Search NASA⌕ Search

SEARCH · Search NASA

Results for “STELLAR ATMOSPHERE”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 109 records · Page 6

Model atmosphere analysis of selected luminous B stars

The general scientific goal of this program has been to determine whether the atmospheric structure of the B-type stars can be represented by the current generation of plane parallel, line-blanketed, LTE stellar atmosphere models sufficiently well to allow accurate effective temperatures and surface gravities to be deduced. The B stars cover a wide range of temperature and luminosity. For the hottest such stars (with T approximately 30,000 K) the applicability of the models may be compromised by departures from LTE in the stellar atmospheres ('non-LTE effects'). At the highest luminosities (the B 'super giants'), the models may be invalidated by departures from plane parallel geometry. Thus we seek to identify the temperature and luminosity range within which these effects are unimportant and where the models may be relied upon.

Fitzpatrick, Edward L.↗

The structure of Neptune's upper atmosphere - The stellar occultation of 24 May 1981

Observations of the 24 May 1981 occulation of an uncatalogued star by Neptune made at the Cerro Tololo Inter-American Observatory have been analyzed to yield temperature profiles of Neptune's upper atmosphere for number densities near 5 x 10 to the 13th per cu cm. The mean temperatures at immersion (latitude - 56 deg) and emersion (latitude - 16 deg) obtained by numerical inversion were 140 + or - 10 K and 154 + or - 10 K, respectively. The immersion and emersion profiles are remarkably similar in overall shape, suggestive of global atmospheric layering. From the astrometry of the event, precise relative positions of Neptune and the occulted star were obtained.

French, R. G.↗

Predicted magnitudes and colors from cool-star model atmospheres

An intercomparison of model stellar atmospheres and observations of real stars can lead to a better understanding of the relationship between the physical properties of stars and their observed radiative flux. In this spirit we have determined wide-band and narrow-band magnitudes and colors for a subset of models of K and M giant and supergiant stars selected from the grid of 40 models by Johnson, Bernat and Krupp (1980) (hereafter referred to as JBK). The 24 models selected have effective temperatures of 4000, 3800, 3600, 3400, 3200, 3000, 2750 and 2500 K and log g = 0, 1 or 2. Emergent energy fluxes (erg/ sq cm s A) were calculated at 9140 wavelengths for each model. These computed flux curves were folded through the transmission functions of Wing's 8-color system (Wing, 1971; White and Wing, 1978) and through Johnson's (1965) wide-band (BVRIJKLM) system. The calibration of the resultant magnitudes was made by using the absolute calibration of the flux curve of Vega by Schild, et al. (1971).

Johnson, H. R.↗

The Lyman-Continuum Fluxes and Stellar Parameters of O and Early B-Type Stars

Using the results of the most recent stellar atmosphere models applied to a sample of hot stars, we construct calibrations of effective temperature (T(sub eff)), and gravity (log(sub g)) with a spectral type and luminosity class for Galactic 0-type and early B-type stars. From the model results we also derive an empirical relation between the bolometric correction and T(sub eff) and log g. Using a sample of stars with known distances located in OB associations in the Galaxy and the Large Magellanic Cloud, we derive a new calibration of M(sub v) with spectral class. With these new calibrations and the stellar atmosphere models of Kurucz, we calculate the physical parameters and ionizing photon luminosities in the H(0) and He(0) continua for O and early B-type stars. We find substantial differences between our values of the Lyman- continuum luminosity and those reported in the literature. We also discuss the systematic discrepancy between O-type stellar masses derived from spectroscopic models and those derived from evolutionary tracks. Most likely, the cause of this 'mass discrepancy' lies primarily in the atmospheric models, which are plane parallel and hydrostatic and therefore do not account for an extended atmosphere and the velocity fields in a stellar wind. Finally, we present a new computation of the Lyman-continuum luminosity from 429 known O stars located within 2.5 kpc of the Sun. We find the total ionizing luminosity from this population ((Q(sub 0)(sup T(sub ot))) = 7.0 x 10(exp 51) photons/s) to be 47% larger than that determined using the Lyman continuum values tabulated by Panagia.

