Goals: The student will learn|
Terms: frame of reference, relative velocity, stellar aberration, solar wind, principle of relativity, classical mechanics, relativistic mechanics, quantum mechanics.
- The concept of frames of reference in physics.
- That two frames of reference, each moving with respect to the other with a constant velocity v (constant speed, constant direction), observe the same accelerations and therefore Newton's laws are the same in both.
- (Optional section #22a) About the aberration of starlight and its explanation by James Bradley, including the story of the flag on a moving boat, which gave the essential clue.
- About the aberration of the solar wind, and how
the proposed "Solar Probe" space mission plans to take advantage of it.
- (optional sect. 22b) What Einstein's theory of relativity is about, in qualitative terms.
Stories and extensions: The story of the aberration of starlight and of Bradley's observation on a boat in the river. About the solar wind and magnetosphere, and how aberration foiled a clever idea of downloading satellite data using a passive laser reflector.
Note to the teacher:
This lesson is closely related to the one on vectors (section #14 of "Stargazers", lesson plan #23). Some ideas expanded here were already introduced in #14--for instance, the motion of an airplane flying with velocity v1 relative to the air, which itself (because of the blowing wind) has a velocity v2 relative to the ground. In that example, the air and the ground represent two frames of reference moving with respect to the other, and we have already shown that the velocity of the airplane with respect to the ground is the vector sum v1+v2 .
Here, however, two additional aspects come into play. One, we are also concerned with accelerations and forces. These are the simplest cases, where all velocities are constant in magnitude and direction, so that shifting from one frame to the other adds no new forces or accelerations (That will no longer hold when we come to discuss rotating frames). And two, we study the changes created by the motion of the observer's own frame of reference.
Section (22a) is optional. It contains interesting stories, illustrating the lesson, but can be omitted (and perhaps assigned to some advanced students) if time runs short. It is also possible to teach only the first example, on the aberration of starlight and on its explanation by James Bradley.
Starting the Lesson
The starting paragraphs of Section #22 are quite appropriate for starting the lesson. After that, bring up the questions below, and continue with Section #22a.
Questions and tidbits:
--What is meant by a "frame of reference"?
A frame of reference is a set of reference points with respect to which motion is measured. These points move together and keep their relative distances and angles of view.
--Can you give examples of frames of reference?
--Interior of a house, a ship, airplane, car, railway carriage or spaceship.
--Surface of the Earth, the Moon or Mars.
--A moving elevator, merry-go-round, roller coaster car or other ride.
--The frame of the wind carrying a run-away balloon, or of a river carrying a swimmer.
--Also, in certain contexts, the frame of the distant stars.
We have two frames of reference: A is the inside an elevator rising with constant velocity u, B is the frame of the building in which the elevator is located. A rider drops a penny inside the elevator. Is the velocity of the penny the same as seen from A and B?
No, in A its velocity includes u, in B it does not. It may be falling with respect to the elevator cabin but actually rising with respect to the building.
In the preceding example, is the acceleration of the penny the same viewed inside the elevator and outside it?
--Yes, it is equal to g in both frames--as long as the elevator's velocity is constant.
You are the passenger in a car driving with velocity u on a rainy night. On the street outside, through the side window of the car, you see raindrops falling. They fall with a constant velocity v (because of air resistance, they no longer accelerate). As you watch them in the light of streetlights, how do they appear to move? What is their apparent velocity w? In what direction do they streak the windows?
--The outside of the car appears to be moving with velocity -u (like u but opposite direction) to the rear. Raindrops as viewed from the moving car seem to have a velocity -u in addition to their falling velocity v, causing them to slant backwards, and their streaks on the windows slant similarly. (Draw on the board)
Their velocity vector w has a vertical downwards component v (magnitude of v) and a horizontal component u (magnitude of –u) to the rear: in vector notation w = v–u = v+(–u). Since v and u are perpendicular to each other, by Pythagoras, w = SQRT(v2 + u2). Their streaks on the window are in the direction of w and the angle A between those streaks and the vertical satisfies sinA = u/w or tanA = u/v.
About the Aberration of Starlight
How are distances to stars measured by the parallax method?
The Earth's orbit around the Sun is approximately a circle whose radius is about 150,000,000 km ("one astronomical unit" or 1 AU). Therefore, on two dates 6 months apart, the Earth occupies positions (A,B) separated by 300,000,000 km (or 2 AU). Say the star is at point C, and assume the diameter AB of the Earth's orbit was chosen in such a way that AC is perpendicular to it (always possible!).
[Section #9a describes how Aristarchus, around 200 BC,first estimated the distance of the Sun, which led him to propose the Earth moved around the Sun. His value was actually only 1/20 of the true AU. Still, Greek astronomers (in particular, Ptolemy, around AD 150) felt that if the Earth orbited the Sun, its displacement every 6 months should have revealed some shift in the positions of the stars. Since they observed none, they concluded that the Sun must actually go around the Earth. They could not imagine how enormously distant the stars actually were!]
