Lectures

Lectures recommended

 

Fall 2019:

1) prof. Bozena Czerny “Accretion processes in Astrophysics”,

Copernicus Astronomical Center. Tuesdays, 11.15 -12.45.

2) prof. Tomasz Sowiński “Special Relativity”

Institute of Physics, Fridays, 12.00-13.00

Journal Club papers:

2019/2020

“Wave heating from proto-neutron star convection and the core-collapse supernova explosion mechanism”. Specialist article about state-of-the art core collapse supernova simulation and proto-neutron star convection, by S. Gossan, J. Fuller and L. Roberts

1910.07599

“The hole picture. ” Popular review by Victor Cardoso about black holes and recent discoveries. PDF here.

V_Cardoso

2017/2018

Lessons from the Short GRB 170817A – the first gravitational wave detection of a binary neutron star merger

GRB 170817A’s relatively low isotropic equivalent γ-ray energy-output may suggest a viewing angle slightly outside the jet’s sharp edge.

The authors also consider the type of remnant that is produced by the NS-NS merger and find that a relatively long-lived (>2s) massive NS is strongly disfavored, while a hyper-massive NS of lifetime ∼1s appears to be somewhat favored over the direct formation of a black hole.

arxiv link, Granot, Guetta and Gill.

Launching of Active Galctic Nuclei Jets

Recent advances in computer simulations demonstrated that accretion disks can accumulate large-scale magnetic flux on the black hole, until the magnetic flux becomes so strong that it obstructs gas infall and leads to a magnetically-arrested disk (MAD). Recent evidence suggests that central black holes in jetted active galactic nuclei and tidal disruptions are surrounded by MADs. Since in MADs both the black hole magnetic flux and the jet power are at their maximum, well-defined values, this opens up a new vista in the measurements of black hole masses and spins and quantitative tests of accretion and jet theory.

Review by A. Tschekhovskoy, available under this link

A peculiar low-luminosity short gamma-ray burst from a double neutron star merger progenitor

On August 17, 2017, coinciding with a double neutron star merger gravitational wave event detected by the LIGO-Virgo gravitational wave detector network, a short-duration GRB 170817A was detected by both Gamma-Ray Monitor (GBM) on board NASA’s Fermi Gamma-Ray Observatory and INTEGRAL. The paper discusses the properties of this GRB. arxiv link, B.-B. Zhang et al. 2017

Reisner-Nordstrom black holes

The materials from the R. Townsend lecture notes

The Magnetorotational Instability and its implications to the simulations

A series of papers were presented focusing on specific conclusions and the applications that the MRI imposes on the General Relativistic MHD simulations. Initially, we presented the newtonian theory of the Balbus A. and Hawley J., 1991, ApJ, 376, 214 , Balbus A., 2003, Annu. Rev. Astron. Astrophys.. 41, 555

The General Relativistic generalization using the mechanical spring model and the formal linear stability analysis was performed  by Gammie C., 2004, ApJ, 614, 309 and was our sequential presentation. We pinpointed the differences in the dispersion relationship and the resulting growth rate for an observer in infinity.

Finally. we were concerned for the conditions applying in a 3D GRMHD simulation so that the MRI is properly resolved following the complicate and technical diagnostics that Hawley J. et al, 2011, ApJ, 738, 84   developed to quantify the proper instability resolution.

The Nobel Prize in Physics 2017

Observation of Gravitational Waves from a Binary Black Hole Merger.

2016/2017

In this paper, authors present the results from a controlled numerical experiment investigating the efect of stellar density gas on the coalescence of binary black holes (BBHs), and the result of gravitational waves (GWs). Authors perform full numerical relativity coupled with general-relativistic hydrodynamics and set up a 30+30 Siolar Masses (motivated by GW150914) inside gas with realistic stellar densities. The results show that at densities ρ≳106−107gcm−3 dynamical friction between the BHs and gas changes the coalescence dynamics and the GW signal in an unmistakable way. Authors show that for GW150914, LIGO observations conclusively rule out BBH coalescence inside stellar gas of ρ≳107gcm−3. Typical densities in the collapsing cores of massive stars are in excess of this density. This excludes the fragmentation scenario for the formation of GW150914. (SCh).

