GWs energy and aLIGO detection

[RockyMarciano]

In a recent campus talk by one of the LIGO memebers authoring the last february announced GW detection paper(and that was very proud because her small team's analytical data had actually made it to the finally published paper) explained to the audience something I didn't fully get. She said that they were monitoring a number of binary pulsars systems in our galaxy close enough so that they should have been producing a detectable signal with the sensitivity acquired in the observing run 1 of last autumn, but the fact no signal was detected meant that only 10 to 1% of their orbital decaying energy was being radiated in the form of GWs and thus went undetected.
However all the models of gravitational radiation I know start from the premise that all(100%) of the energy lost by the binary sistems is radiated as GWs. That's the case in the Hulse-Taylor indirect proof of GWs and I thought was the premise used by LIgo to model the BH merger of the direct detection but I'm not so sure now if instead of 3 solar masses it could haven more or less energy radiated if one is not demanding that it must be 100% of the energy lost by the system. So I don't exactly know what to make of the non-detection in the case of nearby pulsars. If the energy radiated is not necessarily the one one gets from the quadrupole formula or from the computed total energy at lightlike infinity, how does one compute it, or is it an adjustable parameter that must be fitted according to what is observed or not observed?

 

[mfb]

She said that they were monitoring a number of binary pulsars systems in our galaxy close enough so that they should have been producing a detectable signal with the sensitivity acquired in the observing run 1 of last autumn, but the fact no signal was detected meant that only 10 to 1% of their orbital decaying energy was being radiated in the form of GWs and thus went undetected.

"She said" is not a reliable source. Is it published? I think I would have heard of such a publication.

If yes, please give the source. This should be the next post, otherwise I'll close the thread.
If not: we cannot discuss rumors here.

 

[RockyMarciano]

"She said" is not a reliable source. Is it published? I think I would have heard of such a publication.

If yes, please give the source. This should be the next post, otherwise I'll close the thread.
If not: we cannot discuss rumors here.

Oh, my bad. I've checked and I misremembered, the monitoring was not of binary systems but single pulsars(google Hough search or look up Crab pulsar in wikipedia). It's late to edit now, but the question remains for spindown of pulsars instead of decaying rate of binaries. My apologies.

 

[pervect]

Oh, my bad. I've checked and I misremembered, the monitoring was not of binary systems but single pulsars(google Hough search or look up Crab pulsar in wikipedia). It's late to edit now, but the question remains for spindown of pulsars instead of decaying rate of binaries. My apologies.

Google finds http://arxiv.org/abs/1104.2712 and http://www.ligo.org/science/Publication-S6VSR24KnownPulsar/. From the GR standpoint, it seems that we can say that the spindown for several observed pulsars can't be due to gravitational radiation. However, I haven't seen any sources which claim to tell us what does cause the spindown. Perhaps moving the thread to the astrophysics forum would help, perhaps not.

 

[RockyMarciano]

Google finds http://arxiv.org/abs/1104.2712 and http://www.ligo.org/science/Publication-S6VSR24KnownPulsar/. From the GR standpoint, it seems that we can say that the spindown for several observed pulsars can't be due to gravitational radiation.

Rather that at least it can't be entirely due to gravitational radiation. Also from wikipedia:
"The Crab Pulsar was the first pulsar for which the spin-down limit was broken using several months of data of the LIGO observatory. Most pulsars do not rotate at constant rotation frequency, but can be observed to slow down at a very slow rate (3.7e-10 Hz/s in case of the Crab). This spin-down can be explained as a loss of rotation energy due to various mechanisms. The spin-down limit is a theoretical upper limit of the amplitude of gravitational waves that a pulsar can emit, assuming that all the losses in energy are converted to gravitational waves. No gravitational waves being observed at the expected amplitude and frequency (after correcting for the expected Doppler shift) is therefore a proof that other mechanisms must be responsible for the loss in energy. The non-observation so far is not totally unexpected, since physical models of the rotational symmetry of pulsars puts a more realistic upper limit on the amplitude of gravitational waves several orders of magnitude below the spin-down limit. It is hoped that with the improvement of the sensitivity of gravitational wave instruments and the use of longer stretches of data, gravitational waves emitted by pulsars will be observed in future.[19] The only other pulsar for which the spin-down limit was broken so far is the Vela Pulsar."And from the Ligo public pages https://www.lsc-group.phys.uwm.edu/ligovirgo/cw/public/ which seems to be the information the lecturer was referring to :
"http://mr.caltech.edu/media/Press_Releases/PR13154.html is a limit on the strength of gravitational radiation emitted by the Crab Pulsar, a young neutron star (created in a supernova reported by Chinese astronomers in 1045 A.D.) with a radius of only ~10 km, but more massive than the Sun, and spinning on its axis 30 times per second! The Crab's rotation frequency is decreasing perceptibly, implying a significant energy loss. Our most recent limits indicate that no more than two percent of that energy loss can be attributed to gravitational wave emission."

