•First core and main accretion phase
Stahler & Palla: Chapter 11.1 & 8.4.1 & Appendices F & G
Early growth and collapse
In a magnetized
until the central Σ / B
has surpassed the
of the cloud.
The structure that arises from the contraction is not yet a protostar but a temporary
configuration known as the first core. To describe its growth and rapid demise, let’s
neglect (for now) the important element of rotation and magnetic support.
•The isothermal approximation breaks down! As its density climbs, the central
lump becomes opaque to its own cooling radiation, and further compression
causes its internal temperature to rise steadily.
•The enhanced pressure decelerates
material drifting inward, which gently
settles onto the hydrostatic structure.
•The settling gas radiates, removing
energy from the outer skin and further
•The core eventually stops expanding
and begins to shrink.
M ~ 5x10-2 M! , R ~ 5 AU
" ρ ~ 10-10 g cm-3
The first core mostly consists of H2 molecules and this ensures its early collapse
2T + 2U + W + M = 0
The object builds up from the least rotationally
and magnetically supported portion of the
cloud, so we can ignore these terms for now
in the virial equilibrium equation.
(% R (−1
= 850 K '
With the addition of mass and shrinking of
the radius, T soon surpasses 2000 K and
collisional dissociation of H2 begins " T
starts to level off:
H2 dissociation begins
•the number of H2 molecules in the
core is XM/2mH (X = 0.70);
•the thermal energy per molecule is
3kBT/X = 0.74 eV (< 4.48 eV) @
T=2000 K .
During the transition epoch, even a
modest rise in the fraction of
dissociated hydrogen absorbs most
of the compressional work of gravity,
without a large increase in
As the density of the first core keeps climbing (whereas the T rise is damped by the
dissociation process), the region containing atomic H spreads outwards from the
center and increase the mass until the entire configuration becomes unstable and
collapses: recall the isothermal Bonnor-Ebert sphere becomes unstable when the
center to edge density ratio is ~14. This marks the end of the first core.
The collapse of the partially dissociated gas takes the central region to much
higher density and temperature " collisional ionization of the hydrogen.
The true protostar is born.
With a radius of several R!, a protostar of 0.1 M! has T > 105 K and density
~10-2 g cm-3.
The gas approaching the protostellar surface now travels at free-fall velocities >>
local sound speed. The steady rise in the protostellar mass gradually inflates this
supersonic infall region and the cloud collapse proceeds inside-out " main
'$ M* '$ R* '−1
= 61 Lsun & −5
% 10 M sun yr (% 1M sun (% 5Rsun (
This is the energy per unit time released by infalling gas that converts all its kinetic
energy into radiation as it lands on the stellar surface. Throughout the main
phase, Lacc is very nearly equal to Lrad , the average luminosity escaping.
The accretion luminosity is mostly generated close to the protostar’s surface. The
contribution from additional radiated energy (nuclear fusion + quasi-static
contraction) are typically minor compared to Lacc .
The protostar is gaining mass with its luminosity coming
mainly from external accretion.
This radiation is able to escape the
cloud because it is gradually degraded
into the infrared regime as it travels
outward. Observationally, then,
protostars are optically invisible
objects that should appear as compact
sources at longer wavelengths.
Most of the radiation is generated at
the accretion shock, which constitutes
the protostar’s outer boundary (matter
further inside is settling with relatively
The gas raining down on the protostar
originates in the outer envelope,
where gas cooling is efficient.
As the infalling gas continues to be
compressed, the radiation eventually
becomes trapped. The temperature
quickly rises inside the dust
photosphere (Rphot ~ 1014 cm), the
effective radiative surface of the
The dust envelope is the region bounded
by Rphot that is opaque to the protostar’s
Once the temperature climbs past about 1500 K, the grains vaporize. Inside the dust
destruction front (Rd ~ 1013 cm), the opacity is greatly reduced (also the gas, which
dissociates above 2000 K, is nearly transparent to the radiation field): opacity gap.
Immediately outside the accretion shock itself, the gas is collisionally ionized and the
opacity rises (radiative precursor).
Gas approaches R* with speeds close to the surface free-fall value Vff:
" 2GM* %
V ff = $
# R* &
" M* % " R* %
= 280 km s $
# 1 M sun & # 5 R sun &
Using the Rankine-Hugoniot jump
conditions (equivalent to the conservation
€ momentum and energy in J-type
shocks; see Appendix F in S&P) in a
perfect gas with γ = 5/3 and
M1 (≡ (u1-u0) / a1) >> 1 , the immediate
postshock temperature T2 ≥ 106K.
