Evolution of Spin Period and Magnetic Field of the Crab Pulsar: Decay of the Braking Index by the Particle Wind Flow Torque

Round 1

Reviewer 1 Report

The paper is concerned with the spin down evolution of pulsars, and in particular with the Crab pulsar, the only neutron star for which we can definitely pinpoint its age. The authors present a modification to the classical magnetic dipole radiation model (MDR), which predicts that with the passage of time the pulsar slowdown rate decreases, yielding a value of the so called "braking index" of n=3. In this classical model, as the pulsar slows down, the amount of dipole emission gets lower, which slows the amount of the rotational energy loss, in turn decreasing the slowdown rate.

In their paper the authors modify the model, by introducing well known mechanism of particle outflow (which is observed for several pulsars, powering pulsar wind nebulae), a phenomenon that carries away some of the angular momentum of a neutron star but is usually considered to be too weak to constitute a significant source of additional energy loss. However, the authors show, that as the time passes, and the dipole radiation gets weaker, the particle outflow can take over as the main source of the pulsar slow down, especially that authors assume that this outflow remains at basically the same level during the spin-down evolution. As a result, the evolution of the pulsar spin-down deviates vastly from the MDR model prediction, leading to much more rapid slow-down. Additionally, this much faster spin-down also influences the observational estimates of the pulsar magnetic field and age, which are usually calculated using the braking index prediction based on pure MDR model. According to authors these may be the source of the discrepancies between the pulsar age estimates (between the slow-down related characteristic pulsar age and estimates based on the physical properties of supernova remnants they reside in), and also the pulsar magnetic fields - in the model presented in the paper the observationally inferred magnetic field strengths may be up to two orders of magnitude larger than the real magnetic field of a pulsar. As the authors claim this may explain the magnetic field strengths of magnetars - neutron stars with extremely strong fields, as according to their model  some of these may be just a regular magnetic field pulsars for which the filed was hugely overestimated due to using the incorrect pure MDR model. 

The paper is well written, the introduction part is appropriate, giving enough reference to the current state of knowledge in the field of pulsar astrophysics to easily understand the problem and the model presented. The model itself is outlined clearly and in a simple manner (with additional, extended explanations in the appendix). The math of the model is relatively simple to follow. Predictions of the model are clearly outlined and explained, and the authors check their predictions (like the evolution of the braking index) against the observational data for the Crab pulsar.

The model, in its simplicity, is compelling and in their conclusions the authors draw a picture that would explain a lot of observed properties of some pulsars and magnetars if their model is correct. However, that is where most of the weaknesses of the paper lie: the correctness of their model assumptions and overinterpretation of the results. For these reasons I cannot recommend publication of the manuscript in its current form, and I would recommend a moderate revision - not to the model itself, but to the way the model is presented, and the paper conclusions. My specific concerns and comments are listed below.

 

Major concerns:

1. Presented model is simple and elegant, and while authors mention the limitations of the model very briefly in its introduction (mentioning the presence of other phenomena that may affect pulsar spin-down evolution), in the later sections authors seem to treat their results as if the model had 100% level of certainty, drawing very bold conclusions. To be clear: I have nothing about the conclusions themselves. I see the paper, and the model presented, as a form of a thought experiment, a "what if" kind of scenario. The authors just assume that their model is correct and other phenomena that may influence pulsar spin-down do not exist. This is perfectly fine for a scientific paper, and may be very useful for understanding the reality of the situation. If several factors can influence the experiment (in this case a slow-down evolution of a neutron star) then the understanding what each of these factors may do to the experiment on its own may be a real help when trying to understand the full picture. This is fine, as long as the authors present it in a proper way, and they show they understand the limitations brought by their assumptions. In the presented manuscript I don't see that, however. The conclusions are outlined clearly and are soundly (which is a good thing) based on the results the authors obtained, the problem lies in the way they are presented: in most cases - with a huge dose of certainty. 

Therefore I would recommend some "softening" of the language the later parts of the paper are written, especially in the conclusion section. Adding something like "if our model is correct then... ", or "if there were no other factors influencing pulsar spin down then our model shows us that...". The results and implications of the model presented are valid, as long as they are presented as the results of this hugely simplified "thought experiment" the MDRW model in my opinion is.

