resolution $\widetilde{M_{l}}$ of $M_{l}$ by lifting the Monge–Ampere equation
(4.11) to (4.12), we thus obtain uniform estimates as in subsection $5.1.$ (cf
Theorem 8 and Proposition 3) so that the Sasaki-Ricci flow (4.12) performs a
canonical surgical contraction of floating foliation $(-1)$-curves

$\widetilde{\psi_{l}}:\widetilde{M_{l}}\rightarrow\widetilde{N_{l}}$

by Corollary 1. Furthermore, the canonical surgical contraction maps $V_{i}$
onto $V_{i}^{\prime}$ so that

$V_{i}^{\prime}\cdot V_{i}^{\prime}=-b_{i}$

for all $i\neq j$ and

$V_{j}^{\prime}\cdot V_{j}^{\prime}=-(b_{j}-1).$

Finally, by (5.43) and (A.4), it descends to perform foliation extremal
contractions of foliation $K_{M_{l}}^{T}$-negative curves

$\psi_{l}:M_{l}\rightarrow M_{l+1}$

with at worst singularity type
$\frac{1}{r_{\widetilde{N_{l}}}}(1,a_{\widetilde{N_{l}}})$ so that the
Hirzebruch Jung continued fraction

$\frac{r_{\widetilde{N_{l}}}}{a_{\widetilde{N_{l}}}}=[b_{1},\cdot\cdot\cdot,(b_{j}-1),\cdot\cdot\cdot,b_{m}]$

and it will be
$\frac{r_{\widetilde{N_{l}}}}{a_{\widetilde{N_{l}}}}=[b_{1},\cdot\cdot\cdot,b_{j-1},b_{j+1},\cdot\cdot\cdot,b_{m}]$
if $(b_{j}-1)=1.$

Now take $\widetilde{\psi_{l}}=\widetilde{\psi}$ and $\psi_{l}=\psi$ into
(5.42) with $N=M_{l+1}$ and $\widetilde{N}=\widetilde{N}_{l}$, it follows that
the following diagram is commutative :

$\begin{array}[c]{lcl}\widetilde{M_{l}}\supset\widetilde{\Gamma},\text{
}V_{j}&\overset{\widetilde{\psi_{l}}}{\longrightarrow}&\widetilde{N_{l}}\supset
V_{j}^{\prime}\\\
\downarrow\varphi&\curvearrowright&\downarrow\widetilde{\varphi_{l}}\\\
M_{l}\supset\Gamma,\text{
}\mathbf{S}_{p}^{1}&\overset{\psi_{l}}{\longrightarrow}&M_{l+1}.\end{array}$

Then we are done.

Finally, it follows that the divisoral contractions end with either

$T_{k}=\infty$

and $M_{k}$ is nef which has no foliation $K_{M_{k}}^{T}$-negative curves or

$L_{k}^{T}\text{ \ {is \ not \ big} }$

on $M_{k}$.

Note that similar method can be applied to a finite number of cyclic quotient
foliation singularities.

(II) If $L_{k}^{T}$ is not big : In this situation, we have

$VolM=\int_{M}\omega^{2}(t)\wedge\eta_{0}\rightarrow(c_{1}^{B}(L_{k}^{T}))^{2}=0$

as $t\rightarrow T_{k}^{-}$ and we can not have a canonical surgical
contraction. However since it is semi-ample, as in Proposition 11, we have

(i) there exists a transverse morphism

$\phi:M_{k}\rightarrow pt,$

then $K_{M_{k}}^{T}<0$ and thus $M_{k}$ is transverse minimal Fano and the
foliation space $M_{k}/\mathcal{F}_{\xi}$ is minimal log del Pezzo surface of
at worst $\frac{1}{r}(1,a)$-type singularities. Or

(ii)

$\phi:M_{k}\rightarrow\Sigma_{h},$

then $M_{k}$ is an $S^{1}$-orbibundle of a rule surface over Riemann surfaces
$\Sigma_{h}$ of genus $h$. ∎

##  Appendix A  Foliation Minimal Model Program

###  A.1. Basic Holomorphic Line Bundles, Basic Divisors on Sasakian Manifolds

For a completeness, we will address basic holomorphic line bundles, basic
divisors over Sasakian manifolds. We refer to [BG], [M], and references
therein for some details.

Let $(M,\eta,\xi,\Phi,g)$ be a compact Sasakian $(2n+1)$-manifold $M.$ We
define $D=\ker\eta$ to be the holomorphic contact vector bundle of $TM$ such
that

$TM=D\oplus<\xi>=T^{1,0}(M)\oplus T^{0,1}(M)\oplus<\xi>.$

Then its associated strictly pseudoconvex CR $(2n+1)$-manifold to be denoted
by $(M,T^{1,0}(M),\xi,\Phi).$

######  Definition 13.

( [Ta]) Let $(M,T^{1,0}(M))$ be a strictly pseudoconvex CR $(2n+1)$-manifold
and $E\rightarrow M$ a $C^{\infty}$ complex vector bundle over $M.$ A pair
$(E,\overline{\partial}_{b})$ is a CR-holomorphic vector bundle if the
differential operator

$\overline{\partial}_{b}:\Gamma^{\infty}(E)\rightarrow\Gamma^{\infty}(T^{0,1}(M)^{\ast}\otimes
E)$

is defined by

(i)

$\overline{\partial}_{\overline{Z}}(fs)=(\overline{\partial}_{b}f)(\overline{Z})\otimes
s+f\overline{\partial}_{\overline{Z}}s,$

(ii)

$\overline{\partial}_{\overline{Z}}\overline{\partial}_{\overline{W}}s-\overline{\partial}_{\overline{W}}\overline{\partial}_{\overline{Z}}s-\overline{\partial}_{[\overline{Z},\overline{W}]}s=0,$

for any $f\in C^{\infty}(M)\otimes\mathbf{C}$, $s\in\Gamma^{\infty}(E)$ and
$Z,W\in\Gamma^{\infty}(T^{1,0}(M))$.

The condition (ii) of the definition means that $(0,2)$-component of the
curvature operator $R(E)$ is vanishing when $E$ admits a connection $D$ whose
$(0,1)$-part is the operator $\overline{\partial}_{b}$ as in the following
Lemma.

######  Lemma 12.

Let $(M,T^{1,0}(M),\theta)$ be a strictly pseudoconvex CR $(2n+1)$-manifold
and $(E,\overline{\partial}_{b})$ a CR-holomorphic vector bundle over $M$. Let
$h=<,>_{h}$ be a Hermitian structure in $E$. Then there exists a unique
(Tanaka) connection $D$ in $E$ such that

(i)

$D_{\overline{Z}}s=(\overline{\partial}_{b}s)\overline{Z},$

(ii)

$Z<s_{1},s_{2}>_{h}=<D_{Z}s_{1},s_{2}>_{h}+<s_{1},D_{\overline{Z}}s_{2}>_{h},$

(iii) The $(0,2)$-component of the curvature operator $\Theta(E)$ is
vanishing. Here $\Theta(E):=D^{2}s.$

######  Definition 14.

Let $(M,\eta,\xi,\Phi,g)$ be a compact Sasakian $(2n+1)$-manifold. A CR-
holomorphic vector bundle $(E,\overline{\partial}_{b})$ over $M$ is a basic
transverse holomorphic vector bundle over $(M,T^{1,0}(M))$ if there exists an
open cover $\\{U_{\alpha}\\}$ of $M$ and the trivializing frames on
$U_{\alpha}$ , such that its transition functions are matrix-valued basic CR
functions. The trivializing frames is called the basic tansverse holomorphic
frame.

######  Example 4.

Let $(M,\eta,\xi,\Phi,g)$ be a compact Sasakian $(2n+1)$-manifold. Then, with
respect to the trivializing frames

$\\{\frac{\partial}{\partial x},Z_{j}=\left(\frac{\partial}{\partial
z^{j}}+ih_{j}\frac{\partial}{\partial x}\right),\ \ \
j=1,2,\cdot\cdot\cdot,n\\}$

the transition functions of such frames are basic transverse holomorphic
functions, that is $h$ is basic. Thus $T^{1,0}(M)$ is a basic transverse
holomorphic vector bundle. Moreover, the canonical (determinant) bundle
$K_{M}^{T}$ of $T^{1,0}(M)$ is a basic transverse holomorphic line bundle
whose transition functions are given by $t_{\alpha\beta}=\det(\partial
z_{\beta}^{i}/\partial z_{\alpha}^{j})$ on $U_{\alpha}\cap U_{\beta}$, where
$(x,z_{\alpha}^{1},...,z_{\alpha}^{n})$ is the normal coordinate on
$U_{\alpha}.$

######  Definition 15.

(i) Let $(M,\eta,\xi,\Phi,g)$ be a Sasakian $(2n+1)$-manifold and $L$ be a
basic transverse holomorphic bundle over $M$. A basic transverse holomorphic
section $s$ of $L$ is a collection $\\{s_{\alpha}\\}$ of CR-holomorphic maps
$s_{\alpha}:U_{\alpha}\rightarrow\mathbf{C}$ satisfying the transformation
rule $s_{\alpha}=t_{\alpha\beta}s_{\beta}$ on $U_{\alpha}\cap U_{\beta}$. The
transition function $t_{\alpha\beta}$ is basic. A basic Hermitian metric $h$
on $L$ is a collection $\\{h_{\alpha}\\}$ of smooth positive functions
$h_{\alpha}:U_{\alpha}\rightarrow\mathbf{R}$ satisfying the transformation
rule

$h_{\alpha}=|t_{\beta\alpha}|^{2}h_{\beta}$

on $U_{\alpha}\cap U_{\beta}$. Given a basic transverse holomorphic section
$s$ and a Hermitian metric $h$, we can define the pointwise norm squared of
$s$ with respect to $h$ by

$|s|_{h}^{2}=h_{\alpha}s_{\alpha}\overline{s_{\alpha}}$

on $U_{\alpha}$. The reader can check that $|s|_{h}^{2}$ is a well-defined
function on $M$.

(ii) A Hermitian metric is called a basic hermitian metric if $h_{\alpha}$ is
basic. It always exists if $L$ is a basic transverse holomorphic line bundle.

(iii) We define the curvature $R_{h}^{T}$ of a basic Hermitian metric $h$ on
$L$ to be the basic closed $(1,1)$-form on $M$ given by

$R_{h}^{T}=-\frac{\sqrt{-1}}{2\pi}\partial_{B}\overline{\partial}_{B}\log
h_{\alpha}$

on $U_{\alpha}$. This is well-defined. The basic first Chern class
$c_{1}^{B}(L)$ of $L$ to be the cohomology class $[R_{h}^{T}]_{B}\in
H_{\overline{\partial}_{B}}^{1,1}(M,R)$. Since any two basic Hermitian metrics
$h,h^{\prime}$ on $L$ are related by $h^{\prime}=e^{-\phi}h$ for some smooth
basic function $\phi$, we see that
$R_{h^{\prime}}^{T}=R_{h}^{T}+\frac{\sqrt{-1}}{2\pi}\partial_{B}\overline{\partial}_{B}\phi$
and hence $c_{1}^{B}(L)$ is well-defined,independent of choice of basic
Hermitian metric $h.$ We say that $(L,h)$ is positive if the curvature
$R_{h}^{T}$ is positive definite at every $p\in M.$

######  Example 5.

Let $(M,\eta,\xi,\Phi,g)$ be a compact Sasakian $(2n+1)$-manifold. If $g^{T}$
is a transverse Kaehler metric on $M,$ then
$h_{\alpha}=\det((g_{i\overline{j}}^{\alpha})^{T})$ on $U_{\alpha}$ defines a
basic Hermitian metric on the canonical bundle $K_{M}^{T}$. The inverse
$(K_{M}^{T})^{-1}$ of $K_{M}^{T}$ is sometimes called the anti-canonical
bundle. Its basic first Chern class $c_{1}^{B}((K_{M}^{T})^{-1})$ is called
the basic first Chern class of $M$ and is often denoted by $c_{1}^{B}(M).$Then
it follows from the previous result that $c_{1}^{B}(M)$ that
$c_{1}^{B}(M)=[Ric^{T}(\omega)]_{B}$ for any transversal Kaehler metric
$\omega$ on a Sasakian manifold $M$.

