# Microscopic study of K-Selection Rule Violation in 182W K=10+ Isomer Decay

C. R. Praharaj  Institute of Physics, Bhubaneswar - 751005, India  Swagatika
Bhoi  School of Physics, Sambalpur University, Jyoti Vihar, Burla-768019,
India  Z. Naik  School of Physics, Sambalpur University, Jyoti Vihar,
Burla-768019, India  S. K. Ghorui  School of Physics and Astronomy, S.J.T
University, Shanghai 200240, China  S. K. Patra  Institute of Physics,
Bhubaneswar - 751005, India

###### Abstract

In this paper we study the microscopic mechanism for the retarded decay of
K-isomers to lower K bands. We do angular momentum projection from suitable
intrinsic states. The retardation arises from poor overlap between the low K
and high K bands in the integral over Euler angles. Deformed HF and angular
momentum projection calculations are done for the decay of the $K=10^{+}$
isomer band to the ground band of ${}^{182}W$. K-mixing is unimportant and the
K quantum numbers of the bands are quite good. There is significant difference
in the reduced matrix elements of transition operators in our formalism and
that in the rotational model. Angular momentum projection gives J-selection
rule but there is no K selection rule for reduced matrix elements of
electromagnetic multipole operators. Thus, E2 and M1 transitions from the K
isomer to the ground band of ${}^{182}W$ are finite but retarded in angular
momentum projection theory, as in experiments. This provides a theoretical
basis for the study of K-isomers and their decay modes. Quantitative results
are presented. The microscopic model gives J-selection rule and angular
momentum conservation for combined matter and radiation systems.

###### pacs:

21.10.-K, 21.10.Ky, 23.35.+g, 23.20.Js, 21.60.Jz

11footnotetext<EMAIL_ADDRESS><EMAIL_ADDRESS>
### Introduction:

The problem of the decay of K isomers and the associated K selection rule and
its violation in deformed nuclei have been subjects of much discussion. Many
cases of K violating gamma-ray transitions are known experimentally kondev ;
regan ; chowdhury ; cullen ; hayes ; walker . No consistent explanation is
available for the finite but retarded transitions from K isomers to lower K
bands seen experimentally. The rotational model forbids such transitions (the
K-selection rule) bohr ; alder  and in many discussions various ad hoc
assumptions are made to explain the violation of K selection rule cullen . But
a proper quantitative understanding of the mechanism of the K-selection rule
violation in the framework of nucleon-nucleon residual interaction is still
lacking. One needs a refined model at the microscopic level to understand
properly the low multipole decay modes of K isomers.

Figure 1: Energy of HF orbits of 182W in MeV. The m value of an orbit is
indicated by its length. Each orbit is doubly degenerate. The Fermi levels of
neutrons and protons are indicated (dotted). The +ve parity orbits are in
dashed red and the -ve parity orbits are in solid black.

Peierls and Yoccoz peierls  showed more then 60 years ago that one needs
angular momentum projection from deformed intrinsic states by applying
rotation operator and integrating over the Euler angles in order to obtain
physical states of good angular momentum J peierls ; ripka ; praharaj  (An
intrinsic state of a deformed nucleus being a wave packet and a superposition
of various J states, angular momentum projection is essential). We show in
this work that a consistent application of the Peierls-Yoccoz procedure gives
J selection rule for reduced matrix elements for electromagnetic transitions
among states of various bands, but there is no K selection rule forbidding J
allowed transitions between bands. In the following we briefly describe the
deformed HF and angular momentum projection procedures and present results for
deformations, energy spectra and electromagnetic transitions for the ground
and the $\mu$-second K=10+ band of ${}^{182}W$. It is found that E2 and M1
transitions from K=10+ bandhead to the ground band of 182W are finite but
retarded, in violation of K-selection rule, and in agreement with experimental
trends kondev ; regan . We thus obtain a theoretical explanation of K-isomer
decay at the microscopic level, without phenomenological assumptions or K
mixing.

### HF Theory and Angular Momentum Projection:

We use two major shells for protons and neutrons each with a surface delta
residual interaction faessler  in our study. The model space consists of
1g7/2, 2d5/2, 2d3/2, 3s1/2, 1h11/2, 1h9/2, 2f7/2, 1i13/2, and 2g9/2 with
single-particle energies -6.92, -5.30, -3.58, -3.298, -4.376, 1.0, 2.0, 3.0
and 5.5 MeV respectively for protons. For neutrons we have 1h9/2, 2f7/2,
2f5/2, 3p3/2, 3p1/2, 1i13/2, 1i11/2, 2g9/2, and 1j15/2 with single particle
energies -10.943, -11.629, -8.407, -8.739, -7.776, -9.494, -4.049, -3.485 and
-0.95. The force strength of the residual interaction (surface delta) is taken
as 0.15 MeV which is appropriate for this calculation in extended model space.
The band structure for a broad range of rare-earth nuclei are well described
with this model space and interaction. 182W has neutron and proton numbers
above the midshell and we use a fairly large model space to describe its
spectra. The HF procedure is based on variational principle ripka ; thouless .
In applications in light nuclei, the HF and J projection are known to account
for 2-body correlation and give results close to shell model diagonalization
macfarlane ; khadkikar . For heavy deformed nuclei, the model is known to give
quantitative results for the band structures and deformations ghorui ; naik ;
praharaj ; patra ; ghorui1 ; praharaj1 ; praharaj2 ; joshi ; yokoyama ;
praharaj3 ; ghorui2 .

### Hartree-Fock equations with axial symmetry:

The Hamiltonian consists of single-particle energy term and 2-body residual
interaction:

$\displaystyle H=H_{S.P}+V$ (1)

For axially symmetric HF field, an orbit is

$\displaystyle|\alpha m\big{>}=\sum_{j}C_{jm}^{\alpha}|jm\big{>}$ (2)

The HF equations are

$\displaystyle(\epsilon_{j}-E_{\alpha})C_{jm}^{\alpha}+\sum_{j_{1}j_{2}j_{4}m_{2}}V(j_{1}mj_{2}m_{2};jmj_{4}m_{2})$
$\displaystyle\rho_{j_{4}m_{2}j_{2}m_{2}}C_{j_{1}m}^{\alpha}=0$ (3)

and

$\displaystyle\rho_{j_{4}m_{2}j_{2}m_{2}}=\sum_{\alpha(occupied)}C_{j_{4}m_{2}}^{\alpha}C_{j_{2}m_{2}}^{\alpha}$
(4)

The HF Hamiltonian and self energy are:

$\displaystyle h=\epsilon+\Gamma$ (5) $\displaystyle\Gamma=V\rho$ (6)

Equations (3) and (4) are solved by iteration for the amplitudes
$C_{jm}^{\alpha}$ and energies of HF orbits $E_{\alpha}$. During the iteration
procedure the residual interaction V is included in each step in building the
HF Hamiltonian. The prolate HF solution for 182W is the lowest in energy and
its orbits are shown in Fig. 1. Some of the results of the HF solution
(energies, quadrupole moments etc.) are given in Table I. From Table I one
sees that the residual interactions among the nucleus (24 active protons and
26 active neutrons) are substantial. We emphasize that the quadrupole moments
etc. follow dynamically from the HF solutions and are not externally imposed:

$\displaystyle\big{<}Q_{20}\big{>}_{HF}=Tr(Q_{20}\rho)$ (7)

Thus, the converged HF solution provides a self-consistent deformed basis
where the effect of residual interaction is included macfarlane ; khadkikar .
Suitable intrinsic states are obtained as products of proton and neutron
Slater determinants built from the HF orbits.

In the intrinsic state frame (the body frame) we have the deformed intrinsic
state. This is not a state of unique angular momentum, rather it is a
superposition of various J states. For rotational symmetry we need good
angular momentum states. As pointed out by Peierls and Yoccoz (PY) peierls
the intrinsic frame is not conductive to states of good angular momentum. The
PY procedure peierls  consists of rotating the intrinsic state (through Euler
angles) and integration over the angles representing the orientation of the
intrinsic frame. This is angular momentum projection.

Table 1: Prolate HF solution of ${}^{182}W$ where substantial interaction
energies are seen. Energies are in MeV. Mass quadrupole moment $Q_{20}$ and
hexadecapole moment $Q_{40}$ are shown in power units of oscillator length
parameter b. $b^{2}\approx(0.9A^{1/3}+0.7)$ fm2 blomquist .

