# Supersymmetry and Convection-Diffusion-Reaction equations

Choon-Lin Ho  Department of Physics, Tamkang University, Tamsui 25137, Taiwan

###### Abstract

In this work we are concerned with generating solutions of a class of
Convection-Diffusion-Reaction equation from the solutions of another CDR
equation through the Darboux transformations. The method is elucidated by
cases with certain types of the reaction coefficients. We have also discussed
briefly supersymmetric pairs of Convection-Diffusion-Reaction equations
connected through similarity solutions.

##  I Introduction

The Convection-Diffusion-Reaction (CDR) equation is an important type of
second order differential equation which has found many important applications
in physics, chemistry, astrophysics, engineering, and biology. It is widely
employed to model stochastic phenomena that involve the change of
concentration/population of one or more substances/species distributed in
space under the influence of three processes: local reaction which modify the
concentration/population, diffusion which causes the substances/species to
spread in space, and convection/drifting under the influence of external
forces GK1 ; GK2 ; HM ; CP ; Mur . The well-known Fokker-Planck equation Ris ;
Sau  and the reaction-diffusion equation Mur  are special cases of the CDR
equations.

As with any equation in science, exact solutions of CDR equations are not easy
to obtain in general. As such, it is worthwhile to look for any method that
helps find exact solutions of CDR equations. In this work we are concerned
with generating solutions of a class of CDR equation from the solutions of
another CDR equation through the Darboux transformations.

The $(1+1)$-dimension CDR equation equation to be studied in this paper is
taken to have the following general form

$\frac{\partial P(x,t)}{\partial t}=-\frac{\partial}{\partial
x}\left(C(x,t)\,P(x,t)\right)+\frac{\partial}{\partial
x}\left(D(x,t)\frac{\partial}{\partial x}P(x,t)\right)+R(P,x,t),$ (1)

where $P(x,t)$ is the particle number function, $D(x,t),C(x,t)$ and $R(P,x,t)$
are the diffusion coefficient, the convection coefficient , and the reaction
term, respectively. The domains we shall consider in this paper are the real
line $x\in(-\infty,\infty)$, or the half lines $x\in[0,\infty)$.

In Ref. [8] we have considered finding solutions of a FPE from the solutions
of another FPE by means of the Darboux transformations Dar ; Cru ; MS . In
physics literature the time-independent Darboux transformation is usually call
the supersymmetric (SUSY) method SUSY . In Sect. 2 we extend this approach to
the CDR equations. In Sect. 3, we briefly discuss how SUSY pairs of CDR
equations can be determined through the similarity forms of the CDR equations
given in Ref. [13].

##  II CDR equations related by time-dependent Darboux transformation

In this section, we consider the class of CDR equation equations with
$D(x,t)=1$ and $R(P,x,t)=r(x,t)P(x,t)$, i.e.,

$\frac{\partial P(x,t)}{\partial t}=-\frac{\partial}{\partial x}\left(C(x,t)\
P(x,t)\right)+\frac{\partial^{2}}{\partial x^{2}}P(x,t)+r(x,t)P(x,t).$ (2)

The diffusion coefficient $D$ is set to unity in order to make connection with
supersymmetry. The real function $r(x,t)$ is the reaction coefficient.

As we have in mind two CDR equations to be connected by the Darboux
transformations, we shall denote all the defining quantities of the original
CDR with a subscript “0”, and those of the partner CDR with a subscript “1”.

Let the convection coefficient of the original CDR equation be written as a
space derivative of a function $W_{0}(x,t)$,
$C_{0}(x,t)=-2W_{0}^{\prime}(x,t)$, where the prime indicates space-
derivative. For a solution $P_{0}(x,t)$ of the CDR equation, we make the
ansatz

$P_{0}(x,t)=e^{-W_{0}(x,t)}\Psi_{0}(x,t).$ (3)

From Eq.(2) one finds that the function $\Psi_{0}(x,t)$ satisfies a
Schrödinger-like equation

$-\dot{\Psi}_{0}=-\Psi_{0}^{\prime\prime}+\left(W_{0}^{\prime
2}-W_{0}^{\prime\prime}-\dot{W}_{0}-r_{0}\right)\,\Psi_{0}.$ (4)

Here the dot represents time-derivative. It is seen that $W_{0}(x,t)$
determine the potential of this equation, and thus will be called the
prepotential.

We shall apply the Darboux transformation to (4). Consider the Schrödinger
equation

$-\dot{\Psi}_{0}=-\Psi_{0}^{\prime\prime}+V_{0}(x,t)\Psi_{0}.$ (5)

Suppose $\psi_{0}(x,t)$ is a solution of this equation. Then it can be checked
that, for any solution $\Psi_{0}(x,t)$ of (5), the Darboux transformed
functions

$\displaystyle V_{1}(x,t)$ $\displaystyle=$ $\displaystyle
V_{0}-2\,(\ln\psi_{0})^{\prime\prime},$ $\displaystyle\Psi_{1}(x,t)$
$\displaystyle=$
$\displaystyle\left(\partial_{x}-(\ln\psi_{0})^{\prime}\right)\Psi_{0},$ (6)

also satisfy the same form of Schrödinger equation MS ,

$-\dot{\Psi}_{1}=-\Psi_{1}^{\prime\prime}+V_{1}(x,t)\Psi_{1}.$ (7)

The set of equations in (6) is the (time-dependent) Darboux transformation.
The function $\psi_{0}(x,t)$ is called the auxiliary function of the Darboux
transformation.

Our strategy is as follows. Suppose for Eq. (4) we can determine the Darboux
transformed equations (6) and (7), with

$V_{1}(x,t)=W_{1}^{\prime 2}-W_{1}^{\prime\prime}-\dot{W}_{1}-r_{1}$ (8)

for some functions $W_{1}$ and $r_{1}$, then the SUSY partner CDR is defined
by the convection coefficient $C_{1}(x,t)=-2W_{1}^{\prime}(x,t)$ and the
reaction coefficient $r_{1}(x,t)$. And for a solution $P_{0}(x,t)$ in (3) of
the original CDR, a corresponding solution of the SUSY partner is

$\displaystyle P_{1}(x,t)$ $\displaystyle=$ $\displaystyle e^{-W_{1}}\Psi_{1}$
$\displaystyle=$ $\displaystyle
e^{-W_{1}}\left(\partial_{x}-(\ln\psi_{0})^{\prime}\right)\Psi_{0}$
$\displaystyle=$ $\displaystyle
e^{-W_{1}}\left(\partial_{x}-(\ln\psi_{0})^{\prime}\right)\left(e^{W_{0}}P_{0}(x,t)\right).$

Below we illustrate this construction for certain types of the reaction
coefficient $r(x,t)$.

###  II.1 Case A: $r_{0}(x,t)=-2W^{\prime\prime}_{0}(x,t)$

In this case Eq. (4) is

$-\dot{\Psi}_{0}=-\Psi_{0}^{\prime\prime}+\left(W_{0}^{\prime
2}+W_{0}^{\prime\prime}-\dot{W}_{0}\right)\,\Psi_{0}.$ (10)

The function $\psi_{0}=e^{W_{0}}$ is a solution of this equation, and thus can
be used as an auxiliary function for the Darboux transformation giving

$\displaystyle\Psi_{1}$ $\displaystyle=$
$\displaystyle(\partial_{x}-W_{0}^{\prime})\Psi_{0},$
$\displaystyle-\dot{\Psi}_{1}$ $\displaystyle=$
$\displaystyle-\Psi_{1}^{\prime\prime}+\left(W_{0}^{\prime
2}-W_{0}^{\prime\prime}-\dot{W_{0}}\right)\,\Psi_{1}.$ (11)

Note the change of sign in front of the $W_{0}^{\prime\prime}$ term.

Now if we can find a function $W_{1}(x,t)$ that solves the generalized Riccati
equation

$W_{0}^{\prime 2}-W_{0}^{\prime\prime}-{\dot{W}}_{0}=W_{1}^{\prime
2}+W_{1}^{\prime\prime}-{\dot{W}}_{1},$ (12)

then a SUSY partner CDR equation is obtained, with
$C_{1}(x,t)=-2W_{1}^{\prime}(x,t)$ and
$r_{1}(x,t)=-2W_{1}^{\prime\prime}(x,t)$. Given a solution
$P_{0}=e^{-W_{0}}\Psi_{0}$ of the original CDR equation, the corresponding
solution of the partner CDR equation is given by Eq. (II),

$P_{1}(x,t)=e^{-W_{1}}(\partial_{x}-W_{0}^{\prime})\left(e^{W_{0}}P_{0}(x,t)\right).$
(13)

In general it is not easy to solve the generalized Riccati equation (12) for
$W_{1}(x,t)$ given $W_{0}(x,t)$. However, solution of (12) is rather easy if
the prepotential $W_{0}(x,t)=W_{0}(x;a_{n}(t))$, where $a_{n}(t)$ represents
collectively a set of parameters in $W_{0}$, satisfies the condition of shape
invariance SUSY , namely,

$\displaystyle
W_{0}^{\prime}(x,;a_{n}(t))^{2}+W_{0}^{\prime\prime}(x;a_{n}(t))$
$\displaystyle=$ $\displaystyle
W_{0}^{\prime}(x;a_{n+1}(t))^{2}-W_{0}^{\prime\prime}(x;a_{n+1}(t))+R(a_{n}(t)).$

Here $a_{n+1}(t)$ is a function of $a_{n}(t)$, and $R(a_{n}(t))$ is an
$x$-independent shift function.

Shape invariance turns out to be a sufficient condition for the exact-
solvability of all the well-known one-dimensional analytically solvable
quantum models, for which the prepotentials and the shift functions are time-
independent. Interestingly, for the known solvable quantum systems, the
parameters $a_{n}$ and $a_{n+1}$ are related simply by a shift of constant.
For example, $a_{n+1}=a_{n}+1$ for the radial oscillator ($a_{n}$= angular
momentum), and $a_{n+1}=a_{n}$ (unchanged) for the simple harmonic oscillator
($a_{n}$= angular frequency) SUSY .

Using (II.1) we can rewrite (12) as

$\displaystyle
W_{0}^{\prime}(x,;a_{n}(t))^{2}-W_{0}^{\prime\prime}(x;a_{n}(t))-{\dot{W}}_{0}(x,;a_{n}(t))$
$\displaystyle=$ $\displaystyle
W_{0}^{\prime}(x;a_{n-1}(t))^{2}+W_{0}^{\prime\prime}(x;a_{n-1}(t))-{\dot{W}}_{0}(x,;a_{n}(t))-R(a_{n-1}(t))$
$\displaystyle=$ $\displaystyle W_{1}^{\prime
2}(x,t)+W_{1}^{\prime\prime}(x,t)-{\dot{W}}_{1}(x,t).$

From the last equality it is seen that a solution of (12) can be chosen to be

$W_{1}(x,t)=W_{0}(x;a_{n-1}(t))+\int^{t}R(a_{n-1}(t))\,dt,$ (16)

as long as we have

${\dot{W}}_{0}(x;a_{n-1}(t))={\dot{W}}_{0}(x;a_{n}(t)).$ (17)

As pointed out in Ho , if $a_{n}(t)$ and $a_{n-1}(t)$ differ only by a
constant, the condition (17) is true if $a_{n}(t)$ appears in $W_{0}(x,t)$
only as a multiplicative factor of a function of $x$. Of the ten solvable one-
dimensional quantum models, such condition is true for: the 1d oscillator, the
3d oscillator, the Morse, the Scarf I and II, and the Pöschl–Teller potential.

