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 core spins at 2 to 10 kilometers/second 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. The neutron star produces 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.
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 densest parts of the disk will form stars.
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last updated: 28 May 2001