Numerical Studies of Low Mass Star Formation

1
Numerical Studies of Low Mass Star Formation

• Stella Offner
• U.C. Berkeley, Physics Department

2
Outline

• Introduction background on star formation.
• Numerical methodology Orion AMR code.
• Molecular cloud simulations movies and results.
• Multigroup radiation transfer work and proposed
  research.

3
Molecular Clouds

• Star formation takes place in high density
  regions of cold molecular gas, mostly H2.
• Number densities n 102 -106 cm-3.
• Temperatures are 10 - 20K.
• Masses range from small clumps (102 Msun) to
  giant molecular clouds (106 Msun).
• Cloud sizes range from 10 - 150 pc.
• These regions also contain 1 dust, which is
  micron size icy grains made of C, Si, Fe, and O.

NICER (Near IR Color Excess Revisited) extinction
maps, Lada et al PPV, 2006
4
Molecular Clouds

• Spectra of these regions also show a much higher
  than thermal linewidth, indicating large
  velocities of a few km/s characteristic of
  supersonic turbulence.
• Long dense filaments form at the intersection of
  planar shocks, so turbulence is likely
  responsible for the filamentary appearance of the
  gas in star forming regions.

Simulation showing log column density of
hydrodynamic supersonic turbulence for a Mach
number (vrms/cth) 5.
5
Jeans Instability

• Gravitational instability plays a key role in the
  contraction of gas of ? 10-20 g cm-3 to stellar
  densities of ? 1 g cm-3.
• Collapse occurs when thermal pressure is no
  longer sufficient to support against gravity.
  This occurs for perturbations greater than a
  characteristic length scale, the Jeans length
• Turbulent compressions from shock waves may play
  an important role in creating such over dense
  regions.

?J 2 ?/kJ (? cs2 / G ? )1/2
6
Molecular Cloud Cores (Stage 1)

• Regions with high enough densities and cold
  enough temperatures will become gravitationally
  unstable and collapse.
• The collapse occurs in an inside out fashion,
  so that the central region begins to fall
  inward first.
• Cloud cores are typically 0.1pc in size and have
  ratios of rotational to gravitational energy of
  ?0.02.
• Multiple stars can form from a single core.
• Such cloud cores are observed to follow Larsons
  Laws
• They are gravitationally bound.
• They follow the linewidth-size relation
  ?v?R0.38.
• They have constant column density.

7
Protostar (Stage 2)

• Rotation causes the core to flatten into a disk.
• The gas loses angular momentum until it can be
  accreted by the protostar.
• Protostar is heavily obscured by gas and dust.
• Luminosity is powered by accretion.

8
Outflows /T Tauri (Stages 3 4)

• Deuterium burning begins in the center.
• Outflows overcome ram pressure.

• Accretion and outflows weaken.
• High variability in luminosity and accretion.

3
9
Important Questions

• What are the origin and characteristics of the
  observed turbulent motions?
• What determines the distribution of cloud core
  sizes?
• What determines the distribution of initial
  stellar masses? How does this depend on the core
  IMF?
• Given that many stars have at least one
  companion, what influences the multiplicity of a
  system?
• What is the dominant formation mechanism of brown
  dwarfs?
• What role do magnetic fields play?

10
Important Questions

• What are the origin and characteristics of the
  observed turbulent motions?
• What determines the distribution of cloud core
  sizes?
• What determines the distribution of initial
  stellar masses? How does this depend on the core
  IMF?
• Given that many stars have at least one
  companion, what influences the multiplicity of a
  system?
• What is the dominant formation mechanism of brown
  dwarfs?
• What role do magnetic fields play?

11
Numerical Methods AMR

• Orion solves the multi-fluid equations of
  compressible gas dynamics using a conservative
  higher-order Godunov scheme (Truelove et al
  1998). These equations are discretized and solved
  on a set of 3D Cartesian grid points.
• Orion uses an approximate linear Riemann solver
  to solve the hydrodynamics (Pember et al 1998).
• The Poisson equation is solved using a multigrid
  method provided by hypre (Truelove at al 1998).

