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**Explain how we can determine the ages of star clusters**

Explain the relationship between temperature, luminosity, and size of stars Describe the general path a 1 and a 25 solar mass star traces on a H-R diagram from protostar to end state and the changes that happen in the core

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**Draw H-R diagram labeling different types of stars and explain what is going on inside of them**

Describe the different end states of different mass stars Explain how mass transfer in binary stars can alter stars’ evolution

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**Exam Next Tuesday Chapters 14, 15, 16, 17 & sections 18.1 and 18.2**

2 essay questions, 25 multiple choice You’re allowed one sheet of notes with writing on one side only.

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**Einstein’s postulates**

The laws of nature are the same for everyone. 2. The speed of light in a vacuum is constant for all observers.

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Space fleet maneuvers Consider three space ships moving in formation. The flag ship is midway between the lead ship and the rear ship.

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The “go” order The flag ship now signals to the other two ships to fire their engines for exactly one minute. The signal travels at the speed of light to each ship which obeys the command. The flag ship simply adjusts its own speed to remain at the midpoint between the two ships.

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Because both ships are the same distance from the flag ship, both ships fire their engines at the same time. Because they are all moving together at the same speed, they all stop firing their engines at the same time and the formation is preserved.

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Eat lunch The flag ship then signals to the other two ships that it is time to eat lunch. From the moving ships perspective they are all at rest with the flagship at the midpoint. Thus both ships received the “lunch” command at the same time and lunch takes place on both ships simultaneously.

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The relativity of time For someone at rest with respect to the moving ships, the signals also travel at the speed of light, but during the time it takes to reach the back ship it moves forward, toward the signal and thus receives the command a little early. Also during the travel time for the light the lead ship has moved forward, away from the light and receives the signal a little late.

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Not simultaneous Thus, from someone at rest with respect to the moving ships, the last ship eats lunch first and the lead ship eats lunch last.

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Discussion After lunch, the flagship again gives an order for another engine burn. From the perspective on someone at rest which ship receives the order first? What happens to the formation of the ships?

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**Desynchronized motion**

Again the rear ship receives the order first and fires its engine before the other two. Thus, the rear ship for a time is going faster than the other two ships and gets closer to the flag ship. The lead ship gets the message last and starts its engine burn later, allowing the other two ships to catch up.

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**The formation length has shrunk**

After the lead rocket shuts down its engine, all ships will again be going the same velocity, but the distance between them is now smaller than it was before. Desynchronized time causes length contraction

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Discussion How will this appear to the captains of the ships flying in formation? Will they see their ships as being closer? Why or why not?

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Are the captains blind? The ships can consider themselves at rest before the second engine burn. Therefore, from their perspective all the ships fired their engines at the same time and the distance between them did not decrease.

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Discussion How could the ships get closer to each other without the captains knowing it? Consider doing another engine burn. The ships will get closer still. Eventually, even if the fool captains cannot see the difference won’t they feel it as the run into each other?

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Length contraction No. The ships will never run into each other. From the perspective of someone at rest the distance between the ships will decrease to 0 as the speed approaches the speed of light. But the lengths of the ships also decrease in size so that they never hit.

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Space is contracting It is not just objects that contract, but space itself is contracting. The spaceships contract because the space between their atoms is contracted.

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**Width and height contraction?**

Can’t happen because it is logically inconsistent Example – firing a cannon

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A fist fight The force of a punch depends on how fast you can throw a punch (the speed) and the mass of your fist.

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A high speed fight Suppose two twin brothers get into a fight. Each is traveling on a train going in opposite directions with a constant velocity and each tries to punch the other as the trains pass.

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Time dilation again But it takes time to throw a punch. Each twin will consider himself at rest and he will see the other twin as moving in slow motion. Each will also conclude that the other’s slow motion punch will have little effect on them, and that their fast punch is going to knock the other’s fist back in his mouth.

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Who is right? Which twin wins the fight?

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Discussion Neither. The punches will hit with equal force and neither is able to shove the other’s fist back into his brother’s mouth. How can the other twin’s slow motion punch have as much force as their own fast punch?

