e^{-\frac{\rho_{\text{const}}}{\rho}}-b)$ (204)

One has to be cautious as this relation of Eq. 204 only applies to within a
single solid phase. If the isentrope encounters a phase boundary, such as a
Fluid-Solid phase transition, or a Solid-Solid phase transition, then there
will be a discontinuity in the specific entropy, and a discontinuity in the
density (or equivalently specific volume) as well. Then this calculation needs
to be carried out separately for each phase.

The isentropes of the superionic ice (Ice-eighteen) can be calculated from the
ab initio simulation data from [64], as shown in Fig. 23. The local slope of
isentrope, i.e., the Grüneisen parameter $\gamma$, is calculated from Eq. 13.

Figure 23: Isentropes of fluid-H2O and superionic-H2O. The arrows (orange for
superionic and purple for high-pressure fluid phase) show the local slope of
an isentrope, that is, the Grüneisen parameter $\gamma$ which is calculated
from applying Eq. 13 to the ab initio simulation data of [64]. Pressure
contours are shown and labelled with its value in GPa. The density
discontinuity between the fluid and superionic phases is not shown in this
plot, but will be elaborated in more detail in the next section.

It is also quite interesting to see how the isentropes behave below the
density of critical point (Fig. 24). This regime people often refer to as
_Real Gas_ , as it is still considered a gas phase, while the inter-molecular
interaction is quite strong and cannot be neglected. One such type of EOS that
we will introduce later on is called the _Berthelot EOS_.

Figure 24: Isentropes (colored) and Isobars (black) of H2O below and near its
critical point.

Sometimes, it is useful to view these isentropes in a linear-plot of density
$\rho$ rather logarithmic. Thus, we come up with the following Fig. 25:

Figure 25: Isentropes of H2O in linear-$\rho$ and $\ln{T}$ plot.

####  4.4.2 Calculating Melting Curve of Superionic Ice

The _melting curve_ here defines the phase boundary between the liquid and the
superionic ice (also known as Ice-eighteen [30]). It has a shallower slope
compared to the isentropes in the same regime in the temperature-density
($T$-$\rho$) phase diagram. It can be calculated according to Eq. 34.

With this knowledge in mind, we refine the High-Pressure Phase Diagram of H2O
with the density jump at the phase transition between the fluid and
superionic. See Fig. 26.

Figure 26: Eight isentropes of fluid-H2O and superionic-H2O, plus the density
discontinuity at the phase boundary, are chosen here for the purpose of
calculating mass-radius curves of pure-H2O planets. Pressure contours are
shown in black. Their smooth behavior suggests a possible approximation by
power-law for future simplification of calculations. That is $\gamma\approx$
constant.

These eight isentropes can also be viewed in the $\rho$-$P$ (density-pressure)
plot (Fig. 27), in order to calculated the mass-radius curve corresponding to
each isentrope.

Figure 27: Density-Pressure ($\rho$-$P$) plot of isentropes of H2O. The
discontinuity in each curve corresponds to the density jump at the phase
boundary between fluidic and superionic. The color scheme of the curves are
the same as Fig. 26. At high-pressure or high-density, all isentropes converge
and have smooth behavior. This suggests that the EOS in this regime can be
approximated by
$\Gamma\equiv\bigg{(}\frac{\partial\ln{P}}{\partial\ln{\rho}}\bigg{)}_{s}\approx$
constant. The achieve higher fidelity, one could add a temperature-correction
term.

The mass-radius curves of these eight different isentropes, as a result, are
shown in a small-scale mass-radius plot (Fig. 28) and a large-scale mass
radius plot (Fig. 29).

Figure 28: Mass-radius curves of eight H2O isentropes below 4 Earth radii
(R⊕). These mass-radius curves suggest that with the appropriate consideration
of interior temperature/entropy sustained by planet’s own energy source, it
could help explain some of the exoplanets observed in this regime as water-
rich planets.  Figure 29: Mass-radius curves of pure H2O isentropes compared
with the pure Hydrogen isentropes. The color of each mass-radius curve
corresponds to different assumed specific entropy. The black mesh provides
another set of contours of the central densities of planets.

###  4.5 EOS of Ice VI and Ice VII

According to [25], the EOS parameters for Ice VI and Ice VII are as follows.
Note that the underscript ”0” suggests extrapolation of the parameter under
ambient conditions ($P_{0}=$1 bar and $T_{0}=$300 Kelvin). These parameters
are obtained by fit to BM2 EOS with temperature-corrections as:

Density-correction:

$\rho(P,T)=\rho_{0}(P,T_{0})\cdot\exp{\bigg{(}-\alpha_{0}\cdot(T-T_{0})\bigg{)}}$
(205)

Bulk-modulus correction:

$K(T)=K_{0}+(T-T_{0})\cdot\bigg{(}\frac{\partial K}{\partial T}\bigg{)}_{P}$
(206)

The melting curve of both Ice VI and Ice VII is fit to a _Simon-Glatzel_ form
of Equation [75].

$P_{M}=P_{0}+a\cdot\Bigg{(}\bigg{(}\frac{T_{M}}{T_{0}}\bigg{)}^{c}-1\Bigg{)}$
(207)

where $a$ and $c$ are constants to be determined. More recent development
involves the _Kechin_ Equation [80] [54].

####  4.5.1 Ice VI

Thermodynamic parameters of Ice VI [25]:

$\displaystyle\rho_{0}$ $\displaystyle\approx 1.27\text{~{}g/cc}$
$\displaystyle K_{0}$ $\displaystyle\approx 14\text{~{}GPa}$
$\displaystyle\alpha_{0}$ $\displaystyle\approx 14.6\cdot
10^{-5}\text{K}^{-1}$ $\displaystyle c_{v}$ $\displaystyle\approx
2.5~{}\frac{\text{J/K}}{g}$ $\displaystyle\gamma$
$\displaystyle=\frac{\alpha\cdot K_{T}}{c_{v}\cdot\rho}\approx 0.65$

Melting Curve of Ice VI [49]:

$P_{M}\approx
0.63+0.68\cdot\Bigg{(}\bigg{(}\frac{T_{M}}{273\text{K}}\bigg{)}^{4.6}-1\Bigg{)}~{}\text{GPa}$
(209)

####  4.5.2 Ice VII

Thermodynamic parameters of Ice VII [25]:

$\displaystyle\rho_{0}$ $\displaystyle\approx 1.45\text{~{}g/cc}$
$\displaystyle K_{0}$ $\displaystyle\approx 20\text{~{}GPa}$
$\displaystyle\alpha_{0}$ $\displaystyle\approx 11.6\cdot
10^{-5}\text{K}^{-1}$ $\displaystyle c_{v}$ $\displaystyle\approx
2~{}\frac{\text{J/K}}{g}$ $\displaystyle\gamma$
$\displaystyle=\frac{\alpha\cdot K_{T}}{c_{v}\cdot\rho}\approx 0.81$

Melting Curve of Ice VII [41], fit to experimental data up to 60 GPa:

$P_{M}\approx
2.17+0.764\cdot\Bigg{(}\bigg{(}\frac{T_{M}}{355\text{K}}\bigg{)}^{4.32}-1\Bigg{)}~{}\text{GPa}$
(211)

and the density of Ice VII along the melting curve is [41]:

$\rho_{M}\approx
1.45+0.4\cdot\Bigg{(}1-\exp{\bigg{(}-0.0743\frac{P_{M}}{\text{GPa}}\bigg{)}}\Bigg{)}+2.8\cdot\Bigg{(}1-\exp{\bigg{(}-0.0061\frac{P_{M}}{\text{GPa}}\bigg{)}}\Bigg{)}~{}\text{g/cc}$
(212)

Thus, a good rule of thumb for the Grüneisen parameter in the regime of both
Ice VI and Ice VII is that $\gamma\approx 0.7\sim 0.8$.

###  4.6 3D EOS of H2O

Finally, in order to aid our global understanding of the landscape of H2O EOS,
we _visualize_ the isentropes in a 3D $\rho TP$ EOS of H2O in Fig. 30, Fig.
31, Fig. 32, and Fig. 33.

