Friction of the vacuum could slow the rotation of pulsars

Discussion in 'Physics & Math' started by Plazma Inferno!, Aug 1, 2016.

  1. expletives deleted Registered Senior Member

    Messages:
    410
    paddoboy:

    You keep saying that, but you don't realize that only the gravitational and mass-transfer dynamics as affecting individual NS magnetic fields was considered. Such dynamics occur during the evolutionary stages where much mass and gravitational balance changes according to the mass exchanges and their effects on the individual NS magnetic spin and magnetic field strengths, and how these respective changes in spin/field due to gravitational and mass-transfer factors affects each NS's spin-orbital momentum coupling. That is all known already and not in question in my point. OK?

    At no stage does the interaction BETWEEN the TWO NSs' magnetic fields (whatever strengths they may be) ever come into the calculations and considerations as to potential for braking effect on the BINARY period decay rate. Do you understand?

    Your saying they did consider it is just repeating your own misunderstanding. They did not consider what I pointed out. As I again point out re your own referenced paper. You are misreading those other magnetic aspect considerations as somehow being the same as what I point out. It is not. OK?

    Please don't bring DM into this. It is difficult enough to keep you relevant on point as it is. Thanks.

    And whatever "evidence" exists or claimed, it is open to scientific challenge when some important aspect has not been included in the process. As I pointed out seems to be the case in these Hulse-Taylor etc cases used as "evidence" for GR GW claims as main case for period decay rate observed. Hence any evidence is only as good as the scientific reliability of the process leading to interpretations presented as "evidence" for the Hulse-Taylor etc gravitational waves hypothesis/claims.

    It would help a lot if you read and understood what you read without bringing your personal beliefs, opinions, misconceptions and preconceptions into your "reading and understanding". Maybe then you would not keep repeating your assertions that they covered what I pointed out they have not actually covered in this case. If you do find papers that show they have done so (actually done so, and not you misunderstanding that they did so when they haven't) then I would be very grateful if you linked it, paddoboy. Until that happens, thanks anyway. Best.
     
    Last edited: Aug 18, 2016
    dumbest man on earth likes this.
  2. Google AdSense Guest Advertisement



    to hide all adverts.
  3. paddoboy Valued Senior Member

    Messages:
    27,543
    I keep saying that because its fact and you keep continually repeating yourself without any citation, link or reference to support your stance....
    As with the aLIGO confirmation/s all contingencies were considered.
    The Hulse Taylor Binary Pulsar system was supported by overwhelmingly evidence of gravitational radiation, not withstanding your apparently "lone" unsupported, hypothetical suggesting any other possibility.

    Please Register or Log in to view the hidden image!

    I suggest that [1] you were not a party to any of the research over 25 years since this momentous discovery was awarded the Nobel prize, and the 20 odd years investigative/research before that.
    Of course it would have been and was considered!

    Please Register or Log in to view the hidden image!


    It certainly is despite your opposition. This is the same sort of logic you applied in denying DM.
    Again as much as you chose to ignore, support your claim...links, citations, references...

    DM actually illustrates your misunderstandings re standard 21st century cosmology. I could raise many other issues also supporting your continued misunderstandings...cosmological redshift comes to mind.
    I certainly do not want to put words in your mouth [god forbid!!!!] but you seem to be saying the expert professionals since this discovery are incompitent.
    How about the incompitence of public science forum members with agendas?
    A possibility worth considering, don't you agree?

    Please Register or Log in to view the hidden image!




    Please Register or Log in to view the hidden image!

    You keep repeating yourself and you keep doing it without any links of support, citations or references.

    Let me repeat myself....
    The Pulse Taylor Binary Pulsar system was discovered in 1974: That's more than 40 years ago.
    They were awarded the Nobel prize in 1993, near 20 years of reputable scientific research later.
    Many papers have been written up on the discovery, and many papers have considered all contingencies.
    Those are the facts my friend.
     
  4. Google AdSense Guest Advertisement



    to hide all adverts.
  5. expletives deleted Registered Senior Member

    Messages:
    410
    paddoboy:

    I only had to point out that the papers you link and depend on have not treated the point I raised. No more is required from me. If you want to dispute my point, as observed even re your linked references, then please just show where any relevant papers actually properly and exhaustively quantifiably treated the actual point I raised. Unless and until you can do that, and not just repeatedly post your own irrelevant misunderstood readings, beliefs and assertions, then this conversation is not going anywhere useful. In which case maybe we should call it a day between us on this matter, paddoboy? If so, then thanks again for your attention and efforts to date. Best.
     
