# Is the Sun invisible at relativistic speeds?

mfb
Mentor
Samshorn said:
Right, but the question is, which of those effects wins?
In terms of the total light you get, the smaller size wins.

We cannot get a black disk, but we can get a disk that is so small that you don't see it any more, even with the increased brightness per area.

Some more mathematics: If I consider the ultra-relativistic limit here,
$$n(\omega,\Omega)d\omega d\Omega \approx \frac{\omega T_{eff}}{2\pi^2}d\omega d\Omega$$ and
$$T_{eff} \approx 2T_* \gamma$$

Therefore, the brightness increases with the relativistic gamma-factor, while the area scales with ##\gamma^{-2}## (?). This would give a total luminosity (in the visible range) which scales with ##\gamma^{-1}## for very high speeds.

In terms of the total light you get, the smaller size wins. We cannot get a black disk, but we can get a disk that is so small that you don't see it any more, even with the increased brightness per area.
I agree, and this same conclusion can be reached simply by applying a Lorentz transformation to the light energy impinging on the observer, as explained in post #61, given that the star's spectrum in its own rest frame drops off more rapidly than can be compensated by the Doppler intensification.

tom.stoer
I agree, and this same conclusion can be reached simply by applying a Lorentz transformation to the light energy impinging on the observer, as explained in post #61, given that the star's spectrum in its own rest frame drops off more rapidly than can be compensated by the Doppler intensification.
Sorry to repeat myself, but your reasoning
As our speed toward the Sun increases, the visible light comes from the lower frequency range of the Sun's spectrum (Doppler shifted up to the visible frequency range). Remember that the energy of a pulse of light under a Lorentz transformation increases in exactly the same ratio as the frequency. So, for example, when our speed toward the Sun doubles the frequencies (i.e., when we are seeing light emitted by the Sun at half the visible frequencies), it will also be doubling the energies. However, at half the frequency, the spectral energy is extremely low, so doubling it doesn't make it very big. The blackbody spectrum eventually drops exponentially, and this drop prevails over the Doppler energy increase. Remember that as the temperature of a blackbody increases (as we approach the Sun at higher and higher speed), the peak frequency of the spectrum increases, and it will eventually pass out of the visible range. So instead of seeing the light, we'll just be getting fried with x-rays, etc..
cannot be correct.

You are talking about spectrum and Doppler shift only w/o taking into account geometric effects. But the shrinking of the emitter IS a geometric effect. By your reasoning from #61 even isotropic bb radiation would appear darker to the approaching observer, but we know that it appears brighter. So even if the geometric reasoning is correct, it is not contained in #61. Or could you please tell me how to change the reasoning from #61 to get the behavior for CMB? Where does the difference hide?

Sorry to repeat myself, but your reasoning cannot be correct. You are talking about spectrum and Doppler shift only w/o taking into account geometric effects.
You only need to worry about the size of the source if you have taken the approach of first working out the transformed intensity of the source per unit area, in which case you then need to determine the transformed area of the source. You might be tempted to take that approach, thinking that the theorem about black bodies transforming to black bodies provides a shortcut to the answer, but it actually is the long way around. It's better to just look at the energy impinging on the observer from the direction of the star, and apply the Lorentz transformation, and note that the energy content drops off faster than can be compensated by the Doppler intensification.

By your reasoning from #61 even isotropic bb radiation would appear darker to the approaching observer, but we know that it appears brighter. So even if the geometric reasoning is correct, it is not contained in #61. Or could you please tell me how to change the reasoning from #61 to get the behavior for CMB? Where does the difference hide?
The CMB is a different scenario. With a star we have energy impinging on the observer from a specific direction so that it all shares essentially the same Doppler shift and can be analyzed (for its energy content) on that basis. But for the CMB we have a surrounding impingement from all directions, some from behind, all being swept forward by aberration, but with a whole range of Doppler shifts. So the simple reasoning based on a single Doppler shift for the impinging energy that works for a star would not be applicable to that case.

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OK. Here is Prof. Thorne's reply: I thought you were traveling away from the sun...
That's puzzling, because his original reply said "The sun emits infrared radiation, which will get shifted into the visible part of the spectrum...", and so on. This sure makes it sound like he was talking about someone traveling toward (not away from) the sun. Why would infrared get shifted to visible if you were traveling away from the sun?

