How does beaming affect distance perception in special relativity?

[PAllen]

This topic has been dealt with many threads, in various aspects. I want, in this thread, to set up an example focusing in on all related issues and, hopefully, centralizing best answers and explanations.

Example:

We have a large black hole with a star orbiting it with an substantially relativistic orbital speed, e.g. .9c. Please ignore questions of orbital stability, and also do pretend that there is no black hole accretion disk, for simplicity. Imagine that this system has its orbital plane on edge to Earth so we see the star alternately approaching and receding at .9c. Imagine it is close enough for powerful telescopes to image the star's disc, and also we are able to do VLBI parallax measurements of distance (directly on earth; not using opposite sides of the Earth's orbit). (This relative closeness also means we need not worry about cosmological redshift). Imagine one more thing: we are actually able to bounce radar off the star.

I think the following are things we would see (please add as many corrections and amplifications as you want):

1) The star would look (through a telescope) small, blue and bright when approaching.
2) The star would look large, red, and dim when receding.
3) The parallax distance to the star would appear large when approaching and small when receding.
4) The radar ranging distance would be identical whether the star was approaching or receding, and would be in between the two parallax distances.
5) Based on the star's total luminosity measured when moving tangentially, a raw luminosity distance computation would show the star closer when approaching and further when receding.
6) A luminisity computation corrected for blue/red shift would give the same distance whether receding or approaching.

[Actually, to keep this really SR, let's imagine removing the black hole and just imagining that some magical propulsion system is driving the star in circular motion; then we don't need to worry about the gravity well of the black hole.]

 

[PAllen]

A follow on question: suppose you want to Lorentz transform from the Earth frame to an inertial frame comoving with the star. Which distance do you use as the coordinate distance in the Earth's frame (that you then transform)? I would guess it is the (4) or (6) distance.

 

[yuiop]

I think the following are things we would see (please add as many corrections and amplifications as you want):

1) The star would look (through a telescope) small, blue and bright when approaching.
2) The star would look large, red, and dim when receding.

In another thread I think I sowed the seed that a star would change apparent size in an orbiting situation, but thinking more about it in the light of your question I have come to the conclusion that while an object moving at constant velocity in a straight line towards you would appear smaller, this does not apply to the orbiting case. This is slightly complex to explain in words why this is the case, but I will try later.

3) The parallax distance to the star would appear large when approaching and small when receding.

For similar reasons I now think the parallax distance will be the same for the orbiting case.

4) The radar ranging distance would be identical whether the star was approaching or receding, and would be in between the two parallax distances.

Agree with the first part "The radar ranging distance would be identical whether the star was approaching or receding".

5) Based on the star's total luminosity measured when moving tangentially, a raw luminosity distance computation would show the star closer when approaching and further when receding.
6) A luminisity computation corrected for blue/red shift would give the same distance whether receding or approaching.

Not sure about these as more work is needed in the orbiting case. In the straight line case it would be broadly true that an approaching object appears brighter and a receding object appears dimmer.

[Actually, to keep this really SR, let's imagine removing the black hole and just imagining that some magical propulsion system is driving the star in circular motion; then we don't need to worry about the gravity well of the black hole.]

Good idea, because the black hole would introduce gravitational lensing problems.

I was coincidently about to start a thread on a detailed examination of the brightness of moving objects in SR but your thread will do

 

[yuiop]

This is my initial attempt to explain why I think the orbiting case differs from the straight line case. Let us say we have two stars that are physically the same size when at rest wrt each other. Now imagine one (A) is going away you at +v and the other (B) some distance d away and heading towards you at -v. At some time t they narrowly glance of each other at distance d/2 away from you with some fireworks and change of trajectory. At time t you do not see the collision, due to light travel delays. You see A nearer than d/2 and B as further away than d/2 at time t and at this time A appears larger than its apparent size at d/2 and B smaller than its apparent size when at d/2 simply because of where they were, when the light your receiving was emitted.