    Last edited: Aug 19, 2016
    dumbest man on earth likes this.
  6. Google AdSense Guest Advertisement



    to hide all adverts.
  7. paddoboy Valued Senior Member

    Messages:
    27,543
    Wrong on two counts. [1] The paper/s, both of them certainly do discuss, and thus aware of magnetic fields, [2] Plenty more is required of you...links, citations, references supporting whatever it is you are suggesting.

    That's twice now you have called it a day.

    Please Register or Log in to view the hidden image!


    Your assertions, beliefs and hypothetical claims, are irrelevant, as the papers do cover magnetic fields, one in particular, more so than the other.
    As PhysBang had to comment in the DM thread, you can keep denying something as much as you like..it won't change the facts though.
    Now again, please supply a citation supporting any concept of magnetic fields possibly being the cause of the orbital degradation of the binary pulsars.
    The onus is certainly on you, as it was in the DM thread and other threads also.
     
  8. paddoboy Valued Senior Member

    Messages:
    27,543
    http://www.jb.man.ac.uk/distance/frontiers/pulsars/section5.html

    extract:
    There is also a very remarkable interaction between the radio emissions of the two pulsars, which provides the most striking example of magnetospheric physics apart from the terrestrial magnetosphere itself. A powerful wind from one pulsar distorts the magnetosphere surrounding the other, forming a comet-like shape closely resembling the terrestrial magnetosheath and with many of its characteristics (see Figure 5). The diameter of the orbit, determined through radio timing, is 2.9 light-seconds, and the orbital plane lies very nearly in the line of sight; the timing-derived dynamics give an inclination within 2 degrees of 90 degrees. The line of sight to each pulsar therefore passes within 3000 km of its partner. Although the radius of a neutron star is only 10 or 11 km, every pulsar is surrounded by a co-rotating ionised and very energetic magnetosphere, which for an isolated pulsar extends to the velocity-of-light cylinder radius at which co-rotation would require a velocity of c. The magnetosphere of the millisecond pulsar, A, extends to a radius of 1084 km, while that of the long-period pulsar, B, should extend to 132,000 km, well beyond the line of sight to A at superior conjunction. A should therefore be eclipsed by B for several minutes, and an eclipse is indeed seen in every orbit. But the eclipse lasts only 30 seconds, corresponding to an 18,000 km movement of the line of sight to A across pulsar B. This was the first observation of the magnetosphere of any pulsar, but with a radius of only 9,000 km it was much smaller than expected.

    We therefore see a strong influence of pulsar A on pulsar B, distorting its magnetosphere and preventing us from seeing its pulses over much of its orbit, while B's distorted magnetosphere obstructs the line of sight to A at superior conjunction. A's magnetosphere is smaller and does not cross the line of sight; it appears to be unaffected by the proximity of B. This is not surprising; the rapidly rotating A is in fact the more powerful of the two, since its energy loss by spin down is 3000 times greater (the spin down rate of both pulsars is easily measured). The detailed mechanism of energy loss from pulsars has been under discussion for many years. Originally it was thought to be simple magnetic dipole radiation at the rotation period, together with an approximately equal energy density in particles flowing out from the magnetic poles. Observations of energetic nebulae near some pulsars, notably the Crab pulsar, can however only be explained as the result of a pulsar wind streaming out along the rotation axis. The outflow apparently changes character at some distance from the pulsar, but up to now the composition close to the pulsar has been unobservable. The double pulsar binary offers the first opportunity of observing the outflow close to a pulsar.