It's also a bit strange that he says he agrees with Gott, because it isn't obvious what Gott's answer really is. Gott just said there are two factors (increased intensity per unit area, and decreased area), but didn't say which one prevailed. So, when Thorne says he agrees with Gott, it isn't clear (to me) what he is agreeing to, i.e, is he saying the star fades from sight (as the FAQ says and as some of us have concluded here in this discussion), or is he saying it becomes more visible (meaning the energy received in the visible range increases)?

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Gold Member
I have forwarded your question :
Right, but the question is, which of those effects wins? The intensity per unit area of the emitter goes up, but the area goes down... so in the limit as we approach c, do we receive more energy per second from the star (in the visible range), or less? In other words, does the star fade from sight, or become more and more visible?
to Dr. Gott, Samshorn.

Evo
Mentor
A reminder to members - if you post e-mails, please remove all personal contact information first.

Thanks.

tom.stoer
@Samshorn, pervect:
I'm not sure if you ever looked at the thread I referenced previously, https://www.physicsforums.com/showthread.php?t=681172 ...
In the meantime I did, and I worked out the null-dust and the plane wave example for the Poynting vector i.e. the T0i components. After a Lorentz trf. I find - as expected - the factor

##r = \frac{1-v}{1+v}##

But this does not help for the case of a star b/c
1) in the time averaged Poynting vector the Doppler shift is not visible
2) the homogeneous plane wave or null-dust do not show any effect due to geometry

I understand the concerns regarding isotropic bb radiation not being an appropriate model for a (nearly) pointlike star. But I sill do have concerns not taking the Planck spectrum into account. Even for a pointlike star the spectrum is thermal for one single direction of wave propagation.

pervect
Staff Emeritus
I checked http://arxiv.org/pdf/physics/0701200v1.pdf but there is not enough math to see what they are really doing. http://cartan.e-moka.net/content/download/248/1479/file/Astronave relativistica.pdf is damaged and does not open on my computer.
That's really the best paper. :-( It computes just what you're looking for - or very close.

The digital object identifier (doi) for this paper is: http://dx.doi.org/10.1119/1.11834, but it may be behind a paywall.

I don't think I can do the paper justice in a post, but I can provide a quick summary:

The total photon arrival from the source, integrated over the entire view, scales as r. So the photon arrival rate increases if you're moving towards the object.

You seem to have missed a square root in your presentation of r, I'll assume it's just a typo unless otherwise argued about.

See http://en.wikipedia.org/wiki/Relativistic_Doppler_effect for example, for the derivation of r Just In Case.

The total energy delivered scales as r^2, because ##E = h \nu## and ##\nu## gets doppler shifted. The usual defintion of intensity is via delivered energy, so the intensity scales as r^2.

Relativistic aberation makes the angle subtended by the object smaller - this causes the object's apparent area to shrink by a factor of r^2. This would make the surface brightness scale as r^4 (the same energy is delivered in a smaller area). But if you are far enough away so that the object is smaller than the optical resolution of the telescope that you look at it through, this effect won't matter. YOu'll be limited by your optics, and you'll only see a r^2 increase.

Usually, stars don't show a disk , the telescope can't resolve the surface, and we talk about the "stellar magnitude" based on the total energy received, the entire visual field maps to what's effectively a point. So that's my starting assumption. With this assumptoin we get a r^2 brightness enhancent - and a doppler shift.

The authors actually work out the received brightness in the human visual range, by using a crude model of the eye's black and white frequency response.

I'll upload a few screenshot, which I think constitutes "fair use" for educational purposes, so you can at least get some information.

One post graphs their results.

Doppler factors > 1 represent motion towards the star. You can see that eventually the brightness starts to decline.

The other screen snapshot represents the equation for the unweighted spectrum S(r). You multiply this by your "response function", to filter out invisible frequencies, and that integral gives your brightness. The authors used an gaussian weighting function to represent the sensitivity of the human eye, rather than a crude square step function (which has a value of 1 if the frequency is visual and 0 if it isn't).

But I don't feel up to presenting it at the level of detail that includes their approximate visual weighting function specifically, you'll need to track down the original paper for that.

You might be wondering why the factor in front of the integral is r^-2, if the intensity scales as r^2. If you perform the integral though (the paper does it), you'll see that the end result DOES scale as r^2.