######  Definition 16.

(i) Let $(L,h)$ be a basic transverse holomorphic line bundle over a Sasakian
manifold $(M,\eta,\xi,\Phi,g)$ with the basic hermitian metric $h.$ We say
that $L$ is very ample if for any ordered basis
$\underline{s}=(s_{0},...,s_{N})$ of $H_{B}^{0}(M,L)$, the map
$i_{\underline{s}}:M\rightarrow\mathbf{CP}^{N}$ given by

$i_{\underline{s}}(x)=[s_{0}(x),...,s_{N}(x)]$

is well-defined and an embedding which is $S^{1}$-equivariant with respect to
the weighted $\mathbf{C}^{\ast}$action in $\mathbf{C}^{N+1}$ as long as not
all the $s_{i}(x)$ vanish. We say that $L$ is ample if there exists a positive
integer $m_{0}$ such that $L^{m}$ is very ample for all $m\geq m_{0}.$

(ii) $L$ is a semi-ample basic transverse holomorphic line bundle if there
exists a basic Hermitian metric $h$ on $L$ such that $R_{h}^{T}$ is a
nonnegative $(1,1)$-form. In fact, there exists a foliation basepoint-free
holomorphic map

$\Psi:M\rightarrow(\mathbf{CP}^{N},\omega_{FS})$

defined by the basic transverse holomorphic section
$\\{s_{0},s_{1},...s_{N}\\}$ of $H_{B}^{0}(M.L^{m})$ which is
$S^{1}$-equivariant with respect to the weighted $\mathbf{C}^{\ast}$action in
$\mathbf{C}^{N+1}$ with $N=\dim H^{0}(M.L^{m})-1$ for a large positive integer
$m$ and

$0\leq\frac{1}{m}\Psi^{\ast}(\omega_{FS})=\widehat{\omega}_{\infty}\in
c_{1}^{B}(L).$

There is a Sasakian analogue of Kodaira embedding theorem on a compact quasi-
regular Sasakian $(2n+1)$-manifold due to [RT], and [HLM] :

######  Proposition 7.

Let $(M,\eta,\xi,\Phi,g)$ be a compact quasi-regular Sasakian
$(2n+1)$-manifold and $(L,h)$ be a basic transverse holomorphic line bundle
over $M$ with the basic hermitian metric $h.$ Then $L$ is ample if and only if
$L$ is positive.

######  Definition 17.

(i) First we say that a subset $V$ of a (quasi-regular) Sasakian manifold
$(M^{2n+1},\eta,\xi,\Phi,g)$ an invariant (Sasakian) submanifold (with or
without singularities) of dimension $2n-1$. if $\xi$ is tangent to $V$ and
$\Phi TV\subset TV$ at all points of $V$ and is locally given as the zero set
$\\{f=0\\}$ of a locally defined basic CR holomorphic function $f$. In
general, $V$ may not be a submanifold. Denote by $V^{reg}$ the set of points
$p\in V$ for which $V$ is a submanifold of $M$ near $p$. We say that $V$ is
irreducible if $V^{reg}$ is connected. A transverse divisor $D^{T}$ on $M$ is
a formal finite sum $\sum_{i}a_{i}V_{i}$ where $a_{i}\in\mathbf{Z}$ and each
$V_{i}$ is an irreducible invariant submanifold of dimension $2n-1$. We say
that $D^{T}$ is effective if the $a_{i}$ are all nonnegative. The support of
$D^{T}$ is the union of the $V_{i}$ for each $i$ with $a_{i}\neq 0.$

(ii) Given a transverse divisor $D^{T},$ we define an associated line bundle
as follows. Suppose that $D^{T}$ is given by local defining basic functions
$f_{\alpha}$ (vanishing on $D^{T}$ to order $1$) over an open cover
$U_{\alpha}$. Define transition functions
$f_{\alpha}=t_{\alpha\beta}f_{\beta}$ on $U_{\alpha}\cap U_{\beta}$. These are
basic CR holomorphic and nonvanishing in $U_{\alpha}\cap U_{\beta}$, and
satisfy

$t_{\alpha\beta}t_{\beta\alpha}=1;t_{\alpha\beta}t_{\beta\gamma}=t_{\alpha\gamma}.$

Write $[D^{T}]$ for the associated basic line bundle, which is well-defined
independent of choice of local defining functions.

(iii) One can define

(A.1) $L_{M}\cdot V=\int_{V}R_{h}^{T}\wedge\eta$

for all invariant Sasakian $3$-manifold $V$ in $M.$ $h$ is a basic Hermitian
metric on the basic line bundle $L_{M}$ From (ii), for a compact Sasakian
$5$-manifold $M,$ a transverse divisor $D^{T}$ defines an element of
$H_{\overline{\partial}_{B}}^{1,1}(M,R)$ by $D^{T}\rightarrow[R_{h}^{T}]\in
H_{\overline{\partial}_{B}}^{1,1}(M,R)$ for a basic Hermitian metric on the
associate basic line bundle $[D^{T}]$, and we define

$\alpha\cdot\beta=\int_{M}\alpha\wedge\beta\wedge\eta$

for $\alpha,\beta\in H_{\overline{\partial}_{B}}^{1,1}(M,R).$ Then for an
invariant $3$-manifold $V_{i}$ which is both a foliation curve and a
transverse divisor, the $V\cdot V$ is well-defined and we may write $V^{2}$
instead of $V\cdot V.$

######  Remark 2.

([Gei]) Note that the Sasakain $3$-manifold $V$ is either canonical,
anticanonical or null. $V$ is up to finite quotient a regular Sasakian
$3$-manifold, i.e., a circle bundle over a Riemann surface of positive genus.
In the positive case, $V$ is covered by $S^{3}$ and its Sasakian structure is
a deformation of a standard Sasakian structure.

######  Definition 18.

(i) We say that a basic line bundle $L$ is nef if $L\cdot V\geq 0$ for any
invariant Sasakian $3$-manifold $V$ in $M.$ In particular if $M$ is quasi-
regular, then $V$ is the $\mathbf{S}^{1}$-oribundle over the curve $C$ in $Z$
so that

(A.2) $L_{Z}\cdot C=\int_{C}R_{h_{Z}}\geq 0.$

Here $c_{1}^{B}(L_{M})=\pi^{\ast}c_{1}^{orb}(L_{Z})$ and $h_{Z}$ is the
hermitian metric in the corresponding line bundle $L_{Z}.$ Define

(A.3) $C_{M}^{B}=\\{[\alpha]_{B}\in H_{B}^{1,1}(M,\mathbf{R})|\text{
}\exists\text{ }\omega>0\text{ such that }[\omega]_{B}=[\alpha]_{B}\\}.\text{
}$

Then we can also define a class $[\alpha]_{B}$ called nef class if
$[\alpha]_{B}\in\overline{C_{M}^{B}}$ and a class $[\alpha]_{B}$ called big if

$\int_{M}\alpha^{n}\wedge\eta>0.$

(ii) If the Sasakian manifold $(M^{2n+1},\eta,\xi,\Phi,g)$ has the canonical
basic line bundle $K_{M}^{T}$ nef, then we say that $M$ is a smooth transverse
minimal model. If $M$ has $K_{M}^{T}$ big, then we say that $M$ is of general
type.

###  A.2. Minimal Model Program on Compact Quasi-Regular Sasakian
$5$-Manifolds

We will focus on the proof of foliation minimal model program on a compact
quasi-regular Sasakian $5$-manifold with the foliation singularitie of type
$\frac{1}{r}(1,a).$

Let $(M,\eta,\xi,\Phi,g)$ be a compact quasi-regular Sasakian $5$-manifold and
the space $Z$ of leaves be normal projective orbifold with finite cyclic
quotient singularities of type $\frac{1}{r}(1,a)$ which are klt singularities.
It follows from the first strcture theorem for Sasakian structures that

$K_{M}^{T}=\pi^{\ast}(K_{\emph{Z}}^{orb}).$

On the other hand, the leave space $Z$ is well-formed and the orbifold
canonical divisor $K_{\emph{Z}}^{orb}$ and canonical divisor $K_{Z}$ are the
same, then via the $S^{1}$-orbibundle

$\pi:(M,g)\rightarrow(Z,\omega),$

we have the following Sasaki analogue of basepoint-free theorem, rationality
theorem, cone and contraction theorem ([KM], [KMM], [M]) :

######  Proposition 8.

(Foliation Base-point free Theorem) Let $(M,\eta,\xi,\Phi,g)$ be a compact
quasi-regular Sasakian $5$-manifold and the space $Z$ of leaves be normal
projective orbifold with finite cyclic quotient singularities of type
$\frac{1}{r}(1,a)$ which are klt singularities. Suppose that $L^{T}$ is a nef
Cartier basic line bundle and $aL^{T}-K_{M}^{T}$ is nef and big Cartier basic
line bundle for some $a\in\mathbf{N}$. Then $|mL^{T}|$ is transverse
basepoint-free for $m>>0.$ More precisely, there exists a $S^{1}$-equivariant
foliation basepoint-free holomorphic map

$\Psi_{|mL^{T}|}:M\rightarrow\mathcal{P(}H_{B}^{0}(M,(L^{T})^{m}))$

which is $S^{1}$-equivariant with respect to the weighted
$\mathbf{C}^{\ast}$action in $\mathbf{C}^{N+1}$ with $N=\dim
H^{0}(M.L^{m})-1.$

###### Proof.

Note that the leave space $Z$ is a normal projective orbifold surface with
finite cyclic quotient singularities of type $\frac{1}{r}(1,a)$ which are klt,
Suppose that $L^{T}$ is a nef Cartier basic line bundle and $aL^{T}-K_{M}^{T}$
is nef and big Cartier basic line bundle for some $a\in\mathbf{N}$. On the
other hand by applying Proposition 1, there exists a Riemannian submersion,
$S^{1}$-orbibundle $\pi:M\rightarrow Z,$ such that

$K_{M}^{T}=\pi^{\ast}(K_{\emph{Z}}^{orb})=\pi^{\ast}(K_{\emph{Z}})$

and

$\pi^{\ast}(L)=L^{T}.$

Then $L$ is a nef Cartier line bundle and $aL-K_{Z}$ is nef and big Cartier
line bundle. Therefore by Kawamata base-point free theorem, there is a
basepoint-free holomorphic map

$\psi_{|mL|}:Z\rightarrow\mathcal{P(}H^{0}(Z,L^{m})).$

Define

$\Psi_{|mL^{T}|}=\psi_{|mL|}\circ\pi$

such that

$\Psi_{|mL^{T}|}:M\rightarrow\mathcal{P(}H_{B}^{0}(M,(L^{T})^{m})).$

It follows that $\Psi_{|mL^{T}|}$ is a $S^{1}$-equivariant foliation
basepoint-free holomorphic map with respect to the weighted
$\mathbf{C}^{\ast}$action in $\mathbf{C}^{N+1}$ with $N=\dim H^{0}(M.L^{m})-1$
for a large positive integer $m$. ∎