So how does this verification affect the general case that includes binary systems as seen from the GR point of view? In the sense that in the binary system case we only consider the energy loss of the system converted to gravitational radiation in its entirety(quadrupole formula used in i.e. Hulse-Taylor binary) while for single star systems it seems it is not the case. Is there some general relativistic important issue to consider between binary and single systems when accounting for energy at null infinity that explains this asymetry?

 

[mfb]

since physical models of the rotational symmetry of pulsars puts a more realistic upper limit on the amplitude of gravitational waves several orders of magnitude below the spin-down limit.

In other words, an observation would have been surprising.

Binary systems have a huge changing quadrupole moment which allows them to radiate gravitational waves. Individual stars don't have such a large quadrupole moment, and other mechanisms are more important. In addition, a pulsar is not a black hole - light and matter with angular momentum can escape from them. Supernova remnants are also surrounded by other matter.

 

[RockyMarciano]

Binary systems have a huge changing quadrupole moment which allows them to radiate gravitational waves. Individual stars don't have such a large quadrupole moment, and other mechanisms are more important.

Right. Isolated pulsars even when rotating hundreds of times per second have a much smaller quadrupole moment than a binary system inspiraling with similar orbit rate. But my question is about the general relativistic assumption that in the binary case assumes that the whole quadrupole moment is radiated as GWs(that's the assumption in the Hulse-Taylor case where the whole quadrupole moment is thought to be radiated as GWs as made clear by a decaying orbital rate that matches exactly the loss of energy predicted by GR in the forma of GWs), while in the single star case the quadrupole can be mostly radiated electromagnetically independently of the fact that it is smaller in absolute terms.
I was wondering if there was some explanación for this from GR.

 

[mfb]

You don't "radiate away the quadrupole moment". You radiate away energy and angular momentum (only those have relevant conservation laws). In the binary star/BH case this angular momentum is associated to a large quadrupole moment, in the single pulsar case you have a large angular momentum with a negligible quadrupole moment.

 

[RockyMarciano]

You don't "radiate away the quadrupole moment". You radiate away energy and angular momentum (only those have relevant conservation laws).

Of course, I'll try not to be so loose with the terms.

In the binary star/BH case this angular momentum is associated to a large quadrupole moment, in the single pulsar case you have a large angular momentum with a negligible quadrupole moment.

Yes, I understand this. In both cases energy and angular momentum are radiated away, in the binary case one uses the quadrupole formula which is the time-varying quadrupole moment and the angular momentum is conserved which is visualized as orbital decay. In the single pulsar case we have spin-down instead of orbital decay but this is not in one to one correspondence with the angular momentum lost via the quadrupole formula for the single pulsar as it happened with the orbital decay in the binary case. Am I the only one that sees the asimetry here?

 

[mfb]

The setups are completely different, right. So why are you surprised that you see different results?

Binary system: large quadrupole moment -> huge energy and angular momentum loss due to gravitational waves, other things negligible for close binary black holes
Single object: negligible quadrupole moment -> negligible emission of gravitational waves, spin-down occurs on a much larger timescale and nearly exclusively from other effects.

 

[RockyMarciano]

The setups are completely different, right. So why are you surprised that you see different results?

