energy."
Dark energy is an all-pervading
anti-gravity;
it's a repulsive force that
turns out to occupy essentially
all of the Universe,
and 75% or more of the entire
mass energy of the Universe
turns out to be in the form of
this mysterious dark energy.
The evidence for this comes
largely, but not entirely,
from observations of a certain
kind of supernova.
And so what I'm going to focus
on is the observations of the
supernovae and how they
demonstrated that,
in fact, all ordinary matter
and energy and so forth is a
tiny fraction of what's actually
going on in the Universe,
and what's really happening out
there is something we totally
don't begin to understand.
So, that will be the third
topic of the course.
These topics have something in
common.
All of them involve observing
something that you can't
actually see directly.
We don't see these planets
directly because they're too
faint and too far away.
We don't see black holes
directly.
By the definition of black hole
you can't see these things
directly.
And of course,
dark energy,
by its very name,
is also undetectable.
So, how do we know that these
things are there?
The answer is we know that
they're there because of their
influence on other objects that
we can see,
and in particular,
their gravitational influence
on other objects that we can
see.
And so, what binds these three
topics together,
are first of all,
the fact that the observational
techniques to discover them are
actually quite similar to each
other.
And second, that they all
involve different manifestations
of gravity.
And so, we'll be talking in the
first part of the course about
Newtonian gravity.
In the second part of the
course when we get to black
holes, that's relativistic
gravity,
general relativity,
Newton's--Einstein's theory
which supplanted Newton's
theory.
And then by the time we get to
dark energy, it may not even be
correctly described by
Einstein's work,
and we may be in the area of
whole new kinds of physics that
the theorists haven't even
thought about yet.
So, there will be a progression
to more and more sophisticated
theories of gravity underlying
these observations.
There's another feature that
these topics have in common,
and that is that they can be
understood in some detail
without particularly
sophisticated mathematics.
Now, let me pause here and say
some things about math.
Astronomy is a mathematical
topic.
There will be math in this
course, there ought to be math
in any astronomy course or it
isn't really an astronomy
course, it's just a slide show.
Now, the math in this course
has been kept at a deliberately
low level.
That is to say,
the kind of math we'll be doing
is stuff you did in ninth and
tenth grade.
Introductory high school
algebra, high school geometry,
I think we take the sine of an
angle a couple of times,
but it's the one case it
cancels out almost immediately,
so don't let that scare you.
It's the kind of thing that you
all did on the math SATs and
since you're all sitting in this
room you must have done okay.
Having said that,
I have discovered that saying
that is misleading.
And the reason it's misleading
is cast your mind back to ninth
grade;
ninth grade math is hard.
Remember?
In particular,
word problems are hard.
You remember word problems.
This is where you drive from
here to Cleveland and you fill
your tank up with gas,
and the gas costs so much per
gallon, and the question is what
is your shoe size or something.
[laughter]
The way one approaches that is
through a kind of common sense
approach which involves the fact
that many of us have been in a
car,
driving from City A to City B,
perhaps not Cleveland,
but somewhere else,
and so you have a kind of
intuition to fall back on.
When you do math problems that
are logically the same,
but apply to astrophysical
systems,
for which you have absolutely
no common sense to back you up,
then you have to reason purely
from the internal logic of the
problem and that's hard to do.
It's a skill that can be
learned;
it's a skill that's worth
learning;
it's a skill that I'm sure many
of you already have to a large
extent, but it isn't an easy
thing.
So, the fact that the level of
the math is low doesn't mean
that the problems are easy.
We do have a lot of help
mechanisms, which I'll describe
perhaps on Thursday,
to keep you up to speed if you
start having trouble with these
things.
So, I should say something
about course requirements here.
Let's see, we have sections in
this class.
The sections are not just
problem solving sections,
these are actually required.
The fact that we're dealing in
topics for which the answer
isn't fully known means that one
can actually have discussion
sections unlike many science
courses,
so we're going to do that.
And so, the structure of the
course is like a history course.