Gravitational Waves from Binary Black Hole Mergers Inside of Stars

In this paper, authors study the baryon-loading processes of a GRB jet launched from a black hole central engine andconsider a relativistic jet powered by \nu \bar{\nu }-annihilation or by the Blandford-Znajek (BZ) mechanism. We consider baryon loading from a neutrino-driven wind launched from a neutrino-cooling-dominated accretion flow. For a magnetically dominated BZ jet, they consider neutron drifting from the magnetic wall surrounding the jet and subsequent positron capture and proton-neutron inelastic collisions. The minimum baryon loads in both types of jet are calculated. They find that in both cases a more luminous jet tends to be more baryon poor. A neutrino-driven “fireball” is typically “dirtier” than a magnetically dominated jet, while a magnetically dominated jet can be much cleaner. Both models have the right scaling to interpret the empirical Γ-L iso relation discovered recently. Since some neutrino-driven jets have too much baryon loading as compared with the data, they suggest that at least a good fraction of GRBs should have a magnetically dominated central engine. (KS)

Hyperaccreting Black Hole as Gamma-Ray Burst Central Engine. I. Baryon Loading in Gamma-Ray Burst Jets

In case of typical densities and temperatures for the AGN accretion disks, due to iron atomic transition, the value of opacity can be much larger the electron opacity.

The authors study the effects of this iron opacity “bump” on the thermal stability and vertical structure of radiation pressure dominated accretion disks, utilizing three dimensional radiation magneto-hydrodynamic simulations in the local shearing box approximation.  For a 5 × 10^8 solar mass black hole with ∼ 3% of the Eddington luminosity, a model including the iron opacity bump maintains its structure for more than 10 thermal times without showing significant signs of thermal runaway. In contrast, if only electron scattering and free-free opacity are included as in the standard thin disk model, the disk collapses on the thermal time scale.

These results suggest that the iron opacity bump may have a strong impact on the stability and structure of AGN accretion disks, and may contribute to a dependence of AGN properties on metallicity. (MG)

 

 Iron opacity bump changes the stability and structure of accretion disksin Active Galactic Nuclei

 

Through 3D numerical simulations with HARM, the authors obtain a critical power, which depends on the galaxy core mass and radius, below which jets become kink-unstable within the core, stall, and inflate cavities filled with relativistically-hot plasma. Jets above the critical power stably escape the core and form powerful backflows. Thus, the kink instability controls the jet morphology and can lead to the FR dichotomy. The model-predicted dependence of the critical power on the galaxy optical luminosity agrees well with observations. (AJ)

Three-dimensional Relativistic MHD Simulations of Active Galactic Nuclei Jets: Magnetic Kink Instability and Fanaroff-Riley Dichotomy

 

2015/2016

A broad overview of the basic theoretical paradigms of black hole accretion flows, alpha stress prescription of Shakura & Sunyaev (1973), and the properties of both radiatively efficient and inefficient models, including their local properties, their expected stability to secular perturbations, and how they might be tied together in global flow geometries. The alpha stress is a prescription for turbulence, for which the only existing plausible candidate is that which develops from the magnetorotational instability (MRI) (AJ)

General Overview of Black Hole Accretion Theory

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It is commonly believed that the ringdown signal from a binary coalescence provides a conclusive proof for the formation of an event horizon after the merger. This expectation is based on the assumption that the ringdown waveform at intermediate times is dominated by the quasinormal modes of the final object. We point out that this assumption should be taken with great care, and that very compact objects with a light ring will display a similar ringdown stage, even when their quasinormal-mode spectrum is completely different from that of a black hole. In other words, universal ringdown waveforms indicate the presence of light rings, rather than of horizons. Only precision observations of the late-time ringdown signal, where the differences in the quasinormal-mode spectrum eventually show up, can be used to rule out exotic alternatives to black holes and to test quantum effects at the horizon scale. (SCh)

Is the gravitational-wave ringdown a probe of the event horizon?