Well, then choose a setup with a binary system and a single pulsar in which the change in mass quadrupole moment for both as computed by the quadrupole formula ($\bar{h}_{ij}(t,r) = \frac{2 G}{c^4 r} \ddot{I}_{ij}(t-r)$ where $I_{ij}$ is the quadruple moment) happen to be the same.
Now in the binary case the quadruple formula is equivalent to the rate of GWs emission by definition of the quadrupole formula in linearized GR, while in the single pulsar it doesn't, instead it gives us the spin-down limit of the pulsar, and the actual rate of GWs emission may be many orders of magnitude smaller. And yet the quadrupole formula is a fundamental tool in GR for computing the gravitational radiation output according to any GR textbook(its matching the observed orbital decay in binary pulsars is called an indirect proof of GWs). Or is it not?

 

[mfb]

Well, then choose a setup with a binary system and a single pulsar in which the change in mass quadrupole moment for both as computed by the quadrupole formula ($\bar{h}_{ij}(t,r) = \frac{2 G}{c^4 r} \ddot{I}_{ij}(t-r)$ where $I_{ij}$ is the quadruple moment) happen to be the same.

Please show the existence of those systems, and their relevance for astronomy in case their parameters are far off from real systems (they will be).

Now in the binary case the quadruple formula is equivalent to the rate of GWs emission by definition of the quadrupole formula in linearized GR

It is not. A constant quadrupole moment doesn't lead to emission of gravitational waves.
But if you fix that error here, then the same also applies for the pulsar.

 

[RockyMarciano]

Please show the existence of those systems, and their relevance for astronomy in case their parameters are far off from real systems (they will be).

I find quite trivial to think up a binary system and a pulsar whose rates of change of quadrupole are comparable, I can even think of milisecond pulsars with change of quadrupole moment big in comparison to two body systems like the earth-sun. In any case it is just an example to pinpoint that my question is not about "huge versus negligible quadrupole".

It is not. A constant quadrupole moment doesn't lead to emission of gravitational waves.
But if you fix that error here, then the same also applies for the pulsar.

I wasn't implying that a constant quadrupole moment leads to emission at all, I simply used the definition of quadrupole formula as it appears in wikipedia:"In general relativity, the quadrupole formula describes rate at which gravitational waves are emitted from a system of masses based on the change of the (mass) quadrupole moment."

 

[RockyMarciano]

To put it as simply as I can I just want to confirm or reject whether the quadrupole formula is valid only for binary systems.

 

[mfb]

I find quite trivial to think up a binary system and a pulsar whose rates of change of quadrupole are comparable, I can even think of milisecond pulsars with change of quadrupole moment big in comparison to two body systems like the earth-sun.

The orbit of Earth changes due to tons of different things, gravitational waves are negligible compared to several other effects.

To put it as simply as I can I just want to confirm or reject whether the quadrupole formula is valid only for binary systems.

It is valid for all systems (to first order). Otherwise it would be called "quadrupole formula for binary systems". This special case has indeed its own formula, which is a special case of the quadrupole formula.

 

[PeterDonis]

I find quite trivial to think up a binary system and a pulsar whose rates of change of quadrupole are comparable

How physically realistic is the pulsar you are thinking of? Can you give some numbers?

 

[RockyMarciano]

How physically realistic is the pulsar you are thinking of? Can you give some numbers?

Sure, expressed in Watts from the Ligo page ligo.org/science/Publication-S6VSR24KnownPulsar a physically realistic pulsar would lose in angular momentum energy due to its spindown around $10^{31}$ watts using the assumption in the quadrupole formula that all this energy is radiated as GWs this gives us the spindown limit GW amplitude. For comparison in the case of the Hulse-Taylor system the loss in angular momentum energy observed as decay in orbital distante is around $10^{25}$ watts. In the latter the application of the quadrupole formula is not considered an upper limit like in the pulsar but the actual GW emission and the possibility of orbit change due to other possible factors is dismissed.

 

[mfb]

In the latter the application of the quadrupole formula is not considered an upper limit like in the pulsar

The 10³¹ W upper limit is not from the quadrupole formula. It is from the observed loss of angular momentum.

 

[PeterDonis]

using the assumption in the quadrupole formula that all this energy is radiated as GWs

Why would you expect all of the energy and angular momentum loss from spindown to be through quadrupole gravitational radiation? A single pulsar can emit dipole EM radiation--that's exactly what the "pulses" are that give them their name and by which we detect them.

