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Lecture 2.02: Measuring density
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Del curso dictado por Caltech
The Science of the Solar System
274 calificaciones
De la lección
Unit 2: The insides of giant planets (week 1)
Conoce a los instructores
Mike BrownProfessor
Planetary Astronomy
We have a lot of tools available,
if we want to try to understand what's inside of the Earth.
We can't go there, but we can do a lot of stuff.
We can dig holes and
the holes don't go very far down compared to the center of the Earth.
But you learn some things like what's the composition of the outer part of
the Earth.
You learn about heat flux, which you realize,
as you dig a hole down a deep mine is that it's warmer as you get further down.
That's telling you about the heat coming out of the interior of the Earth.
And we can do things like use seismometers to use
earthquakes to monitor what's going on inside of the Earth.
None of these techniques works on Jupiter.
So we're going to have to resort to a lot of even more indirect measurements to try
to understand what's going on.
But the very first measurements that you want to do if you wanted to understand
what a planet was like on the inside, what a planet is made out of is you'd want to
measure the density of that planet.
And let me just remind you of densities, of a few things that matter,
in rough numbers that we'll be using.
Density, which is always written as a row.
Density of rock is about 3 grams per cubic centimeter.
I'm going to use this unit because, well, I'll show you why.
I'm going to use this unit because that's rock.
Because ice or water is 1 gram per cubic centimeter and
that's a really easy thing to remember.
The one other thing you might want to keep in mind is the density of iron.
Density of iron is closer to 8 grams per cubic centimeter.
Now, I can't just go measure the density of a planet and say the density is this,
therefore it's got this, this, and this.
Because materials get compressed by the big pressure of the insides,
their densities go up.
You take a rock, and you smoosh it, and it gets more and more dense.
So you have to understand that.
Remember, we did that for Mars a little bit.
But in general, you can tell the difference between icy, watery planets,
rocky planets.
And even things that have maybe a lot of iron in them,
like the core of the Earth is, just by knowing what their density is.
So how do you know the density of a planet?
Well, the density is of course mass divided by volume.
You can figure out the volume, if you see how big it is in the sky.
You measure its diameter, its radius, and you can measure the volume very easily.
How do you get the mass?
The only good way to get the mass is if the planet has a satellite.
If a planet has a satellite like Jupiter does, we saw, and
you see that satellite going around and
you can measure both the distance away of the orbit.
The semi-major axis, we'll call it, or we can just call it the radius.
As long as it's a circle, we can just call it the radius.
And you measure the amount of time it takes for
that orbit to happen, you get the mass of the thing on the inside.
We'll go through the maths here in a minute, but
let me just show you, Galileo had that from the first moment.
He had these moons going around Jupiter, he could track each one.
He could figure out how far away it got, how long it took to go around, and
he could determine the mass of Jupiter.
Well, almost he could.
What he did determine is that these moons obeyed Kepler's Laws.
And Kepler determined just empirically by looking at the planets, one is,
he figured out that the planets go in elliptical orbits around the Sun.
But he also found that the period, the square of the period of that orbit was
proportional to the radius of that orbit cubed.
Really, he figured out that it was proportional to the semi-major axis cube.
Because if you're in an ellipse,
the semi-major axis is half the distance of this major axis.
But as long you're going to circle the orbit, that's the same as the radius.
Galileo found the same thing held for the Galilean satellites.
Their periods were proportional to the cube of their distances away from Jupiter.
Newton came along later and explained why that was the case.
The force of gravity between any two objects is equal to G,
the gravitational constant.
The mass of one object, the mass of the other object / by r squared,
the distance between those object.
And when an object is in orbit, that force, which is pulling
only in this direction, is balanced by the centrifugal force of the object,
which is pulling it in this direction because of the curvature of the orbit.
And that centrifugal force, as you might remember, is mv squared / r.
v is the velocity that this object is moving along through here.
We can figure out the velocity, the velocity of course is related to how long
it takes to go around this full orbit.
