Graziani F. (editor) Computational Methods in Transport

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Radiation Hydrodynamics in Astrophysics 7

have mostly been run on shared memory machines with few nodes so it is

unclear how well they scale to many processors with limited bandwidth.

3.2 Core-Collapse Supernovae

In the past, the supernovae “transport experts” have worked on supernova

light-curves. However, the transport schemes used in light-curve calculations

are almost entirely limited to solutions of a steady state or uniformly varying

medium. Only recently have radiation hydrodynamics codes (generally 1-

dimensional ﬂux-limited diﬀusion hydrodynamics codes) been applied to this

problem [H¨oﬂich et al. (1993), Blinnikov et al. (2000)]. The results from these

calculations are then post-processed to get detailed spectra.

The supernova “radiation-hydrodynamics experts” are actually the mod-

ellers of stellar collapse (here the neutrino is the transport particle). Core-

collapse supernova theorists came to radiation hydrodynamics by ﬁrst fo-

cusing on hydrodynamics and including increasingly sophisticated transport

techniques. These simulations have been at the cutting edge of computational

astrophysics due to a boost from codes built at national laboratories in the

sixties [Colgate & Johnson (1960), Colgate & White (1966)]. Core collapse

supernovae are powered by the potential energy released during the collapse

of the iron core of a massive star down to a ∼50 km proto-neutron star [Fryer

(2003)]. This energy leaks out of the proto-neutron star in the form of neutri-

nos. The neutrino mean free path ranges from a few cm in the proto-neutron

star to >100 km in the convective region where neutrino energy is deposited.

Figure 1 shows a slice from a 3-dimensional simulation of a core-collapse su-

pernovae. The three circles show the position where the optical depth out

of the star is roughly 0.05 for the µ/τ, anti-electron and electron neutrinos

(the innermost circle denotes µ/τ; the outermost circle denotes the electron

neutrinos).

A number of techniques have been used to study radiation hydrodynamics

in core-collapse supernovae. Tony Mezzacappa has developed some very nice

diagrams that diﬀerentiate these codes. Using a simpliﬁed version of such a

diagram dividing codes by spatial hydrodynamics dimension and tranpsort

technique (Fig. 2), we describe some of the key codes developed and their

relevant citation (recall that we limit our discussion to codes that have been

applied to real problems):

1: In one dimension, a range of transport techniques have been added to

both Lagrangian and Eulerian codes. Multi-group, ﬂux-limited diﬀusion

has long been used in stellar collapse [Bruenn et al. (1978)].

2: Very sophisticated have been used in 1-dimensional codes. Discrete tech-

niques (S

method) have been around for over a decade and are said to be

parallel although I have not seen scaling numbers [Mezzacappa & Bruenn

(1993)]. The variable Eddington factor technique has also been used in

8C.L.Fryer

Fig. 1. A slice in the x-z plane of the 3-dimensional collapse of a rotating (rotation

axis is the z-axis) star. The shading denotes entropy and the vectors denote velocity

(the vector length is the magnitude of the velocity and the direction denotes the

direction of ﬂow). The 3 circles denote the positions of roughly the τ =0.05 surface

for µ and τ neutrinos (inner circle), anti-electron neutrinos (intermediate circle)

and electron neutrinos (outer circle). The neutrinos are in the diﬀusive limit in the

proto-neutron star but pass through the transport regime into the free-streaming

limit in the convective region where neutrino heating drives the explosion

1-dimension [Thompson et al. (2003)]. It has only been used in one paper,

so it is not clear how well this technique truly works and it is not parallel.

3: Two-dimensional, single-energy ﬂux-limited diﬀusion has been done for

over a decade. The ﬁrst simulation using such codes to model collapse

through explosion was in 1994 [Herant et al. (1994)].

4: Two-dimensional, multi-group, ﬂux-limited diﬀusion is less common than

in 1-dimension(2:). Indeed, the transport that has been done was not true

2-dimensional transport. The diﬀusion was followed along rays (no lateral

transport) [Burrows et al. (1995)]

Radiation Hydrodynamics in Astrophysics 9

Gray Flux−

Limited

Diffusion

Fully Parallel

II III

Variable

Discrete

Eddington

Limited Diffusion

Gray Flux−

1.5 Multi−Group

Flux−Limited

Diffusion Along

Single Energy Multi Energy

Multi Angle

Multi Energy

Neutrino/Spatial Dimension

Multi−Group,

Flux−Limited

Diffusion

1.5 Variable

Eddington Factor

Discrete

Rays

Factor

Methods

Along Rays

Methods

Fig. 2. Simpliﬁed Mezzacappa diagram of the radiation-hydrodynamics models

used in core-collapse studies. The y axis shows the transport technique (single-

energy ﬂux-limited diﬀusion, multi-group ﬂux-limited diﬀusion, and more sophisti-

cated techniques that include angular eﬀects). The x axis denotes level of spatial

