Vol.0 (20xx) No.0, 000–000 11institutetext: CAS Key Laboratory of Planetary
Sciences, Purple Mountain Observatory, Chinese Academy of Sciences, Nanjing
210023, China<EMAIL_ADDRESS>
22institutetext: School of Astronomy and Space Science, University of Science
and Technology of China, Hefei 230026, China
33institutetext: CAS Center for Excellence in Comparative Planetology, Hefei
230026, China
44institutetext: CAS Key Laboratory of Radio Astronomy, Purple Mountain
Observatory, Chinese Academy of Sciences, Nanjing 210023, China
Received 2022 November 19; accepted 2022 December 21

# A Two-limb Explanation for the Optical-to-infrared Transmission Spectrum of
the Hot Jupiter HAT-P-32Ab

Xin-Kai Li (李馨凯) 1122 Guo Chen (陈果) 1133 Hai-Bin Zhao (赵海斌) 1133 Hong-Chi Wang
(王红池) 44

###### Abstract

We present a new optical transmission spectrum of the hot Jupiter HAT-P-32Ab
acquired with the Carnegie Observatories Spectrograph and Multiobject Imaging
Camera (COSMIC) on the Palomar 200 inch Hale Telescope (P200). The P200/COSMIC
transmission spectrum, covering a wavelength range of 3990–9390 Å, is composed
of 25 spectrophotometric bins with widths ranging from 200 to 400 Å and
consistent with previous transit measurements obtained in the common
wavelength range. We derive a combined optical transmission spectrum based on
measurements from five independent instruments, which, along with the 1.1–1.7
$\mu$m spectrum acquired by the Hubble Space Telescope and two Spitzer
measurements, exhibits an enhanced scattering slope blueward of a relatively
flat optical continuum, a water absorption feature at 1.4 $\mu$m, and a carbon
dioxide feature at 4.4 $\mu$m. We perform Bayesian spectral retrieval analyses
on the 0.3–5.1 $\mu$m transmission spectrum and find that it can be well
explained by a two-limb approximation of $134^{+45}_{-33}\times$ solar
metallicity, with a strongly hazy morning limb of $1134^{+232}_{-194}$ K and a
haze-free evening limb of $1516^{+33}_{-44}$ K. This makes HAT-P-32Ab a
promising target for James Webb Space Telescope to look for asymmetric
signatures directly in the light curves.

###### keywords:

techniques: spectroscopic — planets and satellites: atmospheres — planets and
satellites: individual (HAT-P-32Ab)

##  1 Introduction

The study of exoplanets is one of the fastest growing sub-disciplines in
astronomy and planetary science. Observations and studies of exoplanet
atmospheres have sprung up, and we have now discovered over 5200 exoplanets,
among which over 3900 were discovered by the transit method (according to NASA
Exoplanet Archive 111https://exoplanetarchive.ipac.caltech.edu/, as of 2022
November). During a transit, some of the stellar light will pass through the
optically thin part of the planetary atmosphere, resulting in wavelength-
dependent planetary radii with potential imprints of absorption and scattering
features of planetary atmosphere at the terminator (Seager & Sasselov 2000).
It is feasible to retrieve the atmospheric properties from the observed
transmission spectrum under certain model assumptions, e.g., line-by-line
radiative transfer 1D model with parameterized temperature structure, chemical
compositions, and clouds or hazes properties (Madhusudhan & Seager 2009).

Close-in hot Jupiters are the most favorable targets for transmission
spectroscopy with current instrumentation, which are gas giants with high
temperatures, short orbital periods and extended atmospheres. The high
atmospheric temperatures of hot Jupiters make them fantastic laboratories for
unveiling the chemical abundances of giant planets. Through the analysis of
the transmission spectrum of hot Jupiters, various species have been
identified (Madhusudhan 2019), such as atomic metals including Na and K in the
optical wavelengths due to their particularly prominent features at $\sim$589
and $\sim$768 nm (e.g., Nikolov et al. 2018a; Chen et al. 2018), and water
vapor in the near infrared wavelengths for a water absorption feature centered
at 1.4 $\mu$m (e.g., Deming et al. 2013). Among species found in the hottest
hot Jupiters, gaseous TiO and VO may drive a temperature inversion in these
planets (Hubeny et al. 2003). The role of TiO/VO in these hottest atmospheres
is still not clear, which could be depleted due to mechanisms such as deep-
atmosphere or nightside cold trap, gravitational settling, photodissociation,
thermal dissociation, and high C/O chemistry (Showman et al. 2009; Spiegel et
al. 2009; Knutson et al. 2010; Madhusudhan 2012; Parmentier et al. 2013,
2018). The 3D global circulation models have been used to investigate the
compositions, distributions, and formation of clouds and how they shape the
transmission and emission spectra (Parmentier et al. 2016; Helling et al.
2019, 2021). Potential observational evidence for asymmetries resulting from
atmospheric circulation has started to emerge through high-resolution Doppler
spectroscopy (Ehrenreich et al. 2020; van Sluijs et al. 2022; Cont et al.
2022).

One target of special interest is the highly inflated hot Jupiter HAT-P-32Ab,
transiting a late-F-type star with a period of 2.15 days at a distance of
0.0343 AU discovered by Hartman et al. (2011). The planet has a mass of
$0.585\pm 0.031M_{\mathrm{Jup}}$, a radius of $1.789\pm
0.025R_{\mathrm{Jup}}$, and an equilibrium temperature of $1801\pm 18$ K,
while the host star has a mass of $1.160\pm 0.041M_{\odot}$, a radius of
$1.219\pm 0.016R_{\odot}$, and an effective temperature of $6269\pm 64$ K
(Hartman et al. 2011; Czesla et al. 2022). There is a resolved M1.5V
companion, HAT-P-32B, at an angular separation of $2\farcs 9$ (Adams et al.
2013). HAT-P-32Ab is one of the best targets for transmission spectroscopy
because of its large transit depth of more than 2%, its large atmospheric
scale height of about 1500 km, and a relatively bright host star ($V=11.4$
mag).

Several observational studies have been conducted to reveal the atmospheric
property of HAT-P-32Ab. Ground-based binned spectrophotometry, from GMOS on
Gemini North telescope (Gemini-N/GMOS) (Gibson et al. 2013), MODS on Large
Binocular Telescope (LBT/MODS) (Mallonn & Strassmeier 2016), and OSIRIS on
Gran Telescopio Canarias (GTC/OSIRIS) (Nortmann et al. 2016), and multi-color
broad-band photometry (Mallonn et al. 2016; Mallonn & Wakeford 2017; Tregloan-
Reed et al. 2018) all come to similar conclusions that HAT-P-32Ab has a flat
featureless optical transmission spectrum, with a possible scattering slope at
the blue-optical, indicative of a bimodal cloud distribution that consists of
a Rayleigh-like haze and a gray cloud deck. The space observations, carried
out with STIS and WFC3 on Hubble Space Telescope (HST), not only confirmed the
enhanced Rayleigh scattering and the thick cloud deck (Alam et al. 2020), but
also revealed the presence of a water absorption feature at $\sim$1.4 $\mu$m
(Damiano et al. 2017; Alam et al. 2020). However, the full optical-to-infrared
retrieval analysis performed by Alam et al. (2020) cannot account for the
shallower transit depths measured by Spitzer at 3.6 and 4.5 $\mu$m. In
addition to transmission spectrum, dayside emission spectrum has also been
acquired using ground-based $H$\- and $K_{S}$-band photometry, Spitzer 3.6 and
4.5 $\mu$m photometry, and HST/WFC3 spectroscopy (Zhao et al. 2014; Nikolov et
al. 2018b), which shows no evidence of the $\sim$1.4 $\mu$m water feature but
agrees with an isothermal atmosphere of $1995\pm 17$ K or an atmosphere with a
modest thermal inversion.

In this work, we present a new optical transmission spectrum of HAT-P-32Ab
observed by the Carnegie Observatories Spectrograph and Multiobject Imaging
Camera (COSMIC; Kells et al. 1998) on the Palomar 200 inch Hale Telescope
(P200) and attempt to reconcile the existing discrepancy between data and
model in the infrared. In Section 2, we introduce the details of the
observation and data reduction processes. In Section 3, we present the
analyses on the white and spectroscopic light curves. In Section 4, we perform
the Bayesian spectral retrieval analyses on HAT-P-32Ab’s transmission
spectrum. Finally, we discuss the implications of the retrieval results in
Section 5 and draw conclusions in Section 6.

##  2 Observations and Data Reduction

We obtained a transit time series of HAT-P-32Ab on the night of 2013 October
10 from 07:35 UT to 12:21 UT. The transit was observed with COSMIC installed
at the prime focus of P200 located atop Palomar Mountain in north San Diego
County, California. The spectroscopic mode of COSMIC has a field of view of
$13\farcm 65\times 13\farcm 65$, equipped with a thinned, back-illuminated
SITe $2048\times 2048$ CCD ($0\farcs 4$ per pixel). A long-slit mask with a
slit width of 12′′ was created to simultaneously monitor the flux of HAT-P-32A
($V=11.3$ mag) and the reference star TYC 3281-957-1 ($V=11.0$ mag, $3\farcm
2$ away). The 300 lines per mm grism was used to acquire the spectra, covering
a wavelength range of 340–970 nm at a dispersion of $\sim$0.31 nm per pixel.
An exposure time of 90 s was adopted, except for the first two taken with 60
and 120 s, resulting in a total of 85 frames. The long readout time of
$\sim$117 s strongly reduced the duty cycle to 43%. The mercury arc lamp was
observed through a longslit mask with a slit width of 2′′ for wavelength
calibration.

The raw spectral images were reduced following the methodology described in
Chen et al. (2021) based on IRAF (Tody 1993) and customized IDL scripts,
including corrections for overscan, bias, flat fields, sky background, and
cosmic rays. The 1D spectra were extracted using the optimal extraction
algorithm (Horne 1986) with an aperture diameter of 21 pixels ($8\farcs 4$),
which minimized the scatter in the white-light curve. The white-light curve
was created by summing the flux between 399 and 939 nm, while the
spectroscopic light curves were created by binning the flux in a step of 20
nm, except for two 30 nm channels and one 40 nm channel at the longest
wavelengths. The time stamp was extracted from the fits header and converted
to Barycentric Julian Dates in Barycentric Dynamical Time
($\mathrm{BJD_{TDB}}$; Eastman et al. 2010).