One can iterate this process to obtain a SUSY hierarchy of CDR equations
defined by

$W_{k}(x,t)=W_{0}(x;a_{n-k}(t))+\int^{t}\sum_{s=n-k}^{n-1}\,R(a_{s}(t))\,dt,,~{}~{}k=1,2,\ldots$
(18)

up to the lowest $a_{n}$ allowed by the model. Generalizing (13), the
solutions $P_{k}(x,t)$ of these CDR equations are related by

$\displaystyle
P_{k}(x,t)=e^{-W_{k}}\left(\partial_{x}-W_{k-1}^{\prime}\right)\left(e^{W_{k-1}}P_{k-1}(x,t)\right),~{}~{}k=1,2,\ldots$
(19)

As an example, let us take $W_{0}$ to be the prepotential of the 1d
oscillator, $W_{0}(x,t)=\gamma(t)x^{2}/4$. This prepotential has the
distinctive characteristics that the parameters $a_{n}(t)=\gamma(t)$ are all
the same, and $R(a_{n}(t))=\gamma(t)$. This means all the solutions
$P_{k}(x,t)$ in the SUSY hierarchy are the solutions of the same CDR equation
with $D^{(1)}=-\gamma(t)x$.

Following the discussion in Ho , for $\gamma(t)=-1/(t+C)$, we find

$P_{0}(x,t)=\sqrt{\frac{t+C}{4\pi t}}\,e^{-\frac{Cx^{2}}{4t(t+C)}},$ (20)

to be a solution of the original CDR equation. Then from Eq. (19), the next
two solutions in the hierarchy are

$P_{1}(x,t)\propto\frac{Cx}{t}\sqrt{\frac{t+C}{4\pi
t}}\,e^{-\frac{Cx^{2}}{4t(t+C)}},$ (21)

and

$P_{2}(x,t)\propto\frac{C(2Ct+2t^{2}-Cx^{2})}{t^{2}}\sqrt{\frac{t+C}{4\pi
t}}\,e^{-\frac{Cx^{2}}{4t(t+C)}}.$ (22)

###  II.2 Case B: $r(x,t)=-2\dot{W}_{0}(x,t)$

Eq. (4) in this case becomes

$-\dot{\Psi}_{0}=-\Psi_{0}^{\prime\prime}+\left(W_{0}^{\prime
2}-W_{0}^{\prime\prime}+\dot{W}_{0}\right)\,\Psi_{0}.$ (23)

Note that the last two terms in (23) differ in the signs from those in (10).

This time the function $\psi_{0}=e^{-W_{0}}$ is a solution of eq. (23), and
the Darboux transformations based on $\psi_{0}$ are

$\displaystyle\Psi_{1}$ $\displaystyle=$
$\displaystyle(\partial_{x}+W_{0}^{\prime})\Psi_{0},$
$\displaystyle-\dot{\Psi}_{1}$ $\displaystyle=$
$\displaystyle-\Psi_{1}^{\prime\prime}+\left(W_{0}^{\prime
2}+W_{0}^{\prime\prime}+\dot{W}\right)\,\Psi_{1}.$ (24)

The generalized Riccati equation one needs to solve for the new prepotential
$W_{1}(x,t)$ is

$W_{0}^{\prime 2}+W_{0}^{\prime\prime}+{\dot{W}}_{0}=W_{1}^{\prime
2}-W_{1}^{\prime\prime}+{\dot{W}}_{1},$ (25)

Once $W_{1}$ is found, the SUSY partner CDR equation is then defined by
$C_{1}(x,t)=-2W_{1}^{\prime}(x,t)$ and $r_{1}(x,t)=-2{\dot{W}}_{1}(x,t)$.
Given a solution $P_{0}=e^{-W_{0}}\Psi_{0}$ of the original CDR equation, the
corresponding solution of the partner CDR equation is given by

$P_{1}(x,t)=e^{-W_{1}}(\partial_{x}+W_{0}^{\prime})\left(e^{W_{0}}P_{0}(x,t)\right).$
(26)

As in Case A, if the prepotentials satisfy the condition of shape invariance,
then solution of (25) is easy. The generalized Riccati equation is

$\displaystyle
W_{0}^{\prime}(x,;a_{n}(t))^{2}+W_{0}^{\prime\prime}(x;a_{n}(t))+{\dot{W}}_{0}(x,;a_{n}(t))$
$\displaystyle=$ $\displaystyle
W_{0}^{\prime}(x;a_{n-1}(t))^{2}-W_{0}^{\prime\prime}(x;a_{n-1}(t))+{\dot{W}}_{0}(x,;a_{n}(t))+R(a_{n}(t))$
$\displaystyle=$ $\displaystyle W_{1}^{\prime
2}(x,t)-W_{1}^{\prime\prime}(x,t)+{\dot{W}}_{1}(x,t).$

A solution of (25) can be chosen to be

$W_{1}(x,t)=W_{0}(x;a_{n+1}(t))+\int^{t}R(a_{n}(t))\,dt,$ (28)

as long as we have

${\dot{W}}_{0}(x;a_{n+1}(t))={\dot{W}}_{0}(x;a_{n}(t)).$ (29)

Similarly, iterating this process one obtains a SUSY hierarchy of CDR
equations defined by

$W_{k}(x,t)=W_{0}(x;a_{n+k}(t))+\int^{t}\sum_{s=n}^{n+k-1}\,R(a_{s}(t))\,dt,,~{}~{}k=1,2,\ldots$
(30)

The solutions $P_{k}(x,t)$ of these CDR equations are related by

$\displaystyle
P_{k}(x,t)=e^{-W_{k}}\left(\partial_{x}+W_{k-1}^{\prime}\right)\left(e^{W_{k-1}}P_{k-1}(x,t)\right),~{}~{}k=1,2,\ldots$

Example of this case can be easily constructed following the description in
Case A, and so will be be discussed here.

###  II.3 Case C: $r(x,t)=2W_{0}^{\prime}S_{0}^{\prime}-S_{0}^{\prime
2}-S_{0}^{\prime\prime}-\dot{S}_{0}$

Consider the following way to link a FPE and a CDR equation.

The FPE with a drift coeeficient $C(x,t)=-2\omega_{0}^{\prime}(x,t)$ is

$\frac{\partial P_{F,0}(x,t)}{\partial t}=\frac{\partial}{\partial
x}\left(2\omega^{\prime}_{0}(x,t)P_{F,0}(x,t)\right)+\frac{\partial^{2}}{\partial
x^{2}}P_{F,0}(x,t).$ (31)

Setting $P_{F,0}(x,t)=e^{S_{0}(x,t)}P_{0}(x,t)$ , where $S_{0}(x,t)$ and
$P_{0}(x,t)$ are smooth functions of $x$ and $t$, we find $P_{0}(x,t)$
satisfies a CDR equation

$\frac{\partial P_{0}(x,t)}{\partial t}=\frac{\partial}{\partial
x}\left(2W_{0}(x,t)^{\prime}\ P_{0}(x,t)\right)+\frac{\partial^{2}}{\partial
x^{2}}P_{0}(x,t)+r_{0}(x.t)P_{0}(x,t),$ (32)

with

$W_{0}=\omega_{0}+S_{0},~{}~{}~{}~{}r_{0}=2W_{0}^{\prime}S_{0}^{\prime}-S_{0}^{\prime
2}-S_{0}^{\prime\prime}-\dot{S}_{0}.$ (33)

Thus every FPE in (31) corresponds to a CDR equation with $W_{0}(x,t)$ and
$r_{0}(x,t)$ given by (33) for a given $S_{0}$. By choosing different
functions $S_{0}$, one generates different CDR equations, all corresponding to
the same FPE (31). This correspondence implies that, if we find a solution of
the $P_{F,0}=e^{-\omega_{0}}\Psi_{0}$, with $\Psi_{0}$ satisfying Ho

$-\dot{\Psi}_{0}=-\Psi_{0}^{\prime\prime}+\left(\omega_{o}^{\prime
2}-\omega_{0}^{\prime\prime}-\dot{\omega}_{0}\right)\,\Psi_{0},$ (34)

then a solution of the corresponding CDR equation (32) is
$P_{0}=e^{-W_{0}}\Psi_{0}=e^{-S_{0}}P_{F,0}$.

It then follows that, if $P_{F,1}=e^{-\omega_{1}}\Psi_{1}$ is a solution of
the SUSY partner FPE, where the drift $\omega_{1}$ is determined by the time-
dependent Darboux transformation discussed in Ho , and
$\Psi_{1}=(\partial_{x}-\omega_{1}^{\prime})\Psi_{0}$, then for a given
$W_{1}(x,t)$,

$\displaystyle P_{1}$ $\displaystyle=$ $\displaystyle e^{-W_{1}}\Psi_{1}$ (35)
$\displaystyle=$ $\displaystyle
e^{-W_{1}}\left(\partial_{x}-\omega_{1}^{\prime}\right)\left(e^{W_{0}}P_{0}\right)$

is a solution of the corresponding SUSY partner CDR equation defined by the
drift and reaction functions

$S_{1}=W_{1}-\omega_{1},~{}~{}~{}~{}r_{1}=2W_{1}^{\prime}S_{1}^{\prime}-S_{1}^{\prime
2}-S_{1}^{\prime\prime}-\dot{S}_{1}.$ (36)

As an example, let us take

$\omega_{0}=\gamma(t)x^{2}/4,~{}~{}~{}\gamma(t)=-\frac{1}{t+C},~{}~{}C={\rm
real\ constant.}$ (37)

From Ref. [8], a solution of the FPE is

$P_{F,0}(x,t)=\frac{1}{\sqrt{4\pi t(t+C)}}e^{-\frac{Cx^{2}}{4t(t+C)}}.$ (38)

CDR equation corresponding to this FPE is defined by $W_{0},S_{0}$ and $r_{0}$
in (33). As mentioned before, there could be many possible choices of $W_{0}$
and $S_{0}$. For illustration purpose, let us take $W_{0}=0$, then

$S_{0}(x,t)=x^{2}/[4(t+C)],~{}~{}~{}r_{0}(x,t)=-\frac{1}{2(t+C)}.$ (39)

From $P_{0}=e^{-S_{0}}P_{F,0}$, we get a solution of this CDR equation to be

$P_{0}(x,t)=\frac{1}{\sqrt{4\pi t(t+C)}}e^{-\frac{x^{2}}{4t}}.$ (40)

Now we consider the SUSY partner of this CDR equation. A choice of
$\omega_{1}$ related to $\omega_{0}$ by the time-dependent Darboux
transformation is $\omega_{1}=ax+ax^{2}-\ln(t+C)/2$ ($a$ a real constant) Ho .
To define a SUSY partner with $\omega_{1}=W_{1}-S_{1}$, we take $W_{1}=ax$ and
$S_{1}=\ln(t+C)/2-at^{2}$. This leads to

$r_{1}(x,t)=-\dot{S}_{1}=-\frac{1}{2(t+C)}+a^{2},$ (41)

and by (35), a solution of the partner CDR equation,

$P_{1}(x,t)=\frac{x+2at}{4\sqrt{\pi(t+C)}t^{3/2}}\,e^{-\frac{1}{4t}(x^{2}+4axt)}.$
(42)

##  III SUSY pairs by similarity form

Let us now discuss briefly SUSY pair of CDR equations connected through
similarity solutions.