Some time later than time t you actually see the collision and fireworks and deflection, but the two stars really were at d/2 at the time the light was emitted and so the star going away from you and the star coming towards you appear the same size at the time they are alongside each other.

Now change the situation slightly. Imagine the stars hooked onto each other at the time of collision by a strong tether and went into a tight spiral. You would continue to see them the same size, whether they are approaching or receding in their tight orbit.

A curious aspect is that an oscillating motion of the receiver and a stationary source, is not the same as an oscillating source and a stationary receiver, although I made made that mistaken assumption in the earlier thread.

Does any of this make any sense at all? I am having trouble describing what I intuitively know :tongue:

[EDIT]I thought I might add, that in the linear case, the difference in apparent sizes comes about due to differences in actual coordinate location and perceived location due to light travel time delays and this visual aberration is not present in the orbital case, because the difference in location within the orbit is small compared to the distance of the system from the observer. The cyclic nature of the orbital case means we would probably not notice.

 

[Dale]

Do you really want a black hole and a star with the corresponding GR issues or do you just want a light-emitting ball in uniform circular motion at relativistic speeds in flat spacetime?

 

[PAllen]

Do you really want a black hole and a star with the corresponding GR issues or do you just want a light-emitting ball in uniform circular motion at relativistic speeds in flat spacetime?

Yes, this is what I want. I thought about that and edited into the end of the original post.

 

[yuiop]

Yes, this is what I want. I thought about that and edited into the end of the original post.

Two equal size, equal mass stars (or light emitting balls) that are tethered to each would be one way to achieve the flat space scenario, as long as the density of the stars is low enough that GR issues are not significant.

 

[PAllen]

I still haven't had time to fully digest yuiop's argument about orbital sources versus inertial sources, but I have the following initial comments:

1) I certainly agree there is no symmetry between orbiting source, inertial observer versus inertial source, orbiting observer. This is exactly the same type of mistake when people try to claim a false symmetry for the twin differential aging.

2) I am suspicious of the claim about major difference between receding/approaching orbital motion of source versus inertial sources. Imagine the stellar orbit is the size of Pluto's orbit, and we are 5 light years away. Then for hours at time the star will be approaching / receding in a way whose divergence from inertial is vanishing.

3) If small orbit and big orbit are really different, we should consider both cases.

 

[yuiop]

2) I am suspicious of the claim about major difference between receding/approaching orbital motion of source versus inertial sources. Imagine the stellar orbit is the size of Pluto's orbit, and we are 5 light years away. Then for hours at time the star will be approaching / receding in a way whose divergence from inertial is vanishing.

If we have two orbiting sources that always opposite each when looking from above the centre of rotation, then view edge on cause distortions so that they do not appear to be opposite each other at all times due to light travel times so where they are in orbit is out of sync. However when a line joining the sources is exactly perpendicular to the line of sight the two sources appear the same size. This is true in the linear inertial case too, as I alluded in post #4. If one source is going away from you and one is coming towards you, they will appear the same size when you receive the light that was emitted at the time they were alongside each other. In this aspect, the liner and orbiting cases are not that much different. In the linear case the change in size is due to where it appears to be when the light is received relative to where it actually is when the light is received, but this difference is not significant in the orbital case and just appears as an anomalous rotation rate. I am working on an animation that takes light travel times into account, that I hope will make clear what I talking about.

 

[PAllen]

If we have two orbiting sources that always opposite each when looking from above the centre of rotation, then view edge on cause distortions so that they do not appear to be opposite each other at all times due to light travel times so where they are in orbit is out of sync. However when a line joining the sources is exactly perpendicular to the line of sight the two sources appear the same size. This is true in the linear inertial case too, as I alluded in post #4. If one source is going away from you and one is coming towards you, they will appear the same size when you receive the light that was emitted at the time they were alongside each other. In this aspect, the liner and orbiting cases are not that much different. In the linear case the change in size is due to where it appears to be when the light is received relative to where it actually is when the light is received, but this difference is not significant in the orbital case and just appears as an anomalous rotation rate. I am working on an animation that takes light travel times into account, that I hope will make clear what I talking about.