12
AMR

• The domain consists of a Cartesian base grid,
  level 0 additional grids, typically a factor of
  2 or 4 smaller, are added based on a user
  specified refinement criterion.
• The domain is advanced in time using recursive
  timestepping, e.g., level 0 is advanced one
  timestep dt0, then level 1 is advanced dt1 until
  it catches up to level 0. After the first level 1
  advance level 2 advances dt2 until it reaches a
  time dt1and so on
• To enforce conservation between levels, the code
  performs a sync solve, which matches fluxes
  across the level boundaries.

gcm-3
Recursive timestepping. Howell Greenough 1998.
gcm-3
13
AMR
Simulation showing log column density with 9
levels of refinement. The grid boundaries are
given in black.
gcm-3
gcm-3
14
Additional Physics

• Turbulence Injection
• Apply kinetic energy in the form of velocity
  perturbations to the gas velocity at every time
  step.
• The scale of the perturbations corresponds to
  wavenumbers of 1 ? k ? 2.
• Sink Particles (Protostars)
• New AMR levels are added when the Jeans density
  is exceeded on a given level.
• For collapsing gas that exceeds the refinement
  criterion on the maximum level, we insert a sink
  particle.
• The sink particles move, gravitationally
  interact, accrete surrounding gas, and merge with
  one another.
• Critical Jeans density ? 0.252 (? cs2 /
  G ?x2 )

15
Molecular Cloud Simulations

• Investigate differences in decaying vs. driven
  turbulence on system characteristics
• Short (Ballesteros-Paredes et al. 2006) vs. long
  (Tan et al. 2006) cloud lifetime.
• High (dynamic) vs. low (equilibrium) star
  formation rate.
• Investigate correlations between core properties
  and stellar multiplicity
• High (Goodwin et al. 2006) vs. low (Lada 2006)
  incidence of multiplicity.
• Brown dwarf formation
• Turbulent compression
• Core fragmentation
• Disk fragmentation
• Ejection and truncated accretion
• Investigate accretion as a function of time
• Core Accretion (Krumholz et al. 2005)
• Stars are formed from gravitationally bound
  cores.
• Once the core mass is accreted or expelled, the
  accretion is negligible.
• Competitive Accretion (Bonnell et al. 2001)
• Mass segregation with the largest mass objects in
  the region of highest gravitational potential.
• Initial masses are small and the final mass is
  determined by the location in the clump.
• Mass falling onto the core causes it to contract
  (increases stellar density).
• Brown dwarfs are formed by ejection.

16
Molecular Cloud Simulations

• Investigate differences in decaying vs. driven
  turbulence on system characteristics
• Short vs. long cloud lifetime.
• High vs. low star formation rate.
• Investigate correlations between core properties
  and stellar multiplicity
• High vs. low incidence of multiplicity.
• Brown dwarf formation
• Turbulent compression
• Core fragmentation
• Disk fragmentation
• Ejection and truncated accretion
• Investigate accretion as a function of time
• Core Accretion
• Stars are formed from gravitationally bound
  cores.
• Once the core mass is accreted or expelled, the
  accretion is negligible.
• Competitive Accretion
• Mass segregation with the largest mass objects in
  the region of highest gravitational potential.
• Initial masses are small and the final mass is
  determined by the location in the clump.
• Mass falling onto the core causes it to contract
  (increases stellar density).
• Brown dwarfs are formed by ejection.

17
Molecular Cloud Simulations

• Investigate differences in decaying vs. driven
  turbulence on system characteristics
• Short vs. long cloud lifetime.
• High vs. low star formation rate.
• Investigate correlations between core properties
  and stellar multiplicity
• High vs. low incidence of multiplicity.
• Brown dwarf formation
• Turbulent compression
• Core fragmentation
• Disk fragmentation
• Ejection and truncated accretion
• Investigate accretion as a function of time
• Core Accretion
• Stars are formed from gravitationally bound
  cores.
• Once the core mass is accreted or expelled, the
  accretion is negligible.
• Competitive Accretion
• Mass segregation with the largest mass objects in
  the region of highest gravitational potential.
• Initial masses are small and the final mass is
  determined by the location in the clump.
• Mass falling onto the core causes it to contract
  (increases stellar density).
• Brown dwarfs are formed by ejection.