######  Proposition 9.

(Rationality Theorem) Let $(M,\eta,\xi,\Phi,g)$ be a compact quasi-regular
Sasakian $5$-manifold and the space $Z$ of leaves be normal projective
orbifold with finite cyclic quotient singularities of type $\frac{1}{r}(1,a)$
which are klt singularities. Suppose that $K_{M}^{T}$ is not nef. Let $a(Z)$
be an integer such that $a(Z)K_{M}^{T}$ is Cartier. Let $H^{T}$ be a nef and
big Cartier basic divisor and define

$r=\sup\\{t\in\mathbf{R:}H^{T}+tK_{M}^{T}\text{ \ \ {is nef} }\\}.$

Then $r$ is a rational number of the form $\frac{p}{q}$ with

$0<q\leq a(Z)(\dim Z+1).$

A basic $1$-cycle $V$ on $M$ is a formal finite sum $V=\sum a_{i}V_{i}$, for
$a_{i}\in\mathbf{Z}$ and $V_{i}$ is the irreducible invariant Sasakian
$3$-manifold. We denote by $N_{1}(M)_{\mathbf{Z}}$ the space of $1$-cycles
modulo numerical equivalence. Write

$N_{1}(M)_{\mathbf{Q}}=N_{1}(M)_{\mathbf{Z}}\otimes_{\mathbf{Z}}\mathbf{Q}\text{
\ \ \ {and} \ \ \
}N_{1}(M)_{\mathbf{R}}=N_{1}(M)_{\mathbf{Z}}\otimes_{\mathbf{Z}}\mathbf{R.}$

Then write $NE(M)$ for the cone of effective elements of
$N_{1}(M)_{\mathbf{R}}$ and $\overline{NE(M)}$ for its closure. Furthermore, a
basic divisor $D^{T}$ is ample if and only if

$D^{T}\cdot V>0$

for all nonzero $V\in\overline{NE(M)}$. It is the Kleiman criterion for the
ample line bundle. Also we define the Picard number

$\rho(M):=\dim N_{\mathbf{R}}^{1}(M)\leq\dim H_{B}^{2}(M,\mathbf{R})<\infty.$

######  Proposition 10.

(Foliation Contraction Theorem) Let $(M,\eta,\xi,\Phi,g)$ be a compact quasi-
regular Sasakian $5$-manifold and the space $Z$ of leaves be normal projective
orbifold surface with finite cyclic quotient singularities of type
$\frac{1}{r}(1,a)$ which are klt singularities. Then there exists a countable
collection of Sasakain $3$-spheres $\\{V_{i}\\}$ such that

$0<-K_{M}^{T}\cdot V_{i}\leq 2\dim Z$

and

$\overline{NE(M)}=\overline{NE(M)}_{K_{M}^{T}\geq 0}+\sum a_{i}|V_{i}|,\text{
\ \ }a_{i}\geq 0.$

The rays $a_{i}|V_{i}|$ are locally discrete in the half space
$\\{K_{M}^{T}<0\\}.$ If $R\in\overline{NE(M)}$ is a $K_{M}^{T}$-negative
foliation extremal ray such that

$R=\overline{NE(M)}_{K_{M}^{T}<0}\cap(L^{T})^{\perp}$

for some nef basic line bundle $L^{T}$ which can be chosed by

$\pi^{\ast}L=L^{T}$

for a nef line bundle $L$ over $Z$. Then there is a unique foliation extremal
ray contraction

$\psi_{(L^{T})^{m}}=cont_{V}:M\rightarrow N$

for some $m>>1$ such that an irreducible foliation curve $V\subset M$ is
mapped to a leave $\mathbf{S}^{1}$ by $\psi$ if and only if $[V]_{B}\in R$.
Furthermore, $L^{T}=\psi^{\ast}A$ for some basic ample line bundle on $N.$

###### Proof.

Since $Z$ is a normal projective orbifold surface with finite cyclic quotient
singularities of type $\frac{1}{r}(1,a)$ which are klt singularities, it
follows ([KM]) that there exists a nef and semi-ample line bundle $L$ over $Z$
such that if $R_{Z}\in\overline{NE()}$ is a $K_{Z}$-negative extremal ray with

$R_{Z}=\overline{NE(Z)}_{K_{Z}<0}\cap(L)^{\perp}.$

Then there is a unique extremal ray contraction

$\psi_{(L)^{m}}=cont_{C}:Z\rightarrow Y\subset\mathcal{P(}H^{0}(Z,mL))$

for some $m>>1$ such that an irreducible curve $C\subset Z$ is mapped to a
point by $\psi_{(L)^{m}}$ if and only if $[C]\in R_{Z}$.

Now for a Riemannian submersion, $S^{1}$-orbibundle $\pi:M\rightarrow Z,$ we
choose $L^{T}$ such that

$\pi^{\ast}L=L^{T}$

ann thus $L^{T}$ is a basic nef and semi-ample line bundle over $M$ such that
there is a unique $S^{1}$-equivariant foliation extremal ray contraction

$\psi_{(L^{T})^{m}}=cont_{V}:M\rightarrow
N\subset\mathcal{P(}H_{B}^{0}(M_{k},(L^{T})^{m}))$

with

$\psi_{(L^{T})^{m}}=\psi_{(L)^{m}}\circ\pi$

for which an irreducible foliation curve $V\subset M$ is mapped to a leave
$\mathbf{S}^{1}$ by $\psi_{(L^{T})^{m}}$ if and only if

$[V]_{B}\in R=\overline{NE(M)}_{K_{M}^{T}<0}\cap(L^{T})^{\perp}.$

∎

Now we are ready to prove the following foliation minimal model program :

######  Proposition 11.

Let $(M,\eta,\xi,\Phi,g)$ be a compact quasi-regular Sasakian $5$-manifold
with finite cyclic quotient foliation singularities of type
$\frac{1}{r}(1,a).$ Then there exists a finite sequence of foliation extremal
ray contractions

$\psi_{i}:M_{i-1}\rightarrow M_{i}\text{ , \ }i=1,...k$

such that every $M_{i}$ is a Sasakian manifold having at worst foliation
cyclic quotient singularities and for every $\psi_{i}$ one of the following
holds :

  1. (1)

Foliation divisorial contraction : (The locus of the foliation extremal ray is
an irreducible basic divisor) : $\psi_{i}$ is a foliation divisorial
contraction of a foliation curve $V$ with $V^{2}<0$, and the Picard number
satisfies $\rho(M_{i})=\rho(M_{i-1})-1.$ or

  2. (2)

Foliation fibre contraction (transverse Mori fibre space) : (The locus of the
foliation extremal ray is $M_{i-1}$) : $\psi_{i}$ is a singular fibration such
that either

    1. (a)

there is a map

$f:M_{k}\rightarrow pt,$

then $K_{M_{k}}^{T}<0$ and thus $M_{k}$ is transverse minimal Fano and the
leave space $Z_{k}$ is minimal log del Pezzo surface of
$\frac{1}{r}(1,a)$-type singularities and Picard number one. Or

    2. (b)

$f:M_{k}\rightarrow\Sigma_{h},$

then $Z_{k}$ is a rule surface over Riemann surfaces $\Sigma_{h}$ of genus $h$
and thus $M_{k}$ is an $S^{1}$-orbibundle over $Z_{k}.$ Or

  3. (3)

$M_{k}$ is nef : $\psi_{i}=\psi_{k}$, and $M_{k}$ has at worst foliation
cyclic quotient singularities and has no foliation $K_{M}^{T}$-negative
curves.

###### Proof.

Let $\mathbf{S}_{p}^{1}$ be the singular fibre of type $\frac{1}{r}(1,a),p\in
M.$ Let $\psi:M\rightarrow N$ be a proper transverse birational morphism. By
applying Theorem 6 and [Cu], we consider the minimal resolution of foliation
singularities of $M$ to be $\varphi:\overline{M}\rightarrow M$ and
$\overline{\varphi}:\overline{N}\rightarrow N$ such that the following diagram
is commutative :

$\begin{array}[c]{lcl}(\overline{M},\overline{\Gamma})&\overset{\overline{\psi}}{\longrightarrow}&\overline{N}\\\
\downarrow\varphi&\circlearrowright&\downarrow\overline{\varphi}\\\
(M,\Gamma,\mathbf{S}_{p}^{1})&\overset{\psi}{\longrightarrow}&(N,\mathbf{S}_{q}^{1}).\end{array}$

Since $\overline{M}$ and $\overline{N}$ are regular Sasakian manifolds, by
Theorem 2, the transverse birational morphism $\overline{\psi}$ can be
factored into a sequence of ordinary blow ups. Suppose $\overline{M}$ and
$\overline{N}$ are not isomorphic, there must exist a foliation $(-1)$-curve
$\overline{\Gamma}$ such that
$\overline{\varphi}\circ\overline{\psi}(\overline{\Gamma})=\mathbf{S}_{q}^{1},$
for $q\in N.$ Let $\Theta$ be a set of foliation curves such that $\Theta=$
$\overline{\psi}\circ\overline{\varphi}^{-1}(\mathbf{S}_{q}^{1})$ and $D^{T}=$
$\\{V_{i}\\}$ be the exceptional locus of $\varphi$ at the singular fibre
$\mathbf{S}_{p}^{1}.$ Then the Hirzebruch Jung continued fraction

$\frac{r}{a}=[b_{1},\cdot\cdot\cdot,b_{l}]$

say that

$V_{i}^{2}=-b_{i}.$

(i) If $V_{i}\cdot\overline{\Gamma}=\emptyset$, for all $i,$ then
$\overline{\Gamma}$ is not exceptional for $\varphi$ and its birational
transform $\Gamma$ is a floating foliation $(-1)$-curve on $M$. Then by
Theorem 2, we obtain a transverse birational morphism
$\psi^{\prime}:M\rightarrow M_{1}$ corresponding to the blow up of a smooth
point with exceptional locus given by $\Gamma.$ Thus $\psi$ factors through
$\psi^{\prime}$ and we obtain a transverse birational morphism
$M_{1}\rightarrow N.$

(ii) If $V_{j}\cdot\overline{\Gamma}\neq\emptyset$ for some $j$ ( at most one
such a foliation exceptional curve) , then $\ V_{j}^{2}=-b_{j}$ and $\Gamma$
is passing the singular fibre $\mathbf{S}_{p}^{1}$ of the foliation
singularities of type $\frac{1}{r}(1,a)$ and $K_{M}^{T}\cdot\Gamma<0.$
Moreover, $\ \overline{\Gamma}$ is also a foliation $(-1)$-curve on
$\overline{M}$. Again by Theorem 2, we obtain a transverse birational morphism
$\psi^{\prime\prime}:M\rightarrow M_{1}$ corresponding to the transverse
extremal ray contraction of $\Gamma$ into a singular fibre of foliation
singularities of at worst type $\frac{1}{r_{M_{1}}}(1,a_{M_{1}})$ such that

(A.4)
$\frac{r_{M_{1}}}{a_{M_{1}}}=[b_{1},\cdot\cdot\cdot,(b_{i}-1),\cdot\cdot\cdot,b_{l}].$

Again $\psi$ factors through $\psi^{\prime}$ and we obtain a transverse
birational morphism $M_{1}\rightarrow N.$ Similar method can be applied to a
finite number of cyclic quotient foliation singularities.

On the other hand, since for divisoral extremal contraction, we have
$\rho(M_{i})=\rho(M_{i-1})-1,$ then after a finite number of foliation
extremal ray contractions, we end up

$M=M_{0}\overset{\psi_{1}}{\rightarrow}M_{1}\overset{\psi_{2}}{\rightarrow}M_{2}\overset{\psi_{3}}{\rightarrow}\cdot\cdot\cdot\overset{\psi_{k}}{\rightarrow}M_{k}$

with $M_{k}$ is nef or

by [L, Lemma 2.1.27] and $L^{T\text{ }}$is semi-ample over $M_{k}$ (see
Proposition 10) to see that there exist a morphism

$\lambda:Z_{k}\rightarrow\mathcal{P(}H^{0}(Z_{k},mL))$

such that $f=\lambda\circ\pi$ is a $S^{1}$-equivariant transverse morphism

$f:M_{k}\rightarrow\mathcal{P(}H_{B}^{0}(M_{k},mL^{T})),$

with $L^{\text{ }}$is semi-ample over $Z_{k}$ for $\pi^{\ast}L=L^{T}$.

