This system is of particular interest. Its host star is a slightly metal-poor
late F-type star. It has been pointed out that metal-poor stars, if their
chemistry only reflects the nucleo-synthetic contributions of Big Bang and
perhaps the pp chain and carbon cycle, would mostly include the elements
representing the early steps of the _alpha-process_ : Carbon, Nitrogen,
Oxygenn, but not much the next two _alpha-process_ elements beyond Neon:
Magnesium and Silicon which are important rock-building elements. Thus, the
planetary systems formed around such metal-poor stars may only contain gas-
giant and icy planets (Lewis, 2004, p. 105).

This idea seems to be applicable to many other Transiting-Timing-Variation
(TTV) planetary systems as well. These systems overlap greatly with the so-
called compact multi-planet systems which often feature adjacent planets near
2:1 mean motion resonance or other small integer mean motion resonances, in
order to allow significant TTV effect to be detectable. It has been noted that
there is a positive correlation between the occurrence of such TTV systems
(sometimes also called compact multi-planet systems) with lower host-stellar
metallicity ([Fe/H]) (Brewer et al., 2018).

If the planets in such a compact multi-planet system are H2-gas-rich or used
to be H2-gas-rich, and puffy, and are or have been experiencing significant
mass loss through H2-envelope escape, then we may explain their compact
orbital configuration as orbital evolution due to mass loss. In such models,
significant mass loss often leads to orbital decay/drift, or the shrinkage of
orbital size (Thom, 2012) (Kurokawa & Nakamoto, 2014), that is, the decrease
of orbital semi-major axis ($a$). If this decay happens to all planets in the
same system, then as planets’ orbits evolve and become more compact together,
it would be natural for adjacent ones to be caught in mean motion resonance.

Another prominent example which happens to be shown in Figure 12 is the
Kepler-29 system. This system features two adjacent planets in 7:9 mean motion
resonance (very adjacent indeed, with one planet orbiting the host star at
0.09 AU and the other orbiting at 0.11 AU!) (Borucki et al., 2011) (Fabrycky
et al., 2012). They (K-29b and K-29c) are the two un-labelled dots in between
planet d & planet c of HIP41378 in Figure 12 with slightly different colors.
They are of similar masses and radii and densities, and from the mass-radius
relations they possess a small but non-trivial (10%$\sim$20% by mass fraction)
amount of H2-envelope.

It seems that they (HIP41378 b,c,&d, K-29b&c), along with some other
exoplanets, all lie roughly along this mass-radius line with negative slope in
log-log plot. It has a slope of $\sim$ (-2/3) in lgM-lgR space. It indicates
that M${}_{p}^{2}\times$ R${}_{p}^{3}$ $\sim$ const, or equivalently,
M${}_{p}^{2/3}\times$ Rp $\sim$ const. Moreover, this line serves as a
dividing line between two groups of exoplanets on the mass-radius diagram:
Group A to the left which are hotter and poorer in H2 envelope; and Group B to
the right which are colder and richer in H2 envelope. This dividing line is
also evident in Figure 5 and Figure 6, and there it shows up as a curve.
Therefore, we hypothesize it as a boundary separating the planets with
H2-envelope-largely-retained versus H2-envelope-lost over billion-year
timescale orbiting sun-like Main-Sequence star. Since we observe mostly mature
planet systems, their ages are generally similar to that of our own solar
system in terms of the order-of-magnitude.

Then, a natural explanation for the observed bi-modal exoplanet radius is that
there exist generally two classes of planet cores: rocky versus icy, and the
evaporation of the primordial H2 help to reveal their true nature to us the
observers. This returns to the argument that metal-poor stars which lack the
elements beyond Neon would only be capable of forming icy cores (made
primarily of elements Carbon, Oxygen, Nitrogen, plus some Hydrogen) and gas
giants, while the stars which have enough Magnesium, Silicon, and/or iron,
would be capable of forming rocky cores (Lewis, 2004, p. 105), and on top of
that forming icy cores and gas giants also. This idea can be tested in future
when more precise measurements of host-stellar elemental abundances become
available.

##  Appendix J On Smaller Host-Stars and Younger Age

###  J.1 TOI-560

It is worthwhile to extend this discussion to younger planetary systems
($\lesssim 1$Gyr), as well as systems with colder-and-smaller host-stars
($\lesssim$0.7R⊙). We expect that the factors of (1) younger age and (2)
smaller host stellar size act in a similar way on planets: (1) Younger
planetary systems would tend to harbor more volatile-rich and puffier planets
at a given host-stellar flux level since there is not enough time yet for the
volatiles to completely escape from the planets; Similarly, (2) Planets around
smaller host stars tend to retain more volatile envelopes at a given age since
the total luminosity of the host star is much less, and the energy of each
individual photon emitted by the host star is less.

