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Kinetic and radiative efficiencies in accretion flows

# Kinetic and radiative power from optically thin accretion flows

## Abstract

We perform a set of general relativistic, radiative, magneto-hydrodynamical simulations (GR-RMHD) to study the transition from radiatively inefficient to efficient state of accretion on a non-rotating black hole. We study ion to electron temperature ratios ranging from to , and simulate flows corresponding to accretion rates as low as , and as high as . We have found that the radiative output of accretion flows increases with accretion rate, and that the transition occurs earlier for hotter electrons (lower ratio). At the same time, the mechanical efficiency hardly changes and accounts to of the accreted rest mass energy flux, even at the highest simulated accretion rates. This is particularly important for the mechanical AGN feedback regulating massive galaxies, groups, and clusters. Comparison with recent observations of radiative and mechanical AGN luminosities suggests that the ion to electron temperature ratio in the inner, collisionless accretion flow should fall within , i.e., the electron temperature should be several percent of the ion temperature.

###### keywords:
accretion, accretion discs – black hole physics – relativistic processes – methods: numerical

## 1 Introduction

Accreting black holes (BH) are behind some of most energetic and fundamental phenomena in the Universe. In the supermassive regime (SMBH), they are often known as active galactic nuclei (AGN) and quasars. When the rate at which gas is lost below the BH horizon is low (normalized to the Eddington unit), accretion flows are optically thin and radiatively inefficient (an extreme case is the accretion flow in the center of the Galaxy). Radiatively inefficient BHs do however inject significant amount of mechanical energy into the environment, preventing catastrophic cooling flows and star formation rates via large-scale outflows, cavities and shocks (McNamara & Nulsen, 2012 for a review), at the same time establishing the celebrated BH mass - velocity dispersion relation (Kormendy & Ho, 2013 for a review). The output mode is expected to change when gas is accreted at rates exceeding a percent of the Eddington rate. Under such conditions the previously radiatively inefficient sources become quasars dominated by the radiative output.

The transition from the radiatively inefficient to the efficient regime is not well understood. Neither the physics of the electron heating in optically thin collisionless plasmas. On the other hand, there is ample amount of observational data which probes accreting systems at very different luminosities and accretion rates. One key observational study attempting to constrain the kinetic versus radiative efficiencies in massive galaxies detected with Chandra telescope (mainly brightest cluster galaxies; BCGs) is Russell et al. (2013). The radiative output is constrained via the nuclear X-ray luminosity, while the kinetic power is estimated by measuring the X-ray bubble enthalpy divided by the buoyancy time. The data corroborates that the mechanical output always dominate at low and intermediate accretion rates (more in Section 1.1). In terms of evolution, in the nearby Universe, only a few percent of systems host a radiatively efficient source (or quasar). In other words, below redshift , AGN feedback is expected to proceed at a sub-Eddington rate, i.e., the AGN feedback ‘maintenance’ phase is dominated by kinetic input of energy – typically via massive outflows (e.g., see Gaspari, 2016 for a brief review).

In this work, we perform state-of-the-art, 3-dimensional (3D) GR-RMHD simulations of accretion flows below and near the transitional regime, and calculate the corresponding mechanical and radiative efficiencies. Moreover, comparing the observed and simulated values of the luminosity at which the radiative efficiency becomes comparable with the mechanical one, we aim to constrain the range of possible values of the electron to ion temperature ratio in accreting systems.

In a parallel work, Gaspari & Sa̧dowski (2017), we apply the mechanical efficiency constrained here, to a model of self-regulated AGN feedback which couples the small (horizon) scale to the large (kpc - Mpc) scale properties of cool-core systems (as massive galaxies, groups, and clusters). The macro feeding properties are mediated by the recently probed chaotic cold accretion (CCA), i.e., the condensation of multiphase clouds out of the turbulent plasma halo, which rain toward the nuclear region and are efficiently funnelled via chaotic inelastic collisions cancelling angular momentum (see Gaspari et al., 2013; Gaspari et al., 2015; Gaspari et al., 2017). The proposed unification model provides the BH growth and AGN outflow properties as a function of the macro properties (e.g., hot halo temperature), and can be used as an effective sub-grid scheme in structure formation simulations and analytic studies, without resorting to the fine-tuning of a free efficiency.

The presented work is structured as follows. Below we review the observational work we base our comparison on. In Section 2, we introduce the numerical method and main assumptions. In Section 3, we discuss the performed 3D GR-RMHD simulations. In Section 4, we discuss the key implications. In Sections 5 and 6, we discuss the caveats and summarize the main results, respectively.