Two lectures a week plus
required section,
and so 10% of your grade comes
from sections.
A large fraction of that is
just showing up,
but there will also be
something in terms of saying
something intelligent once you
get there.
That's 10% of the course;
30% of the course is problem
sets.
We will hand these things out
once a week.
The first problem set will show
up on Thursday,
and if you have any question
about whether this course is
appropriate for you,
the right thing to do is to
look at that problem set and ask
yourself is this reasonable.
I will say that students on
their evaluations have pointed
out that it does--the course
does get harder.
It's not that the math gets
more complicated,
but the situations get more
complicated.
So, if you have serious trouble
with the first problem set
that's probably a warning sign.
As I say, that will be handed
out on Thursday.
These things come about once a
week;
it's 30% of the grade.
I'll say more about problem
sets later on Thursday.
Thirty percent comes from two
midterm exams.
The way we do this is the one
where you get the better score
counts 20%.
The one that you get the worst
score counts 10%.
So, that gives you a little bit
of a break.
And then there will be the
Final exam, that's the last 30%
of the class.
There's also an optional paper.
If you choose to do that,
that will count 15% of your
grade, and what it will do is it
will de-weight whichever the
worst of your 30% parts of your
grade are back down to 15%.
So, if you're a word person
rather than a number person,
you get this opportunity to
augment your score and de-weight
some other part of the class in
which you may have done less
well.
All of this stuff is on the
classes server [Yale's online
course tool].
I should say that the syllabus
that I've put out here is just a
direct copy off of what's on the
classes server,
so feel free to take that.
But all the information,
and actually more information
is online.
Let me pause now and ask
whether there are questions
about the course and the course
procedures.
Yes?
Student:
This may be a silly question,
but I saw on the web that right
below the times listed for this
course was a "to be determined"
or some sort of notation that
could indicate that there is
another of this class at a
different time?
Professor Charles
Bailyn: No,
no this class is going to meet
now.
I'll have to check and see what
you were thinking of,
but it may be that what that
was referring to was section
times,
and actually this is something
that I haven't mentioned.
Sections are required.
They're all going to be on
Mondays.
We're going to have a wide
range of times,
all of them on Mondays from
12:30 until I think 8:00 at
night.
But you do have to sign up for
a section.
Let me also say,
I've mentioned here,
I don't think this
is--;actually,
looking at the number of people
here,
I think we're going to be able
to accommodate everyone,
including juniors and seniors.
But I did set it up in such a
way that freshmen and sophomores
get first crack.
The way that's going to work is
the online sectioning form opens
up on Monday and juniors and
seniors won't be allowed to
officially register for the
class until Tuesday.
So, the freshmen and sophomores
get to fill up the sections
first.
My guess is,
again, looking at the number of
people here today that we won't
have any problem,
and that if you're a junior or
a senior you'll get in just
fine.
So, we'll be picking sections
through what is now the standard
online sectioning thing,
which is going to open for
business next Monday.
I'll check the website and see
if that's actually what you
meant, but it may have been
something else.
Other questions?
Let me, in general,
encourage you to ask questions.
I know that that's hard to do
in a big lecture setting,
but we have an advantage over
other courses,
particularly science courses.
We're not trying to prepare you
for the astronomy part of the
MCATs, so we don't have to cover
a specific syllabus.
We're not even trying to follow
a textbook.
And so we have a little more
leeway than is ordinarily true
to ask questions and go in weird
directions, so please feel free
to do that.
I reserve the right to put a
question off into the future or
into discussion section or
something, but do by all means
ask.
We have some freedom of action.
Yes?
Student:
Is it possible to take an early
final?
Professor Charles
Bailyn: An early final?
Let me think about that.
I prefer to avoid it because
then I have to invent another
final.
The problem with that is trying
to make them come out even.
I will say this,
that if I do an early final,
I'm probably going to err on
the side of making it hard.
But it's very hard to make them
come out even,
but let me think about that.
Other questions?
Yes.
Student:
In discussion sections,
is it just going to be like
discussing things or is it going
to be working on the problem
sets?