Figure 30: A simplified 3-D $\rho-T-P$ EOS of H2O EOS with isentropes (red-
thin curves and colored dots) and isobars (black curves) visualized. The
region of _fluid_ H2O is colored golden. The cyan region above is _super-
ionic_ H2O, and green region below is the gas-liquid two-phase equilibrium
(co-existence).  Figure 31: 3-D $\rho-T-P$ EOS of H2O EOS with isentropes
(red-thin curves and colored dots) and isobars (black curves) visualized.
Recall that along an isentrope,
$\Gamma\equiv\bigg{(}\frac{\partial\ln{P}}{\partial\ln{\rho}}\bigg{)}_{s}$,
and thus, $\Gamma$ are just the tangential slopes of these 3-D isentropes when
viewed from the perspective of the P-$\rho$ plane, c.f. Fig. 27. Figure 32:
3-D $\rho-T-P$ EOS of H2O EOS with isentropes (red-thin curves and colored
dots) and isobars (black curves) visualized. Viewed from the perspective of
the T-P plane, the tangential slopes of isentropes are
$\frac{\gamma}{\Gamma}=\bigg{(}\frac{\partial\ln{T}}{\partial\ln{P}}\bigg{)}_{s}=\bigg{(}\frac{\partial\ln{T}}{\partial\ln{\rho}}\bigg{)}_{s}\Bigg{/}\bigg{(}\frac{\partial\ln{P}}{\partial\ln{\rho}}\bigg{)}_{s}$.
Figure 33: 3-D $\rho-T-P$ EOS of H2O EOS extending down to lower density
($\rho$) below the critical point (big red dot). This shows how quickly it
approaches the ideal gas EOS below the critical point: the contours of
isochors (red thin curves), isobars (black thin curves), and isotherms (green
thin curves) all become _linear_ in this low density regime in the lg-lg-lg
plot. The gas-liquid equilibrium (co-existence) area below the critical
density ($\rho_{c}$) is colored purple. This observation reinforces the idea
to use a _modified_ ideal gas EOS to describe the entire _fluid_ regime of
H2O.

##  5 EOS of NH3, N2, CH4,…

###  5.1 Ammonia (NH3)

According to our earlier discussions, the Temperature-Density (T-$\rho$) phase
diagram of Ammonia should resemble that of water. To the first-order
approximation, we just need to re-scale the H2O phase diagram _properly_ to
obtain the one for Ammonia.

Then, we will implement the ab initio calculation results from Rostock Group,
in particular, Mandy Bethkenhagen [24] [23]. Their calculations show that NH3
also contains a super-ionic phase.

####  5.1.1 Critical Point (C.P) for Ammonia

The analytical formulae of NH3 are anchored at its critical point ($c$) [9]:

$\displaystyle\rho_{c}$ $\displaystyle=0.226\text{~{}g/cc}$ $\displaystyle
T_{c}$ $\displaystyle=405.5\text{~{}K}=132.3^{\circ}\text{C}$ $\displaystyle
P_{c}$ $\displaystyle=1.128\cdot 10^{7}\text{~{}Pa}=112.8\text{~{}bar}$
$\displaystyle n_{c}$ $\displaystyle=0.226/17=0.013\text{~{}mol/cc}$

Here $n_{c}$ is the molar density of ammonia (NH3) at its critical point.

####  5.1.2 Gas-Liquid-Ice Triple Point (t1) for Ammonia

The Gas-Liquid-Ice triple point ($t1$) has the following properties [9] [5]:

$\displaystyle\rho_{t1}$ $\displaystyle=0.733\text{~{}g/cc for liquid}$
$\displaystyle T_{t1}$ $\displaystyle=195.4\text{~{}K}=-77.75^{\circ}\text{C}$
$\displaystyle P_{t1}$ $\displaystyle=6060\text{~{}Pa}\approx
0.06\text{~{}bar}$ $\displaystyle n_{t1}$
$\displaystyle=0.733/17=0.043\text{~{}mol/cc for liquid}$

Here $n_{t1}$ is the molar density of ammonia (NH3) at its triple point.

####  5.1.3 Re-Scaling

Now, let’s look at the _ratio_ of densities at the C.P.:

$\frac{\rho_{c,\text{NH}_{3}}}{\rho_{c,\text{H}_{2}\text{O}}}=\frac{0.226}{0.322}\approx
0.702$ (215)

and the _ratio_ of densities at the Gas-Liquid-Ice Triple Point (t1):

$\frac{\rho_{t1,\text{NH}_{3}}}{\rho_{t1,\text{H}_{2}\text{O}}}=\frac{0.733}{1.0}\approx
0.733$ (216)

These two _ratios_ are close to one another.

###  5.2 Nitrogen (N2)

####  5.2.1 Critical Point (C.P) for Nitrogen

The analytical formulae of N2 are anchored at its critical point ($c$) [12]:

$\displaystyle\rho_{c}$ $\displaystyle=0.314\text{~{}g/cc}$ $\displaystyle
T_{c}$ $\displaystyle=126.19\text{~{}K}=-146.96^{\circ}\text{C}$
$\displaystyle P_{c}$ $\displaystyle=3.4\cdot
10^{6}\text{~{}Pa}=34\text{~{}bar}$ $\displaystyle n_{c}$
$\displaystyle\approx 0.0112\text{~{}mol/cc}$

Here $n_{c}$ is the molar density of nitrogen (N2) at its critical point.

####  5.2.2 Gas-Liquid-Ice Triple Point (t1) for Nitrogen

The Gas-Liquid-Ice triple point ($t1$) has the following properties [8]:

$\displaystyle\rho_{t1}$ $\displaystyle=0.867\text{~{}g/cc for liquid}$
$\displaystyle T_{t1}$
$\displaystyle=63.14\text{~{}K}=-210.01^{\circ}\text{C}$ $\displaystyle
P_{t1}$ $\displaystyle=1.25\cdot 10^{4}\text{~{}Pa}=0.125\text{~{}bar}$
$\displaystyle n_{t1}$ $\displaystyle\approx 0.031\text{~{}mol/cc}$

Here $n_{t1}$ is the molar density of nitrogen (N2) at its triple point.

####  5.2.3 Re-Scaling

Now, let’s look at the _ratio_ of densities at the C.P.:

$\frac{\rho_{c,\text{N}_{2}}}{\rho_{c,\text{H}_{2}\text{O}}}=\frac{0.314}{0.322}\approx
0.975$ (219)

and the _ratio_ of densities at the Gas-Liquid-Ice Triple Point (t1):

$\frac{\rho_{t1,\text{N}_{2}}}{\rho_{t1,\text{H}_{2}\text{O}}}=\frac{0.867}{1.0}\approx
0.867$ (220)

These two _ratios_ are close to one another.

###  5.3 Methane (CH4)

####  5.3.1 Critical Point (C.P) for Methane

The analytical formulae of CH4 are anchored at its critical point ($c$) [11]:

$\displaystyle\rho_{c}$ $\displaystyle\approx 0.0101\cdot 16\approx
0.162\text{~{}g/cc}$ $\displaystyle T_{c}$
$\displaystyle=190.6\text{~{}K}=-82.55^{\circ}\text{C}$ $\displaystyle P_{c}$
$\displaystyle=4.61\cdot 10^{6}\text{~{}Pa}=46.1\text{~{}bar}$ $\displaystyle
n_{c}$ $\displaystyle\approx 0.0101\text{~{}mol/cc}$

Here $n_{c}$ is the molar density of methane (CH4) at its critical point.

####  5.3.2 Gas-Liquid-Ice Triple Point (t1) for Methane

The Gas-Liquid-Ice triple point ($t1$) has the following properties [7]:

$\displaystyle\rho_{t1}$ $\displaystyle\approx 0.439\text{~{}g/cc for liquid}$
$\displaystyle T_{t1}$
$\displaystyle=90.67\text{~{}K}=-182.48^{\circ}\text{C}$ $\displaystyle
P_{t1}$ $\displaystyle=1.169\cdot 10^{4}\text{~{}Pa}=0.1169\text{~{}bar}$
$\displaystyle n_{t1}$ $\displaystyle\approx 0.439/16\approx
0.0275\text{~{}mol/cc}$

Here $n_{t1}$ is the molar density of methane (CH4) at its triple point.

####  5.3.3 Re-Scaling

Now, let’s look at the _ratio_ of densities at the C.P.:

$\frac{\rho_{c,\text{CH}_{4}}}{\rho_{c,\text{H}_{2}\text{O}}}=\frac{0.162}{0.322}\approx
0.503$ (223)

and the _ratio_ of densities at the Gas-Liquid-Ice Triple Point (t1):

$\frac{\rho_{t1,\text{CH}_{4}}}{\rho_{t1,\text{H}_{2}\text{O}}}=\frac{0.439}{1.0}\approx
0.439$ (224)

These two _ratios_ are close to one another.

###  5.4 Discussion

The comparison among all these gas species reveal their similarities in terms
of their molar densities at their critical points and their triple points.
Generally speaking, we have for all of them (H2, H2O, NH3, N2, CH4,…):

$\displaystyle n_{c}$ $\displaystyle\sim 0.01\text{~{}mol/cc}$ $\displaystyle
n_{t1}$ $\displaystyle\sim 0.03\text{~{}mol/cc}$ $\displaystyle n_{t1}/n_{c}$
$\displaystyle\sim 3$

This approximation is generally good to within a factor of 2. Even more so for
the ratio of $n_{t1}/n_{c}$.