##  3 Light-curve Analysis

###  3.1 White-light curve

Table 1: Parameters Estimated from the White-light Curve

Parameter | Prior | Posterior Estimate
---|---|---
$P$ [days] | 2.15000815(fixed) | –
$i$ [∘] | $\mathcal{U}(80,90)$ | $88.57_{-0.60}^{+0.81}$
$a$/$R_{*}$ | $\mathcal{U}(3.0,9.0)$ | $6.202_{-0.069}^{+0.054}$
$R_{p}$/$R_{*}$ | $\mathcal{U}(0.10,0.20)$ | $0.1510_{-0.0011}^{+0.0011}$
$u_{1}$ | $\mathcal{N}(0.3520,0.0340^{2})$ | $0.286_{-0.019}^{+0.021}$
$u_{2}$ | $\mathcal{N}(0.2991,0.0182^{2})$ | $0.291_{-0.018}^{+0.018}$
$T_{\mathrm{mid}}$ $\mathrm{[MJD^{\color[rgb]{0,0,1}a}]}$ | $\mathcal{U}(76.8852,76.9252)$ | $76.90416_{-0.00011}^{+0.00011}$
$\sigma_{\mathrm{w}}$ $[10^{-6}]$ | $\mathcal{U}(0.1,5000)$ | $287_{-27}^{+30}$
$\ln A$ | $\mathcal{U}(-10,-1)$ | $-5.0_{-1.1}^{+1.3}$
$\ln\tau_{1}$ | $\mathcal{U}(-6,5)$ | $0.0_{-1.1}^{+1.2}$
$\ln\tau_{2}$ | $\mathcal{U}(-5,5)$ | $2.7_{-1.1}^{+1.1}$
$c_{0}$ | $\mathcal{N}(1.00587,0.00039^{2})$ | $1.00587_{-0.00038}^{+0.00038}$
$c_{1}$ | $\mathcal{N}(-0.0431,0.0018^{2})$ | $-0.0431_{-0.0018}^{+0.0018}$
$c_{2}$ | $\mathcal{N}(-0.400,0.054^{2})$ | $-0.400_{-0.039}^{+0.040}$

${}^{\color[rgb]{0,0,1}a}\mathrm{MJD=BJD_{TDB}-2,456,500.}$

Figure 1: White-light curve of HAT-P-32 observed by P200/COSMIC on the night
of 2013 October 10. From top to bottom are (i) raw flux time series of
HAT-P-32 and its reference star, (ii) raw white-light curve (i.e., normalized
target-to-reference flux ratios), (iii) white-light curve corrected for
systematics, and (iv) best-fit light-curve residuals. The best-fit models are
shown in black. Figure 2: Inclination and semi-major axis derived in this work
compared to those in the literature.

To model the transit light curve, we used the Python package batman (Kreidberg
2015) configured with the quadratic limb-darkening law, which implements the
analytic formalism from Mandel & Agol (2002). A circular orbit was assumed.
The transit model $\mathcal{M}$ was parameterized by orbital period ($P$;
fixed to 2.15000815 days from Fowler et al. 2021), orbital inclination ($i$),
scaled semi-major axis ($a/R_{\star}$), radius ratio ($R_{p}/R_{\star}$), mid-
transit time ($T_{\mathrm{mid}}$), and limb-darkening coefficients ($u_{1}$
and $u_{2}$). Since the close companion HAT-P-32B was not spatially resolved
in our COSMIC observation, we revised the transit model as
$\mathcal{M}^{\prime}=(\mathcal{M}+f_{d})/(1+f_{d})$ to account for its
dilution. The dilution flux ratio $f_{d}=F_{B}/F_{A}$ was calculated from the
best-fit PHEONIX stellar template retrieved from the GTC/OSIRIS measurements
presented by Nortmann et al. (2016) (see Table 3 in Appendix A).

To account for the correlated systematic noise in the observed light curve, we
used the Python package george (Ambikasaran et al. 2015) to implement the
Gaussian processes (GPs; Rasmussen & Williams 2006; Gibson et al. 2012). For
the GP mean function, we adopted the transit model multiplied by a polynomial
baseline, i.e., $\mathcal{M}^{\prime}(c_{0}+c_{1}t+c_{2}t^{2})$. For the GP
covariance matrix, we used the product of two Matérn $\nu=3/2$ kernels, with
time ($t$) and spatial FWHM ($s$) as the input vectors, parameterized by an
amplitude ($A$) and two characteristic length scales ($\tau_{t}$ and
$\tau_{s}$). To account for potential underestimation of white noise, a jitter
parameter $\sigma_{j}$ was added in the quadrature sum to the nominal flux
uncertainties.

To estimate the posterior distributions of the 13 free parameters ($i$,
$a/R_{\star}$, $T_{\mathrm{mid}}$, $R_{p}/R_{\star}$, $u_{1}$, $u_{2}$,
$c_{0}$, $c_{1}$, $c_{2}$, $A$, $\tau_{t}$, $\tau_{s}$, $\sigma_{j}$), we used
the Python package emcee (Foreman-Mackey et al. 2013) to implement the affine
invariant Markov Chain Monte Carlo (MCMC) ensemble sampler. In practice, the
natural logarithmic values $\ln A$, $\ln\tau_{t}$, and $\ln\tau_{s}$ were used
in the MCMC process. A total of 32 walkers were initialized and two short
chains of 2000 steps were used for the “burn-in” phase. The final production
was created after running a long chain of 50,000 steps that were thinned by
every ten steps. We adopted uniform priors for all the parameters except for
the baseline polynomial coefficients and the limb-darkening coefficients,
which were controlled by normal priors. For the baseline polynomial
coefficients, a second-order polynomial function was fitted to the out-of-
transit flux and the resulting best-fit values and uncertainties were adopted
as the normal priors. For the limb-darkening coefficients, the prior mean and
sigma values were calculated from the ATLAS stellar models using the code
developed by Espinoza & Jordán (2015).

The white light curve and best-fit model are shown in Figure 1. The best-fit
light-curve residuals have a standard deviation of 270 ppm that is 3.6 times
photon noise. The posteriors of free parameters are listed in Table 1. The
derived transit parameters are in a broad agreement with those in the
literature. Figure 2 presents the comparison for $i$ and $a/R_{\star}$ between
this work and the other transmission spectroscopy studies along with the
discovery paper.

###  3.2 Spectroscopic light curves

Figure 3: The top panel shows the stellar spectra of HAT-P-32 and its
reference star, along with passbands used in this work marked in shaded
colors. The bottom panel presents the transmission spectrum of HAT-P-32Ab
acquired with P200/COSMIC, compared to a 1800 K 1$\times$solar cloud-free
fiducial model with Na and K but without TiO and VO.  Figure 4: Spectroscopic
light curves of HAT-P-32 acquired with P200/COSMIC. From left to right are the
original ones, the ones after removing the systematic models, and
corresponding best-fit light-curve residuals. The black solid lines show the
best-fit models.

Previous studies (Alexoudi et al. 2018, 2020) found that the shape of
transmission spectrum could vary with the adopted orbital parameters due to
the impact parameter degeneracy. In the case of HAT-P-32Ab, Alexoudi et al.
(2020) found negligible slope changes introduced by the impact parameter
degeneracy. To compare with the results from HST, we adopted the nonlinear
limb-darkening law, and fixed the four limb-darkening coefficients to the
values interpolated from the Table 3 of Alam et al. (2020).

We modeled each individual spectroscopic light curve using the same method as
that of the white-light curve, with the number of free parameters being
reduced to eight ($R_{p}/R_{\star}$, $c_{0}$, $c_{1}$, $c_{2}$, $A$,
$\tau_{t}$, $\tau_{s}$, $\sigma_{j}$) for each passband. We fixed $i$ and
$a/R_{\star}$ to the values from Hartman et al. (2011) as Alam et al. (2020)
did. We fixed $T_{\mathrm{mid}}$ to the value derived from the white-light
curve (see Table 1). Since there was no significant common-mode noise, we did
not apply the widely adopted common-mode removal technique to avoid potential
underestimation of transit depth uncertainties (Jiang et al. 2022). In the
MCMC processes of the spectroscopic light curves, we also used 32 walkers and
ran two short chains of 2000 steps for the “burn-in” phase, but created the
final production after a long chain of 5000 steps without thinning.

The adopted spectroscopic passbands are illustrated in the top panel of Figure
3, while the derived wavelength dependent planet-to-star radius ratios, i.e.,
the transmission spectrum, are shown in the bottom panel and listed in Table
3. The spectroscopic light curves and best-fit models are shown in Figure 4.
The standard deviations of the best-fit light-curve residuals for all the
passbands are 0.8–2.4$\times$ photon noise, with a median value at
1.3$\times$.

###  3.3 Transmission spectrum

Figure 5: Transmission spectra of HAT-P-32Ab acquired by Gemini-N/GMOS (blue
lower-triangles; Gibson et al. 2013), LBT/MODS (purple left-triangles; Mallonn
& Strassmeier 2016), GTC/OSIRIS (olive upper-triangles; Nortmann et al. 2016),
HST/STIS (green squares; Alam et al. 2020), and P200/COSMIC (red diamonds).
The combined optical transmission spectrum, i.e., weighted average of the five
spectra resampled to the HST/STIS passbands, is shown in black circles.

Our P200/COSMIC transmission spectrum is consistent with a flat line at a
confidence level of 0.3$\sigma$ ($\chi^{2}=6.1$ for 24 degree of freedoms,
hereafter dof). When compared to a sloped line in the form of
$a_{0}+a_{1}\ln\lambda$, we obtain $\chi^{2}=6.0$ for 23 dof at a confidence
level of 0.3$\sigma$. We further utilize the nested sampling algorithm (Feroz
& Hobson 2008) configured with 1000 live points to estimate the model evidence
$\mathcal{Z}$ for model comparison. The Python package PymultiNest (Buchner et
al. 2014) is used to implement the MULTINEST library (Feroz et al. 2009). We
interpret the model preference following the Trotta (2008) criteria, which
categorizes the value of $\ln B_{10}=\ln\mathcal{Z}_{1}-\ln\mathcal{Z}_{0}$
into inconclusive ($|\ln B_{10}|<1$), weak ($1\leq|\ln B_{10}|<2.5$), moderate
($2.5\leq|\ln B_{10}|<5$), and strong ($|\ln B_{10}|\geq 5$). We find that the
flat line has a model evidence higher than that of the sloped line
($\Delta\ln\mathcal{Z}=2.4$), indicating that it is not necessary to consider
a non-flat model to explain the P200/COSMIC transmission spectrum.

Since the optical transmission spectrum of HAT-P-32Ab has also been acquired
by Gemini-N/GMOS (Gibson et al. 2013), LBT/MODS (Mallonn & Strassmeier 2016),
GTC/OSIRIS (Nortmann et al. 2016), and HST/STIS (Alam et al. 2020), we further
investigate the consistency of transmission spectrum in the common wavelength
range of different instruments. Figure 5 shows the comparison between these
five sets of optical transmission spectra. We downsample the other
transmission spectra to the P200/COSMIC passbands with uncertainties
propagated, and estimate a difference of $\chi^{2}=6.8$ (18 dof) for
Gemini-N/GMOS, $\chi^{2}=13.7$ (24 dof) for LBT/MODS, $\chi^{2}=10.5$ (19 dof)
for GTC/OSIRIS, $\chi^{2}=8.3$ (24 dof) for HST/STIS when compared to
P200/DBSP. This reveals a good consistency between our spectrum and the
others.

Given that different instruments might host specific systematics, we decide to
combine the five spectra to alleviate the potential impact of specific
systematics. We resample the other spectra onto the HST/STIS passbands and
average these five spectra in the common passbands using the inverse square of
uncertainties as weight. The combined optical transmission spectrum is
presented in Table 4. It features an enhanced slope, driven by HST/STIS and
LBT/MODS, at $\lambda<4000$ Å, and a relatively flat continuum at
$4000<\lambda<9000$ Å in the common wavelength range of all five spectra.

##  4 Atmospheric retrieval

In order to explore the atmospheric properties of HAT-P-32Ab, we perform
Bayesian spectral retrieval analyses on the combined optical transmission
spectrum derived in Sect. 3.3 along with the HST/WFC3 and Spitzer measurements
from Alam et al. (2020). We configure PLATON (Zhang et al. 2019, 2020) to
implement forward modeling in the subsequent retrieval analyses, which
conducts 1D radiative transfer to calculate the transmission spectrum of a
hydrostatic atmosphere with equilibrium chemistry.