Vacca, William D.↗

Hubble Space Telescope Eclipse Observations of the Nova Like Cataclysmic Variable UX Ursae Majoris

We present and analyze Hubble Space Telescope observations of the eclipsing nova-like cataclysmic variable UX UMa obtained with the Faint Object Spectrograph. Two eclipses each were observed with the G160L grating (covering the ultraviolet waveband) in 1994 August and with the PRISM (covering the near-ultraviolet to near-infrared) in November of the same year. The system was about 50% brighter in November than in August, which, if due to a change in the accretion rate, indicates a fairly substantial increase in Mass accretion by about 50%. The eclipse light curves are qualitatively consistent with the gradual occultation of an accretion disk with a radially decreasing temperature distribution. The light curves also exhibit asymmetries about mideclipse that are likely due to a bright spot at the disk edge. Bright-spot spectra have been constructed by differencing the mean spectra observed at pre- and posteclipse orbital phases. These difference spectra contain ultraviolet absorption lines and show the Balmer jump in emission. This suggests that part of the bright spot may be optically thin in the continuum and vertically extended enough to veil the inner disk and/or the outflow from UX UMa in some spectral lines. Model disk spectra constructed as ensembles of stellar atmospheres provide poor descriptions of the observed posteclipse spectra, despite the fact that UX UMa's light should be dominated by the disk at this time. Suitably scaled single temperature model stellar atmospheres with T(sub eff) approximately equals 12,500-14,500 K actually provide a better match to both the ultraviolet and optical posteclipse spectra. Evidently, great care must be taken in attempts to derive accretion rates from comparisons of disk models to observations. One way to reconcile disk models with the observed posteclipse spectra is to postulate the presence of a significant amount of optically thin material in the system. Such an optically thin component might be associated with the transition region ("chromosphere") between the disk photosphere and the fast wind from the system whose presence has been suggested by Knigge and Drew. In any event, the wind/ chromosphere is likely to be the region in which many, if not most, of the UV lines are formed. This is clear from the plethora of emission lines that appear in the mideclipse spectra, some of which appear as absorption features in spectra taken at out-of-eclipse orbital phases.

Knigge, Christian↗

On Wave Processes in the Solar Atmosphere

This grant was awarded by NASA/MSFC to The University of Alabama in Huntsville (UAH) to investigate the physical processes responsible for heating and wind acceleration in the solar atmosphere, and to construct theoretical, self-consistent and time-dependent solar wind models based on the momentum deposition by finite amplitude and nonlinear Alfven waves. In summary, there are three main goals of the proposed research: (1) Calculate the wave energy spectra and wave energy fluxes carried by magnetic non- magnetic waves. (2) Find out which mechanism dominates in supplying the wave energy to different parts of the solar atmosphere. (3) Use the results obtained in (1) and (2) to construct theoretical, self-consistent and time- dependent models of the solar wind. We have completed the first goal by calculating the amount of non-radiative energy generated in the solar convection zone as acoustic waves and as magnetic tube waves. To calculate the amount of wave energy carried by acoustic waves, we have used the Lighthill-Stein theory for sound generation modified by Musielak, Rosner, Stein & Ulmschneider (1994). The acoustic wave energy fluxes for stars located in different regions of the Hertzsprung-Russell (H-R) diagram have also been computed. The wave energy fluxes carried by longitudinal and transverse waves along magnetic flux tubes have been calculated by using both analytical and numerical methods. Our analytical approach is based a theory developed by Musielak, Rosner & Ulmschnelder and Musielak, Rosner, Gall & Ulmschneider, which allows computing the wave energy fluxes for linear tube waves. A numerical approach has been developed by Huang, Musielak & Ulmschneider and Ulmschneider & Musielak to compute the energy fluxes for nonlinear tube waves. Both methods have been used to calculate the wave energy fluxes for stars located in different regions of the HR diagram (Musielak, Rosner & Ulmschneider 1998; Ulmschneider, Musielak & Fawzy 1998). Having obtained the wave energy fluxes for acoustic and magnetic tube waves, we have investigated the behavior of these waves in the solar and stellar atmospheres. The results of our extensive studies have been published in many papers and presented at numerous scientific meetings. In these studies we have investigated different aspects of propagation of acoustic and magnetic waves, the efficiency of energy transfer along magnetic structures in the solar atmosphere, and behavior of Alfven waves in stgeady and expanding solar and stellar atmospheres. Recently, we have used some of these results to construct first purely theoretical, two component and time-dependent models of solar and stellar chromospheres. Finally, to address the third goal, we have constructed first fully theoretical, self-consistent and time dependent wind models based on the momentum deposition by non-linear Alfven waves. The full set of single-fluid MHD equations with the background flow has been solved by using a modified version of the ZEUS MHD code. The constructed wind models are radially symmetric with the magnetic field decreasing radially and the initial outflow is described by the standard Parker wind solution. In contrast to previous studies, no assumptions regarding wave linearity, wave damping, and wave-flow interaction are made; the models thus naturally account for the backreaction of the wind on the waves as well as for the nonlinear interaction between different types of MHD waves. The models have been used to explain the origin of fast speed streams in solar coronal holes. The obtained results clearly demonstrate that the momentum deposition by Alfven waves in the solar wind can be sufficient to explain the origin of fast stream components of the solar wind. The range of wave amplitudes required to obtain the desired results seems to be in good agreement with recent observations.