A CDR equation is said to possess scaling symmetry, if its functional form is
unchanged under the scale transformation

$x=\epsilon^{a}\,\bar{x}\;\;\;,\;\;\;t=\epsilon^{b}\,\bar{t},~{}~{}~{}\epsilon,a,b:{\rm\
real\ constants}.$ (43)

In this case, the functions $C,D,R$ and $P$ have the following scaling forms
in terms of the similarity variable $z$:

$\displaystyle P(x,t)$ $\displaystyle=$ $\displaystyle
t^{\mu}y(z),~{}~{}C(x,t)=t^{\gamma}\tau(z),$ $\displaystyle~{}~{}D(x,t)$
$\displaystyle=$ $\displaystyle
t^{\delta}\sigma(z),~{}~{}R(P,x,t)=t^{\rho}\rho(z).$ (44)

where

$z\equiv\frac{x}{t^{\alpha}},~{}~{}\mbox{where}~{}\alpha=\frac{a}{b}\;\;\;,\;b\neq
0\;,$ (45)

and $y(z),\tau(z),\sigma(z)$ and $\rho(z)$ are functions of $z$. Scaling
symmetry requires that the exponents are linked by HY

$\gamma=\alpha-1,~{}~{}\delta=2\alpha-1,~{}~{}\rho=\mu-1.$ (46)

Hence $\alpha$ and $\mu$ are the only two independent scaling exponents of the
CDR equation. With these, the CDR equation is reduced to an ordinary
differential equation HY

$\sigma
y^{\prime\prime}+(\sigma^{\prime}+\alpha\,z-\tau)\,y^{\prime}-(\tau^{\prime}+\mu)\,y+\rho=0.$
(47)

Here “prime” represents derivative with respect to $z$.

Let us now consider the special case where $\sigma(z)=1,\tau(z)=\alpha z$ and
$\rho(z)=-\Phi(z)y(z)$. Eq. (47) then reduces to

$-y^{\prime\prime}+(\Phi+\mu+\sigma)y=0.$ (48)

If we set $V-E\equiv\Phi+\mu+\sigma$, where $E$ is a real constant, then (48)
is recast into the form of a Schrödinger equation

$-y^{\prime\prime}+(V-E)y=0.$ (49)

It is now clear that this form permits us to derive a SUSY partner of the CDR
equation by the time-independent version of the Darboux transformation (6).
For any solutions $y_{0}(z)$ and $y(z)$ of (49), the functions

$\displaystyle{\widetilde{V}}(z)$ $\displaystyle=$ $\displaystyle V-2\,(\ln
y_{0})^{\prime\prime},$ $\displaystyle{\widetilde{y}}(z)$ $\displaystyle=$
$\displaystyle\left(\partial_{x}-(\ln y_{0})^{\prime}\right)y,$ (50)

also satisfy the same form of Schrödinger equation (49) with the same value of
$E$. Here $y_{0}(z)$ serves as the auxiliary function of the Darboux
transformation. The SUSY partner CDR equation is then defined by the functions
(recall $\sigma(z)=1,\tau(z)=\alpha z$)

$\displaystyle{\widetilde{P}}(x,t)=t^{\mu}{\widetilde{y}}(z),~{}~{}{\widetilde{C}}(P,x,t)=t^{\gamma}(\alpha
z),~{}~{}{\widetilde{D}}(P,x,t)=t^{\delta},$
$\displaystyle~{}~{}{\widetilde{R}}(P,x,t)=-t^{\rho}\,\widetilde{\Phi}(z)\,{\widetilde{y}}(z),~{}~{}~{}~{}~{}\widetilde{\Phi}(z)={\widetilde{V}}(z)-E-\mu-\alpha.$
(51)

It is not hard to work out examples by matching (49) with the solvable SUSY
quantum systems listed in [12].

##  IV Summary

In this work we have discussed a way to obtain solutions of a class of CDR
equation from those of another CDR equation, by means of the Darboux
transformation. Cases with certain types of the reaction coefficients are
presented. We have also discussed briefly SUSY pairs of CDR equation connected
through similarity solutions.

We have not considered the case with $x$-independent reaction coefficient,
i.e., $r(x,t)=r(t)$, as this case can be transformed into a FPE by a phase
transformation. In fact, letting

$P(x,t)=e^{\int^{t}r(t)dt}\,P_{F}(x,t),$ (52)

one finds $P_{F}(x,t)$ satisfies the FPE. Thus one can follow the discussion
in Ref. [8] to determine the SUSY partners of the CDR equation.

###### Acknowledgements.

The work is supported in part by the Ministry of Science and Technology (MoST)
of the Republic of China under Grant MOST 110-2112-M-032-011.

## References

  * (1) B. H. Gilding and R. Kersner, Travelling Waves in Nonlinear Diffusion Convection Reaction, Birkhäuser, Springer, 2004.
  * (2) B. H. Gilding and R. Kersner, The characterization of reaction-convection-diffusion processes by travelling waves, Journal of differential equations 124 (1996) 27.
  * (3) T. Harko and M. K. Mak, Exact travelling wave solutions of non-linear reaction-convection-diffusion equations: an Abel equation based approach, J. Math. Phys. 56 (2015) 111501.
  * (4) R. Cherniha and O. Pliukhin, New conditional symmetries and exact solutions of nonlinear reaction-diffusion-convection equations. I, II, and III. arXiv:math-ph/0612078, arXiv:0706.0814, arXiv:0902.2290.
  * (5) J.D. Murray, Mathematical Biology, 2nd Ed., Springer-Verlag, Berlin, 1993.
  * (6) H. Risken, The Fokker-Planck Equation, 2nd. ed., Springer-Verlag, Berlin, 1996.
  * (7) Sau Fa Kwok, Langevin and Fokker-Panck Equations and Their Generalizations, World Scientific, Singapore, 2018.
  * (8) C.-L. Ho, Time-dependent Darboux transformation and supersymmetric hierarchy of Fokker-Planck equations, Chin. J. Phys. 77 (2022) 1903.
  * (9) G. Darboux, Sur une proposition relative aux équation linéaires, Compte Rendu Acad. Sci. Paris 94 (1882) 1456.
  * (10) M. M. Crum, Associated Sturm-Liouville systems, Quart. J. Math. Oxford Ser. 6 (1955) 121.
  * (11) V. B. Matveev and M. A. Salle, Darboux transformations and Solitons, Springer, Berlin, 1991.
  * (12) F. Cooper, A. Khare, and U. Sukhatme, Supersymmetry and quantum mechanics, Phys. Rep. 251 (1995) 267.
  * (13) C.-L. Ho and C.-M. Yang, Convection-Diffusion-Reaction equation with similarity solutions, Chin. J. Phys. 59 (2019) 117.


11institutetext:  Department of Physics, Chungbuk National University,
Cheongju 28644, Republic of Korea, 22institutetext: Korea Astronomy and Space
Science Institute, Daejon 34055, Republic of Korea 33institutetext: Institute
of Natural and Mathematical Science, Massey University, Auckland 0745, New
Zealand 44institutetext: Center for Astrophysics $|$ Harvard & Smithsonian, 60
Garden St., Cambridge, MA 02138, USA 55institutetext: Department of Astronomy,
Tsinghua University, Beijing 100084, China 66institutetext: University of
Canterbury, Department of Physics and Astronomy, Private Bag 4800,
Christchurch 8020, New Zealand 77institutetext: Max-Planck-Institute for
Astronomy, Königstuhl 17, 69117 Heidelberg, Germany 88institutetext:
Department of Astronomy, Ohio State University, 140 W. 18th Ave., Columbus, OH
43210, USA 99institutetext: Korea University of Science and Technology, Korea,
(UST), 217 Gajeong-ro, Yuseong-gu, Daejeon, 34113, Republic of Korea
1010institutetext: Department of Particle Physics and Astrophysics, Weizmann
Institute of Science, Rehovot 76100, Israel 1111institutetext: School of Space
Research, Kyung Hee University, Yongin, Kyeonggi 17104, Republic of Korea
1212institutetext: Institute for Space-Earth Environmental Research, Nagoya
University, Nagoya 464-8601, Japan 1313institutetext: Code 667, NASA Goddard
Space Flight Center, Greenbelt, MD 20771, USA 1414institutetext: Department of
Astronomy, University of Maryland, College Park, MD 20742, USA
1515institutetext: Komaba Institute for Science, The University of Tokyo,
3-8-1 Komaba, Meguro, Tokyo 153-8902, Japan 1616institutetext: Instituto de
Astrofísica de Canarias, Vía Láctea s/n, E-38205 La Laguna, Tenerife, Spain
1717institutetext: Department of Earth and Space Science, Graduate School of
Science, Osaka University, Toyonaka, Osaka 560-0043, Japan 1818institutetext:
Department of Physics, The Catholic University of America, Washington, DC
20064, USA 1919institutetext: Department of Astronomy, Graduate School of
Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033,
Japan 2020institutetext: Sorbonne Université, CNRS, UMR 7095, Institut
d’Astrophysique de Paris, 98 bis bd Arago, 75014 Paris, France
2121institutetext: Department of Physics, University of Auckland, Private Bag
92019, Auckland, New Zealand 2222institutetext: University of Canterbury Mt.
John Observatory, P.O. Box 56, Lake Tekapo 8770, New Zealand

# KMT-2022-BLG-0475Lb and KMT-2022-BLG-1480Lb: Microlensing ice giants
detected via non-caustic-crossing channel

Cheongho Han 0101 Chung-Uk Lee 0202 Ian A. Bond 0303 Weicheng Zang
(Leading authors)
Sun-Ju Chung 040405050202 04 04 Michael D. Albrow 0606 Andrew Gould 07070808
Kyu-Ha Hwang 0202 Youn Kil Jung 02020909 Yoon-Hyun Ryu 0202 In-Gu Shin 0404
Yossi Shvartzvald 1010 Hongjing Yang 0505 Jennifer C. Yee 0404 Sang-Mok Cha
02021111 Doeon Kim 0101 Dong-Jin Kim 0202 Seung-Lee Kim 0202 Dong-Joo Lee 0202
Yongseok Lee 02021111 Byeong-Gon Park 0202 Richard W. Pogge
(The KMTNet collaboration)
Shude Mao 08080505 Wei Zhu
(Microlensing Astronomy Probe Collaboration)
Fumio Abe 05051212 Richard Barry 1313 David P. Bennett 13131414 Aparna
Bhattacharya 13131414 Hirosame Fujii 1212 Akihiko Fukui 15151616 Ryusei Hamada
1717 Yuki Hirao 1717 Stela Ishitani Silva 13131818 Yoshitaka Itow 1212 Rintaro
Kirikawa 1717 Iona Kondo 1717 Naoki Koshimoto 1919 Yutaka Matsubara 1212 Shota
Miyazaki 1717 Yasushi Muraki 1212 Greg Olmschenk 1313 Clément Ranc 2020
Nicholas J. Rattenbury 2121 Yuki Satoh 1717 Takahiro Sumi 1717 Daisuke Suzuki
1717 Taiga Toda 1717 Mio Tomoyoshi 1717 Paul J. Tristram 2222 Aikaterini
Vandorou 13131414 Hibiki Yama 1717 Kansuke Yamashita
(The MOA Collaboration)
1717

###### Abstract

Aims.  We investigate the microlensing data collected in the 2022 season from
the high-cadence microlensing surveys in order to find weak signals produced
by planetary companions to lenses.