EHF | $\left<V_{pp}\right>$ | $\left<V_{pn}\right>$ | $\left<V_{nn}\right>$ |  | Q20 in $b^{2}$ |  | Q40 in $b^{4}$
---|---|---|---|---|---|---|---
|  |  |  |  | Proton | Neutron |  | Proton | Neutron
-577.674 | -33.222 | -119.902 | -43.475 |  | 15.368 | 26.339 |  | -36.361 | -52.920

### Angular Momentum Projection formalism:

It is to be emphasized that an intrinsic state $|\phi_{K}\big{>}$, is a
superposition of various angular momentum states:

$\displaystyle|\phi_{K}\big{>}=\sum_{J}C_{JK}|\psi_{K}^{J}\big{>}$ (8)

The Peierls-Yoccoz procedure of angular momentum projection to a state of good
angular momentum peierls ; ripka  consists of applying a rotation operator
$R(\Omega)$ ($\Omega$ stands for the Euler angles $\alpha$,$\beta$,$\gamma$)
to $|\phi_{K}\big{>}$ and integrating over the Euler angles with the D
function:

$\displaystyle|\psi_{K}^{J}\big{>}=\frac{2J+1}{8\pi^{2}}\int d\Omega
D_{MK}^{J*}(\Omega)R(\Omega)|\phi_{K}\big{>}$ (9)

where, $R(\Omega)$ is the rotation operator $e^{-i\alpha J_{z}}e^{-i\beta
J_{y}}e^{-i\gamma J_{z}}$.

Energies and electromagnetic transition operators are evaluated by calculating
their matrix elements, which involves integration over Euler angles. There is
axial symmetry of the Hartree-Fock field and the intrinsic states have good K
quantum numbers. Thus, in the evaluation of the matrix elements between
intrinsic states $|\phi_{K_{1}}\big{>}$ and $|\phi_{K_{2}}\big{>}$ the
integration for two of the Euler angles $\alpha$ and $\gamma$ are done
analytically and only the integration for Euler angle $\beta$ remains to be
done numerically. We use 64 point Gauss-Legendre formula for the numerical
integration of kernels over the angle $\beta$. Extended precision is used in
the evaluation of the kernels. The matrix elements of the Hamiltonian and an
electromagnetic operator of multipolarity L are:

$\displaystyle
H_{K_{1}K_{2}}^{J}=\frac{(J+1/2)}{(N_{K_{1}K_{1}}^{J}N_{K_{2}K_{2}}^{J})^{1/2}}\int_{0}^{\pi}d\beta
sin(\beta)d_{K_{1}K_{2}}^{J}(\beta)\big{<}\phi_{K_{1}}|He^{-i\beta
J_{y}}|\phi_{K_{2}}\big{>}$ (10a)

$\displaystyle\big{<}\psi_{K_{1}}^{J_{1}}||T^{L}||\psi_{K_{2}}^{J_{2}}\big{>}=\frac{(J_{2}+1/2)(2J_{1}+1)^{1/2}}{(N_{K_{1}K_{1}}^{J_{1}}N_{K_{2}K_{2}}^{J_{2}})^{1/2}}\sum_{\mu(\nu)}C_{\mu\;\nu\;K_{1}}^{J_{2}\;L\;J_{1}}\int_{0}^{\beta}d\beta
sin(\beta)d_{\mu
K_{2}}^{J_{2}}(\beta)\big{<}\phi_{K_{1}}|T_{\nu}^{L}e^{-i\beta
J_{y}}|\phi_{K_{2}}\big{>}$ (10b)

where,

$\displaystyle N_{K_{1}K_{2}}^{J}=\int_{0}^{\pi}d\beta
sin(\beta)d_{K_{1}K_{2}}^{J}(\beta)\big{<}\phi_{K_{1}}|e^{-i\beta
J_{y}}|\phi_{K_{2}}\big{>}$ (10c)

is the overlap integral.

We mention here about the possibility of K-selection rule violating
transitions in this theoretical formalism. The Clebsch-Gordan coefficient in
Eqn.(10b) for the reduced matrix element contains the J selection rule for
electromagnetic transitions and there is no K selection rule to prohibit a
J-allowed transition. (As example, for E2 transition from J=10+ K=10+ to J=8+
K=0+, the coefficient $C_{2\;-2\;0}^{10\;2\;8}$ in Eqn.(10b) is non-zero and
thus the transition is allowed by J selection rule). Because of the presence
of the rotation operator $R(\Omega)$ in eqn.(10), there is a summation over
$\mu(\nu)$ in the Clebsch-Gordan coefficient in Eqn.(10b). Thus, the angular
momentum projection theory has J selection rule for the reduced matrix
elements, but there is no K-selection rule in this theory.

Spectroscopic Quadrupole moment and Magnetic moment are

$\displaystyle
Q_{S}(J)=\frac{1}{\sqrt{(2J+1)}}\sqrt{\frac{16\pi}{5}}C_{J\;0\;J}^{J\;2\;J}\big{(}\sum_{i_{(p,n)}}\big{<}\psi_{K}^{J}||Q_{2}^{i}||\psi_{K}^{J}\big{>}\big{)}\;\;\;\;\;\;\;$
(11)
$\displaystyle\mu(J)=\frac{1}{\sqrt{(2J+1)}}C_{J\;0\;J}^{J\;1\;J}\big{(}\sum_{i_{(p,n)}}\big{<}\psi_{K}^{J}||g_{l}^{i}l_{i}+g_{s}^{i}s_{i}||\psi_{K}^{J}\big{>}\big{)}.\;\;\;\;\;\;\;$
(12)

Table 2: BE(2) and BM(1) values for transitions from $K=10^{+}\rightarrow
K=0^{+}$.

Transitions |  |  | BE(2) in $e^{2}fm^{4}$ |  |  | Transitions |  |  | BM(1) in $\mu_{N}^{2}$
---|---|---|---|---|---|---|---|---|---
$10^{+}_{iso}\rightarrow 8^{+}_{gr}$ |  |  | $0.4106\times 10^{-4}$ |  |  | $10^{+}_{iso}\rightarrow 10^{+}_{gr}$ |  |  | $0.3\times 10^{-7}$
$10^{+}_{iso}\rightarrow 10^{+}_{gr}$ |  |  | $0.1963\times 10^{-3}$ |  |  | $11^{+}_{iso}\rightarrow 10^{+}_{gr}$ |  |  | $0.3070\times 10^{-6}$
$11^{+}_{iso}\rightarrow 10^{+}_{gr}$ |  |  | $0.2353\times 10^{-2}$ |  |  | $11^{+}_{iso}\rightarrow 12^{+}_{gr}$ |  |  | $0.4614\times 10^{-6}$
$12^{+}_{iso}\rightarrow 10^{+}_{gr}$ |  |  | $0.8295\times 10^{-2}$ |  |  | $12^{+}_{iso}\rightarrow 12^{+}_{gr}$ |  |  | $0.7755\times 10^{-5}$
|  |  |  |  |  | $13^{+}_{iso}\rightarrow 12^{+}_{gr}$ |  |  | $0.3115\times 10^{-4}$

Figure 2: Plot of J overlaps for K=0+ and K=10+ intrinsic states.

### Results and Discussions:

The Hartree-Fock orbits of the prolate deformed solution of the Eqns.(3) and
(4) are shown in Fig. 1 for 182W with 24 active protons and 26 active
neutrons. The neutron orbit $\Omega$=9/2+ is just below the Fermi level and
the $\Omega$=11/2+ orbit is above. One particle-one hole excitation across the
neutron Fermi level gives K=10+ configuration
$\big{(}9/2^{+}n,11/2^{+}n\big{)}$. This configuration for K=10+ band agrees
with that in kondev . The Hartree-Fock energy, pp, pn, and nn interaction
energies and the deformation obtained in the HF solution are given in Table 1.
The interaction energies $\big{<}V_{pp}\big{>}$, $\big{<}V_{pn}\big{>}$,
$\big{<}V_{nn}\big{>}$ are substantial in the deformed HF and the residual
interactions are well taken into account macfarlane ; khadkikar  in this
theoretical procedure.

Figure 3: Energy spectra of K=0+ and K=10+ bands of 182W obtained by J
projection from respective intrinsic configurations. E2 and M1 transitions
from the 10+ band-head to the ground band are shown and compared with
experimental data. Figure 4: (a) J to J-2 BE(2) values for K=0+ band, (b) J to
J-1 BE(2) values for K=10+ band, (c) J to J-2 BE(2) values for K=10+ band, (d)
J to J-1 BM(1) values for K=10+ band of 182W. Comparision is made with
available experimental BE(2). Figure 5: Values of (a) BE(2)(J to J-1), (b)
BE(2)(J to J-2), (c) BM(1)(J to J-1) for transitions from K=10+ $\rightarrow$
K=0+ of 182W. Figure 6: Single nucleon transition in K-isomer decay
($\Omega_{2}$ to -$\Omega_{1}$). $\Omega_{2}$=11/2+ and $\Omega_{1}$=9/2+ for
K=10+ isomer of 182W.

We have done angular momentum projection from K=0+ (ground band) and K=10+
(isomer) intrinsic states and the results of the spectra are shown in Fig. 3.
The overlap integrals $N_{0^{+}0^{+}}$ and $N_{10^{+}10^{+}}$ Eqn.(10c) are
shown in Fig. 2, indicating well-spread wave-packets and these suggest the
need for angular momentum projection to get states of good J. It is to be
noted that the theoretical spectra shown in Fig. 3 have good K quantum numbers
(K=0+ and K=10+).