In the case of binary pulsar systems, the energy and angular momentum loss that is attributed solely to gravitational waves, and for which the quadrupole formula is used to estimate the rate, is in the orbital parameters; it has nothing to do with the spindown of the individual pulsars. Each pulsar itself also emits EM radiation, as above, since each pulsar individually is spinning down at the same time that their mutual orbit is gradually becoming closer due to GW emission. The two things are separate phenomena, and the fact that the quadrupole formula gives a good estimate for the orbital parameter decay does not in any way mean that it also has to be assumed to be the sole mechanism for spindown of the individual pulsars. The reason GW emission is the sole mechanism for orbital decay is that the pulsars overall are electrically neutral, so there is no EM radiation associated with their orbital motion.

 

[pervect]

The Wiki article on pulsars is , I think, illuminating. A couple of excerpts that I think are particularly important.

The beam originates from the rotational energy of the neutron star, which generates an electrical field from the movement of the very strong magnetic field, resulting in the acceleration of protons and electrons on the star surface and the creation of an electromagnetic beam emanating from the poles of the magnetic field. This rotation slows down over time as electromagnetic power is emitted.

Though the general picture of pulsars as rapidly rotating neutron stars is widely accepted, Werner Becker of the Max Planck Institute for Extraterrestrial Physics said in 2006, "The theory of how pulsars emit their radiation is still in its infancy, even after nearly forty years of work."

So, given that it's generally accepted that the pulsar's EM radiation is powered by it's rotational energy, one would expect that this emission slows the pulsar down. What's missing is a complete list of possible other mechanisms, and an approximate idea of their relative importance. I'd categorize the general possibilities as energy/angular momentum carried away by electromagnetic fields, by matter, and by gravitational fields (i.e. gravitational radiation). I can't think of any other possibilities offhand, but that doesn't mean they don't exist.

 

[RockyMarciano]

The 10³¹ W upper limit is not from the quadrupole formula. It is from the observed loss of angular momentum.

They coincide, as they should if as you claimed the quadrupole formula is valid for single pulsars also. This is confirmed for instance at www.spala2014.p.lodz.pl/talks/thursday/Bejger.pdf slides 3 to 9, specifically when in slide 9 it says that the time derivative of $E_{GW}$ is equal to the time derivative of $E_R$

Why would you expect all of the energy and angular momentum loss from spindown to be through quadrupole gravitational radiation? A single pulsar can emit dipole EM radiation--that's exactly what the "pulses" are that give them their name and by which we detect them.

But I don't expect that, I'm just saying that in accord with what mfb told me, the quadrupole formula(obviously adapted to a single rotating mass) is valid for single pulsars. And the quadrupole formula happens to have been formulated (as far back as 1916 by Einstein) for calculating the output of gravitational radiation in linearized GR. I know perfectly that in the case of single pulsar what one obtains is an upper limit(the spin-down limit. I'm simply pointing out how in the binary system case it is not supposed to be an upper limit, otherwise it wouldn't have served as indirect evidence of the existence of GWs, in other words the strength of the Hulse-Taylor system decay matching the quadrupole formula as indirect proof of the existence of GWs rests on that assumption.
It is not a very hard exercise to imagine that in alternative universe we hadn't found a binary system in 1974, but only recently coinciding with the mounting of the first sensitive interferometer, and that we had been lucky enough to find a binary pulsar in the vecinity with orbit with a period that would have fallen inside of the frequencies detected by our instrument but no GW detection, we would be basically in the same situation we are with single pulsars now, and talking about the quadrupole formula giving just an upper limit of GW amplitude, and looking very hard for alternative ways in which the rotational energy loss observed as orbital decay could be produced

In the case of binary pulsar systems, the energy and angular momentum loss that is attributed solely to gravitational waves, and for which the quadrupole formula is used to estimate the rate, is in the orbital parameters; it has nothing to do with the spindown of the individual pulsars. Each pulsar itself also emits EM radiation, as above, since each pulsar individually is spinning down at the same time that their mutual orbit is gradually becoming closer due to GW emission. The two things are separate phenomena, and the fact that the quadrupole formula gives a good estimate for the orbital parameter decay does not in any way mean that it also has to be assumed to be the sole mechanism for spindown of the individual pulsars. The reason GW emission is the sole mechanism for orbital decay is that the pulsars overall are electrically neutral, so there is no EM radiation associated with their orbital motion.