18
Initial Conditions

• Drive initially constant density field of n2500
  cm-3 with velocity perturbations having
  wavenumbers between 1? k ?2 and 1D vrms 5 cs for
  3.5 crossing times without gravity.
• Decaying (left) and Driven (right) turbulent runs
  after 1 dynamical time with gravity.

gcm-2
Log Column Density
Log Column Density
pc
pc
19
Stage 1 Protostellar core properties

• A core is defined by the mass around a sink
  particle that exceeds a minimum density, is
  gravitationally bound, and has mass above the
  local Jeans mass.
• The linear fits in (a) show that Mdriven R1.04
  and Mdecay R1.17.
• The linear fits in (b) give jdriven R1.21 and
  jdecay R1.15.
• The cores for both are mostly prolate or
  triaxial.
• The linewidth-size fit gives vdriven R0.26 and
  vdecayR0.22.

Decaying Turbulence Simulation
Driven Turbulence Simulation
20
Stage 2 Selecting Cores

• Look at interesting cores at higher resolution
  with a barotropic equaton of state.
• For the first case, we chose a long filament that
  forms a total of three objects in the decaying
  turbulence simulation, the smallest merge with
  the largest object at about 1Msun. The final
  object obtains 22.1 Msun.
• For the second case, we chose a 13.6 Msun core,
  which forms a single object that accretes all the
  mass in the original clump. It is the second
  collapsing object in the simulation.
• In the decaying simulation, this core becomes
  24.6 Msun, which likewise forms one object.

Log column density plots of cores forming in a
filament within the decaying turbulent box.
21
Case 1 Filament Properties

• The core is very prolate (filamentary) with
  aspect ratios of 0.30 and 0.07.
• There is 16.8 Msun in bound mass when the sink
  particle is 2.2 Msun.
• The maximum length is 0.28pc.

g/cm3
cm
cm
22
Filament Collapse
Case 1 T 17,500 years Size 0.7
pc Resolution 3AU 1D Machtot 2.4 Dt0 216
yrs Dt10 77 days
23
Resolution Comparison

• Case 1
• At 23,000 yrs after the formation of the first
  sink particle, all the original objects in the
  filament have drained onto the central disk,
  which the largest and first object was formed.
• The chart of the left shows the sink masses at
  two times for 3AU (subscript 10) and
  6AU(subscript 9).

Log column density at 23,200 years with 3AU
(left) and 6AU(right) grid resolution.
24
IMF and Accretion History
Case 1
At 23,000 years, the sink masses have a 89
agreement with a Kroupa (2001) universal IMF for
a cluster of 6 stars.
Mass and accretion rates of the sinks as a
function of time. A total of 6 sinks were formed
with lifetimes of more than 1 coarse time step.

25
Case 2 Properties
Decaying (top graph) and driven (bottom graph)
core properties are summarized in the table to
the right. The cores are analyzed when the
first sink particle formed has a mass of 0.1 Msun
( 0.1Myr).

g/cm3
cm/s
cm
cm
26
Core Collapse

Case 2 T 9,400 years Size 2,000
AU Resolution 6AU 1D Machtot 4.87 Dt0 348
yrs Dt10 248 days
27
Driven vs. Decaying Masses
Case 2 The driven core remains
more filamentary and forms fewer stars due to the
larger turbulent support. The decaying core
forms more stars, 5 of which are ejected from the
system by dynamical interactions (marked with an
x). Once the bound gas is stripped away, the star
is deprived of accretion matter and its mass is
basically fixed. At this time, the decaying run
has 4 brown dwarfs and 1 large planet.
Log Column Density with Velocity Vectors,
t74,000yrs Left Driven Right Decaying
gcm-2
cm
cm
28
Accretion History
Case 2 Mass and average accretion rates of
the sink particles as a function of time.
Driven(top), Decaying (bottom) Offner, Klein
McKee, in prep.
Driven turbulence
Decaying turbulence
29
IMF
Case 2
Driven Agreement with a Kroupa IMF is 69 with
5 cores at a time 75,000 yrs after the first
sink particle formation.
Decaying Agreement with a Kroupa IMF is 46 with
9 cores at a time 80,000 yrs after the first
sink particle formation.
30
Conclusions

• The simulations are converged and hence can
  sufficiently resolve the core fragmentation.
• The driven turbulence IMF is in agreement with
  the IMF observed by Kroupa (2001), despite the
  small number of statistics.
• Driven turbulence supports against large scale
  gravitational collapse and so fewer and less
  massive objects form.
• Most of the mass of the objects comes from
  accretion rather than merging of sink particles.
• The decaying turbulence simulation exhibits
  competitive accretion and promotes brown dwarf
  formation by ejection.

