What is pluto's speed at the most distant point in its orbit, where it is 7.30×109km from the sun?

Orbital Motion

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With the acceleration caused by gravity changing, Ug = mgh is not true for every situation

Your text describes a new form of potential energy which works on larger scales:

Some notes:

Zero potential energy is located at infinity

The negative sign is important, and expresses that gravity is attractive

The potential energy varies as 1/r, not 1/r2

Gravitational Potential Energy

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Circular Orbits

In the PhET lab, you may have calculated the speed of an object in a circular orbit…

From this, we can calculate circular orbital speed:

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Free Body Diagram of m:

For uniform circular motion:

Orbital Period

The text describes a law which describes how long an orbit takes:

The square of the orbital period is proportional to the cube of the semimajor axis of the orbit.

We can do better than proportional

So Newton’s form of Kepler’s third law:

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The above equations work when m << M. If that is not the case, both masses orbit about their common center of mass.

Whiteboard Problem 13-4

Consider two stars with masses M1 and M2 separated by a distance d and orbiting about their center of mass in circular orbits.

Find an expression for the orbital speed of M1 in terms of M1, M2, r1, and r2. (LC)

What is the orbital speed of M2?

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Elliptical Orbits

The speed of an object in an elliptical orbit changes in response to the changing force

The force varies with the distance from M

Since there are no nonconservative forces acting, the total mechanical energy of the system is conserved:

This information can connect any two points in an orbit:

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Orbital Angular Momentum

Gravity between M and m always acts toward the center, so there is no torque

So angular momentum is conserved

Calculating this at any point is cumbersome, but there are two points where the angle β is 90°

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perihelion

aphelion

Whiteboard Problem 13-5

The dwarf planet Pluto moves in a fairly elliptical orbit. At is closest approach to the sun of 4.45 x 109 km, Pluto’s speed is 6.12km/s.

What is Pluto’s speed at aphelion, which is 7.30 x 109km? (LC)

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Pluto as seen by New Horizons

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Clouds of Mass

What if instead of a star, there was just a cloud of mass?

Like a dust cloud, or Dark Matter

For spherically symmetric clouds, you can calculate the effective mass by finding the mass “inside” of the orbit.

All of the mass inside the orbit is treated as a point mass in the middle.

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r > R

m

m

Mass, M

R

r < R

The Dark Matter Problem

We’ve seen that for circular orbits, the orbital speed is:

If we consider our Solar System, we see this rotation curve:

If we consider the whole Milky Way, we would anticipate something similar…

Even though the composition is similar, the curve is different, why?

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The Dark Matter Problem

As we saw in the video after the last exam, the standard explanation is that there is some “dark” matter which changes the mass of the system.

The issue is, there is a lot of the stuff, and we haven’t been able to detect it.

An alternate explanation is that Newton’s law of gravitation changes at sufficiently small accelerations,

There’s an extra credit assignment on Canvas exploring this problem, you can work with up to 3 other people on it, and it is due on Monday.

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Exploring Orbits

To better prepare for calculating orbits, there is a PhET assigned where you can play with various parameters of a solar system.

Use a computer with Flash (Version 8 or newer, but really just update your flash player) to run the simulation. You can find a link to the simulation in the google doc under the assignment on Canvas.

Finish the assignment as a group, submitting 1 copy per group. If you finish early, you’re free to go. If you don’t be sure to submit the finished document by today at six.

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