Neutron Stars—Thermal Emitters

Abstract

Confronting theoretical models with observations of thermal radiation emitted by neutron stars is one of the most important ways to understand the properties of both, superdense matter in the interiors of the neutron stars and dense magnetized plasmas in their outer layers. Here we review the theory of thermal emission from the surface layers of strongly magnetized neutron stars, and the main properties of the observational data. In particular, we focus on the nearby sources for which a clear thermal component has been detected, without being contaminated by other emission processes (magnetosphere, accretion, nebulae). We also discuss the applications of the modern theoretical models of the formation of spectra of strongly magnetized neutron stars to the observed thermally emitting objects.

Appendix: The Effects of Finite Atomic Masses

In this Appendix, we give a brief account of the effects of motion of atomic nuclei in strong magnetic fields on the quantum-mechanical characteristics of bound species and the ionization equilibrium of partially ionized plasmas (for a more detailed review, see Potekhin 2014)

1.1 A.1 The Finite-Mass Effects on Properties of Atoms

An atomic nucleus of finite mass, as any charged particle, undergoes oscillations in the plane (xy) perpendicular to B, which are quantized in the ion Landau levels. In an atom or a molecule, these oscillations are entangled with the electron motion. Therefore the longitudinal projections of the orbital moments of the electrons and the nucleus are not conserved separately. Different atomic quantum numbers correspond to different oscillation energies of the atomic nucleus, multiple of its cyclotron energy ħω _ci. As a result, the energy of every level gets an addition, which is non-negligible if the magnetic parameter γ is not small compared to the nucleus-to-electron mass ratio m _i/m _e.

For the hydrogen atom and hydrogenlike ions, a conserved quantity is ħs, which corresponds to the difference of longitudinal projections of orbital moments of the atomic nucleus and the electron, and the sum N+s plays role of a nuclear Landau number, N being the electron Landau number. For the bound states in strong magnetic fields, N=0, therefore the nuclear oscillatory addition to the energy equals sħω _ci. Thus the binding energy of a hydrogen atom at rest is

$$ E_{s\nu} = E_{s\nu}^{(0)} - \hbar \omega_\mathrm{ci}s, $$

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where $E_{s\nu}^{(0)}$ is the binding energy in the approximation of non-moving nucleus. It follows that the values of s are limited for the bound states. In particular, all bound states have s=0 at B>6×10¹³ G.

The account of the finite nuclear mass is more complicated for multielectron atoms. Al-Hujaj and Schmelcher (2003) have shown that the contribution of the nuclear motion to the binding energy of a non-moving atom equals ħω _ci S(1+δ(γ)), where (−S) is the total magnetic quantum number of the atom, and |δ(γ)|≪1.

The astrophysical simulations assume finite temperatures, hence thermal motion of particles. The theory of motion of a system of point charges in a constant magnetic field was reviewed by Johnson et al. (1983). The canonical momentum P is not conserved in a magnetic field. A relevant conserved quantity is pseudomomentum

$$ \boldsymbol{K}=\boldsymbol{P}+\frac{1}{2c}\boldsymbol{B}\times\sum_i q_i \boldsymbol{r_i}. $$

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If the system is electrically neutral as a whole, then all Cartesian components of K can be determined simultaneously (i.e., their quantum-mechanical operators commute with each other). For a charged system (an ion), one can determine K ² simultaneously with either K _x or K _y , but K _x and K _y do not commute. The specific effects related to collective motion of a system of charged particles are especially important in NS atmospheres at γ≫1. In particular, so called decentered states may become populated, where an electron is localized mostly in a “magnetic well” aside from the Coulomb center.

For a hydrogen atom, K=P−(e/2c)B×r, where r connects the proton and the electron. Early studies of this particular case were done by Gor’kov and Dzyaloshinskiĭ (1968), Burkova et al. (1976), Ipatova et al. (1984). Numerical calculations of the energy spectrum of the hydrogen atom with an accurate treatment of the effects of motion across a strong magnetic field were performed by Vincke et al. (1992) and Potekhin (1994). Bound-bound radiative transitions of a moving H atom in a plasma were studied by Pavlov and Potekhin (1995), and bound-free transitions by Potekhin and Pavlov (1997).

