Angular Momentum
Angular Momentum
Momentum depends on speed and mass. A train moving at 20 mph has more momentum
than a bicyclist moving at the same speed. A car colliding at 5 mph does not
cause as much damage as that same car colliding at 60 mph. For things moving in
straight lines momentum is simply mass 
speed. In astronomy most things move in curved paths so we generalize the idea
of momentum and have angular momentum.
angular momentum = mass
velocity distance
(from point object is spinning or orbiting around)
Very often in astronomy, the object (or group of objects) we're observing has
no outside forces acting on it in a way to produce torques that would disturb
the angular motion of the object (or group of objects). In these cases, we have
conservation of angular momentum.
conservation of angular momentum - the amount of angular momentum does
not change with time.
Area swept out by line connecting orbiting object and central point
(radius vector) is the same for any two equal periods of times. The rate of
change of swept out area with time is
constant. The line along which gravity acts is parallel to the radius vector so
there are no torques disturbing the
angular motion and, therefore, angular momentum conserved. The part of orbital velocity
(v-orbit) perpendicular to radius vector
(r) is 
. Rate of
change of area = r
/2.
To calculate orbital ang. mom. use
for velocity. Ang. mom. = mass 
r = mass 
rate of
change of area = constant. So if r decreases, v-orbit (and
) must increase! If r
increases, v-orbit (and
)
must decrease. This is just what Kepler observed for the planets!
Total angular momentum = spin ang. mom. + orbital ang. mom. = CONSTANT. To find
spin ang. mom., subdivide object into small pieces of mass and find that small
mass' ang. mom. Add up ang. mom. for all the mass pieces. Earth's spin speed
is decreasing so its spin ang. mom. is decreasing so the Moon's orbital ang.
mom. must compensate by increasing. It does this by increasing the Earth-Moon
distance.
Originally, a big star has a core
10,000's - 100,000's km in radius (the whole star is even bigger!) and
spinning at 2-10 km/sec at the core's equator. If no external forces produce
torques, the angular momentum is constant. Supernova blows off the outer layers
and the core shrinks to only 10 km in radius! The core ang. mom. is
approximately = 0.4
and the mass 
has
stayed approximately the same. Radius 
has shrunk by factors of 10,000's so the spin speed 
must increase by 10,000's of times. Sometimes the neutron star suddenly
shrinks slightly (by a millimeter or so) and it spins faster. Over time,
though, the neutron star has been producing radiation from its strong magnetic
field. This radiation is produced at the expense of rotational energy and the
ang. mom. is not strictly conserved - it decreases. Therefore, the neutron
star spin speed decreases.
Gas flowing from one star falls toward its compact companion into an orbit around
it. Orbital ang. mom. is conserved so as the gas' distance from the compact
companion decreases, its orbital speed must increase. It forms a rapidly rotating
disk-like whirlpool called an accretion disk. Over time parts of the disk gas give
torques to other parts of the disk's orbital motions through friction, causing
their ang. mom. to decrease. Some of that gas, then, eventually falls onto the
compact companion.
Huge slowly spinning gas cloud collapses. Parts of
the roughly spherical gas cloud break up into small chunks to form stars and
globular clusters. As the rest of the gas cloud collapses, the inner more
denser parts collapse more rapidly than the less dense parts. Stars are forming
in the inner denser parts before the outer less dense parts. All the time as
the cloud collapses (radius shrinks), spin speed must increase since no outside
forces produce torques so the angular momentum is conserved. Rapidly spinning
part of gas cloud eventually forms a disk. Dense enough parts of disk will form
stars.
last update 29 Aug 95
Nick Strobel --
Email:
strobel@astro.washington.edu
(206) 543-1979
University of Washington
Astronomy
Box 351580
Seattle, WA 98195-1580