2.2.4 Derived thermodynamic coefficients

First, the missing derivative $\left(\frac{\partial T}{\partial \rho}\right)_{e}$ can be found from the relation:

$$\fcolorbox{blue}{yellow}{$\displaystyle \left(\frac{\partial... ...p}{\rho^2}\left(\frac{\partial T}{\partial e}\right)_{\rho} $}$$ (22)

which is obtained from the equality of the mixed derivatives in Eq.4, written as:

$${\rm d} s = \frac{1}{T} {\rm d} e - \frac{p}{T \rho^2} {\rm d} \rho$$ (23)

Then

$$\frac{\partial^{2}s}{\partial e\partial \rho} = \frac{\parti... ...{\partial }{\partial e}\left( -\frac{p}{T \rho^2} \right)_\rho$$ (24)

First adiabatic exponent:

$$\fcolorbox{blue}{yellow}{$\displaystyle \Gamma_1 \equiv \lef... ...{1}{\rho} \left(\frac{\partial p}{\partial e}\right)_{\rho} $}$$ (25)

This relation is obtained by combining Eq.4 with the identity

$${\rm d} p = \left(\frac{\partial p}{\partial \rho}\right)_{e... ... + \left(\frac{\partial p}{\partial e}\right)_{\rho} {\rm d} e$$ (26)

The adiabatic sound speed is then obtained as

$$\fcolorbox{blue}{yellow}{$\displaystyle c_s \equiv \sqrt{\le... ...\partial \rho}\right)_{s}} = \sqrt{\Gamma_1 \frac{p}{\rho}} $}$$ (27)

Third adiabatic exponent:

$$\fcolorbox{blue}{yellow}{$\displaystyle \Gamma_3 \equiv 1 + ... ...{1}{\rho} \left(\frac{\partial p}{\partial e}\right)_{\rho} $}$$ (28)

This relation is obtained by combining Eq.4 with the identity

$${\rm d} T = \left(\frac{\partial T}{\partial \rho}\right)_{e... ... + \left(\frac{\partial T}{\partial e}\right)_{\rho} {\rm d} e$$ (29)

and then using Eq.22.

Adiabatic temperature gradient:

$$\fcolorbox{blue}{yellow}{$\displaystyle \nabla_{\rm ad} \equ... ...T}{\partial \ln p}\right)_{s} = \frac{\Gamma_3-1}{\Gamma_1} $}$$ (30)

since

$$\left(\frac{\partial \ln T}{\partial \ln p}\right)_{s} = \le... ...rho}{\partial \ln p}\right)_{s} = \frac{\Gamma_3-1}{\Gamma_1}.$$ (31)

Adiabatic energy changes:

$$\fcolorbox{blue}{yellow}{$\displaystyle \rho \left(\frac{\pa... ...l \ln \rho}{\partial \ln p}\right)_{s} = \frac{1}{\Gamma_1} $}$$ (32)

or

$$\fcolorbox{blue}{yellow}{$\displaystyle \left(\frac{\partial... ..._{s} = \frac{1}{\Gamma_1}\left(1 + \frac{\rho e}{p}\right) $}$$ (33)

We define the coefficients $c_v^\prime$ and $c_p^\prime$ through the relation

$${\rm d} s = c_v^\prime {\rm d} \ln p - c_p^\prime {\rm d} \ln \rho$$ (34)

Entropy change at constant density:

$$\fcolorbox{blue}{yellow}{$\displaystyle c_v^\prime = \left(... ... \ln T}\right)_{s} = \frac{p}{\rho T}\frac{1}{\Gamma_3-1} $}$$ (35)

This relation is obtained from the equality of the mixed derivatives in Eq.4 together with Eq.28.

Entropy change at constant pressure:

$$\fcolorbox{blue}{yellow}{$\displaystyle c_p^\prime = -\left... ...\right)_{s} = \frac{p}{\rho T}\frac{\Gamma_1}{\Gamma_3-1} $}$$ (36)

This relation is obtained from the equality of the mixed derivatives in Eq.5 together with Eq.30.