The atmosphere is assumed to have an isothermal temperature of
$T_{\mathrm{iso}}$, which is divided into 100 layers, log-equally spaced from
103 to 10-9 bar, with the reference planet radius ($R_{\mathrm{p,1bar}}$) set
at 1 bar. The gas absorption, collisional absorption, and scattering
absorption are taken into account. The gas abundances are obtained from the
equilibrium chemistry abundance grid pre-calculated by GGChem (Woitke et al.
2018), which is a function of species name, temperature, pressure, metallicity
($Z$), and C/O ratio222To generate the input element abundances for GGChem,
the abundances of elements above helium were scaled to $Z$ times their solar
abundances and then the C abundance was varied to meet the requested C/O ratio
(M. Zhang 2022, private communication).. A total of 34 atomic and molecular
species are considered, including: H, He, C, N, O, Na, K, $\mathrm{H}_{2}$,
$\mathrm{H}_{2}$O, $\mathrm{CH}_{4}$, CO, $\mathrm{CO}_{2}$,
$\mathrm{NH}_{3}$, $\mathrm{N}_{2}$, $\mathrm{O}_{2}$, $\mathrm{O}_{3}$, NO,
$\mathrm{NO}_{2}$, $\mathrm{C}_{2}\mathrm{H}_{2}$,
$\mathrm{C}_{2}\mathrm{H}_{4}$, $\mathrm{H}_{2}$CO, $\mathrm{H}_{2}$S, HCl,
HCN, HF, MgH, OCS, OH, $\mathrm{PH}_{3}$, SiH, SiO, $\mathrm{SO}_{2}$, TiO,
and VO. The gas opacities are calculated at a resolution of
$\lambda/\Delta\lambda=10,000$, with line lists coming from ExoMol (Tennyson &
Yurchenko 2018), HITRAN 2016 (Gordon et al. 2017), CDSD-4000 (Tashkun &
Perevalov 2011), Rey et al. (2017), and NIST. The collisional absorption
coefficients are taken from HITRAN (Richard et al. 2012; Karman et al. 2019).
The clouds are described as an optically thick cloud deck with a cloud-top
pressure of $P_{\mathrm{cloud}}$. The hazes are parameterized to have a
Rayleigh-like scattering with a slope of $\gamma$ at an amplitude of
$A_{\mathrm{scatt}}$, i.e.,
$\sigma(\lambda)=A_{\mathrm{scatt}}\sigma_{\mathrm{Rayleigh}}\lambda^{\gamma}$.

We adopt a planet mass of $0.585M_{\mathrm{J}}$ (Czesla et al. 2022) in the
forward model and use a stellar radius of $1.225R_{\odot}$ (Tregloan-Reed et
al. 2018) to obtain the transit depth. We consider three model hypotheses in
the following subsections and perform the retrieval analyses in the Bayesian
framework. We configure PymultiNest with 250 live points to implement the
nested sampling algorithm to estimate the model evidence and to explore the
posterior distributions of free parameters. As summarized in Table 2, we adopt
uniform or log-uniform priors for all the parameters.

Figure 6: Retrieved transmission spectra of HAT-P-32Ab from hypotheses of (i) uniform limb with uniform clouds (top panel), (ii) uniform limb with patchy clouds (middle panel), and (iii) two limbs with uniform clouds (bottom panel). The optical combined data are shown in circles, while the HST/WFC3 and Spitzer data from Alam et al. (2020) are shown in squares and triangles, respectively. The median, 1$\sigma$, and 2$\sigma$ confidence intervals of retrieved models are shown in solid lines with shaded areas. Table 2: Parameter Estimation and Statistics from the Atmospheric Retrievals Parameter | Description | Prior | Posterior
---|---|---|---
|  |  | Hypothesis 1 | Hypothesis 2 | Hypothesis 3
$T_{\mathrm{iso}}$ | Atmospheric temperature (K) | $\mathcal{U}(500,2300)$ | $1130^{+139}_{-134}$ | $1288^{+121}_{-119}$ | –
$T_{\mathrm{iso}}^{\mathrm{morn}}$ |  $T_{\mathrm{iso}}$ of morning limb | $\mathcal{U}(500,2300)$ | – | – | $1134^{+232}_{-194}$
$T_{\mathrm{iso}}^{\mathrm{even}}$ |  $T_{\mathrm{iso}}$ of evening limb | $\mathcal{U}(500,2300)$ | – | – | $1516^{+33}_{-44}$
$\log P_{\mathrm{cloud}}$ | Log of cloud-top pressure (bar) | $\mathcal{U}(-6,2)$ | $-0.5^{+1.6}_{-1.7}$ | $-1.1^{+2.0}_{-2.3}$ | –
$\log P_{\mathrm{cloud}}^{\mathrm{morn}}$ |  $\log P_{\mathrm{cloud}}$ of morning limb | $\mathcal{U}(-6,2)$ | – | – | $-1.8^{+2.5}_{-2.6}$
$\log P_{\mathrm{cloud}}^{\mathrm{even}}$ |  $\log P_{\mathrm{cloud}}$ of evening limb | $\mathcal{U}(-6,2)$ | – | – | $-0.5^{+1.6}_{-1.6}$
$\gamma$ | Scattering slope | $\mathcal{U}(-20,2)$ | $-1.1^{+0.2}_{-0.4}$ | $-3.0^{+0.8}_{-1.4}$ | –
$\gamma^{\mathrm{morn}}$ |  $\gamma$ of morning limb | $\mathcal{U}(-20,2)$ | – | – | $-17.1^{+3.4}_{-2.0}$
$\gamma^{\mathrm{even}}$ |  $\gamma$ of evening limb | $\mathcal{U}(-20,2)$ | – | – | $-15.7^{+4.5}_{-2.7}$
$\mathrm{log}A_{\mathrm{scatt}}$ | Log of scattering factor | $\mathcal{U}(-4,12)$ | $3.5^{+0.5}_{-0.5}$ | $6.3^{+0.6}_{-0.8}$ | –
$\mathrm{log}A_{\mathrm{scatt}}^{\mathrm{morn}}$ |  $\mathrm{log}A_{\mathrm{scatt}}$ of morning limb | $\mathcal{U}(-4,12)$ | – | – | $11.0^{+0.5}_{-0.7}$
$\mathrm{log}A_{\mathrm{scatt}}^{\mathrm{even}}$ |  $\mathrm{log}A_{\mathrm{scatt}}$ of evening limb | $\mathcal{U}(-4,12)$ | – | – | $-2.5^{+1.7}_{-1.0}$
$\mathrm{C/O}$ | Carbon-to-oxygen ratio | $\mathcal{U}(0.05,2)$ | $0.30^{+0.21}_{-0.17}$ | $0.29^{+0.18}_{-0.15}$ | –
$\mathrm{C/O}^{\mathrm{morn}}$ |  $\mathrm{C/O}$ of morning limb | $\mathcal{U}(0.05,2)$ | – | – | $0.82^{+0.76}_{-0.47}$
$\mathrm{C/O}^{\mathrm{even}}$ |  $\mathrm{C/O}$ of evening limb | $\mathcal{U}(0.05,2)$ | – | – | $0.73^{+0.03}_{-0.04}$
$R_{\mathrm{1bar}}$ | Planet radius at 1 bar ($R_{\mathrm{J}}$) | $\mathcal{U}(0.5,2)$ | $1.704^{+0.016}_{-0.021}$ | $1.686^{+0.028}_{-0.028}$ | $1.704^{+0.014}_{-0.014}$
$\mathrm{log}Z$ | Log of metallicity ($Z_{\odot}$) | $\mathcal{U}(-1,3)$ | $0.96^{+0.78}_{-0.38}$ | $1.89^{+0.25}_{-0.28}$ | $2.13^{+0.13}_{-0.12}$
$\phi$ | Fraction of cloud coverage | $\mathcal{U}(0,1)$ | – | $0.61^{+0.05}_{-0.07}$ | –
$\ln\mathcal{Z}$ | Model evidence | – | $366.1\pm 0.3$ | $369.7\pm 0.3$ | $374.2\pm 0.3$
$\chi^{2}_{\mathrm{MAP}}$ |  $\chi^{2}$ of maximum a posterior model | – | 66.7 | 58.3 | 41.7
dof | Degree of freedom | – | 44 | 43 | 39

###  4.1 Hypothesis 1: uniform limb with uniform clouds

By default, PLATON assumes an isothermal 1D atmosphere with uniform clouds
(hereafter Hypothesis 1). The forward model consists of seven free parameters:
$T_{\mathrm{iso}}$, $\log P_{\mathrm{cloud}}$, $\gamma$, $\log
A_{\mathrm{scatt}}$, $\mathrm{C/O}$, $R_{\mathrm{1bar}}$, and $\log Z$. From
the retrieval of Hypothesis 1, we obtain an isothermal temperature of
$1130^{+139}_{-134}$ K, a subsolar C/O ratio of $0.30^{+0.21}_{-0.17}$, and a
supersolar metallicity of $0.96^{+0.78}_{-0.38}$ dex that is slightly lower
than the value retrieved from the HST and Spitzer measurements using the
default setup of PLATON (Alam et al. 2020). Our scattering slope of
$-1.1^{+0.2}_{-0.4}$ is much shallower than the value of $-9.0^{+1.0}_{-0.6}$
reported by Alam et al. (2020), while our scattering amplitude of $10^{3.5\pm
0.5}$ is much stronger than their $10^{1.0\pm 0.4}$. This is mainly owing to
the fact that the combined optical transmission spectrum is flatter than the
HST/STIS spectrum within the wavelength range of 0.4–0.9 $\mu$m. However, we
also note that the adopted priors and sampling algorithms might also introduce
differences in the posterior estimates. While it is not clear what priors and
sampling algorithms were used in Alam et al. (2020), the comparisons between
parameters derived from our work and Alam et al. (2020) should be taken with a
grain of salt. We retrieve a loosely constrained cloud-top pressure with a 90%
lower limit of $\sim$2.6 mbar. The joint posterior distributions of all the
free parameters are presented in Figure 8.

The retrieved maximum a posterior (MAP) model can fit all the data at a
2.4$\sigma$ confidence level ($\chi^{2}_{\mathrm{MAP}}=66.7$ for 44 dof). As
shown in the top panel of Figure 6, the retrieved models of Hypothesis 1 agree
well with most of the data within 0.4–0.9 and 1.1–1.7 $\mu$m, but fail to fit
a few data points, in particular those at the blue end and those in the mid
infrared. The blue end shows an enhanced slope toward shorter wavelengths. The
data point at 0.985 $\mu$m could be an outlier of the overall spectral shape
given that the neighboring wavelength range of 0.5–0.9 $\mu$m is the average
spectrum from five independent instruments while the one at 0.985 $\mu$m comes
from the average of only two. In the mid infrared, the two Spitzer data points
show clear offsets from the model predictions, and are also deviating from the
overall optical-to-infrared trend.

###  4.2 Hypothesis 2: uniform limb with patchy clouds

Three dimensional general circulation models (GCM) have predicted that the
atmospheric circulation of hot Jupiters could induce asymmetries in
temperature structure, chemistry, and clouds between the morning and evening
limbs (e.g., Showman et al. 2009; Parmentier et al. 2016; Helling et al.
2019), resulting in different spectral signatures in the limb transmission
spectra (Fortney et al. 2010). While it is possible to directly measure such
asymmetries in the light curves if ultra-high photometric precision can be
achieved (von Paris et al. 2016; Powell et al. 2019; Espinoza & Jones 2021),
we seek solutions through approximating the atmosphere by linear combinations
of multi-sector 1D atmospheric models (Line & Parmentier 2016; Kempton et al.
2017; Welbanks & Madhusudhan 2022).

