Rotation and Magentic Fields

Question 1

Angular Momentum

Definition

Answer 1

L = mvr = mωr² = Iω
-where I = inertia is given by:
I = ∫ r²dm, ω=dθ/dt

Question 2

Cloud Rotation

Description

Answer 2

-conservation of specific angular momentum along the equator means that:
ωr² = ΩR²
-where R is the radius of the cloud and Ω is the angular momentum at the surface of the cloud
-and ω is the angular momentum at radius r within the cloud

Question 3

Cloud Rotation

Centrifugal Force

Answer 3

Fc = mω²r ∝ r^(-3)

Question 4

Cloud Rotation

Gravitational Force

Answer 4

Fg = GMm/r² ∝ r^(-2)

Question 5

Cloud Rotation

Balancing of Forces

Answer 5

-centrifugal force will eventually win out over the gravitational force and halt collapse along the equatorial plane where it is largest

Question 6

Cloud Rotation

Centrifugal Radius

Answer 6

-the radius at which centrifugal and gravitational forces are balanced:
Rc = Ω²R^4 / GM
-typically 100-1000au

Question 7

Typical Angular Velocity for Surface of Molecular Clouds

Answer 7

Ω = 10^(-14) rad/s

Question 8

How can material continue to infall if a point is reached where centrifugal and gravitational forces balance?

Answer 8

-the balancing of centrifugal and gravitational forces only considers how rotation at the equator impedes inward motion, the cloud can still contract parallel to the rotation axis leading to a flattened structure
-the rotational velocity decreases with increasing latitude:
vo = ωrosin(θo)
-the centrifugal force decreases and only acts perpendicular to the roation axis, material can continue to infall

Question 9

Cloud Rotation

Disc-Like Geometry Formation

Answer 9

• infalling material flows along streamlines
• there is a pile up of material at the centrifugal radius, rc
• material continues to infall to the centre from other angles
• the resulting density distribution shows the material concentrated in a disc-like geometry, with rdisk~2*Rc

Question 10

The Angular Momentum Problem

Answer 10

-in order to conserve angular momentum during collapse, the rotational frequency must increase:
ωf = (ri/rf)²*ωi OR vf = ri/rf * vi
-the collapse of even very slowly rotating clouds will result in a massive amplification of the rotational velocity
-for a typical cloud this gives a final rotational frequency of 0.1rad/s and a final rotational velocity of 10^8 m/s
-these are relativistic speeds, much faster than we observe stars to move
-the angular momentum must be transferred elsewhere, i.e. it is no conserved within the cloud during collapse

Question 11

How does a molecular cloud lose angular momentum during collapse?

Answer 11

• fragmentation/fission, transfer of angular momentum to a cluster or a binary or planets
• transfer of angular momentum through cloud-cloud interactions
• magnetic braking of the star, charged particles couple with the magnetic field and resist angular motion
• mass loss through outflows, the mass lost carries with it angular momentum

Question 12

Angular Momentum Transfer

Magnetic Breaking of a Star

Answer 12

• the cloud exists in a charged plasma
• this charge fluid velocity bends the magnetic field and creates a resisting tension force
• any spin up during collapse twists the field and increases the local magnetic tension, this tension creates a braking torque on the element that counter acts the spin up and lowers the specific angular momentum

Question 13

How can angular momentum be transferred locally?

Answer 13

• for this to happen we need a force linking the rapidly rotating inner core to the slowly rotating outer cloud
• there are two possibilities:
• -turbulent viscosity
• -magnetic fields

Question 14

Local Angular Momentum Transfer

Turbulent Viscosity

Answer 14

• friction between neighbouring annuli will create torques that act to bring the annuli into co-rotation
• the outer annulus will try to speed up (i.e. gather angular momentum) angular momentum is transferred outwards
• an exchange of particles bringing differing angular momenta to the annuli to which they are transferred

Question 15

Local Angular Momentum Transfer

Magnetic Fields

Answer 15

• friction between neighbouring annuli will create torques that act to bring the annuli into co-rotation
• the outer annulus will try to speed up (i.e. gather angular momentum) angular momentum is transferred outwards
• inner and outer annuli are attached by ‘elastic spring’ which becomes stretched due to shear creating a restoring force (magnetic braking)

Question 16

The Magnetic Flux Problem

Answer 16

-molecular clouds are threaded with magnetic fields
Fb ∝ R²B²
-charged particles spiral around field lines effectively freezing the magnetic field into the material
-if the magnetic field is strong and uniform we can expect flattened clouds, as the material cannot cross the field lines, it can only move along them
-because the field lines are frozen into the material, we expect conservation of magnetic flux: ϕ∝BR²
-since both Fb and Fg are ∝R^(-2), magnetic force cannot overcome gravitational force once collapse has started
-conservation of magnetic flux gives relation:
B* R² = Bc Rc²
-giving a typical star magnetic field:
B = 10^3 T
-this is significantly higher than observed, the system must lose magnetic flux at some point

Question 17

How is magnetic flux lost?

Answer 17

-ambipolar diffusion

Question 18

Ambipolar Diffusion

Answer 18

• magnetic fields can only act on the ions/elecrons, not the neutral particles in the cloud
• neutrals can drift relative to the magnetic field opposed only by collisions with ions, i.e. friction slows down the neutrals
• slowly a cloud supported by a B field will expel the field and contract
• eventually it will no longer be able to support itself and collapse
• the timescale for ambipolar diffusion is typically longer than the free-fall timescale so it must occur before collapse