    The geometry shown in Figure 5 might apply equally well (with some changes in parameters) to the terrestrial magnetosphere and magnetopause. Around the orbit, this cometary tail points away from A, so that our line of sight to B passes through the compressed head when it is furthest away from us, and through the tail when it is closest. The orbital phases of B at which it is observable are between these two positions, when it is seen from either side of the tail. At the eclipse, pulses from A are blocked by the nose and the front half of the 'comet'. The front half of the magnetosheath is therefore opaque to radio waves, probably through synchrotron absorption. The shape of the magnetosheath seems to be well represented as a comet, although rotation of B within the magnetosheath distorts the surface, creating cusps similar to those found in the terrestrial magnetosheath, and giving the complex variations of absorption seen during the eclipse of A.
    ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::


    Obviously magnetic fields and their relationship with Pulsars are well known and particularly that would doubly apply with binary systems.
    In essence the effects of interacting magnetic fields re binary pulsars and orbital degradation is minimal at best and totally insignificant at worst.
    Perhaps our friend disputing this point is unaware that light/photons will warp/curve spacetime, due soley to their momentum....very rarely mention, as the warping/curving effect is insignificant and infinitesemal, and just not worth mentioning.

    The weird thing in this exchange is ed has mentioned the scientific method a couple of times, yet he [1] refuses to accept the onus of responsibility as to his alternative claim as his, and [2] suggests the incumbent theory as invalid while refusing to supply links, references and citations supporting his hypothetical.

     
  9. paddoboy Valued Senior Member

    Messages:
    27,543
  10. expletives deleted Registered Senior Member

    Messages:
    410
    paddoboy:

    No one ever said that magnetic fields were not considered. I already said magnetic fields were covered as usual as part of the individual NS parameters.

    That was not the point I raised.

    The point I raised was that the mutual interactions between the two NSs magnetic fields was not considered/quantified as to braking potential for the Binary orbital period.

    To stress the difference: Consideration of the individual NS' magnetic fields themselves is not at issue; rather it is the interaction between the two NSs' magnetic fields that is at issue because it hasn't been covered in any relevant paper I have seen.

    They are different considerations. I can't say it any clearer to you. Do you understand now what is the actual issue, paddoboy?

    ON EDIT: Please stop trying to irrelevantly bring PhysBang and/or Dark Matter into this Hulse-Taylor etc discussion we are having. Thanks.
     
    Last edited: Aug 19, 2016
    dumbest man on earth likes this.
  11. paddoboy Valued Senior Member

    Messages:
    27,543
    Do you understand that you are continually avoiding the points I made?
    Do you understand the point that magnetic fields were considered? along with all other contingencies?
    Do you understand the point that as yet you refuse to support your stance and hypothetical claim?
    Do you understand how that reflects on the credibility of your posts?
    Do you understand that the onus is on you to support your hypothetical? Because at this stage, its just a story.
    Do you understand that the scientists that researched this discovery are professional? Are you?

    Please Register or Log in to view the hidden image!

     
  12. paddoboy Valued Senior Member

    Messages:
    27,543

    Please Register or Log in to view the hidden image!


    Do you really believe that they can discuss magnetic fields and yet ignore how they interact?
    We are dealing with professionals here and all contingencies were considered and they had 20 years to do it in before the Nobel was awarded.
    Other issues as I chose to raise them are to purely illustrate the nature of your claims in general and your refusal to follow the scientific method. Sorry if that offends you.
     
  13. expletives deleted Registered Senior Member

    Messages:
    410
    paddoboy:

    Answering above respectively:

    You have made no points, merely repeated your assertions and opinions which contradict the content of the referenced paper.

    The only magnetic field considerations were for the individual NSs in the Binary system, not the braking effect of said magnetic fields interacting between the two NSs in that Binary system.

    My posts have been on point and on science and supported by the content of your own linked paper, whereas your posts have been in contradiction to your own linked paper contents; the credibility lays not in your posts so far on this.

    What hypothetical? Neutron Star Magnetic fields exist (as was already understood); and any mutual interaction between such magnetic fields in Hulse-Taylor etc Binaries must have some EM friction/retardation etc effect on their Binary orbital period. All the papers I have so far seen do not exhaustively and quantifiably treat this aspect (as your own linked paper again confirms). So all I have done is point that out, based on known science and expected effects from same. Where is the hypothetical in that scientifically tenable real magnetic phenomena based observation and the lack of its treatment in the relevant papers?

    Paddoboy, please try to separate your personal beliefs and misunderstood misattributions from the actual facts in the paper and in my observation as made, based on known science not "just a story". Thanks. Best.
     
    Last edited: Aug 19, 2016
    dumbest man on earth likes this.
  14. expletives deleted Registered Senior Member

    Messages:
    410
    paddoboy:

    Please show where they have done so.