(and purportedly neutrino dimension). Note that some 2-dimensional techniques

only follow transport along rays. The numbers correspond to more detailed de-

scriptions in the text

5: Some of the most intense research has been focused on sophisticated tech-

niques intwo-dimensions. A variable Eddington technique has been devel-

oped [Scheck et al. (2004)] that follows transport along rays (no lateral

transport). This technique is not extremely parallelizable (diﬀerent nodes

are given diﬀerent energy groups to model). A discrete method has also

been developed [Livne et al. (2004)]. Similarly, it has only been parallelized

by putting diﬀerent energy groups on diﬀerent processors. Also, the one

paper using this technique did not appear to run the simulation for many

timesteps, and I suspect this technique will not be usable to do the full

collapse problem (requiring 100,000 timesteps).

6: Three-dimensional, single-energy ﬂux limited diﬀusion modelling the col-

lapse through explosion was done using SNSPH [Fryer & Warren (2002)].

This code is fully parallel and has scaled nearly linearly up to 512 proces-

sors.

This list of techniques far outstrips any other astrophysics ﬁeld in radiation-

hydrodynamics codes. But the limitations are already evident. First note that

there are few working codes able to model transport in more than 1 spatial

10 C.L. Fryer

dimension. Those that have passed our relaxed criterion for a “working code”

are either not parallelizable, not truly multi-dimensional transport schemes or

both. Indeed, the only true multi-dimensional radiation hydrodynamics code

is limited to single-energy ﬂux-limited transport. Astrophysicists have de-

scribed much more sophisticated techniques (even at this meeting), but they

have not put them into working codes, let alone parallelizable codes. There

is a lot astrophysicists can gain from the transport community in making

working codes.

4 SPH Radiation Transport

One of the few working multi-dimensional radiation hydrodynamics codes in

core-collaspe is a technique coupling smooth particle hydrodynamics (SPH)

to ﬂux-limited diﬀusion. The coupling, developed in 2-dimensions in 1994

[Herant et al. (1994)] and parallelized in 3-dimensions in 2002 [Fryer & War-

ren (2002)] has some twists to it and it is worth describing it in some detail.

Much of this description is taken from a paper submitted to the Astrophysics

Journal by Fryer, Rockefeller, and Warren (2005).

4.1 Transport Scheme

In core-collapse supernovae, we model the transport of 3 neutrino species

(l = ν

, ¯ν

,ν

where ν

corresponds to the τ,µ neutrinos and their anti-

particles that are all treated equally). Because neutrino number is the con-

served quantity, we transport neutrino number and then determine the en-

ergy transport by using the mean neutrino energies. The radiation transport

scheme in our SPH code is modeled after the technique to calculate forces

in SPH: we calculate symmetric interactions between all neighbor particles.

Hence, our ﬂux-limited diﬀusion scheme calculates the radiation diﬀusion in

or out of a particle by summing the transport over all neighbors (the equiv-

alent of all bordering cells in a grid calculation). The neutrino transport for

particle i is given by:

νl

/dt =



νl



νl

i→j

νl

− n

νl

j→i

νl



∇W

/ρ

(1)

and the corresponding energy transport is:

νl

/dt =



νl





νl

i→j

νl

− ξ

j→i



νl

j→i

νl



∇W

/ρ

(2)

where n

νl

are, respectively, the neutrino density and energy density in

particle i for species νl,

νl

is the mean neutrino energy, and ξ

j→i

is the

redshift correction for 

νl

as seen by particle i.

Radiation Hydrodynamics in Astrophysics 11

νl

is the limiter for the ﬂux-limited transport scheme. The simplest such

scheme for 3-dimensions is:

νl

=min(c, D

νl

)(3)

where c is the speed of light, D

νl

=2D

νl

/(D

νl

+ D

νl

) is the harmonic

mean of the diﬀusion coeﬃcients for the species νl of particles i and j and r

is the distance between particles i and j. This limiter was used by [Herant

et al. (1994)] and, for comparison with that work, by [Fryer & Warren (2002)],

but we have used a number of other ﬂux-limiters, all of which are valid under

this transport scheme [Fryer et al. (1999)].