So that the period of the orbit is equal to the circumference
of the circle, 2 pi r / by the velocity.
And if we solve this for velocity,
put it here under the equation,
we got the GmM / r squared = 2 pi r / p squared m / r.
See that the masses are going to cancel out the mass of the actual object itself,
the mass of the planet.
In this case, it doesn't matter, it's only the mass of the central object.
That's only true as long as the central object is much more massive.
But that's the case in all these cases here.
And solving through these,
we get that p squared GM = 4 pi squared r cubed.
Look at this, we got that p squared is proportional to r cube,
that's just what we had over here.
So this recovers Kepler's Law, but
it shows us also what those proportionality factors are.
Those proportionality factors are, well, there's just some numbers over here.
But G and M, M is the mass of that central object, the mass of the Sun.
If it's a planet, the mass of Jupiter, if it's a moon going around.
And so we can solve for the mass, the mass = 4 pi squared / G r cubed / p squared.
So all we have to do is figure out the radius of the orbit, period of the orbit,
and we get the mass of the thing in the middle.
So that point, Newton could go back and figure out the radius of those orbits
from observations of Galileo or the many observations after that.
Periods were very easy to determine.
And the mass, well, we're not quite there yet because there are two problems.
One is G, this is a gravitational constant, but
it was not well known at the time.
And actually, the other is R, we'll talk about that a minute.
First, lets talk about G, how do you measure what G is?
Newton simply said that the force was proportional to this
product of the mass divided by r squared and so that proportionality constant is G.
How do you measure G?
First, really measurement for G was in about 1797, by Cavendish.
And it's so famous, it's actually called the Cavendish experiment.
And it's a pretty simple idea, which is of course if any two masses attract
each other, if you can put two masses next to each other and
see the force that they exert towards each other, you've measured G.
And he did exactly this, he put a pair of
weights on a, it's a torsion spring.
Imagine like a long strip that is allowed to rotate one direction or
rotate the other direction.
And then he would take larger weights on either the front and
back, or he would switch their positions front and back and
watch this thing ever so slightly deflect.
And he could calibrate how much it took to deflect that.
And he measured the value for G,
that's something within 1% of the value that we know, today.
He actually did it, he didn't think of himself as measuring G.
He thought of himself as measuring the density of the Earth.
Nobody really knew what the density of the Earth was, but we knew what G,
the gravitational constant was, something like 10 meters per second squared.
And we knew how big the Earth was and
so, Cavendish's question was, what is the density of the Earth?
How much mass does it take to give this
amount of gravitational pull for the Earth?
And the only way to know that was to know how mass and gravity related,
which is to know G.
But at the time, Cavendish didn't think of it as G.
But in the end, if you take his density measurement inverted for G,
he had a very precise measurement of G.
So now, we can go measure the density of Jupiter, right?
Still not quite, and the reason still not
quite is because when we look at Jupiter in the sky, we can see its radius.
We can see how far away the moons are.
But all of that is just an angular distance on the sky.
And what we don't know at this point is how far away Jupiter is.
If Jupiter is really far away, these distances are huge,
Jupiter is really close, these distances are kind of small.
What we do know is the relative scale of the Solar System.
We know the Sun's in the middle,we know the Earth is here at 1 AU.
We know that Venus is over here at point 0.7 AU.
AU, of course, remember, is astronomical unit, where we've simply defined
the distance from the Earth to the Sun to be one astronomical unit.
Then we know that Jupiter is out here at 5.2 AU.
What we don't know, is what is an AU?
What is an AU in terms of real units like kilometers.
All of Kepler's Laws and Newton's equations worked really well for
predicting the positions of planets, but
they worked equally well if the AU was really small or the AU was really large.
And a good resolution to this finally came with something that seems a little bit
obscure, which is the transit of Venus.
Now, I hope that some of you got a chance to see the transit of
Venus that occurred a couple of years ago.
They don't happen very frequently and this one was one of the first ones that was
well publicized, that a lot of people got a chance to see it.