Musielak, Z. E.↗

On the physics of waves in the solar atmosphere: Wave heating and wind acceleration

This paper presents work performed on the generation and physics of acoustic waves in the solar atmosphere. The investigators have incorporated spatial and temporal turbulent energy spectra in a newly corrected version of the Lighthill-Stein theory of acoustic wave generation in order to calculate the acoustic wave energy fluxes generated in the solar convective zone. The investigators have also revised and improved the treatment of the generation of magnetic flux tube waves, which can carry energy along the tubes far away from the region of their origin, and have calculated the tube wave energy fluxes for the sun. They also examine the transfer of the wave energy originated in the solar convective zone to the outer atmospheric layers through computation of wave propagation and dissipation in highly nonhomogeneous solar atmosphere. These waves may efficiently heat the solar atmosphere and the heating will be especially significant in the chromospheric network. It is also shown that the role played by Alfven waves in solar wind acceleration and coronal hole heating is dominant. The second part of the project concerned investigation of wave propagation in highly inhomogeneous stellar atmospheres using an approach based on an analytic tool developed by Musielak, Fontenla, and Moore. In addition, a new technique based on Dirac equations has been developed to investigate coupling between different MHD waves propagating in stratified stellar atmospheres.

Musielak, Z. E.↗

On the physics of waves in the solar atmosphere: Wave heating and wind acceleration

This paper presents work performed on the generation and physics of acoustic waves in the solar atmosphere. The investigators have incorporated spatial and temporal turbulent energy spectra in a newly corrected version of the Lighthill-Stein theory of acoustic wave generation in order to calculate the acoustic wave energy fluxes generated in the solar convective zone. The investigators have also revised and improved the treatment of the generation of magnetic flux tube waves, which can carry energy along the tubes far away from the region of their origin, and have calculated the tube energy fluxes for the sun. They also examine the transfer of the wave energy originated in the solar convective zone to the outer atmospheric layers through computation of wave propagation and dissipation in highly nonhomogeneous solar atmosphere. These waves may efficiently heat the solar atmosphere and the heating will be especially significant in the chromospheric network. It is also shown that the role played by Alfven waves in solar wind acceleration and coronal hole heating is dominant. The second part of the project concerned investigation of wave propagation in highly inhomogeneous stellar atmospheres using an approach based on an analytic tool developed by Musielak, Fontenla, and Moore. In addition, a new technique based on Dirac equations has been developed to investigate coupling between different MHD waves propagating in stratified stellar atmospheres.

Musielak, Z. E.↗

Reflection and trapping of transient Alfven waves propagating in an isothermal atmosphere with constant gravity and uniform magnetic field

A time-dependent linear magnetohydrodynamic numerical model was used to investigate the propagation of Alfven waves in an isothermal and stratified atmosphere with constant gravity and uniform vertical magnetic field. Results show that the Alfven wave transit time for the wave source to infinity is finite and that the wave exhibits continuous partial reflection which becomes total reflection as the front approaches infinity. The total reflection causes the waves to be trapped in the cavity that extends from the wave source to infinity and in which the wave energy is stored. The results suggest that the reflection of Alfven waves (of sufficiently long period) from the outer corona is an intrinsic phenomenon for any stellar atmosphere stratified by gravity and an open magnetic field, and that, therefore, such waves may be trapped in the stellar atmosphere.

