How Does the Aberration of Light Explain the Apparent Position of Stars?

[ehrenfest]

[SOLVED] Marion and Thornton 14-16

Homework Statement

A photon is emitted at an angle \theta' by a star (system K') and then received at an angle \theta on Earth (system K). The angles are measured from a line between the start and Earth. The star is receding at speed v with respect to Earth. Find the relation between \theta and \theta'; this effect is called the aberration of light.

Homework Equations

The Attempt at a Solution

I already read the Wikipedia article so DO NOT just post a link.
This is a special relativity question but I do not understand how you can answer it without GR. In special relativity, either theta=theta'=0 or theta is not 0 and the photon misses the earth.

 

[dynamicsolo]

This is a special relativity question but I do not understand how you can answer it without GR. In special relativity, either theta=theta'=0 or theta is not 0 and the photon misses the earth.

GR is not needed because this phenomenon appears even when there is a constant relative velocity between the two rest frames. (In fact, aberration of starlight was known to 18th-Century astronomers, as the Wiki article shows, and was already understood classically. There is simply a modification to the equation that becomes necessary for relativistic velocities.)

The condition of the problem is that the star moves parallel to the x-axis (placing Earth at the origin), with the light ray reaching the Earth making an angle theta' to the star's line of motion in the star's reference frame. In the Earth's frame, what does that angle become?

I see the possible source of your confusion: the description for the light ray is poorly phrased... (In fact, this is not even the way this problem is generally posed. See, for instance, Taylor and Wheeler, 2nd ed., problem 3-9, pp. 80-81; D'Inverno, problem 3.4, p. 41; etc.)

 

[ehrenfest]

So you are saying that for example, Earth is at (0,0,0) and the star is moving along the line (t,0,1)? I thought "receding" meant that the star was moving along the line (0,0,t).

 

[dynamicsolo]

That's why I'm saying the problem is poorly stated. If the star were receding directly away from us, there would be no 'stellar aberration' at all (as you say, theta = theta' = 0). The only way the problem would make sense in that formulation would be to put the star's motion on a line not passing through the Earth and measure the angle relative to that line.

 

[ehrenfest]

Cool. Please confirm that this is what the question "means":

for example, Earth is at (0,0,0) and the star is moving along the line (t,0,1).

OK. Actually this is not homework. Since the problem statement does not make sense, I think I will just skip it. Thanks for trying to help.

 

[dynamicsolo]

for example, Earth is at (0,0,0) and the star is moving along the line (t,0,1).

That would be one way of expressing the situation.

The version of the problem I've usually seen is this. A star is at rest at a point (x,0) [the distance is irrelevant, since we are concerned with a ray of light from it]. In the star's rest frame, a light ray travels to the origin of coordinates. The Earth is at the origin at the moment of reception, traveling with speed v along the y-axis. In the Earth's rest frame, what angle does the light ray makes to the x'-axis? (Thus, in what direction does the star appear to lie?)

Classically, Galilean velocity addition predicts that the star would appear to be at an angle given by tan(phi) = v/c "ahead" (in the direction of Earth's motion) of the x'-axis. In fact, this is pretty much what is observed for stellar observations from Earth, owing to the Earth's orbital motion plus the Sun's velocity relative to the star (so you get a sort of cycloidal apparent motion of the star, superimposed on the actual change in the star's direction due to differences in the Galactic orbits of Sun and star; as you can imagine, this complicates unraveling the dynamical behavior of Galactic stars a little...). The classical result means that the greatest possible aberration (angle) would be 45º. [The classical aberration is also used to explain the Poynting-Robertson effect, which imposes a 'drag' on orbiting dust particles in a planetary system, for instance.]

Using the relativistic addition relations, the prediction changes considerably, with the limit for aberration becoming 90º (that is, as v -> c , the star asymptotically approaches appearing straight ahead of you). Generalizing this to all directions, stars that are behind you in the stellar rest frame can appear to be ahead of you; as relative velocity increases, increasingly large fractions of the sky appear compressed into decreasingly smaller cones in the direction of 'flight'.

This is also related to the 'Terrell rotation' (concerning the visual appearance of objects in relative motion), a consequence of SR that was only recognized around 1963...