In this paper, we discuss two approximate methods previously suggested for modeling hyperfine spectral line emission for molecules whose collisional transition rates between hyperfine levels are unknown. Hyperfine structure is seen in the rotational spectra of many commonly observed molecules such as HCN, HNC, NH3, N2H+, and C17O. The intensities of these spectral lines can be modeled by numerical techniques such as Λ-iteration that alternately solve the equations of statistical equilibrium and the equation of radiative transfer. However, these calculations require knowledge of both the radiative and collisional rates for all transitions. For most commonly observed radio frequency spectral lines, only the net collisional rates between rotational levels are known. For such cases, two approximate methods have been suggested. The first method, hyperfine statistical equilibrium, distributes the hyperfine level populations according to their statistical weight, but allows the population of the rotational states to depart from local thermal equilibrium (LTE). The second method, the proportional method, approximates the collision rates between the hyperfine levels as fractions of the net rotational rates apportioned according to the statistical degeneracy of the final hyperfine levels. The second method is able to model non-LTE hyperfine emission. We compare simulations of N2H+ hyperfine lines made with approximate and more exact rates and find that satisfactory results are obtained.

We model the X-ray properties of millisecond pulsars (MSPs) by considering hot-spot emission from a weakly magnetized neutron star (NS) covered by a hydrogen atmosphere. We investigate the limitations of using the thermal X-ray pulse profiles of MSPs to constrain the mass-to-radius (M/R) ratio of the NS. The accuracy is strongly dependent on the viewing angle and magnetic inclination, but is ultimately limited only by photon statistics. We demonstrate that valuable information regarding NSs can be extracted, even from data of fairly limited photon statistics through modeling of archival observations of the nearby isolated PSRs J0030+0451 and J2124–3358. The X-ray emission from these pulsars is consistent with the presence of an atmosphere and a dipolar field configuration. For both MSPs, the favorable geometry allows us to place limits on the allowed M/R of NSs. Assuming 1.4 M☉, the stellar radius is constrained to be R > 9.4 km and R > 7.8 km (68% confidence) for PSRs J0030+0451 and J2124–3358, respectively. We explore the prospects of using future observatories such as Constellation-X and XEUS to conduct X-ray-timing searches for MSPs not detectable at radio wavelengths due to unfavorable viewing geometry. We are also able to place strong constraints on the magnetic field evolution model proposed by Ruderman. The pulse profiles indicate that the magnetic field of an MSP does not have a tendency to align itself with the spin axis or migrate toward one of the spin poles during the low-mass X-ray binary phase.

X-ray spectrum shows: blackbody-like emission from surface; high-energy emission (nature unknown); photoelectric absorption at low energyBlackbody-like emission modeled as radiation of whole NS surface through hydrogen atmosphereBlackbody-like radiation from either release of internal heat, or continued low-level accretionX-ray spectrum of Cen X-4 in quiescence, Rutledge 01

We present a model of thermal X-ray emission from hot spots on the surface of a rotating compact star with an unmagnetized light-element atmosphere. An application to ROSAT, Chandra, and XMM-Newton X-ray observations of the nearest known rotation-powered millisecond pulsar (MSP) PSR J0437-4715 reveals that the thermal emission from this pulsar is fully consistent with such a model, enabling constraints on important properties of the underlying neutron star. We confirm that the observed thermal X-ray pulsations from J0437-4715 are incompatible with blackbody emission and require the presence of an optically thick, light-element (most likely hydrogen) atmosphere on the neutron star surface. The morphology of the X-ray pulse profile is consistent with a global dipole configuration of the pulsar magnetic field but suggests an off-center magnetic axis, with a displacement of 0.8-3 km from the stellar center. For an assumed mass of 1.4 M☉, the model restricts the allowed stellar radii to R = 6.8-13.8 km (90% confidence) and R > 6.7 km (99.9% confidence), which is consistent with standard NS equations of state and rules out an ultracompact star smaller than its photon sphere. Deeper spectroscopic and timing observations of this and other nearby radio MSPs with current and future X-ray facilities (Constellation-X and XEUS) can provide further insight into the fundamental properties of neutron stars.

Current X-ray missions are providing high-quality X-ray spectra from neutron stars (NSs) in quiescent low-mass X-ray binaries (qLMXBs). This has motivated us to calculate new hydrogen atmosphere models, including opacity due to free-free absorption and Thomson scattering, thermal electron conduction, and self-irradiation by photons from the compact object. We have constructed a self-consistent grid of neutron star models covering a wide range of surface gravities, as well as effective temperatures, which we make available to the scientific community. We present multiepoch Chandra X-ray observations of the qLMXB X7 in the globular cluster 47 Tuc, which is remarkably nonvariable on timescales from minutes to years. Its high-quality X-ray spectrum is adequately fitted by our hydrogen atmosphere model without any hard power-law component or narrow spectral features. If a mass of 1.4 M☉ is assumed, our spectral fits require that its radius be in the range Rns = 14.5 km (90% confidence), which is larger than that expected from currently preferred models of NS interiors. If its radius is assumed to be 10 km, then a mass of Mns = 2.20 M☉ is required. Using models with the appropriate surface gravity for each value of the mass and radius becomes important for interpretation of the highest quality data.