### 1.1 Observational constraints

Accreting black holes are known to both emit radiation and inject significant amounts of mechanical energy in the surroundings. The latter effect often leads to the inflation of cavities in the medium surrounding the BH, and could happen both on parsec scales (for stellar mass BHs), as well as on kpc scale (for AGN). From the volume and pressure of a given cavity () and its expansion (sound-crossing) time one may infer the mechanical luminosity of the central source.

Several studies aiming at comparing the radiative and kinetic properties have been recently performed (e.g., King et al., 2012; Mezcua & Prieto, 2014; Hlavacek-Larrondo et al., 2015; Hogan et al., 2015; Shin et al., 2016) In this work, we mainly use results by Russell et al. (2013), one of the most comprehensive investigations based on observations of nuclear X-ray sources and the corresponding cavities in a large sample of 57 BCGs. These authors have found that the nuclear radiation exceeds the mechanical energy output of the outflow when the mean accretion rate rises above a few percent of the Eddington luminosity, corresponding to the onset of the quasar mode. The two components become comparable already at and the radiative component is very weak, sometimes undetectable, for the lowest accretion and emission rates (cf. Fig. 12 in Russell et al., 2013). For all such reasons, below we use the range as reference threshold below which the mechanical luminosity of an accreting black hole system equals or exceeds the corresponding radiative output.

### 1.2 Units

In this work we adopt the following definition for the Eddington mass accretion rate,

 ˙MEdd=LEddηc2, (1)

where is the Eddington luminosity for a BH of mass , and is the radiative efficiency of a thin disk around a black hole with a given spin . For a zero-spin BH, (Novikov & Thorne, 1973) and . We denote the gravitational radius and time as and , respectively.

## 2 Numerical method

For the purpose of simulating optically thin and marginally optically thin accretion flows we adopt the general relativistic, radiative magneto-hydrodynamical solver KORAL (Sa̧dowski et al., 2013, 2014) which is capable of evolving gas and radiation in parallel for arbitrary optical depths. The most recent version of KORAL is capable of evolving two-temperature plasma (Sadowski et al., 2016). This approach, however, is not free from arbitrary choices and numerical issues (see Discussion section below). For this reason we decided to solve a simplified problem where we evolve only the mean temperature of the gas (mixture of electrons and ions), but for the purpose of calculating radiative emission, we split the gas temperature into electron and ion temperatures, and , respectively, following the prescribed ratio and . For pure hydrogen gas the species temperature satisfy,

 Tgas=1/2(Te+Ti). (2)

We account for free-free and synchrotron emission and absorption, as well as Compton heating and cooling. The exact formulae for the opacities are given in Sadowski et al. (2016). In this work we use only grey opacities and we do not calculate electromagnetic spectrum of generated radiation.

### 2.1 Problem setup

We performed a set of 9 three-dimensional simulations of accretion flows on a non-rotating BH. The simulations cover accretion rates from to - enough to cover both the low-luminosity limit and the observed transition to radiatively-efficient mode1. We tested three values of ion to electron temperature ratios, , and . This choice reflects the fact that the expected electron temperature is lower than the ion’s (only electrons cool emitting photons), and, as discussed below, is wide enough to put limits on the temperature ratio consistent with observations.

Initially, we performed a single, non-radiative, simulation with an equilibrium torus (same as in Narayan et al., 2012) threaded by a magnetic field forming set of poloidal loops (again, same as in Narayan et al., 2012, but flipping the polarity also across the equatorial plane) on a grid of 336 cells in radius and polar angle, and 32 cells in azimuth spanning wedge. Such a setup was evolved for , long enough for establishing a now-standard, scale-free solution of a thick and hot accretion flow with converged region extending up to .

We then used the final state of this non-radiative simulation as a starting point for the radiative runs being the subject of this paper. When starting a radiative run we rescaled the density to a desired value (which ultimately determined the final accretion rate) keeping the magnetic to gas pressure ratio and the gas temperature fixed. At the same time we turned on the radiative evolution allowing for emission and absorption of radiation with efficiency determined by the choice of species temperature ratio. In such a way we simulated the 9 models described in this work until the final time (Table 1), typically .