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According to the general-relativistic no-hair theorem, astrophysical black holes depend only on their masses and spins and are uniquely described by the Kerr metric. Mass and spin are the first two multipole moments of the Kerr spacetime and completely determine all other moments. The no-hair theorem can be tested by measuring potential deviations from the Kerr metric which alter such higher-order moments. In this review, I discuss tests of the no-hair theorem with current and future observations of such black holes across the electromagnetic spectrum, focusing on near-infrared observations of the supermassive black hole at the Galactic center, pulsar-timing and very-long baseline interferometric observations, as well as X-ray observations of fluorescent iron lines, thermal continuum spectra, variability, and polarization. (SCh)

Testing the No-Hair Theorem with Observations of Black Holes in the Electromagnetic Spectrum

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We report on a set of long-term general-relativistic three-dimensional (3D) multi-group (energy-dependent) neutrino-radiation hydrodynamics simulations of core-collapse supernovae. We employ a full 3D two-moment scheme with the local M1 closure, three neutrino species, and 12 energy groups per species. With this, we follow the post-core-bounce evolution of the core of a nonrotating 27-M⊙ progenitor in full unconstrained 3D and in octant symmetry for ≳380ms. We find the development of an asymmetric runaway explosion in our unconstrained simulation. We test the resolution dependence of our results and, in agreement with previous work, find that low resolution artificially aids explosion and leads to an earlier runaway expansion of the shock. At low resolution, the octant and full 3D dynamics are qualitatively very similar, but at high resolution, only the full 3D simulation exhibits the onset of explosion. (SCh)

General Relativistic Three-Dimensional Multi-Group Neutrino Radiation-Hydrodynamics Simulations of Core-Collapse Supernovae

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Our data show that optical oscillations on timescales of 100 seconds to 2.5 hours can occur at mass-accretion rates more than ten times lower than previously thought. This suggests that the accretion rate is not the critical parameter for inducing inner-disk instabilities. Instead, we propose that a long orbital period is a key condition for these large-amplitude oscillations (MG)

Repetitive patterns in rapid optical variations in the nearby black-hole binary V404 Cygni

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Accretion disk coronae are hot and radiatively compact, lying close to the boundary of the region in the compactness – temperature diagram which is forbidden due to runaway pair production. The coincidence suggests that pair production and annihilation are essential ingredients in the coronae of AGN and BHB and that they control the shape of the observed spectra. (BCz)

Properties of AGN coronae in the NuSTAR era

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Paper handles the problem of viscous stationary axisymmetric accretion flow in vertical equilibrium in the pseudo-Newtonian framework. There exist a critical value of viscosity parameter alpha, above which two critical points in the flow do not exist, so that the shock solution cannot create. Paper studies the dependence of the critical value of alpha on two other parameters – the specific angular momentum at the horizon and specific energy (computed from the chosen position of the inner critical point). The highest values of alpha_crit reaches 0.3, but for very narrow parameter space. Reasonable volume of parameter space corresponds to alpha_crit = 0.15. (PS)

Upper limit of the viscosity parameter in accretion flows around a black hole with shock waves

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Coupled TVD 2D hydro simulation of low angular momentum flow + Compton scattering of soft photons seeded by Keplerian Shakura-Sunyaev disc. The hydro step is followed by Monte Carlo simulation of scattering of photons at some (random) optical depth. Energy exchange is computed, then energy intent in each grid cell is updated. The light curve of outgoing radiation is constructed together with spectral information. The parameters are the accretion rate in Keplerian disk and in sub-Keplerian component. Change of spectral shape and presence of QPO is discussed with dependence on these parameters. (PS)

Quasi-periodic oscillations in a radiative transonic flow: results of a coupled Monte Carlo–TVD simulation

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On the hydro- and magnetohydrodynamical general relativistic simulations of a tidal disruption of a red dwarf star, approaching a supermassive non-rotating black hole on a close elliptical orbit. The study connects the SPH hydrodynamics with GR MHD. The main results are that the effective viscosity in the debris disc stems from the original hydrodynamical turbulence, which dominates over the magnetic component. The radiative efficiency is very low because of low energetics of the gas crossing the horizon and large optical depth that results in photon trapping. (AJ)

Magnetohydrodynamical simulations of a tidal disruption in general relativity

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Discovery of a supernova (SN 2011kl) associated with the ultra-long-duration burst GRB 111209A, at a redshift z of 0.677. This supernova is more than three times more luminous than type Ic supernovae associated with long-duration GRB, and its spectrum is distinctly different. The slope of the continuum resembles those of super-luminous supernovae, but extends further down into the rest-frame ultraviolet implying a low metal content. The light curve evolves much more rapidly than those of super-luminous supernovae. This combination of high luminosity and low metal-line opacity cannot be reconciled with typical type Ic supernovae, but can be reproduced by a model where extra energy is injected by a strongly magnetized neutron star (a magnetar), which has also been proposed as the explanation for super-luminous supernovae. (AJ)