Professor Charles
Bailyn: It's going to be
some--So, the question is,
"What are the discussions
sections going to be like?"
Are there going to be
discussion of the problem sets
or is it going to sort of
general discussion of the course
material?
The answer is both.
I think there will be both,
in any given discussion
section, there will probably
both be an opportunity to talk
about the previous problem set
and to clarify things about the
next problem set,
and also some kind of activity
that sort of extends and
advances what we've been talking
about in class.
So, I'm hoping to do some of
both.
If we veer too much in either
one direction that's probably
not a good thing.
There will be other ways of
getting help as well,
if you start to have trouble on
the problem sets or in the
course generally.
I'll talk about those a little
bit on Thursday.
Yes sir?
Student:
How are problem sets graded?
Professor Charles
Bailyn: How are problem sets
graded?
Very carefully.
Let's see, I think we'll
probably--it'll probably be on a
kind of zero to twenty-point
schedule.
But let me say this about the
problem sets.
There are going to be two kinds
of things on the problem sets.
One are kind of quantitative
problems which have a right
answer.
Those are relatively easy to
grade on some kind of a point
scale;
you give partial credit and so
forth.
But we will also--because this
is a course that's not only
about the specific of this topic
but also about science in
general,
we're also going to have things
that look kind of like essay
questions on the problem sets.
Those are a little harder to
grade in this way,
but we've got to grade them in
the same way so that we can add
the points up.
And I'll talk a little bit more
about how those are graded.
I will say one thing;
one thing that we do is we make
sure that each problem or essay
is graded by one T.A.
or by myself,
so that we don't have different
people--so that if you're in a
section it's not like your--all
the problem sets for that
section are all graded by your
section leader and some other
section leader grades all the
other problems,
because that leads to
imbalances of various kinds.
So, we assign each problem to a
specific person for the whole
class.
It's basically a zero through
twenty scale,
although what that means varies
depending on what kind of a
problem it is.
I'll say a little bit more
about that.
I will also say there is a
rather detailed lateness policy
that's linked to the classes
server, please read that.
We're going to stick to it.
And one of the features of that
is that there will be answer
sheets.
Problem sets are typically due
Thursday, there will be an
answer sheet up the following
Tuesday,
so if you don't get it done by
five days after it's due,
you're toast because the
answers are posted.
Other questions?
Great, let's start.
This is very cool.
All right, this is going to be
all kinds of fun.
Planets, planets around other
stars, but planets in general.
So, let's start by talking a
little bit about orbits,
planetary orbits.
You probably know some of this
story, originally in the old
days, people used to think that
the Earth was the center of the
Universe.
So, the Earth was at the middle
and planets went around them in
circles.
That's not much of a circle
[drawing on overhead],
but you know what I mean.
And so, everything was circles
around the Earth.
And that's what planets did,
where planets also included to
their way of thinking,
the Sun and the Moon as well,
and so you had these circles
around the Earth.
This is what's called the
geocentric model;
Earth at the middle.
It's associated with the name
of a Greek astronomer named
Ptolemy.
The problem with this model is
very simple.
Namely, that if you actually go
out and observe where the
planets, and the Sun,
and the Moon are night after
night after night it doesn't
work very well.
So, this doesn't fit the
observations.
Doesn't fit observations.
So, they said,
all right well maybe that
doesn't work all that well,
so what we'll do is instead of
imagining that the planets are
on circles around the Earth,
we'll imagine that there are
circles on circles around the
Earth, and the planets go on
those.
So, you add a kind of extra
circle here, so the circle goes
around the Earth and the planet
goes around on that circle.
These circles were called
epicycles.
So, add epicycles.
And what happened is they would
add an epicycle and then they'd
go out and observe some more,
and in particular,
the Arab astronomers a thousand
years ago.
A thousand years ago the center
of all science was in the Arab
countries;
they gave us all their--all our
star names by the way are in
Arabic, so are mathematical
techniques such as algebra;
it all comes from the Arabs.