Specific heat at constant density:

$$\fcolorbox{blue}{yellow}{$\displaystyle c_v = c_v^\prime ... ...} = 1 / {\left(\frac{\partial T}{\partial e}\right)_{\rho}} $}$$ (37)

To derive the specific heat at constant pressure, we start from the relation

$${\rm d} \ln T = \left(\frac{\partial \ln T}{\partial \ln \rh... ...left(\frac{\partial \ln T}{\partial s}\right)_{\rho} {\rm d} s$$ (38)

from which we get

$$\left(\frac{\partial s}{\partial \ln \rho}\right)_{T} = -\le... ..._{s} / \left(\frac{\partial \ln T}{\partial s}\right)_{\rho}$$ (39)

Using Eqs. 28 and 37, we obtain

$$\fcolorbox{blue}{yellow}{$\displaystyle \left(\frac{\partial s}{\partial \ln \rho}\right)_{T} = -c_v (\Gamma_3-1) $}$$ (40)

Now

$${\rm d} s = \left(\frac{\partial s}{\partial \ln p}\right)_{... ...ac{\partial s}{\partial \ln \rho}\right)_{p} {\rm d} \ln \rho$$ (41)

or

$${\rm d} s = \left(\frac{\partial s}{\partial \ln p}\right)_{... ...c{\partial \ln \rho}{\partial s}\right)_{T} {\rm d} s \right\}$$ (42)

hence

$${\rm d} s \left\{1- \left(\frac{\partial s}{\partial \ln \rh... ...ac{\partial \ln \rho}{\partial \ln T}\right)_{s} {\rm d} \ln T$$ (43)

and finally

$$\left(\frac{\partial \ln T}{\partial s}\right)_{p} = \left\{... ...ft(\frac{\partial \ln \rho}{\partial \ln T}\right)_{s}\right\}$$ (44)

or

$$\fcolorbox{blue}{yellow}{$\displaystyle \left(\frac{\partial... ...eft(\frac{\partial \ln \rho}{\partial s}\right)_{T}\right\} $}$$ (45)

Using Eqs. 11, 28, 36, 40, we finally obtain the relation for the specific heat at constant pressure:

$$\fcolorbox{blue}{yellow}{$\displaystyle \frac{1}{c_p} = \fra... ... = \frac{1}{c_v} - T \left(\frac{\Gamma_3-1}{c_s}\right)^2 $}$$ (46)

Alternatively, $c_p$ can be obtained from Eq.16

$$\fcolorbox{blue}{yellow}{$\displaystyle c_p = c_v + \frac{p}{\rho T} \delta \chi_{T} $}$$ (47)

or from

$$\fcolorbox{blue}{yellow}{$\displaystyle c_p = \delta c_p^... ... = \frac{p}{\rho T} \delta \frac{\Gamma_1}{\Gamma_3-1} $}$$ (48)

once $\delta$ and $\chi_{T}$ are known (see below).

We can now express the thermodynamic coefficients provided by CO5BOLD in terms of $c_v$, $\Gamma_1$, $\Gamma_3$, and $\nabla_{\rm ad}$:

$$\fcolorbox{blue}{yellow}{$\displaystyle \left(\frac{\partial p}{\partial e}\right)_{\rho} = \rho (\Gamma_3 - 1) $}$$ (49)

$$\fcolorbox{blue}{yellow}{$\displaystyle \left(\frac{\partial... ...ho}\right)_{e} = \frac{p}{\rho} (1 - \Gamma_3 +\Gamma_1) $}$$ (50)

$$\fcolorbox{blue}{yellow}{$\displaystyle \left(\frac{\partial T}{\partial e}\right)_{\rho} = \frac{1}{c_v} $}$$ (51)

$$\fcolorbox{blue}{yellow}{$\displaystyle \left(\frac{\partial... ...T}{\rho} (\Gamma_3-1) - \frac{p}{\rho^2} \frac{1}{c_v} $}$$ (52)

$$\fcolorbox{blue}{yellow}{$\displaystyle \left(\frac{\partial... ...^2}\left\{1 - \frac{\rho T}{p} c_v (\Gamma_3-1)\right\} $}$$ (53)

$$\fcolorbox{blue}{yellow}{$\displaystyle \left(\frac{\partial... ...= \frac{p}{\rho^2}\frac{\nabla_{\rm ad}-1}{\nabla_{\rm ad}} $}$$ (54)