31
Radiation Transfer Motivation

• Equations of state are suitable when there are no
  substantial radiation sources. In reality,
  forming stars heat their surroundings,
  influencing the fragmentation and star formation
  in the nearby gas.
• Since both the Jeans mass, which dictates the
  stability of dense gas against gravitational
  collapse, and the Toomre Q, which determines the
  stability of disks against fragmentation, are
  sensitive to the sound speed, and hence the gas
  temperature, proper radiation treatment is
  necessary to accurately treat feedback effects.
• Boss et al. (2000) show significant differences
  at high density in calculations using flux
  limited diffusion vs. a barotropic equation of
  state.
• Krumholz et al. (2006) show that radiation
  transfer suppresses fragmentation in high mass
  clumps containing a high mass protostar when
  compared to the isothermal and barotropic cases.
• Work in 2D by Yorke and Sonnhalter (2002)
  determined that multifrequency had a significant
  effect on the accretion and final mass of massive
  stars.

32
Grey Flux Limited Diffusion Radiation Transfer
Scheme

• Equations for flux limited grey radiation
  transfer.
• The radiation energy Er represents the total
  radiation energy integrated over all frequencies.
• B is the Planck emission function integrated over
  all frequencies.
• ? is the flux limiter.

33
Multigroup Radiation Transfer Scheme

• Multigroup is an approximation of the exact
  multifrequency radiation transfer equations.

34
Multigroup Radiation Transfer Scheme

• Replaces the grey flux limited diffusion equation
  with the multigroup flux limited diffusion
  problem.
• This does not affect the solution of the gas
  dynamics equations since the radiation-gas
  coupling term includes only the integrated
  radiative flux.
• Solves for the radiation temperature fully
  implicitly, where the emission term is fixed as a
  function of T in the inner loop and corrected
  until T converges.
• Conservation is enforced with sync-solve.

35
Multigroup Radiation Transfer Test Problem

• Due to the complexity of multifrequency radiative
  transfer, no analytic or semi-analytic tests
  exists for the exact frequency dependent
  radiation transfer. However, test problems do
  exist for linearized multifrequency diffusion
  (Shestakov and Bolstad, 2004).
• The solution is obtained by substituting the Wien
  function for the less tractable Planck emission
  function
• 8?h?3 c-3 (eh?/kT-1)-1 ? 8?h?3c-3e-h?/kT
• The nonlinearity in the Wien exponent is
  surmounted by fixing the temperature, T to a
  reasonable value for the problem, T0.

36
Multigroup Radiation Transfer Test Solution

• Initial dimensionless parameters Tmatter 1.0
  for 0?x ?0.5, Tmatter 0 for 0.5?x?4.0 rho
  1.0 and u 0 everywhere (no hydrodynamics), Trad
  0.
• 64 frequency groups.
• Tf 0.1Tmatter is the fixed temperature.
• To the right, is the solution at t 1.0t0
  graphed with the semi-analytic solution
  calculated in Shestakov Bolstad (2004).

Shestakov Offner 2006, submitted to JCP.
37
Proposed Simulations

• Goals
• Characterize the accuracy of grey radiation
  transfer in comparison to multigroup radiation
  transfer.
• Determine the number of groups necessary for a
  multigroup advantage -- is this affordable?
• Determine the effect of radiation feedback on
  fragmentation, accretion rates, and final stellar
  mass.
• Parameter space compare different realizations
  in core mass, shape, mach number, dust model.
• Simulations
• Group convergence study of core collapse with 2,
  4, 8 groups at low resolution.
• 5 Msun low mass turbulent clump with grey flux
  limited diffusion, driven turbulence, averaged
  dust model.
• with 4 groups multigroup flux
  limited diffusion, frequency dependent dust
  model.
• with 8 groups multigroup flux
  limited diffusion
• with barotropic equation of state
  for comparison.

Total time 450,000 hrs.
38
Objectives Timeline

• Self-gravitational hydrodynamic simulations of a
  large cluster comparing the effects of driven and
  decaying turbulence on SF.
• AMR MGFLD implementation and testing.
• Integration of MGFLD into Orion.
• Low resolution core collapse test with MGFLD to
  compare group requirements (non-turbulent).
• Study of radiative feedback effects on low mass
  star formation using GFLD and MGFLD with
  identical initial conditions (turbulent).
  Include a barotropic EOS comparison case.

