Prof: Welcome back to more of Astronomy 160.
The syllabus, if you haven't got it yet,
is on the classes server.
We have extra copies but I have to say I don't know where they
are, and in any case, everything that you need to
know is on the classes server and there's actually more than
is on the printed syllabus there anywhere.
Let me say a couple things; we're going to stick to the
rule that there should be no science majors taking this
class.
And from the looks of the enrollment, there's going to be
no problem for juniors and seniors.
I think we've got plenty of sections to accommodate everyone
in here, provided you're not a science major.
But the way it's going to work in terms of signing up for the
class, I think, as I understand it,
is as follows: You need a section,
sections are mandatory for this course.
So, you have to go through the online section sign up process.
That'll start on Monday but it'll only start for freshman
and sophomores on Monday.
Juniors and seniors will be able to sign up for the course
and sign for sections on Tuesday, and so please do that.
Sections will start--sections are going to be on Mondays,
so they obviously won't start next week they'll start the
following week.
I have no doubt looking around the room that we're going to be
able to accommodate juniors and seniors if they want to take the
class.
All right, other things, yes, the first problem set is
assigned as of today.
When I get back to my office I'll stick that on the classes
server as well, and you can pick it up from
there.
It's due one week from today.
The way I like to do this, I like to have problem sets due
at the start of class because it bugs me to look out and watch
people copying each other's problem sets.
So, you have to hand it in at the beginning of class otherwise
it's late.
On the classes server, there's a whole little thing
about lateness policy which you ought to read.
And also something--let me make a couple general remarks about
problem sets.
The purpose of problem sets is different from the purpose of a
take-home test.
This is not something where the major purpose is to evaluate how
much you know.
The point of problem sets is to get you to actually do those
particular problems because that's how one learns stuff in
this kind of subject matter.
It's kind of like doing the reading;
if you don't do it, you're just going to have
trouble.
And so, the reason it's a substantial fraction of your
grade is just to make sure that you do it.
So, this has implications for working in groups.
Working in groups is strongly encouraged.
Please do that.
It's a very good thing, make friends with the other
people in the class, work together,
talk to each other, start early,
and make use of each other's intelligence.
However, at the same time, we want to make sure that you
actually do things.
So, we want the work you hand in to be your own.
This sounds contradictory.
On the one hand, I'm tell you to work in groups;
on the other hand, I'm telling you,
you have to hand in your own work.
What does that mean?
So, as is written down on the classes server,
the way we try and work this is when you are actually writing
down the thing you're going to hand in,
do that alone.
So, if you work in a group, figure out what you're doing,
get everything all set up, then split up and write down
the answer, because the process of actually writing the thing
down yourself will make sure that you've actually understood
what you were doing.
We'll check on this.
We'll look for identical wording or identical
multiplication mistakes and so forth, and so we'll keep a
little bit of an eye on it.
I do encourage you to work in groups.
There's other ways of getting help if you need it.
The primary source of help are the sections,
but of course, those won't be meeting this
week so not before the first problem set is due.
Office hours; I'll be in Starbucks Monday
from 9:30 to 11:00; please come and say hello,
and ask any questions you might have.
The teaching fellows are going to have office hours on
Wednesdays; conveniently located the day
before the problem sets are due you'll notice.
And let me strongly recommend, don't send us email with
questions about it.
Send things into the forum on the classes server.
That way--and look first on the forum to see if somebody has
sent your questions.
This works really well.
We'll keep an eye on that all the time.
We'll be answering it--not all the time, not probably after
around 8:00 at night on Wednesday.
If you have a question at 2:00 in the morning the day before
something is due you're kind of on your own.
Yes?
Student: Is that Monday 9:30 to 11:00 in the morning or
the evening?
Prof: Excellent question, thank you so much.
Morning, I'm afraid, sorry.
Yeah, I'm not as young as I used to be.
Let me say, if you do want to meet at some ghastly hour of the
evening or something, send me email.
We can find a time, same with the teaching
assistants.
Send stuff into the classes server at all times,
we just don't guarantee to look at it after 8:00 the night
before it's due.
Start the problem sets early, that way if you have a question
you can answer it.
Not only that, if you even just look at the
problem set, your brain will work on it in your subconscious
and you won't freak out quite so much when you get past midnight
the night before it's due.
Let's see, in general, I'm going to try and make it so
that everything you need to know to do a particular problem set
has been covered in class by the day the problem set is handed
out.
That would be today for this one.