X-ray emission from many rotation-powered millisecond pulsars (MSPs) is observed to be of predominantly thermal nature. In PSR J0437-4715, the nearest MSP known, an additional faint power-law tail is observed above 2.5 keV, commonly attributed to nonthermal magnetospheric radiation. We propose that the hard emission in this and other similar MSPs is instead due to weak Comptonization of the thermal (blackbody or hydrogen atmosphere) polar cap emission by energetic electrons/positrons of small optical depth presumably in the pulsar magnetosphere and wind. This spectral model implies that all soft X-rays are of purely thermal origin, which has profound implications in the study of neutron star structure and fundamental pulsar physics.

A new Fokker-Planck equation is developed for treating resonance-line scattering, which is especially relevant to the treatment of Lyα in the early universe. It is a "corrected" form of the equation of Rybicki & Dell'Antonio that now obeys detailed balance, so the approach to thermal equilibrium is properly described. The new equation takes into account the energy changes due to scattering off moving particles, the recoil term of Basko, and stimulated scattering. One result is a surprising unification of the equation for resonance-line scattering and the Kompaneets equation. An improved energy exchange formula due to resonance-line scattering is derived. This formula is compared to previous formulas of Madau and coworkers and Chen & Miralda-Escudé.

The majority of X-ray-detected rotation-powered millisecond pulsars (MSPs) appear to exhibit predominantly thermal emission, believed to originate from the heated magnetic polar caps of the pulsar. In the nearest MSP, J0437--4715 a faint PL is also observed at >3 keV, usually associated with magnetospheric emission processes. However, the hard emission in this and other similar MSPs may instead be due to weak Comptonization of the thermal polar cap emission by energetic electrons/positrons of small optical depth most likely in the pulsar magnetosphere. This spectral model implies that all soft X-rays are of purely thermal origin, which has important implications in the study of neutron stars.

A soft component of thermal emission is very commonly observed from the surfaces of quiescent, accreting neutron stars. We searched with Chandra for such a surface component of emission from the dynamical black hole candidate XTE J1118+480 (=J1118), which has a primary mass M1 ≈ 8 M☉. None was found, as one would expect if the compact X-ray source is a bona fide black hole that possesses an event horizon. The spectrum of J1118 is well fitted by a simple power-law model that implies an unabsorbed luminosity of LX ≈ 3.5 × 1030 ergs s-1 (0.3-7 keV). In our search for a thermal component, we fitted our Chandra data to a power-law model (with slope and NH fixed) plus a series of nine hydrogen atmosphere models with radii ranging from 9/8 to 2.8 Schwarzschild radii. For the more compact models, we included the important effect of self-irradiation of the atmosphere. Because of the remarkably low column density to J1118, NH ≈ 1.2 × 1020 cm-2, we obtained very strong limits on a hypothetical thermal source: kT∞ < 0.011 keV and L∞,th < 9.4 × 1030 ergs s-1 (99% confidence level). In analogy with neutron stars, there are two possible sources of thermal radiation from a hypothetical surface of J1118: deep crustal heating and accretion. The former mechanism predicts a thermal luminosity that exceeds the above luminosity limit by a factor of ≳25, which implies that either one must resort to contrived models or, as we favor, J1118 is a true black hole with an event horizon. In addition to neutron stars, we also consider emission from several exotic models of compact stars that have been proposed as alternatives to black holes. As we have shown previously, accreting black holes in quiescent X-ray binaries are very much fainter than neutron stars. One potential explanation for this difference is the larger and hence cooler surface of an 8 M☉ compact object that might be masked by the interstellar medium. However, our upper limit on the total luminosity of J1118 of 1.3 × 1031 ergs s-1 is far below the luminosities observed for neutron stars. This result strengthens our long-held position that black holes are faint relative to neutron stars because they possess an event horizon.

We develop a method of analyzing radio-frequency spectral line observations to derive data on the temperature, density, velocity, and molecular abundance of the emitting gas. The method incorporates a radiative transfer code with a new technique for handling overlapping hyperfine emission lines within the accelerated Λ-iteration algorithm and a heuristic search algorithm based on simulated annealing. We apply this method to new observations of N2H+ in three Lynds clouds thought to be starless cores in the first stages of star formation and determine their density structure. A comparison of the gas densities derived from the molecular line emission and the millimeter dust emission suggests that the required dust mass opacity is about κ1.3 mm = 0.04 cm2 g-1, consistent with models of dust grains that have opacities enhanced by ice mantles and fluffy aggregrates.