Figure 5 shows the energies, oscillator strengths, and photoionization cross-sections of a hydrogen atom moving in a magnetic field with γ=1000. The reference point is taken to be the sum of the zero-point oscillation energies of free electron and proton, (ħω _c+ħω _ci)/2.Therefore the negative energies in Fig. 5 a correspond to bound states (E _sν =−E>0). At small transverse pseudomomenta K _⊥, the energies of low levels in Fig. 5a exceed the binding energy of the field-free hydrogen atom (1 Ry) by an order of magnitude. However, the binding energy decreases with increasing K _⊥, and it can become negative for the states with s≠0 due to the term ħω _ci s in Eq. (41). Such states are metastable. In essence, they are continuum resonances. Note that the transverse atomic velocity equals ∂E/∂ K, therefore it attains a maximum at the inflection points (K _⊥=K _c) on the curves in Fig. 5a and decreases with further increase of K _⊥, while the average electron-proton distance continues to increase. At K _⊥>K _c the atom goes into the decentered state, where the electron and proton are localized near their guiding centers, separated by distance $r_{*} = (a_{\mathrm{B}}^{2}/\hbar)K_{\perp}/\gamma$.

Fig. 5

(a) energies, (b) oscillator strengths, and (c) photoionization cross-sections for a hydrogen atom moving across magnetic field B=2.35×10¹² G. Energies of states |s,0〉 (solid curves) and |0,ν〉 (dot-dashed curves) are shown as functions of the transverse pseudomomentum K _⊥ (in atomic units). The heavy dots on the solid curves are the inflection points at K _⊥=K _c. The K _⊥-dependence of oscillator strengths (b) is shown for transitions from the ground state to the states |s,0〉 under influence of radiation with polarization α=+1 (solid curves) and α=−1 (dashed curves), and also for transitions into states |0,ν〉 for α=0 (dot-dashed curves). Cross sections of photoionization (c) under the influence of radiation with α=+1 (solid curves), α=−1 (dashed curves), and α=0 (dot-dashed curves) are shown for the ground state as functions of the photon energy in Ry (the upper x-scale) and keV (the lower x-scale) at K _⊥=20 a.u. (the right curve), K _⊥=200 a.u. (the middle curve), and K _⊥=1000 a.u. (the left curve of every type)

Figure 5b shows oscillator strengths for the main transitions from the ground state to excited discrete levels. Since the atomic wave-functions are symmetric with respect to the z-inversion for the states with even ν, and antisymmetric for odd ν, only the transitions that change the parity of ν are allowed for the polarization along the field (α=0), and only those preserving the parity for the orthogonal polarizations (α=±1). For the atom at rest, in the dipole approximation, due to the conservation of the z-projection of the total angular momentum of the system, absorption of a photon with polarization α=0,±1 results in the change of s by α. This selection rule for a non-moving atom manifests itself in vanishing oscillator strengths at K _⊥→0 for s≠α. In an appropriate coordinate system (Burkova et al. 1976; Potekhin 1994), the symmetry is restored at K _⊥→∞, therefore the transition with s=α is the only one that survives also in the limit of large pseudomomenta. But in the intermediate region of K _⊥, where the transverse atomic velocity is not small, the cylindrical symmetry is broken, so that transitions to other levels are allowed. Thus the corresponding oscillator strengths in Fig. 5b have maxima at K _⊥≈K _c. Analytical approximations for these oscillator strengths, as well as for the dependences of the binding energies E _sν (K _⊥), are given in Potekhin (1998).

Figure 5c shows photoionization cross-sections for hydrogen in the ground state as functions of photon energy at three values of K _⊥. The leftward shift of the ionization threshold with increasing K _⊥ corresponds to the decrease of the binding energy that is shown in Fig. 5a, while the peaks and dips on the curves are caused by resonances at transitions to metastable states |s,ν;K〉 with positive energies (see Potekhin and Pavlov 1997, for a detailed discussion).

Quantum-mechanical calculations of the characteristics of the He⁺ ion that moves in a strong magnetic field are performed by Bezchastnov et al. (1998), Pavlov and Bezchastnov (2005). The basic difference from the case of a neutral atom is that the ion motion is restricted by the field in the transverse plane, therefore the values of K ² are quantized (Johnson et al. 1983). Clearly, the similarity relations for the ions with nonmoving nuclei (Sect. 4.2.1) do not hold anymore.

