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152. Problem 20.42 (RHK) The period of a pulsating star may be estimated by considering the star to be executing radial longitudinal pulsations in the fundamental standing wave mode; that is, the radius varies periodically with time, with a displacement antinode at the surface. (a) We have to answer whether the centre of the star will be a displacement node or antinode. (b) By making an analogy with the open organ pipe, we have to show that the period of pulsation is given by
where R is the equilibrium radius
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is the
average speed of sound. (c) Typical white dwarf stars are composed of
material with a bulk modulus of
and a
density of
. They
have radius of the order of 0.009 solar radius. We have to
estimate the approximate pulsation period of a white dwarf.|
Solution: Click For PDF Version The period of a pulsating star may be estimated by modelling it to be a closed pipe executing longitudinal pulsations in the fundamental standing wave mode. The centre of the star has to be a pressure node as the pressure there is very large. In the fundamental standing wave mode the pulsations will look like as shown in the diagram.
Wavelength
Let
Frequency of oscillation in the fundamental mode will be
the period T of pulsations of the star will be
Data for the white dwarf star is
the speed of sound will be
It is given that the typical radius of a white dwarf star is 0. 009 solar
radius. Solar radius is
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and the
radius R are related as
or
.
be the
speed of longitudinal pressure oscillations of the material of the star. The
speed
is the
density of the star, the average speed of sound waves is given by
.
So, we estimate its period of pulsation to be