    So far I have only seen the magnetic considerations in the context of mass-exchange and gravitational effects on a individual NS's magnetic field and spin evolutions.

    I have not seen any paper yet that treats and quantifies and considers the mutual interaction between the magnetic fields of the TWO NS in that binary as regards the potential for EM-braking/radiation as possible cause for system energy loss or for orbital period decay rate observed in the relevant Binary system.

    Paddoboy, you repeating that many yrs have passed, does not answer the question of: where in those papers, or in all that time since, have they considered the aspect I raised?

    Please show me where that has happened and there will be no need to discuss this further. Thanks. Best.
     
    Last edited: Aug 19, 2016
    dumbest man on earth likes this.
  15. paddoboy Valued Senior Member

    Messages:
    27,543
    False:
    [1] The paper/s discusses magnetic fields, and rotational aspects.
    [2] The onus is on you to show something, any citation, any link, any references supporting your stance: You cannot do that because you are wrong.
    [3] Since you refuse to entertain the fact that the onus is on you to support your claim, you are also ignoring the scientific method and peer review: But you did that also in the DM and redshift thread/s.
    Your posts [1] misinterpret the paper/s, [2]Ignore the scientific method and the fact that the onus is on you to support your hypothetical.
    Yep, minimal effect if any, as I explained to you: No where near the effects of that which you so fervently disgree with, and that which has been confirmed: Gravitational waves.
    All the paper/s so far have researched and discussed professionally without burden of an agenda, magnetic fields.
    Your highly questionable hypothetical is that you seem to believe they did not allow contingencies for that, and that it would make any markable difference.
    My personal beliefs do not come into it. And there you go again being so hypocritical.

    Please Register or Log in to view the hidden image!


    I adhere to the scientific method and peer review while you disregard both it seems. Again, any misattributions are yours along with your obtuseness in ignoring requests to support your hypothetical claim.
    Yes, all you have is a story...and until you can supply some reputable citation, from a professional, it remains a "story".
     
  16. expletives deleted Registered Senior Member

    Messages:
    410
    paddoboy:

    Your misunderstanding subsists in the fact you cannot discern the difference between the aspects considered and the aspects not yet considered. What you refer to is just the ordinary NS parameters. What I refer to is the Binary system parameters as affected by mutual interaction of the Binary NS magnetic fields. Just repeating all the known magnetic field, spin-orbital momentum and other individual NS parameters doesn't address what I am talking about. Until you realize where your misunderstanding lay, you will misunderstand what you are reading. Take a break. Think about it all overnight. Then see if your subconscious can wean you off that misunderstanding which makes all your other comments on this just so much irrelevant repetition. Good night. Best.
     
    dumbest man on earth likes this.
  17. paddoboy Valued Senior Member

    Messages:
    27,543
    What makes a mockery even more of the unsupported claims being made, is the following from
    https://www.parkes.atnf.csiro.au/people/sar049/papers/JMS_17.pdf

    2. A unique test-bed for Relativistic Gravity Due to their strong gravitational fields and rapid motions, the double neutron star binaries exhibit large relativistic effects. PSR J0737−3039A/B promises to ensure the best tests of General relativity and other theories of gravity than ever before. Tests can be performed when a number of relativistic corrections to the Keplerian description of the orbit (the so-called post-Keplerian, hereafter PK, parameters) can be measured.
    and this.....
    Such tests have been possible to date in only two double neutron star systems, PSR B1913+16 (Taylor & Weisberg 1989) and PSR B1534+12 (Stairs et al. 2002). For PSR B1913+16, the relativistic periastron advance, ˙ω, the orbital decay due to gravitational wave damping, P˙ b, and the gravitational redshift/time dilation parameter, γ, have been measured, providing a total of three PK parameters. For PSR B1534+12, Shapiro delay, caused by passage of the pulses through the gravitational potential of the companion, is also visible, since the orbit is seen nearly edge-on. This results in two further PK parameters, r (range) and s (shape) of the Shapiro delay. However, the observed value of P˙ b requires correction for kinematic effects, so that PSR B1534+12 provides four PK parameters usable for precise tests (Stairs et al. 2002). With a intense campaign of regular timing observations started immediately after the discovery, we have measured A’s ˙ω and γ and have also detected the Shapiro delay in the pulse arrival times of A due to the gravitational field of B. This provides four measured PK parameters, resulting in a MA-MB plot (Fig. 1) through which we can test the predictions of general relativity. In particular the current data, spanning only 10 months of observations, indicate an agreement of the observed with the expected Shapiro parameter s of sobs/sexp= 1.00007±0.00220 where the uncertainties are likely to decrease quickly.
    """"""""""""""""""""""""""""""""""""""""
     
  18. paddoboy Valued Senior Member

    Messages:
    27,543

    Please Register or Log in to view the hidden image!


