[Mesa-users] rsp_Cepheid Questions

[Radek Smolec]

Dear Alex,

July 19 2019
>
> Hello, Dr. Smolec and/or anyone else who can help,
>
> Thank you for the instructions the last time you spoke; they were very
> informative and helped me to understand the simulator.
>
> I read the relevant sections of the MESA Instrument Paper V, but I have a
> few more questions after having ran rsp_Cepheid according to your
> instructions.
>
> In your instructions for rsp_Cepheid, you said to ensure that the
> simulation was at constant pulsation/large-amplitude pulsation. I was able
> to do so and generated a light curve by converting luminosity to bolometric
> magnitude and phasing part of the constant-pulsation data using outside
> software. However, I started examining the data before the large-amplitude
> pulsation section.
>
> What exactly is happening in this section, physically speaking? If I
> understand the instrument paper correctly, is RSP checking to see whether a
> set of initial conditions or combination of eigenvectors unique to each
> pulsation cycle stabilizes to a constant period of pulsation/large
> amplitude pulsation? Could a stable Cepheid star exist with these
> conditions (before the models converge to large amplitude pulsation), or is
> it unlikely/impossible?
>

As a first step, RSP constructs a static model and then conducts a linear
stability analysis. As a result, linear periods and growth rates for a few
lowest order radial modes are reported. For typical models , either only
one mode is linearly unstable, or two radial modes are simultaneously
ustable. This is linear analysis however; you perturb a model and ask what
will happen with the perturbation: will it grow (positive growth rate) or
will it decay (negative growth-rate). At the linear level, the growth or
decay are exponential. Linear analysis, unless only one mode is linearly
unstable, tells little about the final, large-amplitude pulsation state.
Here non-linear effects set in. So, there is no simple answer what will
happen when, say, both F and 1O mode s are linearly unstable. There are
several possibilities. You may end up with single-mode F-mode pulsation, or
single-mode 1O pulsation, or stable double-mode pulsation, or the answer
may depend on the initial kick you have applied at the beginning of the
simulation (hysteresis; physically, for given location on the HR diagram,
the pulsation state may depend e.g. on the direction of evolution). You
must run the simulation to see what happens. What you see before the
pulsations reach stable large-amplitude pulsation is beating of the two (or
more pulsation modes). You may apply time dependent Fourier analysis to
trace the amplitudes of the modes involved. You may even describe it
analytically, using the amplitude equations formalism. You may want to take
a look at the review I wrote some time ago about the mode selection problem
(which I briefly outlined above)
https://ui.adsabs.harvard.edu/abs/2014IAUS..301..265S/abstract
and then go deeper along the references provided there.


> Is there any way to determine how long MESA will take to produce a model
> with a steady period of pulsation, or what set of initial conditions would
> give a steady period of pulsation?
>
When only one mode is linearly unstable, inverse of the growth rate
provides a good estimate for a time-scale to reach large amplitude
pulsation.


> Lastly, how exactly do the unconverged models differ from each other and
> from the models where large amplitude pulsations have been reached?
>
I'm not sure what you ask about. You have all the quantities in the history
file to see the differences between the consecutive models/pulsation
cycles. You are interested in large-amplitude pulsation, with stable
amplitude and stable pulsation period. Note however that:
- period at a large amplitude pulsation, non-linear period, differs a bit
from the linear one, reported by linear analysis. Usually it is longer.
When you compare with the observations, you should use a non-linear period
(but the difference is not large). When you are interesetd in pulsation
period change along stellar evolutionary track, using a linear periods is
an excellent proxy.
- with RSP, even if it seems the pulsation is stable, the amplitude and
period still vary at a very tiny rate. With direct time integration we
never reach so-called limit cycle: finite amplitude, strictly periodic
pulsation (relaxation technique may be used for that, but it is not
implemented in RSP; see the review and refernces given there). It is up to
you to decide when to stop the integration and whether the still ongoing
tiny changes are at a sufficiently small level for your applications.

Hope it helps a bit.
Cheers,
Radek







>
> I do not know if this may help you answer any of my questions, but the
> reason I am asking for all this information is that I am trying to develop
> a simulator for Cepheid (both types 1 and 2) light curves for my professor,
> who is trying to calibrate his telescope at an observatory to detect minute
> changes in Cepheid periods as they evolve. The first step in this
> calibration would be to simulate real light curves, a process which
> involves physics that is more sophisticated than I know (as a 2nd year
> undergraduate) to start from scratch, which is why I am using MESA.
>
> Many thanks for your patience,
>
> Alex
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