Rotational effects I

1
Rotational effects I

• Centrifugal force Fcen j2/w3
  Gravitational force Fgrav Gm/w2
• (w radius on cylindrical coordinates j angular
  momentum jmw2v)
• Since the initial ratio of rotational to
  gravitational energy is
• Trot/W 10-3
• Rotation gets important after shrinkage of the
  core by a factor 1000
• (from 0.1 pc to 10 AU) --? important in protostar
  formation, if j is
• conserved dominates dominates on gravity at some
  point!
• However, this is only valid in non-magnetic
  cloud.
• B-field anchored to rarified outer cloud. Spin-up
  twists
• the field during collapse because field still
  coupled
• to ionized outer region (so tries to keep
  original
• direction). This increases magn. tension and
• creates magnetic torque counteracting spin-up
• (angular momentum lost, not conserved)

2
Rotational effects II

• In dense core centers magnetic braking fails
  because neutral matter
• and magnetic field decouple with decreasing
  ionization fraction.
• matter within central region can conserve
  angular momentum.
• Since Fcf grows faster than Fgrav fluid elements
  at some point will miss the geometrical center.
  --gt Formation of disk

• The larger the initial angular momentum j of a
  fluid element, the further
• away from the center it ends up --gt centrifugal
  radius wcen
• wcen m03atW02t3/16 0.3AU (T/10K)1/2
  (W0/10-14s-1)2 (t/105yr)3
• wcen can be identified with disk radius.
  Increases with time because in inside-out
• collapse rarefaction wave moves out j of
  collapsing gas increases with time
• because initial j(r) increasing function of
  radius (for rotational stability)

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Derivation of centrifufal radius wcen

• Gas is in free-fall region, so we can neglect
  magnetic fields and pressure.
• Only gravity and rotation determine the force
  balance.
• Gas flows towards the protostar on parabolic
  orbit (zero energy orbit from
• noting the fact that initial kinetic and
  gravitational potential energy of fluid
• element very small plus conservation of energy)
• If y is angle between instantaneous distance of
  fluid element from protostar
• and req is the distance at which fluid element
  encounters the disk then
• r req/ (1
  cos(y))
• At the pericenter of the parabola, rmin, it is
  y0 so that rminreq/2 and the
• specific angular momentum of the fluid element
  normal to the orbital plane of
• the fluid is jn2 rmin2Vmax2 2GMrmin
  (neglecting the mass of the disk) --?
• --? req jn/GM

4

• - In general the orbital plane of a fluid element
  will be tilted by some angle q0 with respect to
  the orbital plane of the disk. It is the
  component of jn around the main rotation axis z
  that contributes to increasing wcen, so
• we need to consider jn(q0, t). For an isothermal
  cloud rotating with uniform angular velocity W0
  (j(r) r) --?
• ---? jn R2W0sinq (note
  velocity component around z-axis)
• and R varies with time as the rarefaction wave
  expands as RaTt, where t is the time at which
  the fluid element begins to fall.
• Calculating the mass swept up between time t and
  time t (at which the
• fluid element crosses the disk orbital plane)
  with the assumption that the density is that of a
  singular isothermal sphere and that dM/dt m0
  aT3/G (asymptotic accretion rate, m0 1) one
  obtains req (q0, t) and then
• wcen (qo, t) by setting q0p/2