We have evaluated (Eqn. 10a) the energy overlap of the 10+ states:

$\displaystyle\big{<}10^{+}_{isomer}|H|10^{+}{gr}\big{>}=0.342\;eV$ (13)

whereas the experimental energy difference between the two 10+ states is

$\displaystyle\Delta(10^{+}_{isomer}-10^{+}{gr})=0.519\;MeV.$ (14)

Thus the interaction energy in Eqn.(13) is smaller than the energy difference
by a factor of about 1.5$\times 10^{6}$ and so there is practically no mixing
between the two J=10+ states of the K=0+ and K=10+ bands. This rules out the
possibility of K mixing and thus the K quantum numbers of the two bands of
182W are quite robust.

Table 3: Spectroscopic Quadrupole Moments and Magnetic moments of K=0+ and
K=10+ isomer bands from angular momentum projection.

Q${}_{S}^{gr}$(2+) |  |  $\mu_{gr}$ (2+) |  | Q${}_{S}^{iso}$(10+) |  |  $\mu_{iso}$(10+)
---|---|---|---|---|---|---
(in e fm2) |  | (in $\mu$N) |  | (in e fm2) |  | (in $\mu_{N}$)
-409.78 |  | 1.494 |  | 475.861 |  | -3.344

The spectroscopic quadrupole moments and magnetic moments of the ground and
K=10+ bands are listed in Table 3. In evaluating the magnetic moments and the
BM(1) we use g${}_{l}^{p}$=1.0, g${}_{l}^{n}$=0.0, g${}_{s}^{p}$=5.5857,
g${}_{s}^{n}$=-3.8261 and a spin quenching factor of 0.6.

In Fig. 3 the energy spectra of the ground band and the K=10+ bandhead compare
well with experiment. The level spacings of the K=10+ band are a bit
exaggerated in our model. In Fig. 4 (a), we plot the BE(2; J$\rightarrow$J-2)
values for the K=0+ band. In our calculations we have taken effective charges
as 0.4 and 1.4 e for neuton and proton. The Weisskopf single-particle estimate
for BE(2) for 182W is$\;\approx 61.3$ e2fm4 and we have BE(2;
2+$\rightarrow$0+) as 8181.4 e2 fm4 in our calculation, an enhancement of
about 140 over single-particle value. The experimental BE(2;
2+$\rightarrow$0+) is 8342.93 e2fm4 raman . Similarly, the experimental BE(2;
4+$\rightarrow$2+) =12014.8 e2fm4, BE(2; 6+$\rightarrow$4+) =12321.3 e2fm4 and
BE(2; 8+$\rightarrow$6+) =12811.7 e2fm4 where as our theoretical BE(2) values
for the same states are 11672, 12825 and 13379 e2fm4 respectively as shown in
Fig. 4(a).

The BE(2) values of the K=10+ band are plotted in Fig. 4 (b)
(J$\rightarrow$J-1) and in Fig. 4 (c) (J$\rightarrow$J-2). They are
collective, but the (J$\rightarrow$J-2) BE(2)’s are less collective of the
two. The BM(1; J$\rightarrow$J-1) for the K=10+ band are shown in Fig. 4 (d).
The BE(2) and BM(1) values for transitions from the K=10+ band to the K=0+
ground band are given in Table 2 and Fig. 5. These values for
K=10+$\rightarrow$K=0+ transitions are finite and about $10^{-8}$ of the
values of collective transitions within a band (Fig. 4) and are thus severely
retarded. (For example, the BE(2) value for 10${}^{+}_{gr}$
$\rightarrow$8+gr=13680 e2 fm4 and for 10${}^{+}_{iso}$
$\rightarrow$8+gr=0.4106$\times$10-4 e2 fm4). There is no K-selection rule in
our theoretical model prohibiting E2 and M1 transitions from K=10+ to K=0+
band- there is only J selection rule (Eqn. 10b). Thus after careful
calculation we find that the K-selection rule violating E2 and M1 transitions
from the K=10+ to K=0+ band are finite, but retarded. This is in agreement
with experimental trends kondev ; regan ; raman . Our theory describes both
the collective transitions within a band and the severely retarded (but non-
vanishing) ones from K-isomers to lower K band.

Figure 7: Overlap kernels $\big{<}\phi_{K}|e^{-i\beta J_{y}}|\phi_{K}\big{>}$
of 182W for K=0+ and K=10+ intrinsic states. (Black dots) for K=0+ and (red
squares for K=10+).

The overlap kernels $\big{<}\phi_{K}|e^{-i\beta J_{y}}|\phi_{K}\big{>}$ for
the K=0+ and the K=10+ bands are plotted in Fig. 7. K=0+ configuration has
time-reversal symmetry and its overlap kernel has very sharp peaks near
$\beta$=0∘ and 180∘ falling off sharply within Cos$\beta$ interval of 0.05.
The K=10+ configuration has sharp peak near $\beta$=0∘ and falls off very fast
with $\beta$ value. Fig. 8 shows the overlap kernel between K=0+ and K=10+
bands. The overlap kernel $\big{<}\phi_{0^{+}}|e^{-i\beta
J_{y}}|\phi_{10^{+}}\big{>}$ is very small ($\sim$10-8)and vanishes for
$\beta$=0∘ and 180∘ and thus does not match with the kernels of either of the
bands. The severe retardation of BE(2) and BM(1) values for transitions from
K=10+ to the ground band (Table 3) and the small overlap kernel between the
two configurations (Fig. 8) are reflections of the fact that there is a large
change of single neutron configuration (11/2+$\rightarrow$ -9/2+) in the
transition from K=10+ to the ground band (depicted in Fig. 6).

In the microscopic picture, a collective transition is caused by the coherent
contributions of many nucleons. This is reflected in the fact that the overlap
kernel is peaked near 0∘/180∘ in Euler angle $\beta$. In the case of K-isomer
decay to lower K band only a single-nucleon contributes (and that by changing
its $\Omega$ quantum number by a large amount, leading to further
retardation). The process conserves angular momentum (rotational symmetry) for
the combined system of nucleons and the $\gamma$-ray photon, given by the
J-selection rule.

Figure 8: K=0+ and K=10+ overlap kernel $\big{<}\phi_{0^{+}}|e^{-i\beta
J_{y}}|\phi_{10^{+}}\big{>}$.

### Interaction between the two bands and mixings:

The Hamiltonian overlap (interaction energy) between the K=10+ and the K=0+
bands are plotted in Fig. 9 for various J values. It is seen that the
interaction between the two bands are quite small at low J, (J=10+ for
example) and becomes quite higher as J increases. Thus, the mixing between the
two bands is insignificant for K=10+ bandhead and states just above. But at
higher J the mixing can be more significant. Such large increase in
interaction energy between the ground band and the isomeric band at higher J
values may be a general feature of K-isomers for other nuclei too.

Figure 9: Plot of interaction energy between the ground band and the K-isomer.

### Summary and Conclusions:

We have shown in this work that one does not need K-mixing for transitions
from K-isomer to lower K bands. Two K bands of good but widely different K
values can be connected by low order multipoles, provided the J and parity
selection rules are satisfied. It is essential to implement rotational
symmetry by angular momentum projection from intrinsic states.

The spectra and transition matrix elements, including the K-selection rule
violating E2 and M1 transitions from the K=10+ bandhead of 182W to the ground
band, have been studied by deformed Hartree-Fock and angular momentum
projection. The spectra and the collectivity of transitions within bands and
the retardations in K=10+ to K=0+ transitions are well reproduced. There is no
K mixing. The transitions allowed by J selection rule
(J=10${}^{+}_{iso}$$\rightarrow$8${}^{+}_{gr}$ by E2 radiation, for example)
are finite, but retarded, in agreement with experimental trends. The K quantum
numbers of the respective bands are robust and K mixing is negligible in our
microscopic calculation.

Thus, a theoretical understanding of the decay of a K-isomer to lower K bands
is possible if rotational symmetry is implemented by angular momentum
projection from the deformed intrinsic states. All transitions allowed by J
and parity selection rules are in general possible. The retardation of such
transitions from the K isomer is explained naturally by the poor overlaps
between the K-isomer and the lower K configurations.

The constraint on the transition matrix elements imposed by the rotational
model is violated in the microscopic model at the single-nucleon level
(neutron $\Omega$=11/2+$\rightarrow$ -9/2+ for K=10+ decay in 182W), in
agreement with experimental trends. With this theoretical insight one can
study the decay of K isomers in many other cases and this will help in the
quantitative spectroscopy involving K-isomer for its many possible
applications.

## Acknowledgments

We thank Prof. R. K. Bhowmik, Prof. Partha Chowdhury and Prof. S. B. Khadkikar
for discussions and suggestions. CRP was partly supported by SERB grant
SB/S2/HEP-06/2013.