In the very first week this week I'm not quite going to make
it and so there's one problem, it's noted on the problem set,
there's one problem that you'll need stuff that I'll talk about
on Tuesday.
There are going to be a bunch of help sheets for various
topics that are going to also be linked to the classes server.
By all means, take a look at the problem set
and start thinking about it as soon as you can.
Okay.
Let me remind you what we had started to talk about.
The class is organized into three fairly specific topics.
The first of which is extrasolar planets.
Planets around stars other than the Sun.
Exoplanets, so-called.
That's our topic.
One of the things that I pointed out last time is that,
surprisingly enough, very--until ten years ago none
of these were known.
And it's only a very recent development that there's any
actual evidence that these things exist.
So, one question you might ask is why are these things so hard
to find?
The science fiction folks seem to have no trouble;
they just sort of go around in their spaceships and find these
things all over the place.
And to consider that question let me show you a picture.
Here's a picture of a star.
This is the star Sirius.
It's the brightest.
It's a blowup obviously of a photograph plus a little
Photoshopped [computer program to edit photos]
arrow.
That's not a celestial object [laughter].
So, this is a blowup of a photograph of the star Sirius.
Sirius is the brightest star in the sky to the--easily visible
with the naked eye.
In fact, as I say, the brightest star.
It's one of the closest stars.
It's a little bit brighter than the Sun intrinsically,
but it's ten light years away or so.
And let me comment on a couple features of this picture.
First of all, you can see the star covers a
fair amount of area on this blown up picture,
but that has nothing to do with the actual size or shape of the
star itself.
It's not like it's a big round ball with spikes coming off it.
That's not what's happening.
In fact, if you had perfect optics and perfect vision,
this star would be a point basically so small you couldn't
resolve it right in the middle here.
That's the physical extent of the star at this scale.
It's something like this [pointing to photo],
and you wouldn't be able to see it at all.
The reason it's extended is for two reasons;
first of all, the atmosphere distorts--this
was a ground-based picture, the atmosphere distorts how you
see the star, and in particular,
it's like looking up from the bottom of a swimming pool.
There's distortion in the atmosphere and it makes the
stars seem to jump around.
And so if you take a long photographic exposure of
something the star has been jumping around during the
exposure and it smears out the image.
Plus also, the optics of the telescope aren't perfect,
and these spiky things here are due to the optics of the
telescope.
This sort of--there are all these spikes and that has to do
with how one of the mirrors of the telescope is held up.
So, the combination of optics and atmosphere sort of spreads
out the light by a whole bunch.
This is always the case even when you're doing observations
from space.
Now, the little arrow here is pointing at a little bump off in
the side, but you might not have thought anything more of than
you did about these spikes.
But in fact, that's not a property of
optics; there's actually something
there.
That's an object in orbit around Sirius.
And you might think that this is the way you would go about
finding planets orbiting other stars.
It would look kind of like this.
Here's a great big star; there's a planet.
And if you watch this thing over a course of many years,
you actually see the position of this point move around.
It's because it's in orbit.
It's in about a forty-year orbit.
Here's the thing.
This isn't a planet.
It's another star.
This is another star somewhat fainter than this [pointing to
the photo of Sirius].
A planet would be 10,000 times fainter than this thing,
but probably in about the same relative position to this.
Now, picture all this light, all this mess from this star,
and try to find something right here that's 10,000 times fainter
than this.
That's the problem.
The problem is that planets are relatively faint things.
That, in itself isn't catastrophic.
We have big telescopes; we can see pretty faint things.
It's just they are faint things that are really near to things
that are bright, and so basically,
the problem with finding planets around other stars is
the same as trying to find stars in the sky during the daytime.
Why can't you see the stars during the daytime?
Because the background of the sky is much, much brighter than
the stars are.
Similarly, if you had a little planet here you wouldn't be able
to distinguish it from the atmospheric distortions and the
optical distortions caused by the telescope that takes this
picture.
So, the problem is you've got stuff that's faint near things
that are bright, and the bright things have all
kinds of distortions on them anyway, and so you wouldn't be
able to figure out what actually is a planet.
So, just to summarize this, the problem is not that the
planets are too faint, although they are quite faint.
The problem is that the planets are too close to the star.
That prompts--we've got to quantify this statement I think
and so the question arises, "Well how close is it?"
That leads into a discussion, which we already started last
time, of planetary orbits.
If you want to answer that question in any quantitative
sort of way.
