Chapter 2:
Protostellar collapse and star
formation
One of 3 branches of proton-proton chain
CNO cycle:
C, N O atoms act as
catalysts
T-dependence of pp chain and CNO cycle
Hydrostatic equilibrium: negative feedback loop
If core T drops,
•fusion rate drops,
•core contracts
•heats up
If core heats up,
•fusion rate rises
•core expands
•cools down
Main sequence stars are
modeled as concentric
spherical shells in hydrostatic
equilibrium
Mass element dm
Constant density
Inward force = outward force
The Main
Sequence
L = A sT4
Demographics of Stars
• Observations of star clusters show that star formation
makes many more low-mass stars than high-mass stars
Giant molecular clouds are the sites of star formation
GMC:
Length scale ~ 10-100 pc
T = 10 – 20 K
Mass ~ 105 – 106 Msun
Clumps:
Length scale ~ 2-5 pc
T = 10 – 20 K
Mass ~ 103 – 104 Msun
Cores:
Length scale ~ 0.1 pc
T = 10 K
Mass ~ 1 Msun
Clouds exhibit a clumpy structure
Star forming regions in Orion
What supports Cloud Cores from collapsing under their
own gravity?
• Thermal Energy (gas pressure)
• Magnetic Fields
• Rotation (angular momentum)
• Turbulence
Gravity vs. gas pressure
• Gravity can create stars only if it can
overcome the forces supporting a cloud
• Molecules in a cloud emit photons:
– cause emission spectra
– carry energy away
– cloud cools
– prevents pressure buildup
What happens when a cloud core collapses?
Virial theorem:
2K + U = 0
If 2K > |U|, then
•
•
Force due to gas pressure dominates over gravity
Cloud is supported against collapse
Assume a spherical cloud with constant density
Gravitational potential energy
Kinetic energy
where
r0
3 GM
U »
5 Rc
3
K = NkT
2
Mc
N=
mm H
2
c
In order for the cloud to collapse under its own gravity,
3M c kT 3GM
<
mm H
5Rc
2
c
æ 3M c ö
Rc = ç
÷
è 4 pr 0 ø
1/3
where
Using the equality and solving for M gives a special mass,
MJ, called the Jeans Mass, after Sir James Jeans.
æ 5kT ö æ
ö
3
MJ » ç
÷ ç
÷
è Gmm H ø è 4pGmm H r0 ø
3/2
1/2
Jeans Criterion
When the mass of the cloud contained within radius Rc
exceeds the Jeans mass, the cloud will spontaneously
collapse:
Mc > MJ
You can also define a Jeans length, RJ
æ 15kT ö
RJ » ç
÷
è 4 pGmm H r 0 ø
1/2
Figure from
Jeff Hester
& Steve
Desch, ASU
Figure from
Jeff Hester
& Steve
Desch, ASU
“protoplanetary disks”
HH Objects
Collapse slows before fusion begins:
Protostar
• Contraction --> higher density
• --> even IR and radio photons can’t
escape
• --> Photons (=energy=heat) get trapped
• --> core heats up (P ~ nT)
• --> pressure increases
• Protostars are still big --> luminous!
• Gravitational potential energy --> light!
What supports Cloud Cores from collapsing under their
own gravity?
• Thermal Energy (gas pressure)
• Magnetic Fields
• Rotation (angular momentum)
• Turbulence
Angular momentum problem
• A protostellar core has to rid itself of 1000x
Jsolar system
• Core collapse produces a disk whose j
increases with r
• To redistribute (and/or lose) J takes >>
orbital timescale
• The disk is stable over ~106 years
Homework for Wednesday Sept. 14
• Problem 2-5 from book
• One paragraph on a possible topic for your
semester project (for topics, check out the
author’s blog or astrobites; then find a peerreviewed paper on the subject from NASA ADS)
• Estimate how the angular momentum is currently
distributed in the solar system (sun & planets).
Compare to the angular momentum of a uniform
spherical gas cloud with ‘typical’ properties for a
collapsing molecular cloud core.
Protostellar evolution onto the main sequence
Protostellar evolution for Different
Masses
• Sun took ~ 30
million years from
protostar to main
sequence
• Higher-mass stars
evolve faster
• Lower-mass stars
evolve more
slowly
Hayashi Track
Physical cause:
at low T (< 4000 K), no
mechanisms to
transport energy out
Such objects cannot
maintain hydrostatic
equilibrium
4000 K
They will rapidly
contract and heat until
closer to being in
hydro. eq.
Mass accretes onto the star
via an accretion disk
(Krumholtz et al 2009)
Necessary to build stars > 8
Msun
Phases of star formation
Spectral energy distribution
http://feps.as.arizona.edu/outreach/sed.html
æ R ö- p
T(R) = 2000Kç ÷
è R* ø
dust sublimes at
~2000 K
p depends on
grain
properties,
0<p<2
Smaller grains =
flatter T(R)
=smaller p
Comparing disk observations to models:
Modeling SED’s with some simplifying assumptions:
1. Dust grains are perfect blackbody emitters/absorbers
2. Disk is optically thick
3. Disk is geometrically thin
Reality:
1. Radiation absorption and emission depends on
size, composition, shape, orientation (!) of grains
(more so for optically thin disk)
2. Optically thick = disk grains absorb only on the
outside of disk, we only see emission from these
grains
3. Geometrically thick = disk self-gravity, etc
continuous disk that extends
out from the surface of the
star to 100 AU
same disk with an inner hole of 0.3 AU
A gap = cleared by a
planet?
Class 0 Protostar:
Earliest stage of collapse, no star
visible, no disk visible
Class I: bipolar outflow, jets ~100
km/s, still embedded in infalling
material heated by star + disk
Class II:
“Classical T Tauri star”
SED = star + disk, disk lifetime~ 106
yr
Class III:
PMS star w/ debris disk
http://ssc.spitzer.caltech.edu/documents/compendium/galsci/
T Tauri : the prototype protostar
http://ssc.spitzer.caltech.edu/documents/compendium/galsci/
http://vinkovic.org/Projects/Protoplanetary/
http://vinkovic.org/Projects/Protoplanetary/
Anatomy of a flared accretion disk (Kenyon & Hartmann 1987)
Star surface
Kenyon & Hartmann 1987: disks w/ “reprocessed” radiation
H=0.1R5/4
H=0.1R9/8
Addt’l energy from
accr
H=0
MS
Effect of the
‘photospheric’ scale
height
H=0.1R5/4
H=0.1R9/8
H=0
Effect of observing angle
SED’s for accretion disks
with H=0.1R9/8
M=10-8 Msun/year
M=0
Chiang and Goldreich 1997
“interior”
Dust is hotter than
gas
IPS = iron poor silicates
IRS = iron rich silicates
Debris disks are found around 50% of sunlike stars
up to 1 Byr old
0