## 3 Results

### 3.1 General properties

The simulations initialized as described in the previous section evolve into turbulent, magnetorotational instability (MRI) driven, optically thin accretion flows. The inflow of matter takes place in the bulk of the disk where turbulent stresses take angular momentum out of the gas. The bulk of the flow is relatively strongly magnetized due to the non-zero radial component of the initial magnetic field at the equatorial plane, with magnetic pressure contributing ultimately to of the total pressure in the region . The magnetocentrifugal forces drive outflow from the surface layers of the disk. Due to its low density, the outflowing gas does not contribute significantly to the total radiative emission.

Different densities at the onset of the simulations determined the efficiency of radiative cooling – the larger the density, the higher the relative bremsstrahlung cooling rate. Similarly, the larger the electron temperature for a given gas temperature, the more efficient is the cooling. The latter effect is clearly visible in Fig. 1 which shows the poloidal slice of the gas density (left panels) and temperature (right panels) for three models with roughly the same accretion rate of a few : f5t100, f5t30, and f5t10. While the densities are of the same order, the temperatures are not. The simulation with the coldest electrons (f5t100, , top panel) is as hot as it gets with gas temperature close to the virial temperature, as expected for a non-radiative accretion flow. As soon as electrons get hotter relative to ions (model f5t30, middle panel), the radiative losses are significant enough to affect the gas properties – the gas temperature (the mean of electron and ion temperatures) noticeably decreases. Gas at the equatorial plane, at is now at , compared to for model f5t100. Increasing the electron relative temperature increases further the cooling effect. For the lowest ion-to-electron temperature ratio (, bottom panel, model f5t10), the gas temperature at this location falls down to . In addition to the gas temperature, the radiative cooling have an effect on the disk thickness - the two runs affected by cooling have noticeably smaller thickness (see Table 1), but they are far from collapsing to a thin disk.2 At the lowest accretion rates, the radiative emission is dominated by synchrotron, while at the largest, for which the radiative cooling has significant impact on gas properties, it is predominantly Compton cooling (contributing, e.g., to of cooling at the equatorial plane for model f5t30) and, to lesser extent, bremsstrahlung (see, e.g. Narayan & Yi, 1995; Esin et al., 1997).

### 3.2 Mass and energy transfer rates

Quasi-stationary, i.e., accreting at on average constant rate, accretion flows show mass and total energy transport rates independent of radius. Under such condition, neither gas or energy accumulates at any radius, but preserves their average radial profiles. We measure the accretion rate and luminosity in total energy in simulated accretion flows by integrating the mass and energy fluxes over the BH horizon.

The mean accretion rate is therefore defined through,

 ˙M=∫π0∫2π0√−g⟨ρur⟩dϕdθ, (3)

where stands for gas density, denotes the radial velocity, and is the metric determinant.

The luminosity in total energy (binding, kinetic, radiative, magnetic, and thermal, see Sa̧dowski et al., 2016 for the related discussion) is given as,

 Ltot=−∫π0∫2π0√−g⟨Trt+Rrt+ρur⟩dϕdθ, (4)

where and are components of the matter and radiation stress energy tensors, respectively.

The radiative-only luminosity is given by the integral of the latter,

However, because radiative energy is not conserved (radiation can be generated and absorbed by the gas), we need to carefully choose the radius at which to perform the integration. Ideally, we would like to measure the luminosity at infinity. However, we are limited by the region where the accretion flow has settled to the converged solution. We decided to perform the integrals at , which is not infinity, but encompassed the region where most of the radiation in optically thin disks is formed.

Assuming that the energy liberated in an accreting system can ultimately escape only either by radiation or mechanical energy, we can calculate the kinetic component as,

### 3.3 Radiative and kinetic luminosities

The efficiency associated with given luminosity is defined as,

 η=L˙Mc2, (7)

where is the accretion rate through the BH horizon and is the speed of light.

Table 1 lists the nine simulations we performed and gives the accretion rates and efficiencies obtained by applying formulae from the previous section to time- and azimuth-averaged data spanning the last of each simulation. These efficiencies are presented graphically in Figs. 2 and 3.

The bottom panel in Fig. 2 shows the kinetic efficiency calculated from the difference between the total and radiative luminosities (Eq. 6). In all cases the kinetic efficiency is close to 3%. This value is consistent with that observed in the non-radiative simulation described in-depth in Sa̧dowski et al. (2016)4. It is interesting to note, that the kinetic efficiency does not go down with increasing accretion rate and radiative luminosity. This is in part different from previous idealized AGN models (Churazov et al., 2005). It reflects the fact that radiation comes from the innermost part of the accretion flow and the underlying gas cannot efficiently respond to cooling in the last phase of accretion, crossing the horizon with intact dynamical properties (the total efficiency is determined by how bound such horizon crossing gas is). For the lowest accretion rates in each subset, this kinetic energy extraction rate dominates the total luminosity. In the case of the largest accretion rates, for which the radiative output is significant, the kinetic and radiative luminosities have comparable magnitudes.