We consider again Eq.23, replacing ${\rm d} e$ by

$${\rm d} e = \left(\frac{\partial e}{\partial T}\right)_{\rho... ...\left(\frac{\partial e}{\partial \rho}\right)_{T} {\rm d} \rho$$ (55)

so

$${\rm d} s = \frac{1}{T} \left(\frac{\partial e}{\partial T}\... ...l \rho}\right)_{T} - \frac{p}{T \rho^2}\right\} {\rm d} \rho$$ (56)

The requirement that the mixed derivatives must be equal then yields

$$\frac{\partial }{\partial \rho}\left( \frac{1}{T} \left(\fra... ...e}{\partial \rho}\right)_{T} - \frac{p}{T \rho^2} \right)_\rho$$ (57)

or

$$0 = -\frac{1}{T^2}\left(\frac{\partial e}{\partial \rho}\rig... ...\partial p}{\partial T}\right)_{\rho} - \frac{p}{T^2}\right\}$$ (58)

Finally,

$$\fcolorbox{blue}{yellow}{$\displaystyle \left(\frac{\partial... ... T}\right)_{\rho}\right\} = \frac{p}{\rho^2}(1 - \chi_{T}) $}$$ (59)

Comparison with Eq.53 implies

$$\fcolorbox{blue}{yellow}{$\displaystyle \chi_{T} = \frac{\rho T}{p} c_v (\Gamma_3-1) $}$$ (60)

Similarly, replacing ${\rm d} e$ by

$${\rm d} e = \left(\frac{\partial e}{\partial p}\right)_{\rho... ...\left(\frac{\partial e}{\partial \rho}\right)_{p} {\rm d} \rho$$ (61)

in Eq.23, we get

$${\rm d} s = \frac{1}{T} \left(\frac{\partial e}{\partial p}\... ...l \rho}\right)_{p} - \frac{p}{T \rho^2}\right\} {\rm d} \rho$$ (62)

and the requirement that the mixed derivatives must be equal then yields

$$\frac{\partial }{\partial \rho}\left( \frac{1}{T} \left(\fra... ...e}{\partial \rho}\right)_{p} - \frac{p}{T \rho^2} \right)_\rho$$ (63)

or

$$\left(\frac{\partial e}{\partial p}\right)_{\rho} \left(\fr... ...rho}\right)_{p} - \frac{p}{\rho^2}\right\} + \frac{T}{\rho^2}$$ (64)

or

$$\left(\frac{\partial e}{\partial p}\right)_{\rho} \left(\fr... ...al \rho}\right)_{p} - \frac{1}{\rho}\right\} + \frac{1}{\rho}.$$ (65)

Since

$$\left(\frac{\partial \ln T}{\partial \ln \rho}\right)_{p}/\l... ...c{\partial \ln p}{\partial \ln \rho}\right)_{T} = -\chi_{\rho}$$ (66)

we finally obtain, using Eqs.13, 49 and 54,

$$\fcolorbox{blue}{yellow}{$\displaystyle \chi_{\rho} = \Gamma_1 - \chi_{T} (\Gamma_3 -1) $}$$ (67)

and

$$\fcolorbox{blue}{yellow}{$\displaystyle \delta = \chi_{T} / \left\{\Gamma_1 - \chi_{T} (\Gamma_3 -1)\right\} $}$$ (68)

The isothermal sound speed is then obtained as

$$\fcolorbox{blue}{yellow}{$\displaystyle c_T \equiv \sqrt{\le... ...mma_1}\right\}} = c_s \sqrt{(1 - \chi_{T} \nabla_{\rm ad})} $}$$ (69)