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MULTI-BAND TRANSIT FOLLOW UP OBSERVATIONS OF FIVE HOT-JUPITERS WITH CRITICAL NOISE TREATMENTS: IMPROVED PHYSICAL PROPERTIES

0000-0001-8018-0264]Suman Saha
Indian Institute of Astrophysics, II Block, Koramangala, Bengaluru, India
Pondicherry University, R.V. Nagar, Kalapet, Puducherry, India

Suman Saha

0000-0001-6703-0798]Aritra Chakrabarty
Indian Institute of Astrophysics, II Block, Koramangala, Bengaluru, India
University of Calcutta, Salt Lake City, JD-2, Kolkata, India

0000-0002-6176-3816]Sujan Sengupta
Indian Institute of Astrophysics, II Block, Koramangala, Bengaluru, India

for publication in The Astronomical Journal

The most challenging limitation in transit photometry arises from the noises in the photometric signal. In particular, the ground-based telescopes are heavily affected by the noise due to perturbation in the Earth's atmosphere. Use of telescopes with large apertures can improve the photometric signal-to-noise ratio (S/N) to a great extent. However, detecting a transit signal out of a noisy light curve of the host star and precisely estimating the transit parameters call for various noise reduction techniques. Here, we present multi-band transit photometric follow-up observations of five hot-Jupiters e.g., HAT-P-30 b, HAT-P-54 b, WASP-43 b, TrES-3 b and XO-2 N b, using the 2m Himalayan Chandra Telescope (HCT) at the Indian Astronomical Observatory, Hanle and the 1.3m J. C.  Bhattacharya Telescope (JCBT) at the Vainu Bappu Observatory, Kavalur. Our critical noise treatment approach includes techniques such as Wavelet Denoising and  Gaussian Process regression, which effectively reduce both time-correlated and time-uncorrelated noise components from our transit light curves. In addition to these techniques, use of our state-of-the-art model algorithms have allowed us to estimate the physical properties of the target exoplanets with a better accuracy and precision compared to the previous studies.

§ INTRODUCTION

The transit photometry serves as one of the most important methods in the context of exoplanet detection and characterisation. This method helps us to determine several physical parameters of a transiting exoplanet, e.g., the radius, the inclination angle of the planetary orbit with respect to our line of sight (LOS), and the semi-major axis. However, a prior knowledge of the stellar radius is necessary for the estimation of these parameters. Modelling the transit light curves also allows us to determine the limb-darkening properties of the parent stars. Moreover, the orbital inclination angle estimated by using the transit method can be combined with the radial-velocity measurements, if available, and a prior knowledge of stellar mass to precisely estimate the mass of those exoplanets [Chakrabarty & Sengupta, 2019, Southworth et al., 2007].

Adopted physical properties of the Exoplanets and their host-stars
HAT-P-30 b     HAT-P-54 b     WASP-43 b     TrES-3 b     XO-2 N b
P [days]     2.810595±0.000005     3.799847±0.000014       0.813475±0.000001     1.30618608±0.00000038     2.61585922±0.00000028

K_RV [m s^-1]     88.1±3.3     132.6±4.9     551.0±3.2     378.4±9.9     90.17±0.82

R_⋆ [R_]     1.215±0.051     0.617±0.013     0.6506±0.0054     0.829±0.022     0.998±0.033

M_⋆ [M_]     1.242±0.041     0.645±0.020     0.688±0.037     0.928±0.048      0.96±0.05

T_eff⋆ [K]     6304±88     4390±50     4500±100     5650±75     5332±57

Sources     Johnson et al., 2011     Bakos et al., 2015     Hellier et al., 2011,     Sozzetti et al., 2009,     Damasso et al., 2015

Esposito et al., 2017     Christiansen et al., 2011

The photometric observations obtained from the ground-based telescopes are heavily affected by the turbulence in the Earth's atmosphere. This significantly adds to the overall noise in the observed signal. Moreover, if the ground-based survey telescopes used for the detection of new transiting exoplanets is small, it gives rise to a poor signal-to-noise ratio (S/N) in the observed transit signals. Therefore, repeated follow-up observations  are very important in order to estimate the physical properties of the confirmed exoplanets with a good accuracy and precision. Repeated follow-up observations with telescopes of larger aperture can result in high S/N in the transit light curves causing small error-bars in the light-curves. Further, in order to achieve an improved accuracy and precision in the values of the estimated transit parameters, application of critical noise reduction techniques is essential to reduce the fluctuations prevailing in the transit light curves. The results from transit follow-ups spanning over a large period of time can be used for the studies of planetary dynamics and may reveal the presence of any undiscovered planetary mass objects in those systems [Nesvornỳ et al., 2012, Johnson et al., 2015, Gillon et al., 2017, Patra et al., 2017, Maciejewski et al., 2018].

Our ongoing project involves the photometric follow-up of transiting exoplanets using multiple photometric bands and analysis using noise reduction techniques for an improved estimation of the physical properties. For photometric follow-up, we use the 2m Himalayan Chandra Telescope (HCT) at the Indian Astronomical Observatory, Hanle and the 1.3m J. C. Bhattacharya Telescope (JCBT) at the Vainu Bappu Observatory, Kavalur. [Chakrabarty & Sengupta, 2019] (hereafter, CS19) have reported the results from their observations of five hot Jupiters, namely WASP-33 b, WASP-50 b, WASP-12 b, HATS-18 b and HAT-P-36 b using the same telescopes as a part of this project and have addressed to the different sources of noise to improve the accuracy and precision of the estimated parameters. CS19 have segregated the noises according to their sources and spatial scales of span, and employed different techniques for the treatment of different kinds of such noises, including Wavelet Denoising and Gaussian Process regression.

As a continuation of this project, we have followed up five more hot Jupiters using the same telescopes but in multi-wavelength bands. These are  HAT-P-30 b <cit.>, HAT-P-54 b <cit.>, WASP-43 b <cit.>, TrES-3 b <cit.> and XO-2 N b <cit.>. The multi-band observations of transits enable us to estimate the wavelength dependent physical parameters, such as planetary radius, with a better accuracy corresponding to each photometric band.

One of the most prominent noise component in the photometric light-curves is the time-uncorrelated noise (white noise), which consists of both photon noise and the fluctuations in the light curve due to the small spatial scale variability in the transparency of Earth's atmosphere (CS19), such as atmospheric scintillation [Osborn et al., 2015, Föhring et al., 2019]. Most of the pre-processing techniques (such as binning and Gaussian smoothing) that can reduce the effect of these time-uncorrelated noise components also tend to distort the shape of the transit signal. CS19 have demonstrated that the Wavelet Denoising technique [Donoho & Johnstone, 1994, Quan Pan et al., 1999, Luo & Zhang, 2012, del Ser et al., 2018, Cubillos et al., 2017, Waldmann, 2014] can be used to reduce the time-uncorrelated fluctuations in the light-curves without distorting the transit signal and improves the precision of the estimated physical parameters (see Table 3 and Table 4 of CS19) to a great extent. Wavelet denoising technique also reduces the outliers in the light-curves due to cosmic ray events. In the present work, we have used the same technique to pre-process the transit light curves. However, while applying this technique in the present studies, we have made some appropriate upgradation. This was essential in order to handle multi-band data sets and a greater number of free parameters.

Details of the photometric observations
Target Name     Date     Telescope     Instrument     Filter     No. of frames     S/N (Mean)
HAT-P-30 b     2020-02-05     JCBT     UKATC     V     42     747.92

2020-02-05     HCT     HFOSC     I     315     2002.55

2020-02-22     HCT     HFOSC     R     233     1508.76

2020-03-10     JCBT     UKATC     I     50     482.48

2020-03-24     JCBT     UKATC     V     22     747.37

HAT-P-54 b     2020-01-15     JCBT     UKATC     I     24     570.82

2020-02-03     JCBT     UKATC     I     56     410.26

2020-02-03     HCT     HFOSC     V     160     1001.14

2020-02-22     HCT     HFOSC     R     93     1544.17

WASP-43 b     2019-03-06     JCBT     UKATC     V     85     379.06

2019-04-02     JCBT     UKATC     V     93     390.22

2019-04-11     JCBT     UKATC     R     54     606.31

2019-12-31     HCT     HFOSC     I     261     1264.00

2020-02-05     HCT     TIRSPEC     J     232     226.06

2020-02-14     JCBT     UKATC     R     31     662.59

TrES-3 b     2019-02-18     JCBT     UKATC     I     96     283.19

2019-05-03     JCBT     UKATC     R     50     337.95

2020-03-31     JCBT     UKATC     I     26     537.23

2020-09-21     HCT     HFOSC     V     203     1447.40

XO-2 N b     2020-01-28     JCBT     UKATC     I     54     710.21

2020-02-05     HCT     HFOSC     V     292     1789.22

2020-02-18     JCBT     ProEM     I     40     782.06

2020-02-18     HCT     HFOSC     R     622     1261.56

2020-03-23     JCBT     UKATC     I     27     699.05

Another important noise component in the photometric signals is the time-correlated noise (red noise). We have reduced the large temporal scale red noise due to various instrumental and astrophysical effects by using the baseline correction method. On the other hand, the time-correlated fluctuations in the light curves of short temporal scale are due to the small spatial scale variations that affect each object on a frame differently (CS19). The major sources of this kind of red noise is the small-scale activity and pulsation of the host stars. Following CS19, we address this red noise component by using Gaussian Process (GP) regression method  [Rasmussen & Williams, 2006, Johnson et al., 2015, Pereira et al., 2019, Barros et al., 2020] to model it simultaneously while modelling for the transit signal.