Sorry, none of this is clear. If aberration of an inertial source approaching at near lightspeed causes it to subtend a smaller angle than a source receding at lightspeend, then I see no conceivable reason this wouldn't apply to a a large orbit viewed from a very large distance.

It would help to provide either caclulation or animation. So far I don't follow you at all.

So far as I understand the basis of aberation, it would definitely apply to a large orbit.

 

[PAllen]

If we have two orbiting sources that always opposite each when looking from above the centre of rotation, then view edge on cause distortions so that they do not appear to be opposite each other at all times due to light travel times so where they are in orbit is out of sync. However when a line joining the sources is exactly perpendicular to the line of sight the two sources appear the same size. This is true in the linear inertial case too, as I alluded in post #4. If one source is going away from you and one is coming towards you, they will appear the same size when you receive the light that was emitted at the time they were alongside each other. In this aspect, the liner and orbiting cases are not that much different. In the linear case the change in size is due to where it appears to be when the light is received relative to where it actually is when the light is received, but this difference is not significant in the orbital case and just appears as an anomalous rotation rate. I am working on an animation that takes light travel times into account, that I hope will make clear what I talking about.

I disagree with a substantive part of this. For an approaching souce that subtends an angle, the light received from the edges was emitted earlier and further away compared to the center of the object; for a receding source it was emitted ealier and closer. This causes the approaching object to subtend a smaller angle and the receding object a larger angle. I just don't see how a large orbit doesn't have the same effects.

 

[PAllen]

Let me state my current thinking:

- when the star's center shows maximum blue shift it will subtend a smaller angle then when its center shows maximum redshift. Can anyone refute this with mathematics?

 

[yuiop]

Here is the animation I was working on. The two luminous objects are joined by a tether and are exactly opposite each other at all time according to an observer at the centre of their orbit. The animation is the view of an observer seeing the objects orbit edge from a distance of 4 units which compares to a radius of one unit for the orbit of the objects. The objects have a tangential velocity of 0.785c in this example. On the left of the animation the objects are coming towards the observer and appear to be moving much faster than when they are on the right of the line of sight joining the observer to the centre of the orbiting pair. It can also be seen that the objects eclipse each far to the right of centre on the receding side of the orbit. Note that the animation only demonstrates visual size and does not take redshift of the light wavelength or changes in apparent brightness into account. All these visual effects are a simple result of regular Newtonian optics and finite light travel times. Relativistic time dilation and length contraction play no part in the apparent size of the objects. In the animation the objects pause briefly and are coloured blue to mark the time when a line connecting the two objects is exactly perpendicular to the observers's line of sight. At this point the objects appear the same size. This is just how I see it from my calculations (and a bit of light ray tracing geometry) but it probably does not answer the questions you have raised. I will come back to those.

 

[PAllen]

Here is my reasoning:

The time when the center of the star is maximum blue shift is when it's motion fastest toward the observer, which will be when its motion is tangent to the line of sight on the approach. At this point, it will have been moving at near constant speed and near constant direction for hours (given e.g. pluto orbit size and 5 or 10 light years away). The light from the edges of the star won't reach Earth at the same time as the light from the center. The perceived edge of the disc when the center is at maximum blue shift will be from light emitted earlier, thus farther away, thus subtending a smaller angle.

The reverse argument establishes my claim for the redshift side.

I don't see any way out of this reasoning.

 

[PAllen]

The effect I described above (light arrival time from limb versus center of star) is very small. The more significant issue is whether relativistic aberration applies to this example. yuiop is saying it doesn't, even in the non-orbital case. I think he is right and propose a really simple justification.

Re-reading derivations of relativistic aberration, it's root cause seems to be conversion from rest frame of source(s) to rest frame of rapidly moving observer. If you are already describing things in terms of distances and angles seen by the observer, there appears to be no need to adjust for motion of the source. Though it seems, at first glance, a counter-intuitive asymmetry, I think it is true that the star (in our example) will see the Earth small when approaching and big when receding, but the Earth observer will see no such thing about the star. yuiop alluded to this, but I thought he was referring to the fact that one frame was inertial and the other wasn't. Actually that is irrelevant; the star in a large orbit may be treated as inertial over very substantial distances and times. The difference boils down to whether or not you are translating measurements made in one frame to a different frame. If you are, then you get the aberration effect; if not, you don't.