Even my daughter and her friends got to watch it through a little solar telescope
at a local children's museum.
And you can see, there's the Sun, the disk of the Sun is right there and
Venus is just about to get right onto the limb of the Sun, right there.
And it's going to go across and drag a shadow across Venus.
It was a pretty spectacular thing to watch.
Let me show you a very quick NASA video just because it's kind of awesome.
You can watch the whole thing yourself here on YouTube and
it's worth doing just because it's spectacular.
They have a little bit better view than we do.
You can see, there it is, just in the regular light,
this looks like the same color as the one my daughter was looking at.
And that's Venus, blocking part of the Sun as it goes across.
And then they go and show it to you with all the different wavelengths that their
satellite is, or space satellites, making it easier to see.
These are in things like X-rays where they get to see spectacular things.
And some of these are just really, really fun to watch.
Anyway, that was just an aside to show you how cool it looks.
And so it's important to ask yourself okay,
what does this have to do with the density of Jupiter?
So let me show you.
At the time of the early transits of Venus, remember, we didn't know how long
the AU was, but we knew that Venus was 0.72 AU from the Earth.
There is Venus right there.
The Earth is over here at 1 AU.
And you could predict when the transit of Venus was going to occur,
when Venus moves right across the surface of the Sun, the face of the Sun.
And it doesn't occur very often because usually, Venus goes a little high or
Venus goes a little low.
Because the Earth and Venus are not precisely lined up.
But every once in a while, it's very rare, but it happens.
And what was realized is that if somebody was standing on this side of the Earth,
and somebody what standing on this side of the Earth,
they would see it at slightly different times, here's why.
The transit occurs when Venus just hits the limb of the Sun.
And from here, that happens when Venus is right here.
From here, that happens when Venus is right here.
If you think about the geometry here, this is a triangle, 0.72 AU on one side.
A distance on this side that's equal to the time between here and
here times the velocity that this is going.
The velocity depends on the period, 255 days, I think, 252,
the period of Venus going around the Earth.
And so this whole triangle is in units of AU.
This triangle has units of one AU here, and an absolute distance here,
the diameter of the Earth, 12,700 kilometers.
And so we won't go through the math, but
you can see that depending on how big the AU is, this triangle,
all of the legs of this triangle will grow to shrink, except for this one.
We have one absolute tie-in, since we know the diameter of the Earth.
Knowing what we know now, what the distance of the AU is,
you can realize at this time is about four minutes.
So somebody on one side of the Earth would see it four minutes earlier than somebody
on the other side of the Earth.
And backing that out, you can use that fact to figure out exactly, or
very precisely what the AU is.
AU is 1.5 x 10 to the 8th kilometers.
And once you know that an AU is 1.5 x 10 to the 8th kilometers and
you know that Jupiter is 5.2 AU away, you can figure out the distance to Jupiter and
you can figure out how big that disk in the sky really is.
And that disk in the sky, the radius of Jupiter is, I always remember,
is 7 x 10 to the 7 meters.
Although we tend to do things in kilometers, that's just easy for
me to remember, 7 times 10 to the 7.
You can figure out the distance to the moons,
you can figure out the times of the moons.
And you get both the volume of Jupiter because you know the radius of Jupiter,
and you get the mass of Jupiter very directly.
And critically for us here, today, you get the density.
In the next lecture, we'll talk about the density of Jupiter.
We'll also talk about densities of the other planets.
Saturn, Uranus, Neptune all have moons, you could do the same experiment.
Mars has two moons, that's how we figured out the density of Mars.
Mercury and Venus, no moons.
How do you think we figured out the densities of Mercury and Venus?
They turn out to be massive enough that they make perturbations on things that
either fly by them or
even on the other planets that we could estimate their masses that way.
But that's only in the modern era.
At the time, it was very difficult to know what those really were.
But we'll talk about the densities of all the planets,
talk about the density of Jupiter and
see what we can learn about what might be on the inside of this giant planet.
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