Then $Y:=h(Z_{k})$ is either a point or a nonsingular curve.

(a). If $Y$ is a point, then $K^{orb}(Z_{k})<0$ and $K_{M_{k}}^{T}<0$. Hence
$M_{k}$ is transverse Fano.

(b). If $Y=\Sigma_{h}$ is a non-singular curve, then a generic fiber $C$ of
$\lambda$ is a smooth curve and then foliation curve $V$ over $C$ satisfying
$L_{k}\cdot C=0$ and $K_{Z_{k}}^{orb}\cdot C<0$ as well as $L_{k}^{T}\cdot
V=0$ and $K_{M_{k}}^{T}\cdot V<0$. It follows from the adjunction formula that
$C$ is a rational curve and thus $V$ is covered by the three sphere
$\mathbf{S}^{3}$. Hence $Z_{k}$ is a rule surface over Riemann surfaces
$\Sigma_{h}$ of genus $h$ and thus $M_{k}$ is an $S^{1}$-orbibundle over
$Z_{k}.$ ∎

######  Remark 3.

Let $(Z,\emptyset)$ be a log del Pezzo orbifold surface with no branch
divisors (well-formed). It follows from [BG, Theorem 4.7.14] that the total
space of an $S^{1}$-orbibundle over $Z$ is diffeomorphic to some
$k(\mathbf{S}^{2}\mathbf{\times S}^{3})$ for some $k=0,1,...$ ($k=0$ means
$\mathbf{S}^{5}$).

######  Corollary 4.

Let $(M,\eta,\xi,\Phi,g)$ be a compact regular Sasakian $5$-manifold. Then
there exists a transverse morphism $\psi:M\rightarrow M^{\prime},$ which is a
composition of blowing down foliation $(-1)$-curves and a morphism
$\varphi:M^{\prime}\rightarrow N$ such that one of the following holds :

  1. (1)

Nef : $M^{\prime}\simeq N$ is a compact regular Sasakian $5$-manifold with
$K_{N}^{T}$ nef;

  2. (2)

Mori fibre space :

    1. (a)

$N$ is a Riemann surface $\Sigma_{h}$ and $M^{\prime}$ is
$\Sigma_{h}\mathbf{\times S}^{3}$ or
$X_{\infty,h}=\Sigma_{h}\widetilde{\times}\mathbf{S}^{3}$ or

    2. (b)

$N$ is a point and $M^{\prime}$ is isomorphic to $\mathbf{S}^{5}$.

As a consequence, we have

######  Corollary 5.

Let $(M,\eta,\xi,\Phi,g)$ be a compact simply connected regular Sasakian
$5$-manifold.

  1. (1)

If the Sasakian structure is positive, then $M=\mathbf{S}^{5}$ $(k=0)$; or
$k(\mathbf{S}^{2}\mathbf{\times S}^{3})$ or
$M=X_{\infty,0}\\#(k-1)(\mathbf{S}^{2}\mathbf{\times S}^{3}),1\leq k\leq 8.$

  2. (2)

If the Sasakian structure is indefinite ($K_{M}^{T}$ is nef), then
$M=k(\mathbf{S}^{2}\mathbf{\times S}^{3})$; or
$M=X_{\infty,0}\\#(k-1)(\mathbf{S}^{2}\mathbf{\times S}^{3}),1\leq k$.

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11institutetext: Center for Theoretical Astrophysics & Cosmology
Institute for Computational Science, University of Zurich, Zurich, Switzerland
22institutetext: Department of Planetary Sciences, The University of Arizona,
USA 33institutetext: International Space Science Institute, Bern, Switzerland
44institutetext: Department of Geosciences, Tel-Aviv University, Tel-Aviv,
Israel

# Why do more massive stars host larger planets?

M. Lozovsky , E-mail<EMAIL_ADDRESS>R. Helled 11 I. Pascucci 22 C. Dorn
11 J. Venturini 33 R. Feldmann 11

###### Abstract

Aims. It has been suggested that planetary radii increase with the stellar
mass, for planets below 6 R⊕ and host below 1 M⊙. In this study, we explore
whether this inferred relation between planetary size and the host star’s mass
can be explained by a larger planetary mass among planets orbiting more
massive stars, inflation of the planetary radius due to the difference in
stellar irradiation, or different planetary compositions and structures.

Methods. Using exoplanetary data of planets with measured masses and radii, we
investigate the relations between stellar mass and various planetary
properties for G- and K- stars, and confirm that more massive stars host
larger planets and more massive. We find that the differences in the planetary
masses and temperatures are insufficient to explain the measured differences
in radii between planets surrounding different stellar types. We show that the
larger planetary radii can be explained by a larger fraction of volatile
material (H-He atmospheres) among planets surrounding more massive stars.

Results. We conclude that planets around more massive stars are larger most
probably as a result of larger H-He atmospheres. Our findings imply that
planets forming around more massive stars tend to accrete H-He atmospheres
more efficiently.

Conclusions.

###### Key Words.:

planets and satellites: formation – planets and satellites: fundamental
parameters – planets and satellites: interiors – planets and satellites:
composition

##  1 Introduction

The last couple of decades of exoplanetary exploration has led to the
discovery of several thousands of exoplanets. The interior characterization of
these planets has become a key objective in exoplanetary science (e.g., Rogers
& Seager, 2010; Dorn et al., 2015; Unterborn et al., 2016; Brugger et al.,
2017; Owen & Wu, 2017a; Baumeister et al., 2020; Otegi et al., 2020b). We are
now in a stage at which exoplanetary characterization is becoming more
detailed, allowing us to understand the various planetary populations within
our Galaxy, to make the links to planet formation and evolution models, and to
better understand the uniqueness of our own planetary system.

The detected planets with measured radii from the Kepler mission (Borucki et
al., 2010), whose also have mass determination using radial velocity (RV)
follow-ups or transit timing variations (TTV) can be used to infer the
planetary average density, and therefore, to estimate the interior
composition. By now, we have learned that planets are very diverse
astronomical objects, and that they do not follow a single mass-radius
(hereafter M-R) relation, but instead, show an intrinsic scatter (Wolfgang et
al., 2015; Chen & Kipping, 2016; Ning et al., 2018; Neil & Rogers, 2020). This
scatter suggests a large diversity in the planetary formation and evolution,
as well as the interior structure and bulk composition (e.g., Weiss & Marcy,
2014; Turrini et al., 2015; Petigura et al., 2018; Plotnykov & Valencia,
2020). While determining the planetary structure and composition from measured
mass and radius is a highly degenerate problem, because very different
compositions and structures can yield similar masses and radii (e.g., Rogers &
Seager, 2010; Dorn et al., 2015, 2017), some trends and limits on the
planetary compositions can be found using statistical analysis (e.g., Rogers,
2015; Owen & Wu, 2017a; Lozovsky et al., 2018; Neil & Rogers, 2020; Otegi et
al., 2020a).

The relatively large database of exoplanets also allows to investigate the
dependence of the planetary populations on the stellar mass (e.g., Fulton &
Petigura, 2018; Neil & Rogers, 2018; Mulders et al., 2015; Pascucci et al.,
2018) and metallicity (e.g., Mulders et al., 2016; Mulders, 2018; Adibekyan,
2019; Kutra & Wu, 2020). There are contradictory claims in previous studies on
how the Kepler mini-Neptunes and super-Earths depend on stellar mass. For
example, Wu (2019) argued for a linear relationship between planetary mass and
stellar mass, and no correlation between planet radius and host star
metallicity. In contrast, Owen & Murray-Clay (2018) suggested that planetary
radii are correlated to stellar metallicity: Stars of higher metallicity host
larger planets.

Furthermore, Neil & Rogers (2018) found no strong evidence for host star mass
dependence in the M-R relation, using the hierarchical Bayesian modeling
approach presented by Wolfgang et al. (2016).

Given the result above, Pascucci et al. (2018) applied Chen & Kipping (2016)
M-R relation to convert the planetary radii measured by Kepler into planetary
masses. When computing completeness-corrected occurrence rates versus stellar
mass, it was found that the mass of the most common planet for relatively
small stars (with masses $<$ 1 M⊙) scales almost linearly with stellar mass.
Pascucci et al. (2018) showed that the most common planetary mass depends on
the mass of the stellar host. More specifically, a universal (i.e.,
independent on the stellar type) peak in the occurrence rate of the planet-to-
star mass ratio for the Kepler planets was found.

Table 1:  Planet samples mean values, divided by stellar types

Stellar type | Stellar T (K) | No. |  $\mu$(R) |  $\sigma$(R) |  $\mu$(M) |  $\sigma$(M) |  $\mu$ (Teq) |  $\sigma$(Teq)
---|---|---|---|---|---|---|---|---
F | 5200-6000 | 8 | 3.32 | 1.84 | 8.03 | 4.17 | 1085.80 | 489.67
G | 5200-6000 | 51 | 2.84 | 1.19 | 9.41 | 4.42 | 943.09 | 339.35
K | 3700-5200 | 32 | 2.09 | 0.61 | 7.33 | 4.10 | 822.02 | 315.55
M | 2400-3700 | 16 | 1.72 | 0.92 | 4.13 | 3.17 | 529.49 | 241.09

Planet samples mean values (as described in section 2.1), divided by stellar
types. Teq is equilibrium planet temperature that assumes zero albedo (see
section 2.1.1). No. stands for number of planets in the sample, $\mu$ stands
for mean value and $\sigma$ for standard deviation, assuming normal
distribution. Planetary mass, radius and temperatures are given in M⊕, R⊕ and
K, respectively.

The study of Pascucci et al. (2018) focuses on MKG stars and suggests that the
most common planet around a lower mass stars is smaller in mass. For planets
in the radius range of 1-6 R⊕, Chen & Kipping (2016) M-R relation are
presented by a power-law. Therefore, Pascucci et al. (2018) data indicate that
the most common planet around a lower mass stars is smaller in radius. Most of
the planets in the sample used by Pascucci et al. (2018) have radii above 1.6
R⊕, and therefore are likely to have some volatile materials. The addition of
such low-density materials (even if very small in terms of mass) can lead to
much larger radii for small mass planets (Rogers, 2015; Lozovsky et al.,
2018). M-R relations of volatile-rich planets are more complex and can vary
depending on the planetary temperature and the actual distribution of the
material in the interior (e.g., Lozovsky et al., 2018; Mousis et al., 2020;
Müller et al., 2020). In addition, it was recently shown that intermediate-
size planets ($\sim$5 M⊕– 20 M⊕) can follow two different M-R relations since
they can belong to two different planetary populations (Otegi et al., 2020a).

In this work we explore potential reasons why more massive stars tend to host
larger planets. In principle, there are at least three possible explanations
for the identified correlation between the planetary radius and stellar mass:

  * •

Planets are larger due to thermal inflation: since more massive stars are more
luminous, larger stellar irradiation could inflate the planetary radius
(Case-1).

  * •

Planets around more massive stars are more massive and are in consequence
larger for any given composition (Case-2).

  * •

Planets around more massive stars tend to be more volatile-rich and are
therefore larger in size (Case-3).

Below we investigate these three different cases. In Case-1, we would expect
to find a correlation between planetary equilibrium temperature and radius.
Case-2 can be examined by testing whether the observed differences in
planetary radius can be explained by a variation of planetary mass, and Case-3
by testing the effect of planetary composition on planetary radius. In this
work, we discuss the three cases and find that Case-3 is the favorable
interpretation.