In order to model the transit light curves, we have used the analytical formalism provided by Mandel & Agol, 2002, which also incorporates the limb-darkening effect of the host-star using the quadratic limb darkening law. Following CS19, we have used the Markov Chain Monte Carlo (MCMC) method with the Metropolis-Hastings algorithm [Hastings, 1970] while modelling the transit light curves simultaneously along with the GP regression of the red noise. Although, the MCMC sampling technique is computationally expensive, it is extremely effective for modelling any noisy signal with a large number of free parameters.

Unlike the work by CS19, in the present work, we have set the radius of a planet as a free parameter for each wavelength band corresponding to each filter used. This allows us to get a coarse estimation of the band-dependent radius of the planets. This can be helpful in characterising the planets by studying the broad atmospheric features, if any, in those ultra low-resolution transit spectra detected with the help of the model transmission spectra in optical and in near infrared regions [Sengupta et al., 2020, Chakrabarty & Sengupta, 2020].

For the simultaneous handling of multiple observed photometric data from different nights of observation and a smooth and streamlined implementation of all the state-of-the-art techniques for reduction, photometry, processing, and modelling of the light curves, we have used  the software package developed by us. This Python-based package uses a semi-automated approach and is functionally not much different from the pipeline developed and used by CS19. All the steps in this package aim at a precise estimation of the planetary properties from the transit light curves, which are at the same time robust, accurate, and reliable.

In section <ref>, we have described our observational details. In section <ref>, we have detailed our analysis and modelling techniques. In section <ref>, we have discussed the significance of our results and in section <ref>, we have concluded our study.

Observational and modelled light curves for HAT-P-30 b. For each observed transit event (see Table <ref>), the observation date, the instrument, and the photometric filter used are mentioned. Top: the unprocessed light curve (cyan), light curve after Wavelet Denoising (magenta), the best-fit transit model (orange). Middle: the residual after modelling without GP regression (magenta), the mean (orange) and 1-$\sigma$ interval (cyan) of the best-fit GP regression model. Bottom: mean residual flux (orange).

Same as Figure <ref>, but for HAT-P-54 b

Same as Figure <ref>, but for WASP-43 b

Same as Figure <ref>, but for TrES-3 b

Same as Figure <ref>, but for XO-2 N b

§ TARGET SELECTION AND OBSERVATIONS

In this paper we report the follow-up of five hot-Jupiters, e.g., HAT-P-30 b, HAT-P-54 b, WASP-43 b, TrES-3 b and XO-2 N b. The adopted physical properties of these exoplanets and their host-stars are listed in Table <ref>.

Our photometric observations are conducted using the 2m Himalayan Chandra Telescope (HCT) at Indian Astronomical Observatory, Hanle and the 1.3m J. C. Bhattacharya Telescope (JCBT) at Vainu Bappu Observatory, Kavalur. At HCT, the Hanle Faint Object Spectrograph Camera (HFOSC) has been used for photometric observations in V, R and I bands (Bessel), whereas the TIFR Near Infrared Spectrometer and Imager (TIRSPEC) has been used for the observations in J band. At JCBT, the UKATC optical CCD and the ProEM imagers have been used for the photometric observations at V, R and I bands. We have also made simultaneous observations of a few similar magnitude stars present within the photometric field of view of the target host-stars. These are  used as reference stars for the differential photometry as described in the next section. The details of our observations have been listed in Table <ref>. Our observations have been optimised for mid-high cadence and high S/N. As can be seen from the table, the observations from JCBT have mean S/N$>$250. On the other hand, the observations from HCT/HFOSC have very-high cadence and mean S/N$>$1000.

§ DATA REDUCTION AND ANALYSIS

We have reduced the raw photometric data to obtain the transit light curves, which are then processed through several techniques to reduce the noises from various sources. We have then modelled the processed light curves in order to obtain the transit parameters. These parameters are then used to derive a few other physical properties of the exoplanets. For the whole process of data reduction and analysis, we have used our software package, which is written in Python programming language and uses several open-source software libraries. A detailed explanation of our methodology is given in the following subsections.

§.§ Data reduction and differential photometry

Our observational data obtained at each night, consists of a large number of photometric frames which are reduced through our automatic pipeline that uses the standard PyRAF (IRAF) libraries in the back-end. The raw photometric data are first calibrated using the bias and flat frames obtained during the respective nights of observations. Since, all the instruments used in our observations are cooled up to -70oC, the dark noise is found to be negligible. The calibrated frames are used to obtain the flux from our target host-stars and the reference field stars using aperture photometry. We have calculated the photometric noise precisely using the formula:

\begin{equation}
N = \sqrt{f / g + a \times s^2 + a^2 \times s^2 / k}
\end{equation}

where, $f$ is sky-subtracted flux of the target object, $s$ is the standard deviation of the counts on the region of the sky surrounding the object, $g$ is gain of the instrument, $a$ is the area of aperture of the object chosen in square pixels, and $k$ is number of sky pixels. We have converted the time scale to BJD-TDB using the utc2bjd online applet [Eastman et al., 2010].

The observations from the ground-based telescopes are heavily affected by the varying atmospheric transparency and air-mass effect which can be reduced using the differential photometry method. We have used the flux of the reference field stars with the best S/N and minimum differential fluctuations for the differential photometry of our target host-stars and obtained the photometric transit light-curves.

§.§ Baseline correction

The red noise of large temporal scale (i.e., more than the characteristic scale of transit durations), which are mainly due to various instrumental effects and long-term stellar variability, results in a non-flat baseline for the photometric signals. One way to address this noise component is to model it using simple polynomial fits alongside the transits and subtract from the light-curves. However, it risks a potential bias in the estimated parameters due to the large scale nature of these noise components, and an increment in the parameter load in the already heavily populated model parameter space for MCMC sampling (see <ref>).

We have, therefore, performed the baseline correction before modelling for the transit signals. We have used linear and quadratic polynomials of time to model the out-of-transit part of the light-curves and chosen the one with the least BIC (Bayesian Information Criterion) [Schwarz, 1978] for the baseline correction. Use of only the out-of-transit portion of the light curves for baseline modelling removes the risk of the low-mid temporal scale variations in transit signal influencing the determination of baseline coefficients. Hence, it removes the risk of potential manipulation in the estimated transit parameters. The normalised baseline corrected transit light-curves from our observations are shown in Figure <ref>-<ref>.

§.§ Wavelet Denoising

The time-uncorrelated noise (white noise) in the photometric signal also consists of the fluctuations in the light curve due to the small spatial scale variability in the transparency of the Earth's atmosphere. The presence of such fluctuations are more evident in the high-cadence photometric observations due to a better temporal resolution. They can severely affect both accuracy and precision of the estimated transit properties from those light curves. The white noise component cannot be totally removed from a signal and should be only reduced cautiously without distorting the informative part of the signal. Hence, instead of smoothing the light curves with some low-pass filter, we used a more robust technique, namely the Wavelet Denoising [Donoho & Johnstone, 1994, Quan Pan et al., 1999, Luo & Zhang, 2012]. Although wavelet based denoising techniques have been widely  used in image processing and remote sensing in various fields of science and engineering, it is a recent addition in the context of transit photometry and other light curve analysis <cit.>. CS19 applied this technique on the light curves obtained from their observational data and demonstrated that this technique produces no distortion in the transit light curves, but yields better MCMC posterior distributions for the fitted transit parameters.

Wavelet Denoising consists of mainly three steps: deconstruction of the original signal into wavelet coefficients using discrete wavelet transform, thresholding, and reconstruction of the signal from the thresholded coefficients. We have used the PyWavelets [Lee et al., 2019] python package to perform the single level discrete wavelet transform of our photometric light-curves. In this process, we have used the Symlet family of wavelets, which are the least asymmetric modified version of Daubechies wavelets. A single level transform removes the risk of excess denoising. We have calculated the threshold value using the universal thresholding law [Donoho & Johnstone, 1994] given as;

\begin{equation}
Th = \sigma \sqrt{2log_e(N)}
\end{equation}

where $\sigma = |median(D_x)| / 0.6745$, and performed the hard thresholding, where the wavelet coefficients with absolute values less than the threshold value are replaced with it. The advantage behind universal threshold is that the risk of thresholding is small enough to satisfy the requirement of most applications. The threshold coefficients are then used to construct the denoised signal. The transit light curves after Wavelet Denoising are shown in Figure <ref>-<ref>.