An, C.-H.↗

TEM studies of a circumstellar rock

Reported here is the discovery of crystals of titanium carbide in a grain of silicon carbide which formed as a circumstellar dust particle in the atmosphere of a carbon-rich star. Just as in the case of terrestrial rocks, whose assemblage of minerals gives us clues to the composition and conditions of the environment in which they formed, the titanium carbide crystals and their textural relationship to the silicon carbide give us important clues to the nature of the stellar atmosphere in which they formed. From microscopic studies of the relationships between the atomic planes of the silicon carbide and the titanium carbide, we can show that the titanium carbide cannot have existed as already-formed crystals in a gas around which silicon carbide subsequently condensed. An alternative possibility is that both minerals grew quickly and simultaneously from condensing gas in the rapidly cooling and expanding stellar atmosphere. Other microscopic features of the silicon carbide, such as abundant atomic layer disorder and crystal twinning, similarly suggest rapid grain growth. However, another possibility is that the titanium carbide grew inside of the silicon carbide by diffusion of titanium atoms. Our calculations suggest that this scenario is less likely, given the relatively short times (a year or less) for which stellar condensates can be expected to be exposed to temperatures high enough to make diffusion sufficiently rapid.

Bernatowicz, Thomas J.↗

CO fundamental bands in late-type stars. II - Spectrum simulations for F-K stars

The procedure of Ayres and Wiedemann (1989) was applied to a range of stellar-atmosphere models to study the CO Delta-v = 1 spectrum and to establish its use as a remote sensor of thermal conditions in late-type stars. Spectra were computed to examine the sensitivity of the CO Delta-v = 1 to fundamental stellar parameters and to assess the errors introduced into the spectrum intepretation by uncertain input parameters and non-LTE effects. Results of the sensitivity study demonstrate that CO fundamental spectra are useful probes for the temperature structure of the outer layers of cool stellar atmospheres, but that their value is limited by the uncertainties introduced by non-LTE effects. However, in stars with surface gravities of log g of about 1.5 and greater, the values of these uncertainties are reasonably small.

Wiedemann, Guenter↗

Space telescope observations of the earth's upper atmosphere by stellar occultations

Stellar occultations provide a useful means of measuring the trace gas composition of the earth's mesosphere with a sensitivity of order one part per billion. The operational details will differ from those of other astronomical observations by ST, because of the difficulties in guiding near the earth's limb. Two specific trace gases of interest to atmospheric studies, Cl and ClO, are discussed in this paper.

Caldwell, J.↗

Recovery of 29 Second Oscillations in Hubble Space Telescope Eclipse Observations of the Cataclysmic Variable UX Ursae Majoris

Low-amplitude (approximately 0.5%) 29 s oscillations have been detected in Hubble Space Telescope Faint Object Spectrograph eclipse observations of the nova-like cataclysmic variable UX UMa. These are the same dwarf nova-type oscillations that were originally discovered in this system in 1972. The 29 s oscillations are seen in one pair of eclipse sequences obtained with the FOS/PRISM in 1994 November but not in a similar pair obtained with the FOS/GI60L grating in August of the same year. The oscillations in the PRISM data are sinusoidal to within the small observational errors and undergo an approximately - 360' phase shift during eclipses (i.e., one cycle is lost). The amplitudes are highest at pre-eclipse orbital phases and exhibit a rather gradual eclipse whose shape is roughly similar to, although perhaps slightly narrower than, LTX UMa's overall light curve in the PRISM bandpass (2000-8000 A). Spectra of the oscillations have been constructed from pre-, mid, and post-eclipse data segments of the November observations. The spectra obtained from the out-of-eclipse segments are extremely blue, and only lower limits can be placed on the temperature of the source that dominates the modulated flux at these orbital phases. Lower limits derived from blackbody (stellar atmosphere) model fits to these data are >or equal to 95,000 K (> or equal to 85,000 K); the corresponding upper limits on the projected area of this source are all less than 2% of the white dwarf (WD) surface area. By contrast, oscillation spectra derived from mid- eclipse data segments are much redder. Fits to these spectra yield temperature estimates in the range 20,000 K approximately greater T and T approximately less than 30,000 K for both blackbody and stellar atmosphere models and corresponding projected areas of a few percent of the WD surface area. These estimates are subject to revision if the modulated emission is optically thin. We suggest that the ultimate source of the oscillations is a hot, compact region near disk center, but that a significant fraction of the observed, modulated flux is due to reprocessing of the light emitted by this source in the accretion disk atmosphere. The compact source is occulted at orbital phases near mid-eclipse, leaving only part of the more extended reprocessing region(s) to produce the weak oscillations that persist even at conjunction. The highly sinusoidal oscillation pulse shape does not permit the identification of the compact comonent in this model with emission produced by a rotating disturbance in the inner disk or in a classical, equatorial boundary layer. Instead, this component could arise in a bright spot on the surface of the WD, possibly associated with a magnetic pole. However, a standard intermediate polar model can also be ruled out since UX UMa's oscillation period has been seen to change on timescales much shorter than the minimum timescale required to spin up the WD by accretion torques. A model invoking magnetically controlled accretion onto differentially rotating WD surface layers may be viable, but needs more theoretical work.

