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I wonder whether GWs can be produced and emitted by single neutron stars, since it is known that typically they must be emitted by a binary system of them.

If so, can the source be an isolated cool neutron star or, for instance, a milisecond pulsar? I am an early learner on Neutron Star Physics, so I don't know if any neutron star itself can emit GWs. I once saw an open question (with no answer in the book) which asked if a single person would emit GWs by waving its own arms, and I couldn't help getting more confused.

I am also a bit aware with the quadrupole formula for compact objects, so I cannot see how a single star could contribute nonzero terms to it.

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    $\begingroup$ If the neutron star is spherically symmetric then no. No spherically symmetric distribution/movement of matter can produce gravitational waves. This is due to Birkhoff's theorem. Neutron stars are not perfectly symmetric - but they are very nearly so (due to their immense gravity). So single neutron stars are probably not good sources of gravitational waves. $\endgroup$ – enumaris Jun 20 '18 at 21:51
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    $\begingroup$ @enumaris See the article linked in the answer below. Gravitational waves have not yet been detected from single neutron stars, but researchers are at least trying. $\endgroup$ – Grayscale Jun 20 '18 at 22:50
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    $\begingroup$ So, think about the analogy to electromagnetism. Waves are generated whe things change. Similarly with gravitational waves, there must not just be steady motion, but acceleration, massive ones, like neutron stars spiralling in to each other. Or when eutron stars form, in supernovae: phys.org/news/2017-09-gravitational-stars-supernovae.html Also crucial, is not simply what generates gravitional waves, but detectable ones.. It seems whether there is a threshold energy to emit gravitational waves will depend whether the graviton exists. $\endgroup$ – CriglCragl Jun 20 '18 at 22:53
  • $\begingroup$ @enumaris Just finished Janna Levin's book "Black Hole Blues". Good read - mostly about the people that made LIGO happen. But she did suggest the possibility that neutron "mountains" might exist that for a spinning neutron star could shed off gravitational pulses. I was curious if anyone had done the calculations to support this hypothesis, or if it's total malarkey. What force besides gravity itself could hold neutrons together? The mass of the mountain would be competing with the whole mass of the star ... one would expect the mountain to 'melt'. Right? $\endgroup$ – docscience Jul 31 '18 at 20:13
  • $\begingroup$ @docscience this article will be of interest to you: ligo.org/science/Publication-S6VSR24KnownPulsar (and perhaps also the OP). A "mountain" on a neutron star would probably be sizes of a couple of cm assuming neutron matter (no exotic matter). A "mountain" of this size is not going to emit too much gravitational waves, and no such waves have been so far detected for the known pulsars. $\endgroup$ – enumaris Jul 31 '18 at 20:47
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Yes, single neutron stars can emit gravitational waves if they have sufficient asymmetries.

For some background, an object symmetric about its axis of rotation does not produce gravitational waves. A quote from Hartle's An Introduction to Einstein's General Relativity in the example "A Little Rotation" on page 497 of the textbook:

Axisymmetric rotation in general is an example of a highly symmetric motion that does not produce gravitational radiation.

For clarity, "axisymmetric" means "symmetric about an axis."

To view the details of the calculation, that section of Hartle is a good resource.

However, a rotating neutron star which does not have sufficient symmetry would emit gravitational waves. What's more, as pointed out by @Harti, pulsars must exhibit some sort of asymmetry in order to emit the radiation they do - but the question is whether these asymmetries are such that they will produce detectable gravitational waves.

As a concrete mathematical example of a neutron star that could emit gravitational waves, consider this problem written by an old professor of mine...

A neutron star of mass M and uniform mass density $\rho = 3M/4\pi\bar{R}^3$ has the shape of a slightly nonspherical ellipsoid, and rotates around its shortest $(z^\prime)$ axis with rotation period $P$. The lengths of the three principal axes $x^\prime, y^\prime, z^\prime$ (fixed in the body of the star) are in the ratios $(1+\epsilon) : 1 : (1+\epsilon)^{-1}$, where $\epsilon \ll 1$.

