For this experiment, consider a highly unrealistic model of a
pulsating star consisting of a central point mass equal to the entire
mass of the star, M, surrounded by a single thin spherical shell of mass m and radius R that represents the surface layer of the star. Within this shell is a massless gas of pressure P
that supports the shell against the gravitational pull of the central
mass. Newton's second law applied to the forces on the shell yields
mdv/dt = -GMm/R^2 + 4πR2P
The left-hand side of this equation represents the total force on the
system, or the equilibrium state; the right side sums the force upon
the gas shell due to the gravitational influence of the central mass and
the shell's own mass, and the force upon the shell's internal surface
area applied by the internal massless gas.
Assuming the expansion and contraction of the gas are adiabatic and
using the definition of velocity, it is possible to find the velocity of
the outer shell, its radius, and the pressure of the inner massless gas
after an interval of time Δt given the central point mass M and the mass of the gas shell m.
In this example, we are going to assume that M = 1 × 1031 kg and that m = 1 × 1026
kg; the first value is typical of the mass of a classical Cepheid
variable star, and the second value is arbitrarily assigned. Given
Ri = 1.7 × 1010 m
vi = 0 m s-1
Pi = 5.6 × 104 N m-2
using a time interval of Δt of 104 s, it is possible to show that the system visibly pulsates, as seen in Figures 1 through 3.
|Figure 1: The plot of the velocity v of the outer shell over time t.|
|Figure 2: The plot of the radius R of the outer shell over time t.|
|Figure 3: The plot of the pressure P of the inner massless gas over time t.|
Using the values for the radius R, it is possible to visibly demonstrate what this pulsation actually looks like; this is done in Figure 4.
|Figure 4: An animated visualization of the one-zone model with a central point mass M = 1 × 1031 kg.|
Using Figure 2 as a reference, we can see that R0, the radial equilibrium value, is about 1.52 × 1010
m; using this, we can calculate that the period of this oscillation is
roughly 5.28 days. The period for this oscillation is observed in δ
Cephei to be roughly 5.37 days. Considering the highly idealized nature
of this experiment, the obtained value for this is remarkable, and
tells us something of the inner workings of the Cepheid class of