Knigge, Christian↗

VIII. New Zr IV-VII, Xe IV-V, and Xe VII Oscillator Strengths and the Al, Zr, and Xe Abundances in the Hot White Dwarfs G191-B2B and RE0503-289

Context: For the spectral analysis of high-resolution and high-signal-to-noise spectra of hot stars, state-of-the-art non-local thermodynamic equilibrium (NLTE) model atmospheres are mandatory. These are strongly dependent on the reliability of the atomic data that is used for their calculation. Aims: To search for zirconium and xenon lines in the ultraviolet (UV) spectra of G191−B2B and RE 0503−289, new Zr iv-vii, Xe iv-v, and Xe vii oscillator strengths were calculated. This allows, for the first time, determination of the Zr abundance in white dwarf (WD) stars and improvement of the Xe abundance determinations.Methods: We calculated Zr iv-vii, Xe iv-v, and Xe vii oscillator strengths to consider radiative and collisional bound-bound transitions of Zr and Xe in our NLTE stellar-atmosphere models for the analysis of their lines exhibited in UV observations of the hot WDs G191−B2B and RE 0503−289. Results: We identified one new Zr iv, 14 new Zr v, and ten new Zr vi lines in the spectrum of RE 0503−289. Zr was detected for the first time in a WD. We measured a Zr abundance of −3.5 +/- 0.2 (logarithmic mass fraction, approx. 11 500 times solar). We identified five new Xe vi lines and determined a Xe abundance of −3.9 +/- 0.2 (approx. 7500 times solar). We determined a preliminary photospheric Al abundance of −4.3 +/- 0.2 (solar) in RE 0503−289. In the spectra of G191−B2B, no Zr line was identified. The strongest Zr iv line (1598.948 Å) in our model gave an upper limit of −5.6 +/- 0.3 (approx. 100 times solar). No Xe line was identified in the UV spectrum of G191−B2B and we confirmed the previously determined upper limit of −6.8 +/- 0.3 (ten times solar). Conclusions: Precise measurements and calculations of atomic data are a prerequisite for advanced NLTE stellar-atmosphere modeling. Observed Zr iv-vi and Xe vi-vii line profiles in the UV spectrum of RE 0503−289 were simultaneously well reproduced with our newly calculated oscillator strengths.

Rauch, T.↗

BD+44°493: Chemo-dynamical Analysis and Constraints on Companion Planetary Masses from WIYN/NEID Spectroscopy

In this work, we present high-resolution (R ~ 100,000), high signal-to-noise ratio (S/N ~ 800) spectroscopic observations for the well-known, bright, extremely metal-poor, carbon-enhanced star BD+44°493. We determined chemical abundances and upper limits for 17 elements from WIYN/NEID data, complemented with 11 abundances redetermined from Subaru and Hubble data, using the new, more accurate, stellar atmospheric parameters calculated in this work. Our analysis suggests that BD+44°493 is a low-mass (0.83 M ⊙ ), old (12.1–13.2 Gyr) second-generation star likely formed from a gas cloud enriched by a single metal-free 20.5 M ⊙ Population III star in the early Universe. With a disk-like orbit, BD+44°493 does not appear to be associated with any major merger event in the early history of the Milky Way. From the precision radial-velocity NEID measurements (median absolute deviation = 16 m s –1 ), we were able to constrain companion planetary masses around BD+44°493 and rule out the presence of planets as small as $m\sin i=2$ M J out to periods of 100 days. This study opens a new avenue of exploration for the intersection between stellar archaeology and exoplanet science using NEID.