Recent observations reveal that a young ($\sim 500-600$ Myr) mini-Neptune HD
73583 b (TOI 560.01) (Barragán et al., 2021; Zhang et al., 2022, 2023)
orbiting a K-dwarf star is currently losing its gas in a form of an outflow at
velocity comparable in scale to its escape velocity with a mass-loss rate
estimated to be $\sim 0.2$M⊕ Gyr-1 (Zhang et al., 2022). If such outflow is
confirmed as the main physical mechanism of atmospheric loss on other
exoplanets, then it can be thought of as a nozzle which drains the H2-envelope
over age.

Thus, we decide to make additional plots (see Figure 13 and Figure 14) to
include this particular system, and also, include other systems around
slightly smaller host stars by lowering radius threshold from earlier plots
(lower limit from 0.7 R⊙ in Figure 6 down to 0.6 R⊙ in this plot), to see the
effect. As expected, the CHILL#1 is somewhat blurred but is still existent, if
one counts the ratio of hot (orange) planets versus cold (blue-green) planets
on either side of this boundary–CHILL#1. Moreover, there seems to be a pile-up
of planets along this boundary. Thus, we may think of planets not as fixed
points on the mass-radius diagram, but rather, as slowly (or occasionally
fast) moving dots tracking along each of their evolutionary trajectory. One
possibility is that the CHILL#1 is not impenetrable, but can be regarded as a
_stagnation line_ where the evolving planet stagnate there for a while before
moving further along its evolutionary trajectory to lose envelope entirely.
This feature needs to be explored in detail in future.

Figure 13: CHILL#1 for Small Exoplanets: Super-Earths/Mini-Neptunes/Cores with
H2-Envelopes. Linear-Linear Plot in Planet Masses ($M_{p}$) and Radii
($R_{p}$), Zoom-in. Similar to Figure 6 but with slightly different and wider
parameter’ selection reaching down to lower mass, lower radius (lower limit
from 0.7 R⊙ in Figure 6 down to 0.6 R⊙ in this plot), and less luminous host
stars; and also colder and less-irradiated exoplanets; and also allowing
bigger errorbars in planet radius. One prominent example of a twin-planet
system: HD73583 b and HD73583 c are shown (Barragán et al., 2021). HD73583 b
has been studied in detail from ground-based observation to show an
atmospheric outflow directed its host-star (Zhang et al., 2022) at high
velocity. A subplot of planet-radius-versus-flux is shown on the upper right
corner, with host stellar luminosity as colorcoding, whereas planet HD73583 c
is too cold and receives too little flux
($f_{\text{HD73583c}}/f_{\oplus}\approx 10$) so it lies outside the range of
this subplot. Planet HD73583 b receives about 50 times Earth’s insolation
($f_{\text{HD73583b}}/f_{\oplus}\approx 43$) and lies in the upper left region
of the subplot. Again, according to the $R_{p}$ histogram on the right-hand-
side, there is a possibility to further sub-divide the planets which are in
between 2-3 $R_{\oplus}$ into two sub-populations. Also, smaller host-stars
(0.6-0.7 R⊙) seem to host more volatile-rich planets, because the host stellar
bolometric luminosity decreases rapidly with its mass according to the mass-
luminosity relationship ($L_{\star}\propto M_{\star}^{3.5}$) and also each
photon emitted is less energetic on average. Although, there might be
observational biases involved. Thus, the CHILL#1 may be blurred somewhat due
to less-irradiated planets around smaller colder host-stars.  Figure 14:
CHILL#1 for Small Exoplanets: Super-Earths/Mini-Neptunes/Cores with Envelopes.
Log-Log Plot in Planet Masses ($M_{p}$) and Radii ($R_{p}$), Zoom-out. Similar
to Figure 7 but zoom out in planet mass and radius. Again, HD73583 b and
HD73583 c are shown (Barragán et al., 2021). Again, a cut-off flux of
($f_{\text{crit}}/f_{\oplus}\approx 200$) is identified in the
$(f/f_{\oplus})-R_{p}$ subplot on the upper-right corner. Above this cut-off
flux level, no planets bigger than $\sim 3R_{\oplus}$ exist. Dashed line in
M-R plot corresponds to
$(M_{p}/M_{\oplus})^{2}\times(R_{p}/R_{\oplus})^{3}\sim 800$, or
alternatively, $(M_{p}/10M_{\oplus})^{2}\times(R_{p}/2R_{\oplus})^{3}\sim 1$.
It can be viewed as the boundary dividing two groups of exoplanets: Group A to
the left, which are hotter and poorer in H2 envelope, but with some diversity;
and Group B to the right which are uniformly colder and richer in volatile
including H2. Shaded region above pure ices mass-radius curve shows contours
of $z$ from 0 to 1 in step of 0.1.