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Discussion Suppose you have two identical space ships, one is at rest with respect to you, the other is moving at high velocity. As the ship passes you, you give each ship an identical push for the same duration of time. How do passengers on the moving ship perceive the duration of push you give them as compared to the duration of the push you gave the stationary ship?

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Discussion If the moving ship observers see you giving the stationary ship a push for a longer duration, how does this change the acceleration they feel as compared to the acceleration they observe you giving the stationary ship? Explain.

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Discussion According to Newton’s second law F = ma. If the force F applied is the same in both cases and the acceleration is less in the case of the moving ship, what does this mean for the mass of the moving ship? Explain.

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Mass increase From your point of view, objects moving by you have a greater mass than they have while at rest with respect to you. The bigger the velocity, the greater the effective mass

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E = mc2 As the speed of an object approaches the speed of light, its mass tends toward infinity, resisting any further increase in speed. Kinetic energy added to a high speed object gets converted into mass.

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**Special Theory of Relativity**

No matter can travel at or faster than the speed of light in a vacuum Length contraction Time dilation Mass increase

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No magnetic fields The Special theory of Relativity shows that the magnetic field is just an electric field which arises due to the relative motions of charged particles.

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Discussion Given the results of the Special Theory of Relativity, why did Einstein conclude that Newton’s law of gravity needed to be revised?

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**Principle of relativity**

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Accelerated motion

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Discussion Jackie claims that she is at rest and you are moving. But you claim that you are at rest and she is accelerating away from you. Assuming Jackie cannot tell your engines are running, how can she prove that you are accelerating?

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Inertial forces Fictitious forces felt by an object trying to move in a straight line at a constant velocity but unable to do so because its reference frame is being accelerated.

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**Einstein’s happiest thought of his life**

If you fall in a gravitational field, you will not feel gravity. You will feel weightless.

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Discussion If the space shuttle orbits at a distance of 20 miles from the Earth’s surface, according to Newton’s universal law of gravity, Earth’s gravitational force is a little less (because the distance from Earth’s center is 20 miles larger) than on the Earth’s surface. Yet, astronauts are weightless. How can this be?

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**Accelerated motion in a gravitational field (freefall) can cancel gravity.**

Accelerated motion far from any gravitational field, feels just like gravity.

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**The equivalence principle**

The effects of gravity are exactly equivalent to the effects of acceleration.

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Discussion You are driving in a car with a bunch of helium balloons in the back seat. When you accelerate at a stop light, which way to the balloons move? Use the equivalence principle to explain.

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Discussion If you planted some seeds in a pot that you rotated on a turntable with the light source directly above the rotating pot, in which direction would the plants grow?

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Discussion How does the equivalence of accelerated motion and gravity make even accerated motion relative?

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Spacetime The constancy of the speed of light, connects the three dimensions of space with time. The speed of light (measured in m/s) can be considered a conversion factor between space and time.

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**Spacetime is not relative**

All observers will agree on distances measured in the 4-dimensions of spacetime.

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4-dimensions We live in a 4-dimensional universe and all the objects we see have 4-dimensions.

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**Imagining the fourth dimension**

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Discussion What would a 3-dimensional sphere look like to a 2-deminsional being as it passed through a 2-dimensional plane?

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Discussion What would a 4-dimensional sphere look like as it passed through our 3-dimensional space?

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Discussion What does a person look like in 4-dimensional spacetime?

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**But, time is different, or is it?**

Consider what a universe with 4 equal space dimensions and no time dimension would be like if you and me and everything else were moving very close to the speed of light in some direction.

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**Moving the speed of light**

Lengths in the direction of motion would shrink to 0. We wouldn’t be able to move in this fourth dimension. We would be forever stuck at a single point.

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Spacetime diagrams We can try to visualize spacetime by collapsing the 3 dimensions of space onto one or two axes and plotting time on the remaining axis.

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Spacetime diagrams time future present y past x

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Discussion Draw a spacetime diagram of your path through spacetime during this class.

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Spacetime diagrams time future You sitting in class present y past x

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**Me on the way to work time Traveling the speed limit**

Getting speeding ticket hours Speeding y miles x

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Stopping at a red light

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Discussion What does the Earth’s orbit look like on a three dimensional spacetime diagram?

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