##  2 Methods

In this work we apply a simple statistical analysis of exoplanetary data
combined with internal structure models to explore the potential explanations
for the observed trend that planets surrounding more massive stars are larger.

The planetary parameters that are used in the analysis are the planetary mass,
radius and insolation flux; the stellar parameters are stellar mass, radius
and effective temperature.
First, we build an exoplanetary sample based on the NASA Exoplanet Archive.
The planetary sample is used to quantify the differences between different
planetary physical properties, among exoplanets surrounding different stellar
populations.

We next construct a series of planetary models using structure equations in
order to quantify the effect of different parameters, such as planetary mass,
temperature and composition on the planetary radius (section 2.3) and explore
whether the difference in the observed distributions of equilibrium
temperatures (Case-1) and masses (Case-2) is sufficient to explain the
difference in the observed radii. Finally, we investigate the effect of
various assumed compositions (Case-3) on the inferred radii.

###  2.1 Planetary Samples

Figure 1: The M-R diagram for planetary sample used in this study. The dots
with error bars correspond to F-planets (purple crosses), G-planets (blue
squares), K-planets (red diamonds), and M-planets (yellow pluses). Only
G-planets and K-planets are used in the analysis. The colored solid curves are
M-R relations for various theoretical models with different mass fractions of
H-He (see Section 2.3 for details) and constant H2O mass fraction in the
envelope. The presented curves correspond to models with Teq of 700 K (showed
as examples).

The planetary sample we use in this study is based on the online publicly
available NASA Exoplanet Archive. Since we focus on relatively small planets,
we select transiting planets, with RV or TTV follow-ups that have masses Mpl
up to 20 M⊕, radii Rpl up to 8 R⊕ and uncertainties on mass and radius up to
50% of the measured value. We divide the planet samples according to stellar
type, which is derived from the stellar effective temperatures (see Table 1).
We determine the average radius, mass, and equilibrium temperature (see
section 2.1.1 for details) of the planets for each stellar type using the
catalogue.

Figure 2: Histogram of planet-to-star mass ratio for the two sub-samples. It
can be compared to Figure 1 of Pascucci et al. (2018). The value of qbr of
Pascucci et al. (2018) of $\log(M_{pl}/M_{*})=-4.55\pm 0.03$ is presented as a
vertical black line and can be compared with mean values we calculated:
$-4.59\pm 0.03$ (G-planets, red line) and $-4.58\pm 0.04$ (K-planets, blue
line).

Table 1 lists the mean values ($\mu$) and the corresponding standard
deviations ($\sigma$) calculated assuming a normal distribution. In this work
we consider only planets around K and G stars. Hereafter, we refer to planets
surrounding G and K stars as ”G-planets” and ”K-planets”, respectively. We do
not include planets around F stars in our study, because F-planets do not
follow the ”universal planet-to-star mass ratio peak” found by Pascucci et al.
(2018) and because of the rather small number of F-planets in the planetary
sample. M-planets are also not included because of their small number in the
planetary database. The planets we use are listed in the Appendix.

The database used in this study differs from one used by Pascucci et al.
(2018), since the NASA catalogue is inhomogeneous in terms of detection
methods, measurement uncertainties, and selection effects. While Pascucci et
al. (2018) used Kepler planets with well-studied data incompleteness, the
NASA’s database lists planets detected by various methods, such as Radial
Velocity (RV), transit, and direct imagining, and therefore the dataset
suffers from different selections effects and data incompleteness. In
addition, by selecting planets with measured masses and radii with relatively
low uncertainties, we may introduce our own selection. This topic is addressed
in Section 2.2. The uncertainties of our sample are discussed in Section
2.1.2.

Given the potential biases of our planetary catalog, We explore whether the
planet-to-star mass ratio inferred by Pascucci et al. (2018) persists when
using our sample. In order to investigate the existence of a universal peak in
planet-to-star mass ratio we first calculate the mass ratios for two sub-
samples: G-planets and K-planets. The distributions of planet-to-star mass
ratio of the two sub-samples is shown in Figure 2, with the mean values
(dashed horizontal lines) and corresponding uncertainties of the mean value.
This can be compared to the peak value qbr from Pascucci et al. (2018) which
is given by the solid black line. The figure clearly shows that G-planets and
K-planets have a similar mean value, and that this value is in agreement with
qbr inferred by Pascucci et al. (2018). Our results confirms that the mean
planetary mass scales with stellar mass.

It is encouraging that despite the large differences in the data and different
selection effects, a similar behavior is found and supports the use of our
sample to further investigate the correlation origin.

####  2.1.1 Planetary Equilibrium Temperature

Figure 3: Distributions of planetary radius, mass and equilibrium temperature
of G- and K- planets. The left panels show histograms with the mean values and
the corresponding uncertainties (dashed lines). The right panels present the
corresponding CDFs.

The planetary equilibrium temperatures of the sample are calculated assuming
an albedo $A=0$. The planetary albedo depends on various parameters, such as
atmospheric composition, thermal state, cloud layers etc.; by assuming an
albedo of zero, corresponding to full absorption, we infer the highest
possible equilibrium temperature. The planetary temperature is calculated via:

${T}_{eq}={T}_{*}{\left(1-A\right)}^{1/4}\sqrt{\frac{{R}_{*}}{2D}},$ (1)

where Teq is planetary equilibrium temperature, ${T}_{*}$ is the stellar
temperature, $R_{*}$ is the stellar radius, and $D$ is the orbital separation.
The values for these parameters are taken from the planetary database.

Planets with equilibrium temperatures above 2000 K are not considered in this
study, because these ultra-hot planets are rare outliers and are likely to be
subject to other processes, such as atmospheric loss by photo-evaporation
(e.g. Rosotti et al., 2013; Jin & Mordasini, 2018), that could affect their
basic planetary properties. The mean values of Teq, mass and radius of the
samples are listed in Table 1, with the corresponding standard deviations. The
distributions of radii, masses and equilibrium temperatures are presented in
Figure 3.

We performed a K-S test on the sub-samples. The test rejects the null
hypotheses that G-planets’ and K-planets’ masses and radii come from the same
distribution at the 5% significance level and asymptotic p-values of 0.03 and
0.04, respectively. The test could not reject the null hypothesis that the
equilibrium temperatures of G-planets and K-planets come from the same
distribution, with asymptotic p-value of 0.18.

####  2.1.2 Measurement uncertainties

The planets we include in the samples have small measurement uncertainties,
and therefore the planetary mass, radius and orbital period are rather well-
constrained. In order to examine how the measurement uncertainties affect the
main tendency we perform the following test:
For each planet in the sample we produce 10,000 synthetic data-points using
the measured radii and their uncertainties. The simulated values are probed
assuming a normal distribution $R_{pl}\sim\mathcal{(}R,R_{err})$, where $R$ is
the measured radius and $R_{err}$ is the corresponding measurement
uncertainty. The simulated values are presented in Figure 4. As can be seen
from the figure, the main trend of larger stars hosting larger planets
persists.

Figure 4: Simulated values of planetary radii accounting for the measurement
uncertainties. For each planet in the sample we simulated 10,000 radii, taking
into account the uncertainty. The mean values are presented by the dashed
lines. The histogram of the simulated populations show the same tendency as
the original one (see Figure 3).

###  2.2 Selection effects

Most of the detected exoplanets with measured radii were observed with the
Kepler mission using the transit method, where the transit depth is given by:

$\frac{dF}{F_{st}}=\left(\frac{R_{pl}}{R_{st}}\right)^{2}.$ (2)

Here $dF$ is the dimming due to the transit and $F_{st}$ is the stellar flux
(e.g., Bozza et al., 2016). As a result, smaller planets are easier to detect
around smaller stars but are easier to miss around larger stars, thus,
potentially introducing an artificial correlation between planetary size and
stellar size (or stellar mass).

In order to test whether our sample suffers from a strong selection effect of
this kind we compare our sample with the wider (and homogeneous) sample of
Kepler and K2 planets, using the NASA catalog, without restricting the
planetary mass. Specifically, we select planets with sizes up to 8 R⊕ and
uncertainties below 50% of the measured values, surrounding G-stars and
K-stars with measured stellar radii and temperatures.

Figure 5: Planetary radius versus the square of the planet-to-star radius
ratio $(R_{pl}/R_{st})^{2}$ for Kepler G-planets (blue squares) and K-planets
(red diamonds). The planets shown by filled symbols is the sub-sample used in
this study.

Figure 5 shows the planetary radius as a function of the planet-to-star radius
ratio $(R_{pl}/R_{st})$. In this figure, the full sample consisting of 1267
G-planets and 643 K-planets is shown by empty symbols while the sample we use
in this study is shown by the filled symbols. In contrast to the expectation
from a transit depth related selection bias, neither the full Kepler sample
nor our sub-sample show a lack of detected planets below some critical transit
depth, i.e., a vertical cut-off in $(R_{pl}/R_{st})^{2}$. In fact, both G- and
K- stars in our sub-sample appear to host planets as small as 1.2-1.4 R⊕. In
addition, our sub-sample has planet-to-star radius ratios significantly larger
than the minimal value of $\sim{}3\times{}10^{-5}$ in the full sample. This
suggests that selection effects due to small transit depths are unlikely to
affect the main results of this study.

###  2.3 Interior Modeling

In order to explain the observed differences in planetary properties of
different populations, we infer the relation between the planetary mass and
radius (M-R relations) for planets with different assumed compositions and
internal structures. It is reasonable to model the planets in the samples with
volatile-rich interiors, since most of them have radii larger than the ones
expected for an Earth-like composition (e.g., Rogers, 2015; Lozovsky et al.,
2018). The planetary structure models used consist of a rocky core surrounded
by a hydrogen-helium (H-He) and a water (H2O) envelope. For simplicity, the
composition of the core is set to be pure MgSiO3. More complex core structures
are expected to have a negligible effect on the M-R relation for volatile-rich
planets (Howe et al., 2014; Otegi et al., 2020a). The H to He ratio is assumed
to be protosolar, i.e., 72% H to 28% He by mass, and the water mass fraction
is taken to be 20% in the envelope (water + H-He). The equation of state for
H-He is taken from Saumon et al. (1995), for MgSiO3 from Seager et al. (2007)
and for H2O from Vazan et al. (2013). The atmospheric models are derived using
the standard structural equations of hydrostatic equilibrium, mass
conservation, and heat transport. We follow the irradiation model of Guillot
(2010) for the H-He envelope. Further details on the structure models can be
found in Venturini et al. (2015), Dorn et al. (2017), Lozovsky et al. (2018)
and references therein. We consider two cases of envelope structure: a
homogeneously mixed envelope consisting of H-He and water and a differentiated
structure, where the water and H-He are separated, and the pure water layer is
below pure H-He layer. The first case (homogeneous envelope) is used as
default.

The structure models are constructed assuming various percentages of H-He
between 0.5% and 25% of the total planetary mass and different equilibrium
temperatures Teq (see section 2.1.1 for details). Models with H-He mass
fractions larger than 25% were not considered since we focus on planets up to
20 M⊕, which are unlikely to be H-He-rich. The M-R relations, together with
the planetary sample are shown in Figure 1; the curves presented in this plot
are examples assuming Teq=700 K. The calculated M-R relations allow us to
derive the planetary radius corresponding to any given mass and Teq. These
models are used in the following analysis in order to investigate the
potential reason(s) for the trend of ”larger planets around more massive
stars”.