The kinetic and radiative energy extraction rates are compared in Fig. 3. As already discussed, the kinetic energy dominates when radiative emission is negligible, i.e., for the lowest accretion rates. Once mass transfer rate increases, so does the radiative efficiency, which for the hottest electron case can even become comparable or exceed the kinetic component.

Table 2 puts together the estimates for the luminosities, (either kinetic or radiative), at which the transition to the luminous regime () takes place. For the coldest electrons, we have shown that the accretion flows remain radiatively inefficient up to and the radiative output never reaches the level of the kinetic one. The intermediate case behaves in similar way. For the hottest electron case (), however, the accretion flow starts generating radiative luminosity comparable to the mechanical one already at . We discuss the implications of these findings in the next Section.

## 4 Implications

Observational studies of AGN radiative luminosities and mechanical output (Section 1.1) have shown that the radiative component becomes comparable with the kinetic one at the luminosity (e.g., Fig. 12 in Russell et al., 2013). In this work we have shown that this critical luminosity is sensitive to the temperature of the electrons, which determines the efficiency of radiative cooling. Therefore, we can use the observational result to constrain the properties of the electron population in optically thin black hole accretion flows.

In Table 2 we listed the rough estimates of the critical luminosities at which the kinetic and radiative luminosities become comparable. For the hottest electrons case we considered (ion to electron temperature ratio, ), this transition occurred for luminosity as low as (see Fig. 3), significantly below the observational constraint. For both the intermediate () and the coldest () cases, the radiative output is significantly below the mechanical one even for kinetic luminosities , well above the constraint.

This argument implies that to satisfy the observational constraints one should expect the temperature ratio to fall between the hottest and intermediate case, i.e., . We therefore suggest that astrophysical plasmas in optically thin accretion flows (at least in the nuclear region) produce electrons with temperature between 3% and 10% of the ion temperature.

The physics of electron heating in accretion flows depends on the phenomena taking place on the smallest lengthscales of collisionsless plasmas (see, e.g., Quataert & Gruzinov, 1999) and is still debated. Howes (2010) proposed a prescription for the fraction of heating going into the electron population based on theoretical models of the dissipation of MHD turbulence in almost collisionless plasmas and which have been recently applied to simulations of two-temperature accretion flows (Ressler et al., 2015; Sadowski et al., 2016). This model predicts that the efficiency of electron heating increases in highly magnetized regions, and the equilibrium species temperature ratio is determined by the magnetization of local plasma. Putting the range of temperature ratios favored by this work, , into the formula for the equilibrium temperature ratio (Sadowski et al., 2016), one recovers the corresponding range of gas-to-magnetic pressure ratio, . This level of magnetization is in agreement with the level expected from the saturated state of magnetorotational instability (e.g., Davis et al., 2010; Shi et al., 2010), what further supports the arguments made, and suggests that the accretion flows of interest are not threaded by significant net poloidal magnetic fluxes which would imply much stronger magnetization (Bai & Stone, 2013; Salvesen et al., 2016).

A key result worth to remark is the robustness of the kinetic efficiency, which remains stable around 3 percent value regardless of the Eddington ratio, at most varying by 1 percent. While broadly accepted that the maintenance mode of AGN feedback resides in the kinetic regime, given the ubiquitous imprints of AGN outflows via X-ray cavities, shocks, and gas uplift (Section 1), the kinetic output in the quasar regime is still debated5. Previous models assumed a strong decrease of the kinetic power at increasing Eddington ratios (e.g., Churazov et al., 2005), however recent observational data has started to detect the presence of cavities even in systems with a quasar source. E.g., McDonald et al. (2015) show the presence of X-ray cavities in Phoenix cluster which also hosts an X-ray bright quasar emitting at a few percent of the Eddington rate. Albeit the very high Eddington ratios remain to be tested, our work corroborates the importance of mechanical AGN feedback also in the transition to the quasar-like regime, and thus even at high redshifts (as suggested by the observational sample in Hlavacek-Larrondo et al., 2015).

