We thank the referees for their constructive criticism and comments. 
We have responded to all the comments as detailed below. The corresponding 
changes to the paper have been marked in blue.

> Referee: 2
> The manuscript is greatly improved now. I suggest making one more
> minor correction before the paper can be accepted for publication. In
> the last sentence of the first paragraph in Section 3.3, the authors
> should not state explicitly that the value of c_gamma exceeds the speed
> of light. From the physical point of view, it would be more reasonable
> to say that the cooling time, tau_cool ~ l / c_gamma, is much smaller
> than the sound wave crossing time, tau_s ~ \delta z / c_s, so that the
> acoustic time step is unimportant under considered parameters.

Is the referee suggesting to drop our mentioning of c_gamma altogether?
We felt it is not useful to suppress this background information, since it
would obscure the reproduceability, but we have now added the sentence:
"Thus, in the optically thin case, the cooling time l/c_gamma is much 
smaller than the sound wave crossing time, \delta z/c_s."

> Referee: 3
> I have reviewed the manuscript by Brandenburg and Upasana. Overall I
> think that this manuscript presents and interesting results that is
> worthy of publication, but that the authors need to think a bit more
> broadly about where their result is and isn't relevant, and put their
> work in a bit more context. I will organise this review into general
> conceptual and stylistic points, and then go section by section on
> more minor details.
> My first conceptual point is that I think that the time step
> constraint that the authors have derived cannot be valid in general
> when radiation pressure is very significant. In fact, to be precise, I
> think that their time step constraint is only valid for systems that
> are not in the dynamic diffusion regime (as discussed for example in
> Mihalas & Mihalas's textbook, or in the astrophysical literature for
> example by Krumholz et al. 2007, ApJ, 667, 626). This may not actually
> be a constraint compared to the assumptions they have already made,
> since the time-independent transfer equation they adopt is probably
> not valid in the dynamic diffusion regime in any case, but the point
> should be clarified nonetheless.

We agree with the referee and have added the following on page 6 in the
now penultimate paragraph of Section 2.
  
  Nevertheless, when the radiation pressure becomes extremely large, 
  it can in principal also restrict the time step. Although we have not 
  found realistic examples where this additional constraint becomes 
  limiting, we discuss  this possibility at the end of our penultimate 
  section 5.

We have now also added that we consider the nonrelativistic radiation 
hydrodynamics equations and refer to the work of Krumholz et al. (2007) 
for the discussion of the dynamic diffusion case, where the flow speeds 
become comparable to the speed of light; see the beginning of Section 3.

> To see why this is the case, consider a simple thought experiment:
> imagine a fluid that is incredibly opaque, so that the optical depth
> is cross any physical scale of interest is huge. Mathematically, this
> corresponds to taking the limit chi -> infinity and l -> infinity. In
> this case, from the authors' equation 7, we have dt_rad -> infinity,
> and the time step should be constrained only by the ordinary fluid CFL
> number. However, suppose that our system is completely dominated by
> radiation pressure, P_rad >> P_gas. (A physical example of such a
> system might be deep in the interior of a near-Eddington star.) In the
> case where we are only concerned with wave modes that are very large
> compared to diffusion scales, one can reasonably approximate this
> system as just an ordinary gas, but with an equation of state with
> gamma = 4/3 instead of 5/3, and completely ignore diffusive transport
> of radiation. The fundamental wave modes in this case are just
> radiation-acoustic waves, propagating at speed sqrt(4 P_rad / 3 rho),
> and the CFL time step should just be the ordinary CFL step with c set
> equal to the radiation-acoustic mode speed. The proposed time step
> constraint clearly gets this case wrong.

See page 24 for these changes, particularly Eq. 39 and table 4. 
We have chosen to demonstrate the new time step constraint, as 
suggested by the referee, by lowering the speed of light, which 
is referred to as the reduced speed of light approximation or RSLA.

> I think that the fundamental reason for the error is easily understood
> -- the analysis in equations 2 and 3 ignores advection of radiation by
> the gas in comparison to diffusive transport. The regime where this
> assumption breaks down is precisely the dynamic diffusion regime. Thus
> my first recommendation is tha the authors think through this case,
> see if they agree with me, and, assuming they do, go through this
> analysis in the paper to make it clear in which regimes their time
> step constraint can be applied. If they disagree with my analysis,
> they should explain why it is wrong, and supply their own analysis of
> what assumptions have to hold for their scheme to work.

We agree with the referee; see therefore our additions on page 24.

> My second conceptual point is that I am skeptical that a purely local
> time step constraint such as the one the authors propose can really
> work for arbitrary boundary conditions, or if one requires some
> additional assumptions. Again, I have a simple though experiment:
> consider the standard Marshak wave problem (frequently used as a test
> for RHD codes -- see Pomraning, 1979, JSQRT, 21, 249 for the original
> solution, but treatments can also be found in textbooks such as
> Zel'dovich and Razier). In this problem, one start with an initially
> cold medium with a surface, and at time zero a radiative flux is
> applied to the surface, generating a wave of rising temperature that
> travels into the medium. My concern is that, at least at the initial
> stages, equation 7 seems like it can't be correct in this
> case. The medium initially has chi ~= 0 because the opacity is
> temperature-dependent, and since T ~= 0 initially one also has c_gamma
> ~ 0. Thus equation 7 would seem to suggest that one could take
> arbitrarily large timesteps. That clearly can't be right. So can the
> authors clarify what additional assumptions they need to make to deal
> with the Marshak wave problem? Does their method only work if the
> boundary conditions are zero flux in, so that there can't be heating
> from outside the domain? Or do they need chi = constant? I'm not sure
> what is going wrong in the Marshak wave case, but it seems clear to me
> that something is. Again, it would be helpful to work through that
> case, which has an exact analytic solution, to see where the
> assumptions of the method break down.