The quadrupole formula for single pulsar obviously doesn't use orbital parameters, it uses the quadrupole moment for a mass spinning , and by the way the pulsar in a binary pulsar is spinning up not down as the orbit decays.

 

[mfb]

They coincide

No they do not, and I think there is your misconception that lead to the whole discussion.

specifically when in slide 9 it says that the time derivative of $E_{GW}$ is equal to the time derivative of $E_R$

For the spin-down limit which is far away from reality.

For a pulsar, you have two different power values:
- power loss due to the observed slowdown of rotation
- power of gravitational wave emission, given by the quadrupole formula

The second one is orders of magnitude smaller than the first one. You can ask "what would happen if they are the same?" - this hypothetical question is the spin-down limit. The non-observation of gravitational waves can then show that the two power values are not the same, as expected.

 

[RockyMarciano]

No they do not, and I think there is your misconception that lead to the whole discussion.For the spin-down limit which is far away from reality..

And I meant exactly that, they coincide for the spin-down limit. So that cannot lead to any misconception.
.

For a pulsar, you have two different power values:
- power loss due to the observed slowdown of rotation
- power of gravitational wave emission, given by the quadrupole formula
The second one is orders of magnitude smaller than the first one.

Not exactly. The actual emission of GWs is orders of magnitude smaller than the total energy due to rotational energy loss. That's understood. But you have agreed that quadrupole formula gives you the spin-down limit, and the spin-down limit is coincident with the loss due to the observed slowdown of rotation(spin-down), it's just that for physical reasons pulsars don't emit in GWs all that energy, that's why what's given by the quadrupole formula(the spindown limit) is called an upper limit of GW amplitude and it wasn't expected to be observed, since at least 99% of that rotational energy was expected to be lost in the form of a magnetic dipole.

.

You can ask "what would happen if they are the same?" - this hypothetical question is the spin-down limit.

Exactly.

The non-observation of gravitational waves can then show that the two power values are not the same, as expected.

It shows that the actual GW emission must be different(much smaller) than the coincident quantity you are calling different power values(in the above reference:time derivative of$E_R$ for the slowdown equated to the time derivative of $E_{GW}$.for the quadrupole formula for the spin-down limit).

 

[mfb]

But you have agreed that quadrupole formula gives you the spin-down limit

No I have not. Stop misinterpreting my posts please.

and the spin-down limit is coincident with the loss due to the observed slowdown of rotation(spin-down)

That is reversing cause and effect.

that's why what's given by the quadrupole formula(the spindown limit) is called an upper limit of GW amplitude

That does not make sense at all.

It shows that the actual GW emission must be different(much smaller) than the coincident quantity you are calling different power values

"Different power values" are two values. You cannot compare a single value to two values in that way.

 

[RockyMarciano]

No I have not. Stop misinterpreting my posts please

Ok, you haven't. Then I must tell you that you disagree with all textbooks and papers where the spindown limit for pulsars is explained and calculated from the quadrupole formula for pulsars.
Let's see how(taken verbatim from "Physics, astrophysics and cosmology with gravitational waves" by B. Schutz et al. pages 17-18:
The luminosity derived from the quadrupole formula in terms of the quadrupole moment for pulsars with asymetry("bump") $\epsilon$ in watts units is $L_{GW spindown}≈(16/125)(2πf)^6 \epsilon^2 (MR^2)^2$ this is the radiated power that would presumably come from the rotational energy of the pulsar(with a rotational energy of $Mv^2/5$) that would lead to an observed spindown in the timescalte: $t_{spindown}≈\frac{1/5Mv^2}{L_{GW}}$(wich is indeed observed). This is of course an upper limit for power lost in the form of GWs by a pulsar $\frac{dE_{GW}}{dt}$ that is not expected to be observed because it is known that not all the rotational energy of pulsars is radiated as GWs, and that is why it is called an spindown limit(the limit at which $\frac{dE_{GW}}{dt}=\frac{dE_{rot}}{dt}$ which again it is not physicallly realized) from which we can obtain the upper limit on the asymetry or "bump" $\epsilon$ and the upper limit in the amplitude $h_o$ of GWs for pulsars.