Methods.  From these searches, we find that two lensing events
KMT-2022-BLG-0475 and KMT-2022-BLG-1480 exhibit weak short-term anomalies.
From the detailed modeling of the lensing light curves, we identify that the
anomalies are produced by planetary companions with a mass ratio to the
primary of $q\sim 1.8\times 10^{-4}$ for KMT-2022-BLG-0475L and a ratio $q\sim
4.3\times 10^{-4}$ for KMT-2022-BLG-1480L.

Results.  It is estimated that the host and planet masses and the projected
planet-host separation are $(M_{\rm h}/M_{\odot},M_{\rm p}/M_{\rm
U},a_{\perp}/{\rm
au})=(0.43^{+0.35}_{-0.23},1.73^{+1.42}_{-0.92},2.03^{+0.25}_{-0.38})$ for
KMT-2022-BLG-0475L, and
$(0.18^{+0.16}_{-0.09},1.82^{+1.60}_{-0.92},1.22^{+0.15}_{-0.14})$ for
KMT-2022-BLG-1480L, where $M_{\rm U}$ denotes the mass of Uranus. Both
planetary systems share common characteristics that the primaries of the
lenses are early-mid M dwarfs lying in the Galactic bulge and the companions
are ice giants lying beyond the snow lines of the planetary systems.

###### Key Words.:

planets and satellites: detection – gravitational lensing: micro

##  1 Introduction

The microlensing signal of a planet usually appears as a short-term anomaly on
the smooth and symmetric lensing light curve generated by the host of the
planet (Mao & Paczyński, 1991; Gould & Loeb, 1992). The signal arises when a
source approaches the perturbation region formed around the caustic induced by
the planet. Caustics represent the positions on the source plane at which the
lensing magnification of a point source is infinite, and thus source crossings
over the caustic result in strong signals with characteristic spike features.

The region of planetary deviations extends beyond caustics, and planetary
signals can be produced without the caustic crossing of a source. Planetary
signals produced via the non-caustic-crossing channel are weaker than those
generated by caustic crossings, and the strength of the signal diminishes as
the separation of the source from the caustic increases. Furthermore, these
signals do not exhibit characteristic features such as the spikes produced by
caustic crossings. Due to the combination of these weak and featureless
characteristics, the planetary signals generated via the non-caustic channel
are difficult to be noticed. If such signals are missed despite the fact that
they meet the criterion of detection, the statistical studies based on the
incomplete planet sample would lead to erroneous results on the demographics
of planets. In order to prevent this, the Korea Microlensing Telescope Network
(KMTNet: Kim et al., 2016) group has regularly conducted systematic inspection
of the data collected by the survey experiments in search of weak planetary
signals and has published detected planets in a series of papers (Zang et al.,
2021a, b, 2022, 2023; Hwang et al., 2022; Wang et al., 2022; Gould et al.,
2022b; Jung et al., 2022, 2023; Han et al., 2022a, b, c, d, e, 2023a; Shin et
al., 2023).

In this work, we present the analyses of the two microlensing events
KMT-2022-BLG-0475 and KMT-2022-BLG-1480, for which weak short-term anomalies
were found from the systematic investigation of the data collected from the
high-cadence microlensing surveys conducted in the 2022 season. We investigate
the nature of the anomalies by carrying out detailed analyses of the light
curves.

The organization of the paper for the presentation of the analyses and results
are as follows. In Sect. 2, we describe the observations and data used in the
analyses. In Sect. 3, we begin by explaining the parameters used in modeling
the lensing light curves, and we then detail the analyses conducted for the
individual events in the following subsections: Sect. 3.1 for
KMT-2022-BLG-0475 and in Sect. 3.2 for KMT-2022-BLG-1480. In Sect. 4, we
explain the procedure of constraining the source stars and estimating the
angular Einstein radii of the events. In Sect. 5, we explain the procedure of
the Bayesian analyses conducted to determine the physical lens parameters and
present the estimated parameters. We summarize results and conclude in Sect.
6.

##  2 Observations and data

We inspected the microlensing data of the KMTNet survey collected from the
observations conducted in the 2022 season. The total number of KMTNet lensing
events detected in the season is 2803. For the individual events, we first
fitted light curves with a single-lens single-source (1L1S) model and then
visually inspected residuals from the model. From this inspection, we found
that the lensing events KMT-2022-BLG-0475 and KMT-2022-BLG-1480 exhibited weak
short-term anomalies. We then cross-checked whether there were additional data
from the surveys conducted by other microlensing observation groups. We found
that both events were additionally observed by the Microlensing Observations
in Astrophysics (MOA: Bond et al., 2011) group, who referred to the events as
MOA-2022-BLG-185 and MOA-2022-BLG-383, respectively. For KMT-2022-BLG-1480,
there were extra data acquired from the survey observations conducted by the
Microlensing Astronomy Probe (MAP) collaboration during the period from 2021
August to 2022 September, whose primary purpose was to verify short-term
planetary signals found by the KMTNet survey. In the analyses of the events,
we used the combined data from the three survey experiments.111 The Optical
Gravitational Microlensing Experiment (OGLE: Udalski et al., 1994) is another
major microlensing survey, although the two events analyzed in this work were
not detected by the survey because the OGLE telescope was not operational in
the first half of the 2022 season. Besides these surveys dedicated to the
microlensing program, lensing events are detected from other surveys such as
the Zwicky Transient Facility (ZDF) survey (Medford et al., 2023) and the
Asteroid Terrestrial-impact Last Alert System (ATLAS) survey (Tonry et al.,
2018), or observed using space-based instrument such as the Gaia survey
(Kruszyńska et al., 2022; Luberto et al., 2022) and Hubble Space Telescope
(Sahu et al., 2022).

The observations of the events were carried out using the telescopes that are
operated by the individual survey groups. The three identical telescopes used
by the KMTNet group have a 1.6 m aperture equipped with a camera yielding 4
deg2 field of view, and they are distributed in the three continents of the
Southern Hemisphere for the continuous coverage of lensing events. The sites
of the individual KMTNet telescopes are the Cerro Tololo Interamerican
Observatory in Chile (KMTC), the South African Astronomical Observatory in
South Africa (KMTS), and the Siding Spring Observatory in Australia (KMTA).
The MOA group utilizes the 1.3 m telescope at the Mt. John Observatory in New
Zealand, and the camera mounted on the telescope has a 1.2 deg2 field of view.
The MAP collaboration uses the 3.6 m Canada-France-Hawaii Telescope (CFHT) in
Hawaii.

Observations by the KMTNet, MOA, MAP groups were done mainly in the $I$,
customized MOA-$R$, and SDSS-$i$ bands, respectively. A fraction of images
taken by the KMTNet and MOA surveys were acquired in the $V$ band for the
measurement of the source colors of the events. Reduction of data and
photometry of source stars were done using the pipelines of the individual
survey groups. For the data used in the analyses, we readjusted the error bars
estimated from the automated pipelines so that the error bars are consistent
with the scatter of data and $\chi^{2}$ per degree of freedom for each data
set becomes unity following the method described in Yee et al. (2012).

##  3 Light curve analyses

The analyses of the lensing events were carried out by searching for lensing
solutions specified by the sets of lensing parameters that best describe the
observed light curves. The lensing parameters vary depending on the
interpretation of an event. It is known that a short-term anomaly can be
produced by two channels, in which the first is a binary-lens single-source
(2L1S) channel with a low-mass companion to the lens, and the other is a
single-lens binary-source (1L2S) channel with a faint companion to the source
(Gaudi, 1998).

The basic lensing parameters used in common for the 2L1S and 1L2S models are
$(t_{0},u_{0},t_{\rm E},\rho)$. The first two parameters represent the time of
the closest source approach to the lens and the lens-source separation (impact
parameter) scaled to the angular Einstein radius $\theta_{\rm E}$ at $t_{0}$,
respectively. The third parameter denotes the event time scale, that is
defined as the time for the source to transit $\theta_{\rm E}$. The last
parameter is the normalized source radius, that is defined as the ratio of the
angular source radius $\theta_{*}$ to $\theta_{\rm E}$. The normalized source
radius is needed in modeling to describe the deformation of a lensing light
curve caused by finite-source effects (Bennett & Rhie, 1996).

In addition to the basic parameters, the 2L1S and 1L2S models require
additional parameters for the description of the extra lens and source
components. The extra parameters for the 2L1S model are $(s,q,\alpha)$, where
the first two parameters denote the projected separation scaled to
$\theta_{\rm E}$ and the mass ratio between the lens components $M_{1}$ and
$M_{2}$, and the last parameter denotes the source trajectory angle defined as
the angle between the direction of the lens-source relative proper motion
$\mu$ and the $M_{1}$–$M_{2}$ binary axis. The extra parameters for the 1L2S
model include $(t_{0,2},u_{0,2},\rho_{2},q_{F})$, which refer to the closest
approach time, impact parameter, the normalized radius of the source companion
$S_{2}$, and the flux ratio between the source companion and primary
($S_{1}$), respectively. See Table 2 of Han et al. (2023b) for the summary of
lensing parameters that are required to be included under various
interpretations of the lens-system configuration.

For the individual events, we check whether higher-order effects improve the
fits by conducting additional modeling. The considered higher-order effects
are the microlens-parallax effect (Gould, 1992) and the lens-orbital effects
(Albrow et al., 2000), which are caused by the orbital motion of Earth and
lens, respectively. For the consideration of the microlens-lens parallax
effects, we included two extra parameters $(\pi_{{\rm E},N},\pi_{{\rm E},E})$,
which denote the north and east components of the microlens-parallax vector

$\mbox{\boldmath$\pi$}_{\rm E}=\left({\pi_{\rm rel}\over\theta_{\rm
E}}\right)\left({\mbox{\boldmath$\mu$}\over\mu}\right),$ (1)

respectively. Here $\pi_{\rm rel}=\pi_{\rm L}-\pi_{\rm S}={\rm au}(1/D_{\rm
L}-1/D_{\rm S})$ denotes the relative lens-source parallax, while $D_{\rm L}$
and $D_{\rm S}$ denote the distances to the lens and source, respectively. The
lens-orbital effects were incorporated into modeling by including two extra
parameters $(ds/dt,d\alpha/dt)$, which represent the annual change rates of
the binary-lens separation and the source trajectory angle, respectively.

We searched for the solutions of the lensing parameters as follows. For the
2L1S modeling, we found the binary-lens parameters $s$ and $q$ using a grid
approach with multiple seed values of $\alpha$, and the other parameters were
found using a downhill approach based on the MCMC logic. For the local
solutions identified from the $\Delta\chi^{2}$ map on the $s$–$q$ parameter
plane, we then refined the individual solutions by allowing all parameters to
vary. We adopted the grid approach to search for the binary parameters because
it was known that the change of the lensing magnification is discontinuous due
to the formation of caustics, and this makes it difficult to find a solution
using a downhill approach with initial parameters $(s,q)$ lying away from the
solution. In contrast, the magnification of a 1L2S event smoothly changes with
the variation of the lensing parameters, and thus we searched for the 1L2S
parameters using a downhill approach with initial values set by considering
the magnitude and location of the anomaly features. In the following
subsections, we describe the detailed procedure of modeling and present
results found from the analyses of the individual events.