So this is a major correction to my initial list. Hopefully, we will avoid spreading this confusion to other threads, in the future (as has happened in the past).

Note that where we've discussed the case of turning from rapidly receding from an object to rapidly approaching it (e.g. twin scenarios) it is correct that they would see a large change due to aberration (an enlarged, dim image changing to shrunken bright image). Here, one is comparing observations across two different frames of reference.

 

[yuiop]

The effect I described above (light arrival time from limb versus center of star) is very small.

Agree. Thanks. You have saved me a lot of trouble trying to explain why this is so.

The more significant issue is whether relativistic aberration applies to this example. yuiop is saying it doesn't, even in the non-orbital case. I think he is right and propose a really simple justification.

Re-reading derivations of relativistic aberration, it's root cause seems to be conversion from rest frame of source(s) to rest frame of rapidly moving observer. If you are already describing things in terms of distances and angles seen by the observer, there appears to be no need to adjust for motion of the source.

Also agree. If the distance that an object is at at the time the light is emitted is D_e and the distance the object is at the time the light is received is D_r, then the apparent size change the object moving along the line of sight of the observer is proportional to D_r/D_e. This works out at (1-v/c) before length contraction of distances is taken into account. V in this case is positive when coming towards the observer and negative when going away. In the orbital case D_e might well equal D_r if the object completes a half or full orbit in the time the light takes to travel to the observer. The largest the difference between D_r and D_e can be in the orbital case, is the orbital radius and this is normally a small fraction of the distance of the system from the observer in astronomy, D_r/D_e is much less than (1-v/c) in the orbiting case. Even so, in both the linear and orbiting cases, if the light that is received from two objects with different relative velocities was emitted at the time both objects were alongside each other, then both objects will appear the same size in a photograph. When we photograph a distant star, we tend to say the star must be much further away than it looks because it has moved away since the time the light was emitted, but this is a slightly dangerous practice, because we are projecting into the future and assuming that the star continued with constant velocity or accelerating velocity without any real proof.

So this is a major correction to my initial list.

This is the list as of our current understanding in this thread (subject to revision):

1) The star would look (through a telescope) blue and bright when approaching and red and dim when receding.
2) When the star's motion is tangent to the line of sight, its apparent distance as judged by subtended solid angle is the same whether approaching or receding.
3) The parallax distance to the star would appear the same whether approaching or receding.
4) The radar ranging distance would be identical whether the star was approaching or receding, and would be in between the two parallax distances.
5) Based on the star's total luminosity measured when moving tangentially, a raw luminosity distance computation would show the star closer when approaching and further when receding.
6) A luminisity computation corrected for blue/red shift would give the same distance whether receding or approaching.

Would you agree so far? It is worth noting that while distance measurements 2,3 and 4 are all in agreement with each other, most stars in reality are too far away for any of these measurements to be practical with current technology.
I am now interested in looking at the luminosity/brightness/flux aspects.

P.S. We have to be careful in relativity about what we by "when" in expressions such as "when the star's motion is tangent to the line of sight". I propose by "when", we understand this to mean "the time the light was emitted in the rest reference frame of the observer".

 

[PAllen]

P.S. We have to be careful in relativity about what we by "when" in expressions such as "when the star's motion is tangent to the line of sight". I propose by "when", we understand this to mean "the time the light was emitted in the rest reference frame of the observer".

Agree. Here there is a well defined answer. Light emitted when the star's motion is tangent to the line of sight will be the points of maximum blue shift and the point of maximum red shift. Thus it is observable locally by the Earth observer. This is easy to see because we are assuming constant speed of the star, and at the tangent points it has maximum approaching / receding velocity relative to the Earth observer.

