Are we alone? This course introduces core concepts in astronomy, biology, and planetary science that enable the student to speculate scientifically about this profound question and invent their own solar systems.

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来自 Princeton University 的课程

假如其他星球也有生命

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Are we alone? This course introduces core concepts in astronomy, biology, and planetary science that enable the student to speculate scientifically about this profound question and invent their own solar systems.

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Kepler’s Law and Search for Extrasolar Planets

This lecture shows how Kepler’s Law, the relation between a planet’s Period and the radius of its orbit, can be understood in terms of the physics of gravity. We then see how we can use observations of star’s motions (and Kepler’s Law) to detect extrasolar planets and determine their basic properties.

- David SpergelCharles Young Professor of Astronomy on the Class of 1897 Foundation and Chair

Department of Astrophysics

So now, let's turn to Newton's laws. Newton's first law, which really

was due to Galileo, is that the idea that an object in

motion continues in motion. Something moving just keeps moving.

Another way of thinking about that is momentum is conserved.

Momentum is equal

to mass times velocity.

This is a key idea we use when we talked about rockets and talked about rockets

taking astronauts to Mars in order to get to Mars, and to do so and

conserve momentum, you needed to send fuel out at high velocity out the back.

That's one of the applications of Newton's First Law.

Often, this is said as in absence of

a force, a moving object will continue to move.

Given that, we can figure out how a body will move in the absence of a force.

Let's say I have a planet or a rock or a star moving at velocity, v.

The distance that planet will travel in a given time is velocity times time.

So distance is velocity times time.

Newton's second law describes what happens when I exert a force.

And Newton's second law is stated as force is equal to mass times

acceleration. Force is mass times acceleration.

And acceleration is the change in velocity with time.

So if you want to change how something's moving, you have to exert a force on it.

And here's a golfer hitting a golf ball with a given force.

The golf ball has a low mass.

Therefore, when the golfer hits, it gets an enormous acceleration.

Let's take that same golfer.

Now we'll have that golfer hit a large truck, hits it just as hard.

What's going to happen?

Well, the force is the same but the mass is much bigger.

A golfer hitting a large truck will not cause it to move very far.

And that's really the key idea of this equation.

The acceleration you experience will depend upon the force and the mass.

Well since the acceleration is the change in velocity, that's going to make the,

if you have a constant acceleration,

the velocity will grow as acceleration times time.

That's actually a good description of what's going to happen if

I were to take a rock and drop it right now.

It would experience a constant acceleration in the earth's gravitational

field and as it fell downward it's

velocity would increase as acceleration times time.

Since it's accelerating ever more quickly, moving

at ever faster velocity, the distance it travels

would be the initial velocity times time,

plus one half the acceleration times time squared.

So subject to an acceleration,

a particle's velocity can change.

And that's going to be the basic physics behind a planetary orbit.

So now let's apply that to the orbit of a planet moving around a star.

So what we're going to do is ask how we can keep a planet moving around a star.

Well, if the star wasn't there,

the planet would just keep moving. Right?

Newton's first law says a body in motion stays in motion.

So the planet would want to keep moving in this direction here.

In order to deflect it on a circular orbit it's going

to need to experience an acceleration that will change its velocity.

In order to have it move on a circular orbit like this,

it will need to experience an acceleration

equal to the velocity squared divided by R.

And that implies that the gravitational force needed

to keep something moving on a circular orbit,

is the planet is going to have to

experience a force equal to its mass times acceleration.

So, mass times velocity

squared divided by R.

That tells us the strength that gravity will have

to have to keep a planet moving on a circular orbit.

Well, now let's turn to Newton's third law.

Newton's third law says, for every action, there's an equal and opposite reaction.

I like to think of a pair of skaters pushing against

each other. If Sue pushes on Jane, Jane

will experience a force this way, and Sue

will experience a force of equal magnitude that way.

They don't have to experience the same acceleration.

Think the skater's a little heavier, so she'll experience a smaller acceleration.

This one's lighter, she'll experience a larger acceleration.

That's going to apply also to stars and planets.

If a star exerts a force on a planet, that planet

is going to exert an equal force back on the star.

So here's a mathematical expression of that.

The force exerted by the planet on the

star, equals the force exerted by the star on the planet.

The force exerted by one skater on the other skater,

balances the force exerted by the second skater on the first one.

Force is mass times acceleration.

For both forces, we set these equal.

And that implies that the acceleration experienced by,

say the star, is equal to the mass

of the planet divided by the mass of

the star times the acceleration experienced by the planet.

We can combine that with our relationship between velocity acceleration and time to

derive what in fact, is a consequence of conservation of momentum, is that velocity

of planet 1 is going to equal the mass of, or velocity of the star.

It's going to equal the mass of the planet divided by the

mass of the star times the velocity of the planet.

This,

you can think of as conservation of momentum for the system.

Now, we need to add Newton's Law of Universal Gravitation, and

we have all the ingredients we need to describe planetary dynamics.

Newton's law of gravitation states that the gravitational force between

two bodies is proportional to their masses, so this might

be the mass of the star times the mass of

the planet, divided by the distance between them, times a constant.

We call

that constant G, the gravitational constant.

So this tells us the strength of the gravitational force between two bodies.

We can now plug that in to our basic

equations and derive the basic properties of an orbit.

So now we're going to equate

the gravitational acceleration that the planet needs

to stay in a circular orbit, to the gravitational force.

Here M is the mass of the star.

Multiplying by r we find that for a

planet moving on a circular orbit, velocity squared.

Its velocity is equal to the constant G

times the mass of the star, divided by distance.

This implies a planet further away from a star is going to

move slower. You know, a planet at four times the

distance, it's going to move at half the speed, implies

planets orbiting around more massive stars have to move faster.

So given this basic equation, we can now work out what a period of a

planet will be.

The period, the time it takes to go around the orbit.

Is going to equal the distance, the circumference of the orbit, 2 pi r

divided by the velocity that the planet goes around the orbit.

So period's 2 pi r over v. Let's square this.

So period squared is 2

pi squared i squared over v squared. Well we know what v squared is.

We can plug that back into the equation so we take this equation plug it in here.

Let's

do the math, when we do that we see period squared is 2 pi

squared, r cubed over GM. We've just derived Kepler's Laws.

Period squared is distance cubed. We saw a nice graph of that earlier.

We saw how well that relationship worked.

Not only that, we've worked out the constant.

So you can see,

given the mass of the star, and observing

the planet's period, we can infer the distance.

This is a technique that will apply often when we look at extra solar systems.

We actually can't directly measure the distance, but we will be

able to measure the orbital period of a planet around the star.

Given the properties of a star, given its luminosity, how bright it is,

we can use our understanding of stars, something we'll talk about later on in

this course, to go from the star's observed properties to its mass.

So we know this term, we know this term.

That's a constant.

We can solve for distance.

This is going to be a tool that we will use often when we

want to just learn about the properties of some of the newly discovered planets.

So now, let's apply these.

Work out what this would imply, first for our own solar system, and then for

perhaps, for a newly discovered system, want to work through these two problems.

And then we'll come back and talk about how we

use these basic ideas in our search for extra solar planets.