Estimated values of physical parameters for HAT-P-30 b
Parameter     Filter     This work     Johnson et al., 2011
Transit parameters

b         0.8546_-0.0055^+0.0041     0.854_-0.010^+0.008

R_⋆/a         0.1430_-0.0012^+0.0013     0.1348±0.0047

R_p/R_⋆    V     0.10753_-0.00165^+0.00180     0.1134±0.002^ †

R     0.11346_-0.00165^+0.00193

I     0.11683_-0.00104^+0.00110

Limb-darkening coefficients

C_1     V     0.329_-0.020^+0.020     0.1975^ †

R     0.329_-0.020^+0.020

I     0.329_-0.020^+0.020

C_2     V     0.280_-0.020^+0.020     0.3689^ †

R     0.280_-0.020^+0.020

I     0.279_-0.019^+0.021

Derived parameters

T_14  [hr]     V     2.186_-0.022^+0.025     2.129±0.036^ †

R     2.213_-0.021^+0.025

I     2.223_-0.021^+0.024

a/R_⋆        6.994_-0.062^+0.060     7.42±0.26

i  [deg]         82.982_-0.087^+0.082     83.6±0.4

M_p  [M_J]         0.712_-0.031^+0.031     0.711±0.028

M_p  [M_]         226.2_-9.7^+9.8     226.0±8.9

T_eq  [K]         1686.1_-24.7^+24.8     1630±42

a  [AU]         0.03948_-0.00169^+0.00170     0.0419±0.0005

R_p  [R_J]     V     1.2995_-0.0481^+0.0482     1.340±0.065^ †

R     1.3720_-0.0495^+0.0505

I     1.4122_-0.0480^+0.0482

R_p  [R_]     V     14.566_-0.539^+0.540     15.02±0.73^ †

R     15.379_-0.555^+0.566

I     15.829_-0.538^+0.541
$^{\:\bm\dag}$ value doesn't corresponds to the mentioned wavelength filter.

Estimated values of physical parameters for HAT-P-54 b
Parameter     Filter     This work     Bakos et al., 2015
Transit parameters

b         0.7599_-0.0054^+0.0060     0.741_-0.011^+0.010

R_⋆/a         0.07175_-0.00072^+0.00098     0.0697±0.001

R_p/R_⋆    V     0.15616_-0.00214^+0.00204     0.1572±0.002^ †

R     0.16898_-0.00082^+0.00115

I     0.17516_-0.00165^+0.00176

Limb-darkening coefficients

C_1     V     0.627_-0.026^+0.043     0.4324^ †

R     0.685_-0.035^+0.017

I     0.630_-0.028^+0.041

C_2     V     0.080_-0.007^+0.007     0.2457^ †

R     0.080_-0.007^+0.006

I     0.079_-0.006^+0.007

Derived parameters

T_14  [hr]     V     1.819_-0.017^+0.021     1.797±0.017^ †

R     1.855_-0.017^+0.020

I     1.872_-0.018^+0.021

a/R_⋆        13.94_-0.19^+0.14     14.34±0.22

i  [deg]         86.873_-0.057^+0.046     87.040±0.084

M_p  [M_J]         0.761_-0.032^+0.032     0.760±0.032

M_p  [M_]         241.8_-10.1^+10.3     242±10

T_eq  [K]         832.0_-10.8^+10.9     818±12

a  [AU]         0.03994_-0.00098^+0.00098     0.04117±0.00043

R_p  [R_J]     V     0.9796_-0.0330^+0.0329     0.944±0.028^ †

R     1.0611_-0.0332^+0.0333

I     1.0997_-0.0357^+0.0362

R_p  [R_]     V     10.980_-0.370^+0.369     10.6±0.3^ †

R     11.894_-0.372^+0.374

I     12.327_-0.401^+0.406
$^{\:\bm\dag}$ value doesn't corresponds to the mentioned wavelength filter.

Estimated values of physical parameters for WASP-43 b
Parameter     Filter     This work     Esposito et al., 2017
Transit parameters

b         0.6810_-0.0058^+0.0054     0.689±0.013

R_⋆/a         0.2182_-0.0017^+0.0018     0.2012±0.0057

R_p/R_⋆    V     0.16442_-0.00156^+0.00152     0.1588±0.004^ †

R     0.16268_-0.00142^+0.00178

I     0.16214_-0.00121^+0.00123

J     0.1375_-0.0032^+0.0035

Limb-darkening coefficients

C_1     V     0.627_-0.026^+0.041     0.66^ †

R     0.632_-0.029^+0.041

I     0.680_-0.036^+0.021

J     0.643_-0.037^+0.042

C_2     V     0.079_-0.007^+0.007

R     0.079_-0.006^+0.007

I     0.081_-0.007^+0.006

J     0.080_-0.006^+0.007

Derived parameters

T_14  [hr]     V     1.3046_-0.0086^+0.0091     1.164±0.24^ †

R     1.3016_-0.0089^+0.0099

I     1.3010_-0.0094^+0.0091

J     1.2583_-0.0118^+0.0102

a/R_⋆        4.584_-0.037^+0.036     4.97±0.14

i  [deg]         81.46_-0.13^+0.12     82.109±0.088

M_p  [M_J]         1.994_-0.073^+0.072     1.998±0.079

M_p  [M_]         633.9_-23.1^+22.7     635.0±25.1

T_eq  [K]         1486.3_-33.5^+33.5     1426.7±8.5

a  [AU]         0.01387_-0.00016^+0.00016     0.01504±0.00029

R_p  [R_J]     V     1.1006_-0.0598^+0.0599     1.006±0.017^ †

R     1.0900_-0.0593^+0.0595

I     1.0857_-0.0586^+0.0587

J     0.9199_-0.0539^+0.0546

R_p  [R_]     V     12.336_-0.670^+0.671     11.61±0.21^ †

R     12.218_-0.664^+0.667

I     12.169_-0.657^+0.658

J     10.311_-0.604^+0.612
$^{\:\bm\dag}$ value doesn't corresponds to the mentioned wavelength filter.

Estimated values of physical parameters for TrES-3 b
Parameter     Filter     This work     Sozzetti et al., 2009
Transit parameters

b         0.8317_-0.0047^+0.0040     0.84±0.01

R_⋆/a         0.1681_-0.0020^+0.0021     0.1687±0.0016

R_p/R_⋆    V     0.16483_-0.00081^+0.00076     0.1655±0.002^ †

R     0.17226_-0.00215^+0.00192

I     0.16704_-0.00177^+0.00261

Limb-darkening coefficients

C_1     V     0.454_-0.027^+0.024     0.4378

R     0.447_-0.025^+0.028

I     0.448_-0.026^+0.027

C_2     V     0.251_-0.013^+0.013     0.2933

R     0.249_-0.013^+0.014

I     0.250_-0.013^+0.013

Derived parameters

T_14  [hr]     V     1.387_-0.015^+0.015

R     1.405_-0.016^+0.016

I     1.393_-0.015^+0.014

a/R_⋆        5.948_-0.073^+0.072     5.926±0.056

i  [deg]         81.96_-0.13^+0.13     81.85±0.16

M_p  [M_J]         1.954_-0.084^+0.085     1.91_-0.080^+0.075

M_p  [M_]         621.1_-26.6^+26.9     607.030_-25.425^+23.836

T_eq  [K]         1638.3_-24.0^+24.1

a  [AU]         0.02293_-0.00067^+0.00068     0.02282_-0.00040^+0.00023

R_p  [R_J]     V     1.4881_-0.0768^+0.0770     1.336_-0.037^+0.031 †

R     1.5555_-0.0823^+0.0830

I     1.5108_-0.0796^+0.0802

R_p  [R_]     V     16.680_-0.860^+0.863     14.975_-0.415^+0.347 †

R     17.434_-0.922^+0.930

I     16.935_-0.892^+0.899
$^{\:\bm\dag}$ value doesn't corresponds to the mentioned wavelength filter.