Figure 1:  Light curve of KMT-2022-BLG-0475. The lower panel shows the whole
view and the upper panel shows the enlarged view of the region around the
anomaly. The arrow in the lower panel indicates the approximate time of the
anomaly, $t_{\rm anom}$. The curve drawn over the data points is the model
curve of the 1L1S solution. The KMTC data set is used for the reference to
align the other data sets.

###  3.1 KMT-2022-BLG-0475

The source of the lensing event KMT-2022-BLG-0475 lies at the equatorial
coordinates $({\rm RA},{\rm DEC})_{\rm J2000}=$ (18:05:20.56, -27:02:15.61),
which correspond to the Galactic coordinates
$(l,b)=(3^{\circ}\hskip-2.0pt.835,-2^{\circ}\hskip-2.0pt.804)$. The KMTNet
group first discovered the event on 2022 April 19, which corresponds to the
abridged Heliocentric Julian date ${\rm HJD}^{\prime}\equiv{\rm
HJD}-2450000=9688$, when the source was brighter than the baseline magnitude
$I_{\rm base}=18.78$ by $\Delta I\sim 0.6$ mag. Five days after the KMTNet
discovery, the event was independently found by the MOA group, who designated
the event as MOA-2022-BLG-185. Hereafter we use the KMTNet event notation
following the convention of using the event ID reference of the first
discovery survey. The event was in the overlapping region of the two KMTNet
prime fields BLG03 and BLG43, toward which observations were conducted with a
0.5 hr cadence for each field and $\sim 0.25$ hr in combination. The MOA
observations were done with a similar cadence.

Figure 1 shows the light curve of KMT-2022-BLG-0475 constructed from the
combination of the KMTNet and MOA data. The anomaly occurred at around $t_{\rm
anom}=9698.4$, which was $\sim 0.85$ day before time of the peak. The zoom-in
view of the region around the anomaly is shown the upper panel of Figure 1.
The anomaly lasted for about 0.5 day, and the beginning part was covered by
the MOA data while the second half of the anomaly was covered by the KMTS
data. There is a gap between the MOA and KMTS data during $9698.30\leq{\rm
HJD}^{\prime}\leq 9698.42$, and this gap corresponds to the night time at the
KMTA site, which was clouded out except for the very beginning of the evening.

Figure 2 shows the best-fit 2L1S and 1L2S models in the region around the
anomaly. From the 2L1S modeling, we identified a pair of 2L1S solutions
resulting from the close–wide degeneracy. In Table 1, we present the lensing
parameters of the two 2L1S and the 1L2S solutions together with the $\chi^{2}$
values of the fits and degrees of freedom (dof). It was found that the
severity of the degeneracy between the close and wide 2L1S solutions is
moderate, with the close solution being preferred over the wide solution by
$\Delta\chi^{2}=8.4$. For the best-fit solution, that is, the close 2L1S
solution, we also list the flux values of the source $f_{s}$ and blend
$f_{b}$, where the flux values are approximately scaled by the relation
$I=18-2.5\log f$.

Figure 2:  Zoom-in view around the anomaly in the lensing light curve of
KMT-2022-BLG-0475. The lower three panels show the residuals from the finite-
and point-source close 2L1S models and 1L2S model.  Table 1: Model parameters
of KMT-2022-BLG-0475

Parameter | 2L1S | 1L2S
---|---|---
| Close | Wide |
$\chi^{2}/{\rm dof}$ | 7903.4/7918 | 7911.8/7918 | 7930.7/7918
$t_{0}$ (${\rm HJD}^{\prime}$) | $9699.258\pm 0.002$ | $9699.259\pm 0.002$ | $9699.276\pm 0.003$
$u_{0}$ | $0.035\pm 0.001$ | $0.035\pm 0.001$ | $0.035\pm 0.001$
$t_{\rm E}$ (days) | $16.84\pm 0.13$ | $16.77\pm 0.13$ | $17.05\pm 0.13$
$s$ | $0.940\pm 0.011$ | $1.135\pm 0.012$ | –
$q$ (10-4) | $1.76\pm 0.26$ | $1.77\pm 0.25$ | –
$\alpha$ (rad) | $5.692\pm 0.004$ | $5.691\pm 0.005$ | –
$\rho$ (10-3) | $4.06\pm 0.98$ | $3.62\pm 1.09$ | –
$t_{0,2}$ (${\rm HJD}^{\prime}$) | – | – | $9698.425\pm 0.012$
$u_{0,2}$ (10-2) | – | – | $-0.016\pm 0.104$
$\rho_{2}$ (10-3) | – | – | $3.51\pm 0.69$
$q_{F}$ (10-2) | – | – | $0.40\pm 0.10$
$f_{s}$ | $0.4054\pm 0.0003$ |  |
$f_{b}$ | $-0.0215\pm 0.0010$ |  |
222 ${\rm HJD}^{\prime}={\rm HJD}-2450000$.

We find that the anomaly in the lensing light curve of KMT-2022-BLG-0475 is
best explained by a planetary 2L1S model. The planet parameters are
$(s,q)_{\rm close}\sim(0.94,1.76\times 10^{-4})$ for the close solution and
$(s,q)_{\rm wide}\sim(1.14,1.77\times 10^{-4})$ for the wide solution. The
estimated planet-to-host mass ratio is an order of magnitude smaller than the
ratio between the Jupiter and the sun, $q\sim 10^{-3}$, indicating that the
planet has a mass that is substantially smaller than a typical gas giant.
Although the 1L2S model approximately describes the anomaly, it leaves
residuals of 0.03 mag level in the beginning and ending parts of the anomaly,
resulting in a poorer fit than the 2L1S model by $\Delta\chi^{2}=27.3$. It was
found that the microlens-parallax parameters could not be measured because of
the short time scale, $t_{\rm E}\sim 16.8$ days, of the event.

Figure 3:  Lens-system configurations of the close (upper panel) and wide
(lower panel) 2L1S solutions of KMT-2022-BLG-0475. In each panel, the red
cuspy figures are caustics and the line with an arrow represents the source
trajectory. The whole view of the lens system is shown in the inset, in which
the small filled dots indicate the positions of the lens components and the
solid circle represents the Einstein ring. The grey curves surrounding the
caustic represent equi-magnification contours.

In the upper and lower panels of Figure 3, we present the lens-system
configurations of the close and wide 2L1S solutions, respectively. In each
panel, the inset shows the whole view of the lens system, and the main panel
shows the enlarged view around the central caustic. A planetary companion
induces two sets of caustics, with the ”central” caustic indicating the one
lying close to the primary lens, while the other caustic, lying away from the
primary, is referred to as the as the ”planetary” caustic. The configuration
shows that the anomaly was produced by the passage of the source through the
deviation region formed in front of the protruding cusp of the central
caustic. We found that finite-source effects were detected despite the fact
that the source did not cross the caustic. In order to show the deformation of
the anomaly pattern by finite-source effects, we plot the light curve and
residual from the point-source model that has the same lensing parameters as
those of the finite-source model except for $\rho$, in Figure 2.

It is known that the planet separations of the pair of degenerate solutions
resulting from a close–wide degeneracy follow the relation $\sqrt{s_{\rm
close}\times s_{\rm wide}}=1.0$ (Griest & Safizadeh, 1998). For the close and
wide solutions of KMT-2022-BLG-0475, this value is $\sqrt{s_{\rm close}\times
s_{\rm wide}}=1.032$, which deviates from unity with a fractional discrepancy
$(\sqrt{s_{\rm close}\times s_{\rm wide}}-1.0)/1.0\sim 3.2\%$. We find that
the relation between the two planet separations is better described by the
Hwang et al. (2022) relation

$s^{\dagger}_{\pm}=\sqrt{s_{\rm in}\times s_{\rm out}}={\sqrt{u_{\rm
anom}^{2}+4}\pm u_{\rm anom}\over 2},$ (2)

which was introduced to explain the relation between the planet separations
$s_{\rm in}$ and $s_{\rm out}$ of the two solutions that are subject to the
inner–outer degeneracy (Gaudi & Gould, 1997). Here $u_{\rm
anom}^{2}=\tau^{2}_{\rm anom}+u_{0}^{2}$, $\tau_{\rm anom}=(t_{\rm
anom}-t_{0})/t_{\rm E}$, $t_{\rm anom}$ is the time of the anomaly, and the
sign in the left and right sides of Eq. (2) is ”$+$” for a major image
perturbation and ”$-$” for a minor-image perturbation. The terms ”inner” and
”outer” refer to the cases in which the source passes the inner and outer
sides of the planetary caustic, respectively. In the case of KMT-2022-BLG-0475
(and major-image perturbations in general), the close and wide solutions
correspond to the outer and inner solutions, respectively. From the measured
planet separations of $s_{\rm in}=s_{\rm wide}=1.135$ and $s_{\rm out}=s_{\rm
close}=0.940$, we found that

$s^{\dagger}=(s_{\rm in}\times s_{\rm out})^{1/2}=1.033.$ (3)

From the lensing parameters $(t_{0},t_{\rm anom},t_{\rm E},u_{0})=$(9699.25,
9698.40, 16.8, 0.035), we found that $\tau=(t_{\rm anom}-t_{0})/t_{\rm
E}=0.050$, $u_{\rm anom}=0.061$, and

$s^{\dagger}={\sqrt{u_{\rm anom}^{2}+4}+u_{\rm anom}\over 2}=1.038.$ (4)

Then, the fraction deviation of the $s^{\dagger}$ values estimated from Eqs.
(3) and (4) is $\Delta s^{\dagger}/s^{\dagger}=0.5\%$, which is 6.4 times
smaller than the 3.2% fractional discrepancy of the $\sqrt{s_{\rm close}\times
s_{\rm wide}}=1$ relation. Although the $\sqrt{s_{\rm close}\times s_{\rm
wide}}=1.0$ relation is approximately valid in the case of KMT-2022-BLG-0475,
the deviation from the relation can be substantial especially when the
planetary separation is very close to unity, and thus the relation in Eq. (2)
helps to identify correct degenerate solutions.

###  3.2 KMT-2022-BLG-1480

The lensing event KMT-2022-BLG-1480 occurred on a source lying at $({\rm
RA},{\rm DEC})_{\rm J2000}=$ (17:58:54.96, -29:28:23.99), which correspond to
$(l,b)=(1^{\circ}\hskip-2.0pt.015,-2^{\circ}\hskip-2.0pt.771)$. The event was
first found by the KMTNet group on 2022 July 11 (${\rm HJD}^{\prime}=9771$),
when the source was brighter than the baseline magnitude, $I_{\rm
base}=18.11$, by $\Delta I\sim 0.65$ mag. The source was in the KMTNet prime
field BLG02, toward which observations were conducted with a 0.5 hr cadence.
This field overlaps with the BLG42 field in most region, but the source was in
the offset region that was not covered by the BLG42 field. The event was also
observed by the MOA and MAP survey groups, who observed event with a 0.2 hr
cadence and a 0.5–1.0 day cadence, respectively.