    So you still have nothing? No citation? No link? No reference? Just a story?
    Just as it turned out in the DM thread...and the redshift thread...other aspects of your unsupported story telling.

    Please Register or Log in to view the hidden image!


    OK, I'll continue with science.
     
  19. paddoboy Valued Senior Member

    Messages:
    27,543
    The following is just a simple explanation of the mechanics behind the Hulse Taylor binary Pulsar system with an accompanied graph.


    http://www.astro.cardiff.ac.uk/research/gravity/tutorial/?page=3thehulsetaylor

    . The Hulse-Taylor Pulsar - Evidence of Gravitational Waves

    In this current "pre-detection" era it can be difficult to convince those who are not overly familiar with the theory of general relativity that gravitational waves really do exist. Fortunately, the Hulse-Taylor Pulsar (PSR 1913+16) provides firm evidence of a binary system actually emitting gravitational waves!

    In 1974 Russell Hulse and Joseph Taylor discovered the signal of a pulsar using the Arecibo radio telescope. The pulsar had a period of 59 milliseconds. Further measurements showed that the orbital period varied in a repetitive manner over a period of 7.75 hours. This meant the the pulsar must be in orbit with another star.

    Over the years the period of the pulsar has been measured to high accuracy. General relativity tells us that a binary system will emit energy as gravitational waves and eventually the two objects will inspiral towards each other and merge. As the system evolves towards this merger the period of the orbit will gradually decrease.

    Please Register or Log in to view the hidden image!


    The figure (from Weisberg and Taylor (2004)) shows the cumulative shift of periastron time for PSR 1913+16. This shows the decrease of the orbital period as the two stars spiral together. Although the measured shift is only 40 seconds over 30 years, it has been very accurately measured and agrees precisely with the predictions from Einstein's theory of General Relativity. The observation is regarded as indirect proof of the existence of gravitational waves. Indeed, the Hulse-Tayor pulsar is deemed so significant that in 1993 its discoverers were awarded the Nobel prize for their work.
     
  20. The God Valued Senior Member

    Messages:
    3,546
    OP is suggestive of some aether like stuff...then MM experiment also requires re visit. Then I am sure implications on GR also needs to be looked into.
     
  21. expletives deleted Registered Senior Member

    Messages:
    410
    I have been advised by a member of staff to sort out this matter between me and paddoboy in open forum. So I make this relevant summary and appeal for paddoboy's sober consideration and (ever hopefully) relevant scientific and factual response sans personal commentary and attributions of motives, conspiracies, agendas or the like distractions from the actual point. On that understanding, I post the following:

    Paddoboy:

    Plazma Inferno! posted an OP suggesting that a Neutron Star's ROTATIONAL parameters could possibly be affected by a process not involving hypothetical gravitational waves.

    In that non-gravitational wave vein, I posted an observation of my own, based on known science about Neutron Stars' extreme magnetic fields, and suggested that a NS-BINARY system's ORBITAL PERIOD DECAY RATE could also possibly be explained by a process involving strong mutual braking interactions between the two NS's magnetic fields; again not involving hypothetical gravitational waves.

    In my comments I observed that the Hulse-Taylor etc papers have never properly considered let alone quantified the possibly strong braking effect which must arise between any two strong magnetic features in close proximity and extreme motions about each other, in this case the two Neutron Stars and their mutual interactions via their respective extreme magnetic fields, explicitly excluding any mention of hypothetical gravitational waves.

    In response to my comments you, paddoboy, effectively just repeatedly asserted that the papers did consider and quantify the NS extreme magnetic field interactions between the TWO NSs in the Binary for their braking effects and their impact on Binary Orbital Period decay rates.