Finally, as anticipated in Section 1, the robust constraint and stability of the mechanical efficiency allows us to link such micro value to the macro properties of the self-regulated AGN feedback loop (mediated via CCA), which re-heats and preserves the cooling cores of galaxies, groups, and clusters in a state of quasi-thermal equilibrium throughout the cosmic time (e.g., Gaspari, 2015 and refs. within). Such unification model is presented and discussed in-depth in the companion work, Gaspari & Sa̧dowski (2017).

## 5 Caveats

We base our work on simulations performed with a numerical method using some simplifying assumptions. Most importantly, we decided not to track the energy contained in the electron fluid independently of ions, but to fix the ion to electron temperature ratio. This approach effectively arbitrarily prescribes the balance between the electron heating and Coloumb coupling which determines the temperature ratio. We adopted this simplification to get a direct handle on the electron temperatures and to avoid numerical issues that arise when identifying dissipation for the purpose of electron evolution (see discussion in Sadowski et al., 2016). The trade-off is that we force the temperature ratio to be spatially uniform – which is not the case in the whole volume, but is close to constant inside the bulk of the accretion flow (Ressler et al., 2015; Sadowski et al., 2016). Furthermore, we do not track the efficiency of Coulomb coupling, which at the largest accretion rates (and densities) can couple electrons and ions strongly enough to prevent large discrepancy between their temperatures. On the other hand, if a given temperature ratio is feasible, then the above results apply, as the radiative properties depend ultimately on the electron temperature, and not on the processes which determine its value.

We limited our set of simulations to accretion flows on non-rotating BHs. If the BH spin is non-zero, one may expect higher efficiency of extracting both radiative and mechanical energy (Tchekhovskoy et al., 2011; McKinney et al., 2015; Sa̧dowski et al., 2016), and it is likely the transition luminosities identified in this work will change6. However, if BH growth occurs mainly through chaotic accretion, as expected for most massive galaxies, groups, and clusters (Gaspari et al., 2013; Gaspari et al., 2015; Gaspari et al., 2017), one should expect the average value of the BH spin in AGN to be close to zero (King & Pringle, 2006). In particular, a sample as in Russell et al. (2013) should be biased toward the properties of a zero-spin BH population.

It has been understood recently that the level of magnetization and the topology of magnetic field in an accretion flow are not unique, and depend on the large scale properties of the field. Magnetic field may be considered another degree of freedom in determining the flow properties, which we leave to future work. By choosing a particular configuration of the initial seed magnetic field in the equilibrium torus, we chose to study accretion flows which are relatively weakly magnetized, and which do not lead to the saturation of the field at the BH horizon. The opposite limit would be the magnetically arrested disk (MAD, Narayan et al., 2003), which shows properties dramatically different from the weakly magnetized state (SANE – using the nomenclature in Narayan et al., 2012). Whether low luminosity AGN accretion flows are SANE or MAD is still debated. In this work we focused only on the SANE scenario.

## 6 Summary

We performed a set of general relativistic, radiative, magneto-hydrodynamical simulations to study the transition from radiatively inefficient to efficient state of accretion on a non-rotating BH. We studied a range of ion to electron temperature ratios, and simulated flows corresponding to accretion rates as low as , and as high as . We have found the following:

(i) The radiative efficiency increases with accretion rate, and the increase occurs earlier for lower ion to electron temperature ratios (i.e., hotter electrons). For , the radiative output becomes significant (exceeding 1% of the rest mass energy flux) already at -. For colder electrons, this transition takes place only at and for and , respectively (Fig. 2).

(ii) The mechanical output of the accretion flows is insensitive to the accretion rate, with an efficiency remaining stable near of . Such micro efficiency can be exploited to close a unified self-regulated AGN feedback model which shapes massive galaxies, groups, and clusters (see the companion Gaspari & Sa̧dowski, 2017). We remark this is also valid as entering the quasar regime, corroborating the importance of mechanical AGN feedback over a large range of Eddington ratios (and thus redshifts).

(iii) By comparing the simulated radiative and mechanical efficiencies with the observational results constraining the luminosity at which the radiative output becomes comparable with the mechanical power (Russell et al., 2013), we show that our models are consistent with observations when the ion to electron temperature ratio is within a range .