2. Related to the above: from what I understand the only actual observational confirmation of the MDRW model prediction is the observed change in the Crab pulsar braking index from 2.51 to 2.50 over the time span of 45 years. The authors probably realize that, as they bring other, indirect arguments in the form of the properties of other pulsars - see page 7, line 135: the paragraph starting with “Meanwhile some other evidence also supports...". In my opinion these pulsars mentioned afterwards in no way provide any significant "supporting evidence" and should not be considered as such. This is one of the more significant instances where the paper needs some "softening" of the language, I have mentioned in my first comment. It's true, that if the MRDW model would actually be true and was the only mechanism working for the spin-down evolution, that would be the case. In reality, however, and even if we (unrealistically) assume that for the Crab pulsar the model works and is completely responsible for the observed change of braking index, there is no certainty that it should also work for other pulsars.

For these reasons I would soften this paragraph. These low values of the braking index for other pulsars are not an evidence for the model support. They "may" be due to the mechanisms included in the MDRW model, but at the current stage there is no way we can tell, as the spin-down evolution and braking index change over time for these sources remains unknown. Obviously, it will remain unknown for a very long time, and no one would expect holding on the publication of the manuscript for several thousand years (which would be probably required for a proper observational confirmation), the problem is in the interpretation of these values. This paragraph should be written more in a way like: "assuming thought experiment MDRW model has merit, then it can easily explain values like these", rather than - as it reads currently - "these isolated, individual measurements clearly confirm our model is valid".

3. One of the more bold claims the authors draw based on their model is the conclusion (most clearly outlined on page 6) that the proposed evolution of the spin-down rate will influence the estimations of the neutrons stars magnetic fields (and age) in such a way, that the values obtained will be hugely overestimated, placing an aged Crab-like pulsar well within the magnetars with their extremely strong magnetic fields (while the real strength of their fields remain the same as in the beginning). This would mean, that amongst the magnetar population we may have some "impostors" i.e., old neutron stars with overestimated B-fields. A thought that came to me reading this part is that an inexperienced reader may (I emphasize - may) be even considering that all the magnetars may be like these. This is not what the authors wrote, obviously, but since they do not comment on it, the reader may be left wandering.

This fact is brought again in the Conclusion section, i.e. section 4.3 (at the start of page 9), but authors do not comment on this specific problem: are there "impostor" magnetars? Are all magnetars impostors? I would like to se more detailed discussion on this topic, especially on how you can tell a low- to mid-magnetic field old pulsar from a "real" magnetar. Can these "impostor" magnetars mimic other magnetar properties with their weak fields (i.e. X-ray and gamma-ray emission properties etc.).

Minor coments:

a. Abstract: should be softened as well, considering my major comments 1 and 2.

b. page 2, line 51: the estimates of pulsar braking index range from 1 to 3, "which are derived from n=3 by the MDR model" - this part is a bit confusing. For someone not versed in how we estimate braking index that fact that we get values of "n" between 1 and 3 if we assume that n=3 may seem self-contradicting. I would rather state that we get these values from the assumption of the basic MDR model (i.e. not mentioning that that model assumes n-3).

c. page 3, the entirety of Fig.1: is this figure really necessary? It’s a picture of a basic pulsar model, and a simplified one at that (no light cylinder for example, so why are some magnetic field lines closed and some are open). I would presume that any potential readers of this paper will know the most basic "lighthouse" pulsar model, so showing it here in simplified form is unnecessary.

d. page 2, line 83: the beginning of the first paragraph needs rephrasing and clarification: we need to introduce the particle wind outflow to our models because of the fact that we observe Crab pulsar wind nebula. In other words: the introduction of this wind into any models is necessary, not the wind itself.

e. page 4, directly below Eq.5: "which satisfies the condition of d + f = 1" - this is actually the core assumption of the model and it should be stated as such.

f. page 4, Eq. 13: in the formula I would rather use regular "e to the power of 2bt" notation for the exponential function rather than using "exp" to denote such function. It is not an error, rather than preference.

g. page 6, start of section 3.2: "most of them are systematically distributed" - is there a "system" for this distribution? What authors probably mean is that the observed values are just randomly and quite evenly spread in the expected range between 1 and 3.

h. page 7, the start of section 3.3: "The characteristic age is often used to measure the real age of a pulsar" - this is not true. First, we are not "measuring" age, only estimating it, and anyone using the formula the authors implicate probably know its limitations, and assumptions for which it is valid. Characteristic age formula is not a tool for measuring pulsar age, it is only supposed to give one a rough estimation to estimate if one deals with a young object, a regular middle-aged pulsar or an old dying (or recycled) neutron star.

