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Definition
To describe how things move we often use the basic quantities
of length, mass, and time. Quantities such as velocity, acceleration, force and
energy are very powerful ones that help us understand how an object's position
will change over time and how it will interact with other things in the
universe. Momentum and its cousin angular momentum are other
very powerful quantities.
Ordinary momentum is a measure of an object's tendency to move at constant
speed along a straight path. 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 measures an object's
tendency to continue to spin. An ``object'' can be either a single body or two
or more bodies acting together as a single group.
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). A ``torque'' is
simply a force acting along a line that is off the object's spin axis. In these
cases, we have conservation of angular momentum.
conservation of angular momentum---the total amount of angular momentum does
not change with time no matter how the objects interact with one another.
A planet's velocity and distance from the Sun will change but the
combination of speed×distance will not change unless another planet or
star passes close by and provides an extra gravity force.
The area swept
out by a line connecting an orbiting object and the central point is the same
for any two equal periods of times. That line is called a radius vector
in the following discussion. The rate of change of the swept-out area does NOT
change with time. The line along which gravity acts is parallel to the radius
vector. This means that there are no torques disturbing the angular motion and,
therefore, angular momentum is conserved. The part of the orbital velocity
(v-orbit) perpendicular (at a right angle) to the radius vector
(r) is vt. The rate of change of the swept-out area =
r×vt/2.
To calculate the orbital angular momentum use vt for the
velocity. So, the angular momentum = mass × vt × r = mass × 2
× (rate of change of area). That value does not change over time. So if r
decreases, v-orbit (and vt) must increase! If r
increases, v-orbit (and vt) must decrease. This is just what
Kepler observed for the planets!
The total angular momentum = spin
angular momentum + orbital angular momentum. The total angular momentum is
CONSTANT. To find the spin angular momentum, subdivide the object into small
pieces of mass and find the angular momentum for each of the small pieces. Then
add up the angular momentum for all of the pieces. The Earth's spin speed is
decreasing so its spin angular momentum is DEcreasing. Therefore, the Moon's
orbital angular momentum 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!). Here the radius is used instead of the diameter, because what
is important is how far each piece of the core is from the spin axis that goes
through the exact center.
The core spins at 2 - 10 km/sec at the core's equator. If no external forces
produce torques, the angular momentum is constant. During a supernova the outer
layers are blown off and the core shrinks to only 10 kilometers in radius! The
core angular momentum is approximately = 0.4×M×V×R and the mass M
has stayed approximately the same. When the radius R shrinks by factors
of 10,000's, the spin speed V 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 the rotational energy and the angular momentum is not strictly
conserved---it slowly decreases. Therefore, the neutron star spin speed slowly
decreases.
Gas flowing from
one star falls toward its compact companion into an orbit around it. The orbital
angular momentum 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 some
of the gas in the disk gas give torques to other parts of the disk's orbital
motions through friction. This causes their angular momentum to decrease. Some
of that gas, then, eventually falls onto the compact companion.
A 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 denser parts collapse more rapidly than the less dense
parts. Stars form in the inner denser parts before they form in the outer less
dense parts.
All the time as the cloud collapses, the spin speed must increase. Since no
outside forces produce torques, the angular momentum is conserved. The rapidly
spinning part of gas cloud eventually forms a disk. This is because the cloud
can collapse more easily in a direction parallel to the spin axis. The gas that
is orbiting perpendicular to the spin axis has enough inertia to resist the
inward pull of gravity (the gas feels a ``centrifugal force''). The most dense
parts of the disk will form stars.
last updated: 19 August 1997
Is this page a copy of
Strobel's Astronomy Notes?
Author of original content:
Nick Strobel