##  3 Results

###  3.1 Case-1: Larger Planets due to Inflation

As mentioned above, most of the planets in the sample are expected to be
volatile-rich given their large radii and low bulk densities (Figure 6). Radii
of volatile-rich planets are very sensitive to the temperature: planets with
the same mass, chemical composition, and internal structure can have different
radii for different temperatures due to thermal inflation. Planets surrounding
more massive stars seem to be slightly hotter on average (see Figure 3).
Therefore, it is reasonable to expect that the differences in the equilibrium
temperatures for G- and K-planets could lead to differences in radii of the
two populations. Below, we use the planet sample to evaluate any possible
correlation between planetary equilibrium temperatures and radii.

Figure 6: Measured planetary radius (upper panel) and calculated planetary
bulk density (lower panel) versus calculated equilibrium temperature for our
sample of planets.

Figure 6 (upper panel) presents the observed planetary radii versus
equilibrium temperature Teq, calculated from Equation 1. Interestingly, the
observed radii do not increase with temperature. G- and K-planets even tend to
have smaller radii in for higher temperatures. Figure 6 (lower panel) presents
the bulk densities (inferred from the measured masses and radii) versus Teq
for the detected planets. No significant decrease of the bulk density with
temperature is observed. In fact, the bulk densities of both G- and K-planets
is increasing with temperature, suggesting that the very hot planets tend to
be more rocky. This correlation could be a result of atmospheric mass loss of
primordial hydrogen-atmospheres, that depends on stellar insolation (Fulton &
Petigura, 2018; Owen & Wu, 2017b; Jin & Mordasini, 2018).

Thus, the observed radii do not correlate with the temperature, and thermal
inflation does not play a significant role in observed trend. We therefore
conclude that the differences in the equilibrium temperature alone is
insufficient to explain the differences in the observed radii of the two
planetary populations.

###  3.2 Case-2: Larger Planets due to Higher Planetary Mass

Assuming that the planetary radius scales with mass, it is reasonable to
expect that the observed difference in radii is a result of difference in
planetary mass. In this section we investigate whether the observed difference
in average radii is caused by more massive planets around more massive stars.

To investigate this option, we use the theoretical M-R relations of planets
with rocky cores surrounded by H-He + H2O atmospheres, calculated from the
internal structure models described in section 2.3. The typical difference
between masses of G-planets and K-planets in the sample is $\sim$ 2 M⊕. The
mass difference is calculated as a difference between the mean values of the
distributions. This difference in mass distributions (see Figure 3 and Table
1) is insufficient to explain the difference in the observed radii of 0.75 R⊕
, as the M-R curves are very flat in this regime (Lopez & Fortney, 2014; Weiss
et al., 2013; Wolfgang et al., 2015; Chen & Kipping, 2016; Lozovsky et al.,
2018), already implying the weak dependency of the radius for these planetary
masses.

In order to test whether the difference in radii is a result of different
planetary masses, we calculate the radius corresponding to a hypothetical
average mass planet of each stellar type: 9.41 M⊕ for G-planets and 7.33 M⊕
for K-planets, as listed in Table 1. To isolate the effect of mass, we assume
a uniform temperature of Teq= 900K and H-He mass fraction of 2% for both
hypothetical planets. This method compares between the mean values of the
distributions of masses, assuming the average masses to be representative of
the main trend; by calculating a radius corresponding to each average mass and
using the same M-R relation and temperature for all of the masses, we are able
to isolate the influence of the mass alone. The calculated radii of both
G-planets and K-planets are found to be in the range of 2.62-2.76 R⊕ for
G-planets and 2.58-2.81 R⊕ .

These radii ranges are calculated assuming two different atmospheric
structures: The lower radius boundary is calculated assuming a homogeneous
envelope structure (water and H-He are mixed), while the upper limit is
calculated assuming a differentiated structure, where the water and H-He are
separated (see section 2.3). We find that the effect of the mass on the
calculated radius alone is negligible compared to the observed differences.

Therefore, we conclude that the difference in the planetary masses cannot
explain the measured difference in planetary radii.

###  3.3 Case-3: Larger Planets due to Different Compositions

Figure 7: Histogram of simulated f${}_{\text{H-He}}$ for our planetary
samples. Planets that cannot be modeled with any H-He mass fraction are not
presented.

Finally, we investigate the effect of different compositions and envelope
structures on the planetary radius. For planets with thick H-He envelopes, the
planetary structure and exact H-He mass fraction have significant effects on
the M-R relation (see Lozovsky et al. (2018) for details). For the same
planetary mass and temperature the difference between a planet with
f${}_{\text{H-He}}$ of 0.5% and f${}_{\text{H-He}}$ of 10% is between $\sim
1.2$ R⊕ and $\sim 1.8$ R⊕. This range exceeds the difference between the
measured radii of G-planets and K-planets: the difference between the peaks of
the radii distribution is 0.75 R⊕ .

We next compare the radii inferred when assuming various compositions and
internal structures with the measured ones. Using the planetary models
described in section 2.3, we calculate for each planet the corresponding
f${}_{\text{H-He}}$. The calculation uses the planetary effective temperature
and the measured mass. The results are shown in Figure 7. There is a clear
trend where planets around larger stars have higher f${}_{\text{H-He}}$values:
G-planets: $0\lesssim$ f${}_{\text{H-He}}$$\lesssim 0.4$, K-planets:
$0\lesssim$ f${}_{\text{H-He}}$$\lesssim 0.15$) , with mean values of
0.06$\pm$0.01 and 0.03$\pm$0.01 respectively. It should be noted that there
are several small planets which are unlikely to consist of H-He: this
corresponds to 15 G-planets and 15 K-planets. These planets are probably
rocky-water dominated planets without any significant volatile atmosphere, and
therefore do not have a strong dependency on Teq, and thus do not influence
our conclusions. Note that those planets are not presented in Figure 7, as
they are unlikely to possess any H-He envelopes and can be seen from Figure 1,
below the 0.5% H-He line.

As expected, the H-He mass fraction in the planet f${}_{\text{H-He}}$ has a
crucial effect on the inferred radii (e.g Figure 1; Swift et al. (2012); Weiss
& Marcy (2014)). The difference in the assumed f${}_{\text{H-He}}$ leads to a
radius difference that can exceed the difference between G-planets and
K-planets. We therefore suggest that  the difference in observed radii of
planets orbiting stars with difference masses could be caused by different
H-He mass fractions, with the G-planets being more volatile-rich in comparison
to K-planets. In addition, it should be noted that the distribution of
volatiles has a significant effect on M-R relations: the size difference
between fully differentiated and homogeneous envelopes is larger than the
difference due to thermal inflation and larger mass (e.g., Soubiran &
Militzer, 2015).

We conclude that while case-1 (thermal inflation) and case-2 (higher mass)
cannot explain the observed differences, case-3 (different planetary
composition and structure) can explain the observations.

##  4 Discussion

In this study we investigate the potential explanations for the observed
correlation between mean planetary size and a stellar type. We show that
different H-He mass fractions could explain this observations. In the
analysis, we restricted the possible theoretical models to a rather narrow
range: rocky cores surrounded by H-He+H2O atmospheres, with
f${}_{\text{H-He}}$$=0.5\%-25\%$ and water fraction in the envelope of 20% by
mass. Nevertheless, even in this narrow assumption range, we could provide an
interpretation for the reason that the most typical planetary radius scales
with stellar mass. We suggest that this correlation might be a result of
characteristically different planetary composition and/or structure among
planets surrounding different stellar types, while planetary mass and
insolation are unlikely to play a big role in this relation. Planet formation
and evolution models could confirm this idea. In addition, it is unlikely that
the found correlation is caused by data incompleteness. Nevertheless, the
planetary data used in this analysis are somewhat limited and inhomogeneous.
As a result, it is desirable to use larger samples in the future, and repeat
the analysis accounting for a larger range in stellar spectral types, also
including M-dwarfs.

Previous studies have investigated differences in planet populations between
M-dwarfs and FGK- dwarfs. Giant planets are scarce around M-dwarfs (Bonfils et
al., 2013) and there is a higher occurrence of small planets around M-dwarfs
(Dressing & Charbonneau, 2015; Gaidos et al., 2016). The higher planet-
formation efficiency of low-mass stars (Dai et al., 2020) may be in fact
related to the small giant planet occurrence, or the longer disk lifetimes, or
other factors. How the planetary composition distribution differs is yet
unclear. So far, there is no strong evidence for a dependency between the
planetary M-R relation and the host star’s mass for small planets (Neil &
Rogers, 2018). Our findings, if extrapolated to M-dwarfs, would predict a
tendency of H-He-poor planets compared to those around larger stars. This is
somewhat in line with previous studies (Bonfils et al., 2013; Dressing &
Charbonneau, 2015; Gaidos et al., 2016) as planets around M-stars are less
massive. This does not imply that planets are poor in water. In fact, planets
around M-dwarfs can be water-rich due to an efficient inwards migration
(Alibert & Benz, 2017; Burn et al., 2021).

Kepler planets are located relatively close to their stars, and may thus be
affected by planet–star interactions; either by stellar insolation during
planet evolution or during formation in the protoplanetary disk. Here, we are
arguing that stellar insolation alone cannot make up for the different radii
in the observed planetary populations around G- and K-stars. Instead we
provide evidences that the populations are different due to formation
processes, i.e. the efficiency of envelope accretion. This can be a result of
more massive protoplanetary disks (Mohanty et al., 2013; Andrews et al., 2013;
Pascucci et al., 2016; Ansdell et al., 2016) that lead to faster growth
(Hartmann et al., 2006; Alcalá et al., 2014) followed by H-He accretion (even
if in small amounts) and/or due to different disk lifetimes (Ribas et al.,
2015). Essentially, core growth is more efficient both in higher-mass disks
(Mordasini et al., 2012; Venturini et al., 2020) and around higher-mass stars
(Kennedy & Kenyon, 2008) allowing them to to accrete substantial gaseous
envelope before the gas disk dissipates.

The diversity of planetary envelopes of our sample is yet unknown, and
different compositions including H, He, water, and higher mean molecular
species are possible. Besides compositional diversity, these planets may
absorb stellar energy differently which affects their thermal structure and
thus their radii.

In that regard, it would be desirable to investigate the dependence on the
assumed albedo; In this work we assumed a uniform zero albedo for all planets
when calculating the Teq. Since different compositions can lead to different
albedo, the relation between the different parameters is likely to be very
complex. Another important property that could influence the planetary
population is the age of the system. In order to investigate a correlation
between the age and size of the planets, we need better constraints on the
stellar ages, which is not available for most of the stars in current
planetary databases.

It should be noted that planets around F-stars are not included in this
analysis because they do not present the same tendency as G-planets and
K-planets, as was shown by Pascucci et al. (2018). F-stars are more luminous
and as a result have higher irradiation. As a result, close-in planets might
suffer from other physical effects, such as atmospheric loss by photo-
evaporation (e.g., Rosotti et al., 2013; Jin & Mordasini, 2018), that could
affect their basic planetary properties and alter the statistics. It is
therefore clear that more data and as well as theoretical models that account
for the planetary evolution are required before a conclusion can be made
regarding the expected compositions of planets around F-stars.

##  5 Summary and Conclusions

We confirm that planets surrounding larger stars tend to be larger in mass and
radius. In this study we investigated the influence of the planetary mass,
equilibrium temperature, and possible bulk composition on the planetary radius
for planets surrounding G- and K-stars. The larger radii of planets
surrounding more massive stars cannot be explained by inflation due to higher
irradiation of the star, nor by a higher planetary mass: the effect of both
properties on the radius was found to be significantly smaller than the
observed difference. In other words, we show that the difference in planetary
mass, which is given by RV/TTV data, is insufficient to explain the observed
difference in planetary radii, and the planetary temperature is even found to
anti-correlate with the radius.