This supports the idea of the _principle of corresponding states_ [53], and
will aid our future discussion of gaseous envelopes. We can think of
$n_{c}\sim$0.01  mol/cc as a general _critical molar density_ at which
cohesion of molecules transition from being close to zero to non-zero as
pointed out by D. Mendeleev (1870).

$n_{t1}$ can be understood as another _critical molar density_ at which
cohesion of molecules has exerted its full power, and any compression beyond
this point is strongly resisted by the repulsive interactions of molecules,
that is, the Pauli repulsion at shorter ranges due to overlapping electron
orbitals.

To go from $n_{c}$ to $n_{t1}$ only requires a three-fold compression. That
means a mere a factor of $\sqrt[3]{3}=1.44$ reduction in the linear dimension
or linear spacing between neighboring molecules brings about the overturn in
the inter-molecular interaction from cohesion to (strong) repulsion.

###  5.5 An Approximate EOS

Let’s define the density of liquid at ambient conditions as $\rho_{0}\sim 1$
g/cc. In contrast, the density at critical point is typically one-third of
that.

The density under other conditions as $\rho$. Their ratio can be defined as a
dimensionless parameter:

$\eta\equiv\rho/\rho_{0}$ (226)

Furthermore, we define another dimensionless parameter for temperature:

$\tau\equiv T/T_{c}$ (227)

Then, approximately the pressure can be expressed as:

$\frac{P}{\text{GPa}}\approx\eta^{4}-\eta^{2}+\tau\cdot\eta\cdot\exp{\bigg{(}\frac{\eta^{2}-1}{\eta^{2}+1}\bigg{)}}$
(228)

This EOS captures both the increase of pressure due to electronic Coulomb
repulsion, and critical behavior, as well as thermal effects (including both
ideal gas behavior at low-density high-temperature or atomic vibrations in
crystal lattices at high-density).

The reason it works is because at low-density: $P=n\cdot k_{B}\cdot T$, where
$n$ is the number density of _molecules_ , where each molecule contains
several ($\sim$ 3) atoms. At high-density: $P_{\text{thermal}}\approx
3\cdot\tilde{n}\cdot k_{B}\cdot T$, where $\tilde{n}$ is the number of
_atoms_. It can be tuned to higher-accuracy by adding one or two extra
parameters inside the exponential term.

Eq. 228 can reproduce the approximate behavior of H2O, NH3, N2, CH4 from
$\sim$one-tenth g/cc up to $\sim$5 g/cc, which covers the transition regime of
fluid from gas-like behavior at lower density to more crystal-like behavior at
higher density.

Figure 34: Pressure contours (labelled in GPa), calculated from this
approximate EOS (Eq. 228).

###  5.6 Approximate gas-liquid and gas-solid equilibrium
and its relation to atmospheric escape

On the other hand, the pressure of gas-liquid and gas-solid equilibrium can be
crudely represented as:

$P_{\text{eq}}\sim P_{\infty}\cdot\exp{\bigg{(}-\frac{L}{k_{B}\cdot
T}\bigg{)}}$ (229)

This equilibrium can be thought of an _escape process_ , similar to the escape
process occurring in the planetary atmosphere. $L$ is the energy that each
molecule has to overcome in order to go from the condensed phase (solid or
liquid) into the dilute phase (gas) which is more or less close to ideal gas
conditions. As a crude approximation, we treat $L$ as the same for gas-liquid
and gas-solid equilibrium.

If the system is close to thermal equilibrium, then the probability
distribution of molecules (in both the condensed and the gas phases) satisfy
the Boltzmann distribution. The probability of finding the molecules within
the energy range of $E\sim E+\mathrm{d}E$ would be:

$f(E)\cdot\mathrm{d}E\propto\exp{\bigg{(}-\frac{E}{k_{B}\cdot
T}\bigg{)}}\cdot\mathrm{d}E$ (230)

The probability of finding the molecules above a certain energy threshold $L$
would be:

$\int_{L}^{\infty}f(E)\mathrm{d}E\propto\exp{\bigg{(}-\frac{L}{k_{B}\cdot
T}\bigg{)}}$ (231)

In fact, we realize this is exactly the scaling factor multiplying to pressure
in Eq. 229.

On one hand, with regard to _phase equilibrium_ , $L$ is interpreted as the
(nominal) latent heat of phase transition. $P_{\infty}$ is understood as an
_extrapolated_ pressure as $T\rightarrow\infty$. For simplicity, $L/k_{B}$
defines a new Temperature $T_{0}\equiv L/R$. Then, Eq. 229 can be re-written
as:

$P_{\text{eq}}\sim P_{\infty}\cdot\exp{\bigg{(}-\frac{T_{0}}{T}\bigg{)}}$
(232)

For example, for H2O,

$\displaystyle P_{\text{eq}}$ $\displaystyle\sim 1\text{~{}TPa}$
$\displaystyle T_{0}$ $\displaystyle\sim 6000\text{~{}K}$

Interestingly enough, the critical point often occurs around $T_{0}/T\approx
10$ and the triple point often occurs around $T_{0}/T\approx 20$. Thus, for
H2O,

$\displaystyle T_{c}$ $\displaystyle\sim 600\text{~{}K}$ $\displaystyle
T_{t1}$ $\displaystyle\sim 300\text{~{}K}$

From the magnitude of $P_{\text{eq}}\sim 1$ TPa, it indicates that it arises
from electron degeneracy pressure, but as a manifestation of cohesion here.
Thus, the factors of $10$ and $20$ within the exponential seem magical!

$\displaystyle P_{\infty}/P_{\text{critical}}\sim\exp{(10)}$
$\displaystyle\approx 2.2\times 10^{4}$ $\displaystyle
P_{\infty}/P_{\text{triple}}\sim\exp{(20)}$ $\displaystyle\approx 0.5\times
10^{9}$

On the other hand, with regard to _atmospheric escape_ , $L$ can be understood
as the depth of gravitational potential well at planet surface:

$L\equiv\frac{G\cdot M_{p}\cdot\mu}{R_{p}}$ (233)

where $\mu$ is the mass of each molecule or particle participating in the
escaping process. We know that escape velocity $v_{\text{esc}}$ is defined as:

$v_{\text{esc}}\equiv\sqrt{\bigg{(}\frac{2\cdot G\cdot M_{p}}{R_{p}}\bigg{)}}$
(234)

Therefore, $L$ can be re-written as:

$L=1/2\cdot\mu\cdot v_{\text{esc}}^{2}$ (235)

and, $L/k_{B}T$ is again dimensionless.

##  6 Shock-wave Experiments on Ammonia and Methane:
What to be expected

We have put efforts into understanding the Temperature-Density Phase Diagram
of ammonia and methane, in order to guide the shock wave experiments of these
two species to be carried out on Z-machine at Sandia National Laboratories.
These two chemical species are of particular interest in the current age of
exoplanetary science, as they together with water may comprise a significant
fraction of planetary interiors among exoplanets [88, 87]. Therefore, their
physical and chemical properties in the high-pressure and high-temperature are
interesting topic to pursue.

There are existing ab initio calculations of the EOS of ammonia [59] [24], an
analytic EOS of Ammonia up to $\sim$3000 K and $\sim$20 GPa [50], and existing
experimental data [70], to be compared with, and guide the up-coming
experiments. See also Fig. 35.

Figure 35: Schematic phase diagram for water and ammonia, or their mixture.
Also shown are trajectories of isotherm, isentropes, and single/multiple-shock
Hugoniot. A single-shock Hugoniot will likely miss the super-ionic regime. The
detailed differences among the various ice structures/phases have been ignored
as they are all approaching close-packing at sufficient pressures.

The Dissociation, Ionization, Super-Ionization, and Metallization of ammonia
are key features to be expected in its phase diagrams. In particular, the
_super-ionic_ phase has already been experimentally verified for water [64]
[63]. The _super-ionic_ phase of ammonia or water-ammonia mixture is expected
to exist [28] [23] [52].

In order to hit this _super-ionic_ phase of ammonia in terms of the shock-wave
experiments, we need to either:

  * •

use reverberating shocks technique starting with the liquid density at ambient
conditions to approximate an isentrope, or,

  * •

use a single shock starting with a pre-compressed high-pressure ice phase,
which is already 60% or denser than the liquid

For methane, a simple analytic EOS was put forth by G. Kerley that goes up to
$\sim$104 K and $\sim$2.5 g/cc [55], which agrees with the experimental data
from W. Nellis. A detailed experimentally-verified methane EOS table for
relatively low-temperature (up to 625 K) and low-pressure (up to 1 GPa) also
exists [72].

Recent ab initio simulations suggest that methane is expected to dissociate
first into Hydro-carbons (above $\sim$ 100 GPa), and further into Carbon
(Diamond) and Hydrogen (above $\sim$ 300 GPa) [73] [42] [19] [14].