You will recall that in the last class I wrote down an
important equation of--regarding planetary orbits,
that looks like this: a^(3) = P^(2)M,
where this [M] is the mass of the objects in
orbit around each other in units of solar masses.
This [P^(2)] is the orbital period in units
of years.
And this [a^(3)] is--orbits you will recall are
ellipses, and this is the so-called semi-major axis of the
ellipse in units called Astronomical Units,
which is the distance from the Earth to the Sun.
Or the semi-major axis of the Earth-Sun orbit,
more properly.
So, that's in a particular set of units.
Before going on, let me make a couple general
remarks about equations.
First thing to say about equations is there's this idea
that equations, mathematics,
is some terribly different language, not at all.
This is an English sentence.
The sentence reads A cubed is equal to P squared times
M, period.
If you wrote this down in a textbook you'd actually see they
put the period in there.
It's got a verb, is equal to,
it's got a subject, it's got an object.
All this is is a different way of writing stuff down.
So, it's not a different language;
it's just a different writing system, that's all.
So, don't freak out, that's the first thing.
It's just ordinary English.
The second thing to say is that this is a physics equation and
not a math equation, and that's actually a subtler
point.
You will recall back in eighth-grade algebra when they
started writing down X is equal to whatever X was equal to.
And they said, you know, the amazingly cool
thing about algebra is that X could be anything.
That's why it's so powerful, and so you can solve whole
classes of problems all at once just by letting X equal
something unknown.
That's math; physics--it isn't true.
It is not the case that this equation is true regardless of
what a is, or regardless of what P
is.
This is only true for very specific meanings of those
quantities.
And the amazing thing about the Universe is it turns out that
math equations apply in very specific physical situations.
This didn't have to be true.
Either that you could express physical things in mathematical
equations, and that math and physics would have anything to
do with each other.
And it is different to have a physics equation than to have a
math equation because it has to be so specific.
If you substitute something that isn't the semi-major axis
in for a, then that equation isn't true
anymore.
It seems kind of obvious, I know, but worth keeping in
mind.
The other thing is that physics equations tend to have units
associated with them.
Not always, but most of the time.
And so that's why it's so important to say,
well, this is true not only if M stands for mass but
also if mass is expressed in a particular set of units,
in this case solar masses.
You'll recall from last time that this is actually a specific
version of a slightly more general equation which looks
like this: a^(3) = P^(2)GM/4π^(2),
where G is a constant that depends on what units you're
using.
And last time I derived this equation from that.
But another way of saying it is if you choose your units to be
solar masses, years, and astronomical units;
then G is equal to 4π^(2) and so it cancels out.
That's another way of thinking about why this specific version
of the equation works out.
All right.
So, once you've got your physics equation,
what are you supposed to do with it?
Well, the first thing you should do is take a careful look
at it and figure out what kinds of problems you can solve.
This, again, is fairly straight-forward,
obvious stuff.
If you've got a^(3) = P^(2)M,
there are obviously three different problems you can
solve.
If you know a and M you can compute
P, right?
If you know a and P you can compute
M.
And if you know P and M you can compute
a.
So, you know two of these things, you find out the third.
This problem--an example of this problem we did last time.
We had--we solved for the orbital period of Jupiter by
knowing how far away from the Sun it is and what the mass of
the Sun is.
So, like Jupiter last time.
So, let's do another one of these examples just quickly
here.
What is the mass of the Sun?
So, that's obviously this version of the problem [if you
know a and M you can compute P].
And so, what we're going to do is we're going to take a
and P from the Earth's orbit.
Use Earth's orbit to determine the mass of the Sun.
And this is, in fact, how one goes about
determining the mass of the Sun.
So, let's use this in the units we have, a = 1
Astronomical Unit for the Earth.
P is equal to 1 year for the Sun.
1^(3) = 1^(2) x M.
So, M = 1 in units of solar masses.
That's not especially surprising, right?
Not particularly illuminating, right?
That the mass of the Sun comes out to be one in units of solar
masses.
If we use a different set of units this is a more interesting
calculation.
So, let's use a more standard set of units,
this so called mks units, this is meters,
kilograms for mass, and seconds for time.
All right, so now, turns out an Astronomical Unit
is expressed in meters is 150 billion meters,
approximately.
It looks something like this [150,000,000,000].
And what's a year expressed in seconds?
Well, let's see a year is 365.24 days, a day is 24 hours,
an hour is 60 minutes, and a minute is 60 seconds,
so that's a year expressed in seconds.
And at this point we better pause and get our large numbers
straight.
You are all probably familiar I imagine with scientific
notation.