79 ASTRONOMY AND ASTROPHYSICS↗

Theoretical Near-IR Spectra for Surface Abundance Studies of Massive Stars

We present initial results of a study of abundance and mass loss properties of O-type stars based on theoretical near-IR spectra computed with state-of-the-art stellar atmosphere models. The James Webb Space Telescope (JWST) will be a powerful tool to obtain high signal-to-noise ratio near-IR (1-5 micron) spectra of massive stars in different environments of local galaxies. Our goal is to analyze model near-IR spectra corresponding to those expected from NIRspec on JWST in order to map the wind properties and surface composition across the parameter range of 0 stars and to determine projected rotational velocities. As a massive star evolves, internal coupling, related mixing, and mass loss impact its intrinsic rotation rate. These three parameters form an intricate loop, where enhanced rotation leads to more mixing which in turn changes the mass loss rate, the latter thus affecting the rotation rate. Since the effects of rotation are expected to be much more pronounced at low metallicity, we pay special attention to models for massive stars in the the Small Magellanic Cloud. This galaxy provides a unique opportunity to probe stellar evolution, and the feedback of massive stars on galactic evol.ution in conditions similar to the epoch of maximal star formation. Plain-Language Abstract: We present initial results of a study of abundance and mass loss properties of massive stars based on theoretical near-infrared (1-5 micron) spectra computed with state-of-the-art stellar atmosphere models. This study is to prepare for observations by the James Webb Space Telescope.

Sonneborn, George↗

Physics-Based Spectra of Accretion Disks around Black Holes

The purpose of this grant was to begin the process of deriving the light output of accretion disks around black holes directly from the actual processes that inject heat into the accreting matter, rather than from guessed dependences of heating rate on physical parameters. At JHU, the effort has focussed so far on models of accretion onto "intermediate mass black holes", a possible class of black holes, examples of which may have recently been discovered in nearby galaxies. There, Krolik and his student (Yawei Hui) have computed stellar atmospheres for uniformly-heated disks around this class of black holes. Their models serve two purposes: they are the very first serious attempts to compute the spectrum from accreting black holes in this mass range; and a library of such models can be used later in this program as contrasts for those computed on the basis of real disk dynamics. The output from these local disk calculations has also been successfully coupled to a program that applies the appropriate relativistic transformations and computes photon trajectories in order to predict the spectrum received by observers located at different polar angles. The principal new result of these calculations is the discovery of potentially observable ionization edges of H-like C and O at frequencies near the peak in flux from these objects. Most of the grant money at UCSB was spent on supporting graduate student Shane Davis. In addition. some money was spent on supporting two other students: Ari Socrates (now a Hubble Fellow at Princeton), and Laura Melling. Davis spent the year constructing stellar atmosphere models of accretion disks appropriate for the high/soft (thermal) state of black hole X-ray binaries. As with AGN models published previously by our collaboration with NASA support. our models include a complete general relativistic treatment of both the disk structure and the propagation of photons from the disk to a distant observer. They also include all important continuum opacity sources, including Compton scattering and bound-free opacity from abundant metal species. The principal new result is that bound-free opacity is very significant in altering the continuum spectral shape, resulting for example in quite different "color correction factors" compared to those predicted previously. In addition, the models predict a relationship between luminosity and inner disk temperature that is, for the first time, in accord with that observed. The primary purpose of the grant was to incorporate more realistic accretion disk physics, learned largely from simulations, into such spectral models. The Davis et al. paper includes consideration of a vertical dissipation profile computed from radiation magneto-hydrodynamic simulations of MRI turbulence by N. J. Turner (2004). So long as the disk is effectively thick, such dissipation profiles do not affect the predicted spectrum significantly. (More work needs to be done on these simulations, however.) A potentially more serious issue is that MRI turbulence produces substantial inhomogeneities, as do photon bubble instabilities. These inhomogeneities can affect the spectra by enhancing the effects of absorption opacity over scattering opacity. We have done some preliminary Monte Carlo calculations to explore these effects.

Krolik, Julian H.↗