For future purpose, in terms of their mixtures, ab initio calculations show
that a _linear mixing_ model for water-ammonia system is good enough
approximation for a wide range of pressure-temperature phase-space [22],
because these two species are very soluble in each other, and behave
chemically similar even under high-pressure conditions. However, ammonia ice
and water ice may eventually separate above $\sim$ 500 GPa [71].

In terms of the water-ammonia-methane triple mixture, the inclusion of methane
(or carbon) into the water-ammonia mixture is more tricky at high-pressure
high-temperature conditions. Methane/Carbon may separate out from the mixture,
to either rain down as diamonds, or may float upwards as gas as it is somewhat
lighter and more volatile. Some ab initio simulations are also available [62].

##  7 Light Gaseous Envelopes

###  7.1 Envelope-Core Radius Relation:
Introducing _fugacity_ $\mathbf{f}$

This section deals with planets covered with a not-so-massive gaseous
envelope, and its effect on mass-radius relation. For a light (not-so-massive)
envelope, typically less than about 20 percent ($\lesssim 20\%$) the planet
core mass $M_{c}$ (defined as whatever underneath the envelope), the
calculation is simplified with the following assumptions.

  1. 1.

First, the gravitational field within the envelope depends only on the radial
distance $r$ to the planet center, rather than any other conditions (such as
mass distribution or scale height variation) within the envelope itself,
because the planet core mass dominates.

  2. 2.

Second, the energy transport process within the envelope is assumed to be
understood and well-defined. There are three modes of energy transport
process: radiative, convective, and conductive. They together define a unique
path or trajectory that the thermal dynamical parameters
$T$-$\rho$-$P$-entropy… corresponds to one another and to the envelope metric.
It is exactly along this path that any differentiation is regarded in the
following differential equations. It is implicitly path-independent.

Under these two simplifications, the equation governing hydrostatic
equilibrium is written as:

$\frac{dP}{dr}=g\cdot\rho=-\frac{G\cdot M_{c}}{r^{2}}\cdot\rho$ (236)

By re-arrangement of terms, we have:

$\frac{dP}{\rho}=-\frac{G\cdot M_{c}}{r^{2}}\cdot dr=G\cdot M_{c}\cdot
d\Bigg{(}\frac{1}{r}\Bigg{)}$ (237)

The RHS of Eq. 237 can be integrated straightforwardly from the envelope top
to the envelope bottom. Let’s define this integral as parameter $x$:

$\displaystyle x$ $\displaystyle\equiv\int_{\text{top}}^{\text{bottom}}G\cdot
M_{c}\cdot d\Bigg{(}\frac{1}{r}\Bigg{)}=G\cdot
M_{c}\cdot\Bigg{(}\frac{1}{r_{\text{bottom}}}-\frac{1}{r_{\text{top}}}\Bigg{)}$
(238) $\displaystyle=G\cdot
M_{c}\cdot\Bigg{(}\frac{1}{R_{c}}-\frac{1}{R_{p}}\Bigg{)}=\Bigg{(}\frac{G\cdot
M_{c}}{R_{c}}\Bigg{)}\cdot\Bigg{(}1-\frac{R_{c}}{R_{p}}\Bigg{)}$

Here $R_{c}$ is the radius measured at the core surface or the envelope
bottom, and $R_{p}$ is the radius defined at the envelope top, also known as
the total planet radius. We can re-express the relationship between $R_{p}$
and $R_{c}$ in terms of $x$ as:

$R_{p}=R_{c}\cdot\frac{\displaystyle 1}{\displaystyle 1-\frac{x}{(G\cdot
M_{c}/R_{c})}}$ (239)

The cluster $GM_{c}/R_{c}$ has its physical meaning. It is the square of
escape velocity at the core surface. It measures how deep inside a
gravitational potential well does the core surface resides. It has the unit of
specific energy (energy per unit mass).

This Envelope-Core Radius-Relation (Eq. 239) helps understand how the mass of
a planet core $M_{c}$ may affect the extent of its envelope. Let’s perform the
following thought experiment: If $R_{c}$ stays constant, then,

  * •

when $M_{c}\searrow$, $\frac{x}{(GM_{c}/R_{c})}\nearrow$, $R_{p}\nearrow$

  * •

when $M_{c}\nearrow$, $\frac{x}{(GM_{c}/R_{c})}\searrow$, $R_{p}\searrow$

Another corollary of Eq. 239 is that if $x$ becomes too large, then the
denominator turns negative and situation becomes nonphysical. In reality, this
suggests that the envelope becomes unbound.

For the LHS of Eq. 237, we need to consider the integral
$\int\frac{dP}{\rho}$. Without any a priori knowledge of the thermal structure
of the envelope, it is assumed to be at a constant temperature (isothermal)
determined only by the host stellar radiation, i.e., the equilibrium
temperature $T_{\text{eq}}$. This is similar to the situation on Earth surface
where the heat flux from interior is orders-of-magnitude smaller than solar
radiation. The mean temperature of fluids on Earth surface, i.e., the
hydrosphere and atmosphere, is mostly determined by the average solar
radiation.

Then the integral $\int\frac{dP}{\rho}$ is a corollary function of EOS
$P=P[\rho,T_{\text{eq}}]$, specifically for an isothermal process at constant
temperature $T_{\text{eq}}$, with the proper upper and lower limits applied.
This integral also has the unit of specific energy, that is, energy per unit
mass.

Alternatively, the same integral $\int\frac{dP}{\rho}$ can be derived by
equating the _chemical potential_ $\mathbf{\mu}$ at any given height within
this light isothermal envelope. In this manner, this integral is related to
_fugacity_ $\mathbf{f}$, introduced by G. Lewis (1901).

$x=\Bigg{(}\frac{R\cdot
T_{\text{eq}}}{\mu}\Bigg{)}\cdot\ln{\Bigg{(}\frac{\mathbf{f_{\text{bottom}}}}{\mathbf{f_{\text{top}}}}\Bigg{)}}=\int_{\text{top}}^{\text{bottom}}\frac{dP}{\rho}$
(240)

here the molar mass $\mu$ is needed to convert energy per mole to energy per
unit mass.

The _fugacity_ $\mathbf{f}$ is especially useful when considering an envelope
consisting of multiple species, and chemical reactions occurring among the
constituent species. When writing out an equilibrium equation for chemical
reactions, _fugacity_ $\mathbf{f}$ is used often instead of partial pressure.
$\mathbf{f}$ can be viewed as the pressure a given real system must produce to
have the same action as an ideal system, given the proper asymptotic limit
(constant of integration): $\lim_{P\to 0}\frac{\mathbf{f}}{P}=1$.

$\mathbf{f}$ is the corrected (effective) pressure which takes into account
the attractive and repulsive intermolecular interactions.

It should be emphasized that any $\mathbf{f}$ is only specific to a given
temperature $T_{\text{eq}}$. For a different $T_{\text{eq}}$,
$\int\frac{dP}{\rho}$ needs to be re-calculated.

As regards to the limits of this integral:

  * •

The lower limit relates to what pressure-level or density-level a transit
observation defines the rim of a planet. It is typically on the millibars (100
Pa) level or a few tens of millibars level, depending on the chemical and
thermal conditions in the envelope (please see the detailed calculation of the
slant-path optical depth which takes into account planet surface curvature in
[34] and its supplement).

  * •

The upper limit relates to the total mass or mass fraction of the envelope.
This relation is elucidated by the next sub-section.

Once $\int\frac{dP}{\rho}$, that is, $x$ is computed from an EOS, then we can
use it to compute $R_{p}$ according to Eq. 239. To aid our calculation, we
define a dimensionless parameter $y$ as:

$y\equiv\Bigg{(}\int_{\text{top}}^{\text{bottom}}\frac{dP}{\rho}\Bigg{)}\Bigg{/}\Bigg{(}\frac{G\cdot
M_{c}}{R_{c}}\Bigg{)}=x\Bigg{/}\Bigg{(}\frac{G\cdot M_{c}}{R_{c}}\Bigg{)}$
(241)

Then the Envelope-Core Radius-Relation (Eq. 239) becomes:

$R_{p}=R_{c}\cdot\frac{\displaystyle 1}{\displaystyle 1-y}$ (242)

In many cases, it is more convenient to express planet mass and radius in
Earth units. Therefore,to aid our calculation, we introduce yet another
dimensionless parameter $z$:

$z\equiv\Bigg{(}\int_{\text{top}}^{\text{bottom}}\frac{dP}{\rho}\Bigg{)}\Bigg{/}\Bigg{(}\frac{G\cdot
M_{\oplus}}{R_{\oplus}}\Bigg{)}=x\Bigg{/}\Bigg{(}\frac{G\cdot
M_{\oplus}}{R_{\oplus}}\Bigg{)}$ (243)

The denominator equals a constant in unit of specific energy (erg/g):

$\Bigg{(}\frac{G\cdot M_{\oplus}}{R_{\oplus}}\Bigg{)}=\frac{6.674\cdot
10^{-8}\text{cc/g/s}^{2}\cdot 5.972\cdot 10^{27}\text{g}}{6.371\cdot
10^{8}\text{cm}}=6.256\cdot 10^{11}\text{erg/g}$ (244)

Then the Envelope-Core Radius-Relation becomes:

$R_{p}=R_{c}\cdot\frac{\displaystyle 1}{\displaystyle
1-\Bigg{(}z\bigg{/}\bigg{(}\frac{M_{c}/M_{\oplus}}{R_{c}/R_{\oplus}}\bigg{)}\Bigg{)}}$
(245)

Or, more simply if $M_{c}$ and $R_{c}$ are already expressed in Earth units:

$R_{p}=R_{c}\cdot\frac{\displaystyle 1}{\displaystyle
1-\Bigg{(}z\bigg{/}\bigg{(}\frac{M_{c}}{R_{c}}\bigg{)}\Bigg{)}}$ (246)

This relation (Eq. 246) is useful when comparing to _contours_ of
$M_{p}/R_{p}$ (planet mass divided by planet radius, in Earth units) in the
mass-radius plot.