This becomes very important, otherwise the numbers,
as you can clearly see, get completely out of hand.
And because it's so important I'm going to tire you by writing
down some facts about scientific notation.
There's a help sheet on this if you have any trouble but recall
how this works.
You're supposed to express numbers as in the following
form: N times 10^(m), this [N]
is a decimal with one digit out in front of the dot.
This [m] is supposed to be an integer,
a whole number either positive or negative.
And the reason that this is easy to deal with is because of
one of the great limitations of the human brain.
We only deal well with really small numbers,
basically one-digit numbers.
If I take a few coins here and I sort of throw them up you
immediately know that there are four of them,
you don't have to count one, two, three, four to satisfy
yourself that there are four coins on this table because we
have a concept of fourness, all of us in our minds,
and it's clear that this pattern satisfies that concept.
If I threw two coins or six coins, or seven coins you'll
probably be okay.
If I threw thirteen, you'd start to have real
trouble.
If there were 271 of them, there's no way you could guess
just by looking at them how many there were.
You'd have to count them one by one.
So, the human brain really only deals well with one-digit
numbers and you can tell this because of the names we give to
numbers.
When you start getting big numbers we call them millions or
billions, or trillions and they all sound alike,
that's because we can't tell the difference between a million
and a trillion, it's just big.
So, anytime you get up into big numbers we start to completely
lose our grip on what these numbers mean,
and you have to have elaborate metaphors;
a trillion is a million millions.
And if you fill up the Moon with ping-pong balls there
are--I don't know.
You have to do that kind of mental exercise to get a grip on
it.
The nice thing about scientific notation is that it turns big
numbers into small numbers, right?
N is a number between one and nine, m is an integer,
it's one of those numbers four, five, six that we can get our
minds wrapped around.
Recall what this means, right, 10^(m) is 1 with m zeros
after it.
So, that's the point of scientific notation,
it's to take big numbers and allow us to deal with them.
I think this is a really valuable thing and I kind of
wish that budget discussions in the political context would take
place in scientific notation because people are always making
mischief from the fact that all big numbers sound alike.
You have congress critters saying things like,
"oh terrible disaster," there's a million dollars of waste in
the Pentagon budget.
The Pentagon budget is $500 billion dollars.
One million dollars of waste--that's me dropping a
nickel behind the couch on my budget.
So that's--nobody would accuse me of being wastefulness if I
didn't move the couch and get that nickel out,
all right.
It's because a million sounds big, and so does 500 billion.
We have no real sense of how big these things are relative to
each other.
But we do have a sense of the difference between 6 and 11,
which is the difference in scientific notation between one
million dollars and $100 billion dollars,
so that's how this works.
So, arithmetic rules.
Again, there's a help sheet on this if you're rusty.
If you have N times 10^(m) and you multiply it by another
scientific notation number, A times 10^(B),
that is equal to A times N (times 10 to the n+m).
And the other nice thing about scientific notation is it turns
multiplication into addition because the exponents add:
A times N to--times 10^(m) plus B.
Similarly, if you take N times 10^(m) and raise it to the K,
that's N^(k) times 10 to the m times K,
so it turns exponentiation into multiplication.
So, the nice thing about dealing with exponents is it
makes it one kind of arithmetic easier.
So, a specific example of this might be N times 10^(m),
the square root of that which is raising something to the ½
power.
This is equal to N^(1/2) times 10^(m/2).
That works if m is an even number but if m is
an odd number you get a crazy exponent.
This really has to be an integer over here.
So, what do you do then?
You say N times 10^(m) to the ½ is equal to 10 times N times
N^(m-1) to the ½ power.
So, I've just moved a 10 from here over to here,
and then that is equal to 10 N to the ½ times 10^((m-1)/2).
And that's how you recover when you take a square root or
something like that.
That's how you recover an integer on this side.
We'll have lots of practice in this kind of thing.
So, going back to the problem.
1 AU is equal to 1.5 times 10^(11) meters;
1 year is equal to 2.4 times 10^(1).
That's 24 times 6 times 10^(1) times, 6 times 10^(1),
times 3.6524 times 10^(2) seconds.
All right, so let's do that one, two, three,
four, five--2.4 times 6 times, 6 times 3.6 times 10^(5) to the
5; 2.5 times 6 is around 15,15
times 6 is around 90,90 is a little less than 100,3.6 a
little more than 6.
Those two things multiplied together is around 300.
And 300 times 10^(5) is equal to 3 times 10^(7).