 

[PAllen]

1) The star would look (through a telescope) blue and bright when approaching and red and dim when receding.
2) When the star's motion is tangent to the line of sight, its apparent distance as judged by subtended solid angle is the same whether approaching or receding.
3) The parallax distance to the star would appear the same whether approaching or receding.
4) The radar ranging distance would be identical whether the star was approaching or receding, and would be in between the two parallax distances.
5) Based on the star's total luminosity measured when moving tangentially, a raw luminosity distance computation would show the star closer when approaching and further when receding.
6) A luminisity computation corrected for blue/red shift would give the same distance whether receding or approaching.

Would you agree so far?

Yes. Some leftover bad wording on parallax needs fix up. In principle, we could point out the max blue shift size would be theoretically smaller than the max red shift size. An approximate calculation I did suggested this is on the order of one part in 10^7 or 10^8 for a .1 light second size stare observed from 5 light years away. I doubt this could be observed.

I agree 2,3,4 are generally impossible for stars if (3) is limited to base line on earth. I believe parallax across the Earth's orbit can measure distance to 'near by' stars, e.g. up to 100 light years.

So let's move to the luminisity issues. First question: can we assume fixed number of photons in approaching / receding case that just change energy? I have heard that accelerating frames don't necessarily preserve the number of photons, and our star is accelerating. I am hoping this effect is way too small to be significant for this situation. Any one know?

 

[yuiop]

So let's move to the luminosity issues. First question: can we assume fixed number of photons in approaching / receding case that just change energy?

Unfortunately it is not that simple! Let us say we have a reference star which is stationary relative to us, at the same distance as the average distance of the orbiting star. Assume that the reference star is the same as the orbiting star when they are both at rest wrt each other. If the reference star emits x photons in t seconds, then the moving star, whether approaching or receding, emits x*sqrt(1-v^2/c^2) photons in the same time, due to time dilation. On top of this is a "relativistic beaming" or "headlight" effect which focuses more photons per unit area towards us when it approaching and less when receding (basically an aberration effect proportional to the relativistic Doppler effect). Then there is the energy change in the photons due to redshift that you have already mentioned. Finally there is way the radiation spectrum of the star is distributed (there is a proper term for this but I cannot recall it at the moment). The redshift shifts some of the light into a range that is not detectable to our cameras or eyes and shifts another part of the spectrum that we do not normally see into a range that is detected. This is the most complicated part that requires some knowledge of the radiation profile of the star.

 

[PAllen]

Unfortunately it is not that simple! Let us say we have a reference star which is stationary relative to us, at the same distance as the average distance of the orbiting star. Assume that the reference star is the same as the orbiting star when they are both at rest wrt each other. If the reference star emits x photons in t seconds, then the moving star, whether approaching or receding, emits x*sqrt(1-v^2/c^2) photons in the same time, due to time dilation. On top of this is a "relativistic beaming" or "headlight" effect which focuses more photons per unit area towards us when it approaching and less when receding (basically an aberration effect proportional to the relativistic Doppler effect). Then there is the energy change in the photons due to redshift that you have already mentioned. Finally there is way the radiation spectrum of the star is distributed (there is a proper term for this but I cannot recall it at the moment). The redshift shifts some of the light into a range that is not detectable to our cameras or eyes and shifts another part of the spectrum that we do not normally see into a range that is detected. This is the most complicated part that requires some knowledge of the radiation profile of the star.

I was blithely thinking that if aberration didn't apply, then beaming didn't apply (they have the same root cause and derivation). Thinking about this more, can see the following explanation of why it does.

When asking about how many photons we will observe, we need to convert from an angle in the observer's frame to an angle in the star's frame; it is the latter that determines power we will see, subject to all the other corrections you mention. Converting an angle perceived in one frame to and angle perceived in a different frame is exactly when aberration and beaming do apply.

Does this make sense?

