Cyclotron-line variation in accreting magnetic neutron stars is now well understood!
Neutron stars are the densest objects and the strongest magnets in the Universe. The Earth and the Sun are also magnets, with strengths of about one unit (called Gauss in cgs units). The magnets that we have been able to create in the lab are some tens of thousands Gauss. It is therefore impressive to have measured (in 1977) that neutron stars have magnetic fields 12 to 13 orders of magnitude larger than that of the Sun.
The measurement of the magnetic field of neutron stars is equally impressive. It was shown by Landau (1930) that electrons in a strong magnetic field behave like harmonic oscillators and therefore their energy levels are quantized and equidistant in energy. The electrons at or near the surface of a neutron star are in their ground state, as collisions are not strong enough to excite them. If a continuum of X-ray photons shines on them, they absorb photons near the energy that is called “cyclotron energy” and populate the first excited state. These absorbed photons are missing from the observed spectrum and we call this feature a “cyclotron line”. In other words, if we observe the X-ray spectrum from a neutron star and this spectrum exhibits a cyclotron line, we can infer the magnetic-field strength at the place of emission of the spectrum, because the cyclotron-line energy is proportional to the magnetic-field strength and the proportionality constant is known.
We are familiar with spectral lines formed by the chemical elements on Earth and in the Universe, but they are at lower energies. No element can produce an absorption line at several tens of keV, so we are sure that the cyclotron line observed from X-ray pulsars is due to the local magnetic field.
Nick Loudas, now a PhD student at Princeton University, undertook as a second-year undergraduate student in Crete to understand the formation of the cyclotron line in X-ray pulsars and its variation with the X-ray luminosity of the source. An arduous task, but, by the time that he finished his Master’s in Crete, he proved quantitatively what was proposed in 1976, namely that the cyclotron line in X-ray pulsars is formed in the accretion shock above the surface of a neutron star. Then, taking into account the variation of the shock height with luminosity and all the relativistic effects involved (Special and General), he was able to reproduce quantitatively the cyclotron-line variation with luminosity for one X-ray pulsar (V0332+53) for which we have detailed observations. This is shown in the Figure.
The green, dashed line shows the expected variation of the cyclotron-line energy with luminosity, if one considers only the variation of the shock-height with luminosity. The magenta, dot-dashed curve shows the expected variation of the cyclotron-line energy if one also accounts for the gravitational redshift of the line. Finally, the black, solid line shows the expected variation of the cyclotron-line energy, if all the effects due to Special and General Relativity are taken into account. It is evident that the black line fits well the observed data (blue dots) from the source V0332+53. Notice that a true cyclotron-line energy of about 30 keV on the neutron-star surface can be mistaken as about twice this value, if the General and Special Relativistic effects are not taken into account.
Articles: “Cyclotron line formation in the radiative shock of an accreting magnetized neutron star”, N. Loudas, N. Kylafis, J. Truemper, 2024, A&A, 685, 95 – May 2024 “A quantitative explanation of the cyclotron-line variation in accreting magnetic neutron stars of super-critical luminosity”, N. Loudas, N. Kylafis, J. Truemper, 2024, A&A, 689, 75 – September 2024