WHile you can see that the authors integrated in terms of wavelength rather than frequency, it appears that my earlier mistake was a failure to scale ##d\nu## properly under the transform. I scaled what multiplied it properly, but not ##d\nu## itself. The paper doesn't use the same approach, they use ##\lambda## instead.

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tom.stoer
Thanks for the summary.

I will check the references the again. The PDF that seemed to be broken on my notebook is displayed correctly on the iPad ;-)

I agree to all the r and r^2 factors; and I guess we agree on the main ideas.

My key idea is that if you use plane waves or something like that the object always gets darker b/c the visible part of the spectrum is shifted to the UV and there is nothing from the IR to replace that. But in reality there is a Planck-type spectrum and you get an enhancement in the visible part due to replacement from the IR which is shifted and enhanced due to the hotter effective temperature T'(v). I think we agree in that idea.

I found some other references discussing images of nearby stars. The images are not distorted due to length contraction (spheres a mapped to spheres) but they take different Doppler shifts due to the extension of the disc into account. That's quite interesting.

http://www.vis.uni-stuttgart.de/~weiskopf/publications/acmtog99.pdf
http://www.tempolimit-lichtgeschwindigkeit.de/sphere/sphere.pdf

pervect
Staff Emeritus
The IR does shift up to the visible, but when you do the math correctly, if you assume a black body spectrum it just isn't enough to replace what you've lost. So the intensity slowly goes down.

I don't know for sure how good an approximation the black body spectrum is, but I would think it'd be a good approximation at low frequencies.

tom.stoer
The IR does shift up to the visible, but when you do the math correctly, if you assume a black body spectrum it just isn't enough to replace what you've lost. So the intensity slowly goes down.
Can you please give me a hint where exactly I can find this formula

To evaluate the asymptotic behavior at ultra-relativistic speeds, we can focus on just the very low frequency range of the star’s spectrum, so the energy received (from a star assumed to radiate a black body spectrum of a certain temperature) by an eye at a certain location with frequencies near n is proportional to n^2. Hence the ratios of the energies impinging on the eye at a low frequency n1 and an even lower frequency n2 is (n2/n1)^2. If we give the eye some (additional) speed v directly toward the star, such that the n2 frequency is Doppler shifted up to n1, we will scale up the energy by n1/n2 (since energy scales like frequency), so the energy now being received at frequency n1 is n2/n1 times the energy that was being received at that frequency without that increased speed. Thus the energy drops off asymptotically in proportion to the frequency, so the factor in terms of speed v is just the Doppler factor sqrt[(1-v)/(1+v)].

Gold Member
seems to suggest that Kip was right
Yup!

Yup!
Well, again we have the confusion between considering the actual spectrum of the Sun, versus an ideal black body spectrum. Carroll says "There is a lower limit to the frequency of light emitted by the Sun, although it's down in the radio regime." So he is not considering the question for an ideal black body that emits out to infinitely long wavelengths. I think everyone agrees that, if there is a lower limit on the Sun's frequencies, then obviously we could Doppler shift its emissions up above the visible frequency range. That is self-evident. The more challenging question that we've been discussing is what happens in the theoretical case of an ideal black body.

By the way, I think that posing vaguely and ambiguously worded questions to random "experts", without clearly explaining the background and intent of the question, is not a very efficient way of seeking enlightenment. I would guess that all of the "experts" that have been cited would quickly agree on the answer, if only the question was posed to them in a clear way - for example, distinguishing between the actual Sun versus an ideal black body. Again, if we're talking about the actual Sun, which has a lower limit to emitted frequencies, the answer is self-evident.

Gold Member
By the way, I think that posing vaguely and ambiguously worded questions to random "experts", without clearly explaining the background and intent of the question, is not a very efficient way of seeking enlightenment. I would guess that all of the "experts" that have been cited would quickly agree on the answer, if only the question was posed to them in a clear way - for example, distinguishing between the actual Sun versus an ideal black body. Again, if we're talking about the actual Sun, which has a lower limit to emitted frequencies, the answer is self-evident.
What is vague and ambiguous about my question? This is what I've been sending the experts:

We are having a discussion on our forum whether the Sun disappears from
the visible part of the spectrum the closer we get to c. Let's assume we are on a spaceship a
few light-years away traveling toward the Sun at relativistic speed. For simplicity purpose, the universe is devoid of any other light sources (galaxies, stars, the CMB, etc).