In Hypothesis 2, we assume that the atmosphere is effectively composed of one
clear sector and one cloudy sector, with $\phi$ being the faction of cloud
coverage. Following MacDonald & Madhusudhan (2017) and Welbanks & Madhusudhan
(2021), the cloudy sector has a Rayleigh-like scattering haze above a gray
cloud deck. This amounts to eight free parameters: $T_{\mathrm{iso}}$, $\log
P_{\mathrm{cloud}}$, $\gamma$, $\log A_{\mathrm{scatt}}$, $\mathrm{C/O}$,
$R_{\mathrm{1bar}}$, $\log Z$, and $\phi$. From the retrieval of Hypothesis 2,
we obtain a cloud coverage of $61^{+5}_{-7}$ %, an isothermal temperature of
$1288^{+121}_{-119}$ K, a subsolar C/O ratio of $0.29^{+0.18}_{-0.15}$, and a
supersolar metallicity of $1.89^{+0.25}_{-0.28}$ dex. For the haze property,
the scattering slope of $-3.0^{+0.8}_{-1.4}$ is slightly steeper, while the
scattering amplitude increases significantly to $10^{6.3^{+0.6}_{-0.8}}$. The
cloud-top pressure is again loosely constrained, with a 90% lower limit of
$\sim$0.1 mbar. The joint posterior distributions of all the free parameters
are presented in Figure 9.

The retrieved models of Hypothesis 2 are shown in the middle panel of Figure
6. The MAP model of Hypothesis 2 can fit all the data at the 1.9$\sigma$
confidence level ($\chi^{2}_{\mathrm{MAP}}=58.3$ for 43 dof), slightly better
than Hypothesis 1. The major improvement in Hypothesis 2 occurs in the
optical, where the new retrieval suggests the presence of pressure-broadened
line wings of Na and K. The strong scattering amplitude introduced by the haze
in the cloudy sector also slightly moves downwards the model in the mid-
infrared. However, the enhanced slope at the blue end and the Spitzer data
points are still not well explained.

Figure 7: The first row presents the confidence regions of the derived
morning-limb and evening-limb transmission spectra based on the two-limb
retrieval in Hypothesis 3. The second and third rows show the marginalized
posterior distributions of the volume mixing ratios of the major spectral
tracing species retrieved in Hypothesis 3, including H2O, CO2, Na, K, TiO, and
VO.

###  4.3 Hypothesis 3: two limbs with uniform clouds

While the patchy clouds could result in asymmetries in the limb, as assumed in
Hypothesis 2, it is also possible that the morning limb and evening limb are
indeed asymmetric. Similar to the retrieval frameworks adopted in Espinoza &
Jones (2021) and Welbanks & Madhusudhan (2022), we assume in Hypothesis 3 that
the atmosphere can be equivalent to a cooler morning limb and a warmer evening
limb with equal weights, which have separate isothermal temperatures
($T_{\mathrm{iso}}^{\mathrm{morn}}$ and $T_{\mathrm{iso}}^{\mathrm{even}}$).
For both limbs, uniform clouds are adopted individually, each composed of a
Rayleigh-like scattering haze above a gray cloud deck. The 3D GCM studies
suggest that planets with intermediate temperatures (1400–1800 K) could have
homogeneous mean molecular weight but intermittent C/O ratio across observable
planet disk (Helling et al. 2022). Therefore, the C/O ratio is considered to
be limb-dependent, while the metallicity is assumed to be the same for both
limbs. Consequently, there are 12 free parameters:
$T_{\mathrm{iso}}^{\mathrm{morn}}$, $T_{\mathrm{iso}}^{\mathrm{even}}$, $\log
P_{\mathrm{cloud}}^{\mathrm{morn}}$, $\log
P_{\mathrm{cloud}}^{\mathrm{even}}$, $\gamma^{\mathrm{morn}}$,
$\gamma^{\mathrm{even}}$, $\log A_{\mathrm{scatt}}^{\mathrm{morn}}$, $\log
A_{\mathrm{scatt}}^{\mathrm{even}}$, $\mathrm{C/O}^{\mathrm{morn}}$,
$\mathrm{C/O}^{\mathrm{even}}$, $R_{\mathrm{1bar}}$, and $\log Z$.

From the retrieval of Hypothesis 3, we obtain a supersolar metallicity of
$2.13^{+0.13}_{-0.12}$ dex. The retrieved C/O ratios are supersolar and almost
the same for both limbs ($\sim$0.7), although the constraints are tighter for
the evening limb while looser for the morning limb. The retrieved temperatures
are $1134^{+232}_{-194}$ K for the morning limb and $1516^{+33}_{-44}$ K for
the evening limb, resulting in an evening-morning difference of $\Delta
T=373^{+192}_{-224}$ K. This is consistent with the evening-averaged and
morning-averaged temperature profiles within 1–10 mbar derived from cloud-free
GCM simulations for another hot Jupiter with similar physical properties
(WASP-17b; Kataria et al. 2016). The scattering slopes are loosely
constrained, with a 90% upper limit of $\gamma<-12.5$ for the morning limb and
$\gamma<-9.8$ for the evening limb. The scattering amplitudes are much
stronger in the morning limb ($A_{\mathrm{scatt}}=10^{11.0^{+0.5}_{-0.7}}$)
than in the evening limb ($A_{\mathrm{scatt}}=10^{-2.5^{+1.7}_{-1.0}}$),
indicating that the morning limb is dominated by a very strong haze and that
the evening limb is almost haze free. Like Hypotheses 1 and 2, the cloud-top
pressure is also loosely constrained in Hypothesis 3, with a 90% lower limit
of 0.02 mbar in the morning limb and 4 mbar in the evening limb. However, the
lower limit does indicate that the cloud deck in the evening might be deeper.
The joint posterior distributions of all the free parameters are presented in
Figure 10.

The retrieved models of Hypothesis 3 are shown in the bottom panel of Figure
6. The MAP model of Hypothesis 3 can fit all the data at the 0.9$\sigma$
confidence level ($\chi^{2}_{\mathrm{MAP}}=41.7$ for 39 dof), better than both
Hypotheses 1 and 2. The enhanced slope at the blue end and the Spitzer data
points can be reasonably explained by the models of Hypothesis 3. The optical
spectral features in the MAP model are dominated by the opacities of TiO, VO,
MgH, Na, and K in the evening limb. If the data point at 0.985 $\mu$m is
excluded as an outlier, the values of $\chi^{2}_{\mathrm{MAP}}$ are reduced to
61.3, 51.1, and 35.8 for Hypotheses 1, 2, and 3, respectively, corresponding
to goodness of fit at 2.0$\sigma$, 1.3$\sigma$, and 0.5$\sigma$.

##  5 Discussion

Our Bayesian spectral retrieval analyses reveal that the current 0.3–5.1
$\mu$m transmission spectrum data set of HAT-P-32Ab strongly favors the
hypothesis of two limbs with uniform clouds as opposed to the hypotheses of
uniform limb with either uniform clouds or patchy clouds. The retrieved models
from the two-limb hypothesis can fit the current data set reasonably well,
indicating that the atmosphere of HAT-P-32Ab can be equivalent to a warmer
hazy-free component and a cooler hazy component, which we attribute to the
evening and morning limbs. Although the retrieved C/O ratios are almost the
same ($\sim$0.7) on both limbs, it is largely unconstrained in the morning
limb. Figure 7 presents the confidence regions of the derived morning-limb and
evening-limb transmission spectra based on the two-limb retrieval. It also
shows the marginalized limb-dependent posteriors of the major spectral tracers
that are derived from our equilibrium chemistry retrieval, of which Na, K,
TiO, and VO contribute to the optical wavelengths, H2O contributes to the
infrared wavelengths, and CO2 contributes to the Spitzer bands. The most
evident abundance changes occur in TiO and VO, which are likely depleted in
the cooler morning limb due to either condensation or nightside cold trap,
consistent with GCM predictions (Parmentier et al. 2016; Helling et al. 2022).

Based on the assumption of equilibrium chemistry, the atmospheric metallicity
of HAT-P-32Ab is constrained to be $134^{+45}_{-33}$ times solar metallicity,
which is strongly enhanced over its solar-metallicity host star
($\mathrm{[Fe/H]}=-0.04\pm 0.08$; Hartman et al. 2011) and much stronger than
the observed planet mass-metallicity enrichment trend at the mass of 0.585
$M_{\mathrm{J}}$ (e.g., Kreidberg et al. 2014; Wakeford et al. 2017; Welbanks
et al. 2019). The enrichment of atmospheric metallicity toward lower planet
mass has been suggested as a potential link to the core-accretion planet
formation theory (Miller & Fortney 2011; Fortney et al. 2013; Mordasini et al.
2014; Thorngren et al. 2016). However, Welbanks et al. (2019) found that the
enrichment trends could differ if it was based on different species (e.g.,
CH4, H2O), suggesting that the equilibrium chemistry assumption might not work
in general. Future observations with James Webb Space Telescope (JWST) that
cover a variety of molecular species in the infrared wavelengths will enable
us to answer whether or not equilibrium chemistry is reasonable for
HAT-P-32Ab.

The powerful capability of JWST could also enable a direct measurement of the
transmission spectrum for each individual limb, which can independently
confirm whether the morning-evening asymmetries are present. The morning-to-
evening transit depth differences derived from our retrieval have a maximum
value of $\sim$990 ppm within 0.3–5.1 $\mu$m, which could induce an asymmetric
signature as large as $\sim$400 ppm during ingress or egress in the light-
curve residuals when compared to the symmetric light-curve model depending on
the orientation of the semi-circles of the evening and morning limbs.
Asymmetric signatures of such amplitude are easily detectable by JWST
according to the simulations on the hot Jupiter HAT-P-41b orbiting a star of
similar spectral type and brightness (Espinoza & Jones 2021).

We note that the major driver to favor the two-limb hypothesis in this work
could probably come from the enhanced slope at the blue end and the two
Spitzer data points that have a potential CO2 spectral signature but are at a
lower level than the 0.3–1.7 $\mu$m wavelength range. The offset between the
two Spitzer data points and other wavelengths could also come from
contamination of stellar activity or instrumental biases. The five-season
photometric monitoring reveals that HAT-P-32A is constant on night-to-night
timescales within the precision of $\sim$2 mmag and likely to be constant on
year-to-year timescales (Nikolov et al. 2018b; Alam et al. 2020). The
consistency among the optical transmission spectra (0.5–0.9 $\mu$m)
independently acquired by five instruments and the consistency between optical
(0.3–0.9 $\mu$m) and near-infrared (1.1–1.7 $\mu$m) both confirm the inactive
nature of HAT-P-32A. The large instantaneous wavelength coverage of JWST will
further confirm whether the downward mid-infrared spectral signature of CO2 is
of instrumental origin or a sign of morning-evening asymmetry (The JWST
Transiting Exoplanet Community Early Release Science Team et al. 2022).