They knew what they were doing
back then when the Europeans
were kind of in squalor.
And they made these great
observations,
and every time they made more
observations it turned out it
didn't fit.
So, they had to add more
epicycles.
So then, they added one here,
and one here,
and so on until you had
circles,
and circles,
and circles,
and circles in order to explain
the observations.
So, add epicycles repeatedly.
And this is kind of
unsatisfying because it's not a
good thing where every time you
get more or better observations
you have to revise and extend
your theory.
That's not such a great theory.
In fact, the word epicycles has
now become a kind of a swear
word in the scientific
community,
meaning a sort of theory that
has become so complex it's just
ridiculous and you don't want to
believe it anymore.
So, someone will come up with
some really seemingly
sophisticated but very
complicated theory and if you
don't like that you just go
that's just epicycles,
forget about it.
So, this has become a little
bit of a swear word,
and it was unsatisfactory at
the time.
Now, let me pause for a moment
and confess that the story I've
just told you,
which is the standard story
about Ptolemaic epicycles is,
well, it has what I think
Colbert would refer to as
"truthiness."
It's a commonly told story that
people like to believe,
but if you talk to the
historians of science this isn't
actually how it happened.
And, in fact,
this idea of circles on
circles, on circles that isn't
the way epicycles worked,
they had circles and they did
get more complicated every time
they fit the observations,
but not by adding more and more
circles.
They would move the circles
side to side,
they would have things going at
variable speeds around the
circle,
all sorts of things but this
little picture that I've just
drawn here has a kind of
"truthiness" to it.
I would say that this is a
general issue with the way
scientists describe how science
works.
We tell these nice anecdotes
and we put them in the textbooks
too;
in the little bars that go down
the side of the textbook,
where you get the head and
shoulder shot of the famous dead
white male scientist and so
forth.
And then we tell these stories.
And the historians of science
hate this because it isn't
actually what happened.
Nevertheless,
we persist in telling these
stories, and I've been thinking
about why that is.
I think the way to think about
this is what these stories are,
are fables.
And like any fable,
the point is not that the story
is true.
The point is that it vividly
illustrates a moral,
which tells you how to behave
or how not to behave and they're
useful for that reason.
You'll recall the famous fable
of the ant and the grasshopper.
Grasshopper sings and plays and
dances all summer long.
The ant is very industrious,
piles up food,
doesn't have any fun.
But then in the winter,
the grasshopper starves and the
ant does fine.
If an entomologist were to come
along and say but that's not how
ants and grasshoppers behave,
you would correctly say that
he's missed the whole point.
And the point is that it's just
a nice story which illustrates
certain kinds of behaviors and
whether they're good or bad.
So, here's what I'm going to do;
I'm going to tell these
stories, but I'm going to label
them fables and I'm going to
point out the morals explicitly.
And the optional paper is going
to be: go and take any one of
these things and find out what
really happened and comment
somewhat on the implications of
the real story for science.
I should say that the biggest
of these fables is probably the
one about Galileo and the
Catholic Church,
where the Catholic Church
oppresses the pioneering
scientist and the scientist
stands firm against this huge
impersonal bureaucracy,
and the establishment trying to
squelch them and so forth.
The truth of that is actually
very subtle and very interesting
and I can't go into it now,
among other things because I'm
not a historian of science,
I'm not the best person to talk
about it,
but check that out sometime.
Anyway, for this
particular--this is the fable of
the Ptolemaic epicycles and the
moral is that simple theories
are better.
And you particularly don't like
theories which get more and more
complicated, the better and
better your data become.
I should say that the word
simple in there turns out to
have a technical meaning if you
take a statistics course.
What I mean by simple is
something that has relatively
few free parameters.
I'll just leave that at that.
You can go talk to the
statisticians about it.
So, if your theory is getting
overwhelmed by epicycles,
then you'd better go out and
come up with some other better
theory.
And so, people tried to do
that, and the first step along
the way was, of course,
Copernicus.