Estimated values of physical parameters for XO-2 N b
Parameter     Filter     This work     Damasso et al., 2015
Transit parameters

b         0.1993_-0.0295^+0.0239     0.287_-0.055^+0.043

R_⋆/a         0.1204_-0.0008^+0.0010     0.1261±0.0017

R_p/R_⋆    V     0.10702_-0.00062^+0.00064     0.1049_-0.00063^+0.00059 †

R     0.09797_-0.00039^+0.00052

I     0.10164_-0.00148^+0.00153

Limb-darkening coefficients

C_1     V     0.503_-0.032^+0.030     0.474_-0.028^+0.030 †

R     0.466_-0.011^+0.023

I     0.487_-0.026^+0.036

C_2     V     0.201_-0.014^+0.013     0.171_-0.070^+0.067 †

R     0.194_-0.010^+0.015

I     0.199_-0.013^+0.014

Derived parameters

T_14  [hr]     V     2.628_-0.016^+0.018     2.7024±0.006^ †

R     2.606_-0.016^+0.018

I     2.615_-0.017^+0.018

a/R_⋆        8.308_-0.072^+0.059     7.928_-0.093^+0.099

i  [deg]         88.63_-0.17^+0.21     87.96_-0.34^+0.42

M_p  [M_J]         0.595_-0.021^+0.021     0.597±0.021

M_p  [M_]         189.1_-6.8^+6.7     189.737±6.674

T_eq  [K]         1308.4_-14.8^+14.9

a  [AU]         0.0385_-0.0013^+0.0013     0.03673±0.00064

R_p  [R_J]     V     0.9996_-0.0521^+0.0522     1.019±0.031^ †

R     0.9155_-0.0476^+0.0477

I     0.9492_-0.0509^+0.0513

R_p  [R_]     V     11.264_-0.584^+0.585     11.422±0.347^ †

R     10.262_-0.534^+0.534

I     10.640_-0.571^+0.575
$^{\:\bm\dag}$ value doesn't corresponds to the mentioned wavelength filter.

Mid-transit times and GP regression parameters for each observed transit event
Target Name     Date     Instrument/Filter     Mid-transit time [BJD-TDB]     $\alpha$     $\tau$
HAT-P-30 b     2020-02-05     UKATC/V     2458885.401561_-0.001481^+0.001551     0.00054_-0.00020^+0.00019     0.0582_-0.0264^+0.0219

2020-02-05     HFOSC/I     2458885.401169_-0.000772^+0.000696     0.00158_-0.00009^+0.00009     0.0263_-0.0008^+0.0010

2020-02-22     HFOSC/R     2458902.265187_-0.000976^+0.000822     0.00159_-0.00011^+0.00015     0.0240_-0.0011^+0.0009

2020-03-10     UKATC/I     2458919.128141_-0.001620^+0.002098     0.00358_-0.00042^+0.00045     0.0650_-0.0042^+0.0055

2020-03-24     UKATC/V     2458933.175811_-0.001794^+0.001766     0.00036_-0.00021^+0.00024     0.0825_-0.0592^+0.0447

HAT-P-54 b     2020-01-15     UKATC/I     2458864.201261_-0.000410^+0.000374     0.00035_-0.00021^+0.00026     0.1007_-0.0437^+0.0492

2020-02-03     UKATC/I     2458883.204510_-0.000402^+0.000427     0.00095_-0.00039^+0.00039     0.0642_-0.0149^+0.0212

2020-02-03     HFOSC/V     2458883.204556_-0.000437^+0.000331     0.00151_-0.00016^+0.00017     0.0711_-0.0050^+0.0069

2020-02-22     HFOSC/R     2458902.203596_-0.000308^+0.000231     0.00091_-0.00009^+0.00011     0.0564_-0.0048^+0.0044

WASP-43 b     2019-03-06     UKATC/V     2458549.296039_-0.000409^+0.000334     0.00112_-0.00023^+0.00032     0.0581_-0.0129^+0.0178

2019-04-02     UKATC/V     2458576.140878_-0.000322^+0.000294     0.00150_-0.00029^+0.00022     0.0613_-0.0088^+0.0125

2019-04-11     UKATC/R     2458585.087944_-0.000520^+0.000496     0.00096_-0.00029^+0.00043     0.0848_-0.0242^+0.0251

2019-12-31     HFOSC/I     2458849.468659_-0.000166^+0.000188     0.00171_-0.00010^+0.00010     0.0235_-0.0010^+0.0009

2020-02-05     TIRSPEC/J     2458885.259449_-0.000911^+0.000789     0.00407_-0.00027^+0.00029     0.0276_-0.0017^+0.0018

2020-02-14     UKATC/R     2458894.207388_-0.000365^+0.000255     0.00675_-0.00028^+0.00032     0.0744_-0.0022^+0.0029

TrES-3 b     2019-02-18     UKATC/I     2458533.473769_-0.000400^+0.000382     0.00223_-0.00021^+0.00025     0.0281_-0.0032^+0.0035

2019-05-03     UKATC/R     2458607.351900_-0.000402^+0.000405     0.00051_-0.00025^+0.00022     0.0582_-0.0242^+0.0248

2020-03-31     UKATC/I     2458940.426286_-0.000281^+0.000337     0.00030_-0.00018^+0.00021     0.0854_-0.0330^+0.0421

2020-09-21     HFOSC/V     2459114.150846_-0.000244^+0.000216     0.00105_-0.00009^+0.00010     0.0327_-0.0011^+0.0010

XO-2 N b     2020-01-28     UKATC/I     2458877.220952_-0.000626^+0.000550     0.00151_-0.00021^+0.00031     0.0653_-0.0084^+0.0088

2020-02-05     HFOSC/V     2458885.071089_-0.000316^+0.000290     0.00101_-0.00006^+0.00006     0.0288_-0.0008^+0.0008

2020-02-18     ProEM/I     2458898.149337_-0.000425^+0.000431     0.00126_-0.00030^+0.00029     0.0896_-0.0120^+0.0129

2020-02-18     HFOSC/R     2458898.149390_-0.000237^+0.000249     0.00180_-0.00004^+0.00005     0.0238_-0.0005^+0.0005

2020-03-23     UKATC/I     2458932.154930_-0.000723^+0.000927     0.00116_-0.00030^+0.00031     0.1501_-0.0413^+0.0557
The mid-transit times for the same transit event observed by two different telescopes in three different instances (2020-02-05 for HAT-P-30 b, 2020-02-03 for HAT-P-54 b and 2020-02-18 for XO-2 N b) have been calculated independently.

§.§ Gaussian Process Regression

The small-mid temporal scale red noise, which are correlated in time, form the major source of the remaining reducible noise components in our transit light curves after the Wavelet Denoising. This noise is primarily due to the small-scale activity of the host stars. To reduce this noise component, we have used the Gaussian Process (GP) regression [Rasmussen & Williams, 2006, Johnson et al., 2015, Pereira et al., 2019, Barros et al., 2020] technique.

We have modelled the time-correlated noise in the transit light-curves using GP regression alongside modelling for the transit signals, using the Markov Chain Monte Carlo (MCMC) technique (see Section <ref>). In order to treat the photometric noise in our observed light-curves while modelling for time-correlated noise, we have followed the regression formalism as given by Rasmussen & Williams, 2006 for noisy observations. In this process, we have used the Matérn class covariance function with parameter of covariance, $\nu = 3/2$, and two free parameters, e.g., the signal standard deviation $\alpha$ and the characteristic length scale $\tau$, which are used as model parameters in the MCMC sampling.

§.§ Markov Chain Monte Carlo sampling

In the final phase of our analysis of the transit light-curves, we have modelled them to the analytical transit formulation following Mandel & Agol, 2002, which also incorporates the limb darkening effect of the host-star using the quadratic limb-darkening law given by,

\begin{equation}
I(\mu) = 1 - C_1(1-\mu) - C_2(1-\mu)^2
\end{equation}

We have used the MCMC sampling technique for this modelling, and simultaneously modelled for the time-correlated noise in the signal using the GP regression method as explained in section <ref>. The model parameters in MCMC includes the transit parameters: (i) mid-transit time $T$, (ii) the impact-parameter $b$, (iii) the scaled stellar radius $R_\star/a$, (iv) the ratio of planet to stellar radius $R_p/R_\star$, (v) the out-of-transit flux $f_{o}$, (vi) the limb darkening coefficients $C_1$ and $C_2$; and (vii) the GP regression coefficients $\alpha$ and $\tau$. The orbital periods have been kept constant and are adopted from previous studies as given in Table <ref>. We have followed Sing, 2010 for the prior values of the quadratic limb-darkening coefficients.

We have modelled all the transit light-curves corresponding to each of our target exoplanets simultaneously keeping the transit independent parameters the same for all transit events of the same target and keeping the wavelength dependent parameters same across those transit events observed using the same photometric filters. We have used the Metropolis-Hastings algorithm [Hastings, 1970] in MCMC sampling and used the modified marginal likelihood function due to the inclusion of the GP regression technique [Rasmussen & Williams, 2006]. In MCMC sampling for each of our targets, we have used 30 walkers and 100,000 iterations for each independent walker. Such a large volume of sampling is done in order to handle the large number of model parameters and high volume of photometric data under simultaneous modelling of multiple transit events. The aim has been to assess the robustness of the convergence of the sampling parameters such that a high accuracy in the derived parameters is achieved. Such a large computation is practically impossible without the incursion of parallel computing. For this purpose, our software package for modelling and data analysis has been facilitated with parallel computing using the Python-based multiprocessing library. We have used the high performance computing facility (NOVA) at Indian Institute of Astrophysics (IIA) for our numerical computation. We have discarded the first 10,000 iterations for each walkers as burn-in and accepted the rest 90,000 iterations for obtaining the posterior distributions. We have shown the corner diagrams depicting the posterior distributions of the transit parameters and the GP regression coefficients from our MCMC sampling in the appendix <ref>.