## 7 Acknowledgments

The authors thank Sean Ressler for comments on the manuscript. AS and MG acknowledge support for this work by NASA through Einstein Postdoctotral Fellowships number PF4-150126 and PF5-160137, respectively, awarded by the Chandra X-ray Center, which is operated by the Smithsonian Astrophysical Observatory for NASA under contract NAS8-03060. Support for this work was also provided by NASA Chandra award number G07-18121X. The authors acknowledge computational support from NSF via XSEDE resources (grant TG-AST080026N), from the PL-Grid Infrastructure, and the NASA/Ames HEC Program (SMD-16-7251).

### Footnotes

1. Although tempting, simulating accretion flows with accretion rates higher than is extremely challenging because such flows collapse to a geometrically thin disk which accretes very slowly, and, therefore, obtaining converged solution in a reasonable volume requires both extreme numerical resolution and long simulation time.
2. Their properties resemble to large extent the luminous hot accretion flow (LHAF) solutions proposed by Yuan (2001). Detailed study of the collapse would require precise treatment of Coulomb coupling (see Discussion).
3. Our own experiments have shown that axisymmetric accretion flows are characterized by significantly higher radiative efficiency than the three-dimensional counterparts – enforcing axisymmetry changes the nature of turbulence leading to the formation of rapidly radiating, high density filaments in the inner region which are not present in non-axisymmetrical solutions.
4. We remind this 3% characterizes thick accretion flows on non-rotating BHs. For non-zero BH spin this disk-related efficiency may go up; on top of it, a generated relativistic jet may overcome the total energy budget, albeit having difficulty to efficiently couple with the host halo due to the very high collimation.
5. We note that even in the quasar-like regime the radiative power has severe difficulty in coupling with the gas, especially at  kpc scales.
6. Another complication is that it is uncertain how efficiently the chaotic inflow would align with the BH spin axis.

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# Math Help - matrix

1. ## m

Suppose $
{A^T}
$
is transpose of $A$ . Does the basis for the $Col A$ always $=$ basis of $Row({A^T})$.

2. The rows of $A^T$ are precisely the columns of A.

3. However i need to somehow prove or disprove that basis for the $Col A$ always $=$ basis of $Row({A^T})$. So i was wondering if there is a theorem or a definition that could be help me here. i know that if it is linearly independent it does not matter if i transpose them. it would still be linearly independent thus rank of A would also equal to Rank of A transpose. Since the columns of A are the rows of $Row({A^T})$, finding a basis for $Col(A)$ is equivalent to finding a basis for $Row({A^T})$. Is there any case this would not hold, i guess is the question that i am asking.
Thanks,
KC

4. Well I guess the question is a little vague.

What I am saying is clearly any basis for the Col(A) will be a basis for Row( $A^T$) and the other way too. These things are exactly the same thing Row( $A^T$)=Col(A). so if you look at the same thing on each side there is no way they are not the same.

But there is one concern. For instance there are lots of different bases you can pick for Col(A). But this also means there are lots of different bases that can be picked for Row( $A^T$). So on an individual basis you could certainly pick a different basis for each of the spaces, but both of those would be bases for the other space? Do you know what I mean?

like just take the matrix A to be the identity matrix on $\mathbb{R}^2$. It is symmetric so $A^T=A$.

The vectors <1,0> and <0,1> form a basis for Col(A)
The vectors <2,0> and <0,2> form a basis for Row( $A^T$)

these two sets are clearly not the same, but they are still a basis for the other space because the two spaces are the same. Know what I mean?

Your question is not well posed basically is what I am getting at. You say the basis for something, there are lots of choices for a basis.

5. I don't understand what you mean by being able to pick bases. I thought a basis for Row A are the non-zero rows of echelon form or row reduced echelon form. And basis for col A are the pivot columns of A. I think i am missing something.

6. Lets just talk about Col(A), I think you trouble is understanding what a basis is.
The number of pivot positions is how you determine the RANK of Col(A). It tells you the dimension of Col(A) as a vector space. This will tell you the NUMBER of elements in your basis.

DEFINITION: A basis is an ordered set of linearly independent elements which span the space.

With this definition comes many things. In a vector space every element can be uniquely represented as a linear combination of the basis. The trouble is there are uncountably many choices for a basis of a vector space (of degree at least 1).

Take $\mathbb{R}^3$

Standard basis
(1,0,0)
(0,1,0)
(0,0,1)

A DIFFERENT BASIS FOR $\mathbb{R}^3$
(0,2,0)
(9,0,0)
(0,0, $\pi$)

One choice is not THE basis, while most people pick the standard basis there is no reason one should discount any other possible choice of basis.