Copernicus, as you probably
recall, decides that the
geocentric model is wrong,
things ought to be
heliocentric;
the Sun in the middle.
So, you put the Sun in the
middle and everything,
including the Earth,
goes in circles around the Sun.
This was revolutionary,
and in fact,
the title of the book he
published was De
Revolutionibus Orbium
Coelestium,
which means "of the
revolutions" in the sense of
"revolving of the celestial
spheres."
The use of that word revolution
is one of the things that pushed
the word revolution to its
current meaning,
meaning overthrowing authority
in some ways.
Originally, it just meant to
revolve but this was so
revolutionary that people
started to use the word in the
other way.
This wasn't actually as great a
theory as you might think,
because it still needed
epicycles.
Not as big, not as many,
but it didn't get rid of the
problem with epicycles.
And that didn't work itself out
until a generation or two later
when Kepler came along.
Kepler was a famous astronomer
and he had in his possession,
because he stole them,
the best naked-eye results that
had ever been obtained of the
motions of the planets,
in particular, Mars.
He described these motions in
Three Laws of Planetary Motion.
You can look them all up in a
textbook.
In other kinds of courses we
would have you memorize these
things;
I'm not going to do that.
The key point here is that
these are not circles;
they're ellipses around the Sun.
That, it turns out,
gives you a model for planetary
orbits which,
when you take better and better
data, doesn't change.
They're still ellipses;
you don't need little ellipses
on top of these ellipses to
explain everything that's going
on.
So, this now has excellent
descriptive power.
It really describes what's
going on, and when you make
further observations,
it still describes what's going
on.
It does not have any
explanatory power in the sense
that if you say,
"why ellipses?"
Kepler had no idea.
That's just the way God made it.
So, it's not in any particular
way an explanation.
For the explanation you have to
wait another generation or two
until we get Newton.
Newton writes down three laws
of his own, but these are now
three laws of motion,
not planetary motion in
particular.
And again, one could write
these down and memorize them and
learn them, and that would be a
good thing.
Let me write down one of them,
the Second Law,
looks like this:
F = ma,
force equals mass times
acceleration.
And I write this one down
simply to point out that that
equation is the entire
intellectual content of
Introductory Physics for physics
majors.
If you go take Physics 180 this
is all that they do and they
spend the whole time.
It turns out you don't actually
want acceleration,
that doesn't tell you what you
want to know.
What you want to know is the
trajectory, where the object is
as a function of time.
Those of you who have taken
some calculus may recall that if
you take the acceleration,
and you take an integral twice,
you'd come up with the position
as a function of time.
So here's what--so in the next
thirty seconds I'm going to
explain Physics 180 to you.
You substitute in some kind of
a force, you divide by mass,
you take two integrals,
and that gives you the
trajectory of the thing.
That's all you need to know.
Technically,
of course, it's quite hard,
but conceptually pretty
straight-forward.
One of the things that Newton
did with this equation was he
took a particular force,
namely the force of gravity,
which he also wrote down a Law
of Gravity.
That tells you for any given
situation what the force due to
gravity is, substituted it in
here,
and figured out what the
motions of the planets ought to
be.
And it turns out that he could
derive Kepler's Laws.
He derives Kepler's Laws.
Very nice.
Now, of course,
in order to do this he has to
invent calculus,
so it takes a little while.
He was a great genius but even
so, inventing calculus from
scratch, not something you want
to attempt at home.
And that was basically the
start of both modern science and
modern mathematics.
So, this marks the start of
science in the following
sense--that Newton has to make a
couple assumptions along the
way,
sort of deep assumptions about
how the world works.
One is that the Universe is
governed by laws,
and in fact,
by universal laws.
What I mean by universal,
in this sense,
is that they apply everywhere;
that the same law of gravity
that resulted in the top of my
pen falling to the floor over
there also is responsible for
the orbits of the planets and
the motions of the stars.
This was a new idea.
It's very familiar to us by
now, but the idea that the
planets ought to behave
according to the same rules as
stuff down here on Earth was a
whole new concept.
