The Size, Structure, and Variability of Late-Type Stars Measured ...

78
4.1 Implications of Diameter Measurements on Pulsation
Mode and Dynamics in Miras
The relationship between period and radius is defined by the internal structure
of the star. The speed of sound within a Mira (and consequently, its resonant periods)
can be calculated from a model of its density and temperature structure. In this way, the
dependence of radius on period for a given mode of oscillation can be established. Ya’ari
and Tuchman (1999) [115] argue that the uncertainties in such a calculation are small.
Moreover, for a given radius, the periods corresponding to the fundamental mode and to
the first overtone mode are typically a factor of two apart. Hence, if the radius of a variable
star is measured to an accuracy better than about 50%, an unambigous determination of
pulsation mode should be possible. By comparison, the period-luminosity relation does not
serve as a sensitive test of pulsation mode. For a given oscillation mode, the P-L relation
is very dependent on the choice of certain parameters 1 used to model the star and their
uncertainty does not allow determination of oscillation mode from period and luminosity
data alone.
The radius of the Mira model of Bowen (1990) [16] varies between 216R ⊙ and
273R ⊙ for a 320 d period, fundamental mode Mira variable. This is similar to the 332 d
period of o Cet. If the Hipparcos parallax of 7.79 mas is assumed, the ISI measured 11 µm
radius of o Cet, which is believed to be close to the continuum photosphere, is seen to
vary between 643R ⊙ and 761R ⊙ . Taken at face value, this implies that o Cet is necessarily
pulsating in a higher mode. The data would suggest a first or possibly second overtone
oscillation mode for o Cet according to the model of Bowen. That the pulsation mode of
most or all Miras is the first overtone is the conclusion reached by Feast (1999) [27] based
on assorted angular diameter measurements and assuming “classical” pulsation models.
The above conclusion is not universally accepted, however. Bowen (1990) [16]
argues that the power required to drive an overtone pulsator can be much greater than
that for the fundamental mode. As a result, higher modes are preferentially damped. It
is suggested that this mechanism may be responsible for limiting the accessible oscillation
modes to the fundamental (Bowen, private communication). Tuchman (1999) [101] finds
better agreement between fundamental mode radii and observed angular diameters using
non-linear models. The Tuchman model predicts the radius of a 332 d variable such as
1 Most notably, the mixing length describing convection scale is uncertain.