Beyond some radius in a core-collapse simulation, neutrinos are essentially

in the free-streaming regime where transport is not necessary (unless one

wants to truly follow the radiation wave as it progresses through the star). We

do not model transport beyond this “trapping” radius. Instead we sum up all

neutrinos that transport beyond this radius and emit them using a lightbulb

approximation. That is, the material beyond this radius sees a constant ﬂux

and we can determine the amount of energy a particle gains (dE

/dt)from

neutrino interactions simply by using the free-streaming limit:

/dt = L

(1.0 − e

−∆τ

)(4)

where L

is the neutrino luminosity and ∆τ

is the optical depth of a parti-

cle i. This assumption is only valid if the total amount of energy imparted

(



/dt) from the neutrinos onto the matter is much less than the to-

tal neutrino ﬂux (L

). To guarantee this, we determine this trapping radius

by evolving it with time such that (



/dt)/L

is always less than some

value. This value was originally set to 0.1 by Herant et al. (1994), but in

recent calculations, we use 0.05.

Such a scheme can be easily converted into multi-group, but such modiﬁ-

cations have not yet been done. The scheme scales reasonably well on multiple

processors (for a 5 million particle run, the code has scaled nearly linearly

up to 256 processors on the Space Simulator Beowulf cluster and up to 512

processors on the ASC Q machine at Los Alamos National Laboratory). In

part, this scalability is due to the explicit nature of the transport scheme. In

general, explicit transport schemes strongly limits timesteps as the speed of

light, not sound, constrains the duration of the timestep. In core-collapse su-

pernovae, this constraint is not too onerous because the sound speed is nearly

a third the speed of light anyway, so the explicit transport scheme leads to

only a factor of 3 decrease in the timestep. But this explicit ﬂux-limited

transport scheme can be used in a much wider variety of problems where the

mean free path is very short for the smallest particles. Such scenarios occur

in many astrophysics problems.

12 C.L. Fryer

4.2 First Test

Testing radiation transport schemes in general would, and has, comprised

many papers in itself. Here we focus on a simple test comparing the SPH ﬂux-

limited scheme to a 1-dimensional grid-based ﬂux-limited transport scheme.

This test does not prove the applicability of ﬂux-limited diﬀusion to the

supernova problem, but it does show that our scheme for putting ﬂux-limited

diﬀusion into SPH does work.

For the initial conditions of this test, we use a spherically-symmetric neu-

tron star atmosphere (material below 10

gcm

−3

) for a neutron star roughly

130 ms after bounce. We map this structure onto our SPH particles, using the

shell setup described in Fryer & Warren (2002). We make a similar setup in a

1-dimensional grid using one zone per shell of SPH particles. With this setup,

we minimize the diﬀerences between the density and temperature structure

of our 1- and 3-dimensional models. We determine the trapping radius to be

25 km. Below this radius, we set the electron neutrino fraction (Y

νe

) to 0.15

with a mean energy of 10 MeV. Above this radius, Y

νe

is set to zero.

Our test focuses on the neutrino transport alone; we evolve only the elec-

tron neutrino fraction with time and hold the density, temperature, and elec-

tron fraction ﬁxed. We allow no new neutrino emission. The ﬂux arising from

Fig. 3. Neutrino luminosity versus time for a cooling neutron star using a simple 1-

dimensional ﬂux-limited diﬀusion scheme and the technique discussed here coupling

ﬂux-limited diﬀusion to smooth particle hydrodynamics. The initial discrepancy

arises from the fact that diﬀusion occurs over all SPH neighbors, allowing transport

across the eﬀective (in a 1-dimensional case) of several radial zones (instead of 1

zone in the 1-dimensional transport scheme)

Radiation Hydrodynamics in Astrophysics 13

our neutrino trapping radius versus time for both our 1-dimensional grid

simulation and our 3-dimensional SPH calculation is plotted in Fig. 3. The

initial ﬂux of the SPH calculation is higher, since the neighbors of any par-

ticle extend beyond the equivalent of an adjacent cell for the 1-dimensional

calculation. In general, the luminosity for both these calculations agree to

better than 3%.

References

[Blinnikov et al. (2000)] Blinnikov, S., Lundqvist, P., & Bartunov, O., Nomoto, K.,

Iwamoto, K. Radiation Hydrodynamics of SN 1987A. I. Global Analysis of the

Light Curve for the First 4 Months ApJ, 532, 1132 (2000)

[Bruenn et al. (1978)] Bruenn, S.W., Buchler, J. R., & Yueh,W. R. Neutrino Trans-

port in Supernova Models – A Multigroup, Flux-Limited diﬀusion Scheme Ap&

SS, 59, 261–284 (1978).