###  7.2 Envelope-Core Pressure Relation

In order to understand the relation between the mass of the envelope
$M_{\text{env}}=(M_{p}-M_{c})$ and the pressure $P_{\text{env}}$ it exerts
onto the planet core, We now express the hydrostatic equilibrium Eq. 236 in
variable mass $m$ instead of radius $r$:

$\frac{dP}{dm}=-\frac{G\cdot m}{4\pi\cdot r^{4}}=-\frac{1}{4\pi\cdot G\cdot
m}\cdot(g[m])^{2}$ (247)

Notice that here gravitational acceleration $g$ is a function dependent on $m$
or $r$. Its square occurs on the RHS of the above Eq. 247. Through
manipulation, we have:

$dP=-\frac{g^{2}}{4\pi G}\cdot\frac{dm}{m}$ (248)

Eq. 248 tells that Pressure $P$ and Gravity $g$ are essentially two sides of
the same coin. Pressure arises from gravity from matter itself to compress
itself, and thus, the square in the equation.

It is very tempting to integrate Eq. 248:

$P_{\text{env}}\approx\frac{\mkern
1.5mu\overline{\mkern-1.5mug^{2}\mkern-1.5mu}\mkern 1.5mu}{4\pi
G}\cdot\frac{M_{\text{env}}}{M_{p}}$ (249)

where $\mkern 1.5mu\overline{\mkern-1.5mug^{2}\mkern-1.5mu}\mkern 1.5mu$ is
some kind of mass-averaged value within the envelope. It would be nice if we
can somehow evaluate $g$ or $g^{2}$ within the envelope. In fact, this is
feasible, as long as:

  1. 1.

the envelope is light (not-so-massive) ($\lesssim 20\%~{}M_{c}$), compared to
the planet core mass $M_{c}$ or the total planet mass
$M_{p}=(M_{c}+M_{\text{env}})$.

  2. 2.

the envelope mass is concentrated towards its bottom, so that gravity $g$ can
be approximated by its value evaluated at the bottom of the envelope, or
equivalently, on top of the planet core: $g_{c}$.

If both (1) and (2) are satisfied, then the integral from the envelope top to
the envelope bottom gives an approximate pressure $P_{\text{env}}$ that the
envelope exerts onto the planet core due to gravitation:

$\displaystyle P_{\text{env}}\equiv P_{\text{bottom}}$
$\displaystyle\approx\bigg{(}P_{\text{bottom}}-P_{\text{top}}\bigg{)}\approx\frac{\mathinner{\\!\left\lvert
g_{c}\right\rvert}^{2}}{4\pi
G}\cdot\ln{\Bigg{(}\frac{M_{c}+M_{\text{env}}}{M_{c}}\Bigg{)}}$ (250)
$\displaystyle=\frac{\mathinner{\\!\left\lvert g_{c}\right\rvert}^{2}}{4\pi
G}\cdot\ln{\Bigg{(}1+\frac{M_{\text{env}}}{M_{c}}\Bigg{)}}\approx\frac{\mathinner{\\!\left\lvert
g_{c}\right\rvert}^{2}}{4\pi
G}\cdot\Bigg{(}\frac{M_{\text{env}}}{M_{c}}\Bigg{)}$

and $M_{c}+M_{\text{env}}$ is no different than the total mass of the planet
$M_{p}$. $g_{c}$ is of course related to the core mass $M_{c}$ and core radius
$R_{c}$ as:

$g_{c}=\frac{G\cdot M_{c}}{(R_{c})^{2}}$ (251)

Therefore,

$P_{\text{env}}\approx\Bigg{(}\frac{G\cdot M_{c}^{2}}{4\pi\cdot
R_{c}^{4}}\Bigg{)}\cdot\Bigg{(}\frac{M_{\text{env}}}{M_{c}}\Bigg{)}\equiv
P_{c}\cdot\Bigg{(}\frac{M_{\text{env}}}{M_{c}}\Bigg{)}$ (252)

Here we introduce $P_{c}$ as a _characteristic_ pressure in the deep interior
of the planet core. For example, for Earth, $P_{c}=\frac{G\cdot
M_{\oplus}}{4\pi R_{\oplus}^{2}}\approx$ 100 GPa or 1 megabar, which happens
to be the pressure at the core-mantle boundary inside Earth, and about one-
third of the pressure at the Earth’s center. This is a _fundamental_ relation
that relates the envelope pressure at its bottom to the characteristic
pressure deep within the core, regardless of the detailed chemistry and
physics of the envelope or the core themselves:

$\Bigg{(}\frac{P_{\text{env}}}{P_{c}}\Bigg{)}\approx\Bigg{(}\frac{M_{\text{env}}}{M_{c}}\Bigg{)}$
(253)

###  7.3 Real Gas EOS:
Introducing _Compression Factor_ $\mathbf{Z}$

The complexity of the fluid H2O EOS benefits from the multitude of
experimental data available and its application in industry. The accuracy
achieved there is typically less than 0.1% level. For other gases, let’s start
from scratch and _solve a simpler problem first_. As for our purpose in
planetary science, a 10% accuracy is tolerable, so let’s construct a EOS from
a simpler standpoint.

To aid our discussion, let’s first define a dimensionless parameter:
_Compression Factor_ $\mathbf{Z}$ as:

$\mathbf{Z}\equiv\frac{p\cdot V_{m}}{R\cdot T}=\frac{p}{(\rho/\mu)\cdot R\cdot
T}$ (254)

Where $V_{m}$ is the molar volume, and $\mu$ is the molar weight of molecules.
For Ideal Gas, $\mathbf{Z}=1$. Deviations of the _Compression Factor_
$\mathbf{Z}$ from unity are due to attractive and repulsive intermolecular
interactions. By its definition, it is linked to the _Fugacity_ $\mathbf{f}$
as:

$\mathbf{f}=\underbrace{P\cdot\exp{\Bigg{(}\int\frac{\mathbf{Z}-1}{P}\cdot
dP}\Bigg{)}}_{\text{integral performed along an isotherm}}$ (255)

When $\mathbf{Z}=1$, Eq. 255 immediately gives:

$\mathbf{f}=P$ (256)

####  7.3.1 Virial EOS

$\mathbf{Z}$ can also be derived from the first principle through statistical
mechanics, known as the Virial Equation [53]:

$\mathbf{Z}=1+B\cdot\Bigg{(}\frac{\rho}{\mu}\Bigg{)}+C\cdot\Bigg{(}\frac{\rho}{\mu}\Bigg{)}^{2}+D\cdot\Bigg{(}\frac{\rho}{\mu}\Bigg{)}^{3}+...$
(257)

where $B$, $C$, and $D$, … are the second, third, fourth, etc. _virial
coefficients_. These coefficients depend only on temperature. They account for
interactions between successively larger groups of molecules. They reflect
double ($B$), triple ($C$), quadruple ($D$), etc. interactions of molecules.
The detailed calculations are based upon equations establishing the
relationships between the energy of interaction of molecules, and distance
between them, and their arrangement.

For example, for spherical molecules, coefficient $B$ can be calculated as:

$B=\frac{1}{2}\cdot
N_{A}\cdot\int_{0}^{\infty}\Bigg{(}1-\exp{\bigg{(}-\frac{E_{p}(r)}{k_{B}\cdot
T}\bigg{)}}\Bigg{)}\cdot(4\pi r^{2})\cdot dr$ (258)

where $E_{p}(r)$ is the intermolecular-force potential energy expressed as a
function of distance $r$. $N_{A}$ is the Avogadro number and $k_{B}$ is the
Boltzmann constant. $B$ has the unit of volume per mole. One example of
$E_{p}(r)$-potential to plug in is the _Lenard-Jones_ 6-12 potential.