Larger H-He mass fractions on planets around larger stars can easily explain
the trends in the observed data. We therefore suggest that the larger radii of
planets surrounding larger stars are a result of different compositions:
larger stars tend to host planets with larger H-He mass fractions. If this is
correct, it provides important constraints for planet formation models, since
it implies that planets forming around more massive stars tend to accrete H-He
atmospheres more efficiently. This can be a result of more massive
protoplanetary disks that lead to faster core growth; allowing the cores to
accrete substantial gaseous envelopes before the gas disk dissipates. Thus, we
argue that planets around G- and K-stars are different by formation.

In that light, it is desirable to have further constraints available to test
our findings. Further information on the planetary composition would come from
JWST and Ariel missions with constraints on atmospheric composition, which
will greatly shed more light into the diversity of planetary volatile
envelopes, their formation efficiencies and evolutions. In addition, an
important property of the planetary system is the planetary age, which is
typically not very well constrained (e.g., Saffe et al., 2005; Silva Aguirre
et al., 2015; Guillot et al., 2014). This is expected to change with the PLATO
mission which will allow to estimate the stellar ages with accuracies of about
10% (Rauer et al., 2014). With age constraints, we will be able to check if
and how the discussed differences in planet populations are evolving in time.

###### Acknowledgements.

R.H. acknowledges support from SNSF grant 200021_169054. C.D. acknowledges
support from the Swiss National Science Foundation under grant PZ00P2_174028.
This work has been carried out within the frame of the National Center for
Competence in Research PlanetS supported by the SNSF Swiss National
Foundation.

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##  Appendix A Planetary Data Tables