So, there are around 3 times 10^(7) seconds in 1 year.
That's a good number to remember.
In mks units, G is this magical constant,
is something like 7 times 10^(-11).
That's an approximation.
It's actually 6.6 something, something, something times
10^(-11), but we'll call it 7.
So, now we're in a position to calculate what the solar mass is
in kilograms.
So remember a^(3) equal to GMP squared over
4π^(2).
So, let's see, 1.5 times 10^(11);
that's an astronomical unit cubed.
P squared, 3 times 10^(7);
that's 1 year squared; 7 times 10^(-11) that's
G over M over 4π^(2).
I'm going to do the arithmetic really fast.
I would like you to learn how to do this yourself rather than
relying on calculators so this is an example.
Let me blaze through this at warp speed;
then we'll stop and consider what's happened.
All right.
On the right-hand side, 3^(2) is--that's 3^(2) times
10^(7) times 2 is 14, times 7 times 10^(-11).
M over 4 times π, times π.
3^(2) is 10.
10 times 10^(14) is 10^(15).
And so on the top we've got 10^(15) times 7,
times 10^(-11).
15 minus 11 is 4 so that's 7 times 10^(4) on top.
π times π is 10 that's 40.
4 times 10 to the 1 on the bottom, 7 divided by 4 is 2,
and so that's 2 times 10^(3) times m.
Good, left-hand side, all right 1.5 times,
1.5 times, 1.5, that's 1.5^(3) times 10^(33),
why 33?
That's 11 times 3.
1.5 times 1.5 that's a little bit more than 2.
If you multiply something that's a little bit more than 2
by something that's a little bit less than 2 that's 4,
and so this is 4 times 10^(33).
Now we want m, right?
So we've got to divide both sides by 2 times 10^(3);
m is equal to 4 halves times 10^(33) over 10^(3).
That's 10^(-33) minus 3.
That's 2 times 10^(30) kilograms, got to put the units
in.
Okay, how we doing?
Questions, comments, abuse.
Yes sir?
Student: How accurate is that number?
Prof: How accurate is that number?
It's accurate to about one digit, which is why I only wrote
down one digit of accuracy out front.
This is a pro--excellent question thank you very much.
This is appropriate because remember I'm using 7 times 10
minus 11 for G, where it's actually 6.6.
So, I'm already about 10% off from there.
I did my little calculation to come up with one year equals 3
times 10^(7) seconds.
That's about accurate to one digit or so.
And so the whole thing is done to one digit accuracy.
If you're dealing with one digit accuracy,
it is true that 7 divided by 4 is 2.
It really is true, because if that wasn't true,
then you would have to have more digits on your unit for
G or something like that.
In particular, let me make an official rule
for this course.
Three equals π, equals the square root of 10,
all right.
That will solve an enormous amount of arithmetic problems
and it will not get you into any serious trouble.
So, we don't have to worry about the .14159 and however
many more digits you all memorized it to.
And when you multiply it together you get ten.
Yes?
Student: Are you expecting this kind of
calculation for problem sets?
Prof: Yes.
The question was, "Am I expecting this kind of
calculation for problem sets?"
The answer is "yes."
Here's the rule about calculators.
Let me put it this way: You can only use calculators if
I can't tell that you've done it.
So, that means you can check your work to make sure you've it
right or something.
But if you start coming up with numbers like 7.1516397,
that's eight digits of accuracy and I'm pretty sure you haven't
worked it out yourself.
So important, no calculators on the tests,
okay?
So, get some practice doing this kind of thing.
And this will--this I promise you will be useful to you in
everyday life because this is how you catch the politicians
doing screwy things with big numbers.
You do it in your head in scientific notation and you
figure out whether the answer is meaningful or not.
This whole business of significant digits,
I think, is badly distorted; by the way, it's taught in high
school.
In high school you, and also I should say in
laboratory courses sometimes at the college level,
you often get situations where people say--give you a whole
sheet of rules on how to figure out how many significant digits
you have.
This is nonsense.
All you have to do is behave like a human being.
We say to each other, I'll meet you in the dining
hall in ten minutes.
That doesn't mean--that means something different from I'll
meet you in the dining hall in eleven minutes and twenty-six
seconds.
Even if the person happens to show up in the dining hall in
exactly eleven minutes and twenty-six seconds.
Ten minutes means I'll meet you there in ten minutes,
we all know what that means.
I'll meet you there in eleven minutes and twenty-six seconds
means you're a character in a bad spy novel who's just
synchronized his watch.