We present the best-fit transit models, best-fit GP regression model, and the mean residual fluxes in Figure <ref>-<ref>. The estimated values of the physical properties for our target exoplanets with 1$\sigma$ error have been given in Table <ref>-<ref> and the estimated values of mid-transit times along with the GP regression coefficients for each transit light-curves have been given in Table <ref>.

§.§ Derived parameters

The posterior distribution of the transit parameters from the MCMC sampling along with the adopted physical properties of the target exoplanets and the host stars as given in Table <ref> are used to estimate the values of some other physical parameters.

We have estimated the transit duration $T_{14}$ for each of our targets at each photometric band using the relation (CS19)

\begin{equation}
T_{14} = \frac{P}{\pi}\sin^{-1} \begin{pmatrix}
\frac{\sqrt{(1 + R_p / R_\star)^2 - b^2}}{\sqrt{(a / R_\star)^2 - b^2}}.
\end{pmatrix}
\end{equation}

The inclination angle of the planetary orbit $i$ is estimated using the relation

\begin{equation}
i = \cos^{-1} \begin{pmatrix}
\frac{bR\star}{a}
\end{pmatrix}
\end{equation}

We have estimated the mass $M_p$ and the equilibrium temperature $T_{eq}$ of the target exoplanets using the relations (CS19),

\begin{equation}
M_p = M_\star^{2/3} \begin{pmatrix}
\frac{P}{2 \pi G}
\end{pmatrix}^{1/3}
\frac{K_{RV}}{\sin i}
\end{equation}

\begin{equation}
T_{eq} = T_{eff} \sqrt{\frac{R_\star}{2a}}
\end{equation}

For all the cases, we have assumed circular orbits, zero Bond albedo, and full re-circulation of the incident stellar flux over the planetary surface. The estimated values of all the derived physical properties for each of our targeted exoplanets are given in the Table <ref>-<ref>.

§ RESULTS AND DISCUSSION

Our study involves the photometric follow up observations of five hot-Jupiters using a 1.3m (JCBT) and a 2m (HCT) class telescopes. Table <ref> shows that the observations from HCT have a better S/N and time-cadence. This can be attributed to the large aperture of HCT, short readout time of the new CCD installed at the back-end (HFOSC2), and good observational condition at the site of IAO. However, the observations from JCBT also show good S/N and have been extremely useful in complementing the observations from the HCT. Since the number of nights allocated at each telescope is limited, we use both the telescopes for our observations.

Figure <ref>-<ref> demonstrate that the Wavelet Denoising technique is a useful tool for the reduction of the time-uncorrelated noise components and outliers from the transit light curves. Also, we have avoided excessive smoothing of the light-curves by cautiously choosing the decomposition levels and threshold levels while using this denoising technique.

The GP regression method has very effectively modelled the correlated noise components in the transit signal as shown in  Figure <ref> - <ref>. The origin of this correlated noise component is not only attributed to the low-scale variability of the exoplanet host-stars, but also to that of the reference stars used for differential photometry. Since, the reference stars are different in most of the cases, due to various observational reasons, the amplitude of the correlated noises are different for the same target host star. Similarly, the correlated noise components are negligible for some of the low-cadence observations from JCBT. Our analysis shows the efficiency of GP regression technique in reducing the correlated noise components irrespective of their origin and amplitude.

Our MCMC sampling includes a large number of sampling points, which is helpful to remove any bias from the choice of prior values in the MCMC, thereby resulting in an more accurate estimation of the free parameters. This  is evident from the near-normal distribution of most of the model-parameter posterior distributions as shown in the figures in appendix <ref>. As can be seen from the estimated values of the mid-transit times of our observed transit events, the independently estimated mid-transit times for the same transit event observed from the two different telescopes (HCT and JCBT) in three different instances [2020-02-05 for HAT-P-30 b, 2020-02-03 for HAT-P-54 b and 2020-02-18 for XO-2 N b] are in perfect agreement with each other, with the error-bar in the estimated value from the HCT observation being small due to a better S/N. This proves the robustness of our analysis method in the estimation of these properties.

Due to the multi-band observations of the transit events, we have been able to estimate the wavelength dependent radii of our target exoplanets with good accuracy at each band. Even though the site conditions at IAO allows observation in the infrared wavelengths, the S/N of the data observed using the TIRSPEC instrument at the back-end of HCT is significantly low, especially in the context of transit photometry. However, we managed to observe one frame in the J-band for WASP-43 b. Apart from this, all of the targets have been observed in V-, R- and I-bands. We will further analyse these results to check if any information can be excavated about the atmospheres of these planets with the help of the models we have developed for transmission spectra [Chakrabarty & Sengupta, 2020, Sengupta et al., 2020].

We have compared the estimated values of the physical parameters for the target exoplanets in our study with those from the previous studies (given in Tables <ref>-<ref>) to understand the effectiveness of the critical noise reduction and modelling algorithm used in our study, and the improvement in the accuracy and precision in the parameter estimation. While for most of the parameters, the estimated values are in good accordance with those from previous studies, small differences in a few parameter values can be attributed to the improvement in the quality of the transit signals due to the use the noise reduction algorithms, thereby improving the accuracy in parameter estimation. Also, the precision in the estimated values of the transit parameters and the parameters which are directly derived from them have been improved up to 4 times compared to the previous studies, which is significant in the context of transit photometric studies of exoplanets. This improvement in the uncertainties in estimated parameters is due to a better S/N in our follow-up observations and further reduction of various noise components. On the other hand, the estimated values of other derived parameters, which are dependent upon the stellar properties of the host-stars for their estimation (adopted from the previous studies and given in Table <ref>), have seen no significant improvements in their precision. This is due to the large uncertainties embedded with those adopted values of stellar parameters.

§ CONCLUSION

In this study, we have performed new multi-band transit photometric follow-up of five hot-Jupiters, HAT-P-30 b, HAT-P-54 b, WASP-43 b, TrES-3 b and XO-2 N b, using the 2m Himalayan Chandra Telescope (HCT) at IAO, Hanle and the 1.3m J. C. Bhattacharya Telescope (JCBT) at VBO, Kavalur. Taking the advantage of the larger apertures of these telescopes compared to those used for previous studies of these exoplanets, we have obtained transit light-curves with better S/N.

Our critical noise treatment analysis employs the Wavelet Denoising technique for the reduction of time-uncorrelated noise components from the photometric light curves without potential loss of signal due to transit origins and the Gaussian Process regression technique to effectively model and compensate for the time-correlated noise in the photometric signal simultaneously along with the transit modelling. We have used the Markov Chain Monte Carlo (MCMC) sampling technique by adopting the Metropolis-Hastings algorithm.

Due to the high S/N photometric follow-up observations as well as the adopted highly optimised noise reduction and analysis techniques, the estimated physical parameters from our study have a better accuracy and overall precision. Also, the follow-up using multiple photometric bands have enabled us to estimate accurately, the wavelength dependent physical properties which can be used as an outset for the high-resolution atmospheric characterisation of these exoplanets in future.

Our ongoing project of follow-up studies of the exoplanets can be extended for the study of other existing exoplanets in  future. Also, the critical noise analysis algorithm can be used for the analysis of transit light curves from the existing and upcoming global exoplanet survey missions, such as the TESS (Transiting Exoplanet Survey Satellite), for the estimation of physical properties of the detected exoplanets with better accuracy.

We are thankful to the anonymous reviewer for a critical reading of the manuscript and for providing many useful comments and suggestions. We thank to the supporting staffs at the Indian Astronomical Observatory (IAO), Hanle, the Center For Research and Education in Science and Technology (CREST), Hosakote, and the Vainu Bappu Observatory (VBO), Kavalur. Some of the computational results reported in this work were performed on the high performance computing facility (NOVA) of IIA, Bangalore. We are thankful to the computer division of Indian Institute of Astrophysics for the help and co-operation extended for the present project. We have used PyRAF for most of the tasks of reduction and photometry. PyRAF is a product of the Space Telescope Science Institute, which is operated by AURA for NASA.

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§ POSTERIOR DISTRIBUTIONS FROM MCMC SAMPLING

Corner diagram depicting the posterior distributions of the transit parameters and the GP regression coefficients from MCMC sampling for HAT-P-30 b.

Same as Figure <ref>, but for HAT-P-54 b

Same as Figure <ref>, but for WASP-43 b

Same as Figure <ref>, but for TrES-3 b

Same as Figure <ref>, but for XO-2 N b