A very luminous magnetar-powered supernova associated with an ultra-long gamma-ray burst

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In the paper is presented an analysis of ionized X-ray disk winds observed in the Fe K band of four stellar-mass black holes observed with Chandra, including 4U 1630-47, GRO J1655-40, H 1743-322, and GRS 1915+105. High-resolution photoionization grids were generated in order to model the data. The fastest components of detected winds have velocities approaching or exceeding 0.01c, increasing mass outflow rates and wind kinetic power by orders of magnitude over prior single-zone models. Disk winds are rotating with the orbital velocity of the underlying disk, and provides a new means of estimating launching radii — crucial to understanding wind driving mechanisms. Some aspects of the wind velocities and radii correspond well to the broad-line region (BLR) in active galactic nuclei, suggesting a physical connection.
(MG)

Powerful, Rotating Disk Winds from Stellar-mass Black Holes

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If two galaxies each hosting a central BH merge, the BHs would be dragged towards the center of the newly formed galaxy. When the holes get sufficiently close, they coalesce via the emission of gravitational waves. After galaxies have merged, however, the BHs still have a long journey until they touch and coalesce. Their dynamical evolution is radically different in gas-rich and gas-poor galaxies, leading to a sort of “dichotomy” between high-redshift and low-redshift galaxies, and late-type and early-type, typically more massive galaxies. (SCh)

Massive black holes in merging galaxies

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In the frame of a galaxy hosting a supermassive black hole (10^8 Solar Mass) we investigated a mechanism for the production of high velocity stars, which was suggested by the results of N-body simulations of the close interaction between a massive, orbitally decayed, globular cluster and the super massive black hole. Stars could reach a velocity sufficient to travel in the halo and even overcome the galactic gravitational well, while some of them are just stripped from the globular cluster and start orbiting on precessing loops around the galactic centre. (SCh)

High velocity stars from close interaction of a globular cluster and a super massive black hole

2014/2015

A cosmological simulation in which the central 50 solar mass clump breaks up into two cores, which will likely form a binary star system. The cores are optically thick to its own cooling radiation from molecular hydrogen lines, but is still able to cool via collision-induced emission. (AJ)

The Formation of Population III Binaries from Cosmological Initial Conditions

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On the massive Black Hole seeds at redshift z≳10 now thought to explain the presence of the super-massive BHs. By combining radiation-hydrodynamic and spectral synthesis codes, the authors simulate the time-evolving spectrum emerging from the host halo of a MBH seed (Bcz)

Shining in the Dark: the Spectral Evolution of the First Black Holes

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On the estimate of the mass of the microquasar IGR J17091-3624 with three different methods, which are Photon Index (Gamma) – QPO frequency (nu) correlation, QPO frequency (nu) – Time (day) evolution and broadband spectral modelling based on Two Component Advective Flow.
The joint likelihood of these methods gives the estimate 11.8-13.7 solar masses. The 90% confidence interval is 8.7 – 15.6 solar masses. (PS)
Determination of mass of IGR J17091-3624 from “Spectro-Temporal” variations during onset-phase of the 2011 outburst

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Unusual soft state of the black hole low-mass X-ray binary GRO J1655-40, observed during its 2005 outburst by the Rossi X-ray Timing Explorer Chandra X-ray grating observations have revealed a high mass-out ow accretion disc wind in this state, and we show that the broadband X-ray spectrum is remarkably similar to that observed in the so-called `hypersoft’ state of the high mass X-ray binary Cyg X-3,which possesses a strong stellar wind from a Wolf-Rayet secondary. The power-spectral density (PSD) of GRO J1655-40 shows a bending power-law shape, similar to that of canonical soft states albeit with larger  fractional rms. (MG).

The remarkable timing properties of a `hypersoft’ state in GRO J1655-40

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Ultra-compact Dwarf Galaxy discovered to have a supermassive black hole that weighs about 15% of its total mass. M60-UCD1 might be a tidally stripped nucleus of a once larger galaxy. (AJ, BCz)

A supermassive black hole in an ultra-compact dwarf galaxy

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Mapping the velocity fields of the moving subunits in the galaxy clusters would enable a new diagnostics tool for cluster formation studies and unbiased mass estimates. The article discusses detecting and mapping the velocities of the bulk motions using the Doppler shift of the Fe XXV K alpha line with the proposed ATHENA satellite. (AJ)

Bulk motion measurements in clusters of galaxies with ATHENA-like missions

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The strong-field gravitational lensing effect is used to calculate the distortion caused by isolated black holes and by numerically evolved black hole binaries. On small scales the resulting images show complex and in some cases self-similar structure, which for the systems involving the radiation emitting matter may lead to unique optical signature of an ongoing merger. (SCh)

What would a binary black hole merger look like?

