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We have a lot of tools available,

Â if we want to try to understand what's inside of the Earth.

Â We can't go there, but we can do a lot of stuff.

Â We can dig holes and

Â the holes don't go very far down compared to the center of the Earth.

Â But you learn some things like what's the composition of the outer part of

Â the Earth.

Â You learn about heat flux, which you realize,

Â as you dig a hole down a deep mine is that it's warmer as you get further down.

Â That's telling you about the heat coming out of the interior of the Earth.

Â And we can do things like use seismometers to use

Â earthquakes to monitor what's going on inside of the Earth.

Â None of these techniques works on Jupiter.

Â So we're going to have to resort to a lot of even more indirect measurements to try

Â to understand what's going on.

Â But the very first measurements that you want to do if you wanted to understand

Â what a planet was like on the inside, what a planet is made out of is you'd want to

Â measure the density of that planet.

Â And let me just remind you of densities, of a few things that matter,

Â in rough numbers that we'll be using.

Â Density, which is always written as a row.

Â Density of rock is about 3 grams per cubic centimeter.

Â I'm going to use this unit because, well, I'll show you why.

Â I'm going to use this unit because that's rock.

Â Because ice or water is 1 gram per cubic centimeter and

Â that's a really easy thing to remember.

Â The one other thing you might want to keep in mind is the density of iron.

Â Density of iron is closer to 8 grams per cubic centimeter.

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Now, I can't just go measure the density of a planet and say the density is this,

Â therefore it's got this, this, and this.

Â Because materials get compressed by the big pressure of the insides,

Â their densities go up.

Â You take a rock, and you smoosh it, and it gets more and more dense.

Â So you have to understand that.

Â Remember, we did that for Mars a little bit.

Â But in general, you can tell the difference between icy, watery planets,

Â rocky planets.

Â And even things that have maybe a lot of iron in them,

Â like the core of the Earth is, just by knowing what their density is.

Â So how do you know the density of a planet?

Â Well, the density is of course mass divided by volume.

Â You can figure out the volume, if you see how big it is in the sky.

Â You measure its diameter, its radius, and you can measure the volume very easily.

Â How do you get the mass?

Â The only good way to get the mass is if the planet has a satellite.

Â If a planet has a satellite like Jupiter does, we saw, and

Â you see that satellite going around and

Â you can measure both the distance away of the orbit.

Â The semi-major axis, we'll call it, or we can just call it the radius.

Â As long as it's a circle, we can just call it the radius.

Â And you measure the amount of time it takes for

Â that orbit to happen, you get the mass of the thing on the inside.

Â We'll go through the maths here in a minute, but

Â let me just show you, Galileo had that from the first moment.

Â He had these moons going around Jupiter, he could track each one.

Â He could figure out how far away it got, how long it took to go around, and

Â he could determine the mass of Jupiter.

Â Well, almost he could.

Â What he did determine is that these moons obeyed Kepler's Laws.

Â And Kepler determined just empirically by looking at the planets, one is,

Â he figured out that the planets go in elliptical orbits around the Sun.

Â But he also found that the period, the square of the period of that orbit was

Â proportional to the radius of that orbit cubed.

Â Really, he figured out that it was proportional to the semi-major axis cube.

Â Because if you're in an ellipse,

Â the semi-major axis is half the distance of this major axis.

Â But as long you're going to circle the orbit, that's the same as the radius.

Â Galileo found the same thing held for the Galilean satellites.

Â Their periods were proportional to the cube of their distances away from Jupiter.

Â Newton came along later and explained why that was the case.

Â The force of gravity between any two objects is equal to G,

Â the gravitational constant.

Â The mass of one object, the mass of the other object / by r squared,

Â the distance between those object.

Â And when an object is in orbit, that force, which is pulling

Â only in this direction, is balanced by the centrifugal force of the object,

Â which is pulling it in this direction because of the curvature of the orbit.

Â And that centrifugal force, as you might remember, is mv squared / r.

Â v is the velocity that this object is moving along through here.

Â We can figure out the velocity, the velocity of course is related to how long

Â it takes to go around this full orbit.