39
Objectives Timeline

• Self-gravitational hydrodynamic simulations of a
  large cluster comparing the effects of driven and
  decaying turbulence on SF.
• AMR MGFLD implementation and testing.
• Integration of MGFLD into Orion.
• Low resolution core collapse test with MGFLD to
  compare group requirements (non-turbulent).
• Study of radiative feedback effects on low mass
  star formation using GFLD and MGFLD with
  identical initial conditions (turbulent).
  Include a barotropic EOS comparison case.

• Offner, Klein, and McKee in prep.

40
Objectives Timeline

• Self-gravitational hydrodynamic simulations of a
  large cluster comparing the effects of driven and
  decaying turbulence on SF.
• AMR MGFLD implementation and testing.
• Integration of MGFLD into Orion.
• Low resolution core collapse test with MGFLD to
  compare group requirements (non-turbulent).
• Study of radiative feedback effects on low mass
  star formation using GFLD and MGFLD with
  identical initial conditions (turbulent).
  Include a barotropic EOS comparison case.

• Offner, Klein, and McKee in prep.
• Shestakov Offner sub to JCP.

41
Objectives Timeline

• Self-gravitational hydrodynamic simulations of a
  large cluster comparing the effects of driven and
  decaying turbulence on SF.
• AMR MGFLD implementation and testing.
• Integration of MGFLD into Orion.
• Low resolution core collapse test with MGFLD to
  compare group requirements (non-turbulent).
• Study of radiative feedback effects on low mass
  star formation using GFLD and MGFLD with
  identical initial conditions (turbulent).
  Include a barotropic EOS comparison case.

• Offner, Klein, McKee in prep.
• Shestakov Offner sub to JCP.
• Testing phase.

42
Objectives Timeline

• Self-gravitational hydrodynamic simulations of a
  large cluster comparing the effects of driven and
  decaying turbulence on SF.
• AMR MGFLD implementation and testing.
• Integration of MGFLD into Orion.
• Low resolution core collapse test with MGFLD to
  compare group requirements (non-turbulent).
• Study of radiative feedback effects on low mass
  star formation using GFLD and MGFLD with
  identical initial conditions (turbulent).
  Include a barotropic EOS comparison case.

• Offner, Klein, McKee in prep.
• Shestakov Offner sub to JCP.
• Testing phase.
• 6 months setup, computational running time, and
  analysis.

43
Objectives Timeline

• Self-gravitational hydrodynamic simulations of a
  large cluster comparing the effects of driven and
  decaying turbulence on SF.
• AMR MGFLD implementation and testing.
• Integration of MGFLD into Orion.
• Low resolution core collapse test with MGFLD to
  compare group requirements (non-turbulent).
• Study of radiative feedback effects on low mass
  star formation using GFLD and MGFLD with
  identical initial conditions (turbulent).
  Include a barotropic EOS comparison case.

• Offner, Klein, McKee in prep.
• Shestakov Offner sub to JCP.
• Testing phase.
• 6 months setup, computational running time, and
  analysis.
• 24 months setup, computational running time,
  and analysis.

44
Objectives Timeline

• Self-gravitational hydrodynamic simulations of a
  large cluster comparing the effects of driven and
  decaying turbulence on SF.
• AMR MGFLD implementation and testing.
• Integration of MGFLD into Orion.
• Low resolution core collapse test with MGFLD to
  compare group requirements (non-turbulent).
• Study of radiative feedback effects on low mass
  star formation using GFLD and MGFLD with
  identical initial conditions (turbulent).
  Include a barotropic EOS comparison case.

• Offner, Klein, McKee in prep.
• Shestakov Offner sub to JCP.
• Testing phase.
• 6 months setup, computational running time, and
  analysis.
• 24 months setup, computational running time,
  and analysis.

Graduate Summer 2009 !!
45
Questions?

46
On the sky

• Grey Milky Way
• Black Molecular Clouds
• Taurus 30 pc, 140 pc from earth.
• Orion is 120 pc, 500 pc from earth.
• Below Orion in CO 1-0.

