[mesa-users] density partial

[Francis Timmes]

hi ehsan,

> The term, e.g., in the numerator ∂P/∂T must be evaluated at constant mu and rho.

exactly. and the term in the denominator is evaluated at constant temperature and constant mu.
using the symbol “|" to show explicitly what is held constant.

d(rho)/dT |_{P,mu} = - ∂P/∂T |_{rho,mu}  /  ∂P/∂(rho) |_{T,mu}

this is a standard relationship you can find in stellar textbooks
and it is how you calculated it numerically (divide two partials).

mesa uses, as josiah notes, the chi_T and chi_rho formulation - same thing.

fxt





On May 18, 2014, at 3:59 PM, Ehsan Moravveji <e.moravveji at gmail.com> wrote:

> Thanks Josiah and Frank for your replies.
> The term, e.g., in the numerator ∂P/∂T must be evaluated at constant mu and rho. This is the definition of the differential of a multivariate function, say P(rho, T, mu) in this case.
> Then, I still do not know how to carry this out numerically.
> 
> Best regards,
> Ehsan 
> 
> 
> 
> On May 18, 2014, at 2:35 AM, Francis Timmes wrote:
> 
>> hi ehsan,
>> 
>> dP = ∂P/∂T dT + ∂P/∂(rho) d(rho) + ∂P/∂(mu) d(mu) 
>> 
>> for constant pressure dP = 0 and constant composition d(mu) = 0,
>> 
>> ∂P/∂T dT + ∂P/∂(rho) d(rho) = 0
>> 
>> or
>> 
>> d(rho)/dT = - ∂P/∂T / ∂P/∂(rho)
>> 
>> these two partial derivatives should be readily available from an eos.
>> 
>> fxt
>> 
>> 
>> 
>> On May 17, 2014, at 3:27 PM, Josiah Schwab <jwschwab at berkeley.edu> wrote:
>> 
>>> Hi Ehsan,
>>> 
>>>> I need to calculate the partial derivative of density
>>>> w.r.t. temperature at constant pressure and constant mu. looking into
>>>> the public pointers in star_data.inc did not help me.  Can any of you
>>>> kindly help me calculate that?
>>> 
>>> The basic eos (see eos/public/eos_def.f), gives you
>>> 
>>> ,----
>>> | integer, parameter :: i_chiRho = 5 
>>> |       ! dlnP_dlnRho at constant T
>>> | integer, parameter :: i_chiT = 6 
>>> |       ! dlnP_dlnT at constant Rho
>>> `----
>>> 
>>> and then I think you can derive dlnRho_dlnT at constant P from those
>>> using an identity like
>>> 
>>> http://en.wikipedia.org/wiki/Triple_product_rule
>>> 
>>> Hope that helps,
>>> Josiah
>>> 
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