We thank the referees for the constructive criticism. We have responded to all the comments as detailed below. The corresponding changes to the paper have been marked in blue. > Referee: 1 > Major issues: > A) From the article: The authors assume that current radiation > hydrodynamic (RHD) codes compute an optically thick and optically thin > time step constraint individually and everywhere in the computational > domain irrespective of the true optical depth of the region, and that > those constraints are then applied as taking the global minimum of the > two. One example, where this assumption is made in the article is in > the derivation of the scheme on page 3, lines 28/29; but actually this > assumption is described throughout the article, e.g. in Eq. 9 or during > the later data analyses to show that such an approach denotes a major > problem. Based on this assumption, it is postulated that the sum of the > two constraints solves the problem. > Comments: > A1) Frankly, the situation is — from my point of view — exactly > the opposite. I do not know of any RHD code, in which an optically > thick time step constraint would be applied in an optically thin region > or vice versa. That such an approach would cause a problem is quite > obvious. Hence, there is no need for a new time step constraint. Prompted by the referee, we have now looked at many more papers. We have provided a more detailed discussion about the problem in our Introduction, as well as throughout the manuscript as appropriate. We also highlight several misconceptions in the literature pertaining to this issue. If we missed anything important, we would like to know. It seems to us that there is no detailed treatment that is comparable to ours, where a thorough numerical study is performed investigating the time step constraint in *explicit* radiation hydrodynamics. The work by Freytag+12 seems to us the best reference, where they use, like us, the cooling time of Spiegel+57, but they argue (without demonstrating any numerical results) that there should be just one radiative Courant-like coefficient. In Sec.3.3 and Fig.2(b) of our revised paper, we demonstrate that there are clearly 2 different coefficients in the radiative time step constraint. > A2) The postulated solution (using the sum of the optically thick and > optically thin time step constraints) is not original or new. Actually, > this sum was already introduced for computing the cooling timescale in > Rafikov (2004), see Eq. 4. The paper "Fast Accretion of Small Planetesimals by Protoplanetary Cores" doesn't seem to be concerned with radiation, so we are puzzled as to what the referee is referring to exactly. Nevertheless, we have added several new references in the Introduction with appropriate discussion regarding the necessity for a radiative time step constraint. If there is another paper by Rafikov (2004), which is not listed on ADS, we apologize and would be grateful for gaining access to the right paper. > B) From the article: The basic research question is outlined as the > need for a new time step constraint in RHD simulations, especially for > the case of hot stellar atmospheres. As explained by the authors, such > a serious time step constraint can not be derived analytically, but has > been shown empirically in simulations. Comment: Unfortunately, the only > simulations which seems to demonstrate this serious time step problem > are reported in unpublished work by the author(s), namely Brandenburg & > Spiegel (2006, unpublished) ad Brandenburg & Das (2018, unpublished). Even > more, as pointed out in the article, very similar studies by other authors > (Owocki & Sundqvist 2018, Sundqvist et al. 2018) do not encounter such > a time step problem. For the other example of hot and cold accretion > disks, the authors eventually conclude themselves that those disks are > not dominated by the radiative time step. And a similar study by Coleman > et al. (2018) does not report any time step issue as well. Hence, based > on the available literature, the time step problem (which denotes the > basis of the research question of this article!) is not verified. We understand the concern of the referee. However, it is important to note that there are differences in the numerical treatments employed by the aforementioned works, which we now address. We have now made significant modifications in the Introduction regarding our motivation. We discuss a number of different RHD codes in the Introduction. For example, CO5BOLD is explicit; Zeus is implicit; HERACLES is semi-implicit; Athena has various modules with different kinds of radiation transfer solvers. We also discuss issues with the Davis+12 paper regarding the Athena implementation. We agree in principle with the Freytag+12 treatment for CO5BOLD, but highlight a major quantitative difference in setting the radiative Courant coefficients. > C1) The title of the article suggests a time step constraint for RHD > simulations. But actually, the authors only study hydrodynamic simulations > coupled to time-independent radiation transport; this is severely more > restricted than the general title suggests. We do solve the time-dependent equations for velocity and specific entropy. However, radiation propagates with the speed of light, which is fast, so its evolution and time step do not affect our calculations. This is a commonly invoked assumption in many of the RHD codes mentioned