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Protostellar disk formation and evolution I
The disk begins to form when wcen gt R, R being
the radius of the protostar. We can infer the
time of first disk appearance by setting
wcenR. This yields t0 3 x 104 yr
(R /3 Rsun)1/3 (W0/ 10-14 s-1) -2/3 (aT / 0.3
km/s) 1/3 The mass of the protostar at this
epoch, M0, can be determined using the
asymptotic mass accretion rate and setting
M0 dM/dt x t0. We obtain M0 0.2
Msun(R/ 3 Rsun) 1/3 (W0/ 10-14 s-1)-2/3 (aT /
0.3 km/s) 8/3 Therefore the protostar mass at
the time of disk formation increases with
higher infall rate (higher aT within the
rarefaction wave where free-fall applies)
or lower W0 (lower resistance to collapse by
rotation) This formula however neglects that the
protostar itself would be rotating in a spherical
cloud and would be flattened instead of a sphere
with radius R. But youngest pre-main sequence
stars have low rotation speeds ? their J is
partially dissipated by magnetic torques or
dynamical evolution
6
The Angular Momentum Problem formation of
binaries
Characteristic values of specific angular
momentum (Bodenheimer 95)
J distr in protostellar collapse
Molecular clouds
Initial (core)
Molecular cloud cores
100 AU disk
Final (protostar disk)
T Tauri star (spin)
But if the disk has only 10 of the stellar mass,
the distributions of angular momenta j(m) at the
end must be very different.
formation of binaries
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The inner 100 AU of a protostellar collapse
simulation (no magnetic
field)
Phase 1 rapidly rotating bar unstable
protostellar core
T0.02 Myr
T0.022 Myr
Bar shape grows because prostostellar core is
self-gravitating bar mode is the fastest
growing mode in most self-gravitating systems
that undergo a small perturbation, including
Bonnor-Ebert spheres (instability for mass lt MBE))
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Phase II bar fragmentation and merging of
fragments
T0.025 Myr
T0.024 Myr
9
T0.035 Myr
Phase III Formation of a binary system
with protostellar cores and protostellar disks.
Mdisk 0.5 Mstar after one binary orbit (1000
years)
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Disk thickness
In the early phases of disk formation most of the
mass is still in the protostar --? to determine
disk thickness simply consider hydrostatic
equilibrium in the vertical direction where the
gravitating mass is just M. For an ideal gas
equation of state
Dz (aT/Vkep) w Vkep
(GM/w)1/2 The thin-disk approximation, Dz ltlt w,
holds everywhere in the disk because aT ltlt Vkep
(Vkep Vff recall the accretion flow is
supersonic e.g. accretion shock temperature in
radiative precursor 106 K - and aT is
comparable to that of radiative precursor (Teff
7000 K), so aT/Vkep ltlt 1).
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Accretion through the (low-mass) disk
As long as the Mdisk ltlt M (early phase)
accretion pattern modified compared to no-disk
case (gas falls onto disk and then flows towards
the star) but accretion rate not modified
because determined by large scale flow (in early
phase disk still small so crossing time in the
disk negligible) Accretion shock in the disk
because opposing streams collide at
supersonic speed (assuming flow is symmetric
around z-axis in general). After shock uz
component of velocity completely dissipated. uw
and uf components remain, uw gt uf until wcen is
not big enough. At late times large wcen -? uw lt
uf accretion towards star becomes inefficient and
disk begins to grow significantly in mass -?
accretion pattern changes
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Evolutionary disk equations
We neglect disk self-gravity assuming Mdisk ltlt
M in early phase (small wcen) and consider a
thin disk (so neglect vertical disk structure and
internal pressure gradient) We can then derive
(1) mass continuity equation (2) conservation
of momentum in radial and azimuthal direction
including centrifugal acc. radial force per
unit mass due to infalling material In early
phase wcen grows slowly compared to the time it
takes to traverse a fluid element in the disk
because u Vkep -? can assume du/dt dj/dt0 ?
solve the resulting steady-state ordinary
differential equations
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-- Solution is spiral pattern in outer disk
(means trjaectory of infalling material follow
spirals) circular patter in inner disk with
ring between two regions (circular pattern
develops naturally because minimizes energy
dissipation) -- Spiral pattern extends in radius
as t3 like wcen because it is related to the
growth of j in the disk as more distant material
falls in (sequence of steady- states) for
different values of jj(t)) The ring is
turbulent because velocity of flow switches from
quite radial (spiral trajectory) to nearly
circular, hence strong velocity fluctuation In
the inner disk orbits are not exactly circular.
Infalling material exerts a drag on the disk (via
azimuthal force in momentum conservation) and
allows some radial inflow to continue --? infall
drives transport!
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Mass transport in the disk
Mass transport through the disk, dMdisk/dt, from
full integration of differential evolutionary
equations -? time late enough that inner circular
disk and ring have developed. Once the disk
appears this dMdisk dM/dt of the
protostar Can be used also in a temporal sense,
from rightmost point (disk first appearance) to
leftmost point (effective accretion rate of the
star in a well developed disk), and relative to
dM/dt from the infalling cloud (no disk
case) Note that (1) accretion rate through
the disk grows first as the disk intersects more
infalling gas as it becomes larger
(2) it then decreases dramatically as the
circular inner disk develops and trajectories
become more circular
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Angular momentum problem I the problem
Radial distribution of mass transport suggests
that in well developed disk more mass accumulates
in the outer disk than it does in the
protostar--? at some point Mdisk gt M! The
tendency of gravitational collapse is thus to
produce a system in which most of the mass is in
the disk. Instead in young T Tauri stars we
observe the opposite, i.e. Mdisk lt M and in
mature stars there is no disk at all (e.g. the
Sun)! --? The disk might become larger than the
star temporarily but then the accretion rate
onto the star has to increase again and somehow
eat the disk around it. Another possibility
is that the disk is at least partially evaporated
by the radiation from the star or from
neighboring stars (however does not work for
dwarf K,M,F, G stars for which radiating power
is quite low even in early stages)
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Angular momentum problem II the solution
The solution lies in a form of internal
frictional torque between different pieces of
the disk. Disks are differentially rotating, so
an anulus at radius w and angular momentum j
rotates faster than an anulus just outside it
with radius w dw and angular momentum j Dj
(easy to see in a keplerian disk). The inner
anulus slows down as a result of friction with
the outer anulus -? angular momentum is
transported outward and mass inward (inner anulus
sinks). This friction is called shear viscosity
( grad uf) Adding viscous force to momentum
equations gives accretion restart with rate
proportional to viscous force ? pure
phenomenological however!
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Shear viscosity also does something to energy as
well
Disk Viscosity -gt Accretion Disk
? Converts shear to heat (shear comes from
gravity ultimately!)
? Heat radiated away (important, otherwise rsing
pressure would stop accretion again!)
? Energy being lost

• Gas sinks deeper in
• the potential well