So, this shows up in science fiction too.
I don't know how many of you are Star Trek fans,
I certainly am [laughter].
And in all the different Star Trek movies [inaudible
comment]--thank you.
In all the different--a friend [referring to person who made
comment].
In all the different Star Trek movies there's always a second
in command who isn't a human being, right?
A Vulcan or an android or some damn thing or another.
And to emphasize the non-humanness of these
characters, what they do is they make them use too many
significant digits.
And so that makes them inhuman and so the captain will say,
"When are we landing on omicron M?"
The second in command will say, "Well, we should assume
standard orbit in 2.6395 minutes," emphasizing somehow
superior brain power or something.
But it's nonsense because it takes the guy ten seconds to say
that sentence, so what is this time calculated
to a 100th of a second?
Does it start from when he begins the sentence?
From when he ends the sentence?
What's the other end of that time interval?
Can you say you assume standard orbit to the 100th of a second?
What does that even mean?
When you start beaming down?
When you end beaming down?
Also, keep in mind it takes more than a 100th of second for
the sound to travel from his lips to the captain's ears,
so the whole thing is just nonsense.
And so, you don't need any special rules,
just behave like a human being; don't behave like an android.
So, no androids.
And that's the only rule I'm going to give you [laughter].
These two are the only rules I'm going to give you about
significant digits, just do the right thing,
okay.
All right, okay so this is all lovely but we haven't actually
started to answer the question we started the class with.
Let me remind you where we began this disquisition on
numbers.
The question was supposed to be how far away from stars are
planets?
So, we haven't answered the question, when the question was,
"How close are planets to stars?"
This notation will confuse the people who didn't make it to
class today; that's probably just as well
too.
All right, how close to planets--are planets to stars?
Now, we have a problem because closeness--we usually think of
this as being measured with distances.
And distances are a serious problem in astronomy.
Supposing you--Distance, by the way, is more or less the
same thing as size, right?
Size is the distance from one side of something to another
side of something.
So, size and distance are the same.
So, supposing you ask the question, "How big is the Moon?"
What is the size of the Moon?
What is the distance from one side of the Moon to the other?
You go out one night and you look up at the Moon and you do
this [holding fingers apart], one on each side of the Moon.
You say, well, it's about an inch and a half.
Now, obviously that's meaningless, because if you had
done it here, it would have been about a half
inch.
And we all know that isn't--that somehow is not the
right answer.
If you look at the screen over here, for example,
and you hold out your hands about eight inches from
your--close one eye, hold out--you can try this,
hold out your hands about here, you'll find that that
screen--the lit part of the screen is about four or five
inches across.
But if you go closer to your eye, it's only about an inch
across.
If you're at arm's length it's maybe six, eight inches where
I'm standing here.
And if you try it at the back of the class,
you'll get a different answer from what I just got.
So, distance is a problematical thing.
And so--But on the other hand, obviously, the size of the
screen hasn't changed depending on where my arms were or whether
I'm standing at the back of the room or the front of the room.
So, something has to be measurable there,
but it isn't distance.
What it is, is angle.
What you can measure in astronomy are angles.
So, here's what you really want to know.
Here's a star; here's a planet going around
the star, and here is you [draws diagram on overhead].
This is the international symbol for an observer;
it's supposed to be a little stylized eyeball.
So, this observer is looking at this star-planet system.
And looks at the star and looks at the planet.
And what you can measure is the angle, which I'm going to call
α here.
And if you simulate the angle by putting your hands up close
to your eye, you'll get a small distance;
if you do it further away you'll get a bigger distance.
But what you are measuring is the angle.
What you would like is a way of being able to convert from that
angle into a distance.
And now, the ugly specter of trigonometry rears its head.
Because that's what trigonometry is.
It's a way of turning angles into distances and vice versa.
And you may remember constructions that look vaguely
like this; let's put some labels on it.
Here's distance 1, here's distance 2,
here's distance 3.
And this is the definition of sine, the sine of α is
equal to the opposite over the hypotenuse, so that's
D_2 over D_3.
And now--so that's how you do it.
Now, I promised you on Tuesday that the sine was going to
cancel out, and so it is.
Here's how.
If you use small angles it turns out that the sine of
α is equal to α, if α is expressed in
radians, which is a particular kind of measure of angles.
And so for small angles, first of all,
the hypotenuse and the longer side are the same length more or
less.
So, you've got a situation like this where D_2 over
D_1 is equal to α.