Figure 4:  Light curve of the lensing event KMT-2022-BLG-1480. Notations are
same as those in Fig. 1.

In Figure 4, we present the light curve of KMT-2022-BLG-1480. We found that a
weak anomaly occurred about 3 days after the peak centered at $t_{\rm
anom}\sim 9793.2$. The anomaly is characterized by a negative deviation in
most part of the anomaly and a slight positive deviation in the beginning part
centered at ${\rm HJD}^{\prime}\sim 9792.2$. The anomaly, that lasted about
2.7 days, was covered by multiple data sets from KMTS, KMTA, and CFHT. The sky
at the KMTC site was clouded out for two consecutive nights from July 30 to
August 1 ($9791\leq{\rm HJD}^{\prime}\leq 9793$), and thus the anomaly was not
covered by the KMTC data.

In Table 2, we present the best-fit lensing parameters of the 2L1S solution.
We did not conduct 1L2S modeling because a negative deviation cannot be
explained with a 1L2S interpretation. We found a pair of 2L1S solutions, in
which one solution has binary parameters $(s,q)\sim(0.83,4.9\times 10^{-4})$
and the other solution has parameters $(s,q)\sim(1.03,4.7\times 10^{-4})$.
Similar to the case of KMT-2022-BLG-0475L, the estimated mass ratio of order
$10^{-4}$ is much smaller than the Jupiter/sun mass ratio. As we discuss
below, the similarity between the model curves of the two 2L1S solutions is
caused by the inner–outer degeneracy, and thus we refer to the solutions as
”inner” and ”outer” solutions, respectively. From the comparison of the inner
and outer solutions obtained under the assumption of a rectilinear relative
lens-source motion, it was found that the outer model yields a substantially
better fit than the fit of the inner model by $\Delta\chi^{2}=63.9$,
indicating that the degeneracy was resolved. In Figure 5, we present the model
curves of the two solutions in the region around the anomaly. From the
comparison of the models, it is found that the fit of the outer solution is
better than the inner solution in the region around ${\rm HJD}^{\prime}\sim
9792$, at which the anomaly exhibits slight positive deviations from the 1L1S
model.

Table 2: Model parameters of KMT-2022-BLG-1480 Parameter | Inner | Outer
---|---|---
| Standard | Standard |  Higher-order ($u_{0}>0$) |  Higher-order ($u_{0}<0$)
$\chi^{2}/{\rm dof}$ | 5724.8/5678 | 5660.9/5678 | 5654.0/5674 | 5655.3/5674
$t_{0}$ (HJD′) | $9790.133\pm 0.004$ | $9790.132\pm 0.004$ | $9790.134\pm 0.004$ | $9790.138\pm 0.004$
$u_{0}$ | $0.067\pm 0.001$ | $0.069\pm 0.001$ | $0.069\pm 0.001$ | $-0.069\pm 0.001$
$t_{\rm E}$ (days) | $26.37\pm 0.15$ | $26.09\pm 0.15$ | $26.18\pm 0.16$ | $26.08\pm 0.16$
$s$ | $0.826\pm 0.004$ | $1.030\pm 0.018$ | $1.017\pm 0.014$ | $1.011\pm 0.015$
$q$ ($10^{-4}$) | $4.87\pm 0.36$ | $4.68\pm 0.783$ | $4.30\pm 0.69$ | $4.18\pm 0.70$
$\alpha$ (rad) | $0.516\pm 0.004$ | $0.515\pm 0.004$ | $0.525\pm 0.006$ | $-0.532\pm 0.009$
$\rho$ ($10^{-3}$) | $<6$ | $14.18\pm 4.02$ | $14.68\pm 2.51$ | $14.83\pm 2.46$
$\pi_{{\rm E},N}$ | – | – | $0.42\pm 0.29$ | $0.27\pm 0.31$
$\pi_{{\rm E},E}$ | – | – | $-0.02\pm 0.05$ | $0.04\pm 0.05$
$ds/dt$ (yr-1) | – | – | $0.60\pm 0.40$ | $1.14\pm 0.44$
$d\alpha/dt$ (yr-1) | – | – | $-0.57\pm 0.49$ | $1.59\pm 0.73$
$f_{s}$ |  | $0.9222\pm 0.0009$ |  |
$f_{b}$ |  | $0.0921\pm 0.0016$ |  |
Figure 5:  Enlarged view around the anomaly in the lensing light curve of
KMT-2022-BLG-1480. The lower three panels show the residuals form the inner
2L1S, finite- and point-source outer 2L1s models.

Figure 6 shows the lens-system configurations of the inner and outer 2L1S
solutions. Although the fit is worse, we present the configuration of the
inner solution in order to find the origin of the fit difference between the
two solutions. The configuration shows that the outer solution results in a
resonant caustic, in which the central and planetary caustics merge and form a
single caustic, while the central and planetary caustics are detached in the
case of the inner solution. According to the interpretations of both
solutions, the source passed the back-end side of central caustic without
caustic crossings. The configuration of the outer solution results in strong
cusps lying on the back-end side, and this caustic feature explains the slight
positive deviation appearing in the beginning part of the anomaly around ${\rm
HJD}^{\prime}\sim 9792.2$. Similar to the case of KMT-2022-BLG-0475, finite-
source effects were detected although the source did not cross the caustic. We
plot the point-source model in Figure 5 for the comparison with the finite-
source model.

Figure 6:  Lens-system configurations for the inner and outer 2L1S solutions
of KMT-2022-BLG-1480. Notations are same as those in Fig. 3.

We find that the relation in Eq. (2) is also applicable to the two local
solutions of KMT-2022-BLG-1480. With $(s_{\rm in},s_{\rm
out})\sim(0.82,1.03)$, the fractional deviation of the value $\sqrt{s_{\rm
in}\times s_{\rm out}}$ from unity is $(\sqrt{s_{\rm in}\times s_{\rm
out}}-1.0)/1.0\sim 8\%$. On the other hand, the fractional difference between
$s^{\dagger}=\sqrt{s_{\rm in}\times s_{\rm out}}=0.919$ and
$s^{\dagger}=[(u_{\rm anom}^{2}+4)^{1/2}-u_{\rm anom}]/2=0.934$ is $\Delta
s^{\dagger}/s^{\dagger}=1.6\%$, which is 5 times smaller than that of the
$\sqrt{s_{\rm close}\times s_{\rm wide}}=1$ relation. This also indicates that
the two local solutions result from the inner–outer degeneracy rather than the
close–wide degeneracy. We also checked the Hwang et al. (2021) relation
between the planet-to-host mass ratio and the lensing parameters for the ”dip-
type” anomalies,

$q=\left({\Delta t_{\rm dip}\over 4t_{\rm
E}}\right){s\over|u_{0}|}|\sin\alpha|^{3},$ (5)

where $\Delta t_{\rm dip}\sim 1.9$ day is the duration of the dip in the
anomaly. With the lensing parameters $(t_{\rm
E},u_{0},s,\alpha)=(26,0.069,1.03,59^{\circ})$, we found that the mass ratio
analytically estimated from Eq. (5) is $q\sim 5.9\times 10^{-4}$, which is
close to the value $4.6\times 10^{-4}$ found from the modeling.

Figure 7:  Maps of $\Delta\chi^{2}$ on the $(\pi_{{\rm E},E},\pi_{{\rm E},N})$ parameter plane for the higher-order $u_{0}>0$ and $u_{0}<0$ solutions of KMT-2022-BLG-1480. Color coding is set to represent points with $\leq 1\sigma$ (red), $\leq 2\sigma$ (yellow), $\leq 3\sigma$ (green), $\leq 4\sigma$ (cyan), and $\leq 5\sigma$ (blue).  Table 3: Source properties, Einstein radii, and lens-source proper motions Quantity | KMT-2022-BLG-0475 | KMT-2022-BLG-1480
---|---|---
$(V-I,I)_{\rm S}$ | $(2.016\pm 0.005,18.980\pm 0.008)$ | $(2.498\pm 0.005,18.088\pm 0.006)$
$(V-I,I)_{\rm RGC}$ | $(2.108,15.747)$ | $(2.732,16.199)$
$(V-I,I)_{\rm 0,RGC}$ | $(1.060,14.332)$ | $(1.060,14.396)$
$(V-I,I)_{\rm 0,S}$ | $(0.968\pm 0.005,17.566\pm 0.008)$ | $(0.826\pm 0.005,16.284\pm 0.006)$
$\theta_{*}$ ($\mu$as) | $1.29\pm 0.09$ | $1.98\pm 0.14$
$\theta_{\rm E}$ (mas) | $0.32\pm 0.02$ | $0.13\pm 0.01$
$\mu$ (mas/yr) | $6.92\pm 0.49$ | $1.88\pm 0.13$

We check whether the microlens-parallax vector $\mbox{\boldmath$\pi$}_{\rm
E}=(\pi_{{\rm E},N},\pi_{{\rm E},E})$ can be measured by conducting extra
modeling considering higher-order effects. We find that the inclusion of the
higher-order effects improves the fit by $\Delta\chi^{2}=6.9$ with respect to
the model obtained under the rectilinear lens-source motion (standard model).
In Table 2, we list the lensing parameters of the pair of higher-order models
with $u_{0}>0$ and $u_{0}<0$, which result from the mirror symmetry of the
source trajectory with respect to the binary-lens axis (Skowron et al., 2011).
The $\Delta\chi^{2}$ maps of the models on the $(\pi_{{\rm E},E},\pi_{{\rm
E},N})$ parameter plane obtained from the higher-order modeling are shown in
Figure 7. It is found that the maps of the solutions results in a similar
pattern of a classical 1-dimensional parallax ellipse, in which the east
component $\pi_{{\rm E},E}$ is well constrained and the north component
$\pi_{{\rm E},N}$ has a fairly big uncertainty. Gould et al. (1994) pointed
out that the constraints of the 1-dimensional parallax on the physical lens
parameters are significant, and Han et al. (2016) indicated that the parallax
constraint should be incorporated in the Bayesian analysis to estimate the
physical lens parameters. We describe the detailed procedure of imposing the
parallax constraint in the second paragraph of Sect. 5. While the parallax
parameters $(\pi_{{\rm E},N},\pi_{{\rm E},E})$ are constrained from the
overall pattern of the light curve, the orbital parameters
$(ds/dt,d\alpha/dt)$ are constrained from the anomaly induced by the lens
companion. For KMT-2022-BLG-1480, the orbital parameters are poorly
constrained because the duration of the planet-induced anomaly is very short.

Figure 8:  Source locations of the lensing events KMT-2022-BLG-0475 (left
panel) and KMT-2022-BLG-1480 (right panel) with respect to the positions of
the centroids red giant clump (RGC) in the instrumental color-magnitude
diagrams of stars lying around the source stars of the individual events. For
KMT-2022-BLG-1480, the position of the blend is additionally marked.

##  4 Source stars and angular Einstein radii

In this section, we constrain the source stars of the events to estimate the
angular Einstein radii. Despite the non-caustic-crossing nature of the
planetary signals, finite-source effects were detected for both events, and
thus the normalized source radii were measured. With the measured value of
$\rho$, we estimated the angular Einstein radius using the relation

$\theta_{\rm E}={\theta_{*}\over\rho},$ (6)

where the angular radius of the source was deduced from the reddening- and
extinction-corrected (de-reddened) color and magnitude.