47
Core Observations
Lada et al. PPV 2006.
Left B68 IR extinction (dust column density),
1000 AU resolution. Right Azimuthally averaged
extinction profile, 500 AU resolution, with
pressure-confined isothermal sphere fit overlaid
(red dots).
48
Computational Time Estimation of Proposed
Simulations

• Estimate the time for a fiducial low mass
  simulation, whose initial conditions are taken
  from observation (Kirk et al. 2005)
• Observations show that low mass cores have
  density profiles that are well fit by
  Bonnor-Ebert Spheres (Lada et al. 2006).
• Take values for the critical density, nentt
  5x105 cm-3,RBE10,000AU, and T 10K, e.g. L1521D.
• The dynamical time is given by the free-fall
  time, tff 200,000 yrs.
• The simulation will have 6 levels of refinement
  with 643 base grid, where the number of cells on
  each level will be set by the radiation gradient
  criterion and Jeans criterion. General estimates
  give 300,000 cells per level.
• The coarse timestep on the base grid will be
  determined by the largest velocity of the
  problem, in the case it is the Keplerian velocity
  of the accretion disk. vKep 1d6 cm/s.
• The total cost of a run with GFLD is given by
  40,000 hrs.
• Assuming, conservatively, that the cost for N
  frequency groups is NtGFLD, then for 4 and 8
  groups the cost will be 120,000 hrs and 220,000
  hrs, respectively.
• For comparison, the non radiative transfer case
  and the initial test runs with fewer levels of
  refinement will require much less time.
• Total time 450,000 hrs.

49
Accretion Models

• Core Accretion
• Stars are formed from gravitationally bound
  cores.
• Once the core mass is accreted or expelled, the
  accretion is negligible.
• Competitive Accretion
• Mass segregation with the largest mass objects in
  the region of highest gravitational potential.
• Initial masses are small and the final mass is
  determined by the location in the clump.
• Mass falling onto the core causes it to contract
  (increases stellar density).
• Brown dwarfs are formed by ejection.

50
Core 2 Binaries
Driven 1 triplet system with masses (1.14, 0.31,
0.44). The separations are highly variable with
d50, 116 AU.
Decaying 2 binaries with separations of 83 AU
and 44 AU. The mass of the pairs respectively
are (1.17,1.07) and (2.07, 0.029).
51
Radiation Hydrodynamics Equations
52
Multigroup Radiation Transfer Test Problem

• Due to the complexity of multifrequency radiative
  transfer, no analytic or semi-analytic tests
  exists for frequency dependent radiation
  transfer. However, two tests problems do exist
  for linearized multifrequency diffusion
  (Shestakov and Bolstad, 2004).
• The solution is obtained by substituting the Wien
  function for the less tractable Planck emission
  function
• 8?h?3 c-3 (eh?/kT-1)-1 ? 8?h?3c-3e-h?/kT
• The nonlinearity in the Wien exponent is
  surmounted by fixing the temperature, T to a
  reasonable value for the problem, T0.
• The solution assumes an opacity of the form ?
  ?0?-3.
• This problem can be written in nondimensional
  form and can be scaled to the characteristic
  problem parameters x0, ?0,T0, ?0, ?0, t0.
• Thus, the nondimensional multifrequency diffusion
  equations become
• Integrating over each group gives

53
Multigroup Radiation Transfer Test Solution

• Initial dimensionless parameters Tmatter 1.0
  for 0?x ?0.5, Tmatter 0 for 0.5?x?4.0 rho
  1.0 and u 0 everywhere (no hydrodynamics), Trad
  0.
• 64 groups, with smallest group ?0 0 and ?1
  5.5d-4, and group spacing increasing by alpha
  1.1.
• Tf 0.1Tmatter is the fixed temperature.
• To the right is the solution at t 1.0t0
  graphed with the semi-analytic solution
  calculated in Shestakov Bolstad (2004).

Shestakov Offner 2006, submitted to JCP.
54
Multigroup Radiation Transfer Comparison with
Planck

• Initial dimensionless parameters Tmatter 1.0
  for 0?x ?0.5, Tmatter 0 for 0.5?x?4.0 rho
  1.0 and u 0 everywhere (no hydrodynamics), Trad
  0.
• 64 groups, with smallest group ?0 5.5d-4, using
  log spacing and ?max 15.0.
• Tf 1.0Tmatter is the fixed temperature.
• To the right is the solution at t 1.0t0
  graphed with the result using the nonlinear
  Planck emission function.

55
Radiation Transfer Grey Flux Limited Diffusion
with Radiation Pressure

• Explicit treatment, valid only in the static
  diffusion limit.
• 0th order in (v/c).
• Krumholz et al 2006.