For comparatively low pressures (in terms of the reduced pressure
$\pi\equiv(P/{P_{c}})\lesssim 0.5$), we can limit ourselves to the second
virial coefficient $B$, which takes into account only double collisions:

$\mathbf{Z}=1+B\cdot\Bigg{(}\frac{\rho}{\mu}\Bigg{)}$ (259)

If we include not only $B$, but also $C$, then the _cubic_ virial EOS gives
satisfactory results up to reduced density
$\delta\equiv(\rho/{\rho_{c}})\lesssim 2$ and _inverse_ reduced temperature
$\tau\equiv(T_{c}/T)\lesssim 2$:

$\mathbf{Z}=1+B\cdot\Bigg{(}\frac{\rho}{\mu}\Bigg{)}+C\cdot\Bigg{(}\frac{\rho}{\mu}\Bigg{)}^{2}$
(260)

The _cubic_ virial EOS (Eq. 260) has all the nice characteristics of _Van der
Waals_ EOS, but without its fatal singularity. The _cubic_ virial EOS
accurately represents the gas-liquid equilibrium of most substance from the
critical point down to the triple point, where solid phase starts to appear.

####  7.3.2 Berthelot EOS:
For comparatively low pressures ($\lesssim$100 bar)
or molar density up to $\sim$0.01 mol/cc

D. Berthelot (1903) proposed the following EOS [53]:

$\mathbf{Z}=1+\frac{9\cdot\pi\cdot\tau}{128}\cdot\Bigg{(}1-6\cdot\tau^{2}\Bigg{)}$
(261)

Here $\pi$ is the reduced pressure $\pi\equiv\frac{P}{P_{c}}$, and $\tau$ is
the _inverse_ reduced temperature $\tau\equiv\frac{T_{c}}{T}$.

###  7.4 Procedure to Calculate Mass-Radius Relation

When we calculate the mass-radius curves for a fixed envelope-to-core mass
ratio: $\bigg{(}\frac{M_{\text{env}}}{M_{c}}\bigg{)}$, we are essentially
fixing $\bigg{(}\frac{P_{\text{env}}}{P_{c}}\bigg{)}$. However, recall the
definition of $P_{c}$ that it depends on both $M_{c}$ and $R_{c}$, and thus,
it varies along the mass-radius relation ($M_{c}$-$R_{c}$) of a given core
composition.

Therefore, we need to follow this procedure to calculate the mass-radius
relation for a planet core with an envelope:

  * •

First, we need to identify the mass-radius relation ($M_{c}$-$R_{c}$) of the
core itself.

  * •

Second, we need to identify the mass $M_{\text{env}}$ or mass fraction
$M_{\text{env}}/M_{c}$ of the envelope. Since it is a light envelope, we
assume that it does not exert too much pressure onto the core, and change the
core’s mass-radius relation ($M_{c}$-$R_{c}$).

  * •

Third, we compute the pressure at the bottom of the light envelope
$P_{\text{bottom}}$:

$P_{\text{bottom}}=\Bigg{(}\frac{G\cdot M_{c}^{2}}{4\pi\cdot
R_{c}^{4}}\Bigg{)}\cdot\Bigg{(}\frac{M_{\text{env}}}{M_{c}}\Bigg{)}$ (262)

  * •

Fourth, carry out the integral $\int\frac{dP}{\rho}$ from top to bottom of the
envelope with the appropriate EOS, to find the dimensionless parameter $z$:

$z\equiv\Bigg{(}\int_{P_{\text{top}}}^{P_{\text{bottom}}}\frac{dP}{\rho}\Bigg{)}\Bigg{/}\Bigg{(}\frac{G\cdot
M_{\oplus}}{R_{\oplus}}\Bigg{)}=x\Bigg{/}\Bigg{(}\frac{G\cdot
M_{\oplus}}{R_{\oplus}}\Bigg{)}$ (263)

  * •

Fifth, we apply the Envelope-Core Radius-Relation, to calculate $R_{p}$:

$R_{p}=R_{c}\cdot\frac{\displaystyle 1}{\displaystyle
1-\Bigg{(}z\bigg{/}\bigg{(}\frac{M_{c}}{R_{c}}\bigg{)}\Bigg{)}}$ (264)

Here we show an example of adding a 2-percent-by-mass pure-H2-envelope onto a
half-ice-half-Earth-like-rocky core and its effect on the mass-radius curves.
We compare this set of isothermal-envelope mass-radius curves to the Kepler-36
system and K2-36 system (Fig. 36).

Figure 36: The K-36 two-planet system and K2-36 two-planet system happen to
resemble each other by mere coincidence of their designation. The two
planetary systems, each has one rocky-super-Earth, and another planet with
sufficient amount of volatiles. Their formation pathways may be similar [33]
[27]. The planet occurrence rate with Gaia-correction is taken from [20] [21].

A minimum point exists as a common feature for these mass-radius curves of
fixed envelope-to-core mass ratio. It is due to the trade-off between the
extensiveness of the envelope and the size of the core. Since in the mass-
radius plot shown here (Fig. 36), mass coordinate is in logarithmic scale, it
suggests that the overall planet radius $R_{p}$ stays roughly constant over a
significant mass-range.

Using this minimum point, we could separate the envelope mass-radius curve
into a _left branch_ and a _right branch_.

Within the _left branch_ , the planet radius $R_{p}$ increases with decreasing
core mass $M_{c}$ or planet mass $M_{p}$. It quickly crosses over _contours_
of constant surface escape velocity or constant surface gravitational
potential ($M_{p}/R_{p}$=const), contours of constant surface gravity
($M_{p}/R_{p}^{2}$=const), and contours of constant mean planet density
($M_{p}/R_{p}^{3}$=const), in the decreasing way. So it will quickly become
unbound to the planet core and escape.

Within the _right branch_ , the planet radius $R_{p}$ increases with
increasing core mass $M_{c}$ or planet mass $M_{p}$. In the meantime, the
envelope thickness ($R_{p}-R_{c}$) decreases and then tends towards constant
with increasing core mass $M_{c}$ or planet mass $M_{p}$. The planet core
holds the envelope tighter at higher mass. Thus, the envelope becomes more
stably bound to the planet core, and less susceptible to escape.

###  7.5 Example: Separation of Variables

Here we give an example to calculate the parameters $x$, $y$, and $z$ for a
specific EOS–a Modified Ideal Gas EOS, and explain the implications of the
results.

For a generic fluid, including both gas and liquid, its pressure-EOS can often
be treated with the technique of separation of variables:

$P=\underbrace{P_{\text{thermal}}}_{\text{Part
I}}+\underbrace{P_{\text{residual}}}_{\text{Part II}}$ (265)

$P=\underbrace{P_{\text{thermal}}(T,\rho)}_{\text{Part
I}}+\underbrace{P_{\text{residual}}(\rho)}_{\text{Part II}}$ (266)

Part I is, for an example, the Ideal Gas EOS. Part II results from interaction
among molecules through effective pair potential. High-pressure shock wave
experiments show that this effective pair potential is quite universal for
fluid diatomic molecules such as N2, CO, O2 and H2. It is independent of
initial fluid densities at ambient pressure and potential parameters that
scale with the critical points ($\rho_{c}$, $T_{c}$, $P_{c}$) of various
fluids [68].

Thus, it motivates us to re-express pressure $P$ in terms of molecular molar
density $n$ (in mol/cc) instead of mass density $\rho$ (in g/cc), in order to
make our derivations more generic. This is especially useful when treating a
fluid which is a _mixture_ of multiple components rather than a single
component. In order to convert $\rho$ into $n$ and vice versa, we need to
introduce the mean molecular weight $\mkern
1.5mu\overline{\mkern-1.5mu\mu\mkern-1.5mu}\mkern 1.5mu$ as:

$n=\frac{\rho}{\mkern 1.5mu\overline{\mkern-1.5mu\mu\mkern-1.5mu}\mkern
1.5mu}$ (267)

Suppose a generic fluid consists of Hydrogen Molecules H2, Helium Atoms He,
and other heavier species collectively called Z. Assume it is a Linear Mixture
which obeys volume additive law, then the molar density $n$ follows the
additive law as follows:

$n_{\text{total}}=n_{\text{H${}_{2}$}}+n_{\text{He}}+n_{\text{Z}}$ (268)

Eq. 268 is of course violated if chemical reactions occur within or among
species which result in association or dissociation (such as H2 dissociates
into two H-atoms) which change the total number of molecules. However, for now
we assume this is not the case. This is approximately true for H2 until
pressures of the order of $\sim$100 GPa, for high pressure will prohibit the
dissociation of H2 when temperature $T$ is not too high. Furthermore, we
assume each molecule behaves identically and uniformly in this mixture, and
thus, their interaction can be described by a single effective pair potential.