The left and right panels of Figure 8 show the source locations in the
instrumental color-magnitude diagrams (CMDs) of stars lying around the source
stars of KMT-2022-BLG-0475 and KMT-2022-BLG-1480, respectively. For each
event, the instrumental source color and magnitude, $(V-I,I)_{\rm S}$, were
determined by estimating the flux values of the source $f_{s}$ and blend
$f_{b}$ from the linear fit to the relation $F_{\rm obs}=A(t)f_{s}+f_{b}$,
where the lensing magnification is obtained from the model. We are able to
constrain the blend for KMT-2022-BLG-1480 and mark its location on the CMD,
but the blend flux of KMT-2022-BLG-0475 resulted in a slightly negative value,
making it difficult to constrain the blend position. By applying the method of
Yoo et al. (2004), we then estimated the de-reddened source color and
magnitude, $(V-I,I)_{0,{\rm S}}$, using the centroid of the red giant clump
(RGC), for which its de-reddened color and magnitude $(V-I,I)_{0,{\rm RGC}}$
were known (Bensby et al., 2013; Nataf et al., 2013), as a reference, that is,

$(V-I,I)_{\rm 0,S}=(V-I,I)_{\rm 0,RGC}+[(V-I,I)_{\rm S}-(V-I,I)_{\rm RGC}].$
(7)

Here $(V-I,I)_{\rm RGC}$ denotes the instrumental color and magnitude of the
RGC centroid, and thus the last term in the bracket represents the offset of
the source from the RGC centroid in the CMD.

In Table 3, we summarize the values of $(V-I,I)_{\rm S}$, $(V-I,I)_{\rm RGC}$,
$(V-I,I)_{\rm 0,RGC}$, and $(V-I,I)_{\rm 0,S}$ for the individual events. From
the estimated de-reddened colors and magnitudes, it was found that the source
star of KMT-2022-BLG-0475 is an early K-type turnoff star, and that of
KMT-2022-BLG-1480 is a late G-type subgiant. We estimated the angular source
radius by first converting the measured $V-I$ color into $V-K$ color using the
Bessell & Brett (1988) color-color relation, and then deducing $\theta_{*}$
from the Kervella et al. (2004) relation between $(V-K,V)$ and $\theta_{*}$.
With the measured source radius, we then estimated the angular Einstein radius
using the relation in Eq. (6) and the relative lens-source proper motion using
the relation $\mu=\theta_{\rm E}/t_{\rm E}$. The estimated $\theta_{\rm E}$
and $\mu$ values of the individual events are listed in Table 3. We note that
the uncertainties of the source colors and magnitudes presented in Table 3 are
the values estimated from the model fitting, and those of $\theta_{*}$ and
$\theta_{\rm E}$ are estimated by adding an additional 7% error to consider
the uncertain de-reddened RGC color of Bensby et al. (2013) and the uncertain
position of the RGC centroid (Gould, 2014).

##  5 Physical lens parameters

We determined the physical parameters of the planetary systems using the
lensing observables of the individual events. For KMT-2022-BLG-0475, the
measured observables are $t_{\rm E}$, and $\theta_{\rm E}$, which are
respectively related to the mass and distance to the planetary system by

$t_{\rm E}={\theta_{\rm E}\over\mu};\qquad\theta_{\rm E}=(\kappa M\pi_{\rm
rel})^{1/2},$ (8)

where $\kappa=4G/(c^{2}{\rm au})=8.14\leavevmode\nobreak\ {\rm
mas}/M_{\odot}$. For KMT-2022-BLG-1480, we additionally measured the
observable $\pi_{\rm E}$, with which the physical parameters can be uniquely
determined by

$M={\theta_{\rm E}\over\kappa\pi_{\rm E}};\qquad D_{\rm L}={{\rm
au}\over\pi_{\rm E}\theta_{\rm E}+\pi_{\rm S}}.$ (9)

We estimated the physical parameters by conducting Bayesian analyses because
the observable $\pi_{\rm E}$ was not measured for KMT-2022-BLG-0475, and the
uncertainty of the north component of the parallax vector, $\pi_{{\rm E},N}$,
was fairly big although the east component $\pi_{{\rm E},E}$ was relatively
well constrained.

Figure 9:  Bayesian posteriors of the lens mass and distances to the lens and
source for the lens system KMT-2022-BLG-0475L. In each panel, the solid
vertical line represents the median value, and the two dotted vertical lines
indicate the uncertainty range of the posterior distribution. The curves
marked in blue and red represent the contributions by the disk and bulge lens
populations, respectively.

The Bayesian analysis was done by first generating artificial lensing events
from a Monte Carlo simulation, in which a Galactic model was used to assign
the locations of the lens and source and their relative proper motion, and a
mass-function model was used to assign the lens mass. We adopted the Jung et
al. (2021) Galactic model and the Jung et al. (2018) mass function. With the
assigned values of $(M,D_{\rm L},D_{\rm S},\mu)$, we computed the lensing
observables $(t_{{\rm E},i},\theta_{{\rm E},i},\pi_{{\rm E},i}$) of each
simulated event using the relations in Eqs. (1) and (8). Under the assumption
that the physical parameters are independently and identically distributed, we
then constructed the Bayesian posteriors of $M$ and $D_{\rm L}$ by imposing a
weight $w_{i}=\exp(-\chi_{i}^{2}/2)$. Here the $\chi_{i}^{2}$ value for each
event was computed by

$\chi_{i}^{2}=\left[{t_{{\rm E},i}-t_{\rm E}\over\sigma(t_{\rm
E})}\right]^{2}+\left[{\theta_{{\rm E},i}-\theta_{\rm
E}\over\sigma(\theta_{\rm
E})}\right]^{2}+\sum_{j=1}^{2}\sum_{k=1}^{2}b_{j,k}(\pi_{{\rm
E},j,i}-\pi_{{\rm E},i})(\pi_{{\rm E},k,i}-\pi_{{\rm E},i}),$ (10)

where $(t_{\rm E},\theta_{\rm E},\pi_{\rm E})$ represent the observed values
of the lensing observables, $[\sigma(t_{\rm E}),\sigma(\theta_{\rm E})]$
denote the measurement uncertainties of $t_{\rm E}$ and $\theta_{\rm E}$,
respectively, $b_{j,k}$ denotes the inverse covariance matrix of
$\mbox{\boldmath$\pi$}_{\rm E}$, and $(\pi_{{\rm E},1},\pi_{{\rm
E},2})_{i}=(\pi_{{\rm E},N},\pi_{{\rm E},E})_{i}$ denote the north and east
components of the microlens-parallax vector of each simulated event,
respectively. We note that the last term in Eq. (10) was not included for
KMT-2022-BLG-0475, for which the microlens-parallax was not measured.

In the case of the event KMT-2022-BLG-1480, for which the blending flux was
measured, we additionally imposed the blending constraint that was given by
the fact that the lens could not be brighter than the blend. In order to
impose the blending constraint, we computed the lens magnitude as

$I_{\rm L}=M_{I,{\rm L}}+5\log\left({D_{\rm L}\over{\rm
pc}}\right)-5+A_{I,{\rm L}},$ (11)

where $M_{I,{\rm L}}$ represents the absolute $I$-band magnitude of a star
corresponding to the lens mass, and $A_{I,{\rm L}}$ represents the $I$-band
extinction to the lens. For the computation of $A_{I,{\rm L}}$, we modeled the
extinction to the lens as

$A_{I,{\rm L}}=A_{I,{\rm tot}}\left[1-\exp\left(-{|z|\over h_{z,{\rm
dust}}}\right)\right],$ (12)

where $A_{I,{\rm tot}}=1.53$ is the total $I$-band extinction toward the
field, $h_{z,{\rm dust}}=100$ pc is the vertical scale height of dust,
$z=D_{\rm L}\sin b+z_{0}$, $b$ is the Galactic latitude, and $z_{0}=15$ pc is
vertical position of the sun above the Galactic plane (Siegert, 2019). It
turned out that the blending constraint had little effect on the posteriors
because the other constraints, that is, those from $(t_{\rm E},\theta_{\rm
E},\pi_{\rm E})$, already predicted that the planet host are remote faint
stars whose flux contribution to the blended flux is negligible.

Figure 10:  Bayesian posteriors of KMT-2022-BLG-1480. Notations are same as
those in Fig. 9.

Figures 9 and 10 show the Bayesian posteriors of the lens mass and distances
to the lens and source for KMT-2022-BLG-0475 and KMT-2022-BLG-1480,
respectively. In Table 4, we summarize the estimated parameters of the host
mass, $M_{\rm h}$, planet mass, $M_{\rm p}$, distance to the planetary system,
projected separation between the planet and host, $a_{\perp}=s\theta_{\rm
E}D_{\rm L}$, and the snow-line distances estimated as $a_{\rm snow}\sim
2.7(M/M_{\odot})$ (Kennedy & Kenyon, 2008). Here we estimated the
representative values and uncertainties of the individual physical parameters
as the median values and the 16% and 84% range of the Bayesian posteriors,
respectively.

Table 4: Physical lens parameters

Quantity | KMT-2022-BLG-0475L | KMT-2022-BLG-1480L
---|---|---
$M_{\rm h}$ ($M_{\odot}$) | $0.43^{+0.35}_{-0.23}$ | $0.18^{+0.16}_{-0.09}$
$M_{\rm p}$ ($M_{\rm J}$) | $0.079^{+0.065}_{-0.042}$ | $0.083^{+0.073}_{-0.042}$
($M_{\rm U}$) | $1.73^{+1.42}_{-0.92}$ | $1.82^{+1.60}_{-0.92}$
$D_{\rm L}$ (kpc) | $6.58^{+0.82}_{-1.24}$ | $7.79^{+0.00}_{-0.92}$
$a_{\perp}$ (au) | $2.03^{+0.25}_{-0.38}$ | $1.22^{+0.15}_{-0.14}$
$a_{\rm snow}$ (au) | $1.16$ | $0.49$

333 $M_{\rm J}$ and $M_{\rm U}$ denote the masses of Jupiter and Uranus,
respectively.

We find that the two planetary systems KMT-2022-BLG-0475L and
KMT-2022-BLG-1480L are similar to each other in various aspects. According to
the estimated physical lens parameters, the masses of KMT-2022-BLG-0475Lb and
KMT-2022-BLG-1480Lb are $\sim 1.7$ and $\sim 1.8$ times the mass of Uranus in
our solar system. The planets are separated in projection from their hosts by
$\sim 2.0$ au and $\sim 1.2$ au, respectively. The masses of planet hosts are
$\sim 0.43\leavevmode\nobreak\ M_{\odot}$ and $\sim 0.18\leavevmode\nobreak\
M_{\odot}$, which correspond to the masses of early and mid-M dwarfs,
respectively. Considering that the estimated separations are projected values
and the snow-line distances of the planetary systems are $a_{\rm snow}\sim
1.2$ au for KMT-2022-BLG-0475L and $\sim 0.5$ au for KMT-2022-BLG-1480L, the
planets of both systems are ice giants lying well beyond the snow lines of the
systems. The planetary systems lie at distances of $\sim 6.6$ kpc and $\sim
7.8$ kpc from the sun. The planetary systems are likely to be in the bulge
with a probability 70% for KMT-2022-BLG-0475L and with a probability 83% for
KMT-2022-BLG-1480L.