  1. Express the gravitational-wave luminosity of this object $(L_{GW})$ in terms of the quantities given above.
  2. Estimate the root-mean-square value of a component of $h_{ij}^{TT}$ of the strain at distance D from the star in terms of $L_{GW}$. (Ignore the dependence on the angle between the line of sight and the rotation axis.)
  3. Taking $M=1.4M_\odot$, $R = 10 \,\mathrm{km}$, $P = 1 \,\mathrm{ms}$, and $D = 10 \,\mathrm{kpc}$, estimate $\epsilon$ in order that $\langle(h_{ij}^{TT})^2\rangle^{1/2} \approx 10^{-23}$, the approximate current sensitivity of LIGO.

Understanding the solution would require some decent GR knowledge, but the point here is that, in principle, a single rotating body could emit gravitational radiation.

I am not sure if the particular scenario in the above problem is realistic, but it turns out that rotating neutron stars emitting gravitational waves is a realistic thing. See this article on astrobites about hunting for gravitational waves from spinning neutron stars.

A quote from the article that reinforces the above:

However, a neutron star must exhibit certain properties for it to be detectable through gravitational waves. A perfectly spherically symmetric neutron star won’t produce continuous gravitational waves as it rotates – the neutron star needs to maintain a long-lasting asymmetry. More importantly, this asymmetric distortion cannot be aligned with respect to the rotation axis (non-axisymmetric).

One example they mention is if the neutron star has some mountain on its surface.

(Note however that gravitational waves from single neutron stars have not yet been detected.)

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  • $\begingroup$ Please cite the source of the homework problem. Don't just cut and paste random stuff on the internet without attribution. It's rude. $\endgroup$ – Ben Crowell Jun 20 '18 at 23:44
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    $\begingroup$ @BenCrowell It is not from the internet. One of my old professors wrote it, but for purposes of anonymity I did not cite it. $\endgroup$ – Grayscale Jun 21 '18 at 0:16
  • $\begingroup$ @BenCrowell Do you think it adds to the answer or should I just remove it? $\endgroup$ – Grayscale Jun 21 '18 at 0:30
  • $\begingroup$ @Grayscale What was the name of the course you take with that teacher? Do you know from which textbook he could have got inspired for the exercise? I am currently reading Maggiore's and Kokkotas articles. $\endgroup$ – chandrasekhar17 Jun 21 '18 at 18:13
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    $\begingroup$ @chandrasekhar17 It was an intro to GR course, and I think the problem aligns with Hartle's treatment / notation (see chapter 23). $\endgroup$ – Grayscale Jun 22 '18 at 20:59
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In the regime of the quadrupole approximation, a body emits gravitational waves only if there is a change in its quadrupole moment. A perfectly spherical neutron star rotating symmetrically about its axis will not have any change in quadrupole moment and hence will not emit gravitational waves. However if the rotation is not symmetric about the axis, they will certainly emit gravitational waves, the intensity will probably depend on the extent of asymmetry.

Pulsars beam em radiation and hence are not symmetric, therefore expected to produce gravitational radiation. Quoting Schutz (A First Course in General Relativity) - " Stars could radiate gravitational waves if they are not symmetric about the rotation axis. Pulsars are clearly not symmetric, since they beam their radiation somehow. But it is not clear how much mass asymmetry is required to produce the beaming."

Gravitational radiation emitted by binary neutron stars are in the detection range of our ground based detectors (LIGO, VIRGO) and hence the detection was possible. If the frequency of gravitational radiation of a spinning neutron star happens to be in the detection range and if the signal is "loud" enough, we may very well detect them !

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  • $\begingroup$ thanks for summing it up concisely, could you explain in your own words what relation else do pulsar timing array has with detection? I've read a few and that term comes out everywhere $\endgroup$ – chandrasekhar17 Jun 21 '18 at 18:20
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    $\begingroup$ @chandrasekhar17 I am no expert in pulsar timing arrays and hence can give you only a brief overview. Pulsar Timing Array basically monitors the periods of a large number of pulsars spread out in the sky. Note that the pulsars considered are millisecond ones as their period is quite stable which is important for our purpose (being direct detection of GWs). When a GW passes over the earth, there is a deviation from the time when we expect to observe the pulse from a particular pulsar. This time difference is called a residual, a GW will cause correlated residuals in the different pulsars ! $\endgroup$ – Hari Jun 22 '18 at 12:02
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    $\begingroup$ So if we observe correlated residuals in our array of pulsars, that could be because of a GW passing over the earth ! Hence we look for such residuals. Based on the diiference in the values and signs of the residuals for the different pulsars, I think localization of the source should be possible. $\endgroup$ – Hari Jun 22 '18 at 12:05

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