That is--let's circle that one in red;
it's going to be important.
That's called the small angle formula.
And once again, you have to be careful of the
units.
And once again, there's two sets of units in
which you might possibly use this.
So D_2 over D_1 is equal to α.
If D_2 and D_1 are in the same
units, then α has to be radians.
A radian--there are 2π radians in a circle.
You'll recall there are 360 degrees in a circle that's how
you convert from degrees to radian.
But there's a better set of units.
It turns out also that if you do this, use this equation
D_2,/D_1, α, and D_2 is
measured in Astronomical Units, remember those?
That's the Earth-Sun distance.
And D_1 is measured in parsecs, which is equal to 3
times 10^(16) meters; that's about three light years,
for those of you counting at home.
And α is in arc seconds.
Then it also works out.
Let me tell--an arc second, let's see, 60 arc seconds is
equal to an arc minute.
Just so you know how these units work;
60 arc minutes is equal to 1 degree.
So, that's a really small angle.
So, it turns out this set of units is particularly useful for
these kinds of planet calculations,
because distances from planets to stars tend to be a few
astronomical units.
Distances from us to the nearest star.
The nearest star, aside from the Sun,
turns out to be about 1 parsec away.
So, distances to stars are some number of parsecs.
And so, you'll get answers that are a few arc seconds,
and this is yet another way of turning big numbers into small
numbers.
You've used the right set of units so that everything's
approximately one.
Another way of compensating for the weakness of the human brain.
Okay, so now we can solve a problem.
We can actually solve numerically the question of how
far away from a star is our planet?
So, here's an example.
Here's a problem of the kinds that might show up on a problem
set.
I told you that that object orbiting Sirius has about a
forty-year period.
So, let's take a planet with a forty-year period;
that's kind of between Saturn and Uranus in our own Solar
System.
Around a star 3 parsecs away; pc means parsecs,
whatever they may tell you in the humanities classes.
Around a star 3 parsecs away, what is the angular separation?
Angular separation, because that's what we can
actually measure.
We can't measure the distance but we can measure the angle.
Okay, so this is a hard problem, why is it hard?
It's hard for two different reasons.
First of all, well, you guys know how to
solve problems like this.
You all went to high school.
What you do is you say, okay what do I know?
I have P = 40, I have D = 3,
I want α, so I look for an equation with D,
P, D and α in it.
I shove these two things into--you substitute them in for
P and D, and then I calculate α.
And you don't have to actually know anything about anything in
order to do that; you just have to have a good
set of notes.
And you go through all the different equations until you
find the equation that contains all the variables you have,
and the one that you're asked for.
And then you can just plug it in, you don't have to know
anything about anything.
This won't work, right?
Because if you go through the lecture notes thus far,
you will not find an equation with P and D and
α in it because it's two different equations that you
have to put together.
And that requires you to actually know what you're doing.
And that of course is much harder.
So, plug and chug won't work.
Chug fails because there's no one equation.
This is a hard problem for another reason,
which is that I haven't told you something important.
If you try and go through and solve this you will discover
that you don't have all the information you need,
and in particular, you need to know the total mass
of the system, the mass of the star.
Most of the mass is in the mass of the star.
So, there's missing information.
That's one way to make a problem hard,
the other way to make a--if I don't tell you everything you
need to know, it's a somewhat harder problem.
The other way to make a problem hard--this is a classic,
be warned--I will do this at some point,
is to give you one extra piece of information that you don't
need.
This screws students up totally because you go looking for the
equation that uses the extra piece of information,
and there is no such an equation and you don't need it,
then you're cast into doubt.
I must need this somehow or he wouldn't have given it to me.
Nonsense.
In real life it is not the case that all problems come to you
with all the information you need and only the information
that you need.
That's not how things work in real life.
And it shouldn't be how things work in preparation for real
life, which is I guess what classes sort of ought to be.
So, missing information, too much information,
sometimes although not in this case;
I could have told you the temperature of the star,
for example.
I'll do that.
The temperature of Sirius is 10,000 degrees,
surface temperature.
So, now I've given you both; I've left out information and
told you too much.
All right, so what is the mental process that you need to
go through in solving a problem which has these kinds of
difficulties associated with it?
Okay.
So, how to think about such problems?
The first thing to notice is what you would do in the
ordinary plug and chug.
What have you got, what do you need?
You have a value for the orbital period,
you have a value for the distance, and what you want is
an angle.
Okay.
Now, so this doesn't lead you to where you want to go.
The next thing you've got to do is make an assumption.