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Pimordial Black Holes (PBHs) lighter than about 10^12 kg are predicted to be directly detectable by their Hawking radiation. This radiation should produce both a diffuse extragalactic gamma-ray background from the cosmologically-averaged distribution of PBHs and gamma-ray burst signals from individual light black holes. The Fermi, Milagro, Veritas, HESS and HAWC observatories, in combination with new burst recognition methodologies, offer the greatest sensitivity for the detection of such black holes or placing limits on their existence. (AJ)

Primordial Black Holes

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The authours considered problem of the limits concerning the physical information that can be extracted from the analysis of one or more time series. The field of the stochastic dynamics appears to be a useful framework for reconstruction of the lightcurves. For example, the use of SDE permits to make the analysis of a continuous signal independent from the frequency sampling (MG)

Time series analysis in Astronomy: limits and potentials

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The authors obtained six observation of the pulsar PSR J1741-2054 using the Chandra ICIS-S detector. They measured proper motion and spectrum of the pulsar. Spectrum of the pulsar can be fitted as the blackbody with the
power-law. They described also the emission from the nebula and looked for extended features that might represent a jet using image deconvolution and PSF-substraction (MG)

X-Ray analysis of the proper motion and pulsar wind nebula for PSR J1741-2054

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The paper offers new data sets for a systematic exploration of quasar variability. Although the interpretation of this phenomenon is still uncertain, the most plausible mechanisms involve a binary system of two supermassive black holes with a subparsec separation. Such systems are an expected consequence of galaxy mergers and can provide important constraints on models of galaxy formation and evolution (AJ)

A possible close supermassive black-hole binary in aquasar with optical periodicity

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The paper introduce a new class of radio sources named as FR0. The question is whether it is really a new class? (BCz)

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The paper tries to investigate whether the observational inferences (the dissipative efficiency, the equipartition between particle and magnetic energies and the requirement of non-thermal particle acceleration) can constrain the dissipation mechanism responsible for the jet emission. Their claim is that shocks are unlikely to mediate the dissipation of energy in relativistic jets (the conditions are satisfied only for small parameter range), but relativistic magnetic reconnection (the mean magnetic energy per particle is larger than the rest mass energy) can do the job. The numerical PIC (particle in cell) simulations are described in the second paper. (PS)

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The paper shows an analysis of the RXTE/PCA data of neutron stars Sco X-1 and Cyg X-2, along with the black hole Cyg X-1 and the unknown source Cyg X-3. It is shown that while Sco X-1 and Cyg X-2 are low dimensional chaotic systems, Cyg X-1 and Cyg X-3 are stochastic sources. Based on our analysis, the authors argue that Cyg X-3 may be a black hole. (AJ)

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The detailed global structure of black hole accretion flows and outflows is studied through two-dimensional radiation-magnetohydrodynamic simulations. By starting from a torus threaded with weak toroidal magnetic fields and by controlling the central density of the initial torus, ρ0, the authors reproduce three distinct modes of accretion flow. Namely: standard thin disk, slim disk and radiatively inefficient flow (RIAF).

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Study of X-ray timing and spectral evolution of IGR J17091 – 3624. Comparison with GRS 1915+105. Quite detailed study of the outburst in Feb – Apr 2011. (PS)

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A truly Newtonian softening length for disc simulations. The softened point mass model is commonly used in simulations of gaseous discs including self-gravity while the value of associated length remains, to some degree, controversial. This parameter” is however fully constrained when, in a discretized disc, all uid cells are demanded to obey Newton’s law. (MG)

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On the importance of highly accreting quasars. The 4D eigenvector 1 parameter space makes them easily recognizable over a broad range range of redshift and luminosity which goes beyond the understanding of the details of their physics. Their Eddington ratio is expected to saturate toward values of order unity, making them possible cosmological probes. (AJ)

