#22   Frames of Reference:
The Basics   

  Velocity and acceleration always need to be measured relative to some fixed benchmarks, which define a "frame of reference. " Do the laws of physics depend on which benchmarks we use? This section highlights the simple concept, that two frames of reference moving relative to each other with a constant velocity are completely equivalent, and the same laws of mechanics hold in both.

  Each such frame is consistent, but their observations may differ.
Section 22a describes how the motion of the Earth modifies the apparent positions of stars and the arrival direction of the solar wind.
Section 22b contains a brief non-mathematical discussion of what the theory of relativity is about. Both these are optional sections.

"From Stargazers to Starships" home page: ....stargaze/Sintro.htm
Lesson plan home page and index:             ....stargaze/Lintro.htm

• 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.

    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.

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.

  --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.

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.

--Yes, it is equal to g in both frames--as long as the elevator's velocity is constant.

--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.

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!).

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.

It seemed to move in a small ellipse, about 20" wide.

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.

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.

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 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.]

    (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.)

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.

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.

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.

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.

The changes were made to accommodate electric and magnetic, which added complications. forces

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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.

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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

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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.)