above. > C2) Furthermore, in the special case of time-independent radiation > transport, the time step is not directly constrained by the radiation > transport, but can only be influenced indirectly by either a) modifying > the local temperature (which set the local speed of sound) or b) > modifying the pressure slope (which affects the local gas velocity). The > article does not show under what kind of circumstances this would > yield a numerical problem, which cannot be solved by the standard CFL > condition for explicit hydrodynamic simulations. Especially in the case > of equilibrium solutions (such as for the hot stellar atmosphere), it > is unclear, how a time-independent radiation transport scheme can cause > a time step constraint. Due to the fact that this denotes the basis of > the study, this indirect connection should be discussed in detail. We do not fully understand the referee's concern. We would like to point out that, in all our cases, we are talking only about the cooling time; see Eq.(6) of the revised version, which was also used by Freytag+12 (as we now include). We hope that the extensive revisions done throughout the current version addresses the referee's concern. > D) The analytically derived time step constraints include two > pre-factors in front of the optically thin and optically thick time > step constraint. These required pre-factors are found by comparison with > the numerical simulations and evaluate to 0.2 and 4.2 here. It remains > unclear, if these values hold for other physical systems or setups. This > should be discussed in the article at reasonable length. The third paragraph of the conclusions now addresses this point explicitly. The detailed analysis of the new Sec.3.3 shows that the value 4.2 is now closer to 4. This has been corrected throughout the manuscript. > Minor comments: > - The setup of the simulation models should be described in the > associated setup sections completely, i.e. the simulation should be > reproducible by reading the setup section alone. Currently e.g. the > model specification (i)-(v) are only given in the caption of Figure 2, > although these labels are further used throughout the main text and in > other captions. We have now added lots of more detail both in the main part of the text and also in the two new appendices A and B. > - p.3 l.21: It should be noted that the quantity c_gamma is the ratio > of radiation energy density over thermal energy density times the speed > of light times a constant factor divided by the adiabatic index of the > gas (explicitly: for an adiabatic index of gamma = 4/3, the constant > is unity). For cp/cv=4/3, we find c_gamma/c = 4 aT^4 / [(4/3) rho*cvT] = 3, not 1. We have now mentioned this general connection in the revised version below equation (4). > - p.7, l.11/12: “The same opacity prescription will also be used > in our accretion disc models described below.“ Later on in the main > text it is stated that a constant opacity (for electron scattering) > is used. What is the correct description? We have now clarified this in the text, namely, which coefficients and prescriptions are used exactly when; see paragraph below equation (27) and first paragraph of page 12 in the revised version. > - p.13, l.7/8: “DNS“ what is DNS? Abbreviations should not be used > in headings if avoidable. We have now avoided introducing this abbreviation throughout the revised version and do not even mention direct numerical simulations explicitly. > Referee: 2 > > 1. Since the paper is devoted to the study of the stability of > the numerical solution of equations (10-13) using the PENCIL code, > the authors should describe the finite-difference schemes which are > used to solve these equations. In this paper, the authors verify the > proposed stability condition (7) empirically. It should be noted that a > more rigorous approach would be to study the stability of a difference > scheme used to solve the transfer equation with the help of the von > Neumann method, for example (see, Numerical Methods in Astrophysics: > An Introduction. Eds. P. Bodenheimer, G.P Laughlin, M.Rozyczka and H.W > Yorke, Taylor & Francis Group, 2007). We have now added a new Sec.3.2 to address computational aspects. We recall that the goal of the paper is to find out what sets the radiative time step constraint in numerical simulations, especially in the Pencil Code; see the new Fig.2(b) for a very visual demonstration. We verify the validity of our proposed time step constraint with the help of several examples. In this sense, we have achieved our goal. A detailed analytic study using von Neumann analysis would be welcome. We recall, however, that this paper is to appear in a special issue about the Pencil Code. should be > 2. There is no closed description of the problem statements in sections > 4.1, 4.2, 4.3, 4.4. Before describing each simulation, it is necessary to > give the initial and boundary conditions for all independent variables > (density, temperature, pressure, velocity, intensity) in each spatial > direction under consideration (x, y, z). For all simulations, the duration > of the simulation and time instances for which the figures are given > should be presented. In the simulations of the structure of accretion > disks, the distance from the star should be indicated, as well as the > stellar mass, accretion rate, disk height 'z_phot', disk scale-height > 'H_p' should be given. I