[mesa-users] Writing out the atmosphere in the profile

[Anwesh Mazumdar]

Dear Bill,

Thanks for your mail.

On 01/25/2013 01:39 AM, Bill Paxton wrote:
> For the particular example you give, can you help me understand how the outer R is determined for the integral giving the acoustic radius?
> Is it the photosphere?  If so, why do you need the atmosphere for that if the model goes out to the photosphere?
> We currently calculate delta_nu as 1/(2*photosphere_acoustic_r), with photosphere_acoustic_r calculated by a discrete sum of dr/c.
> Is that different that what you suggest?

Well, the issue of the outer boundary of the star relevant for 
seismology ("seismic radius") is a tricky one. In principle, it is a 
fiducial radius that defines the outer phase of the acoustic modes for a 
chosen boundary condition. For the Sun, for example, it should be beyond 
the temperature minimum, about 225 seconds (in terms of sound travel 
time) above the photosphere (see, for example, Houdek & Gough 2007, 
MNRAS, 375, 861).

To quote from the above reference, the seismic radius is "essentially 
the surface on which c ^2 , if extrapolated linearly outwards from the 
outer layers of the adiabatically stratified region of the convection 
zone, would vanish".

In practice, for other stars, it is more difficult to define. But the 
important point is that the seismic radius should be consistent with the 
frequency calculations. Thus, if we calculate frequencies with the 
atmosphere included (for example, with the boundary condition of an 
isothermal atmosphere), then the acoustic radius should not terminate at 
the photosphere, but rather at the appropriate point in the atmosphere 
(the outer turning point of the acoustic modes). Including an atmosphere 
affects the frequencies (and the average large separation), and should 
thus be considered while calculating the acoustic radius as well.

The method of calculation of the acoustic radius is, of course, as you 
describe, but the issue is whether the photosphere is the relevant outer 
boundary or not. From the solar experience, it is not; hence the need to 
include the atmosphere.

Even with the seismic radius defined "correctly", we will not get the 
exact correspondence between the model acoustic radius and the delta_nu. 
But that is due to our (acknowledged) inability to model the physics of 
the outermost layers well -- a different problem altogether.

With best regards,
Anwesh

-- 

Dr. Anwesh Mazumdar
Homi Bhabha Centre for Science Education
Tata Institute of Fundamental Research
V. N. Purav Marg, Mankhurd, Mumbai 400088, India.
Phone: +91 22 2507 2424   Fax: +91 22 2556 6803
E-mail: anwesh at tifr.res.in