[Burrows et al. (1995)] Burrows, A., Hayes, J., & Fryxell, B. A. On the Nature of

Core-Collapse Supernova Explosions, ApJ, 450, 830–850 (1995).

[Colgate & Johnson (1960)] Colgate, S.A., Johnson, H.J., Phys Rev. Letters, 5,

573–576 (1960).

[Colgate & White (1966)] Colgate, S.A., White, R.H. ApJ, 143, 626–681 (1966)

[Fryer et al. (1999)] Fryer, C. L., Benz, W., Herant, M., Colgate, S.A. What Can

the Accretion-induced Collapse of White Dwarfs Really Explain? ApJ, 516,

892 (1999)

[Fryer & Warren (2002)] Fryer, C. L., & Warren, M. S. Modeling Core-Collapse

Supernovae in Three Dimensions ApJ, 574, L65 (2002)

[Fryer (2003)] Fryer, C. L. Stellar Collapse, International Journal of Modern

Physics D, 12, 1795–1835

[Fryer & Warren (2004)] Fryer, C. L., & Warren, M. S. The Collapse of Rotating

Massive Stars in Three Dimensions, ApJ, 601, 391–404

[Herant et al. (1994)] Herant, M., Benz, W., Hix, W. R., Fryer, C. L., & Colgate, S.

A. Inside the Supernova: A Powerful Convective Engine ApJ, 435, 339 (1994)

[H¨oﬂich et al. (1993)] H¨oﬂich, M¨uller, E., & Khokhlov, A. Light Curve Models for

Type Ia Supernovae - Physical Assumptions, their Inﬂuence and Validity A&A,

268, 570–590 (1993)

[Klein et al. (1996)] Klein, R.I., Arons, J., Garrett, J., Hsu, J.J.L.: Photon bubble

Oscillations in Accretion-powered Pulsars. ApJ, 457, L85–89 (1996).

[Kley 1989] Kley, W. Radiation Hydrodynamics of the Boundary Layer in Accre-

tion Disks. I- Numberical models A&A, 208, 98–110 (1989).

[Livne et al. (2004)] Livne, E., Burrows, A., Walder, R., Lichtenstadt, I., & Thomp-

son, T. A. Two-Dimension, Time-Dependent, Multigroup, Multiangle Radia-

tion Hydrodynamics in the Core-Collapse Supernova Context, ApJ, 609, 277–

287

[Mezzacappa & Bruenn (1993)] Mezzacappa, A., & Bruenn, S. W. A Numerical

Method for Solving the Neutrino Boltzmann Equation Coupled to Spherically

Symmetric Stellar Core Collapse ApJ, 405, 669–684 (1993).

[Murray et al. (1995)] Murray, S.D., Woods, D.T., Castor, J.I., Klein, R.I., McKee,

C.F.: Radiation Hydrodynamic Models of Eclipsing Low-Mass X-ray Binaries

ApJ, 454, L133–136 (1995).

14 C.L. Fryer

[Sandford et al. (1982)] Sandford, M.T., Whitaker, R.W., Klein, R.I.: Radiation-

Driven Implosions in Molecular Clouds ApJ 260, 183–201 (1982).

[Scheck et al. (2004)] Scheck, L., Plewa, T., Janka, H.-Th., Kifonidis, K., & M¨uller,

E. Pulsar Recoil by Large-Scale Anisotropies in Supernova Explosions PRL, 92,

0111031–011034

[Stein & Nordlund 2003] Stein, R.F., Nordlund, A.: Stellar Atmosphere Modeling,

ASP Conference Proceedings, Vol 288, Aug. 8–12, 2002, Hubeny, Mihalas,

Werner (eds.). San Francisco: Astronomical Society of the Paciﬁc (2003) p.519

[Thompson et al. (2003)] Thompson, T. A., Burrows, A., Pinto, P. A. Shock Break-

out in Core-Collapse Supernovae and Its Neutrino Signature, ApJ, 592, 434–

456 (2003).

[Turner & Stone (2001)] Turner, N. J., & Stone, J. M. 2001, A Module for Radia-

tion Hydrodynamic Calculations with ZEUS-2D Using Flux-Limited Diﬀusion,

ApJS, 135, 95–107

[Turner et al. (2003)] Turner, N. J., Stone, J. M., Krolik, J. H. & Sano, T. Local

Three-dimensional Simulations of Magnetorotational Instability in Radiation-

Dominated Accretion Disks ApJ, 593, 992 (2003).