So I think your confusion when you say "power of gravitational wave emission, given by the quadrupole formula"is in mixing the spindown limit of GWs that is what the quadrupole formula in the case of single pulsars gives you with the actual power of GW emission by pulsars that is always smaller than this limit, and it's only a fraction of what the quadrupole formula for pulsars gives.

 

[mfb]

You keep misinterpreting both textbooks and my posts in a consistent way. I don't see how continuing this discussion would help anyone.

 

[PeterDonis]

The luminosity derived from the quadrupole formula in terms of the quadrupole moment for pulsars with asymetry("bump") $\epsilon$

Where are they getting $\epsilon$ from?

 

[RockyMarciano]

Where are they getting $\epsilon$ from?

$\epsilon$ is the ellipticity of the pulsar, as you probably know the spherically symmetric case doesn't radiate gravitationally. It is also referred to as a "mountain" or "bump" in the popular account by LIGO that I linked at the beginning.
It is basically model dependent, constrained according to the quadrupole formula for pulsars by observations, in the case of the Crab and Vela pulsar the fact that no gravitational radiation has been observed so far constrains not only the maximum amplitude of the GWs radiated by a pulsar but also the size of the "bump"(I think to 1 meter and 10 meters for Crab and Vela).
That's why when mfb says "power of gravitational wave emission, given by the quadrupole formula", it is obvious it is a confusion with the binary systems since for pulsars the quadrupole formula only gives an upper limit power of emission, not the actual power of emission, otherwise in the cases of the Crab and Vela pulsars we would have detected GWs, the sensitivity was enough to detect GW amplitudes associated to the power given by the quadrupole formula already before shutdown in 2010.

 

[mfb]

That's why when mfb says "power of gravitational wave emission, given by the quadrupole formula", it is obvious it is a confusion with the binary systems since for pulsars the quadrupole formula only gives an upper limit power of emission, not the actual power of emission

Okay, one last time: You are wrong. You can use the quadrupole formula to get an upper limit, but only if you take an upper limit for the change in the quadrupole moment. Take the actual quadrupole moment (which we do not know for pulsars!) and you get the actual emission.

Feel free to ignore it once more if you like, I won't look at this thread again.

 

[PeterDonis]

$\epsilon$ is the ellipticity of the pulsar

Yes, I understand that, but how do you know its value?

The answer I get from looking at the reference you gave is this: we know the actual spindown rate of the pulsar from measurements of its pulse period and how that changes over time. So we know the actual rate of rotational energy loss. The question is, how much of that energy loss is due to GWs, and how much is due to EM radiation (which is basically the only other energy loss mechanism available)?

To know that, we have to know the actual EM dipole moment and gravitational quadrupole moment of the pulsar. However, we have no independent way of measuring either of those things. The best we can do is to use the intensity of the EM radiation we detect from the pulsar (namely, its pulses) to estimate how much energy it is radiating as EM waves, and to try to detect GWs from it to get an estimate how much energy it is radiating as GWs.

As I understand it, the EM intensity we see is consistent with all of the observed rotational energy loss being due to EM wave emission. This would indicate that the pulsar is not radiating GWs to any significant extent, and therefore has a negligible gravitational quadrupole moment (or more precisely a negligible third time derivative of same). This is also consistent with the fact that we have observed no GWs from the pulsars in question.

How does all this relate to the quadrupole formula? Well, the quadrupole formula predicts that a negligible third time derivative of the quadrupole moment gives a negligible energy loss due to GW emission. That's why we deduce a negligible third time derivative of the quadrupole moment based on the fact that all of our observations indicate a negligible energy loss due to GW emission. I assume that $\epsilon$ is basically equivalent to "third time derivative of the quadrupole moment" (the latter is what I'm used to seeing in GW emission power formulas), so these observations would also predict a negligible $\epsilon$.

the spindown limit of GWs that is what the quadrupole formula in the case of single pulsars gives you with the actual power of GW emission by pulsars that is always smaller than this limit, and it's only a fraction of what the quadrupole formula for pulsars gives.