$\mu_{\text{H${}_{2}$}}$=2 g/mol, and $\mu_{\text{He}}$= 4 g/mol.
$\mu_{\text{Z}}$ is the mean molecular weight of the heavier species
themselves, which is typically on the order of 20 or 30.

$\displaystyle\frac{\rho_{\text{H${}_{2}$}}}{\mu_{\text{H${}_{2}$}}}$
$\displaystyle=n_{\text{H${}_{2}$}}\leftarrow\text{molar density of H${}_{2}$
(moles per unit volume)}$
$\displaystyle\frac{\rho_{\text{He}}}{\mu_{\text{He}}}$
$\displaystyle=n_{\text{He}}\leftarrow\text{molar density of He (moles per
unit volume)}$ $\displaystyle\frac{\rho_{\text{Z}}}{\mu_{\text{Z}}}$
$\displaystyle=n_{\text{Z}}\leftarrow\text{molar density of heavier species
(moles per unit volume)}$

However, in astronomy, people often like to component of a gas mixture in mass
fraction instead of number fraction. The relationship can be expressed as
follows:

$\displaystyle X$ $\displaystyle\text{:(mass fraction of
H${}_{2}$)}\equiv\frac{\rho_{\text{H${}_{2}$}}}{\rho_{\text{H${}_{2}$}}+\rho_{\text{He}}+\rho_{\text{Z}}}=\frac{\mu_{\text{H${}_{2}$}}\cdot
n_{\text{H${}_{2}$}}}{\mu_{\text{H${}_{2}$}}\cdot
n_{\text{H${}_{2}$}}+\mu_{\text{He}}\cdot n_{\text{He}}+\mu_{\text{Z}}\cdot
n_{\text{Z}}}$ $\displaystyle Y$ $\displaystyle\text{:(mass fraction of
He)}\equiv\frac{\rho_{\text{He}}}{\rho_{\text{He}}+\rho_{\text{He}}+\rho_{\text{Z}}}=\frac{\mu_{\text{H${}_{2}$}}\cdot
n_{\text{H${}_{2}$}}}{\mu_{\text{H${}_{2}$}}\cdot
n_{\text{H${}_{2}$}}+\mu_{\text{He}}\cdot n_{\text{He}}+\mu_{\text{Z}}\cdot
n_{\text{Z}}}$ $\displaystyle Z$ $\displaystyle\text{:(mass fraction of
Z)}\equiv\frac{\rho_{\text{Z}}}{\rho_{\text{Z}}+\rho_{\text{He}}+\rho_{\text{Z}}}=\frac{\mu_{\text{H${}_{2}$}}\cdot
n_{\text{H${}_{2}$}}}{\mu_{\text{H${}_{2}$}}\cdot
n_{\text{H${}_{2}$}}+\mu_{\text{He}}\cdot n_{\text{He}}+\mu_{\text{Z}}\cdot
n_{\text{Z}}}$

From Eq. 268, the relation between $\mkern
1.5mu\overline{\mkern-1.5mu\mu\mkern-1.5mu}\mkern 1.5mu$ and
$\mu_{\text{H${}_{2}$}}$, $\mu_{\text{He}}$, $\mu_{\text{Z}}$ can be written
as:

$\frac{1}{\mkern 1.5mu\overline{\mkern-1.5mu\mu\mkern-1.5mu}\mkern
1.5mu}=\frac{X}{\mu_{\text{H${}_{2}$}}}+\frac{Y}{\mu_{\text{He}}}+\frac{Z}{\mu_{\text{Z}}}$
(271)

Now, let’s try the following generic form of pressure-EOS as motivated by real
experimental data:

$P=\underbrace{R\cdot T\cdot n}_{\text{Part
I}}+\underbrace{\lambda\cdot\Bigg{(}\frac{n}{\text{1
mol/cc}}\Bigg{)}^{\Gamma}}_{\text{Part II}}$ (272)

$\lambda$ is a constant in unit of pressure. It is about $10^{3}$ GPa or 1
TPa, or $10^{13}$dyn/cm2 in cgs units, for H2 and other molecules alike
approximately. According to our demonstration earlier, we treat $\Gamma$ as
constant, which is $\sim$2 for metallic fluid H2, and H2-He Cosmic Mixture
($X=\frac{3}{4}$, $Y=\frac{1}{4}$), and other molecules alike approximately
[79]. At low number density $n$, the temperature-dependent Part I dominates.
At high number density $n$, the temperature-independent Part II dominates.
Then, the integral $\int\frac{dP}{\rho}$ for an isothermal envelope at
constant temperature $T_{\text{eq}}$ and constant mean molecular weight
$\mkern 1.5mu\overline{\mkern-1.5mu\mu\mkern-1.5mu}\mkern 1.5mu$ can be
performed:

$x=\int_{\text{top}}^{\text{bottom}}\frac{dP}{\rho}=\frac{1}{\mkern
1.5mu\overline{\mkern-1.5mu\mu\mkern-1.5mu}\mkern
1.5mu}\cdot\int_{\text{top}}^{\text{bottom}}\frac{dP}{n}$ (273)

Then, in cgs units,

$dP=\underbrace{R\cdot T\cdot dn}_{\text{Part
I}}+\underbrace{\lambda\cdot\Gamma\cdot n^{\Gamma-1}\cdot dn}_{\text{Part
II}}$ (274)

Then,

$\frac{dP}{n}=\underbrace{R\cdot T\cdot\frac{dn}{n}}_{\text{Part
I}}+\underbrace{\lambda\cdot\Gamma\cdot n^{\Gamma-2}\cdot dn}_{\text{Part
II}}$ (275)

Then,

$\int_{\text{top}}^{\text{bottom}}\frac{dP}{n}=\underbrace{R\cdot
T\cdot\int_{\text{top}}^{\text{bottom}}\frac{dn}{n}}_{\text{Part
I}}+\underbrace{\lambda\cdot\Gamma\cdot\int_{\text{top}}^{\text{bottom}}n^{\Gamma-2}\cdot
dn}_{\text{Part II}}$ (276)

Then,

$\int_{\text{top}}^{\text{bottom}}\frac{dP}{n}=\underbrace{R\cdot
T\cdot\ln{\Bigg{(}\frac{n_{\text{bottom}}}{n_{\text{top}}}\Bigg{)}}}_{\text{Part
I}}+\underbrace{\lambda\cdot\frac{\Gamma}{\Gamma-1}\cdot\Bigg{(}n_{\text{bottom}}^{\Gamma-1}-n_{\text{top}}^{\Gamma-1}\Bigg{)}}_{\text{Part
II}}$ (277)

Then, Parameter $x$, which is a represent of envelope thickness, is:

$x=\underbrace{\frac{R\cdot T_{\text{eq}}}{\mkern
1.5mu\overline{\mkern-1.5mu\mu\mkern-1.5mu}\mkern
1.5mu}\cdot\ln{\Bigg{(}\frac{n_{\text{bottom}}}{n_{\text{top}}}\Bigg{)}}}_{\text{thermal
contribution to envelope thickness}}+\underbrace{\frac{\lambda}{\mkern
1.5mu\overline{\mkern-1.5mu\mu\mkern-1.5mu}\mkern
1.5mu}\cdot\frac{\Gamma}{\Gamma-1}\cdot\Bigg{(}n_{\text{bottom}}^{\Gamma-1}-n_{\text{top}}^{\Gamma-1}\Bigg{)}}_{\text{intrinsic
thickness of envelope}}$ (278)

Or, equivalently, since $n_{\text{bottom}}\gg n_{\text{top}}$,

$x\approx\underbrace{\frac{R\cdot T_{\text{eq}}}{\mkern
1.5mu\overline{\mkern-1.5mu\mu\mkern-1.5mu}\mkern
1.5mu}\cdot\ln{\Bigg{(}\frac{n_{\text{bottom}}}{n_{\text{top}}}\Bigg{)}}}_{\text{thermal
contribution to envelope thickness}}+\underbrace{\frac{\lambda}{\mkern
1.5mu\overline{\mkern-1.5mu\mu\mkern-1.5mu}\mkern
1.5mu}\cdot\frac{\Gamma}{\Gamma-1}\cdot\Bigg{(}n_{\text{bottom}}^{\Gamma-1}\Bigg{)}}_{\text{intrinsic
thickness of envelope}}$ (279)

The interpretation of this result is that Part I gives rise to the _thermal
thickness_ of an envelope, which is characterized by temperature-dependent
scale-height. Part II gives rise to the _intrinsic thickness_ of an envelope,
regardless of the thermal conditions and energy transport process. This result
is very general.

This Separation-of-Variable treatment allows us to understand how a fluidic
envelope expands under increased stellar insolations, applicable to many
short-period exoplanets, such as hot-Jupiters and hot-Saturns.