##  6 Conclusions

We obtained an optical transmission spectrum for the hot Jupiter HAT-P-32Ab
within the wavelength range of 399–939 nm using P200/COSMIC. We derived a
combined optical transmission spectrum by weighted averaging the measurements
from five independent instruments including HST/STIS, Gemini-N/GMOS, LBT/MODS,
GTC/OSIRIS, and P200/COSMIC. We performed Bayesian spectral retrievals on the
combined optical spectrum along with the HST/WFC3 and Spitzer measurements,
with the hypotheses of (i) uniform limb with uniform clouds, (ii) uniform limb
with patchy clouds, and (iii) two limbs with uniform clouds. We conclude that:

  * 1.

The current 0.3–5.1 $\mu$m transmission spectrum of HAT-P-32Ab is
characterized by an enhanced scattering slope at the blue-optical, a
relatively flat continuum but consistent with spectral signatures of TiO, VO,
Na, K, and MgH in the optical band, a water absorption feature at 1.4 $\mu$m,
and a CO2 absorption feature at 4.4 $\mu$m.

  * 2.

The current data set of HAT-P-32Ab reveals an atmosphere of high metallicity
($\mathrm{[Fe/H]}=2.13^{+0.13}_{-0.12}$), and can be well explained by a two-
limb approximation, with the warmer evening limb being haze-free and the
cooler morning limb being strongly hazy. The morning-evening temperature
difference of $373^{+192}_{-224}$ K is consistent with the GCM predictions.

  * 3.

HAT-P-32Ab is a prior target for follow-up observations with JWST transmission
spectroscopy. The inferred morning and evening limbs, if confirmed, will
enable direct measurements of limb spectra through asymmetric light-curve
modeling.

###### Acknowledgements.

G.C. acknowledges the support by the B-type Strategic Priority Program of the
Chinese Academy of Sciences (grant No. XDB41000000), the National Natural
Science Foundation of China (grant Nos. 42075122, 12122308), Youth Innovation
Promotion Association CAS (2021315). H.Z. thanks the Space debris and NEO
research project (grant Nos. KJSP2020020204, KJSP2020020102), Civil Aerospace
pre-research project (grant No. D020304). G.C. and H.Z. also thank the Minor
Planet Foundation. The authors would like to thank the anonymous referee for
the constructive comments on the manuscript, and Carolyn Heffner, Kajsa
Peffer, Kevin Rykoski, and Jennifer Milburn for their great supports during
the observations. This research uses data obtained through the Telescope
Access Program (TAP), which has been funded by the TAP member institutes.
Observations obtained with the Hale Telescope at Palomar Observatory were
obtained as part of an agreement between the National Astronomical
Observatories, Chinese Academy of Sciences, and the California Institute of
Technology.

## References

  * Adams et al. (2013) Adams, E. R., Dupree, A. K., Kulesa, C., & McCarthy, D. 2013, AJ, 146, 9
  * Alam et al. (2020) Alam, M. K., López-Morales, M., Nikolov, N., et al. 2020, AJ, 160, 51
  * Alexoudi et al. (2020) Alexoudi, X., Mallonn, M., Keles, E., et al. 2020, A&A, 640, A134
  * Alexoudi et al. (2018) Alexoudi, X., Mallonn, M., von Essen, C., et al. 2018, A&A, 620, A142
  * Ambikasaran et al. (2015) Ambikasaran, S., Foreman-Mackey, D., Greengard, L., Hogg, D. W., & O’Neil, M. 2015, IEEE Transactions on Pattern Analysis and Machine Intelligence, 38, 252
  * Buchner et al. (2014) Buchner, J., Georgakakis, A., Nandra, K., et al. 2014, A&A, 564, A125
  * Chen et al. (2018) Chen, G., Pallé, E., Welbanks, L., et al. 2018, A&A, 616, A145
  * Chen et al. (2021) Chen, G., Pallé, E., Parviainen, H., et al. 2021, MNRAS, 500, 5420
  * Cont et al. (2022) Cont, D., Yan, F., Reiners, A., et al. 2022, A&A, 668, A53
  * Czesla et al. (2022) Czesla, S., Lampón, M., Sanz-Forcada, J., et al. 2022, A&A, 657, A6
  * Damiano et al. (2017) Damiano, M., Morello, G., Tsiaras, A., Zingales, T., & Tinetti, G. 2017, AJ, 154, 39
  * Deming et al. (2013) Deming, D., Wilkins, A., McCullough, P., et al. 2013, ApJ, 774, 95
  * Eastman et al. (2010) Eastman, J., Siverd, R., & Gaudi, B. S. 2010, PASP, 122, 935
  * Ehrenreich et al. (2020) Ehrenreich, D., Lovis, C., Allart, R., et al. 2020, Nature, 580, 597
  * Espinoza & Jones (2021) Espinoza, N., & Jones, K. 2021, AJ, 162, 165
  * Espinoza & Jordán (2015) Espinoza, N., & Jordán, A. 2015, MNRAS, 450, 1879
  * Feroz & Hobson (2008) Feroz, F., & Hobson, M. P. 2008, MNRAS, 384, 449
  * Feroz et al. (2009) Feroz, F., Hobson, M. P., & Bridges, M. 2009, MNRAS, 398, 1601
  * Foreman-Mackey et al. (2013) Foreman-Mackey, D., Hogg, D. W., Lang, D., & Goodman, J. 2013, PASP, 125, 306
  * Fortney et al. (2013) Fortney, J. J., Mordasini, C., Nettelmann, N., et al. 2013, ApJ, 775, 80
  * Fortney et al. (2010) Fortney, J. J., Shabram, M., Showman, A. P., et al. 2010, ApJ, 709, 1396
  * Fowler et al. (2021) Fowler, M. J. F., Sienkiewicz, F. F., Zellem, R. T., & Dussault, M. E. 2021, Journal of the British Astronomical Association, 131, 359
  * Gibson et al. (2013) Gibson, N. P., Aigrain, S., Barstow, J. K., et al. 2013, MNRAS, 436, 2974
  * Gibson et al. (2012) Gibson, N. P., Aigrain, S., Roberts, S., et al. 2012, MNRAS, 419, 2683
  * Gordon et al. (2017) Gordon, I. E., Rothman, L. S., Hill, C., et al. 2017, J. Quant. Spec. Radiat. Transf., 203, 3
  * Hartman et al. (2011) Hartman, J. D., Bakos, G. Á., Torres, G., et al. 2011, ApJ, 742, 59
  * Helling et al. (2019) Helling, C., Iro, N., Corrales, L., et al. 2019, A&A, 631, A79
  * Helling et al. (2021) Helling, C., Lewis, D., Samra, D., et al. 2021, A&A, 649, A44
  * Helling et al. (2022) Helling, C., Samra, D., Lewis, D., et al. 2022, arXiv e-prints, arXiv:2208.05562
  * Horne (1986) Horne, K. 1986, PASP, 98, 609
  * Hubeny et al. (2003) Hubeny, I., Burrows, A., & Sudarsky, D. 2003, ApJ, 594, 1011
  * Jiang et al. (2022) Jiang, C., Chen, G., Pallé, E., et al. 2022, A&A, 664, A50
  * Karman et al. (2019) Karman, T., Gordon, I. E., van der Avoird, A., et al. 2019, Icarus, 328, 160
  * Kataria et al. (2016) Kataria, T., Sing, D. K., Lewis, N. K., et al. 2016, ApJ, 821, 9
  * Kells et al. (1998) Kells, W., Dressler, A., Sivaramakrishnan, A., et al. 1998, PASP, 110, 1487
  * Kempton et al. (2017) Kempton, E. M. R., Bean, J. L., & Parmentier, V. 2017, ApJ, 845, L20
  * Knutson et al. (2010) Knutson, H. A., Howard, A. W., & Isaacson, H. 2010, ApJ, 720, 1569
  * Kreidberg (2015) Kreidberg, L. 2015, PASP, 127, 1161
  * Kreidberg et al. (2014) Kreidberg, L., Bean, J. L., Désert, J.-M., et al. 2014, ApJ, 793, L27
  * Line & Parmentier (2016) Line, M. R., & Parmentier, V. 2016, ApJ, 820, 78
  * MacDonald & Madhusudhan (2017) MacDonald, R. J., & Madhusudhan, N. 2017, MNRAS, 469, 1979
  * Madhusudhan (2012) Madhusudhan, N. 2012, ApJ, 758, 36
  * Madhusudhan (2019) Madhusudhan, N. 2019, ARA&A, 57, 617
  * Madhusudhan & Seager (2009) Madhusudhan, N., & Seager, S. 2009, ApJ, 707, 24
  * Mallonn & Strassmeier (2016) Mallonn, M., & Strassmeier, K. G. 2016, A&A, 590, A100
  * Mallonn & Wakeford (2017) Mallonn, M., & Wakeford, H. R. 2017, Astronomische Nachrichten, 338, 773
  * Mallonn et al. (2016) Mallonn, M., Bernt, I., Herrero, E., et al. 2016, MNRAS, 463, 604
  * Mandel & Agol (2002) Mandel, K., & Agol, E. 2002, ApJ, 580, L171
  * Miller & Fortney (2011) Miller, N., & Fortney, J. J. 2011, ApJ, 736, L29
  * Mordasini et al. (2014) Mordasini, C., Klahr, H., Alibert, Y., Miller, N., & Henning, T. 2014, A&A, 566, A141
  * Nikolov et al. (2018a) Nikolov, N., Sing, D. K., Fortney, J. J., et al. 2018a, Nature, 557, 526
  * Nikolov et al. (2018b) Nikolov, N., Sing, D. K., Goyal, J., et al. 2018b, MNRAS, 474, 1705
  * Nortmann et al. (2016) Nortmann, L., Pallé, E., Murgas, F., et al. 2016, A&A, 594, A65
  * Parmentier et al. (2016) Parmentier, V., Fortney, J. J., Showman, A. P., Morley, C., & Marley, M. S. 2016, ApJ, 828, 22
  * Parmentier et al. (2013) Parmentier, V., Showman, A. P., & Lian, Y. 2013, A&A, 558, A91
  * Parmentier et al. (2018) Parmentier, V., Line, M. R., Bean, J. L., et al. 2018, A&A, 617, A110
  * Powell et al. (2019) Powell, D., Louden, T., Kreidberg, L., et al. 2019, ApJ, 887, 170
  * Rasmussen & Williams (2006) Rasmussen, C. E., & Williams, C. K. I. 2006, Gaussian Processes for Machine Learning (The MIT Press)
  * Rey et al. (2017) Rey, M., Nikitin, A. V., & Tyuterev, V. G. 2017, ApJ, 847, 105
  * Richard et al. (2012) Richard, C., Gordon, I. E., Rothman, L. S., et al. 2012, J. Quant. Spec. Radiat. Transf., 113, 1276
  * Seager & Sasselov (2000) Seager, S., & Sasselov, D. D. 2000, ApJ, 537, 916
  * Showman et al. (2009) Showman, A. P., Fortney, J. J., Lian, Y., et al. 2009, ApJ, 699, 564
  * Spiegel et al. (2009) Spiegel, D. S., Silverio, K., & Burrows, A. 2009, ApJ, 699, 1487
  * Tashkun & Perevalov (2011) Tashkun, S. A., & Perevalov, V. I. 2011, J. Quant. Spec. Radiat. Transf., 112, 1403
  * Tennyson & Yurchenko (2018) Tennyson, J., & Yurchenko, S. 2018, Atoms, 6, 26
  * The JWST Transiting Exoplanet Community Early Release Science Team et al. (2022) The JWST Transiting Exoplanet Community Early Release Science Team, Ahrer, E.-M., Alderson, L., et al. 2022, arXiv e-prints, arXiv:2208.11692
  * Thorngren et al. (2016) Thorngren, D. P., Fortney, J. J., Murray-Clay, R. A., & Lopez, E. D. 2016, ApJ, 831, 64
  * Tody (1993) Tody, D. 1993, in Astronomical Society of the Pacific Conference Series, Vol. 52, Astronomical Data Analysis Software and Systems II, ed. R. J. Hanisch, R. J. V. Brissenden, & J. Barnes, 173
  * Tregloan-Reed et al. (2018) Tregloan-Reed, J., Southworth, J., Mancini, L., et al. 2018, MNRAS, 474, 5485
  * Trotta (2008) Trotta, R. 2008, Contemporary Physics, 49, 71
  * van Sluijs et al. (2022) van Sluijs, L., Birkby, J. L., Lothringer, J., et al. 2022, arXiv e-prints, arXiv:2203.13234
  * von Paris et al. (2016) von Paris, P., Gratier, P., Bordé, P., Leconte, J., & Selsis, F. 2016, A&A, 589, A52
  * Wakeford et al. (2017) Wakeford, H. R., Sing, D. K., Kataria, T., et al. 2017, Science, 356, 628
  * Welbanks & Madhusudhan (2021) Welbanks, L., & Madhusudhan, N. 2021, ApJ, 913, 114
  * Welbanks & Madhusudhan (2022) Welbanks, L., & Madhusudhan, N. 2022, ApJ, 933, 79
  * Welbanks et al. (2019) Welbanks, L., Madhusudhan, N., Allard, N. F., et al. 2019, ApJ, 887, L20
  * Woitke et al. (2018) Woitke, P., Helling, C., Hunter, G. H., et al. 2018, A&A, 614, A1
  * Zhang et al. (2019) Zhang, M., Chachan, Y., Kempton, E. M. R., & Knutson, H. A. 2019, PASP, 131, 034501
  * Zhang et al. (2020) Zhang, M., Chachan, Y., Kempton, E. M. R., Knutson, H. A., & Chang, W. H. 2020, ApJ, 899, 27
  * Zhao et al. (2014) Zhao, M., O’Rourke, J. G., Wright, J. T., et al. 2014, ApJ, 796, 115