To further investigate the nature of these planets, we invoke the contours of
dimensionless parameter $z$ in mass-radius plot (Figure. 14) as introduced in
(Zeng et al., 2019), and calculate them _only_ for an atmosphere instead of an
envelope:

$z\equiv\frac{\left[\int\frac{dP}{\rho}\right]}{\left(\frac{G\cdot
M_{\oplus}}{R_{\oplus}}\right)}$ (J1)

The numerator corresponds to an integral which runs from the bottom of the
atmosphere to the top. In practicality, without loss of generality, we can
invoke the ideal gas law for atmosphere and show that:

$\left[\int\frac{dP}{\rho}\right]=\frac{\text{R}T}{\overline{\mu}}\cdot\ln{\frac{\overbrace{P_{\text{bottom}}}^{\text{$\sim
10^{8}$Pa }}}{\underbrace{P_{\text{top}}}_{\text{$\sim
10^{2}$Pa}}}}=\frac{\text{R}T}{\overline{\mu}}\cdot\ln{\frac{\overbrace{n_{\text{bottom}}}^{\text{defined
by fluid density:$\sim
10^{22}$cm${}^{-3}$}}}{\underbrace{n_{\text{top}}}_{\text{defined by depth
transit obs. probes:$\sim 10^{16}$cm${}^{-3}$}}}}$ (J2)

$n_{\text{bottom}}$ is defined by the fluid density where the inter-molecular
distance becomes comparable to the dimension (van der Waal radius) of the
molecule itself so inter-molecular interaction becomes strong. Any number
density above this threshold level should be considered as belonging to an
envelope instead of an atmosphere. This typically corresponds to a pressure
level of 0.1$\sim$1 GPa ($10^{8}\sim 10^{9}$ Pa). This level corresponds to a
number density of dense super-critical fluid ($n\sim 10^{22}$cm-3). Any number
density above this threshold level H2 molecules are squeezed strongly against
each other in the dense fluid and is considered an envelope instead of an
atmosphere.

$n_{\text{top}}$ is defined by the number density level where transit
observation probes by a grazing ray passing through the planetary atmosphere.
Depending on the composition and scale height of the atmosphere, it is
typically of the pressure level of 1$\sim$10 millibar ($10^{2}\sim 10^{3}$
Pa). This level corresponds to the number density of von Karman line
previously defined ($n\sim 10^{16}$cm-3). Thus, the von Karman line can be
viewed as the edge of the atmosphere of the planet or the boundary between the
planet atmosphere and the outer space. Any number density below this threshold
level the stellar photon can mostly pass right through without much
absorption.

Thus, from the bottom of the atmosphere to the top, the number density is
rarefied by roughly one-million-fold. That is, the linear separation between
molecules is increased by roughly one-hundred-fold. So,

$\left[\int\frac{dP}{\rho}\right]=\frac{\text{R}T}{\overline{\mu}}\cdot\ln{\left(10^{6}\right)}=13.8\cdot\frac{\text{R}T}{\overline{\mu}}=115\cdot\frac{T}{\overline{\mu}}=1.15\times
10^{5}\cdot\frac{T/10^{3}\text{K}}{\overline{\mu}}$ (J3)

The mean molecular weight $\overline{\mu}$ of a cosmic H2-He mixture with
heavy component mass fraction $Z$ is:

$\overline{\mu}\approx\frac{\mu_{\text{H${}_{2}$-He}}}{(1-Z)}=\frac{2.3\text{g/mol}}{(1-Z)}=\frac{2.3\times
10^{-3}\text{kg/mol}}{(1-Z)}$ (J4)

Therefore, in S.I. unit:

$\left[\int\frac{dP}{\rho}\right]=\left(\frac{1.15\times 10^{5}}{2.3\times
10^{-3}}\right)\cdot\left(\frac{T}{10^{3}\text{K}}\right)\cdot(1-Z)=\overbrace{\left(\frac{1}{2}\cdot
10^{8}\right)}^{\text{S.I. unit}}\cdot T_{3}\cdot(1-Z)$ (J5)

On the other hand, the normalization factor in the denominator expressed in
S.I. unit is:

$\left(\frac{G\cdot
M_{\oplus}}{R_{\oplus}}\right)=\underbrace{\left(0.625\times
10^{8}\right)}_{\text{S.I. unit}}$ (J6)

Thus, the dimensionless parameter $z$ which characterizes the thickness of
planetary atmosphere or envelope is:

$z\equiv\frac{\left[\int\frac{dP}{\rho}\right]}{\left(\frac{G\cdot
M_{\oplus}}{R_{\oplus}}\right)}=\left(\frac{0.5\times 10^{8}}{0.625\times
10^{8}}\right)\cdot
T_{3}\cdot(1-Z)=\underbrace{\left(\frac{1}{1.25}\right)}_{\approx 1}\cdot
T_{3}\cdot(1-Z)$ (J7)

Higher metallicity ($Z$) or Lower Temperature ($T_{3}$) of the atmosphere will
decrease the value of $z$, and thus, will decrease the thickness of
atmosphere.