A large number of neutron stars (NSs), ~109, populate the Galaxy, but only a tiny fraction of them is observable during the short radio pulsar lifetime. The majority of these isolated NSs, too cold to be detectable by their own thermal emission, should be visible in X-rays as a result of accretion from the interstellar medium. The ROSAT All-Sky Survey has, however, shown that such accreting isolated NSs are very elusive: only a few tentative candidates have been identified, contrary to theoretical predictions that up to several thousand should be seen. We suggest that the fundamental reason for this discrepancy lies in the use of the standard Bondi formula to estimate the accretion rates. We compute the expected source counts using updated estimates of the pulsar velocity distribution, realistic hydrogen atmosphere spectra, and a modified expression for the Bondi accretion rate, as suggested by recent MHD simulations and supported by direct observations in the case of accretion around supermassive black holes in nearby galaxies and in our own. We find that, whereas the inclusion of atmospheric spectra partly compensates for the reduction in the counts due to the higher mean velocities of the new distribution, the modified Bondi formula dramatically suppresses the source counts. The new predictions are consistent with a null detection at the ROSAT sensitivity.

A kinetic equation for Compton scattering is given that differs from the Kompaneets equation in several significant ways. By using an inverse differential operator, this equation allows treatment of problems for which the radiation field varies rapidly on the scale of the width of the Compton kernel. This inverse operator method describes, among other effects, the thermal Doppler broadening of spectral lines and continuum edges and automatically incorporates the process of Compton heating/cooling. It is well adapted for inclusion into a numerical iterative solution of radiative transfer problems. The equivalent kernel of the new method is shown to be a positive function and with reasonable accuracy near the initial frequency, unlike the Kompaneets kernel, which is singular and not wholly positive. It is shown that iterations of the inverse operator kernel can be easily calculated numerically, and a simple summation formula over these iterations is derived that can be efficiently used to compute Comptonized spectra. It is shown that the new method can be used for initial-value and other problems with no more numerical effort than the Kompaneets equation and that it more correctly describes the solution over times comparable to the mean scattering time.

A kinetic equation for Compton scattering is given that differs from the Kom-paneets equation in several significant ways. By using an inverse differential operator this equation allows treatment of problems for which the radiation field varies rapidly on the scale of the width of the Compton kernel. This inverse operator method describes, among other effects, the thermal Doppler broadening of spectral lines and continuum edges, and automatically incorporates the process of Compton heating/cooling. It is well adapted for inclusion into a numerical iterative solution of radiative transfer problems. The equivalent kernel of the new method is shown to be a positive function and with reasonable accuracy near the intitial frequency, unlike the Kompaneets kernel, which is singular and not wholly positive. It is shown that iterates of the inverse operator kernel can be easily calculated numerically, and a simple summation formula over these iterates is derived that can be efficiently used to compute Comptonized spectra. It is shown that the new method can be used for initial value and other problems with no more numerical effort than the Kompaneets equation, and that it more correctly describes the solution over times comparable to the mean scattering time.

We investigate the structure of self-gravitating polytropic stellar systems. We present a method that allows us to obtain approximate analytical solutions, psi (n+epsilon)(x) of the nonlinear Poisson equation with the polytropic index n+epsilon, given the solution psi (n)(x) with the polytropic index n, for any positive or negative epsilon, such that \epsilon\ <<1. A similar technique has been developed independently by Seidov & Kuzakhmedov. Application of this method to the spherically symmetric stellar polytropes with n similar or equal to 5 yields the solutions that describe spatially bound systems, if n <5, and the formation of a second core, if n>5. A heuristic approximate expression for the radial profile is also presented. Because of the duality between stellar and gas polytropes, our results are valid for gaseous, self-gravitating, polytropic systems (e.g., molecular clouds) with index gamma similar or equal to 6/5. The stability of such systems and observational consequences for both stellar and gaseous systems are discussed.

In Loeb & Rybicki (hereafter Paper I), it was shown that before reionization, the scattering of Lyα photons from a cosmological source might lead to a fairly compact (~15'') Lyα halo around the source. Observations of such halos could constrain the properties of the neutral intergalactic medium and, in particular, yield the cosmological density parameters of baryons and matter on scales where the Hubble flow is unperturbed. Paper I did not treat the polarization of this scattered radiation but did suggest that the degree of such polarization might be large. In this Letter, we report on improved calculations for these Lyα halos, now accounting for the polarization of the radiation field. The polarization is linear and is oriented tangentially to the projected displacement from the center of the source. The degree of polarization is found to be 14% at the core radius, where the intensity has fallen to half of the central value. It rises to 32% and 45% at the radii where the intensity has fallen to one-tenth and one-hundredth of the central intensity, respectively. At larger radii, the degree of polarization rises further, asymptotically to 60%. Such high values of polarization should be easily observable and provide a clear signature of the phenomenon of Lyα halos surrounding sources prior to reionization.