The other piece of the new
concept is that these laws are
mathematical in nature.
This is why science is hard,
because it's hard for human
beings.
I think it's something to do
with the way our brains are
wired, to accept that this is
true.
It's very easy to imagine a
world in which that's not true.
Go read any fantasy novel.
Any fantasy novel has a
situation where the hero or the
villain, by virtue of their
strength of character influences
the events around them.
So that is a rule governed by
laws, perhaps,
that are not mathematical in
nature, but depend on the moral
character of the individuals
involved.
Every human culture has such
stories including our own.
It's very hard to get away from
it, and the idea that there's
just this sort of mathematical
structure and that your moral
stature has no bearing on what's
going to happen is kind of hard
to accept.
Fortunately,
people turn out to be pretty
good at math,
so we can actually solve these
problems and move forward.
These two ideas were
revolutionary and they are the
basis pretty much of all
science.
So then Newton's laws get
elaborated on for several
centuries.
By the end of the nineteenth
century things are starting to
come apart a little bit.
There are now problems that
show up with Newtonian physics.
It's been a big success on the
whole but there are now
problems.
And in the early twentieth
century what happens is two new
laws of physics are invented.
These are the given the names
quantum mechanics and general
relativity.
And the situation with these is
they don't overturn Newton's
laws, they extend them.
It turns out that in the kinds
of situations that Newton was
looking at, both quantum
mechanics and general
relativity, reduced down to
Newton's law.
So, you have a situation where
here are Newton's laws,
Ns Laws, of which Kepler's laws
are a tiny subset.
And then general relativity;
I'm drawing a kind of Venn
diagram here,
is here, relativity,
occupying Newton's laws but
that's some other stuff.
Quantum mechanics looks kind of
like this;
extends in a different
direction.
Let me make these axes-specific.
I don't like Venn diagrams when
they don't tell you what you're
actually plotting.
This is mass,
so heavy things are when
relativity kicks in.
This is size,
and so small things are when
quantum mechanics kicks in.
But you can see the problem.
We've got two big theories.
You really want those theories
to be encompassed by one yet
bigger theory.
And that is the current goal of
theoretical physics,
to try and find the one great
theory that encompasses both
quantum mechanics and general
relativity,
which contradict each other in
various awkward ways,
particularly in this region up
here.
This is called the Theory of
Everything, or TOE.
And the best current guess as
to what kind of a theory that
will be is that it will be some
kind of string theory.
I won't go into string theories
now, you can go read many
popular books on this;
it's very exciting.
There is currently no string
theory that really works out all
that well but the people who are
studying this kind of thing like
to believe that that's going to
work out sometime in the future.
This is good.
We've gone about forty minutes
from the start of science to the
Theory of Everything,
so we're done.
Everything else is a detail and
so the whole rest of the course
is filling in details.
The first of which--so let's
start on the details.
The first of which,
I want to go back and catch one
of Kepler's Laws.
And I want to write down the
Newtonian Modification of
Kepler's Third Law.
That is an equation that looks
like this: a^(3) =
GMP^(2)/4 π^(2 ) We're
going to circle this in red.
This is something you're going
to want to memorize.
This, it turns out to be,
a basis of a large fraction of
what we're going to do in this
course.
So, let me explain the symbols;
a is the semi-major axis
of an elliptical orbit.
Remember these orbits are going
to be ellipses;
here's an ellipse.
The long side is the major axis;
the short side is the minor
axis.
Half the major axis is the
semi-major axis,
so this is a right here.
P is the orbital period,
how long it takes the planet or
whatever orbiting object you've
got to go around one orbit.
M is the total mass of
the two things in orbit around
each other, of the orbiting
bodies.
And the existence of that
M is why this is Newton's
modification.
In Kepler's law,
it was always planets going
around the Sun,
so the mass was always the
same;
the mass was that of the Sun
and so it cancelled out.
In general, you can use the
same equation to deal with
things orbiting the Earth or
things orbiting the Moon as long
as you put in the right mass
there.