In velocity frame of the shock front:
ρ2 v2 = ρ1 v1
P2 + ρ2 v22 = P1 + ρ1 v12
0.5v22 + ε2 + P2/ρ2 = 0.5v12 + ε1 + P1/ρ1
ε is internal energy per unit mass
Post-shock temperature (strong, adiabatic shock):
T2 = 3 µ mH Vshock2 / (16 kB)
= 2.9 x 105K ( Vshock / 100 km/s)2
[see S&P eq. 8.50 & Appendix F]
Such a hot gas emits photons in the extreme UV and soft X-ray
regimes (λ≈hc/(kBT2) ≤ 100 Å), mainly from highly ionized metallic
species (e.g. Fe IX), but the material in the postshock settling region
and the radiative precursor is opaque.
The protostar radiates into the opacity gap almost if it were a blackbody surface, with
4 πR*2σ B Teff4 ≈ Lacc
˙ &1/ 4 # R* &−3 / 4
&1/ 4 # M
T ≈ 7300 K % -5
$ 10 M sun yr ' $1 M sun ' $ 5 R sun '
€ characterizes the spectral energy
distribution of the radiation field. In the
opacity gap, the characteristic temperature of
the radiation and the gas temperature do not
Radiative Diffusion and the temperature of the envelope
[see S&P Appendix G]
In the dust envelope, the infalling matter is highly opaque to optical radiation. To
determine the temperature, we need to consider the radiative diffusion equation.
In a very optically thick medium, the specific intensity Iν ≈ Bν(T), with T = Tkin
(ntot >> ncrit). But Iν cannot be precisely Bν(T), otherwise Fν would vanish.
Suppose the net flux is locally in the z-direction
and consider the variation of Iν(z,θ) w.r.t. θ.
∫ I µdΩ
The radiative transfer equation becomes:
∂Iν (z, µ)
= −ρκ ν Iν + jν
Assuming thermal emission (Kirchoff’s Law applies):
µ dBν dT
ρκ ν dT dz
Iν (z, µ) ≈ Bν (T) −
Δs = Δz / µ
µ = cosθ
Using this Iν(z,µ) to determine Fν :
€ Fν =
∫ I µdΩ = 2π ∫ dµµ I
4 π dBν dT
3ρκ ν dT dz
The frequency-integrated flux Frad:
4 π dT
= ∫ Fν dν = −
Defining the Rosseland mean opacity κ:
dBν /dT dν
∫ dBν /dT dν
∫ u dν = 4σ
∫ I dΩ
Radiative diffusion equation
16σ B T 3 dT
T 4 /c
..back to the temperature of the envelope:
Frad = Lacc /4 π r 2
3 ρ κ Lacc
64 πσ B r 2
In the temperature regime of interest (100-600 K).
$ T '
κ ≈ κ 0&
% 300 K (
€cm2 g -1 ; α = 0.8
κ 0 = 4.8
Dimensional analysis tell us that:
T(r) ∝ r−γ
2(4 − α )
The steady temperature decline continues until
the gas becomes transparent to the IR
radiation. The transition occurs when the mean
free path of the “average” photon (1/ρκ)
becomes comparable to the radial distance
from the star. At this point, the entire dust
envelope emanates as a blackbody of radius
Rphot and temperature Tphot:
Dust envelope temperature
M = 1 M!
ρκR phot = 1
Lacc = 4 πR phot
σ B Tphot
˙ r−3 / 2
4 π 2GM*
˙ = 10 -5 M yr -1 and M = 1 M :
R phot = 2.1×1014 cm and Tphot = 300 K
( T +α
κ ≈ κ 0*
) 300 K ,
Warning: crude approximations!
It is best to visualize Rphot as the radius where a photon carrying the mean energy
of the spectral distribution escapes the cloud. The wavelength of this photon is
λ ≈ hc /kB Tphot = 49 µm
From detailed modeling, the envelope becomes transparent to outgoing radiation
once its temperature falls below several 100 K. In this regime, the dust
temperature Td follows from a simple energy argument and it is found that:
Td ∝ r−1/ 3
This optically thin profile is generally useful for modeling the observed emission
at far-infrared and millimeter wavelengths from any dust cloud with an
Collapsing, rotating cloud
Lrad = 21 L!
M* = 0.5 M!
dM/dt = 5x10-6 M! yr-1
Ω0 = 1.35x10-14 s-1
But realistic geometries are
Zhang & Tan, in prep.