Table 2: Planets used in the study Name | Mass(M⊕) | Radius(R⊕) | Teq (K) | Host Mass(M⊙) | Host Radius(R⊙) | Host type
---|---|---|---|---|---|---
CoRoT-7 b | 4.08${}^{+1.02}_{-1.02}$ | 1.68${}^{+0.11}_{-0.11}$ | 1807 | 0.82${}^{+0.17}_{-0.17}$ | 0.83${}^{+0.04}_{-0.04}$ | G
EPIC 249893012 b | 8.75${}^{+1.09}_{-1.08}$ | 1.95${}^{+0.09}_{-0.08}$ | 1616 | 1.05${}^{+0.05}_{-0.05}$ | 1.71${}^{+0.04}_{-0.04}$ | G
EPIC 249893012 c | 14.67${}^{+1.84}_{-1.89}$ | 3.67${}^{+0.17}_{-0.14}$ | 990 | 1.05${}^{+0.05}_{-0.05}$ | 1.71${}^{+0.04}_{-0.04}$ | G
EPIC 249893012 d | 10.18${}^{+2.46}_{-2.42}$ | 3.94${}^{+0.13}_{-0.12}$ | 752 | 1.05${}^{+0.05}_{-0.05}$ | 1.71${}^{+0.04}_{-0.04}$ | G
HD 110113 b | 4.55${}^{+0.62}_{-0.62}$ | 2.05${}^{+0.12}_{-0.12}$ | 1371 | 1.00${}^{+0.06}_{-0.06}$ | 0.97${}^{+0.02}_{-0.02}$ | G
HD 136352 b | 4.62${}^{+0.45}_{-0.44}$ | 1.48${}^{+0.06}_{-0.06}$ | 911 | 0.91${}^{+0.06}_{-0.05}$ | 1.01${}^{+0.02}_{-0.02}$ | G
HD 136352 c | 11.29${}^{+0.73}_{-0.69}$ | 2.61${}^{+0.08}_{-0.08}$ | 682 | 0.91${}^{+0.06}_{-0.05}$ | 1.01${}^{+0.02}_{-0.02}$ | G
HD 183579 b | 19.70${}^{+4.00}_{-3.90}$ | 3.55${}^{+0.15}_{-0.12}$ | 769 | 1.03${}^{+0.03}_{-0.03}$ | 0.98${}^{+0.04}_{-0.03}$ | G
HD 219666 b | 16.60${}^{+1.30}_{-1.30}$ | 4.71${}^{+0.17}_{-0.17}$ | 1073 | 0.92${}^{+0.03}_{-0.03}$ | 1.03${}^{+0.03}_{-0.03}$ | G
HD 86226 c | 7.25${}^{+1.19}_{-1.12}$ | 2.16${}^{+0.08}_{-0.08}$ | 1311 | 1.02${}^{+0.06}_{-0.07}$ | 1.05${}^{+0.03}_{-0.03}$ | G
K2-111 b | 5.29${}^{+0.76}_{-0.77}$ | 1.82${}^{+0.11}_{-0.09}$ | 1304 | 0.84${}^{+0.02}_{-0.02}$ | 1.25${}^{+0.02}_{-0.02}$ | G
K2-138 b | 3.10${}^{+1.05}_{-1.05}$ | 1.51${}^{+0.11}_{-0.08}$ | 1299 | 0.94${}^{+0.02}_{-0.02}$ | 0.86${}^{+0.03}_{-0.02}$ | G
K2-138 c | 6.31${}^{+1.13}_{-1.23}$ | 2.30${}^{+0.12}_{-0.09}$ | 1137 | 0.94${}^{+0.02}_{-0.02}$ | 0.86${}^{+0.03}_{-0.02}$ | G
K2-138 d | 7.92${}^{+1.39}_{-1.35}$ | 2.39${}^{+0.10}_{-0.08}$ | 988 | 0.94${}^{+0.02}_{-0.02}$ | 0.86${}^{+0.03}_{-0.02}$ | G
K2-138 e | 12.97${}^{+1.98}_{-1.99}$ | 3.39${}^{+0.16}_{-0.11}$ | 858 | 0.94${}^{+0.02}_{-0.02}$ | 0.86${}^{+0.03}_{-0.02}$ | G
K2-263 b | 14.80${}^{+3.10}_{-3.10}$ | 2.41${}^{+0.12}_{-0.12}$ | 470 | 0.88${}^{+0.03}_{-0.03}$ | 0.85${}^{+0.02}_{-0.02}$ | G
K2-265 b | 6.54${}^{+0.84}_{-0.84}$ | 1.71${}^{+0.11}_{-0.11}$ | 1423 | 0.92${}^{+0.02}_{-0.02}$ | 0.98${}^{+0.05}_{-0.05}$ | G
K2-291 b | 6.49${}^{+1.16}_{-1.16}$ | 1.59${}^{+0.10}_{-0.07}$ | 1398 | 0.93${}^{+0.04}_{-0.04}$ | 0.90${}^{+0.04}_{-0.03}$ | G
K2-38 b | 19.07${}^{+9.53}_{-9.53}$ | 1.91${}^{+0.34}_{-0.34}$ | 1281 | 2.24${}^{+1.67}_{-1.67}$ | 1.38${}^{+0.21}_{-0.21}$ | G
K2-38 c | 9.90${}^{+4.60}_{-4.60}$ | 2.42${}^{+0.29}_{-0.29}$ | 858 | 1.07${}^{+0.05}_{-0.05}$ | 1.1${}^{+0.09}_{-0.09}$ | G
Kepler-107 b | 3.51${}^{+1.52}_{-1.52}$ | 1.54${}^{+0.03}_{-0.03}$ | 1593 | 1.24${}^{+0.03}_{-0.03}$ | 1.45${}^{+0.01}_{-0.01}$ | G
Kepler-107 c | 9.39${}^{+1.77}_{-1.77}$ | 1.60${}^{+0.03}_{-0.03}$ | 1379 | 1.24${}^{+0.03}_{-0.03}$ | 1.45${}^{+0.01}_{-0.01}$ | G
Kepler-107 e | 8.60${}^{+3.60}_{-3.60}$ | 2.90${}^{+0.04}_{-0.04}$ | 955 | 1.24${}^{+0.03}_{-0.03}$ | 1.45${}^{+0.01}_{-0.01}$ | G
Kepler-11 d | 7.30${}^{+0.80}_{-1.50}$ | 3.12${}^{+0.06}_{-0.07}$ | 715 | 0.96${}^{+0.03}_{-0.03}$ | 1.06${}^{+0.02}_{-0.02}$ | G
Kepler-11 e | 8.00${}^{+1.50}_{-2.10}$ | 4.19${}^{+0.07}_{-0.09}$ | 637 | 0.96${}^{+0.03}_{-0.03}$ | 1.06${}^{+0.02}_{-0.02}$ | G
Kepler-11 f | 2.00${}^{+0.80}_{-0.90}$ | 2.49${}^{+0.04}_{-0.07}$ | 562 | 0.96${}^{+0.03}_{-0.03}$ | 1.06${}^{+0.02}_{-0.02}$ | G
Kepler-18 b | 6.90${}^{+3.40}_{-3.40}$ | 2.00${}^{+0.10}_{-0.10}$ | 1290 | 0.97${}^{+0.04}_{-0.04}$ | 1.11${}^{+0.05}_{-0.05}$ | G
Kepler-18 c | 17.30${}^{+1.90}_{-1.90}$ | 5.49${}^{+0.26}_{-0.26}$ | 995 | 0.97${}^{+0.04}_{-0.04}$ | 1.11${}^{+0.05}_{-0.05}$ | G
Kepler-18 d | 16.40${}^{+1.40}_{-1.40}$ | 6.98${}^{+0.33}_{-0.33}$ | 797 | 0.97${}^{+0.04}_{-0.04}$ | 1.11${}^{+0.05}_{-0.05}$ | G
Kepler-20 b | 9.70${}^{+1.41}_{-1.44}$ | 1.87${}^{+0.07}_{-0.03}$ | 1105 | 0.95${}^{+0.05}_{-0.05}$ | 0.96${}^{+0.02}_{-0.02}$ | G
Kepler-20 c | 12.75${}^{+2.17}_{-2.24}$ | 3.05${}^{+0.06}_{-0.06}$ | 772 | 0.95${}^{+0.05}_{-0.05}$ | 0.96${}^{+0.02}_{-0.02}$ | G
Kepler-20 d | 10.07${}^{+3.97}_{-3.70}$ | 2.74${}^{+0.07}_{-0.06}$ | 401 | 0.95${}^{+0.05}_{-0.05}$ | 0.96${}^{+0.02}_{-0.02}$ | G
Kepler-289 d | 4.00${}^{+0.90}_{-0.90}$ | 2.68${}^{+0.17}_{-0.17}$ | 504 | 1.08${}^{+0.02}_{-0.02}$ | 1.00${}^{+0.02}_{-0.02}$ | G
Kepler-29 b | 5.00${}^{+1.50}_{-1.30}$ | 2.55${}^{+0.12}_{-0.12}$ | 761 | 0.76${}^{+0.02}_{-0.03}$ | 0.73${}^{+0.03}_{-0.03}$ | G
Kepler-30 b | 11.30${}^{+1.40}_{-1.40}$ | 3.90${}^{+0.20}_{-0.20}$ | 609 | 0.99${}^{+0.08}_{-0.08}$ | 0.95${}^{+0.12}_{-0.12}$ | G
Kepler-36 c | 7.13${}^{+0.18}_{-0.18}$ | 3.68${}^{+0.10}_{-0.09}$ | 1035 | 1.03${}^{+0.02}_{-0.02}$ | 1.63${}^{+0.04}_{-0.04}$ | G
Kepler-454 b | 6.05${}^{+1.51}_{-1.51}$ | 2.35${}^{+0.22}_{-0.22}$ | 943 | 0.85${}^{+0.22}_{-0.22}$ | 1.05${}^{+0.07}_{-0.07}$ | G
Kepler-538 b | 10.60${}^{+2.50}_{-2.40}$ | 2.21${}^{+0.04}_{-0.03}$ | 380 | 0.89${}^{+0.05}_{-0.04}$ | 0.87${}^{+0.01}_{-0.01}$ | G
Table 3: Table 2, continued Name | Mass(M⊕) | Radius(R⊕) | Teq (K) | Host Mass(M⊙) | Host Radius(R⊙) | Host type
---|---|---|---|---|---|---
Kepler-87 c | 6.40${}^{+0.80}_{-0.80}$ | 6.14${}^{+0.29}_{-0.29}$ | 403 | 1.10${}^{+0.05}_{-0.05}$ | 1.82${}^{+0.04}_{-0.04}$ | G
Kepler-93 b | 4.54${}^{+0.85}_{-0.85}$ | 1.57${}^{+0.11}_{-0.11}$ | 1134 | 1.09${}^{+0.14}_{-0.14}$ | 0.98${}^{+0.04}_{-0.04}$ | G
TOI-125 b | 9.50${}^{+0.88}_{-0.88}$ | 2.73${}^{+0.07}_{-0.07}$ | 1037 | 0.86${}^{+0.04}_{-0.04}$ | 0.85${}^{+0.01}_{-0.01}$ | G
TOI-125 c | 6.63${}^{+0.99}_{-0.99}$ | 2.76${}^{+0.10}_{-0.10}$ | 828 | 0.86${}^{+0.04}_{-0.04}$ | 0.85${}^{+0.01}_{-0.01}$ | G
TOI-125 d | 13.60${}^{+1.20}_{-1.20}$ | 2.93${}^{+0.17}_{-0.17}$ | 638 | 0.86${}^{+0.04}_{-0.04}$ | 0.85${}^{+0.01}_{-0.01}$ | G
TOI-421 b | 7.17${}^{+0.66}_{-0.66}$ | 2.68${}^{+0.19}_{-0.18}$ | 981 | 0.85${}^{+0.03}_{-0.02}$ | 0.87${}^{+0.01}_{-0.01}$ | G
TOI-421 c | 16.42${}^{+1.06}_{-1.04}$ | 5.09${}^{+0.16}_{-0.15}$ | 674 | 0.85${}^{+0.03}_{-0.02}$ | 0.87${}^{+0.01}_{-0.01}$ | G
TOI-561 c | 5.40${}^{+0.98}_{-0.98}$ | 2.88${}^{+0.10}_{-0.10}$ | 821 | 0.79${}^{+0.02}_{-0.02}$ | 0.85${}^{+0.01}_{-0.01}$ | G
TOI-561 d | 11.95${}^{+1.28}_{-1.28}$ | 2.53${}^{+0.13}_{-0.13}$ | 615 | 0.79${}^{+0.02}_{-0.02}$ | 0.85${}^{+0.01}_{-0.01}$ | G
TOI-561 e | 16.00${}^{+2.30}_{-2.30}$ | 2.67${}^{+0.11}_{-0.11}$ | 426 | 0.79${}^{+0.02}_{-0.02}$ | 0.85${}^{+0.01}_{-0.01}$ | G
TOI-763 b | 9.79${}^{+0.78}_{-0.78}$ | 2.28${}^{+0.11}_{-0.11}$ | 1038 | 0.92${}^{+0.03}_{-0.03}$ | 0.90${}^{+0.01}_{-0.01}$ | G
TOI-763 c | 9.32${}^{+1.02}_{-1.02}$ | 2.63${}^{+0.12}_{-0.12}$ | 800 | 0.92${}^{+0.03}_{-0.03}$ | 0.90${}^{+0.01}_{-0.01}$ | G
55 Cnc e | 7.99${}^{+0.32}_{-0.33}$ | 1.88${}^{+0.03}_{-0.03}$ | 1949 | 0.91${}^{+0.01}_{-0.01}$ | 0.94${}^{+0.01}_{-0.01}$ | K
GJ 9827 b | 4.91${}^{+0.49}_{-0.49}$ | 1.58${}^{+0.03}_{-0.03}$ | 1184 | 0.61${}^{+0.02}_{-0.01}$ | 0.60${}^{+0.01}_{-0.00}$ | K
GJ 9827 d | 4.04${}^{+0.82}_{-0.84}$ | 2.02${}^{+0.05}_{-0.04}$ | 686 | 0.61${}^{+0.02}_{-0.01}$ | 0.60${}^{+0.01}_{-0.00}$ | K
HD 15337 b | 7.51${}^{+1.09}_{-1.01}$ | 1.64${}^{+0.06}_{-0.06}$ | 1001 | 0.90${}^{+0.03}_{-0.03}$ | 0.86${}^{+0.02}_{-0.02}$ | K
HD 15337 c | 8.11${}^{+1.82}_{-1.69}$ | 2.39${}^{+0.12}_{-0.12}$ | 642 | 0.90${}^{+0.03}_{-0.03}$ | 0.86${}^{+0.02}_{-0.02}$ | K
HD 219134 b | 4.74${}^{+0.19}_{-0.19}$ | 1.60${}^{+0.06}_{-0.06}$ | 1015 | 0.81${}^{+0.03}_{-0.03}$ | 0.78${}^{+0.01}_{-0.01}$ | K
HD 219134 c | 4.36${}^{+0.22}_{-0.22}$ | 1.51${}^{+0.05}_{-0.05}$ | 782 | 0.81${}^{+0.03}_{-0.03}$ | 0.78${}^{+0.01}_{-0.01}$ | K
HD 219134 d | 16.17${}^{+0.64}_{-0.64}$ | 1.61${}^{+0.02}_{-0.02}$ | 410 | 0.81${}^{+0.03}_{-0.03}$ | 0.78${}^{+0.01}_{-0.01}$ | K
HD 219134 f | 7.30${}^{+0.40}_{-0.40}$ | 1.31${}^{+0.02}_{-0.02}$ | 523 | 0.81${}^{+0.03}_{-0.03}$ | 0.78${}^{+0.01}_{-0.01}$ | K
HD 97658 b | 8.88${}^{+1.23}_{-1.23}$ | 2.35${}^{+0.11}_{-0.11}$ | 741 | 0.89${}^{+0.15}_{-0.15}$ | 0.74${}^{+0.01}_{-0.01}$ | K
HIP 116454 b | 12.71${}^{+3.18}_{-3.18}$ | 2.47${}^{+0.11}_{-0.11}$ | 740 | 0.81${}^{+0.11}_{-0.11}$ | 0.72${}^{+0.02}_{-0.02}$ | K
K2-216 b | 8.00${}^{+1.60}_{-1.60}$ | 1.75${}^{+0.17}_{-0.10}$ | 1103 | 0.70${}^{+0.03}_{-0.03}$ | 0.72${}^{+0.03}_{-0.03}$ | K
K2-266 e | 14.30${}^{+6.40}_{-5.00}$ | 2.73${}^{+0.14}_{-0.11}$ | 490 | 0.69${}^{+0.03}_{-0.03}$ | 0.70${}^{+0.02}_{-0.02}$ | K
K2-285 b | 9.68${}^{+1.21}_{-1.37}$ | 2.59${}^{+0.06}_{-0.06}$ | 1089 | 0.83${}^{+0.02}_{-0.02}$ | 0.79${}^{+0.02}_{-0.02}$ | K
K2-285 c | 15.68${}^{+2.28}_{-2.13}$ | 3.53${}^{+0.08}_{-0.08}$ | 741 | 0.83${}^{+0.02}_{-0.02}$ | 0.79${}^{+0.02}_{-0.02}$ | K
K2-36 b | 3.90${}^{+1.10}_{-1.10}$ | 1.43${}^{+0.08}_{-0.08}$ | 1224 | 0.79${}^{+0.01}_{-0.01}$ | 0.72${}^{+0.01}_{-0.01}$ | K
K2-36 c | 7.80${}^{+2.30}_{-2.30}$ | 3.20${}^{+0.30}_{-0.30}$ | 788 | 0.79${}^{+0.01}_{-0.01}$ | 0.72${}^{+0.01}_{-0.01}$ | K
Kepler-80 b | 6.93${}^{+1.05}_{-0.70}$ | 2.67${}^{+0.10}_{-0.10}$ | 709 | 0.73${}^{+0.03}_{-0.03}$ | 0.68${}^{+0.02}_{-0.02}$ | K
Kepler-80 c | 6.74${}^{+1.23}_{-0.86}$ | 2.74${}^{+0.12}_{-0.10}$ | 641 | 0.73${}^{+0.03}_{-0.03}$ | 0.68${}^{+0.02}_{-0.02}$ | K
Kepler-80 d | 6.75${}^{+0.69}_{-0.51}$ | 1.53${}^{+0.09}_{-0.07}$ | 936 | 0.73${}^{+0.03}_{-0.03}$ | 0.68${}^{+0.02}_{-0.02}$ | K
Kepler-80 e | 4.13${}^{+0.81}_{-0.95}$ | 1.60${}^{+0.08}_{-0.07}$ | 815 | 0.73${}^{+0.03}_{-0.03}$ | 0.68${}^{+0.02}_{-0.02}$ | K
L 168-9 b | 4.60${}^{+0.56}_{-0.56}$ | 1.39${}^{+0.09}_{-0.09}$ | 965 | 0.62${}^{+0.03}_{-0.03}$ | 0.60${}^{+0.02}_{-0.02}$ | K
TOI-1235 b | 6.91${}^{+0.75}_{-0.85}$ | 1.74${}^{+0.09}_{-0.08}$ | 754 | 0.64${}^{+0.02}_{-0.02}$ | 0.63${}^{+0.01}_{-0.01}$ | K
TOI-178 b | 1.50${}^{+0.39}_{-0.44}$ | 1.15${}^{+0.07}_{-0.07}$ | 1040 | 0.65${}^{+0.03}_{-0.03}$ | 0.65${}^{+0.01}_{-0.01}$ | K
TOI-178 c | 4.77${}^{+0.55}_{-0.68}$ | 1.67${}^{+0.11}_{-0.10}$ | 873 | 0.65${}^{+0.03}_{-0.03}$ | 0.65${}^{+0.01}_{-0.01}$ | K
TOI-178 d | 3.01${}^{+0.80}_{-1.03}$ | 2.57${}^{+0.07}_{-0.08}$ | 690 | 0.65${}^{+0.03}_{-0.03}$ | 0.65${}^{+0.01}_{-0.01}$ | K
Table 4: Table 2, continued Name | Mass(M⊕) | Radius(R⊕) | Teq (K) | Host Mass(M⊙) | Host Radius(R⊙) | Host type
---|---|---|---|---|---|---
TOI-178 e | 3.86${}^{+1.25}_{-0.94}$ | 2.21${}^{+0.09}_{-0.09}$ | 600 | 0.65${}^{+0.03}_{-0.03}$ | 0.65${}^{+0.01}_{-0.01}$ | K
TOI-178 f | 7.72${}^{+1.67}_{-1.52}$ | 2.29${}^{+0.11}_{-0.11}$ | 521 | 0.65${}^{+0.03}_{-0.03}$ | 0.65${}^{+0.01}_{-0.01}$ | K
TOI-178 g | 3.94${}^{+1.31}_{-1.62}$ | 2.87${}^{+0.14}_{-0.13}$ | 470 | 0.65${}^{+0.03}_{-0.03}$ | 0.65${}^{+0.01}_{-0.01}$ | K
TOI-776 b | 4.00${}^{+0.90}_{-0.90}$ | 1.85${}^{+0.13}_{-0.13}$ | 514 | 0.54${}^{+0.03}_{-0.03}$ | 0.54${}^{+0.02}_{-0.02}$ | K
TOI-776 c | 5.30${}^{+1.80}_{-1.80}$ | 2.02${}^{+0.14}_{-0.14}$ | 415 | 0.54${}^{+0.03}_{-0.03}$ | 0.54${}^{+0.02}_{-0.02}$ | K
TOI-824 b | 18.47${}^{+1.84}_{-1.88}$ | 2.93${}^{+0.20}_{-0.19}$ | 1253 | 0.71${}^{+0.03}_{-0.03}$ | 0.69${}^{+0.03}_{-0.03}$ | K