G is a constant of
nature, the gravitational
constant, and it equals some
value depending on what units
you use.
And we'll come back to that
later.
Four is 4, π is this obscure
number from elementary
mathematics 3.14159 whatever the
heck it is.
And you can punch it in on your
calculator or whatever.
So, you can use this equation
to find things out.
Now, these numbers tend to be
awkward to work with.
The mass of the Sun is some
huge number of kilograms,
G is a very awkward
number, π is always a mess.
But let me show you a trick.
Consider the Earth's orbit
around the Sun.
The semi-major axis of the
Earth's orbit is a very common
unit in astronomy,
and it's called an Astronomical
Unit.
It's a unit of length, or AU.
The mass of the Sun,
mass of the Earth plus the Sun
is mostly the mass of the Sun;
of Sun, is called the solar
mass obviously,
and it's given this symbol
M with a little circle
with a dot inside,
that's the symbol for the Sun.
What's the orbital period of
the Earth?
A year, thank you very much.
Period of Earth--one year.
That's what a year means;
it takes a year for the Earth
to go around the Sun.
So, it must be the case that
one Astronomical Unit cubed,
is equal to G times the
mass of the Sun,
times one year squared,
that's P^(2) over
π^(2).
Now, let me show you a trick.
Take the general equation,
it's a useful trick,
and divide by the specific
equation.
So a^(3) =
P^(2)GM/ π^(2)
and we're going to divide that
by 1 AU^(3) equals one year
squared,
G mass of the Sun over
4π ^(2).
We can do this because these
two things are equal so we're
dividing both sides of the
equation by the same amount.
G cancels,
4π ^(2) cancels;
that's very nice.
We end up with a over 1
AU^(3) equals P over one
year squared,
M over the solar mass.
This is just saying that
quantity is a in units of
an Astronomical Unit.
This quantity is P in
units of a year.
If this is two years then this
number will come out to 2,
and this is M in units
of the mass of the Sun.
So, you can say a^(3) =
P^(2)M,
providing you're dealing in
units of the mass of the Sun,
units of one year,
and units of an AU.
So, this is now much easier to
work with.
You've got rid of all kinds of
terrible things,
so let me give you the first
numerical example of the course.
This will be the last thing we
do today, namely,
the orbit of Jupiter.
Turns out the distance from
Jupiter to the Sun is about five
times the distance of the Earth
to the Sun.
So, a of Jupiter is
approximately five times
a, a of Earth;
a of Earth you'll recall
is this 1 AU so this is about 5
AU.
So, how does this equation work
out?
You get 5^(3) equals
P^(2)M,
M is the mass of the
Sun, 1 solar mass.
And since Jupiter is going
around the Sun that's equal to
1.
So, you have 5^(3),
5 times 5 is 25,25 times 5 is
125, so you end up with 125
equals P^(2),
so you can answer the question
now.
What is the orbital period of
Jupiter in years?
Obviously, that's going to
equal the square root of 125.
Here's another trick.
What's the square root of 125?
Quickly?
Good, more decimals?
You could type it into your
calculator though and find out,
but let me make a suggestion.
Don't take the square root of
125;
take the square root of 121
instead.
What's the square root of 121?
11.
Much easier, right?
And notice this,
a of Jupiter is
approximately five,
so 5^(3) is approximately 125,
and it's just as good to say
121 is equal to the square
root--the square of the period,
and P equals 11 years.
That's the orbital period of
Jupiter.
All right, so now,
I'm aware that many of you are
shopping the course today and
may not be back for future
lectures.
And so, I want for those people
who have decided against this
that they'll do something far
more worthwhile with their time,
I want to leave you with
something you can carry through
your life from your brief
experience with Astronomy 160.
And that is the following piece
of advice: Don't take the square
root of 125, take the square
root of 121.
It's much easier.
This is what the business
people call thinking outside the
box.
Don't do the stupid hard thing.
Do the thing that is just as
good but requires some thought
first in order to make it easy.
So, I will leave you with that,
the rest of you I'll see you on
Thursday morning.