If I were to look out the window of my ship, would I be able to see the
Sun slowly become invisible as the spectrum tilted toward the blue? In other words, is it possible for the Sun to become totally invisible as I approach c?

Thank you,
I think a child can understand it. You guys are the one making it more complicated with ''ideal blackbodies'' and whatnot lol.

What is vague and ambiguous about my question?

But this very self-evidentness may cause some people to suspect that this can't be what you had in mind, because if it was, the answer is just too obvious. So they might try to give you the benefit of the doubt, and assume you had in mind the possibility that the Sun may emit at all frequencies (like a black body, for example, which actually is a fairly accurate representation of the Sun's spectrum over a wide range of frequencies), in which case the question becomes: What asymptotic spectrum should we assume (in place of the actual astrophysical spectrum of the Sun)? This leads on to the issues related to the blackbody spectrum.

Admittedly, it's entirely possible that none of this is of interest to you, but a random respondent may not be sure what your interest is. Do you just want the answer to the braindead obvious question (which is really an astrophysics question, not a relativity question), or are you getting at something more subtle that involves relativity?

Also I note that in your follow-up to Carroll you quoted some comments about black body radiation, so if you were not intending him to provide an answer for that, it was somewhat misleading to introduce those words into your correspondence with him. But Gott seems to have also had black body spectrum in mind... even though you apparently aren't interested in the answer to that question. And in response, Carroll begins to justify his frequency cutoff for the actual Sun based on quantum considerations, which may or may not be a red herring. (Even classically, could a cavity radiator emit wavelengths longer than the size of the cavity?)

Having said all that, I agree that many of the responses in this thread have been off-point, and your question wasn't so ambiguous as to warrant so much confusion. But that's what you get in an open forum, or a survey of "experts". You always get the right answer to your question - along with all possible wrong answers. lol

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Gold Member

But this very self-evidentness may cause some people to suspect that this can't be what you had in mind, because if it was, the answer is just too obvious. So they might try to give you the benefit of the doubt, and assume you had in mind the possibility that the Sun may emit at all frequencies (like a black body, for example, which actually is a fairly accurate representation of the Sun's spectrum over a wide range of frequencies), in which case the question becomes: What asymptotic spectrum should we assume (in place of the actual astrophysical spectrum of the Sun)? This leads on to the issues related to the blackbody spectrum.

Admittedly, it's entirely possible that none of this is of interest to you, but a random respondent may not be sure what your interest is. Do you just want the answer to the braindead obvious question (which is really an astrophysics question, not a relativity question), or are you getting at something more subtle that involves relativity?

Also I note that in your follow-up to Carroll you quoted some comments about black body radiation, so if you were not intending him to provide an answer for that, it was somewhat misleading to introduce those words into your correspondence with him. But Gott seems to have also had black body spectrum in mind... even though you apparently aren't interested in the answer to that question. And in response, Carroll begins to justify his frequency cutoff for the actual Sun based on quantum considerations, which may or may not be a red herring. (Even classically, could a cavity radiator emit wavelengths longer than the size of the cavity?)

Having said all that, I agree that many of the responses in this thread have been off-point, and your question wasn't so ambiguous as to warrant so much confusion. But that's what you get in an open forum, or a survey of "experts". You always get the right answer to your question - along with all possible wrong answers. lol
Samshorn, no worries. I'm sending your questions to Prof. Carroll. I also don't mind you guys going off on tangents. I just wanna get the best possible accurate answer to my question. And no, I'm not entertaining any 'subtle' agenda or crackpot theory, if that's what you're asking.

Dale
Mentor
Hi Everyone,

After some discussion amongst the mentors we have decided to update the rules to prohibit the posting of personal communications with 3rd parties. This thread contained an extreme amount of such personal communications. The specific posts containing the personal communications have been removed, but as a result the thread has become very disjointed and confused.

Therefore, we are closing this thread. The topic itself is not closed and may be discussed in a new thread which should hopefully be less confusing for all involved.

As a personal recommendation, I would recommend that a follow-up discussion explicitly consider an ideal black-body radiator, rather than the messy complications of absorption bands that are found in the actual sun.