i. page 8, lines 160 to 161: authors mention other sources of the "timing noise", and they are not too precise about it. The main source is supposedly the random walk related to the proposed ~1 per day micro-glitches affecting the pulsar rotation. On the other hand, the interstellar scattering (if that is what the authors mean by the pulsars "scatting") should not affect timing, unless it changes on its own; interstellar scintillation is also a source of very small variations. And pulsar proper motion has nothing to do with noise, it is just a deterministic effect that is usually very small and hard to measure and may be - in some circumstances - mimicking the timing noise (especially of one deals with observational data shorter than a few years). This sentence needs to be rewritten.

j. page 8, line 183: "Meanwhile, if the multiple B-field exists" - not clear what the authors mean. Magnetic field, as any physical "field" always exist, and only in one instance. It may have multiple components (like from different sources), or have a multipole (like quadrupole etc.) i.e. non-dipolar characteristics, but there always be only one field.

k.  page 8, line 186: "meaning that the other mechanism should be possible" - all the mechanisms are possible until they are shown explicitly, by theory or observations that they are not applicable. And with the mounting evidence for magnetic field alignment change and other magnetospheric effects - they are definitely real. What the authors mean is probably that their model does not take those into account, which is fine - that is the core assumption of the model. I would rather see the observational facts the authors present here as a proof that their model is not describing reality, and again - as it was stated in my major comments - should be treated as a thought experiment that may explain some of the observed facts.

l. page 9, line 221: "discrepancy between two ages represents the significant contribution of the particle flow component." - i would rather say that it "may be due to" than "represents".

m. page 9, line 233: "...is there a source to proof this evolution path? The answer is "Yes" - one of the more blatant statements that supposedly coming from a doubtful amount of proof for the model presented in the manuscript. Instead of "The answer is Yes" I would rather see something akin to "The answer is strong "maybe", considering the assumptions of our model".

Author Response

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Author Response File: Author Response.pdf

Reviewer 2 Report

For the simple magnetic dipole model, the braking index of pulsars in n=3.0. In practice,  pulsar radiation mechanism can be far more complex  and the braking index can certainly can be lower than  3.0.

Here is an old paper on the estimates of n for 22 radio pulsars  which indeed found anomalous values:

“Measurements of radio pulsar braking indices” by

A. E. Chukwude A. A. Baiden, C. C. Onuchukwu

https://www.aanda.org/articles/aa/full_html/2010/07/aa11634-09/aa11634-09.html

This paper concluded that “Our data are largely consistent with the prevailing paradigm that the systematic smooth spin-down of most radio pulsars is overshadowed by effects due to rotational instabilities.”

Of course there could be other mechanisms for deviations in the values of n. Yet it any model which claims that n can reduce to 1 from 3 in 50 K yr seems to be erroneous.

In any case, the conclusion by the authors that the magnetic field of a pulsar  may INCREASE by a factor of 100 is simply unphysical. Only in a laboratory, one might try to increase the magnetic field of a magnet by artificial magnetization. What is known in astrophysics is that a pulsar born with B= 10**12 G may end up even with B= 10**8 G  due to accretion and become Millisecond Binary Pulsar.

 

Thus I am afraid I cannot recommend the publication of this manuscript.

Author Response

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Author Response File: Author Response.pdf

Reviewer 3 Report

The paper is well written and the references adequate. The comparison between the MDR and the MDRW models is interesting and well described. The conclusions are in line with the results obtained.

Author Response

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Round 2

Reviewer 1 Report

Dear authors,

I'm happy to see that all of my suggestions were accepted. The revised version of the paper is in my opinion ready to be published.

Author Response

Thank you sincerely for your review, which makes the article more rigorous and accurate in its expression.

Reviewer 2 Report

While this is a significant improvement with regard to their original version, it should still be revised considerably.

1.       Tone down all claims. Say .. our tentative conclusion….

2.       Tone down conclusions about magnetars.  Say it is likely that the magnetic field of some magnetars got overestimated because….

3.       Be more emphatic on the fact that while the actual magnetic field of Crab or any pulsar cannot increase ever, they might appear as much higher if interpreted without consideration likely change of the braking index.

4.       Though there is improvement in English, slightly more to be done. Break long sentences into two, and all blocks should be connected smoothly. The very first sentence in the INTRODUCTION is too long and has wrong syntax. Break it into two.

 

 

Author Response

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Author Response File: Author Response.pdf

Round 3

Reviewer 2 Report

It may be accepted for publication.