##  Appendix A Additional Tables and Figures

Table 3 presents the transmission spectrum of HAT-P-32Ab measured by
P200/COSMIC. Table 4 presents the combined optical transmission spectrum of
HAT-P-32Ab, which is a weighted average of transmission spectra measured by
five independent instruments, including Gemini-N/GMOS, LBT/MODS, GTC/OSIRIS,
HST/STIS, and P200/COSMIC.

Figure 8, 9, and 10 show the joint posterior distributions of Hypotheses 1, 2,
3, respectively.

Table 3: P200/COSMIC Transmission Spectrum of HAT-P-32Ab and the Adopted Dilution Flux Ratios Wavelength (Å) | $F_{B}/F_{A}$ | $R_{p}/R_{\star}$
---|---|---
3990–4190 | 0.00051 | $0.1514^{+0.0030}_{-0.0034}$
4190–4390 | 0.00047 | $0.1510^{+0.0024}_{-0.0025}$
4390–4590 | 0.00066 | $0.1517^{+0.0042}_{-0.0039}$
4590–4790 | 0.00063 | $0.1504^{+0.0022}_{-0.0020}$
4790–4990 | 0.00079 | $0.1486^{+0.0018}_{-0.0018}$
4990–5190 | 0.00077 | $0.1515^{+0.0013}_{-0.0013}$
5190–5390 | 0.00137 | $0.1513^{+0.0019}_{-0.0018}$
5390–5590 | 0.00150 | $0.1499^{+0.0016}_{-0.0016}$
5590–5790 | 0.00169 | $0.1497^{+0.0016}_{-0.0016}$
5790–5990 | 0.00148 | $0.1511^{+0.0015}_{-0.0015}$
5990–6190 | 0.00232 | $0.1520^{+0.0016}_{-0.0017}$
6190–6390 | 0.00183 | $0.1509^{+0.0015}_{-0.0016}$
6390–6590 | 0.00311 | $0.1503^{+0.0013}_{-0.0013}$
6590–6790 | 0.00263 | $0.1504^{+0.0017}_{-0.0018}$
6790–6990 | 0.00281 | $0.1519^{+0.0015}_{-0.0016}$
6990–7190 | 0.00379 | $0.1514^{+0.0017}_{-0.0018}$
7190–7390 | 0.00556 | $0.1506^{+0.0017}_{-0.0018}$
7390–7590 | 0.00846 | $0.1514^{+0.0018}_{-0.0018}$
7590–7790 | 0.00673 | $0.1516^{+0.0025}_{-0.0028}$
7790–7990 | 0.00821 | $0.1514^{+0.0030}_{-0.0030}$
7990–8190 | 0.01062 | $0.1519^{+0.0025}_{-0.0027}$
8190–8390 | 0.01113 | $0.1534^{+0.0022}_{-0.0023}$
8390–8690 | 0.01090 | $0.1503^{+0.0030}_{-0.0031}$
8690–8990 | 0.01286 | $0.1503^{+0.0026}_{-0.0025}$
8990–9390 | 0.01462 | $0.1493^{+0.0022}_{-0.0022}$
Table 4: Combined Optical Transmission Spectrum of HAT-P-32Ab Wavelength (Å) | $R_{p}/R_{\star}$
---|---
3020–3300 | $0.15466^{+0.00189}_{-0.00189}$
3300–3700 | $0.15287^{+0.00076}_{-0.00076}$
3700–3950 | $0.15218^{+0.00061}_{-0.00061}$
3950–4200 | $0.15141^{+0.00040}_{-0.00040}$
4200–4350 | $0.15139^{+0.00056}_{-0.00056}$
4350–4500 | $0.15146^{+0.00043}_{-0.00043}$
4500–4650 | $0.15127^{+0.00054}_{-0.00054}$
4650–4800 | $0.15122^{+0.00043}_{-0.00043}$
4800–4950 | $0.15132^{+0.00041}_{-0.00041}$
4950–5100 | $0.15115^{+0.00035}_{-0.00035}$
5100–5250 | $0.15105^{+0.00035}_{-0.00035}$
5250–5400 | $0.15138^{+0.00028}_{-0.00028}$
5400–5550 | $0.15146^{+0.00027}_{-0.00027}$
5550–5700 | $0.15164^{+0.00029}_{-0.00029}$
5700–6000 | $0.15174^{+0.00018}_{-0.00018}$
6000–6300 | $0.15130^{+0.00023}_{-0.00023}$
6300–6500 | $0.15138^{+0.00037}_{-0.00037}$
6500–6700 | $0.15129^{+0.00030}_{-0.00030}$
6700–6900 | $0.15103^{+0.00031}_{-0.00031}$
6900–7100 | $0.15113^{+0.00027}_{-0.00027}$
7100–7300 | $0.15154^{+0.00026}_{-0.00026}$
7300–7500 | $0.15121^{+0.00021}_{-0.00021}$
7500–7700 | $0.15153^{+0.00027}_{-0.00027}$
7700–8100 | $0.15129^{+0.00019}_{-0.00019}$
8100–8350 | $0.15091^{+0.00024}_{-0.00024}$
8350–8600 | $0.15063^{+0.00025}_{-0.00025}$
8600–8850 | $0.15107^{+0.00030}_{-0.00030}$
8850–9100 | $0.15070^{+0.00035}_{-0.00035}$
9100–9500 | $0.15059^{+0.00038}_{-0.00038}$
9500–10200 | $0.14903^{+0.00080}_{-0.00080}$
Figure 8: Corner plots of the retrieval of Hypothesis 1, where uniform limb
with uniform clouds is assumed. Figure 9: Corner plots of the retrieval of
Hypothesis 2, where uniform limb with patchy clouds is assumed. Figure 10:
Corner plots of the retrieval of Hypothesis 3, where two limbs with uniform
clouds are assumed.


# Jacob’s ladders, new properties of the function
$\arg\zeta\left(\frac{1}{2}+it\right)$ and corresponding metamorphoses

Jan Moser  Department of Mathematical Analysis and Numerical Mathematics,
Comenius University, Mlynska Dolina M105, 842 48 Bratislava, SLOVAKIA
<EMAIL_ADDRESS>
###### Abstract.

The notion of the Jacob’s ladders, reversely iterated integrals and the
$\zeta$-factorization is used in this paper in order to obtain new results in
study of the function $\arg\zeta\left(\frac{1}{2}+it\right)$. Namely, we
obtain new formulae for non-local and non-linear interaction of the functions
$|\zeta\left(\frac{1}{2}+it\right)|$ and
$\arg\zeta\left(\frac{1}{2}+it\right)$, and also a set of metamorphoses of the
oscillating Q-system.

###### Key words and phrases:

Riemann zeta-function

##  1\. Introduction

### 1.1.

Let us denote by $N(T)$ the number of zeroes $\beta+i\gamma$ of the
$\zeta(s)$-function such that

$\beta\in(0,1),\ \gamma\in(0,T).$

We suppose that $T$ is not equal to any $\gamma$. Otherwise, we put

$N(T)=\frac{1}{2}\lim_{\epsilon\to 0^{+}}[N(T+\epsilon)+N(T-\epsilon)].$

It us well-know that

$N(T)=\frac{1}{2\pi}T\ln\frac{T}{2\pi
e}+\frac{7}{8}+S(T)+\mathcal{O}\left(\frac{1}{T}\right),$

where

(1.1) $S(T)=\frac{1}{\pi}\arg\zeta\left(\frac{1}{2}+it\right),$

and the value of $\arg$ is obtained by continuous variation along the straight
lines joining the points

$2,\ 2+iT,\ \frac{1}{2}+iT,$

starting with the value zero. Next, we have the function

(1.2) $S_{1}(T)=\int_{0}^{T}S(t){\rm d}t.$

### 1.2.

Further, let us remind the following facts

$\zeta\left(\frac{1}{2}+it\right)=\left|\zeta\left(\frac{1}{2}+it\right)\right|e^{i\arg\zeta\left(\frac{1}{2}+it\right)},$

i.e. the functions

(1.3) $\left|\zeta\left(\frac{1}{2}+it\right)\right|,\
\arg\zeta\left(\frac{1}{2}+it\right)$

are parts of the Riemann function

$\zeta\left(\frac{1}{2}+it\right).$

The study of these functions have proceeded by isolated ways. Namely:

  * (a)

the first one studied by Hardy-Littlewood

$\int_{T}^{T+U}\left|\zeta\left(\frac{1}{2}+it\right)\right|^{2}{\rm d}t\sim
U\ln T,\dots$

  * (b)

the second one by Backlund, E. Landau, H. Bohr, Littlewood, Titchmarsh - to
the fundamental Selberg’s results (see [3], [4]).

Let us mention two of the Selberg’s results:

(1.4) $\begin{split}&\int_{T}^{T+H}\\{S_{1}(t)\\}^{2l}{\rm
d}t=c_{l}H+\mathcal{O}\left(\frac{H}{\ln T}\right),\\\ &T^{a}\leq H\leq T;\
\frac{1}{2}<a\leq 1,\ l\in\mathbb{N},\end{split}$

where $l$ is arbitrary and fixed, (see [4], p. 130), and

(1.5) $S_{1}(t)=\Omega_{\pm}\left\\{(\ln t)^{1/3}(\ln\ln t)^{-10/3}\right\\},$

(see [4], p.150).