This is a star; all stars are more or less the
same.
So, let us imagine that this star has a mass more or less
similar to that of the Sun.
So, the M is approximately equal to a solar
mass.
How far wrong could you go?
In fact, you go wrong by about a factor of two,
but at a certain point you're going to take the cube root of
two, and so it actually isn't so bad.
So suppose M is equal to the mass of the Sun,
most stars are approximately.
Ah ha, now we're getting somewhere, because now we have
M and P.
And if you have put that equation--Kepler's Third
Law--and interpreted it deeply into your brain,
and it has become part of your heart and soul and very being,
you will immediately realize that if you have M and
P, of course you can calculate
a.
In fact, you probably realized this first before you make the
assumption in step two, because that's why you're
interested in the mass, because you realize that if you
happened to have M, you would be able to determine
a.
Next thing you have to do is, is interpret what this a
is; a is the semi-major axis
of the orbit of the planet around the star.
That's something very close to--very similar to--the
distance between the star and the planet is a.
So, a in this equation is the same as D_2 in
the other equation, all right.
So, a is D_2 and the distance you have up
there is the distance from us to that whole system,
that's D_1.
And now, you can compute α, right?
That's the thought process.
And you may have to go--sort of worry a little bit around here
before you have the "ah ha" moment of how to get the
problem.
This is the moment when you've solved the problem.
The key to solving this is to have an instinctive reaction to
knowing that you've got a period;
namely, if I had a mass I could figure out the semi-major axis.
Or if I had the semi-major axis, I could figure out the
mass.
And that has to click.
And the way that clicks is to have this equation really--I was
joking before--but really become instinctive in a certain way,
and there won't be many equations in this class.
But it will really help you to assimilate them in that kind of
depth.
All right, so now in the few minutes remaining,
let me actually solve this problem numerically.
P is equal to 40 years, D is equal to 3 parsecs.
Here's something not to do: 40 times 3 times 10^(7),
because that'll give you the answer in seconds.
No don't do that, because both of the equations
would rather have things in Astronomical Units and years and
things like that.
So a^(3) equals P^(2)M,
M is 1 by assumption, P is 40 so this is
1,600, that's 40^(2) on this side, a^(3) that's 1.6
times 10 to the 3.
Now, we're going to take the cube root of both sides,
that's the cube root of 1.6 times, the cube root of 1,000.
3 times a third is equal to 1, conveniently enough.
The cube root of 1.6, I don't know that's 1.2 or
something but call it 1.
So, that's 10, and so it's 10 Astronomical
Units.
Remember we're doing this in solar masses,
years, Astronomical Units.
10 Astronomical Units, that's how far away it is.
Now, we can do this [D_2/D_1]
where this is in Astronomical Units, and this down here is in
parsecs, and 10 divided by 3 is 3.
So, the answer is a is equal to 3 arc seconds.
Now, this is an important result because the--let me go
back to this picture.
The angular scale over which light is kind of thrown around
by the atmosphere and by the optics of the kinds of
telescopes we can build is some number of arc seconds.
Good ground-based observations, maybe half the light is inside
1 arc second; the rest of the light is
scattered all the way out to maybe 10 arc seconds.
From space it's a little bit better.
But, light from observations of stars is scattered over angular
sizes of arc seconds.
So, you are guaranteed that there's going to be a bunch of
light from the star right on top of where you're looking for your
planet.
And in fact, it's going to get worse,
because one of the things about this equation is D_1;
the distance to the star.
We did this for Sirius.
Sirius is one of the closest stars;
it's 3 parsecs away.
Most--the center of the galaxy is 8,000 parsecs away.
So, if you take a typical star in our galaxy this number isn't
going to be 3; it's going to be 8,000.
And this number isn't going to be 3, it's going to something
like 1/1000th of an arc second.
And the space telescope scatters light over tenths of
arc seconds, so you have factors of 100 buried inside the light
coming from that star.
So, that's why you don't care about more than one digit.
In fact, you probably don't even care about anything other
than the exponents of this answer because why are you doing
this problem?
You're doing this problem to figure out whether it's
plausible to think that you might be able to see a planet as
a separate dot, separate from the star.
And the fact that your answer comes out in arc seconds or
fractions of an arc second tells you everything you need to know.
You don't need anymore decimal places to realize that the idea
of looking up and seeing a planet is going to fail.
So, that's not the way that we're going to find planets.
So, we have to come up with different scheme and that's what
we're going to talk about on Tuesday.
Thanks.