$x\approx\underbrace{\frac{R\cdot T_{\text{eq}}}{\mkern
1.5mu\overline{\mkern-1.5mu\mu\mkern-1.5mu}\mkern
1.5mu}\cdot\ln{\Bigg{(}\frac{n_{\text{bottom}}}{n_{\text{top}}}\Bigg{)}}}_{\text{Part
I}}+\underbrace{\frac{\lambda}{\mkern
1.5mu\overline{\mkern-1.5mu\mu\mkern-1.5mu}\mkern
1.5mu}\cdot\frac{\Gamma}{\Gamma-1}\cdot\Bigg{(}n_{\text{bottom}}^{\Gamma-1}\Bigg{)}}_{\text{Part
II}}$ (280)

####  7.5.1 Discussion

Now, let’s plug in the numbers and compare the actual results, in cgs units
(erg/g):

$x\approx\frac{1}{\mkern 1.5mu\overline{\mkern-1.5mu\mu\mkern-1.5mu}\mkern
1.5mu}\cdot\Bigg{(}\underbrace{10^{11}\cdot\Bigg{(}\frac{T_{\text{eq}}}{10^{3}\text{K}}\Bigg{)}\cdot\ln{\Bigg{(}\frac{n_{\text{bottom}}}{n_{\text{top}}}\Bigg{)}}}_{\text{Part
I}}+\underbrace{2\cdot
10^{13}\cdot\Bigg{(}\frac{n_{\text{bottom}}}{\text{mol/cc}}\Bigg{)}}_{\text{Part
II}}\Bigg{)}~{}\text{erg/g}$ (281)

$n_{\text{top}}$ corresponds to the number density of the top of the envelope
that the transit observation probes. Let’s treat it as $\sim 10^{-8}$ mol/cc.
The exact value shall depend on the photo-chemistry of the envelope top. This
typically corresponds to pressures of millibar level (please see the detailed
calculation of the slant-path optical depth which takes into account planet
surface curvature in [34] and its supplement).

Note that Part II becomes comparable to Part I in magnitude when
$n_{\text{bottom}}$ approaches $\sim 10^{-2}$ mol/cc or $\sim$0.01 mol/cc.

Recall that D. Mendeleev (1870) pointed out that it is useful to compare the
molar densities among different chemical species, not at their boiling points,
but at their critical points when the cohesion of the molecules transition
from being close to zero to non-zero there [53]. Indeed, in previous sections
we see that:

$\displaystyle n_{\text{c,H${}_{2}$}}$ $\displaystyle\approx
0.015\text{~{}mol/cc}$ $\displaystyle n_{\text{c,H${}_{2}$O}}$
$\displaystyle\approx 0.018\text{~{}mol/cc}$ $\displaystyle
n_{\text{c,NH${}_{3}$}}$ $\displaystyle\approx 0.013\text{~{}mol/cc}$
$\displaystyle n_{\text{c,N${}_{2}$}}$ $\displaystyle\approx
0.0112\text{~{}mol/cc}$ $\displaystyle n_{\text{c,CH${}_{4}$}}$
$\displaystyle\approx 0.010\text{~{}mol/cc}$

Thus, near or above this _critical_ molar density of $\sim 0.01$ mol/cc, the
cohesive interactions among molecules start to take over and turn the
substance into a _condensed_ state. They should be considered liquid or
_super-critical_ fluid from there on, and they should not be considered _gas-
like_ above this critical molar density.

Using Eq. 244, we can obtain the dimensionless parameter $y$ from $x$:

$y\approx\frac{1}{\mkern 1.5mu\overline{\mkern-1.5mu\mu\mkern-1.5mu}\mkern
1.5mu}\cdot\Bigg{(}\underbrace{0.133\cdot\Bigg{(}\frac{T_{\text{eq}}}{10^{3}\text{K}}\Bigg{)}\cdot\ln{\Bigg{(}\frac{n_{\text{bottom}}}{n_{\text{top}}}\Bigg{)}}}_{\text{Part
I}}+\underbrace{16\cdot\Bigg{(}\frac{n_{\text{bottom}}}{\text{mol/cc}}\Bigg{)}}_{\text{Part
II}}\Bigg{)}\Bigg{/}\Bigg{(}\frac{M_{c}}{R_{c}}\Bigg{)}~{}\text{,~{}$M_{c}$,$R_{c}$
in $\oplus$}$ (283)

and recall that:

$R_{p}=R_{c}\cdot\frac{\displaystyle 1}{\displaystyle 1-y}$ (284)

Now, let’s focus on Part I which varies from $n_{\text{top}}\sim 10^{-8}$
mol/cc to $n_{\text{bottom}}\sim 10^{-2}$ mol/cc. This is approximately one-
million-fold contrast in density.

To _visualize_ this contrast, let’s imagine a small volume box containing a
certain number of molecules at the very top of the atmosphere defined by the
transit observation. Now, when we move this box containing the same number of
molecules down to the bottom, each side (linear dimension) of it shrinks by
one-hundred fold, and this gives rise to one-million-fold increase in density.

On the other hand, to go from the bottom of the atmosphere ($\sim 0.01$
mol/cc) all the way to the center of a super-Earth exoplanet ($\sim 0.1$
mol/cc) the density increase is only about one-order-of-magnitude (a factor of
10 or so). This shows how different and difficult it is to compress a
condensed phase compared to a gas phase.

Since $\ln{\Bigg{(}\frac{10^{-2}}{10^{-8}}\Bigg{)}}=\ln{(10^{6})}\approx
13.8$,

$y_{\text{I}}\approx\Bigg{(}\frac{2}{\mkern
1.5mu\overline{\mkern-1.5mu\mu\mkern-1.5mu}\mkern
1.5mu}\Bigg{)}\cdot\Bigg{(}\frac{T_{\text{eq}}}{10^{3}\text{K}}\Bigg{)}\Bigg{/}\Bigg{(}\frac{M_{c}}{R_{c}}\Bigg{)}$
(285)

From Fig. 36, one can see that most interesting super-Earth exoplanets have
$M_{p}/R_{p}\gtrsim 3$. That suggests $M_{c}/R_{c}\gtrsim 3$ for most of these
planets. Moreover, $\mkern 1.5mu\overline{\mkern-1.5mu\mu\mkern-1.5mu}\mkern
1.5mu\gtrsim 2$ and it takes the minimum value only when it is a pure H2
atmosphere. Moreover, most of these interesting super-Earth exoplanets are
$\lesssim 1000$ K equilibrium temperature. Therefore,

$y_{\text{I}}\lesssim
0.3\cdot\Bigg{(}\frac{T_{\text{eq}}}{10^{3}\text{K}}\Bigg{)}$ (286)

From this analysis, we show that though gas has low density, it is also very
compressible under a gravitational potential gradient. It does not take a long
distance for a gas-like species to build up its pressure and molar density
until reaching its critical molar density, where the molecular cohesion takes
over and turns it into a condensed phase.

As a result of that, fundamentally speaking, a gaseous atmosphere can only
account for a surface-layer of a planetary body, a mere skin-effect. This
statement even applies to a gas giant like Jupiter or Saturn, because most of
their interior is considered high-density _super-critical_ molecular fluid and
then metallic fluid rather than a gaseous state, in terms of condensed matter
science.

The fractional boost gained from such a gaseous atmosphere is typically on the
order of 10% to 30% (10-30 percent) of the planet core radius ($R_{c}$) if the
atmosphere composition is dominated by H2-He, and is typically on the order of
1% to 3% (1-3 percent) of the planet core radius ($R_{c}$) if the atmosphere
composition is dominated by Oxygen-Nitrogen-Carbon-species since their
molecular weight is about one order of magnitude higher.

Then in Fig. 36, the group of super-Earth exoplanets with $M_{p}/R_{p}\gtrsim
3$ and with lower-than-rocky density raise an interesting question. They may
be explained by:

  * •

_intrinsic_ lower-than-rocky density in the fluid-layer of these planets, or,

  * •

_intrinsic_ heat source or heat reservoir that maintains a high-entropy state
in the interior of these planets.

Last but not least, due to proximity to host star, the _stability_ of a such
gaseous atmosphere on a short-period exoplanet needs to be examined carefully.
Very plausibly, if such a gaseous atmosphere exists, then it is experiencing
_escape_ and must be continuously replenished by a reservoir such as the
fluid-layer underneath.

##  8 Conclusion

I always thought something was fundamentally uniform with the universe. [82]

##  9 Acknowledgement

The author Li Zeng would like to thank Dr. William J. Nellis for his help on
the science of high-pressure shock wave experiments. The author Li Zeng would
like to thank Dr. Dave Latham for his continuous pursuit of exoplanets through
the _TESS_ mission. The author Li Zeng would like to thank Dr. Ronald Redmer
and his research group, Dr. Stephane Mazevet, Dr. Amit Levi, and Dr. Morris
Podolak for discussion on Hydrogen EOS and H2O EOS.

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