Currently there is no detailed calculation of binding energies, oscillator strengths, and photoionization cross-sections for atoms and ions other than H and He⁺, arbitrarily moving in a strong magnetic field. For such species one usually neglects the decentered states and uses a perturbation theory with respect to K _⊥ (e.g., Mori and Hailey 2002, Medin et al. 2008). This approximation can be sufficient for simulations of relatively cool atmospheres of moderately magnetized NSs. A condition of applicability of the perturbation theory for an atom with mass m _a=Am _u requires T/E ⁽⁰⁾≪m _a/(γm _e)≈4A/B ₁₂ (Potekhin 2014). If B≲10¹³ G and T≲10⁶ K, it is satisfied for low-lying levels of carbon and heavier atoms.

1.2 A.2 The Finite-Mass Effects on the Ionization Equilibrium and Thermodynamics

Since quantum-mechanical characteristics of an atom in a strong magnetic field depend on the transverse pseudomomentum K _⊥, the atomic distribution over K _⊥ cannot be written in a closed form, and only the distribution over longitudinal momenta K _z remains Maxwellian. The first complete account of these effects has been taken in Potekhin et al. (1999) for hydrogen atmospheres. Let p _sν (K _⊥)d² K _⊥ be the probability of finding a hydrogen atom in the state |s,ν〉 in the element d² K _⊥ near K _⊥ in the plane of transverse pseudomomenta. Then the number of atoms in the element d³ K of the pseudomomentum space equals

$$ \mathrm{d}N(\boldsymbol{K}) = N_{s\nu} \frac{\lambda_\mathrm{a} }{2\pi\hbar} \exp \biggl(- \frac{K_z^2}{2m_\mathrm{a} T} \biggr) p_{s\nu }(K_\perp) \mathrm{d}^3K, $$

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where m _a is the mass of the atom, λ _a=[2πħ ²/(m _a T)]^1/2 is its thermal wavelength, and N _sν =∫dN _sν (K) is the total number of atoms with given discrete quantum numbers. The distribution N _sν p _sν (K _⊥) is not known in advance, but should be calculated in a self-consistent way by minimization of the free energy including the nonideal terms. It is convenient to define deviations from the Maxwell distribution with the use of generalized occupation probabilities w _sν (K _⊥). Then the atomic contribution to the free energy equals (Potekhin et al. 1999)

$$ F_\mathrm{id}^\mathrm{at}+F_\mathrm{int}^\mathrm{at} = T \sum_{s\nu} N_{s\nu} \int\ln \biggl[n_{s\nu} \lambda_\mathrm{a}^3 \frac{w_{s\nu}(K_\perp) }{ \exp(1) \mathcal{Z}_{s\nu}} \biggr] p_{s\nu}(K_\perp) \mathrm{d}^2K_\perp, $$

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where

$$ \mathcal{Z}_{s\nu} = \frac{ \lambda_\mathrm{a}^2 }{ (2\pi\hbar^2) } \int_0^\infty w_{s\nu}(K_\perp)\mathrm{e}^{E_{s\nu}(K_\perp)/ T} K_\perp \mathrm{d}K_\perp. $$

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The nonideal part of the free energy that describes atom-atom and atom-ion interactions and is responsible for the pressure ionization has been calculated by Potekhin et al. (1999) with the use of the hard-sphere model. The plasma model included also hydrogen molecules H₂ and chains H _n , which become stable in the strong magnetic fields. For this purpose, approximate formulae of Lai (2001) have been used, which do not take full account of the motion effects, therefore the results of Potekhin et al. (1999) are reliable only when the molecular fraction is small.

This hydrogen-plasma model underlies thermodynamic calculations of hydrogen atmospheres of NSs with strong and superstrong magnetic fields (Potekhin and Chabrier 2003, 2004). Mori and Heyl (2007) applied the same approach with slight modifications to strongly magnetized helium plasmas. One of the modifications was the use of the plasma microfield distribution from Potekhin et al. (2002) for calculation of the K _⊥-dependent occupation probabilities. Mori and Heyl considered atomic and molecular helium states of different ionization degrees. Their treatment included rotovibrational molecular levels and the dependence of binding energies on orientation of the molecular axis relative to B. The K _⊥-dependence of the energy, E(K _⊥), was described by an analytical fit, based on an extrapolation of adiabatic calculations at small K _⊥. The effects of motion of atomic and molecular ions were not considered.