recommend specifying the simulation time in > units of the rotation period at a given distance from the star. We have now added significant detail to all these sections; see the blue highlighted regions of these sections, as well as appendices A and B. > 3. Section 4.2. is devoted to solution of equations (16-23) > of hydrostatic equilibrium. There is no time dependency in these > equations. It is not clear how the problem of choosing a time step has > to do with the study of the equilibrium state? Figures 5 and 6 in Sec.4.2 of the revised version show the "Expected time step constraints for stellar surface layers". These plots are obtained by solving the equations, which are now (20)-(26). We have now added a paragraph in the beginning of this section to explain this. > 4. In their work, the authors repeatedly refer to their unpublished > works (see p.2, line 20, reference to Brandenburg, Spiegel, 2006; line > 21, reference to Brandenburg & Das 2018; as well as p.8, lines 53, 57; > p.13, line 9). I consider this unacceptable for a scientific paper. These > references should be excluded. The purpose was to make the reader understand the background. We have now modified the formulations throughout the revised version. > 5. In section 5, the authors discuss approaches to solving the problem > of a stringent time step constraint, which is due to the fact that the > cooling time is much less than the time of a sound wave crossing a cell of > size ‘dz’. To solve problem ‘A’, the authors suggest artificially > increasing the opacity, and for problem ‘B’, artificially reducing > the speed of light. These suggestions contradict physical principles > and cannot be used. At least, the authors should discuss the limits of > applicability of these assumptions from the point of view of the physics > of the processes under consideration., i.e prove that these artificial > corrections will not lead to incorrect solutions. We fully agree. It was never our intention to "solve" the problem. The idea is to prepare the ground for addressing physically relevant aspects of the full problem. We have now added a clear discussion about this in the last two paragraphs of Sec.5 in the revised version. > > 1. P.3, line 26: expression ‘l -> infinity’ should be replaced by > ‘k*l >> 1’. The authors call the ‘c_gamma’ as a photon diffusion > velocity. However, this quantity enters the second term in equation > (6), which describes the cooling rate in an optically thin medium, in > which the concept of diffusion of photons has no physical meaning. It is > necessary to reformulate the definition of ‘c_gamma’ or explicitly > clarify this contradiction. We have now changed "l -> infinity" by "k*l >> 1" in the concerned sentence of the revised version. We would like to point out that 'c_gamma' makes its first appearance in the cooling rate given by Eq. (4) of the revised version, which is valid for any mean free path l; this is where we define c_gamma as well. We have now added an additional definition of how c_gamma is related to the radiative relaxation time. In fact, c_gamma indeed enters the first term (optically thick cooling time) of Eq. (7) in the revised version, since '\chi = c_gamma l/3'. Hence, the name 'photon diffusion velocity' seems to be justified in want of a better one (also see Barekat & Brandenburg 2014). The fact that c_gamma also appears in the optically thin term of Eq. (7) can be interpreted as it representing a "characteristic photon crossing velocity in a optically thin medium" in this case. We have added a clarification just below Eq. (8) in the revised version, stating the same. > 2. It is necessary to describe in more detail the formulation of > the problem statement, based on the analysis of which the value of the > coefficient C = 4.2 is chosen on p.4, line 50. The sentence on line 48 > should refer to equation (6), not (7). We have now added a new Sec.3.3 and Figs. 1 and 2, to address explicitly how we arrived at the value of the coefficient C_rad^thin. This we now find to be closer to 4 after a more rigorous investigation, which has been corrected throughout the manuscript. We have also corrected the concerned sentence with the correct equation number, which happens to be equation (7) in the revised version. > 3. Quantity ‘g_z’ (p.5, lines 37-38) is not defined in the text. We have now replaced g_z by the scalar g to avoid confusion. > 4. Equality sign (‘=’) is missing in equation (22) on p.6. We have now corrected this in equation (25) of the revised version. > 5. It is necessary to correct the following typos: replace > ‘ionisation’ with ‘ionization’, ‘parametrisation’ > with ‘parametrization’ (p.7, line 34), ‘parallelised’ with > ‘parallelized’ (p.8, line 36). We use this format as we believe the journal uses UK spelling. It can be changed during the proof reading stage if required. > 6. I recommend the authors to remove the empty space in the Figures with > temperature profiles (left panel in Fig. 1, upper left panel in Fig. 5). The empty spaces in these figures (Fig. 3 and Fig. 7 in the revised version) is to clearly bring out how far from zero the photosphere occurs, as well as the gradient in temperature in different systems. We would prefer to retain them if possible. > 7. It is necessary to specify the units of velocity components u_x, > u_y, u_z in Figure 9. The unit is km/s, which we have now added in the caption of Figure 11 in the revised version.