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On the model for the magnetohydrodynamical modeling of the astrophysical plasma including radiation transport (MG)

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On the fitting of the data from GX 339-4 with two component advective flow model, obtaining the position and strength of the shock and the ratio of accretion rates for the two components — Keplerian disk and sub-Keplerian halo. (PS)

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Small note about CAMK Journal Club on recent Nature paper. (PS)

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On a scenario for launching relativistic jets from rotating black holes, in which small-scale magnetic flux loops, sustained by disc turbulence, are forced to inflate and open by differential rotation between the black hole and the accretion flow. (AJ)

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On the behaviour of reprocessed QPOs in optical/IR band by the outer part of the dics from the X-ray QPOs arising from the Lense-Thirring precession of the inner part of the disc. (PS)

2013/2014

On the small black holes with masses in the TeV range (Planck scale) which can be discovered by neutrino observatories

On the Spin Period Distribution in Be/X-ray Binaries

Galaxy Mergers as a Source of Cosmic Rays, Neutrinos, and Gamma Rays

Observational evidence of milliparsec supermassive black hole binary in quiescent galaxy derived from the dips in the X-ray lightcurve of the tidal event flare on the timescale of several weeks to months:

Classification of X-Ray sources – evidence for extreme blazar at the distance around 600 Mpc for source HESS J1943+213 instead of HMXB or young pulsar wind nebulae (PWN) in our Galaxy:

Ultra high energy cosmic rays – debate about the reason for the cut-off in the very high energy (EeV to ZeV) band (GZK-effect or the exhaustion of sources), connection between part of the observed events and a nearby galaxy Centaurus-A (2-3 Mpc), where the distance is too small for the GZK effect to work. (group work)

X-ray bursts – the interaction of the X-Ray burst of neutron star with the accretion disc causing the switch off of low kHz QPOs for several tens of seconds after the burst (depletion of the disc due the enhanced accretion on the neutron star during the burst). (P)

Speculation that the hysteretic cycle of black hole state transitions arises from two established properties of accretion disks: the increase in turbulent stress in disks threaded by a net magnetic field and the ability of thick (but not thin) disks to advect such a field radially

Halos, galaxies and black holes to z=0 in the high resolution hydrodynamical simulation MassiveBlack-II (MBII) which evolves a LCDM cosmology in a comoving volume Vbox=100(Mpc/h)^3. MBII is the highest resolution simulation of this size which includes a self-consistent model for star formation, black hole accretion and associated feedback.

The black hole X-ray binaries GX 339-4 and V404 Cygni have brought evidence of a strong correlation between radio and X-ray emission during the hard spectral state. Several outliers, including the source H1743-322, do not follow this track. The Comptonization of additional photons from a weak, cool inner disk during the hard spectral state can explain the observations.

The Bright Optical Flash and Afterglow from the Gamma-Ray Burst GRB 130427A. (P)

A High-Frequency Doppler Feature in the Power Spectra of Simulated GRMHD Black Hole Accretion Disks:

A New Accretion Disk Around the Missing Link Binary System PSR J1023+0038:

Menus for Feeding Black Holes. (P)

 

 

 

Review papers:

Narrow-Line Seyfert 1 Galaxies:

 

Lectures, “Compact Stars”

Lecture given by Agnieszka Janiuk, spring semester 2017/2018.

Place: room 309, Physics Faculty, PW, Time: Thursday, 16.00-18.00.

Lecture 1. CompactStars_01_2018

Lecture 2. CompactStars_02_2018

Lecture 3. CompactStars_03_2018

Lecture 4. CompactStars_04_2018

Lecture 5. CompactStars_05_2018

Lecture 6. CompactStars_06_2018

Lecture 7. CompactStars_07_2018

Lecture 8. CompactStars_08_2018

Lecture 9. CompactStars_09_2018

Lecture 10. CompactStars_10_2018

Lecture 11. CompactStars_11_2018

Lecture 12. CompactStars_12_2018

Lecture 13. CompactStars_13_2018

Journal Club

Our Journal Club and working meetings are held every Friday, at 11.00 am in the Library room in CFT.

Problems to solve

Problems_1

Problems_2

Problems_3

Relativistic astrophysics and related computational methods

Lecture given by Michał Bejger, spring semester 2013/2014

Place: seminar room, CAMK. Time: Tuesdays, 13.15.

Lectures notes are here.


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