##  6 Summary and conclusion

We analyzed the light curves of the microlensing events KMT-2022-BLG-0475 and
KMT-2022-BLG-1480, for which weak short-term anomalies were found from the
systematic investigation of the 2022 season data collected by high-cadence
microlensing surveys. We tested various models that could produce the observed
anomalies and found that the anomalies were generated by planetary companions
to the lenses with a planet-to-host mass ratio $q\sim 1.8\times 10^{-4}$ for
KMT-2022-BLG-0475L and a ratio $q\sim 4.3\times 10^{-4}$ for
KMT-2022-BLG-1480L. From the physical parameters estimated from the Bayesian
analyses using the observables of the events, it was found that the planets
KMT-2022-BLG-0475Lb and KMT-2022-BLG-1480Lb have masses $\sim 1.7$ and $\sim
1.8$ times the mass of Uranus in our solar system, respectively, and they lie
well beyond the snow lines of their hosts of early and mid-M dwarfs,
indicating that the planets are ice giants.

Ice giants around M dwarf stars are difficult to be detected by other surveys
using the transit and radical-velocity (RV) methods not only because of the
long orbital period of the planet but also because of the faintness of host
stars. The number of low-mass planets increases with the increase of the
observational cadence of microlensing surveys as shown in the histogram of
detected microlensing planets as a function of the planet-to-host mass ratio
presented in Fig. 1 of Han et al. (2022c). Being able to complement the
transit and RV surveys, high-cadence lensing surveys will play an important
role in the construction of a more complete planet sample, and thus for better
understanding the demographics of extrasolar planets.

The two events are also similar in that they have $\rho$ measurements (and
therefore also $\theta_{\rm E}$ measurements) despite the fact that the source
does not cross any caustics. Zhu et al. (2014) predicted that about half of
KMT planets would not have caustic crossings, and Jung et al. (2023) confirmed
this for a statistical sample of 58 planetary events detected during the
2018–2019 period. However, Gould (2022a) showed that about 1/3 of non-caustic-
crossing events nevertheless yield $\theta_{\rm E}$ measurements.

Measurements of $\theta_{\rm E}$ are important, not only because they improve
the Bayesian estimates (see Sect. 5), but also because they allow accurate
prediction of when high-resolution adaptive-optics (AO) imaging can resolve
the lens separately from the source, which will then yield mass measurements
of both the host and the planet (Gould, 2022a). For KMT-2022-BLG-0475, with
proper motion $\mu=6.9$ mas/yr, the separation in 2030 (approximate first AO
light on 30 m class telescopes), will be $\Delta\theta\sim 55$ mas, which
should be adequate to resolve the lens and source. Resolving the lens of this
event would also be important to confirm the planetary interpretation of the
event because it is difficult to completely rule out the 1L2S interpretation.
By contrast, for KMT-2022-BLG-1480, with $\mu=1.9$ mas/yr, the separation will
be only $\Delta\theta\sim 15$ mas, which almost certainly means that AO
observations should be delayed for many additional years. In particular, if
the Bayesian mass and distance estimates are approximately correct, then the
expected contrast ratio between the source and lens is $\Delta K\sim 7$ mag,
which will likely require separations of at least 4 FWHM, that is, 55 mas even
on the 39 m European Extremely Large Telescope. Hence, the contrast between
the two planets presented in this paper underlines the importance of
$\theta_{\rm E}$ measurements.

###### Acknowledgements.

Work by C.H. was supported by the grants of National Research Foundation of
Korea (2019R1A2C2085965). This research has made use of the KMTNet system
operated by the Korea Astronomy and Space Science Institute (KASI) at three
host sites of CTIO in Chile, SAAO in South Africa, and SSO in Australia. Data
transfer from the host site to KASI was supported by the Korea Research
Environment Open NETwork (KREONET). This research was supported by the Korea
Astronomy and Space Science Institute under the R&D program (Project No.
2023-1-832-03) supervised by the Ministry of Science and ICT. The MOA project
is supported by JSPS KAKENHI Grant Number JSPS24253004, JSPS26247023,
JSPS23340064, JSPS15H00781, JP16H06287, and JP17H02871. J.C.Y., I.G.S., and
S.J.C. acknowledge support from NSF Grant No. AST-2108414. Y.S. acknowledges
support from BSF Grant No 2020740. This research uses data obtained through
the Telescope Access Program (TAP), which has been funded by the TAP member
institutes. W.Zang, H.Y., S.M., and W.Zhu acknowledge support by the National
Science Foundation of China (Grant No. 12133005). W.Zang acknowledges the
support from the Harvard-Smithsonian Center for Astrophysics through the CfA
Fellowship. C.R. was supported by the Research fellowship of the Alexander von
Humboldt Foundation.

## References

  * Albrow et al. (2000) Albrow, M. D., Beaulieu, J.-P., Caldwell, J. A. R., et al. 2000, ApJ, 534, 894
  * Bennett & Rhie (1996) Bennett, D. P., & Rhie, S. H. 1996, ApJ, 472, 660
  * Bensby et al. (2013) Bensby, T. Yee, J.C., Feltzing, S. et al. 2013, A&A, 549, A147
  * Bessell & Brett (1988) Bessell, M. S., & Brett, J. M. 1988, PASP, 100, 1134
  * Bond et al. (2011) Bond, I. A., Abe, F., Dodd, R. J., et al. 2001, MNRAS, 327, 868
  * Gaudi & Gould (1997) Gaudi, B. S., & Gould, A. 1997, ApJ, 486, 85
  * Gaudi (1998) Gaudi, B. S., 1998, ApJ, 506, 533
  * Gould (1992) Gould, A. 1992, ApJ, 392, 442
  * Gould & Loeb (1992) Gould, A. & Loeb, A. 1992, ApJ, 396, 104
  * Gould et al. (1994) Gould, A., Miralda-Escudé, J., & Bahcall, J. N. 1994, ApJ, 423, L105
  * Gould (2014) Gould, A. 2014, Journal of the Korean Astronomy Society, 47, 153
  * Gould (2022a) Gould, A. 2022a, arXiv:2209.12051
  * Gould et al. (2022b) Gould, A., Han, C., Weicheng, Z., et al. 2022b, A&A, 664, A13
  * Griest & Safizadeh (1998) Griest, K., & Safizadeh, N. 1998, ApJ, 500, 37
  * Han et al. (2016) Han, C., Udalski, A., Gould, A., et al. 2016, ApJ, 828, 53
  * Han et al. (2022a) Han, C., Gould, A., Albrow, M. D., et al. 2022a, A&A, 658, A62
  * Han et al. (2022b) Han, C. Bond, I. A., Yee, J., et al. 2022b, A&A, 658, A94
  * Han et al. (2022c) Han, C. Kim, D., Gould, A, et al. 2022c, A&A, 664, A33
  * Han et al. (2022d) Han, C. Kim, D., Yang, H., et al. 2022d, A&A, 664, A114
  * Han et al. (2022e) Han, C. Lee, C.-U., Gould, A., et al. 2022e, A&A, 666, A132
  * Han et al. (2023a) Han, C., Udalski, A., Jung, Y. K., et al. 2023b, A&A, 670, A172
  * Han et al. (2023b) Han, C., Gould, A., Jung Y. K., et al. 2023a, A&A, in press
  * Hwang et al. (2021) Hwang, K.-H., Zang, W., Gould, A., et al. 2022, AJ, 163, 43
  * Hwang et al. (2022) Hwang, K.-H., Zang, W., Gould, A., et al. 2022, AJ, 163, 43
  * Jung et al. (2018) Jung, Y. K., Udalski, A., Gould, A., et al. 2018, AJ, 155, 219
  * Jung et al. (2021) Jung, Y. K., Han, C., Udalski, A., et al. 2021, AJ, 161, 293
  * Jung et al. (2022) Jung, Y. K., Zang, W., Han, C., et al. 2022, AJ, 164, 262
  * Jung et al. (2023) Jung, Y. K., Zang, W., Wang, H., et al. 2023, arXiv:2302.13544
  * Kennedy & Kenyon (2008) Kennedy, G. M. & Kenyon, S. J. 2008, ApJ, 673, 502
  * Kervella et al. (2004) Kervella, P., Thévenin, F., Di Folco, E., & Ségransan, D. 2004, A&A, 426, 29
  * Kim et al. (2016) Kim, S.-L., Lee, C.-U., Park, B.-G., et al. 2016, JKAS, 49, 37
  * Kruszyńska et al. (2022) Kruszyńska, K., Wyrzykowski, Ł., Rybicki, K. A., et al. 2022, A&A, 662, A59
  * Luberto et al. (2022) Luberto, J., Martin, E. C., McGill, P., Leauthaud, A., Skemer, A. J., & Lu, J. R. 2022, AJ, 164, 253
  * Medford et al. (2023) Medford, M. S., Abrams, N. S., Lu, J. R., Nugent, P., & Lam, C. Y. 2023, ApJ, 947, 24
  * Mao & Paczyński (1991) Mao, S., & Paczyński, B. 1991, ApJ, 374, 37
  * Nataf et al. (2013) Nataf, D. M., Gould, A., Fouqué, P. et al. 2013, ApJ, 769, 88
  * Sahu et al. (2022) Sahu, K. C., Anderson, J., Casertano, S., et al. 2022, ApJ, 933, 83
  * Shin et al. (2023) Shin, I.-G., Yee, J., Zang, W., et al. 2023, arXiv:2303.16881
  * Siegert (2019) Siegert, T. 2019, A&A, 632, L1
  * Skowron et al. (2011) Skowron, J., Udalski, A., Gould, A., et al. 2011, ApJ, 738, 87
  * Tonry et al. (2018) Tonry, J. L., Denneau, L., Heinze, A. N., et al. 2018, PASP, 130:064505
  * Udalski et al. (1994) Udalski, A., Kubiak, M., Szymański, M., Kałużny, J., Mateo, M., Krzemiński, W. 1994, Acta Astron., 44, 317
  * Wang et al. (2022) Wang, H., Zang, W., Zhu, W., et al. 2022, MNRAS, 510, 1778
  * Yee et al. (2012) Yee, J. C., Shvartzvald, Y., Gal-Yam, A., et al. 2012, ApJ, 755, 102
  * Yoo et al. (2004) Yoo, J., DePoy, D.L., Gal-Yam, A. et al. 2004, ApJ, 603, 139
  * Zang et al. (2021a) Zang, W., Han, C., Kondo, I., et al. 2021a, Research in Astron. and Astroph., 21, 239
  * Zang et al. (2021b) Zang, W., Hwang, K.-H., Udalski, A., et al. 2021b, AJ, 162, 163
  * Zang et al. (2022) Zang, W., Yang, H., Han, C., et al. 2022, MNRAS, 515, 928
  * Zang et al. (2023) Zang, W., Jung, Y.K., Yang, H., et al. 2023, AJ, 165, 103
  * Zhu et al. (2014) Zhu, W., Penny, M., Mao, S., Gould, A., & Gendron, R. 2014,