 

[PAllen]

On the question of the spectrum of a star, we can use two models. The simpler is to say we can detect all frequencies of e.m. radiation. Then since the effect of Doppler is apply a fixed multiplier to frequency, and energy is proportional to frequency, the Doppler effect on luminosity is simply a multiplier. If we want to model a window of observable frequencies, then everything will depend on what choice is made for that and what type of object is being observed.

 

[JDoolin]

(BackLink: From https://www.physicsforums.com/showthread.php?p=3071946#post3071946")

This is the list as of our current understanding in this thread (subject to revision):

1) The star would look (through a telescope) blue and bright when approaching and red and dim when receding.
2) When the star's motion is tangent to the line of sight, its apparent distance as judged by subtended solid angle is the same whether approaching or receding.
3) The parallax distance to the star would appear the same whether approaching or receding.
4) The radar ranging distance would be identical whether the star was approaching or receding, and would be in between the two parallax distances.
5) Based on the star's total luminosity measured when moving tangentially, a raw luminosity distance computation would show the star closer when approaching and further when receding.
6) A luminisity computation corrected for blue/red shift would give the same distance whether receding or approaching.

Would you agree so far? It is worth noting that while distance measurements 2,3 and 4 are all in agreement with each other, most stars in reality are too far away for any of these measurements to be practical with current technology.
I am now interested in looking at the luminosity/brightness/flux aspects.

P.S. We have to be careful in relativity about what we by "when" in expressions such as "when the star's motion is tangent to the line of sight". I propose by "when", we understand this to mean "the time the light was emitted in the rest reference frame of the observer".

I agree with the current list. Another noticeable effect, which shows up in yuiop's animation, is the way the star on the approach appears to be making up for lost time; the star on the way out is going to appear to move less than half the speed of light, then on the way back, it appears to move superluminally.

As for luminosity, I don't have any answers, but I could pose a few questions, in the hopes that by asking several questions, one of them might be a good one.

1) Does the frequency of the light correspond to the frequency of the photons; that is, the number of photons per second?

2) what is luminosity; is it the # of photons per unit time, or Energy per unit time?

3) What is the relationship between the luminosity of a star and its peculiar velocity (non-cosmological redshift)?

4) Has the universe shrunk? Bcause I used to hear about galaxies that were 60 billion light-years away, but when I look today, the most distant galaxy is http://www.wired.com/wiredscience/2010/10/most-distant-galaxy/" at 13.1 billion light years away.

5) If two astronomers use two different assumptions about the universe (1) that the redshift is primarily due to recession velocity, and therefore the luminosity is affected heavily by this phenomenon, and (2) that the redshift is caused by cosmological stretching, so no recession velocity, and thus the luminosity is NOT affected by this phenomenon... The difference between their descriptions of the universe would be dramatic. Is the "standard model" of cosmology based on assumption 2?

 

[JDoolin]

I was blithely thinking that if aberration didn't apply, then beaming didn't apply (they have the same root cause and derivation). Thinking about this more, can see the following explanation of why it does.

When asking about how many photons we will observe, we need to convert from an angle in the observer's frame to an angle in the star's frame; it is the latter that determines power we will see, subject to all the other corrections you mention. Converting an angle perceived in one frame to and angle perceived in a different frame is exactly when aberration and beaming do apply.

Does this make sense?

If you are far enough away from the star that the light can be approximated as a plane wave, can we ignore the aberration, and just worry about the relativistic doppler effect? I'm not altogether sure.

Another way to approach the question, is should we treat the photons conceptually as concentric spheres expanding at the speed of light, or should we treat them as lines coming straight out of the center. I suppose that thinking in terms of concentric circles actually fails to capture the angular effects.

As for the angular aberration, I know of an applet that might (or might not) be helpful: http://www.its.caltech.edu/~phys1/java/phys1/MovingCharge/MovingCharge.html (I'm not entirely sure how this applet calculates the appropriate angles, but it suggests it might be done without switching to the star's reference frame.)

On the other hand, the angles that go from the star to the receiver is going to represent a tiny, tiny, tiny angle. I mean, you're looking straight at an object; it's going to be a very small angle of aberration. It might be better just to use the idea of plane waves.