If the directions to C are slightly different when viewed from A and B, then the difference gives the "parallax" angle between AC and BC. Using that angle one can calculate all other properties of the triangle ABC, including the distances AC abd BC from Earth to the star.
What changes were observed around 1700 in the position of Polaris?
It seemed to move in a small ellipse, about 20" wide.
How did astronomers know that it was not Polaris that did the moving?
The motion around the ellipse took 1 year to complete, and it was highly unlikely that Polaris would match the Earth's orbital period. Also, other stars near Polaris displayed similar motions.
--How did James Bradley know that the shift of Polaris was not a parallax effect?
Because the greatest displacement of Polaris in any direction did not match the greatest displacement in the opposite direction by Earth in its orbit, but occured 3 months afterwards.
--In the end, how did Bradley explain the strange shift in the position of
Polaris and other stars?
The velocity u of the Earth in its orbit made starlight observed from Earth appear to have an extra velocity (–u) added to its own. Since the added velocity had a sideways component, perpendicular to the direction from the star, it shifted the direction from which the light appeared to come.
--The aberration of starlight allows us to deduce that the Earth is indeed
moving. Doesn't that contradict an earlier claim that absolute motion is
The claim was that absolute motion with constant velocity in a straight line was undetectable. The motion of the Earth discussed here is around a circular orbit.
[Optional further discussion by the teacher:
Suppose Earth and the whole solar system did move with constant velocity u along a straight line. The positions of stars would then be shifted, too, but the shift would not change as time went on. Astronomers would see the positions of the stars fixed and not suspect their real direction was different.
Actually, a systematic shift does exist, and from it we know that the solar system is moving at about 20 km/s towards a point known as the solar apex, near the star Vega. But in principle, it could also be that we are at rest and all those stars are moving in our direction, away from the solar apex. The physical effects would be exactly the same. It is only our logic that tells us it is more likely that our sun is moving, rather that a large number of distant suns happen to move on parallel tracks.]
[Harder poser--perhaps to take home]
How do you think would a star on the ecliptic appear to move? Hint: it's not a circle--not even close!
(It's a back-and-forth motion along a straight line. Twice a year the Earth is moving towards the star or away from it, and at those times, the aberration is zero. Halfway between those times, the Earth's motion is transverse to the direction of the star, and the aberration is greatest.)
About the Aberration of the Solar Wind
Why does the solar wind, on the average, appear to come not from the Sun but
from a direction 4 degrees off the Sun?
Because of the orbital velocity u of the Earth. In the frame of the Earth the solar wind appears to move as if it has an extra added velocity –u, and that shifts its direction.
What do you know about the "Solar Probe" mission?
How would instruments aboard the "solar probe" detect solar wind particles,
even though they are shielded from direct sunlight?
The solar probe near its closest approach to the Sun moves almost as fast as the solar wind, but in a direction perpendicular to that of the solar wind (which moves radially). In the probe's own frame of reference, therefore, solar wind ions move along slanting paths that brings them behind the heat shield.
About the Theory of Relativity
What is the principle of relativity?
When one frame of reference moves in a straight line and at a constant velocity relative to another, no physical process can distinguish which one is moving and which one is at rest.
How does the theory of relativity modify Newtonian mechanics?
The mass of any moving material (as seen from some other frame) increases as it approaches the speed of light, and it resists further acceleration more and more. As a result, the speed of light is a limit which no material velocity can cross.
Why did Newton's laws need to be modified? Don't they already satisfy the principle of relativity as they stand--only accelerations can be distinguished, while a constant velocity changes nothing?
The changes were made to accommodate electric and magnetic, which added complications. forces
What does relativity say about time in two moving frames of reference--especially if their relative velocity is close to the velocity of light??
Time does not flow at the same rate in the two frames. Two events which in one frame are a second apart, viewed from another frame may be two seconds apart.
In the late 1930s an unstable particle was discovered, named the muon (originally, "mu-meson"). Muons were fragments of collisions of very fast nuclei, and in the laboratory they decayed radioactively (into an electron and an unseen neutrino) in about 2 millionths of a second (microseconds). How far should muons traveling at the speed of light (300,000 km/s) be able to move, on the average, before decaying?
300,000 x 2/1000,000 = 0.6 kilometers
Muons moving close to the speed of light are produced in the atmosphere by collisions of fast atomic nuclei from space ("cosmic rays") at an altitude of about 12 kilometers. Yet a large fraction of them is still observed at sea level (they form the greater part of the cosmic radiation observed there). If they are so short-lived, how come they are not lost by decaying before reaching the ground?
The lifetime of these muons is 2 microseconds in its own frame of reference. Because of their speed, in the frame of the Earth is it much longer, allowing them to last long enough to reach the ground.
(In the frame of the muons, the lifetime remains 2 microseconds, but the distance from the top of the atmosphere to the ground, which is 12 km in the Earth's frame, may be only 0.6 km in the frame of the muon.)