######  Remark 1.

For our purpose it is sufficient to use the formula (1.4) in the minimal case

$H=T^{1/2+\epsilon},\ \epsilon>0,$

where $\epsilon$ is sufficiently small (non-principal improvements of the
exponent $1/2$ are not relevant for our purpose).

### 1.3.

To this date, there is no result in the theory of the Riemann zeta-function
about the interaction of the functions (1.3), or the functions

(1.6) $\left|\zeta\left(\frac{1}{2}+it\right)\right|,S_{1}(t).$

That is, there is nothing known like

$F\left(\left|\zeta\left(\frac{1}{2}+it\right)\right|,|S_{1}(\tau)|\right)=0$

for a set of values $t,\tau$.

On the other hand, we have developed (see [1]) the method of
$\zeta$-factorization that gives, for example, the following formula (see [1],
(1.7))

$\frac{1}{\sqrt{\left|\zeta\left(\frac{1}{2}+i\alpha_{0}\right)\right|}}\sim\frac{1}{\sqrt{\Lambda}}\prod_{r=1}^{k}\left|\zeta\left(\frac{1}{2}+i\alpha_{r}\right)\right|$

together with the infinite set of corresponding metamorphoses of the main
multiform.

In this paper we use this method to obtain a result of the new type

(1.7)
$|S_{1}(\alpha_{0})|\sim\Phi\left\\{\prod_{r=1}^{k}\left|\frac{\zeta\left(\frac{1}{2}+i\alpha_{r}\right)}{\zeta\left(\frac{1}{2}+i\beta_{r}\right)}\right|\right\\}$

together with the infinite set of metamorphoses of the corresponding Q-system
from [2].

######  Remark 2.

A kind of nonlinear and nonlocal interaction of the functions (1.6) is
expressed by the formula (1.7).

##  2\. Theorem

### 2.1.

We begin with the Selberg’s formula

(2.1) $\begin{split}&\int_{T}^{T+H}\\{S_{1}(t)\\}^{2l}{\rm d}t\sim c_{l}H,\
T\to\infty,\\\ &H=T^{1/2+\epsilon},\ l\in\mathbb{N},\ \epsilon>0,\end{split}$

(comp. (1.4) and Remark 1), where $l$ is arbitrary and fixed, $\epsilon$ is
sufficiently small. Now, if we use our method of transformation (see [2],
(4.1)–(4.19)) in the case of the formula (2.1) then we obtain (see (1.1),
(1.2)) the following

######  Theorem.

Let

(2.2)
$[T,T+H]\longrightarrow[\overset{1}{T},\overset{1}{\wideparen{T+H}}],\dots,[\overset{k}{T},\overset{k}{\wideparen{T+H}}],$

where

$[\overset{r}{T},\overset{r}{\wideparen{T+H}}],\ r=1,\dots,k,\ k\leq k_{0},\
k_{0}\in\mathbb{N}$

be the reversely iterated segment corresponding to the first segment in (2.2)
and $k_{0}$ be an arbitrary and fixed number. Then there is a sufficiently big

$T_{0}=T_{0}(l,\epsilon)>0$

such that for every $T>T_{0}$ and every admissible $l,\epsilon,k$ there are
the functions

(2.3) $\begin{split}&\alpha_{r}=\alpha_{r}(T,l;\epsilon,k),\ r=0,1,\dots,k,\\\
&\beta_{r}=\beta_{r}(T;\epsilon,k),\ r=1,\dots,k,\\\
&\alpha_{r},\beta_{r}\not=\gamma:\
\zeta\left(\frac{1}{2}+i\gamma\right)=0\end{split}$

such that

(2.4)
$\begin{split}&\left|\int_{0}^{\alpha_{0}(T)}\arg\zeta\left(\frac{1}{2}+it\right){\rm
d}t\right|\sim\\\
&\sim\pi(c_{l})^{\frac{1}{2l}}\prod_{r=1}^{k}\left|\frac{\zeta\left(\frac{1}{2}+i\alpha_{r}(T,l)\right)}{\zeta\left(\frac{1}{2}+i\beta_{r}(T)\right)}\right|^{-\frac{1}{l}},\
T\to\infty.\end{split}$

Moreover, the sequences

$\\{\alpha_{r}\\}_{r=0}^{k},\ \\{\beta_{r}\\}_{r=1}^{k}$

have the following properties

(2.5) $\begin{split}&T<\alpha_{0}<\alpha_{1}<\dots<\alpha_{k},\\\
&T<\beta_{1}<\beta_{2}<\dots<\beta_{k},\\\ &\alpha_{0}\in(T,T+H),\
\alpha_{r},\beta_{r}\in(\overset{r}{T},\overset{r}{\wideparen{T+H}}),\\\
&r=1,\dots,k,\end{split}$ (2.6)
$\begin{split}&\alpha_{r+1}-\alpha_{r}\sim(1-c)\pi(T),\ r=0,1,\dots,k-1,\\\
&\beta_{r+1}-\beta_{r}\sim(1-c)\pi(T),\ r=1,\dots,k-1,\end{split}$

where

$\pi(T)\sim\frac{T}{\ln T},\ T\to\infty$

is the prime-counting function and $c$ is the Euler’s constant.

######  Remark 3.

Let us notice that the asymptotic behavior of the sets

(2.7) $\\{\alpha_{r}\\}_{r=0}^{k},\ \\{\beta_{r}\\}_{r=1}^{k}$

is as follows: at $T\to\infty$ the points of every set in (2.7) recede
unboundedly each from other and all together recede to infinity. Hence, at
$T\to\infty$ each set in (2.7) looks like one-dimensional Friedmann-Hubble
universe.

### 2.2.

Let us denote the mean-value of the function

$\arg\zeta\left(\frac{1}{2}+it\right),\ t\in[0,T]$

by the symbol

$\left.\langle\arg\zeta\left(\frac{1}{2}+it\right)\rangle\right|_{[0,T]}.$

Let us mention that the function under consideration has an infinite set of
first-order discontinuities. Since

$\int_{0}^{\alpha_{0}(T)}\arg\zeta\left(\frac{1}{2}+it\right){\rm
d}t=\alpha_{0}(T)\left.\langle\arg\zeta\left(\frac{1}{2}+it\right)\rangle\right|_{[0,\alpha_{0}(T)]},$

then we obtain from (2.4) the following

######  Corollary 1.

(2.8)
$\begin{split}&\left|\left.\langle\arg\zeta\left(\frac{1}{2}+it\right)\rangle\right|_{[0,\alpha_{0}(T)]}\right|\sim\\\
&\sim\frac{\pi(c_{l})^{\frac{1}{2l}}}{\alpha_{0}(T)}\prod_{r=1}^{k}\left|\frac{\zeta\left(\frac{1}{2}+i\alpha_{r}(T,l)\right)}{\zeta\left(\frac{1}{2}+i\beta_{r}(T,l)\right)}\right|^{-\frac{1}{l}},\\\
&\alpha_{0}(T)\in(T,T+H),\ T\to\infty.\end{split}$

Let us remind that the following Littlewood’s estimate (comp. [5], p. 189)

$S_{1}(t)=\mathcal{O}(\ln t),\ t\to\infty$

holds true. Hence, we have (see (1.1), (1.2)) the estimate

(2.9)
$\left.\langle\arg\zeta\left(\frac{1}{2}+it\right)\rangle\right|_{[0,T]}=\mathcal{O}\left(\frac{\ln
T}{T}\right),\ T\to\infty.$

######  Remark 4.

Consequently, we have obtained in the direction of the estimate (2.9) the
explicit asymptotic formula (2.8) for the mean-value

$\left.\langle\arg\zeta\left(\frac{1}{2}+it\right)\rangle\right|_{[0,T]}$

on the infinite subset

$\\{\alpha_{0}(T)\\},\alpha_{0}(T)\in(T,T+T^{1/2+\epsilon}),\ T\to\infty.$

##  3\. Reduction of the integral in (2.4)

### 3.1.

Now, we use the Selberg’s $\Omega$-theorem (1.5) to transform our formula
(2.4). It follows from (1.5) that there are two sequences

$\\{a_{n}\\}_{n=1}^{\infty},\ \\{b_{n}\\}_{n=1}^{\infty},\
a_{n},b_{n}\to\infty$

such that

(3.1) $\begin{split}&S_{1}(a_{n})>A(\ln a_{n})^{1/3}(\ln\ln a_{n})^{-10/3},\\\
&S_{1}(b_{n})<-B(\ln a_{n})^{1/3}(\ln\ln a_{n})^{-10/3};\\\
&A,B>0.\end{split}$

Since

$S_{1}(t),\ t>0$

is the continuous function then by (3.1) there is (Bolzano-Cauchy) the
sequence

(3.2) $\\{\mu_{n}\\}_{n=1}^{\infty}:\ S_{1}(\mu_{n})=0,\ \mu_{n}\to\infty,$

where $\mu_{n}$ is the odd-order root of the equation

(3.3) $S_{1}(t)=0,\ t>0.$

######  Remark 5.

We may suppose, of course, that the sequence (3.2) is _complete_ one in the
usual sense, the interval

$(\mu_{n},\mu_{n+1})$

does not contain any other odd-order root of the equation (3.3).

######  Remark 6.

There is no need to discuss (for our purpose) the question about even-order
roots of the equation (3.3).

Hence, we have: if

(3.4) $\bar{k}=\bar{k}[\alpha_{0}(T)]:\
\mu_{\bar{k}}<\alpha_{0}(T)<\mu_{\bar{k}+1}$

and (of course, see (2.4), (3.3))

$S_{1}[\alpha_{0}(T)]\not=0,$

then

(3.5) $\begin{split}&S_{1}[\alpha_{0}(T)]=\int_{0}^{\alpha_{0}(T)}S(t){\rm
d}t=\int_{0}^{\mu_{\bar{k}}}+\int_{\mu_{\bar{k}}}^{\alpha_{0}(T)}=\int_{\mu_{\bar{k}}}^{\alpha_{0}(T)}S(t){\rm
d}t.\end{split}$

Consequently, we have from (2.4) by (3.4), (3.5) the following

######  Corollary 2.

(3.6)
$\begin{split}&\left|\int_{\mu_{\bar{k}}}^{\alpha_{0}(T)}\arg\zeta\left(\frac{1}{2}+it\right){\rm
d}t\right|\sim\\\
&\sim\pi(c_{l})^{\frac{1}{2l}}\prod_{r=1}^{k}\left|\frac{\zeta\left(\frac{1}{2}+i\alpha_{r}(T,l)\right)}{\zeta\left(\frac{1}{2}+i\beta_{r}(T)\right)}\right|^{-\frac{1}{l}},\
T\to\infty.\end{split}$

### 3.2.

Next, we obtain from (3.6), (comp. (2.8)), the following

######  Corollary 3.

(3.7)
$\begin{split}&\left|\left.\langle\arg\zeta\left(\frac{1}{2}+it\right)\rangle\right|_{[\mu_{\bar{k}},\alpha_{0}(T)]}\right|\sim\\\
&\sim\frac{\pi(c_{l})^{\frac{1}{2l}}}{\alpha_{0}(T)-\mu_{\bar{k}}}\prod_{r=1}^{k}\left|\frac{\zeta\left(\frac{1}{2}+i\alpha_{r}(T,l)\right)}{\zeta\left(\frac{1}{2}+i\beta_{r}(T)\right)}\right|^{-\frac{1}{l}},\end{split}$

and, of course, (see (2.8), (3.7))