On yet another hand; (I think I'm turning into an octopus.) When you calculate the amount of light received, one should really use the reference frame of the emitter.

Sorry, this post doesn't really have a main point--just several (possibly self-contradictory) ideas I thought I'd throw out.

 

[JDoolin]

On yet another hand; (I think I'm turning into an octopus.) When you calculate the amount of light received, one should really use the reference frame of the emitter.

Wow, I botched that. You should use the reference frame of the receiver, not the emitter.

 

[PAllen]

Wow, I botched that. You should use the reference frame of the receiver, not the emitter.

This is actually the origin of beaming. You need to relate what angle of emission corresponds to a given angle for the receiver. When you do this, you find the receiver gets a larger share of emitted power when an emitter relativistically approaches, and a smaller share when the emitter is receding.

[EDIT] In my view, the most conceptually straightforward approach is to simply write down the equation for two light paths separated by some angle (e.g. alpha) in terms of x,y,t and alpha, in the emitter frame. The apply Lorentz transform to the equation, and see what the new angle between the paths is. Thus if alpha becomes alpha/2, than power will be doubled relative to a stationary emitter of the same intensity and momentary location. This effect will be on top of all other effects (e.g. doppler).

 

[PAllen]

I believe all effects other than the spectrum of the source and the detector sensitivity spectrum are easy to account for. I am sorry I am not good (lack tools, as well) at diagrams and latex. However, I have one simple diagram that, combined with the Lorentz transform, shows all effects combine as follows:

1) A beaming effect that amplifies an approaching source and attenuates a receding source. This is a result of Lorentz transform of angles.

2) The relativistic doppler factor applies both to the energy per photon and the number of photons per second received. The same factor that compresses or expands wave fronts for Doppler, would compress or expand a pattern of photons. Thus the same factor applies twice, to power (unshifted) and then energy per photon due to Doppler.

Thus you have: (doppler factor squared) * (beaming factor).

 

[JDoolin]

I believe all effects other than the spectrum of the source and the detector sensitivity spectrum are easy to account for. I am sorry I am not good (lack tools, as well) at diagrams and latex. However, I have one simple diagram that, combined with the Lorentz transform, shows all effects combine as follows:

1) A beaming effect that amplifies an approaching source and attenuates a receding source. This is a result of Lorentz transform of angles.

2) The relativistic doppler factor applies both to the energy per photon and the number of photons per second received. The same factor that compresses or expands wave fronts for Doppler, would compress or expand a pattern of photons. Thus the same factor applies twice, to power (unshifted) and then energy per photon due to Doppler.

Thus you have: (doppler factor squared) * (beaming factor).

I'd like to understand and verify beaming (or what I think you're calling beaming) fully. I made a diagram that may help. Using the diagram attached, say we have a light source traveling along the line AD. The light is falling on on the screen at BC.

The angle in the source-reference-frame is BDC. But the angle in the screen reference frame is not BDC, because D would no longer be simultaneous with BC. We could call the BC-simultaneous event D', and find the angle BD'C, for instance, and then carefully check the assumptions about angles and intensity.

 

[PAllen]

I'd like to understand and verify beaming (or what I think you're calling beaming) fully. I made a diagram that may help. Using the diagram attached, say we have a light source traveling along the line AD. The light is falling on on the screen at BC.

The angle in the source-reference-frame is BDC. But the angle in the screen reference frame is not BDC, because D would no longer be simultaneous with BC. We could call the BC-simultaneous event D', and find the angle BD'C, for instance, and then carefully check the assumptions about angles and intensity.

The way I did it my diagram was to Lorentz transform paths, then measure angles. Thus write the equations for two light paths separated by angle alpha in the emitter frame, transform to receiver frame and then compute the angle between the paths. For small alpha in the emitter frame, with emitter *approaching* receiver at speed v, I get receiver frame angle of alpha*sqrt ((1-v/c)/(1+v/c)) . Thus the 'concentration factor' approaches infinite as v approaches c.