We have difficulty following the argumentation of the referee. The Marshak
wave problem is time-dependent and seems to involve equations different
from those solved in our paper. If c_gamma -> 0, then there is no radiative 
cooling and thus no constraint from this cooling, so the time step must be limited
by other factors such as the CFL condition. We have now mentioned the
Marshak wave problem on page 2.

> Those are my two main conceptual points. More minor issues, in order
> of appearance:
> Section 1, 3rd paragraph: the authors' description of long
> characteristics is somewhat confused, because they assert that "In
> this case there is no time advancement in the radiation field, which
> is propagated instantaneously across all rays without imposing any
> direct time step constraint." While one can certainly drop the
> time-dependent term in a long characteristics method, there is no
> requirement to do so, and in fact there are plenty of long
> characteristics-based solvers where the time dependent term is
> retained, and radiation is followed explicitly in time along rays; the
> most notable example of this is probably in simulations of
> cosmological reionization, where one has to account for the finite
> speed of light. Long characteristics is just a way of discretising the
> angle dependence of the transfer equation, and the choice of whether
> to drop the time-dependent term is independent of the choice of
> angular discretization. Please rewrite this paragraph.

We agree with the referee and have rephrased our statement and have also
included a reference to time-dependent radiative transfer simulations
of cosmological reionization using long characteristics (see page 2):

  Relativistic effects are ignored and the radiation field is propagated
  instantaneously across all rays without imposing any direct time
  step constraint. We refer to Finlator et al. (2009) for a treatment
  of time-dependent radiative transfer simulations of cosmological
  reionization using long characteristics...

> Section 2, equation 1: the authors are solving the time-independent
> transfer equation, which is fine, since it has a wide range of
> applicability. However, please make it clear what assumptions have to
> be satisfied for the time-dependent transfer equation to apply. I
> think the main one is that one is only justified in using the
> time-independent transfer equation if c_gamma is much greater than the
> material velocity, so that the radiation field reaches equilibrium on
> timescales short compared to matter advection timescales.

Yes, this sounds plausible, but we have not investigated the
break-down of the time-independent approximation in this work. Thus, we 
prefer not to explicitly comment about it.

> A separate issue is that equation 1 ignores the distinction between
> scattering and absorption opacity, which related to the next point.

We have indeed ignored scattering, which is now explicitly mentioned on page 2:

  We also assume that the source function is just given by the Planck
  function and thus ignore the possibility that it depends on the mean
  intensity. This implies that scattering is treated as true absorption,
  as is commonly done (Freytag et al. 2012), but see the work of Skartlien
  (2000) for a detailed treatment of scattering.

> Section 3.4: equation 21 does not make sense for electron scattering
> opacity, because electron scattering is a scattering process, meaning
> that there is no exchange of energy between the matter and the
> case. In terms of equation 13, the kappa that appears on the RHS
> should be only the absorption opacity, not the scattering opacity, and
> in equation 15 it is not correct to adopt S = sigma_SB T^4 / 4 for a
> medium where scattering dominates the opacity; instead one should have
> (for isotropic scattering) S = 1/4 pi \int I d\Omega. Since the point
> of this paper is code testing, and in the actual examples that are
> computed later on the scattering opacity is generally sub-dominant
> compared to the Rosseland and H^- opacities (which really are
> absorption opacities) this is not a big deal, but please just fix the
> discussion here to clarify that you are treating electron scattering
> as an absorption rather than a scattering opacity just for the
> purposes of code testing, and that your method does not treat
> scattering.

We hope that the new sentence, mentioned in the point above, addresses 
this issue.

> Section 4.4: I'm a bit confused what is being done with the radiation
> in this test. Are the authors still treating it as just one
> dimensional in z, and ignoring transport in any other direction? If
> so, I'm a bit mystified by the point of this test, since, even though
> the hydrodynamics are multi-dimensional, the radiation is still 1D. Can
> the authors clarify (1) whether even for the 3D test they still
> propagate radiation only in the z direction, (2) if not, now have they
> discretised in angle?

We have now clarified this by writing "... three-dimensional simulation
(of both the gas and the radiation field)..."; see page 20, section 4.4.

> Section 5, after equation 37: the paper says that they best chance of
> studying photoconvection is in the relativistic regime, where c_s and
> c are not so different. However, didn't you throw out the possibility
> of treating relativistic problems by dropping the time dependence of
> the radiation field? The stated goal is to make c_s and c as close to
> one another as possible, so as to maximise the time step, but if c_s ~
> c, don't you have to retain the dI/dt term in equation 1?

We understand the referee's point and, hence, have now added the following 
sentence on page 24: 

"In that case, however, one can no longer neglect the time dependence of the 
radiation field."