Â So that the period of the orbit is equal to the circumference

Â of the circle, 2 pi r / by the velocity.

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And if we solve this for velocity,

Â put it here under the equation,

Â we got the GmM / r squared = 2 pi r / p squared m / r.

Â See that the masses are going to cancel out the mass of the actual object itself,

Â the mass of the planet.

Â In this case, it doesn't matter, it's only the mass of the central object.

Â That's only true as long as the central object is much more massive.

Â But that's the case in all these cases here.

Â And solving through these,

Â we get that p squared GM = 4 pi squared r cubed.

Â Look at this, we got that p squared is proportional to r cube,

Â that's just what we had over here.

Â So this recovers Kepler's Law, but

Â it shows us also what those proportionality factors are.

Â Those proportionality factors are, well, there's just some numbers over here.

Â But G and M, M is the mass of that central object, the mass of the Sun.

Â If it's a planet, the mass of Jupiter, if it's a moon going around.

Â And so we can solve for the mass, the mass = 4 pi squared / G r cubed / p squared.

Â So all we have to do is figure out the radius of the orbit, period of the orbit,

Â and we get the mass of the thing in the middle.

Â So that point, Newton could go back and figure out the radius of those orbits

Â from observations of Galileo or the many observations after that.

Â Periods were very easy to determine.

Â And the mass, well, we're not quite there yet because there are two problems.

Â One is G, this is a gravitational constant, but

Â it was not well known at the time.

Â And actually, the other is R, we'll talk about that a minute.

Â First, lets talk about G, how do you measure what G is?

Â Newton simply said that the force was proportional to this

Â product of the mass divided by r squared and so that proportionality constant is G.

Â How do you measure G?

Â First, really measurement for G was in about 1797, by Cavendish.

Â And it's so famous, it's actually called the Cavendish experiment.

Â And it's a pretty simple idea, which is of course if any two masses attract

Â each other, if you can put two masses next to each other and

Â see the force that they exert towards each other, you've measured G.

Â And he did exactly this, he put a pair of

Â weights on a, it's a torsion spring.

Â Imagine like a long strip that is allowed to rotate one direction or

Â rotate the other direction.

Â And then he would take larger weights on either the front and

Â back, or he would switch their positions front and back and

Â watch this thing ever so slightly deflect.

Â And he could calibrate how much it took to deflect that.

Â And he measured the value for G,

Â that's something within 1% of the value that we know, today.

Â He actually did it, he didn't think of himself as measuring G.

Â He thought of himself as measuring the density of the Earth.

Â Nobody really knew what the density of the Earth was, but we knew what G,

Â the gravitational constant was, something like 10 meters per second squared.

Â And we knew how big the Earth was and

Â so, Cavendish's question was, what is the density of the Earth?

Â How much mass does it take to give this

Â amount of gravitational pull for the Earth?

Â And the only way to know that was to know how mass and gravity related,

Â which is to know G.

Â But at the time, Cavendish didn't think of it as G.

Â But in the end, if you take his density measurement inverted for G,

Â he had a very precise measurement of G.

Â So now, we can go measure the density of Jupiter, right?

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Still not quite, and the reason still not

Â quite is because when we look at Jupiter in the sky, we can see its radius.

Â We can see how far away the moons are.

Â But all of that is just an angular distance on the sky.

Â And what we don't know at this point is how far away Jupiter is.

Â If Jupiter is really far away, these distances are huge,

Â Jupiter is really close, these distances are kind of small.

Â What we do know is the relative scale of the Solar System.

Â We know the Sun's in the middle,we know the Earth is here at 1 AU.

Â We know that Venus is over here at point 0.7 AU.

Â AU, of course, remember, is astronomical unit, where we've simply defined

Â the distance from the Earth to the Sun to be one astronomical unit.

Â Then we know that Jupiter is out here at 5.2 AU.

Â What we don't know, is what is an AU?

Â What is an AU in terms of real units like kilometers.

Â All of Kepler's Laws and Newton's equations worked really well for

Â predicting the positions of planets, but

Â they worked equally well if the AU was really small or the AU was really large.