Radiative Transfer

in Astrophysical Applications

I. Hubeny

Department of Astronomy, University of Arizona, Tucson, AZ 85721, USA

1 Introduction

Radiative transfer is particularly important in astrophysics. One reason is

quite understandable: radiation is in most cases the only information we have

(and will ever have) about distant objects (exceptions are detected neutrinos

from the Sun and supernova SN1987a, and in a near future the gravitational

waves). However, there is an even more compelling reason for the a need to

deal with detailed radiation transport in astrophysics: In many astronomical

objects the radiation is so strong that it signiﬁcantly contributes to the energy

and momentum budget of the medium. Therefore, radiation is not only a

probe of the physical state, but is in fact an important constituent.Inother

words, radiation in fact determines the structure of the medium, yet the

medium is probed only by this radiation.

Unlike laboratory physics, where one can change a setup of the experiment

in order to examine various aspects of the studied structures separately, we do

not have this luxury in astrophysics: we are stuck with the observed spectrum

so we should better make a very good use of it.

All that puts heavy demands on proper radiation transfer techniques in

astrophysics. Also, there are many diﬀerent situations and physical conditions

(e.g., temperature ranges from a few K to more than 10

K; there is a huge

range of microscopic processes that give rise to an absorption or emission

of a photon), so consequently there is a wide range of numerical techniques

used for treating radiation transport in astrophysics. This paper is not aimed

to provide a review of all or even the most important numerical techniques;

instead it aims at outlining the outstanding problems and ideas behind recent

numerical approaches. I also stress that although I talk about transport of

photons, many methods can be used for treating transport of neutrinos, which

is important for instance in numerical simulations of core-collapse supernovae.

Typical objects for which the radiation transport is important are above

all stellar atmospheres. In this ﬁeld, most numerical techniques were devel-

oped that are now being used in other parts of astrophysics. Other objects are

extrasolar planetary atmospheres (the solar system planets as well, but here

we have a lot of direct information thanks to various planetary missions).

Other important radiation-dominated objects are accretion disks of various

kinds—around supermassive black holes (quasars and active galactic nuclei);

16 I. Hubeny

neutron stars (X-ray binaries); white dwarfs (cataclysmic variables), and even

main-sequence close binaries. Transport of photons (and neutrinos at early

stages!) is important in supernovae, and radiative transfer is important in

curcumstellar structures like planetary nebulae and H II regions. Finally,

radiation transport is also important in global cosmological simulations of

large-scale structure of the Universe.

The basic textbook of astrophysical radiative transfer is [1] but it does

not cover modern and eﬃcient numerical transport techniques. Another text-

books, with emphasis more on hydrodynamical aspect of the problems, are [2],

and recent textbook [3]; both contain excellent discussions of radiation trans-

port techniques. Recent proceedings that contains a number of review papers,

and to which the reader is referred to for more information about modern

methods used in astrophysical radiative transfer, is [4].

2 Description of Radiation

In astrophysical formalism, the basic quantity describing the radiation ﬁeld

is the speciﬁc intensity of radiation, I(r,t,n,ν), deﬁned such that it is the

energy transported by radiation at position r, in a unit frequency range at the

frequency ν, across a unit area perpendicular to the direction of propagation,

n, into a unit solid angle, and in a unit time interval. The speciﬁc intensity

provides a complete description of the unpolarized radiation ﬁeld from the

macroscopic point of view. The speciﬁc intensity is related to the photon

distribution function, f,as

I = hνc f (1)

where h and c are the Planck constant and the speed of light, respectively.

We note that the same quantity is called angular ﬂux, and is usually denoted

as ψ(r,t,n,ν), in the neutron transport theory.

The transport (or transfer) equation is essentially the Boltzmann equation

for photons, which can be generally written as

∂f

∂t

+(v ·∇)f +(F ·∇

)f =





coll

, (2)

where v is the velocity, F is the external force, ∇

is the nabla operator in

the momentum space, and the term (Df/Dt)

coll

is the collision term. In case

of photons, v = cn,andF = 0 (in the absence of general relativistic eﬀects).

The transfer equation can then be written as



∂

∂t

+ n ·∇



I(ν, r, n,t)=η(ν, r, n,t) − χ(ν, r, n,t) I(ν, r, n,t) . (3)

where we write the collisional term in terms of the usual absorption coeﬃcient

χ and emission coeﬃcient η. The equation looks simple, but this is deceiving.