$\begin{split}&\left|\left.\langle\arg\zeta\left(\frac{1}{2}+it\right)\rangle\right|_{[\mu_{\bar{k}},\alpha_{0}(T)]}\right|\sim\\\
&\sim\frac{\alpha_{0}(T)}{\alpha_{0}(T)-\mu_{\bar{k}}}\left|\left.\langle\arg\zeta\left(\frac{1}{2}+it\right)\rangle\right|_{[0,\alpha_{0}(T)]}\right|,\
T\to\infty.\end{split}$

##  4\. On infinite set of metamorphoses of the Q-system that is generated by
the factorization formula (2.4)

### 4.1.

Let us remind the Riemann-Siegel formula

(4.1) $Z(t)=2\sum_{n\leq\tau(t)}\frac{1}{\sqrt{n}}\cos\\{\vartheta(t)-t\ln
n\\}+\mathcal{O}(t^{-1/4}),$

where

$\begin{split}&Z(t)=e^{i\vartheta(t)}\zeta\left(\frac{1}{2}+it\right),\
\tau(t)=\sqrt{\frac{t}{2\pi}},\\\
&\vartheta(t)=-\frac{t}{2}\ln\pi+\mbox{Im}\ln\Gamma\left(\frac{1}{4}+i\frac{t}{2}\right),\end{split}$

(see [5], pp. 79, 239). Next, we have introduced (see [2], (2.1)) the
following oscillatory Q-system (based exactly on the Riemann-Siegel formula
(4.1))

(4.2)
$\begin{split}&G(x_{1},\dots,x_{k};y_{1},\dots,y_{k})=\prod_{r=1}^{k}\left|\frac{Z(x_{r})}{Z(y_{r})}\right|=\\\
&=\prod_{r=1}^{k}\left|\frac{\sum_{n\leq\tau(x_{r})}\frac{2}{\sqrt{n}}\cos\\{\vartheta(x_{r})-x_{r}\ln
n\\}+R(x_{r})}{\sum_{n\leq\tau(y_{r})}\frac{2}{\sqrt{n}}\cos\\{\vartheta(y_{r})-y_{r}\ln
n\\}+R(y_{r})}\right|,\\\ &(x_{1},\dots,x_{k})\in M_{k}^{1},\
(y_{1},\dots,y_{k})\in M_{k}^{2},\\\ &R(t)=\mathcal{O}(t^{-1/4}),\ k\leq
k_{0}\in\mathbb{N},\end{split}$

where

(4.3) $\begin{split}&M_{k}^{1}=\\{(x_{1},\dots,x_{k})\in(T_{0},+\infty)^{k},\
T_{0}<x_{1}<\dots<x_{k}\\},\\\
&M_{k}^{2}=\\{(y_{1},\dots,y_{k})\in(T_{0},+\infty)^{k},\
T_{0}<y_{1}<\dots<y_{k}\\},\\\ &x_{r},y_{r}\not=\gamma:\
\zeta\left(\frac{1}{2}+i\gamma\right)=0,\ r=1,\dots,k.\end{split}$

### 4.2.

Next, we have obtained (see [2], (3.1)) the following spectral formula

(4.4)
$\begin{split}&Z(t)=2\sum_{n\leq\tau(x_{r})}\frac{1}{\sqrt{n}}\cos\left\\{t\ln\frac{\tau(x_{r})}{n}-\frac{x_{r}}{2}-\frac{\pi}{8}\right\\}+\\\
&+\mathcal{O}(x_{r}^{-1/4}),\ \tau(x_{r})=\sqrt{\frac{x_{r}}{2\pi}},\\\
&t\in[x_{r},x_{r}+V],\ V\in(0,\sqrt[4]{x_{r}}],\end{split}$

(and similarly for $x_{r}\longrightarrow y_{r}$), where

$T_{0}<x_{r},y_{r},\ r=1,\dots,k.$

######  Remark 7.

The spectral formula (4.4) is, of course, a variant of the Riemann-Siegel
formula (4.1).

######  Remark 8.

We call the expressions

(4.5)
$\frac{2}{\sqrt{n}}\cos\left\\{t\omega_{n}(x_{r})-\frac{x_{r}}{2}-\frac{\pi}{8}\right\\},\dots$

as the local Riemann’s oscillators with:

  * (a)

the amplitudes

$\frac{2}{\sqrt{n}},$

  * (b)

the incohorent local phase constants

$\left\\{-\frac{x_{r}}{2}-\frac{\pi}{8}\right\\},\
\left\\{-\frac{y_{r}}{2}-\frac{\pi}{8}\right\\},$

  * (c)

the non-synchronized local times

$t=t(x_{r})\in[x_{r},x_{r}+V],\dots$

  * (d)

the local spectrum of the cyclic frequencies

$\begin{split}&\\{\omega_{n}(x_{r})\\}_{n\leq\tau(x_{r})},\
\omega_{n}(x_{r})=\ln\frac{\tau(x_{r})}{n},\\\
&\\{\omega_{n}(y_{r})\\}_{n\leq\tau(y_{r})},\
\omega_{n}(y_{r})=\ln\frac{\tau(y_{r})}{n}.\end{split}$

######  Remark 9.

The Q-system (4.2) represents a complicated oscillating process generated by
oscillations of big number of the local Riemann’s oscillators (4.5).

### 4.3.

Now, in connection with the oscillating Q-system (4.2), the following
corollary follows from our Theorem

######  Corollary 4.

(4.6)
$\begin{split}&\prod_{r=1}^{k}\left|\frac{\sum_{n\leq\tau(\alpha_{r})}\frac{2}{\sqrt{n}}\cos\\{\vartheta(\alpha_{r})-\alpha_{r}\ln
n\\}+R(\alpha_{r})}{\sum_{n\leq\tau(\beta_{r})}\frac{2}{\sqrt{n}}\cos\\{\vartheta(\beta_{r})-\beta_{r}\ln
n\\}+R(\beta_{r})}\right|\sim\\\
&\sim\pi^{l}\sqrt{c_{l}}\left|\int_{0}^{\alpha_{0}(T)}\arg\zeta\left(\frac{1}{2}+it\right){\rm
d}t\right|^{-l},\ T\to\infty.\end{split}$

######  Remark 10.

Hence, we have two resp. one parametric sets of control functions (=Golem’s
shem) for admissible and fixed $\epsilon,k$, (see (2.3)),

(4.7)
$\begin{split}&\\{\alpha_{0}(T,l),\alpha_{1}(T,l),\dots,\alpha_{k}(T,l)\\},\\\
&\\{\beta_{1}(T),\dots,\beta_{k}(T)\\},\\\ &T\in(T_{0},+\infty),\
l\in\mathbb{N},\end{split}$

of the metamorphoses (4.6), (comp. [1],[2]).

######  Remark 11.

The mechanism of the metamorphosis is as follows. Let (comp. (4.3), (4.7))

(4.8) $\begin{split}&M_{k}^{3}=\\{\alpha_{1}(T,l),\dots,\alpha_{k}(T,l)\\},\\\
&M_{k}^{4}=\\{\beta_{1}(T),\dots,\beta_{k}(T)\\},\end{split}$

where, of course,

(4.9) $\begin{split}&M_{k}^{3}\subset M_{k}^{1}\subset(T_{0},+\infty)^{k},\\\
&M_{k}^{4}\subset M_{k}^{2}\subset(T_{0},+\infty)^{k}.\end{split}$

Now, if we obtain after random sampling of the points

$(x_{1},\dots,x_{k}),\ (y_{1},\dots,y_{k})$

(see the conditions (4.3)) such that

(4.10)
$\begin{split}&(x_{1},\dots,x_{k})=(\alpha_{1}(T,l),\dots,\alpha_{k}(T,l))\in
M_{k}^{3},\\\ &(y_{1},\dots,y_{k})=(\beta_{1}(T),\dots,\beta_{k}(T))\in
M_{k}^{4},\end{split}$

(see (4.8), (4.9)), then - at the points (4.10) - the Q-system (4.2) changes
its old form (=chrysalis) to the new one (=butterfly), and the last ist
controlled by the function $\alpha_{0}(T)$.

### 4.4.

Now, we rewrite the formula (4.6), (comp. (3.6)), as follows:

(4.11)
$\begin{split}&\left|\int_{\mu_{\bar{k}}}^{\alpha_{0}(T)}\arg\zeta\left(\frac{1}{2}+it\right){\rm
d}t\right|\sim\\\
&\sim\pi(c_{l})^{\frac{1}{2l}}\prod_{r=1}^{k}\left|\frac{\sum_{n\leq\tau(\alpha_{r})}\frac{2}{\sqrt{n}}\cos\\{\vartheta(\alpha_{r})-\alpha_{r}\ln
n\\}+R(\alpha_{r})}{\sum_{n\leq\tau(\beta_{r})}\frac{2}{\sqrt{n}}\cos\\{\vartheta(\beta_{r})-\beta_{r}\ln
n\\}+R(\beta_{r})}\right|^{-\frac{1}{l}}.\end{split}$

######  Remark 12.

The formula (4.11) expresses the metamorphosis in the reverse direction. We
describe the mechanism of this as follows: we begin with the integral

$\left|\int_{0}^{w}\arg\zeta\left(\frac{1}{2}+it\right){\rm d}t\right|$

that is the Aaron staff,

$\longrightarrow\left|\int_{\mu_{\bar{k}}}^{\alpha_{0}(T)}\arg\zeta\left(\frac{1}{2}+it\right){\rm
d}t\right|$

that is the bud of the Aaron staff corresponding to $w=\alpha_{0}(T)$,

$\sim\pi(c_{l})^{\frac{1}{2l}}\prod_{r=1}^{k}\left|\frac{\sum_{n\leq\tau(\alpha_{r})}\frac{2}{\sqrt{n}}\cos\\{\vartheta(\alpha_{r})-\alpha_{r}\ln
n\\}+R(\alpha_{r})}{\sum_{n\leq\tau(\beta_{r})}\frac{2}{\sqrt{n}}\cos\\{\vartheta(\beta_{r})-\beta_{r}\ln
n\\}+R(\beta_{r})}\right|^{-\frac{1}{l}}$

already metamorphosed one into almonds ripened, (motivation: Chumash,
Bamidbar, 17:23).

I would like to thank Michal Demetrian for his help with electronic version of
this paper.

## References

  * [1] J. Moser, ‘Jacob’s ladders, $\zeta$-factorization and infinite set of metamorphosis of a multiform‘, arXiv: 1501.07705.
  * [2] J. Moser, ‘Jacob’s ladders, Riemann’s oscillators, quotient of the oscillating multiforms and set of metamorphoses of this system‘, arXiv: 1506.00442.
  * [3] A. Selberg, ‘On the remainder in the formula for $N(T)$, the number of zeros of $\zeta(s)$ in the strip $0<t<T$‘, Av. Nor. Vid. Acad. Oslo (1944), No. 1, 2-27.
  * [4] A. Selberg, ‘Contributions to the theory of the Riemann zeta-function‘, Arch. for Math. og Naturv. B (48), (1946), No. 5, 89-115.
  * [5] E.C. Titchmarsh, ‘ _The theory of the Riemann zeta-function_ ‘, Clarendon Press, Oxford, 1951.