    I then responded that I had not come across any such papers considering let alone quantifying such mutual NS magnetic Interactions as to their braking potential for Binary Orbit Period decay; so I asked you to prove your blanket repeated assertion to the contrary, by posting a link or quote to/from any such relevant paper which explicitly deals with the aspect I mentioned.

    You responded by posting more links and quotes to/from papers which dealt with everything but what I mentioned.

    So, can you please explain succinctly why you keep repeatedly linking papers and making assertions about irrelevant individual NS magnetic parameters and hypothetical gravitational waves, when the topic and my discussion is strictly about possible explanations involving processes explicitly excluding hypothetical gravitational waves?



    But anyhow:

    Paddoboy, this has gone on far too long when it could have been concluded almost immediately one way or the other if you had been able to support your assertion that it was properly considered etc.

    I now respectfully request again that you please either:

    1) Immediately prove what you asserted; and do so without repeating irrelevant links and quotes about individual NS magnetic parameters and hypothetical gravitational wave claims and verbiage which does not answer what I observed regarding the mutual interactions between the two NSs extreme magnetic fields and its possibly strong braking effects on the Binary Orbital Period decay observed.

    OR

    2) Concede that your assertion was not correct based on what your own references have contained so far.


    Paddoboy, I would sincerely and gratefully appreciate your honest and scientific reply to the point so that we can politely and unambiguously conclude our exchange on the relevant matter one way or the other for the time being.


    Thanks. Best.
     
    Last edited: Aug 19, 2016
    dumbest man on earth likes this.
  22. paddoboy Valued Senior Member

    Messages:
    27,543
    Yep, a speculative proposition akin to hawking radiation, but NOT EXCLUDING GRAVITATIONAL WAVES:.
    Yep, agreed, and obviously discussed and taken into consideration, along with gravitational waves, NOT EXCLUDING GRAVITATIONAL WAVES!
    [1] Obviously you do not know that any braking effect by magnetic fields were not considered:
    [2] Magnetic fields were discussed as evidenced by reputable papers I posted.
    [3] Gravitational waves are not "hypothetical": They have now been observed and categorised twice.
    Yes, and in response to my evidenced back comments, you simply parroted unsupported scenarios that you were unable to reference or supply any citation supporting your hypothetical assumption. [ that being that magnetic fields alone would count for all orbital degradation, and suggesting that gravitational waves are just hypothetical even after aLIGO]
    This is for science. Post any of your other hypothetical stories in alternative section please.
    [1] You are making the alternative assumption: The onus is on you to support your story.
    [2] Magnetic fields were certainly discussed.
    [3] And just as obviously, magnetic fields having some orbital degradation effect along with the more pronounced gravitational waves which are now confirmed.
    I responded with evidence that magnetic fields were discussed, in light of your continued parroting that magnetic fields alone could count for the amount of orbital degradation seen.
    Sure!
    The Hulse Taylor binary Pulsar system was discovered and researched in 1974 and the Nobel prize awarded in 1993: It's reasonably safe to assume that in that 40 odd years research continued by professional scientists, and since then gravitational waves have been confirmed twice by aLIGO.
    For you to suggest that I am totally disregarding your magnetic field argument is a LIE. I'm saying that I agree they are a possibility, and obviously were taken into account also, but that gravitational waves were the main means by which the energy from orbital degradation was taken away.
    You on the other hand, ever so underhandedly are trying to promote your single obsession that gravitational waves are invalid.


    Please Register or Log in to view the hidden image!

    But anyhow they are confirmed. I hope that has removed this bee from your bonnet.
    It's your assertion that needs to be supported as I have told you.
    Please do that now or cease your tiring hypothetical nonsense.
    I conclude that you are obviously looking into a mirror and should apply all your thoughts, requests and demands to yourself.
    As a member of staff has advised me, you are an annoying pedant.
     
  23. Dr_Toad It's green! Valued Senior Member

    Messages:
    2,527
    Aren't y'all aware that this specious discussion relies on cargo-cult popscience misunderstanding of what Einstein really meant in 1905 when he published his paper on the photoelectric quantization of radiation?

    Why isn't the ether responsible for frame-dragging, instead of all this woo about virtual pairs?

    Honest to God, sometimes this site just pisses me off with the sophomoric backstory sideshow bullshit.

    Exchemist, you are more of a physicist than any of the other respondents to this thread, so please don't apologize.
     

Share This Page