Lower metallicity ($Z$) or Higher Temperature ($T_{3}$) of the atmosphere will
increase the value of $z$, and thus, will increase the thickness of
atmosphere.

Therefore, for the range of physical parameters that we are interested in
($T_{3}\sim 1$ and $Z\geqslant 0$), the planets which _only_ have an
atmosphere but not an envelope should have $z\lesssim 1$, because the bottom
pressure is limited by $\sim 0.1$ GPa. In Figure 14, the gray shaded region
immediately above the pure H2O mass-radius curve corresponds to $z$ contours
of values from 0 to 1 in increment of 0.1. So the planets lying in this gray
shaded region of the mass-radius diagram can in principle be explained by
_only_ having an extended/puffed atmosphere but not a H2-envelope.

On the other hand, planets with $z\gtrsim 1$ must have a H2-envelope which has
a bottom pressure higher than $\sim 0.1$ GPa. This includes many planets in
the neighborhood of Uranus and Neptune in Figure 14. We can take this argument
one-step further and argue that if we take the CHILL#1 of
$(M_{p}/10M_{\oplus})^{2}\times(R_{p}/2R_{\oplus})^{3}\sim 1$ as boundary
dividing two groups of exoplanets: Group A to the left, and Group B to the
right. Then, all planets of Group A can be explained by possessing only an
atmosphere (with various amount of metallicity) but not an envelope.
Meanwhile, many planets of Group B certainly or likely possess a H2-envelope.

We expect that as the escape process occurs on these planets, their
atmosphere-envelope will become gradually enriched in heavier components (most
likely H2O vapor and H2O super-critical fluid) than H2 and He, and the escape
process of hydrogen on them will eventually become _diffusion-limited_. They
are like cosmic refinery which selectively keep element Oxygen and its
subsidiary compound: water (H2O), while allowing hydrogen to be lost to space.

Therefore, a future theoretical problem to tackle is to understand the escape
process of hydrogen in an H2O-dominated atmosphere-envelope system (Goldblatt,
2015; Pierrehumbert, 2023).

###  J.2 TOI-1695

Another prominent example of exoplanet around a cooler and less luminous
M-dwarf host-star is the TOI-1695 system (Cherubim et al., 2022; Kiefer et
al., 2022). In this particular case, the host star is very slightly larger
than 0.5 solar radius ($R_{\odot}$). Here we provide four plots and sub-plots
of the TOI-1695 system illustrating different coloring schemes (Fig. 15).
Small host stars such as M-Dwarfs are expected to host a lot of water worlds,
because their bolometric luminosity is weak so unlikely to perturb the states
of planets from after their formation.

Figure 15: Mass-radius plots and flux-planet radius subplots including
TOI-1695 as an example, reaching down to smaller host stars (0.45 R⊕). First
important feature is the contrast between the sequence of rocky planets versus
the volatile-rich planets. This contrast is most evident in (a) color-coding
using $a_{\text{orb}}/R_{\star}$ and (b) color-coding using planet globally-
average equilibrium temperature T${}_{\text{eqavg}}$. Secondly, (c) reveals
that CHILL is a boundary of a group of planets of almost uniform temperature
(green-colored), while blending with another group of planets which are either
hotter or colder in temperature but lie closely with the water-EOS mass-radius
curve. Finally, (d) suggests the abundance of water worlds with almost no
envelopes around smaller host stars.

##  Appendix K More Mass-Radius Subplots of Gas Giants

Figure 16: CHILL#2 for Large Exoplanets: Gas Giants/Hot Jupiters. Each column
corresponds to a selected equilibrium temperature range: 500K-1000K,
1000K-1500K, 1500K-2000K. Each row corresponds to a different color-coding
scheme: $f/f_{\oplus}$, [Fe/H], T${}_{\text{eq}}$. Theoretical mass-curves are
given for pure-hydrogen with its variations in both directions: (1) upward-
direction with increasing internal specific entropy; (2) downward-direction
with progressively mixing with heavier elements/species. Each subplot on the
upper-right corner shows the mass-radius diagram of host stars
($M_{\star}$-$R_{\star}$) color-coded by their luminosities ($L_{\star}$) in
logarithmic-scale. The hosts are mostly Main-Sequence (MS) stars hotter than
M-dwarfs.

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