7. Does this mean that it'd be true only for a standard matrix? and it would not necessarily true for others?

8. What is a standard matrix? You mean the identity matrix? that was just an example since it is already in RREF. The matrix is a representation of a linear map. Would you talk about a basis for f if it were a polynomial?

A basis is talking about a vector space, you can find a basis for the domain or range, but not for a function (unless you are talking about the dual space to a vector space which is the collection of all linear functionals, then in fact you can pick a basis of functions for the dual space, but only because it is actually a vector space in its own right).

Counting the pivot positions simply tells you the rank of the matrix. That is the dimension of the image of the function. If you count 4 pivot positions of A it means col(A) has dimension 4, and so a basis will need to consist of precisely 4 linearly independent vectors which span Col(A). That is all you know.

It is up to you to pick the basis, there are uncountably many choices.

9. Originally Posted by igodspeed
I don't understand what you mean by being able to pick bases. I thought a basis for Row A are the non-zero rows of echelon form or row reduced echelon form. And basis for col A are the pivot columns of A. I think i am missing something.
Perhaps youu are missing the fact that there exist an infinite number of different bases for a vector space. Yes, the non-zero rows of the echelon form of A form a basis but that is not the only basis.

10. Originally Posted by HallsofIvy
Perhaps youu are missing the fact that there exist an infinite number of different bases for a vector space. Yes, the non-zero rows of the echelon form of A form a basis but that is not the only basis.
I got what you mean but i wanted the answer to this question to be centered around that definitional.


text:
# Talk:Digital video

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## Article terminology

I came to Wikipedia to look up some of the terminology in article Digital Video & HDTV.

Digitized Video Standards NTSC PAL SMPTE 259M CCIR 656 EU95 SMPTE 292M

Serial Digital Video Formats SMPTE 259M ITU-R601 CCIR 656 EU95 SMPTE 292M

Are "ITU-R601" and "CCIR 656" the same as CCIR 601? Did EU95 become DVB ?

--65.70.89.241 17:55, 6 September 2006 (UTC)

Hi, I wanted to add along the terminology lines that for someone coming to the digital video page to simply gain a basic understanding of the topic, they would be confused by much of the language used throughout the article. There are advanced mathematical equations and large words that aren't appropriate for all learners. By eliminating some of the article's advanced elements or explaining more complicated terminology within the article, the page would become much more accessible to readers. Avam11 (talk) 19:42, 2 February 2018 (UTC)

## Uncompressed digital video bit rate

The article mentions the bit rate of roughly 400 MBit/s. By my calculation, standard definition digital PAL would have a bit rate of around 160 MBit/s at 8-bit per sample and 4:2:2 color sampling.

Agreed; the "400MB/s" comment probably hinted at the upper limits of those interfaces, rewriting that sentence would be good.--C xong (talk) 12:11, 24 June 2009 (UTC)

## comparison with analogue formats

can someone make a section on this please? --AlexOvShaolin 19:44, 7 January 2007 (UTC)

## digital vs electronic, video vs still

In the first paragraph the article claims Digital video was first used in television cameras and then goes on to explain the first VTR which was of course analogue. The next paragraph claims 1960s lunar probes used digital imaging, but in reality they were also analogue. Furthermore the latter sent still images, not video. Third, the article claims that the Sony Mavica was an example of digital video, but once again this camera stored still image as an analogue signal. It appears the author confuses rastered images and digital, still image and moving image all the time. This article needs a serious cleanup. Anorak2 (talk) 04:44, 17 November 2007 (UTC)

Done :) Anorak2 (talk) 04:56, 17 November 2007 (UTC)

I don't know how (or why), but the article returned to the state that warranted these complaints. I've reverted this wholesale rewrite of the history section because it was worse than uninformative--it was incorrect. As noted above, neither magnetic tape nor electronics imply digital. 98.202.51.77 (talk) 10:50, 17 March 2010 (UTC)

Here's a source that shows that digital video on magnetic tape arrived when Sony introduced the D1 videocassette in 1988. [1] It's counter intuitive given the binary nature of magnetic poles, but magnetic tape can be used to store both analog and digital signals; unless a video or audio source is converted to a digital signal, the magnetic tape is storing analog. Only recently did we have sufficiently sophisticated electronics to encode audio/video into a digital signal. 98.202.51.77 (talk) 04:47, 19 March 2010 (UTC)

## Technical overview

This section needs a lot of work. There's some pov in there, some wrong information and a strange bias toward this one specific camera:

