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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From the course by Princeton University

Imagining Other Earths

243 ratings

Princeton University

243 ratings

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.

From the lesson

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.

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.

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.

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.

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,

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.

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.

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.

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.

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.

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.

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.

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