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C6M.7 Astronomers currently believe that every galaxy has a supermassive black hole at its center and that a typical galaxy has collided with other galaxies several times since the beginning of the universe. Imagine that the central black holes of two colliding galaxies end up, after the collision, orbiting each other in the middle of a cloud of dust and stars. Assume that the black holes have the same mass M (which maybe millions or even billions of times that of the sun) and each is initially in a circular orbit of radius R around the pair’s center of mass (which is halfway between them), and the surrounding cloud has no net angular momentum, as shown in figure C6.14. Over time, frictional interactions with the cloud cause the black holes to slowly spiral in toward each other. If their orbits remain approximately circular as they slowly spiral in, then thespeed of each in its circular orbit will be │V│ = 1/2 (GM/r)^1/2, where G is the universal gravitation constant and r is the distance between each black hole and the pair’s center of mass. (We will prove this result in unit N. For now, accept that this is true.) Assume that after a billion years, the distance between each black hole and the pair’s center of mass is now 9/5 R. If the cloud has not been involved with any significant external interactions during this time, what is the magnitude and direction of the cloud’s angular momentum at this point? Give the magnitude in terms of G, M, and R, and the direction assuming that we view the cloud from the direction implied by figure C6.14.
Transcribed Image Text:C6M.7 Astronomers currently believe that every galaxy has
a supermassive black hole at its center and that a typical
galaxy has collided with other galaxies several times since
the beginning of the universe. Imagine that the central
black holes of two colliding galaxies end up, after the col-
lision, orbiting each other in the middle of a cloud of dust
and stars. Assume that the black holes have the same mass
M (which may be millions or even billions of times that of
the sun) and each is initially in a circular orbit of radius R
around the pair's center of mass (which is halfway between
them), and the surrounding cloud has no net angular
momentum, as shown in figure C6.14. Over time, frictional
interactions with the cloud cause the black holes to slowly
spiral in toward each other. If their orbits remain approxi-
mately circular as they slowly spiral in, then the speed of
each in its circular orbit will be |ö| =}(GM/r)/², where
G is the universal gravitation constant and r is the distance
between each black hole and the pair's center of mass.
(We will prove this result in unit N. For now, accept that
this is true.) Assume that after a billion years, the distance
between each black hole and the pair's center of mass is
now R. If the cloud has not been involved with any sig-
nificant external interactions during this time, what is the
magnitude and direction of the cloud's angular momen-
tum at this point? Give the magnitude in terms of G, M,
and R, and the direction assuming that we view the cloud
from the direction implied by figure C6.14.
Transcribed Image Text:Cloud
Black
R R
Black
hole
СМ
7 hole
Figure C6.14
A pair of equal-mass black holes orbiting their common
center of mass in a cloud of dust and stars.
15
Definition Definition Product of the moment of inertia and angular velocity of the rotating body: (L) = Iω Angular momentum is a vector quantity, and it has both magnitude and direction. The magnitude of angular momentum is represented by the length of the vector, and the direction is the same as the direction of angular velocity.
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