Â And a good resolution to this finally came with something that seems a little bit

Â obscure, which is the transit of Venus.

Â Now, I hope that some of you got a chance to see the transit of

Â Venus that occurred a couple of years ago.

Â They don't happen very frequently and this one was one of the first ones that was

Â well publicized, that a lot of people got a chance to see it.

Â Even my daughter and her friends got to watch it through a little solar telescope

Â at a local children's museum.

Â And you can see, there's the Sun, the disk of the Sun is right there and

Â Venus is just about to get right onto the limb of the Sun, right there.

Â And it's going to go across and drag a shadow across Venus.

Â It was a pretty spectacular thing to watch.

Â Let me show you a very quick NASA video just because it's kind of awesome.

Â You can watch the whole thing yourself here on YouTube and

Â it's worth doing just because it's spectacular.

Â They have a little bit better view than we do.

Â You can see, there it is, just in the regular light,

Â this looks like the same color as the one my daughter was looking at.

Â And that's Venus, blocking part of the Sun as it goes across.

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And then they go and show it to you with all the different wavelengths that their

Â satellite is, or space satellites, making it easier to see.

Â These are in things like X-rays where they get to see spectacular things.

Â And some of these are just really, really fun to watch.

Â Anyway, that was just an aside to show you how cool it looks.

Â And so it's important to ask yourself okay,

Â what does this have to do with the density of Jupiter?

Â So let me show you.

Â At the time of the early transits of Venus, remember, we didn't know how long

Â the AU was, but we knew that Venus was 0.72 AU from the Earth.

Â There is Venus right there.

Â The Earth is over here at 1 AU.

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And you could predict when the transit of Venus was going to occur,

Â when Venus moves right across the surface of the Sun, the face of the Sun.

Â And it doesn't occur very often because usually, Venus goes a little high or

Â Venus goes a little low.

Â Because the Earth and Venus are not precisely lined up.

Â But every once in a while, it's very rare, but it happens.

Â And what was realized is that if somebody was standing on this side of the Earth,

Â and somebody what standing on this side of the Earth,

Â they would see it at slightly different times, here's why.

Â The transit occurs when Venus just hits the limb of the Sun.

Â And from here, that happens when Venus is right here.

Â From here, that happens when Venus is right here.

Â If you think about the geometry here, this is a triangle, 0.72 AU on one side.

Â A distance on this side that's equal to the time between here and

Â here times the velocity that this is going.

Â The velocity depends on the period, 255 days, I think, 252,

Â the period of Venus going around the Earth.

Â And so this whole triangle is in units of AU.

Â This triangle has units of one AU here, and an absolute distance here,

Â the diameter of the Earth, 12,700 kilometers.

Â And so we won't go through the math, but

Â you can see that depending on how big the AU is, this triangle,

Â all of the legs of this triangle will grow to shrink, except for this one.

Â We have one absolute tie-in, since we know the diameter of the Earth.

Â Knowing what we know now, what the distance of the AU is,

Â you can realize at this time is about four minutes.

Â So somebody on one side of the Earth would see it four minutes earlier than somebody

Â on the other side of the Earth.

Â And backing that out, you can use that fact to figure out exactly, or

Â very precisely what the AU is.

Â AU is 1.5 x 10 to the 8th kilometers.

Â And once you know that an AU is 1.5 x 10 to the 8th kilometers and

Â you know that Jupiter is 5.2 AU away, you can figure out the distance to Jupiter and

Â you can figure out how big that disk in the sky really is.

Â And that disk in the sky, the radius of Jupiter is, I always remember,

Â is 7 x 10 to the 7 meters.

Â Although we tend to do things in kilometers, that's just easy for

Â me to remember, 7 times 10 to the 7.

Â You can figure out the distance to the moons,

Â you can figure out the times of the moons.

Â And you get both the volume of Jupiter because you know the radius of Jupiter,

Â and you get the mass of Jupiter very directly.

Â And critically for us here, today, you get the density.

Â In the next lecture, we'll talk about the density of Jupiter.

Â We'll also talk about densities of the other planets.

Â Saturn, Uranus, Neptune all have moons, you could do the same experiment.

Â