This doesn't seem to be stated correctly (as mfb has pointed out several times now). The difference between the spindown limit prediction for pulsar GW emission, and the actual observed pulsar GW emission, is in the value of $\epsilon$ we plug into the formula; the formula itself is the same in both cases. For the spindown limit, we use the observed rate of energy loss to infer what the value of $\epsilon$ would have to be to account for all of that energy loss being due to GWs. To match the actual observations, we calculate what the upper limit on $\epsilon$ would be such that the energy loss rate due to the GW quadrupole formula is less than the smallest value we could have measured, i.e., the upper limit on $\epsilon$ consistent with no GWs actually being observed. But whatever that rate is, it is exactly the rate given by the quadrupole formula for that value of $\epsilon$. There isn't a different formula for the spindown limit case vs. the actual case.

 

[RockyMarciano]

Yes, I understand that, but how do you know its value?

The answer I get from looking at the reference you gave is this: we know the actual spindown rate of the pulsar from measurements of its pulse period and how that changes over time. So we know the actual rate of rotational energy loss. The question is, how much of that energy loss is due to GWs, and how much is due to EM radiation (which is basically the only other energy loss mechanism available)?

To know that, we have to know the actual EM dipole moment and gravitational quadrupole moment of the pulsar. However, we have no independent way of measuring either of those things. The best we can do is to use the intensity of the EM radiation we detect from the pulsar (namely, its pulses) to estimate how much energy it is radiating as EM waves, and to try to detect GWs from it to get an estimate how much energy it is radiating as GWs.

As I understand it, the EM intensity we see is consistent with all of the observed rotational energy loss being due to EM wave emission. This would indicate that the pulsar is not radiating GWs to any significant extent, and therefore has a negligible gravitational quadrupole moment (or more precisely a negligible third time derivative of same). This is also consistent with the fact that we have observed no GWs from the pulsars in question.

How does all this relate to the quadrupole formula? Well, the quadrupole formula predicts that a negligible third time derivative of the quadrupole moment gives a negligible energy loss due to GW emission. That's why we deduce a negligible third time derivative of the quadrupole moment based on the fact that all of our observations indicate a negligible energy loss due to GW emission. I assume that $\epsilon$ is basically equivalent to "third time derivative of the quadrupole moment" (the latter is what I'm used to seeing in GW emission power formulas), so these observations would also predict a negligible $\epsilon$.
This doesn't seem to be stated correctly (as mfb has pointed out several times now). The difference between the spindown limit prediction for pulsar GW emission, and the actual observed pulsar GW emission, is in the value of $\epsilon$ we plug into the formula; the formula itself is the same in both cases. For the spindown limit, we use the observed rate of energy loss to infer what the value of $\epsilon$ would have to be to account for all of that energy loss being due to GWs. To match the actual observations, we calculate what the upper limit on $\epsilon$ would be such that the energy loss rate due to the GW quadrupole formula is less than the smallest value we could have measured, i.e., the upper limit on $\epsilon$ consistent with no GWs actually being observed. But whatever that rate is, it is exactly the rate given by the quadrupole formula for that value of $\epsilon$. There isn't a different formula for the spindown limit case vs. the actual case.

Thanks for the effort to understand. You are basicallly right and I can see now where mfb and I were talking at cross-purposes. But we all actually agree on the math.

I thought he didn't distinguish between the spindown limit and the actual emission, and I admit I might have misled when talking about what I mean by the quadrupole formula. My claim was that upon the assumption that all the rotational energy loss was converted to GWs the quadrupole formula gave the spindown limit, not that the quadrupole formula cannot be applied by tuning the ellipticity according to observation to compute the actual emission.
That's what the tunable parameter $\epsilon$ does. $\epsilon$ is defined from components of the quadrupole moment $\frac{I_{xx}-I_{yy}}{I_{zz}}$ and it is $7.5*10^{-4}$ for the spindown limit case for Crab pulsar, but as I said one can lower it according to the model of pulsar or the observations from the LIGO interferometer. So yes the formula is in that sense the same, the difference comes from the ellipticity one inserts(for a given $I_{zz}$ that in the the Crab case is $10^{38} Kg.m^2$) and the particular spindown observed.

Now that we are on the same page maybe we can leave behing the prejudices and move on so I can formulate my question?

 

[PeterDonis]

so I can formulate my question?

Your question in the OP, as I understand it, is basically, if GW emission isn't what is causing the spindown of single pulsars, what is causing it, correct?

I have not looked in detail at the numbers, but the obvious candidate to me would be EM radiation emission.