• Motion Strobing: This line, "They can both shoot at 24 frames per second, which results in motion strobing (blurring of the subject when fast movement occurs)." is incorrect. That is not the definition of motion strobing (it's the definition of "motion blur"). Motion strobing as defined in Digital Compositing for Film and Video By Steve Wright, "an unpleasant artifact in which an object's motion takes on a jerky and "staccato" appearance due to the lack of appropriate motion blur."
• Motion strobing is not unique to 24 fps or to progressive scan. Neither is motion blur. They're both simply more apparent at slower frame rates.
• The article implies that motion strobing is a reason that "progressive scanning video cameras tend to be more expensive than their interlaced counterparts". The two are unrelated. (Progressive scan cameras tend to be more expensive because the chips have to read more data at a specific moment rather than getting to read half the data and spreading it out).
• Panasonic DVX100: Why mention this specific camera and no other?
• "Progressive scan camcorders such as the Panasonic DVX100 are generally more desirable because of the similarities they share with film." This is pov: The "film look" is not necessarily more desirable.
• "Note that even though the digital video format only allows for 29.97 interlaced frames per second [or 25 for PAL], 24 frames per second progressive video is possible by displaying identical fields for each frame, and displaying 3 fields of an identical image for certain frames." This is a specific case applied incorrectly to the general (I suspect as a result of getting too much technical information from a single source that is specific to the camera mentioned above). There is nothing inherent to dv that requires a frame rate of 29.97. Only to NTSC. It is true that many cameras use a pulldown to record 24 frames with their native 29.97 hardware, but true 24p cameras exist and there's no reason hardware can't be built at any arbitrary frame rate.

If there is no objection, I'll take a crack at fixing this up. Focomoso (talk) 22:15, 18 June 2008 (UTC)

## History

I suggest that this section (really most of the article in general) be more thoroughly referenced. There are numerous assertions that have no references at all. I'm really surprised that someone ranked this as a B-grade article. It's well written, but is it verifiable? I wish I could help, but I stumbled on this looking for an explanation of the various digital video file formats. I think the point of the encyclopedia is to be a resource for folks looking for general info, but it should also should give them the ability to go further by pointing to the sources of information. Best of luck with it MichaelKrobinson (talk) 17:35, 25 June 2008 (UTC)

## A "fact" about PIXELs ?

I'd suggest it is untrue to state that pixels "have only one property - their colour". Surely, their shape is also a property - an important one in that non-square pixels are an essential determinant of the aspect ratio? Johncurrandavis (talk) 10:42, 23 February 2010 (UTC)

## Unit of W and H

In the section, "Overview of basic properties", it is stated that the units of W and H are pixels. Another statement that could be made is that W is in units of pixels/line and H is in units of lines/frame. If W is 480 pixels/line and H is 640 lines/frame it follows that

${\displaystyle pixels\ per\ frame=pixels/frame=480(pixels/line)\times 640(lines/frame)=}$
${\displaystyle 480\times 640(pixels/frame)=307,200\ pixels\ per\ frame}$.

On the other hand if you use pixels as units for W and H, you get the rather meaningless units of pixels square for pixels per frame. RHB100 (talk) 21:33, 30 June 2010 (UTC)

## Towards an amendment...

In order to amend the untrue "fact" I allude to above, perhaps something along the lines of the following will be considered acceptable?..

Digital video comprises a series of orthogonal bitmap digital images displayed in rapid succession at a constant rate. In the context of video, these images are called frames. The rate at which frames are displayed is enumerated in frames-per-second (FPS).

Since each frame is an orthogonal bitmap digital image, it comprises a raster of pixels (short for "picture elements"), areas of the image meant to be too small to be perceived individually, but which determine the colour and brightness information of the whole image. Pixels have two properties: colour and dimension. The colour of a pixel is represented by a fixed number of bits. The more bits defining each pixel, the more subtle the variations in colour that are reproduced; this is called the colour depth (CD) of the video. Pixels are rectangular in shape (possibly square), and the dimensions of the pixels comprising each frame are a determinant of the aspect ratio of the video. For example, the frames of a PAL video 720 pixels across and 576 pixels high will, if the video aspect ratio is 4x3, be composed of rectangular pixels with proportions of 1.0666 to 1; but if the video aspect ratio is 16x9, then the pixels will be rectangular with proportions of 1.4222 to 1. (In other words, "widescreen" is in this case achieved by altering the dimensions rather than the number of pixels.)

Johncurrandavis (talk) 17:10, 5 July 2010 (UTC)